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0704.0743

Diatomic molecule as a quantum entanglement switch

Diatomicmolecule as a quantum entanglement switch Adam Rycerz ∗ Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30–059 Kraków, Poland Abstract We investigate a pair entanglement of electrons in diatomic molecule, modeled as a correlated double quantum dot attached to the leads. The low-temperature properties are derived from the ground state obtained by utilizing the Rejec-Ramšak variational technique within the framework of EDABI method, which combines exact diagonalization with ab initio calculations. The results show, that single-particle basis renormalization modifies the entanglement-switch effectiveness significantly. We also found the entanglement signature of a competition between an extended Kondo and singlet phases. Key words: Correlated nanosystems, Entanglement manipulation, EDABI method PACS: 73.63.-b, 03.67.Mn, 72.15.Qm Quantum entanglement, as one of the most intriguing features of quantum mechanics, have spurred a great deal of scientific activity during the last decade, mainly because it is regarded as a valuable resource in quantum commu- nication and information processing [1]. The question on entanglement between microscopic degrees of freedom in a condensed phase have been raised recently [2], in hope to shed new lights on the physics of quantum phase transitions and quantum coherence [3]. In the field of quantum elec- tronics, a pair entanglement appeared to be a convenient tool to characterize the nature of transport through quan- tum dot, since its vanish when the system is in a Kondo regime [4]. The analogical behavior was observed for two qubits in double quantum dot, for either serial and paral- lel configuration [5]. The latter case is intriguing, since the concurrence [6] at T = 0 changes abruptly from C ≈ 1 to C = 0 when varying the interdot coupling, so a finite An- derson system shows a true quantum phase transition. Here we consider a nanoscale version of such an entangle- ment switch, inspired by conductance measurements for a single hydrogenmolecule [7]. A special attention is payed to electron-correlation effects, in particular the wave-function renormalization [8]. Recent experiment [9] shows the cur- rent through a molecule is carried by a single conductance channel, so serial configuration shown in Fig. 1 seems to be the realistic one. The Hamiltonian of the system is ∗ Corresponding author. Tel: (+48 12) 663–55–68 Fax: (+48 12) 633–40–79 Email address: [email protected] (Adam Rycerz). ���������� �������� �������� ������������������������ �������� Fig. 1. Diatomic molecule modeled as a double quantum dot attached serially to the leads. A cross-section of the single-particle potential along the main system axis is shown schematically. H = HL + VL +HC + VR +HR, (1) where HC models the central region, HL(R) describes the left (right) lead, and VL(R) is the coupling between the lead and the central region. Both HL(R) and VL(R) terms have a tight–binding form, with the chemical potential in leads µ, the hopping t, and the tunneling amplitude V , as depicted schematically in Fig. 1. The central-region Hamiltonian iσcjσ + iσ 6=jσ′ Uijniσniσ′ + (Ze) 2/R (2) (with i, j = 1, 2 and σ =↑, ↓) describes a double quan- tum dot with electron-electron interaction. tij and Uij are single-particle and interaction elements, the last term de- scribes the Coulomb repulsion of the two ions at the dis- tance R. Here we put Z = 1 and calculate all the param- eters tij , Uij as the Slater integrals [10] for 1s-like hydro- genic orbitals Ψ1s(r) = α3/π exp(−α|r|), where α−1 is the orbital size (cf. Fig. 1). The parameter α is optimized to get a minimal ground-state energy for whole the system Preprint submitted to Elsevier 15 November 2018 http://arxiv.org/abs/0704.0743v2 -3 -2 -1 0 1 chemical potential, 0 1 2 3 4 average filling, 〈n1+n2〉 Fig. 2. Entanglement and transport through the system in Fig. 1 as a function of the chemical potential µ (top panel) and the average filling 〈n1+n2〉 (bottom panel). Tick (thin) solid and dashed lines shows the concurrence C (conductance G) for Γ = t/9 and t/4, respectively. The limits Γ → 0 are depicted with dotted lines in the bottom panel. The interatomic distance is R = 1.5a. 1 2 3 4 5 6 interatomic distance, Fig. 3. Concurrence (tick lines) and conductance (thin lines) at the half-filled sector 〈n1+n2〉 = 2 as a function of the interatomic distance R. The remaining parameters are the same as in Fig. 2. described by the Hamiltonian (1). Thus, following the idea of EDABI method [8], we reduce the number of physical parameters of the problem to just a three: the interatomic distance R, the lead-molecule hybridization Γ = V 2/t, and the chemical potential µ (we put the lead hopping t = 1 Ry = 13.6 eV to work in the wide–bandwidth limit). The entanglement between electrons placed on two atoms can be characterized by the charge concurrence [4] C = 2max 0, |〈c iσcjσ〉| − 〈niσnjσ〉〈n̄iσ n̄jσ〉 where n̄iσ ≡ 1−niσ. We also discuss the conductivity cal- culated from the formula G = G0 sin 2(E+−E−)/4tN [11], where G0 = 2e 2/h̄, and E± are the ground-state ener- gies of the system with periodic and antiperiodic boundary conditions, respectively. Either the energies E± or correla- tion functions in Eq. (3) are calculated within the Rejec– Ramšak variational method [11], complemented by the or- bital size optimization, as mentioned above. We use up to N = 104 sites to reach the convergence. In Fig. 2 we show the concurrence and conductance for R = 1.5a0 (where a0 is the Bohr radius) and two values of the hybridization Γ = t/9 and t/4. The conductance spectrum asymmetry, caused by wave-function renormal- ization [8], is followed by an analogical effect on entangle- ment, which changes significantly faster for the upper con- duction band, where the average filling is 〈n1+n2〉 ≈ 3 (one extra electron). The asymmetry vanish when analyz- ing the system properties as a function of 〈n1+n2〉, show- ing it originates from varying charge compressibility χc = ∂〈n1+n2〉/∂µ ≈ 2/(U11 + U12) ∼ 1/α. We also note the convergence of discussed quantities with Γ → 0 to C ≈ 1− |〈n1+n2〉 − 2|/2 and G ≈ G0 sin 2(π〈n1+n2〉/2). Entanglement evolution with R is illustrated in Fig. 3, where we focus on the charge neutral section 〈n1+n2〉 = 2. The abrupt entanglement drop follows the sharp con- ductance peak for Γ = t/9, which is associated with the competition between double Kondo and spin/charge sin- glet phases [12]. For Γ = t/4 both C andG dependence onR become smooth, but the switching behavior is still present. Earlier, we have shown that Γ = t/4 is large enough to cause molecule instability and therefore may allow the in- dividual atom manipulation [8]. In conclusion, we analyzed a pair entanglement of elec- trons in diatomic molecule attached serially to the leads. Entanglement evolution with the chemical potential speeds up remarkably for the negatively charged system, due to electron correlation effects. The switching behavior was also observed when changing the interatomic distance. The work was supported by Polish Science Foundation (FNP), and Ministry of Science Grant No. 1 P03B 001 29. References [1] See review by C.H. Bennet and D.P. Divincenzo, Nature 404, 247 (2000); M.A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge, 2000). [2] A. Osterloh et al., Nature 416, 608 (2002); T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66, 032110 (2002); S.–J. Gu et al., Phys. Rev. Lett. 93, 086402 (2004). [3] J. van Wezel, J. van den Brink, J. Zaanen, Phys. Rev. Lett. 94, 230401 (2005); cond-mat/0606140. [4] A. Rycerz, Eur. Phys. J. B 52, 291 (2006); S. Oh, J. Kim, Phys. Rev. B 73, 052407 (2007). [5] A. Ramšak, J. Mravlje, R. Žitko, J. Bonča, Phys. Rev. B 74, 241305(R) (2006); R. Žitko, J. Bonča, ibid. 74, 045312 (2006). [6] W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [7] R.H.M. Smit et al., Nature 419, 906 (2002). [8] J. Spa lek et al., cond-mat/0610815. [9] D. Djukic, J.M. van Ruitenbeek, Nano. Lett. 6, 789 (2006); M. Kiguchi et al., cond-mat/0612681. [10] J. C. Slater, Quantum Theory of Molecules and Solids, McGraw– Kill (New York, 1963), Vol. 1, p. 50. [11] T. Rejec, A. Ramšak, Phys. Rev. B 68, 033306 (2003). [12] P.S. Cornaglia, D.R. Grempel, Phys. Rev. B 71, 075305 (2005); J. Mravlje, A. Ramšak, T. Rejec, ibid. 73, 241305(R) (2006). References

0704.0745

Weak and Strong Taylor methods for numerical solutions of stochastic differential equations

October 31, 2018 WEAK AND STRONG TAYLOR METHODS FOR NUMERICAL SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS MARIA SIOPACHA AND JOSEF TEICHMANN Abstract. We apply results of Malliavin-Thalmaier-Watanabe for strong and weak Taylor expansions of solutions of perturbed stochastic differential equa- tions (SDEs). In particular, we work out weight expressions for the Taylor coefficients of the expansion. The results are applied to LIBOR market models in order to deal with the typical stochastic drift and with stochastic volatility. In contrast to other accurate methods like numerical schemes for the full SDE, we obtain easily tractable expressions for accurate pricing. In particular, we present an easily tractable alternative to “freezing the drift” in LIBOR market models, which has an accuracy similar to the full numerical scheme. Numerical examples underline the results. 1. Introduction and Setting Let (Ω,F ,P) be a probability space carrying an N -dimensional Brownian motion (Wt)t≥0 with a d × d correlation matrix. We consider smooth curves Fǫ : R → L2(Ω;RN ) of random variables, where ǫ ∈ R is a parameter. We apply Taylor theorems to obtain strong approximations of the curve Fǫ at ǫ = 0 and we apply partial integration on Wiener space to obtain weak approximations of the law of Fǫ for small values of ǫ. We choose the notion Taylor expansion instead of asymptotic expansion in order to point out that the strong method is indeed a classical Taylor expansion with usual conditions for convergence. The weak method represents a truncated converging power series in the parameter ǫ if – for instance – the payoff f : RN → R stems from a real analytic function and some distributional properties are satisfied. 2. Weak and strong Taylor methods - Structure Theorems We introduce in this section two concepts of approximation. Consider a curve ǫ 7→ Fǫ, where ǫ ∈ R and Fǫ ∈ L2(Ω;RN ). Definition 1. A strong Taylor approximation of order n ≥ 0 is a (truncated) power series (2.1) Tnǫ (Fǫ) := such that (2.2) E |Fǫ −Tnǫ (Fǫ)| = o(ǫn), Financial support from the Austrian Science Fund (FWF) under grant P 15889 and the START- prize-grant Y328-N13 is gratefully acknowledged. Furthermore this work was financially supported by the Christian Doppler Research Association (CDG). The authors gratefully acknowledge a fruitful collaboration and continued support by Bank Austria and the Austrian Federal Financing Agency (ÖBFA) through CDG. http://arxiv.org/abs/0704.0745v1 2 MARIA SIOPACHA AND JOSEF TEICHMANN holds true as ǫ→ 0. Remark 1. In our setting a strong Taylor approximation of any order n ≥ 0 of the curve Fǫ can always be obtained, see for instance [KM97]. Let f : RN → R be a Lipschitz function with Lipschitz constant K, then we obtain (2.3) E |f(Fǫ)− f ǫ (Fǫ) ‖Fǫ −Tnǫ (Fǫ)‖ = Ko(ǫn). Equation (2.3) does not hold anymore if f is not globally Lipschitz continuous. In particular, we observe the dependence of the right hand side on the Lipschitz constant K. Hence, truncating an a-priori known Taylor expansion leads to an error term, which contains the Lipschitz constant and is therefore not useful for non-Lipschitz claims. The weak method navigates around this feature by partial integration. Definition 2. A weak Taylor approximation of order n ≥ 0 is a power series for each bounded, measurable f : RN → R, ǫ (f, Fǫ) := E(f(F0)πi), where πi ∈ L1(Ω) denote real valued, integrable random variables, such that f(Fǫ) −Wnǫ (f, Fǫ)| = o(ǫn). Remark 2. The weights πi for i ≥ 1 are called Malliavin weights. Remark 3. If the law of Fǫ is real analytic at ǫ = 0 in the weak sense, i.e. if there exist (signed) measures µi such that for all bounded, measurable f : R N → R the following series converges and the equality f(Fǫ) f(x)µi(dx), holds true, precisely then we do have a converging weak Taylor expansion. We aim for constructing stochastic representations of the following type, for i ≥ 0: f(x)µi(dx) = E(f(F0)πi). For the definition of the weak Taylor approximation to make sense, existence of the Malliavin weights has to hold. The following theorem can be found in a slightly different version in [MT06] and goes back to S. Watanabe. For the definition and notion of D∞(RN ) see [Mal97] or [Nua06]. Theorem 1. Let Fǫ : R → D∞(RN ) be smooth and assume that the Malliavin co- variance matrix γ(Fǫ) is invertible with p-integrable inverse for every p ≥ 1 around ǫ = 0 (i.e. on an open interval containing ǫ = 0). Then there is a weak Taylor approximation of any order n ≥ 0 and there are explicit formulas for the weights πi. If we only know that the Malliavin covariance matrix γ(F0) is invertible with p-integrable inverse, then we can also calculate the Malliavin weights, since they depend only on γ(F0). WEAK AND STRONG TAYLOR METHODS FOR NUMERICAL SOLUTIONS OF SDES 3 Proof. Fix n ≥ 0 and take a smooth test function f : RN → R and assume that γ−1(Fǫ) exists as a smooth curve in D∞ on a open ǫ-interval containing ǫ = 0. By standard arguments we can prove the following formula f(Fǫ) f(Fǫ)δ s 7→ (DsFǫ)Tγ−1(Fǫ) More precisely, by the integration by parts [Nua06, Definition 1.3.1-(1.42)], the chain rule [Nua06, Proposition 1.2.3] and the definition of the Malliavin covariance matrix, [Nua06, page 92], we obtain from the right hand side the desired left hand side. Notice that the ǫ-dependence of the Skorohod integral is smooth due to basic properties of D∞. Hence, we can calculate higher derivatives of the left hand side by iterating the above procedure and differentiating the Skorohod integral. We denote (2.4) π1 := δ s 7→ (DsFǫ)Tγ−1(Fǫ) We write then, pars pro toto, the formula for the second derivative E(f(Fǫ)) = E f(Fǫ)δ s 7→ π1(DsFǫ)Tγ−1(Fǫ) f(Fǫ)δ s 7→ (Ds γ−1(Fǫ) f(Fǫ)δ s 7→ (DsFǫ)Tγ−1(Fǫ) dγ(Fǫ) γ−1(Fǫ) f(Fǫ)δ s 7→ (DsFǫ)Tγ−1(Fǫ) This formula makes perfect sense at ǫ = 0 and – by induction – we see that we can perform this step for any derivative. The general, recursive result is the following: as := (DsFǫ) γ−1(Fǫ) for 0 ≤ s ≤ T, πn := δ(s 7→ asπn−1) + πn−1, π0 := 1. Here we understand the weights πn as ǫ-dependent, whereas in the final formulas we put ǫ = 0. This proves the result for smooth test functions f and under the assumption that the Malliavin covariance matrix is invertible around ǫ = 0. If we approximate a bounded, measurable function f by smooth test functions we obtain the desired assertion by standard arguments, since the weights are integrable. � Remark 4. By Taylor’s theorem and the Faà-di-Bruno-formula we obtain dnf(Fǫ) |α|≤n f (α)(Fǫ)pα, where pα is a well-defined polynomial in derivatives of the curve ǫ 7→ Fǫ, for a multi-index α. Since D∞ is an algebra, see [Mal97], the above expression lies in 4 MARIA SIOPACHA AND JOSEF TEICHMANN Lp(Ω) for each p ≥ 0. The previous result provides a representation of the partial integration result for |α|≤n f (α)(Fǫ)pα) = E(f(Fǫ)πn). The structure of the weights is seen from above. The result can be considered as a dual version of the Faà-di-Bruno-formula. However, the structure of this dual formula is much simpler. We provide an example to demonstrate the strong and weak method of approx- imation. The method works in order to replace time-consuming iteration schemes, like the Euler-scheme, by simulations of “simple” Itô integrals. Example 1. We deal with a generic, real-valued random variable over a one- dimensional Gaussian space, see [Nua06], i.e. where the F i lie in the (i+1)st Wiener chaos Hi+1(Ω) (one can think of a Hermite expansion for instance) and the sum is understood in the L2-sense. From the strong expansion we obtain immediately – for a given Lipschitz function f : R → R – that f(Fǫ) f(F 0 + ǫF 1) | ≤ Ko(ǫ), as ǫ→ 0, where K denotes the Lipschitz constant of f . This simple approximation can be sometimes quite useful. We assume now that F 0 = h(s)dWs has non-vanishing variance in order to calculate the weights, which do depend only on γ(F0). The strong Taylor approxi- mation is given by definition, the weak Taylor expansion can be constructed by the previous recursive formulas and the specifications 0 = h(s), γ(F 0) = h(s)2ds, h(s)2ds In order to obtain a first-order approximation for bounded, measurable random variables we therefore have to calculate f(F 0) f(F 0)π1 where π1 = δ s 7→ asF 1 This amounts to an integration of f times a polynomial with respect to a Gaussian density, since: f(F 0)π1 f(F 0)F 1 asdWs f(F 0)DsF Notice that the strong approximation does not yield such a result for bounded, measurable random variables. Notice also that in the given case the approximation can be calculated in a deterministic way, since we deal with Gaussian integrations. WEAK AND STRONG TAYLOR METHODS FOR NUMERICAL SOLUTIONS OF SDES 5 The second-order weak Taylor approximation is given by f(F 0) f(F 0)π1 + ǫ2E f(F 0)π2 where π2 = δ s 7→ π1asF 1 s 7→ asF 2 3. Applications from Financial Mathematics For applications we want to deal with strong and weak Taylor approximations of a given curve of random variables. We are particulary interested in cases, where the first derivative dFǫ |ǫ=0 is of simple form or – even more important – where the Malliavin covariance matrix γ(F0) is of simple form. In these cases it is easy to obtain first or second order approximations of the respective quantities in the weak or strong sense. In what follows, first we will present one of the most applied interest rate models, namely the LIBOR market model (LMM). Then, we will introduce the commonly used technique of freezing the drift. We will show how to embed the”freezing the drift” technique into our framework of Taylor approximations. We understand freezing the drift as a strong Taylor approximation of order zero in the drift term of the LIBOR SDE. Our goal is to put this technique into a method, where we can in particular improve the order of approximation. We will finally extend the assumption of log normality and develop a stochastic volatility LMM, where we will show how to obtain tractable option prices via our weak Taylor approximations. 3.1. The LIBOR Market Model. We apply our concepts to the LMM, initially constructed by [BGM97], [MSS97] and [Jam97]. Let T denote a strictly positive fixed time horizon and (Ω,FT ,P, (Ft)0≤t≤T ) be a complete probability space, sup- porting an N -dimensional Brownian motion Wt = (W t , ...W t )0≤t≤T . The factors are correlated with dW it dW t = ρijdt. Let 0 = T0 < T1 < T2 < . . . < TN < TN+1 =: T be a discrete tenor structure and α := Ti+1 − Ti the accrual factor for the time period [Ti, Ti+1], i = 0, . . . , N . Let P (t, Ti) denote the value at time t of a zero coupon bond with maturity Ti ∈ [0, T ]. The measure P is the terminal forward measure, which corresponds to taking the final bond P (t, T ) as numéraire. The forward LIBOR rate Lit := Lt(Ti, Ti+1) at time t ≤ Ti for the period [Ti, Ti+1] is given by: Lit = Lt(Ti, Ti+1) = P (t, Ti) P (t, Ti+1) We assume that for any maturity Ti there exists a bounded, continuous, determin- istic function σi(t) : [0, Ti] → R, which represents the volatility of the LIBOR Lit, i = 1, ..., N . The log normal LIBOR market model can be expressed under the measure P as: (3.1) dLit = σ i(t)Lit j=i+1 1 + αL dt+ σi(t)LitdW t , i = 1, ..., N. 3.2. Freezing the Drift. The dynamics of forward LIBORs for i = 1, ..., N − 1 depend on the stochastic drift term , i ≤ j ≤ N , which is determined by LI- BOR rates with longer maturities. This random drift prohibits analytic tractability when pricing products that depend on more that one LIBOR rate, since there is no unifying measure under which all LIBOR rates are simultaneously log normal. 6 MARIA SIOPACHA AND JOSEF TEICHMANN In addition, it encumbers the numerical implementation of the model. Common practice is to approximate this term by its starting value or as it is widely referred to as freezing the drift, i.e. 1 + αL 1 + αL It was first implemented in the original paper [BGM97] for the pricing of swaptions based on the LMM. [BW00] and [Sch02] argue that freezing the drift is justified due to the fact that this term has small variance. However, by freezing the drift there is a difference in option prices with the real and the frozen drift. It has not been examined how big the error is or for which assets it works well or not. Our aim is to investigate such a phenomenon and improve the performance by providing with correction terms of order one. 3.3. Correcting the Frozen Drift. The purpose of this section is to embed the well-known and often applied technique of freezing the drift into the strong and weak Taylor approximations, in order to develop a method to improve the order of accuracy. Specifically for the strong Taylor approximation, the method works well, since we always deal with a globally Lipschitz drift term x 7→ αx+ 1+αx+ with small Lipschitz constant α. Remark 5. As it will be clear later, the strong Taylor correction method can be accommodated with any extension of the log normal LMM, for example with the Lévy LIBOR model by Eberlein and Özkan [EÖ05]. 3.3.1. Strong Taylor Approximation. We first state a useful lemma, asserting that we can indeed freeze the drift under special model formulation and choice parame- ters. Lemma 1. Let ǫ1 ∈ R and consider for i = 1, . . . , N the following stochastic differential equation: (i,ǫ1) t = ǫ1 σi(t)X (i,ǫ1) j=i+1 (j,ǫ1) 1 + αX (j,ǫ1) ρijdt+ dW ,(3.2) defined on the complete probability space (Ω,FT ,P, (Ft)0≤t≤T ) where Wt is an N - dimensional Brownian motion under the measure P with dW it dW t = ρijdt. Then the first-order strong Taylor approximation for X (i,ǫ1) t is given by: (3.3) T1ǫ1(X (i,ǫ1) t ) = X (i,0) t + ǫ1 (i,ǫ1) Proof. By (1) we obtain for n = 1: (i,ǫ1) t ) ≃ X (i,ǫ1) t = X (i,0) 0 + ǫ1Y t + o(ǫ1), since X (i,0) t = X (i,0) 0 and where Y |ǫ1=0X (i,ǫ1) t is the first-order correction term. By differentiating (3.2) with respect to ǫ1, we calculate: (i,ǫ1) = σi(t)X (i,0) j=i+1 (j,0) 1 + αX (j,0) ρijdt+ dW WEAK AND STRONG TAYLOR METHODS FOR NUMERICAL SOLUTIONS OF SDES 7 and derive Y it as the solution to the above linear SDE: (3.4) Y it = −σi(s)X(i,0)0 j=i+1 (j,0) 1 + αX (j,0) σi(s)X (i,0) with Y i0 = 0. � Remark 6. We parametrise the LIBOR market model in terms of the parameter ǫ1 as follows: (i,ǫ1) t = σ i(t)L (i,ǫ1) j=i+1 (j,ǫ1) 1 + αX (j,ǫ1) ρijdt+ dW and assume at t = 0 that L (i,ǫ1) 0 = X (i,ǫ1) 0 for all ǫ1 and all i = 1, ..., N . If ǫ1 = 1, what we obtain is the standard LIBOR market model formulation and in particular (i,1) t = X (i,1) t . For ǫ1 = 0, X (i,0) t equals its starting value and thus the drift term in the following SDE is no longer stochastic: (3.5) dL (i,0) t = σ i(t)L (i,0) j=i+1 (j,0) 1 + αX (j,0) ρijdt+ dW The next proposition provides a way for a pathwise approximation of L (i,ǫ1) t , by means of adjusting its SDE. This is achieved by adding Tnǫ1(X (j,ǫ1) t ) in the frozen drift part. Proposition 1. Assume the setup of Lemma 1 and assume further at t = 0 that (i,ǫ1) 0 = X (i,ǫ1) 0 for all ǫ1 and all i = 1, ..., N . Then the stochastic differential equation for L (i,ǫ1) t with the unfrozen drift: (3.6) dL (i,ǫ1) t = σ i(t)L (i,ǫ1) j=i+1 (j,ǫ1) 1 + αX (j,ǫ1) ρijdt+ dW can be strongly approximated as ǫ1 ↓ 0 by (3.7) dL̂ (i,ǫ1) t = σ i(t)L̂ (i,ǫ1) j=i+1 Tnǫ1(X (j,ǫ1) σj(t) 1 + α Tnǫ1(X (j,ǫ1) ρijdt+ dW Remark 7. For n = 0, we derive the ”freezing the drift” case. For n = 1, we already obtain an improvement. Proof. First step is to interchange X (j,ǫ1) t with (X (j,ǫ1) t )+ in (3.6) to obtain: (i,ǫ1) t = σ i(t)L (i,ǫ1) j=i+1 (j,ǫ1) t )+σ 1 + α(X (j,ǫ1) ρijdt+ dW This yields no change for the dynamics of L (i,ǫ1) t , since X (j,ǫ1) t = (X (j,ǫ1) t )+. 8 MARIA SIOPACHA AND JOSEF TEICHMANN By Taylor’s expansion, we know that as ǫ1 ↓ 0, L̂(i,ǫ1)t → L (i,ǫ1) t P-a.s. The estimate for the error term is given by log L̂ (i,ǫ1) t − logL (i,ǫ1) σi(s) j=i+1 Tnǫ1(X (j,ǫ1) σj(s) 1 + α Tnǫ1(X (j,ǫ1) ρij + j=i+1 (j,ǫ1) t )+σ 1 + α(X (j,ǫ1) α|X(j,ǫ1)s − (Tnǫ1(X (j,ǫ1) s ))+|ds. Remark 8. The SDE for the approximated L̂ (i,ǫ1) t is easier and faster to simulate than (3.1), as it is exhibited by the following example. Notice additionally that (i,ǫ1) t is a continuous functional of the process Y t (3.4) and of the Brownian path W it . Eventually, by using L̂ (i,ǫ1) t as the LIBOR rates, the computational complexity of the drift and thus of the model can be reduced substantially, while maintaining accuracy of prices. Example 2. In this example, we examine the performance of the strong Taylor correction method. Let N = 3 and consider pricing a caplet on the LIBOR rate L1 with strike K. Its price is given by: 0 = αEP L1T1 −K Assume that the volatility functions σi(t) : [0, Ti] → R for i = 1, 2, 3 are given by (cf. Brigo and Mercurio [BM01], formulation (6.12)): σi(t) = a(Ti − t) + d − b(Ti − t) where the constants a, b, d, e are the same for all three LIBOR rates and are equal to a = −0.113035, b = 0.22911, d = −a, e = 0.684784. Thus, we can write the model under the terminal measure P as: (1,ǫ1) t = σ 1(t)L (1,ǫ1) (2,ǫ1) 2(t)ρ12 1 + αX (2,ǫ1) (3,ǫ1) 3(t)ρ13 1 + αX (3,ǫ1) dt+ σ1(t)L (1,ǫ1) (2,ǫ1) t = σ 2(t)L (2,ǫ1) (3,ǫ1) 3(t)ρ23 1 + αX (3,ǫ1) dt+ σ2(t)L (2,ǫ1) dL3t = σ 3(t)L3t dW (1,ǫ1) t = ǫ1 σ1(t)X (1,ǫ1) (2,ǫ1) 2(t)ρ12 1 + αX (2,ǫ1) (3,ǫ1) 3(t)ρ13 1 + αX (3,ǫ1) dt+ σ1(t)X (1,ǫ1) (2,ǫ1) t = ǫ1 σ2(t)X (2,ǫ1) (3,ǫ1) 3(t)ρ23 1 + αX (3,ǫ1) dt+ σ2(t)X (2,ǫ1) (3,ǫ1) t = ǫ1 σ3(t)X (3,ǫ1) with initial values L (i,ǫ1) 0 = X (i,ǫ1) 0 = ci, for i = 1, 2, 3 and for all ǫ1. The Brownian motion vector (W 1t ,W t ) is correlated with correlation coefficient ρij given by: ρij = 0.49 + (1− 0.49) exp (−0.13|i− j|), i, j = 1, 2, 3. WEAK AND STRONG TAYLOR METHODS FOR NUMERICAL SOLUTIONS OF SDES 9 The SDEs for the approximated LIBOR rates L̂ (1,ǫ1) t and L̂ (2,ǫ1) t are given by: (1,ǫ1) t = σ 1(t)L̂ (1,ǫ1) c2 + ǫ1Y σ2(t)ρ12 1 + α c2 + ǫ1Y c3 + ǫ1Y σ3(t)ρ13 1 + α c3 + ǫ1Y + σ1(t)L̂ (1,ǫ1) (2,ǫ1) t = σ 2(t)L̂ (2,ǫ1) c3 + ǫ1Y σ3(t)ρ23 1 + α c3 + ǫ1Y dt+ σ2(t)L̂ (2,ǫ1) The partial derivative terms Y 2t and Y t are equal to: Y 2t = c2 ( ∫ t σ2(s)dW 2s − αc3ρ23 1 + αc3 σ2(s)σ3(s)ds Y 3t = c3 σ3(s)dW 3s . We compare three caplet prices: • benchmark price, underlying L(1,ǫ1)t ; • strong Taylor price, underlying L̂(1,ǫ1)t ; • frozen drift price, underlying L(1,0)t . Numerical results in basis points (bps) are displayed in Table 2 for parameters ǫ1 = 1, N = 3, α = 0.50137, c1 = 3.86777%, c2 = 3.7574%, c3 = 3.8631%, T1 = 1.53151, Ti = T1 + iα, i = 2, 3, 4. We characteristically observe the difference in prices between the benchmark and frozen drift price, whilst our strong Taylor correction method performs very well and is computationally simpler and faster. strikes K=3% K=3.5% K=4% K=5.75% K=6.25% K=8% benchmark 11.1831 8.5897 6.5503 3.0349 2.4423 1.2969 strong Taylor 11.0687 8.5691 6.5867 3.1448 2.5513 1.3926 frozen drift 13.9551 11.1822 8.8803 4.6313 3.8506 2.2524 Table 1: Caplet values in bps for parameters ǫ1 = 1, α = 0.50137, c1 = 3.86777%, c2 = 3.7574%, c3 = 3.8631% and T1 = 1.53151. 3.3.2. Weak Taylor Approximation. In what follows, we provide some results on how to correct option prices obtained by the SDE with the frozen drift (3.5) by adding a correction term involving the appropriate Malliavin weight. Let L i,k,ǫ1 denote the vector of the LIBOR rates (L (i,ǫ1) , . . . , L (k,ǫ1) Proposition 2. Assume the setup of Lemma 1, where the ith LIBOR rate is given (3.8) dL (i,ǫ1) t = σ i(t)L (i,ǫ1) j=i+1 (j,ǫ1) 1 + αX (j,ǫ1) ρijdt+ dW with L (i,ǫ1) 0 = X (i,ǫ1) 0 for all ǫ1 and all i = 1, ..., N . Assume furthermore that the Malliavin covariance matrix γ(L i,k,0 ) is invertible. Then the price of an option with 10 MARIA SIOPACHA AND JOSEF TEICHMANN payoff g(L i,k,ǫ1 ), for i ≤ k ≤ N and g bounded measurable, can be approximated by the weak Taylor approximation of order one: a(g,L i,k,ǫ1 ) = P (0, T ) i,k,0 + ǫ1EP i,k,0 ,(3.9) where the Malliavin weight ζTi is given by: ζTi = δ i,k,0 )Tγ−1(L i,k,0 i,k,ǫ1 ,(3.10) for t ≤ Ti. Proof. The weight ζTi is obtained by (2.4). Notice that we can write: i,k,ǫ1 i,k,0 and hence the result (3.9) by Definition 2 for n = 1. � Example 3. In this example we let N = 3 and we price a payers swaption with strike price K and maturity T1, where the underlying swap is entered at T1 and has payment dates T2 and T3. We assume that the volatility functions σ i(t) : [0, Ti] → R for i = 1, 2, 3 are constant: σ1(t) = σ1, σ 2(t) = σ2, σ 3(t) = σ3, such that we obtain under the terminal measure P: (1,ǫ1) t = σ1L (1,ǫ1) (2,ǫ1) 1 + αX (2,ǫ1) (3,ǫ1) 1 + αX (3,ǫ1) dt+ dW 1t (2,ǫ1) t = σ2L (2,ǫ1) (3,ǫ1) 1 + αX (3,ǫ1) dt+ dW 2t dL3t = σ3L t ,(3.11) (1,ǫ1) t = ǫ1 (1,ǫ1) (2,ǫ1) 1 + αX (2,ǫ1) (3,ǫ1) 1 + αX (3,ǫ1) dt+ dW 1t (2,ǫ1) t = ǫ1 (2,ǫ1) (3,ǫ1) 1 + αX (3,ǫ1) dt+ dW 2t (3,ǫ1) t = ǫ1 (3,ǫ1) with initial values L (i,ǫ1) 0 = X (i,ǫ1) 0 = ci, for i = 1, 2, 3 and for all ǫ1. W t and W 2t are correlated with correlation coefficient ρ12. We freeze the drifts in the above equations to obtain: (1,0) t = c1 exp ( αc2σ2 1 + αc2 αc3σ3 1 + αc3 (2,0) t = c2 exp ( αc3σ3 1 + αc3 L3t = c3 exp Similarly to the previous example, we compare four option prices: • benchmark price; • frozen drift; WEAK AND STRONG TAYLOR METHODS FOR NUMERICAL SOLUTIONS OF SDES 11 • strong Taylor price; • weak Taylor price. The weak correction formula (3.9) adds a correction term to the closed form price of the option. The swaption payoff at Ti can be found for example in [MR98]: swptn k=i+1 (1 + αL if the underlying swap is entered at time Ti and has payment dates Ti+1, ..., T . αk is given by: Kα, k = i+ 1, . . . , N, 1 +Kα, k = N + 1. The payers swaption value at time t = 0 can be written as: swptn 0 = P (0, Ti)EPi swptn = P (0, T )EP (1 + αL )− (1 +Kα) ,(3.12) where αi := −1 and Pi denotes the forward measure corresponding to the bond P (t, Ti) as numéraire. Therefore, its benchmark price is given by the above formula with N = 2 and i = 1: swptn 0 = P (0, T ) αL1T1 + αL + α2L1T1L −Kα2L2T1 − 2Kα Its weak Taylor price is given by (3.9) with i = 1 and k = N = 2. swptn 0 = P (0, T ) (1,0) (2,0) + α2L (1,0) (2,0) −Kα2L(2,0) − 2Kα + ǫ1EP (1,0) (2,0) + α2L (1,0) (2,0) −Kα2L(2,0) − 2Kα The weight ζT1 is given by (3.10). The partial derivative terms C |ǫ1=0L (1,ǫ1) and C2T1 := |ǫ1=0L (2,ǫ1) are given by: C1T1 = L (1,0) σ1ρ12 ( σ3αc3β2 (1 + αc3) t− (β2 + β3)W 2t C2T1 = L (2,0) −σ2β3W 2t dt, with C10 = C 0 = 0 and β2 := (1+αc2)2 , β3 := (1+αc3)2 . The Malliavin covariance matrix of the vector (L (1,0) (2,0) ) is equal to: (1,0) (2,0) (1 + ρ212)(L (1,0) )2T1σ 1 2ρ12(L (1,0) (2,0) )T1σ1σ2 2ρ12(L (1,0) (2,0) )T1σ1σ2 (1 + ρ 12)(L (2,0) )2T1σ ⇒ det (1,0) (2,0) (1,0) (2,0) )2T 21 σ 2(1− ρ212). 12 MARIA SIOPACHA AND JOSEF TEICHMANN The determinant is not zero as long as ρ12 6= 1, which is a natural assumption. Hence under this condition, its inverse is given by: (1,0) (2,0) (1− ρ212) 1+ρ212 (1,0) )2T1σ − 2ρ12 (1,0) (2,0) T1σ1σ2 − 2ρ12 (1,0) (2,0) T1σ1σ2 1+ρ212 (2,0) )2T1σ Write the weight ζT1 = ζ + ζ2T1 , where the first weight ζ is obtained as: ζ1T1 = (1,0) C1T1γ 11 + C γ−112 +D1tL (2,0) C1T1γ 21 + C γ−122 δW 1t , and ζ2T1 similarly: ζ2T1 = (1,0) C1T1γ 11 + C γ−112 +D2tL (2,0) C1T1γ 21 + C γ−122 δW 2t . Performing all necessary calculations, we conclude that: ζ1T1 = ρ12 W 1T1 (σ3αc3β2T1 2(1 + αc3) − (β2 + β3) W 2t dt ρ12(β2 + β3)T1 − ρ12 (ρ12β3T1 W 2t dt Analogously we obtain ζ2T1 as: ζ2T1 = ρ W 2T1 (σ3αc3β2T1 2(1 + αc3) (β2 + β3) W 2t dt (β2 + β3)T1 (β3T1 W 2t dt Notice that the weights are functions of normal variables and thus the calculation of the weak Taylor price amounts just to computation of deterministic integrals. Table 3 gives the swaption prices in bps for parameters N = 3, α = 0.25, σ1 = 18%, σ2 = 15%, σ3 = 12%, c0 = 5.28875%, c1 = 5.37375%, c2 = 5.40%, c3 = 5.40125% and ρ12 = 0.75. strikes K=4% K=4.5% K=4.75% K=5% K=5.15% K=5.25% benchmark 10.2240 6.5386 4.7454 3.1060 2.2599 1.7758 frozen drift 10.2132 6.5326 4.7419 3.1028 2.2582 1.7618 strong Taylor 10.2240 6.5386 4.7454 3.1060 2.2599 1.7758 weak Taylor 10.2266 6.5407 4.7485 3.1064 2.2593 1.7626 Table 2: Swaption values in bps for parameters ǫ1 = 1, α = 0.25, σ1 = 18%, σ2 = 15%, σ3 = 12%, c0 = 5.28875%, c1 = 5.37375%, c2 = 5.40%, c3 = 5.40125% and ρ12 = 0.75. WEAK AND STRONG TAYLOR METHODS FOR NUMERICAL SOLUTIONS OF SDES 13 3.4. The Stochastic Volatility LIBOR Market Model. In this section, we develop a stochastic volatility LMM. The stochastic volatility parameter vt follows a square root process, like in the extensively applied Heston model [Hes93]. The resulting model, called hereafter the stochastic volatility LMM (SVLMM), has the following dynamics under the terminal measure: dLit = σ i(t)Lit j=i+1 1 + αL vtdt+ dW , i = 1, ..., N,(3.13) dvt = κ(θ − vt)dt+ ǫ2 vtdBt, where κ, θ, ǫ2 ∈ R+. The Brownian motions Wt = (W 1t , ...,WNt ) and Bt are ex- pressed under the terminal measure with correlations dW it dBt = ρidt and dW ρijdt for i, j = 1, ...N . We assume additionally that the filtration (Ft)0≤t≤T is gen- erated by both Brownian motions. Observe that the process vt is a time-changed squared Bessel process with dimension δ = 4κθ/ǫ22. If δ ≥ 2, then the point zero is unattainable. So we require 2κθ ≥ ǫ22 for the process vt not to reach zero. 3.4.1. Pricing a multi-LIBOR option. In this section, we aim at approximating the price of an option with payoff depending on the vector L i,k,ǫ1,ǫ2 i,ǫ1,ǫ2 , . . . , L k,ǫ1,ǫ2 We interpret the volatility of the volatility parameter ǫ2 as a parameter on which the LIBOR rates depend. Overall, we parametrise the SVLMM by both ǫ1 and ǫ2 and correct prices in a weak sense introducing Malliavin weights. Proposition 3. Consider the SVLMM (3.13) and assume that the Malliavin co- variance matrix γ(L i,k,0,0 ) is invertible. Then the price of an option with payoff i,k,ǫ1,ǫ2 ), i ≤ k ≤ N , where ψ is a bounded measurable function, can be approx- imated by the weak Taylor approximation of order one: (ǫ1,ǫ2) i,k,ǫ1,ǫ2 )) = P (0, T ) i,k,0,0 + ǫ1EP i,k,0,0 + ǫ2EP i,k,0,0 ,(3.14) where the Malliavin weights ζTi , πTi are given by: ζTi = δ i,k,0,0 )Tγ−1(L i,k,0,0 i,k,ǫ1,0 ,(3.15) πTi = δ i,k,0,0 )Tγ−1(L i,k,0,0 i,k,0,ǫ2 ),(3.16) for t ≤ Ti. Proof. The weights ζTi and πTi are obtained by (2.4). We derive (3.14) by noticing that: i,k,ǫ1,ǫ2 i,k,0,0 i,k,ǫ1,0 i,k,0,ǫ2 i,k,0,0 + ǫ1E i,k,0,0 + ǫ2E i,k,0,0 from Definition 2 for n = 1. � 14 MARIA SIOPACHA AND JOSEF TEICHMANN Example 4. Let N = 2 and consider the SVLMM where the volatility functions σi(t) : [0, Ti] → R for i = 1, 2 are assumed to be constant and in particular σ1(t) = σ1, σ 2(t) = σ2. We derive an approximative formula for the price of a payers swaption with maturity T1 and strike price K. The underlying swap is entered at T1 and has payment dates T2, T3. Under the terminal measure P we can write the SDEs for the LIBOR rates and stochastic volatility as: dvǫ2t = κ θ − vǫ2t dt+ ǫ2 vǫ2t dBt, (1,ǫ1,ǫ2) t = −L (1,ǫ1,ǫ2) t ρ12 (2,ǫ1,ǫ2) 1 + αX (2,ǫ1,ǫ2) t dt+ σ1L (1,ǫ1,ǫ2) vǫ2t dW (2,ǫ2) t = σ2L (2,ǫ2) vǫ2t dW (2,ǫ1) t = ǫ1 (2,ǫ2) vǫ2t dW W 1t and W t are assumed to be correlated, so correlations are as dW t dBt = ρidt and dW 1t dW t = ρ12 for i = 1, 2. The (0, 0)-model is given by: v0t = exp (−κt)(v00 − θ) + θ, (1,0,0) = c1 exp v0t dW ( αc2ρ12 1 + αc2 (2,0) = c2 exp v0t dW (2,0,0) t = c2, with c := v0t dt = θT1 − (exp (−κT1) − 1). As in the previous example, we compare the following option prices: • benchmark price; • frozen drift; • weak Taylor price (3.14). The benchmark price is given by (3.12) with N = 2 and i = 1: swptn 0 = P (0, T )EP (1,ǫ1,ǫ2) (2,ǫ2) (1,ǫ1,ǫ2) (2,ǫ2) −Kα2L(2,ǫ2) The weak Taylor price is obtained by (3.14): swptn 0 = P (0, T ) (1,0,0) (2,0) + α2L (1,0,0) (2,0) −Kα2L(2,0) − 2Kα + ǫ1EP (1,0,0) (2,0) + α2L (1,0,0) (2,0) −Kα2· · L(2,0) − 2Kα + ǫ2EP (1,0,0) (2,0) + α2L (1,0,0) (2,0) −Kα2L(2,0)T1 − 2Kα We calculate the Malliavin weights ζT1 , πT1 as given by (3.15) and (3.16) corre- spondingly. We can express the weight ζT1 as: ζT1 = ζ + ζ2T1 , WEAK AND STRONG TAYLOR METHODS FOR NUMERICAL SOLUTIONS OF SDES 15 with: ζ1T1 = (1,0,0) (1,ǫ1,0) γ−1(L (1,0,0) (2,0,0) (2,0,0) (1,ǫ1,0) γ−1(L (1,0,0) (2,0,0) δW 1t , ζ2T1 = (1,0,0) (1,ǫ1,0) γ−1(L (1,0,0) (2,0,0) (2,0,0) (1,ǫ1,0) γ−1(L (1,0,0) (2,0,0) δW 2t . since ∂ |ǫ1=0 L (2,ǫ1,0) = 0. The partial derivative term with respect to ǫ1 for L given by: (1,ǫ1,0) (1,0,0) (1 + αc2)2 v0sdW = −σ1ρ12β2L(1,0,0)T1 θ(T1 − t)− v00 − θ (exp (−κT1)− exp (−κt)) dW 2t , where β2 = (1+αc2)2 . Similarly the weight πT1 is given by: πT1 = π + π2T1 , with: π1T1 = (l,0,0) (j,0,ǫ2) (1,0,0) (2,0,0) δW 1t , π2T1 = (l,0,0) (j,0,ǫ2) (1,0,0) (2,0,0) δW 2t . Partial derivative terms are equal to: (1,0,ǫ2) (1,0,0) exp (−κt) exp (κs) v0sdBsdW αc2σ1σ2ρ12 1 + αc2 exp (κs) exp (−κT1)− exp (−κs) Doing similar calculations, we derive the second partial derivative: (2,ǫ2) (2,0) σ1σ2Vtdt where Vt = exp (−κt) exp (κs) v0sdBs. 16 MARIA SIOPACHA AND JOSEF TEICHMANN We calculate the Malliavin covariance matrix γ (1,0,0) (2,0,0) and its in- verse. (1 + ρ212)(L (1,0,0) )2σ21 v0t dt ︸ ︷︷ ︸ 2ρ12L (1,0,0) (2,0,0) σ1σ2c 2ρ12L (1,0,0) (2,0,0) σ1σ2c (1 + ρ 12)(L (2,0,0) )2σ22c ⇒ det (1,0,0) (2,0,0) (1,0,0) (2,0,0) )2σ21σ 2(1− ρ212). Hence its inverse is given by, for ρ12 6= 1: γ−1 = (1− ρ212) 1+ρ212 (1,0,0) )2σ21c − 2ρ12 (1,0,0) (2,0,0) σ1σ2c − 2ρ12 (1,0,0) (2,0,0) σ1σ2c 1+ρ212 (2,0,0) )2σ22c If we define Xi = v0t dW t , i = 1, 2 and Y = θ(T1−t)− v (exp (−κT1)− exp (−κt)) dW 2t , we finally obtain the weights as: ζ1T1 = − ρ12β2 X1Y − Cov(X1, Y ) ζ2T1 = ρ212β2 X2Y − Cov(X2, Y ) Moreover, for the weight πT1 we define: exp (−κT1)− exp (−κt) and random variables Di, Zi for i = 1, 2: g(s)dW isdW g(s)dZisdW where the Brownian motions Zit are independent from W t and f(t) = exp (−κt)√ g(s) = exp (κs) v0s . Therefore, we obtain the weights as: π1T1 = X1(ρ1D1 + 1− ρ21Z1) + (αc2(2ρ12σ2 + σ1) + σ1 1 + αc2 (αc2(2ρ12σ2 + σ1) + σ1 1 + αc2 X1(ρ2D2 + 1− ρ22Z2)+ σ1BX1 − σ1ρ1E WEAK AND STRONG TAYLOR METHODS FOR NUMERICAL SOLUTIONS OF SDES 17 where E equals to 1− exp (−κT1) . Similarly we get π2T1 as: π2T1 = X2(ρ2D2 + 1− ρ22Z2) + σ2X2 + 1 − σ2ρ2E − ρ12 X2(ρ1D1 + 1− ρ21Z1) + σ1BX2 (αc2(2ρ12σ2 + σ1) + σ1 1 + αc2 − σ1ρ2E (αc2(2ρ12σ2 + σ1) + σ1 1 + αc2 In this example, the weights are functions of normal variables and double sto- chastic integrals, which are computed via simulation. Table 4 reports the swaption prices in bps with parameters N = 2, α = 1.5, σ1 = 25%, σ2 = 15%, c0 = 5.28875%, c1 = 5.4%, c2 = 5.39%, v0 = 1, ρ1 = −0.75, ρ2 = −0.6, κ = 2.3767, θ = 0.2143, ǫ2 = 25%, ρ12 = 0.63. strikes K=3.5% K=4% K=5% K=6% K=7% K=8% benchmark 3.8984 2.9221 1.2588 0.3858 0.1019 0.0216 (0, 0)-model 3.8951 2.9053 1.2705 0.3966 0.0942 0.0185 weak Taylor 3.8990 2.9159 1.2694 0.3791 0.1042 0.0210 Table 3: Stochastic volatility swaption values in bps for parameters ǫ1 = 1, α = 1.5, σ1 = 25%, σ2 = 15%, c0 = 5.28875%, c1 = 5.4%, c2 = 5.39%, v0 = 1, ρ1 = −0.75, ρ2 = −0.6, κ = 2.3767, θ = 0.2143, ǫ2 = 25%, ρ12 = 0.63. References [BGM97] A. Brace, D. Gatarek, and M. Musiela, The Market Model of Interest Rate Dynamics, Mathematical Finance 7 (1997), no. 2, 127–155. [BM01] D. Brigo and F. Mercurio, Interest Rate Models: Theory and Practice, Springer Finance, Springer, 2001. [BW00] A. Brace and R.S. Womersley, Exact Fit to the Swaption Volatility Matrix using Semi- definite Programming, Working paper, presented at ICBI Global Derivatives Confer- ence, Paris, April 2000, 2000. [EÖ05] E. Eberlein and F. Özkan, The Lévy LIBOR model, Finance and Stochastics 9 (2005), 327348. [Hes93] S. Heston, A Closed-Form Solution for Options with Stochastic Volatility with Applica- tions to Bond and Currency Options, The Review of Financial Studies 6 (1993), no. 2, 327–343. [Jam97] F. Jamshidian, LIBOR and Swap Market Models and Measures, Finance and Stochas- tics 1 (1997), 293–330. [KM97] A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis, American Mathematical Society, 1997. [Mal97] P. Malliavin, Stochastic Analysis, Springer, 1997. [MR98] M. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling, second ed., Springer, 1998. [MSS97] K. Miltersen, K. Sandmann, and D. Sondermann, Closed Form Solutions for Term Structures Derivatives with Log-Normal Interest Rates, Journal of Finance 52 (1997), 409–430. [MT06] P. Malliavin and A. Thalmaier, Stochastic Calculus of Variations in Mathematical Fi- nance, Springer, 2006. 18 MARIA SIOPACHA AND JOSEF TEICHMANN [Nua06] D. Nualart, The Malliavin Calculus and Related Topics, second ed., Springer Verlag, 2006. [Sch02] E. Schlögl, A Multicurrency Extension of the Lognormal Interest Rate Market Models, Finance and Stochastics 6 (2002), 173196. Department of Mathematical Methods in Economics, Vienna University of Technol- ogy, Wiedner Hauptstrasse 8–10/105–1, A-1040 Vienna, Austria. E-mail address: [josef.teichmann,siopacha]@fam.tuwien.ac.at 1. Introduction and Setting 2. Weak and strong Taylor methods - Structure Theorems 3. Applications from Financial Mathematics 3.1. The LIBOR Market Model 3.2. Freezing the Drift 3.3. Correcting the Frozen Drift 3.4. The Stochastic Volatility LIBOR Market Model References

0704.0746

Finite bias visibility of the electronic Mach-Zehnder interferometer

Finite bias visibility of the electronic Mach-Zehnder interferometer Preden Roulleau, F. Portier, D. C. Glattli,∗ and P. Roche† Nanoelectronic group, Service de Physique de l’Etat Condensé, CEA Saclay, F-91191 Gif-Sur-Yvette, France A. Cavanna, G. Faini, U. Gennser, and D. Mailly CNRS, Phynano team, Laboratoire de Photonique et Nanostructures, Route de Nozay, F-91460 Marcoussis, France (Dated: November 28, 2018) We present an original statistical method to measure the visibility of interferences in an electronic Mach-Zehnder interferometer in the presence of low frequency fluctuations. The visibility presents a single side lobe structure shown to result from a gaussian phase averaging whose variance is quadratic with the bias. To reinforce our approach and validate our statistical method, the same experiment is also realized with a stable sample. It exhibits the same visibility behavior as the fluctuating one, indicating the intrinsic character of finite bias phase averaging. In both samples, the dilution of the impinging current reduces the variance of the gaussian distribution. PACS numbers: 85.35.Ds, 73.43.Fj Nowadays quantum conductors can be used to per- form experiments usually done in optics, where electron beams replace photon beams. A beamlike electron mo- tion can be obtained in the Integer Quantum Hall Effect (IQHE) regime using a high mobility two dimensional electron gas in a high magnetic field at low temperature. In the IQHE regime, one-dimensional gapless excitation modes form, which correspond to electrons drifting along the edge of the sample. The number of these so-called edge channels corresponds to the number of filled Lan- dau levels in the bulk. The chirality of the excitations yields long collision times between quasi-particles, mak- ing edge states very suitable for quantum interferences experiments like the electronic Mach-Zehnder interfer- ometer (MZI) [1, 2, 3]. Surprisingly, despite some ex- periments which show that equilibrium length in chiral wires is rather long [4], very little is known about the coherence length or the phase averaging in these ”per- fect” chiral uni-dimensional wires. In particular, while in the very first interference MZI experiment the interfer- ence visibility showed a monotonic decrease with voltage bias, which was attributed to phase noise [1], in a more recent paper, a surprising non-monotonic decrease with a lobe structure was observed [5]. A satisfactory expla- nation has not yet been found, and the experiment has so far not been reported by other groups to confirm these results. We report here on an original method to measure the visibility of interferences in a MZI, when low frequency phase fluctuations prevent direct observation of the peri- odic interference pattern obtained by changing the mag- netic flux through the MZI. We studied the visibility at finite energy and observed a single side lobe structure, which can be explained by a gaussian phase averaging whose variance is proportional to V 2, where V is the FIG. 1: SEM view of the electronic Mach-Zehnder with a schematic representation of the edge state. G0, G1, G2 are quantum point contacts which mimic beam splitters. The pairs of split gates defining a QPC are electrically connected via a Au metallic bridge deposited on an insolator (SU8). G0 allows a dilution of the impinging current, G1 and G2 are the two beam splitters of the Mach-Zehnder interferometer. SG is a side gate which allows a variation of the length of the lower path (b). bias voltage. To reinforce our result and check if low frequency fluctuation may be responsible for that behav- ior, we realized the same experiment on a stable sample : we also observed a single side lobe structure which can be fitted with our approach of gaussian phase averaging. This proves the validity of the results, which cannot be an artefact due to the low frequency phase fluctuations in the first sample. In both samples, the dilution of the impinging current has an unexpected effect : it decreases the variance of the gaussian distribution. The MZI geometry is patterned using e-beam lithogra- phy on a high mobility two dimensional electron gas in a GaAs/Ga1−xAlxAs heterojunction with a sheet density nS = 2.0×1011 cm−2 and a mobility of 2.5×106 cm2/Vs. http://arxiv.org/abs/0704.0746v3 The experiment was performed in the IQHE regime at filling factor ν = nSh/eB = 2 (magnetic field B =5.2 Tesla). Transport occurs through two edge states with an extremely large energy redistribution length [4]. Quan- tum point contacts (QPC) controlled by gates G0, G1 and G2 define electronic beam splitters with transmis- sions T0, T1 and T2 respectively. In all the results pre- sented here, the interferences were studied on the outer edge state schematically drawn as black lines in Fig.(1), the inner edge state being fully reflected by all the QPCs. The interferometer consists of G1, G2 and the small cen- tral ohmic contact in between the two arms. G1 splits the incident beam into two trajectories (a) and (b), which are recombined with G2 leading to interferences. The two arms defined by the mesa are 8 µm long and en- close a 14 µm2 area. The current which is not transmit- ted through the MZI, IB = ID − IT , is collected to the ground with the small ohmic contact. An additional gate SG allows a change of the length of the trajectory (b). The impinging current I0 can be diluted thanks to the beam splitter G0 whose transmission T0 determines the diluted current dID = T0 × dI0. We measure the differ- ential transmission through the MZI by standard lock-in techniques using a 619 Hz frequency 5 µVrms AC bias VAC superimposed to the DC voltage V . This AC bias modulates the incoming current dID = T0 × h/e2 ×VAC , and thus the transmitted current in an energy range close to eV , giving the transmission T (eV ) = dIT /dI0. Using the single particle approach of the Landauer- Büttiker formalism, the transmission amplitude t through the MZI is the sum of the two complex transmission amplitudes corresponding to paths (a) and (b) of the interferometer; t = t0{t1 exp(iφa)t2 − r1 exp(iφb)r2}. This leads to a transmission probabil- ity T (ǫ) = T0{T1T2 + R1R2 + T1R2R1T2 sin[ϕ(ǫ)]}, where ϕ(ǫ) = φa − φb and Ti = |ti|2 = 1 −Ri. ϕ(ǫ) cor- responds to the total Aharonov-Bohm (AB) flux across the surface S(ǫ) defined by the arms of the MZI, ϕ(ǫ) = 2πS(ǫ) × eB/h. The surface S depends on the energy ǫ when there is a finite length difference ∆L = La−Lb be- tween the two arms. This leads to a variation of the phase with the energy, ϕ(ǫ+EF ) = ϕ(EF )+ǫ∆L/(~vD), where vD is the drift velocity. When varying the AB flux, the interferences manifest themselves as oscillations of the transmission; in practice this is done either by varying the magnetic field or by varying the surface of the MZI with a side gate [1, 5, 6]. The visibility of the interfer- ences defined as V = (TMAX−TMIN )/(TMAX+TMIN ), is maximum when both beam splitter transmission are set to 1/2. In the present experiment the MZI is designed with equal arm lengths (∆L = 0) and the visibility is not expected to be sensitive to the coherence length of the source ~vD/max(kBT, eVAC). Thus the visibility pro- vides a direct measurement of the decoherence and/or phase averaging in this quantum circuit. In Ref.[1], 60% visibility was observed at low tem- -0.76 -0.72 -0.68 -0.64 -0.4 -0.2 0.0 0.2 0.4 2 x Visibility δT / T 0.0 0.2 0.4 0.6 0.8 1.0 Transmission T gate voltage (V or V ) (Volt) FIG. 2: Sample #1 a)Transmission T = dIT /dI0 as a func- tion of the gate voltages V 1 and V 2 applied on G1 and G2. (◦) T = T1 versus V 1. (•) T = T2 versus V 2. The solid line is the transmission T obtained with T1 fixed to 1/2 while sweeping V 2 : transmission fluctuations due to interferences with low frequency phase noise appears. b) Stack histogram on 6000 successive transmission measurements as a function of the normalized deviation from the mean value. The solid line is the distribution of transmission expected for a uniform distribution of phases. c)Visibility of interferences as a func- tion of the transmission T2 when T1 = 1/2. The solid line is T2(1− T2) dependence predicted by the theory. perature, showing that the quantum coherence length can be at least as large as several micrometer at 20 mK (and probably larger if phase averaging is the limiting factor). At finite energy (compared to the Fermi en- ergy), the visibility was also found decreasing with the bias voltage[1, 5, 6]. This effect is not due to an increase of the coherence length of the electron source which re- mains determined by eVAC or kBT [7]. In a first exper- iment, a monotonic visibility decrease was found, which was attributed to phase averaging, as confirmed by shot noise measurements [1]. Nevertheless, it remains unclear why and how the phase averaging increases with the bias. In a recent paper, instead of a monotonic decrease of the visibility, a lobe structure was observed for filling factor less than 1 in the QPCs [5]. No non-interacting electron model was found to be able to explain this observation, and although interaction effects have been proposed [8], a satisfactory explanation has not yet been found to ac- count for all the experimental observations. So far, two experiments have shown to two different behaviors, rais- ing questions about the universality of these observations. Here, we report experiments where different samples give consistent results, with a fit to the data clearly demon- strating that our MZI suffers from a gaussian phase av- eraging whose variance is proportional to V 2, leading to the single side lobe structure of the visibility. We have used the following procedure to tune the MZI. We first measure independently the two beam splitters’ transparencies versus their respective gate voltages, the inner edge state being fully reflected. This is shown in figure (2a) where the transmission (T1 or T2) through one QPC is varied while keeping unit transparency for the other QPC. This provides the characterization of the transparency of each beam splitter as a function of its gate voltage. The fact that the transmission vanishes for large negative voltages means that the small ohmic contact in between the two arms can absorb all incom- ing electrons, otherwise the transmission would tend to a finite value. This is very important in order to avoid any spurious effect in the interference pattern. In a sec- ond step we fix the transmission T1 to 1/2 while sweep- ing the gate voltage of G2 (solid line of figure (2a)). Whereas for a fully incoherent system the T should be 1/2× (R2 + T2) = 1/2, we observe large temporal trans- mission fluctuations around 1/2. We show in the fol- lowing that they result from the interferences, expected in the coherent regime, but in presence of large low fre- quency phase noise. This is revealed by the probabil- ity distribution of the transmissions obtained when mak- ing a large number of transmission measurements for the same gate voltage. Figure (2b) shows a histogram of T when making 6000 measurements (each measurement be- ing separated from the next by 10 ms). The histogram of the transmission fluctuations δT = T − Tmean displays two maxima very well fitted using a probability distribu- tion p(δT /Tmean) = 1/(2π 1− (δT /Tmean)2/V2) (the solid line of figure (2b)). This distribution is obtained assuming interferences δT = Tmean×Vsin(ϕ) and a uni- form probability distribution of ϕ over [−π,+π]. Note that the peaks around |δT /Tmean| = V have a finite width. They correspond to the gaussian distribution as- sociated with the detection noise which has to be convo- luted with the previous distribution. Although no regular oscillations of transmission can be observed due to phase noise, we can directly extract the visibility of the interferences by calculating the vari- ance of the fluctuations (the approach is similar to mea- surements of Universal Conductance Fluctuations via the amplitude of 1/f noise in diffusive metallic wires) [10]. As expected when T1 = 1/2, the visibility extracted by our method is proportional to T2(1− T2), definitively showing that fluctuations results from interference: we are able to measure the visibility of fluctuating interfer- ences (see figure (2c)). The visibility depends on the bias voltage with a lobe structure shown in figure (3), confirming the pioneer- ing observation [5]. Nevertheless, there are marked dif- ferences. The visibility shape is not the same as that in ref.[5]. We have always seen only one side lobe, al- though the sensitivity of our measurements would be high enough to observe a second one if it existed. Moreover, -100 -50 0 50 100 = 0.02 = 0.14 Drain-Source Voltage (µV) FIG. 3: (Color online) Sample #1 : Visibility of the inter- ferences as a function of the drain-source voltage I0h/e 2 for three different values of T0. The curves are shifted for clar- ity. The energy width of the lobe structure is modified by the dilution whereas the maximum visibility at zero bias is not modified. Solid lines are fits using equation (1). From top to bottom, T0 = 0.02 and V0 = 31 µV, T0 = 0.14 and V0 = 22 µV, T0 = 1 and V0 = 11.4 µV. the lobe width (see figure (3)) can be increased by di- luting the impinging current with G0, whereas no such effect is seen for G1 and G2. This apparent increase of the energy scale cannot be attributed to the addition of a resistance in series with the MZI because G0 is close to the MZI, at a distance shorter than the coherence length. An almost perfect fit for the whole range of T0 (dilu- tion), is V = V0e−V 2/2V 2 0 |1− V ID dID/dV |, (1) where V0 is a fitting parameter. Equation (1) is obtained when assuming a gaussian phase averaging with a vari- ance < δϕ2 > proportional to V 2 and a length difference ∆L small enough to neglect the energy dependence of the phase in the observed energy range eV ≪ ~vD/∆L. In such a case, the interfering part of the current I∼ is thus proportional to ID sin(ϕ). The gaussian distribu- tion of the phase leads to I∼ ∝ ID sin(< ϕ >)e−<δϕ 2>/2, where < ϕ > is the mean value of the phase distribu- tion. The measured interfering part of the transmission, T∼ = h/e2 dI∼/dV gives a visibility corresponding to for- mula (1) when < δϕ2 >= V 2/V 2 . Such behavior gives a nul visibility accompanied with a π shift of the phase when V ID/(V dID/dV ) = 1. When T0 ∼ 1, ID is pro- portional to V and the width of the central lobe is simply equal to 2V0. However in the most general case, dID/dV varies with V . One can see in figure (3) that the fit with Equation (1) is very good, definitively showing that the -20 0 20 0.4 0.6 0.8 / dI 0.5 0.6 0.7 / dI V (µV) -40 -20 0 20 40 d) T = 0.06 Drain Source Voltage (µV) FIG. 4: (Color online) Sample #2 : a) Gray plot of the transmission T as a function of the bias voltage V and the side gate voltage VSG. Note the π shift of the phase when the visibility reaches 0. b) & c) T as a function of the side gate voltage for two different values of the drain source volt- age corresponding to the dashed line of a) (0 and 16 µV respectively). d) Lobe structure of the visibility fitted using equation (1) for a diluted and an undiluted impinging current. existence of one side lobe, as observed in the experiment of ref.[5] at ν = 2 (for the highest fields) and at ν = 1, can be explained within our simple approach. Concern- ing multiple side lobes, we cannot yet conclude if they do arise from long range interaction as recently proposed by ref.[8]. Our geometry is different from the one used in the earlier experiment [5] and the coupling between counter propagating edge states, thought to be responsible for multiple side lobe [8], should be less efficient here. To check if low frequency fluctuations have an impact on the finite bias phase averaging, we have studied an- other sample, with the same geometry and fabricated simultaneously (sample #2), which exhibits clear inter- ference pattern (see Figure (4a,b,c)). As one can remark on figure (4d), the lobe structure is well fitted with our theory, definitively showing that the gaussian phase av- eraging is not associated with low frequency phase fluc- tuations. It is noteworthy that V0 increases (see figure (5)) with the dilution, namely when the transmission T0 at zero bias decreases. An impact of the dilution was already observed as it suppressed multiple side lobes [9] (arXiv version of Ref.[5]), but the conclusion was that the width of the central lobe was barely affected. Here, dilution plays a clear role whose T0 dependence is the same for the two studied samples, once normalized to the not di- luted case. This dilution effect is nevertheless not easy to explain. For example, mechanisms like screening, in- tra edge scattering and fluctuations mediated by shot noise should have maximum effect at half transmission, in contradiction with figure (5). More generally, it is difficult to determine if the process responsible for the phase averaging introduced in our model is located at the beam splitters, or is uniformly distributed along the interfering channels. However, setting T1 = 0.02 or 0.05, keeping T2 = 0.5, leaves the lobe width unaffected. This shows that, if located at the Quantum Point Contacts, the phase averaging process is independent of transmis- sion. 0.0 0.2 0.4 0.6 0.8 1.0 Sample #1 : V (1) = 13.7 µV Sample #2 : V (1) = 10.6 µV FIG. 5: (Color online) V0 obtained by fitting the visibility with equation (1), normalized to V0 at T0 = 1, as a function of T0 at zero bias. To summarize, we propose a statistical method to mea- sure the visibility of ”invisible” interferences. We observe a single side lobe structure of the visibility on stable and unstable samples which is shown to result from a gaus- sian phase averaging whose variance is proportional to V 2. Moreover, this variance is shown to be reduced by diluting the impinging current. However, the mechanism responsible for such type of phase averaging remains yet unexplained. The authors would like to thank M. Büttiker for fruit- ful discussions. This work was supported by the French National Research Agency (grant n◦ 2A4002). ∗ Also at LPA, Ecole Normale Supérieure, Paris. † Electronic address: [email protected] [1] Y. Ji, Y. Chung, D. Sprinzak, M. Heiblum, D. Mahalu, and H. Shtrikman, Nature 422, 415 (2003). [2] P. Samuelsson, E. V. Sukhorukov, and M. Büttiker, Phys. Rev. Lett. 92, 026805 (2004). [3] I. Neder, N. Ofek, Y. Chung, M. Heiblum, D. Mahalu, and V. Umansky, arXiv:0705.0173 (2007). [4] T. Machida, H. Hirai, S. Komiyama, T. Osada, and Y. Shiraki, Solid State Commun. 103, 441 (1997). mailto:[email protected] http://arxiv.org/abs/0705.0173 [5] I. Neder, M. Heiblum, Y. Levinson, D. Mahalu, and V. Umansky, Phys. Rev. Lett. 96, 016804 (2006). [6] L. V. Litvin, H.-P. Tranitz, W. Wegscheider, and C. Strunk, Phys. Rev. B 75, 033315 (2007). [7] V. S.-W. Chung, P. Samuelsson, and M. Büttiker, Phys. Rev. B 72, 125320 (2005). [8] E. V. Sukhorukov and V. V. Cheianov, cond-mat/0609288 . [9] I. Neder, M. Heiblum, Y. Levinson, D. Mahalu, and V. Umansky, cond-mat/0508024 (2005). [10] All the results on the visibility reported here on sam- ple #1 have been obtained using the following proce- dure : we measured N = 2000 times the transmission and calculated the mean value Tmean and the variance < δT 2 >. It is straightforward to show that the visi- bility is V = < δT 2 > − < δT 2 >0/Tmean, where < δT 2 >0 is the measurement noise which depends on the AC bias amplitude, the noise of the amplifiers and the time constant of the lock-in amplifiers (fixed to 10 ms), measured in absence of the quantum interferences. http://arxiv.org/abs/cond-mat/0609288 http://arxiv.org/abs/cond-mat/0508024 0.2 0.4 0.6 0.6 d) Magnetic Field - 4.6 T (mT) 0.2 0.4 0.6 0.6 c) 0.2 0.4 0.6

0704.0747

A note on higher-order differential operations

arXiv:0704.0747v1 [math.DG] 5 Apr 2007 Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 7 (1996), 105–109. A NOTE ON HIGHER-ORDER DIFFERENTIAL OPERATIONS Branko J. Malešević In this paper we consider successive iterations of the first-order differential operations in space R 1. INTRODUCTION Let C∞(R3) be the set of scalar functions f = f(x1, x2, x3) : R 3 7→ R which have the continuous partial derivatives of the arbitrary order on coordinates xi (i = 1, 2, 3). Let ~C∞(R3) be the set vector functions ~f = f1(x1, x2, x3), f2(x1, x2, x3), f3(x1, x2, x3) : R3 7→ R3 which have the coordinately continuous partial deriva- tives of the arbitrary order on coordinates xi (i = 1, 2, 3). First-order differential operations of the vector analysis of the space R3 are defined on the following set of functions: f : R3 7→ R | f ∈ C∞(R3) and ~F = ~f : R3 7→ R3 | ~f ∈ ~C∞(R3) First-order differential operations of the vector analysis of the space R3 are defined as the following three linear operations [1], denoted here by ∇1,∇2 and ∇3 for a convenience: (1) grad f = ∇1f = ~e1 + ~e2 + ~e3 : F 7→ ~F , (2) curl ~f = ∇2 ~f = ~e1 + ~e2 + ~e3 : ~F 7→ ~F , (3) div ~f = ∇3 ~f = : ~F 7→ F. Let Ω = {∇1,∇2,∇3} be the set of above defined operations and let Σ = F ∪ ~F . Then the first-order differential operations can be considered as partial operations Σ 7→ Σ, i.e. as operations whose domain (and codomain) are subsets F or 01991 Mathematics Subject Classification: 26B12 http://arxiv.org/abs/0704.0747v1 106 Branko J. Malešević ~F of Σ. Second and higher-order differential operations are then defined as products of operations in Ω in the sense of composition of operations. Some of these products might be meaningful, like ∇3 ◦ ∇1, while the others are meaningless, like ∇1 ◦ ∇1. To all meaningless products for any argument we associate the value of nowhere defined function ϑ (Dom (ϑ) = ∅ and Ran (ϑ) = ∅). Nowhere defined function ϑ(f∅) is a concept from the recursive function theory [2]. We do not consider the function ϑ as the starting argument for calculating the value of the higher-order differential operations. In that way we increase set Σ into set Σ = F ∪ ~F ∪ {ϑ}. All meaningful second-order differential operations are: (4) ∆f = div grad f = (∇3 ◦ ∇1) (f), (5) curl curl ~f = (∇2 ◦ ∇2) (~f), (6) graddiv ~f = (∇1 ◦ ∇3)(~f), (7) div curl ~f = (∇3 ◦ ∇2) (~f) = 0, (8) curl grad f = (∇2 ◦ ∇1) (f) = ~0, f, ~f ∈ Σ \ {ϑ}. In this paper we consider higher-order differential operations, search for mean- ingful ones and present some applications. 2. HIGHER-ORDER DIFFERENTIAL OPERATIONS Theorem 1. For arbitrary operations ∇i,∇j ,∇k ∈ Ω (i, j, k ∈ {1, 2, 3}) and argument ξ ∈ Σ \ {ϑ} the associative law holds: (9) ∇i ◦ (∇j ◦ ∇k)(ξ) = (∇i ◦ ∇j) ◦ ∇k(ξ). Proof. Choosing the ∇i,∇j ,∇k from Ω and argument ξ from Σ\{ϑ}, (9) appears in 54 possible cases. It is directly verified that whenever the left side of the equality is meaningless, the right side is also meaningless. Than, all meaningless products have the same value of the nowhere defined function ϑ, so that (9) is true in the following form: ϑ = ϑ. Also, whenever the left side of equality is meaningful, the right side is also meaningful. Then, according to the associative law of the meaningful functions, we conclude that (9) is true. From Theorem 1 it follows (by induction) that the generalized associative law also holds, so we may write the product ∇i1 ◦ ∇i2 ◦ · · · ◦ ∇in without brackets (ij ∈ {1, 2, 3} : j = 1, 2, ..., n). For higher-order differential operations, given as meaningful products, we say that they are the trivial products if they are trivially anullated, i.e. if they are identically the same as the anullating functions 0, ~0 from Σ. Otherwise, we refer to the higher-order differential operations, given as meaningful products, as nontrivial products (if they are nontrivially anullated). Next, we prove the statement: A note on higher-order differential operations 107 Theorem 2. Higher-order differential operations appear as nontrivial products in the following three forms: (grad) div . . . graddiv grad f = (∇1◦)∇3 ◦ · · · ◦ ∇1 ◦ ∇3 ◦ ∇1f, curl curl . . . curl curl curl ~f = ∇2 ◦ ∇2 ◦ · · · ◦ ∇2 ◦ ∇2 ◦ ∇2 ~f, (div) grad . . . div graddiv ~f = (∇3◦)∇1 ◦ · · · ◦ ∇3 ◦ ∇1 ◦ ∇3 ~f, for arbitrary functions f, ~f ∈ Σ \ {ϑ}, where terms in brackets are included for odd number of terms and are left out otherwise. All other meaningful operations are identically zero in their domain. Proof. Meaningful third-order differential operations appear in the form of eight compositions as follows: (10) graddiv gradf = ∇1 ◦ ∇3 ◦ ∇1f, (11) curl curl curl ~f = ∇2 ◦ ∇2 ◦ ∇2 ~f, (12) div graddiv ~f = ∇3 ◦ ∇1 ◦ ∇3 ~f, (13) div curl curl ~f = ∇3 ◦ ∇2 ◦ ∇2 ~f = 0, (14) div curl gradf = ∇3 ◦ ∇2 ◦ ∇1f = 0, (15) curl curl grad f = ∇2 ◦ ∇2 ◦ ∇1f = ~0, (16) curl graddiv ~f = ∇2 ◦ ∇1 ◦ ∇3 ~f = ~0, (17) graddiv curl ~f = ∇1 ◦ ∇3 ◦ ∇2 ~f = ~0, f, ~f ∈ Σ \ {ϑ}. Anullations of the operations (13)–(17) follow directly from the anullations (4)–(5). The statement follows directly from the principle of mathematical induction by means of using the general associative law and formulas (10)–(17). For a given sequence of operations ∇i1 ,∇i2 , . . . ,∇in from the set Ω of func- tions, let define the concept of the collection of functions as a subset of functions Θ ⊆ Σ \ {ϑ} such that all functions ξ from Θ anullate the nontrivial product ∇i1 ◦ ∇i2 ◦ · · · ◦ ∇in (ξ). Let us form some collections. Scalar functions f from Σ, such that ∆nf = 0 is true, define harmonic collection Hn of order n, as the form of the polyharmonic functions. Let us notice that in the case of two dimensions there is a general form of polyharmonic functions f as a solution of the equation ∆nf = 0, [3]. Vector functions ~f from Σ, such that curln ~f = ~0 is true, define curling collection Cn of order n. We can remark that besides the total scalar operation ∆ : F 7→ F (partial scalar operation ∆ : Σ 7→ Σ) we can also consider the total vector operation ~∆ : ~F 7→ ~F (partial vector operation ~∆ : Σ 7→ Σ) defined by: (18) ~∆~f = (∆f1,∆f2,∆f3) = ∆f1 · ~e1 +∆f2 · ~e2 +∆f3 · ~e3. 108 Branko J. Malešević Let set ~Hn be the sign for the vector functions ~f from Σ such that ~∆ n(~f) = ~0, where ~∆n is iteration of order n of the vector operation ~∆ given by (18). The set of vector harmonic functions ~Hn of order n, which is defined in such a way, is not in the list of collections which appear in the previous theorem because it is not obtained through the compositions of operations (1)–(3). For the set ~Hn we shall keep the term collection. Let us notice that for scalar polyharmonic collections, vector polyharmonic collections and curling collections, related to the index-order, the following inclu- sions hold: (19) H ⊂ H2 ⊂ · · · ⊂ Hn−1 ⊂ Hn ⊂ · · · , (20) ~H ⊂ ~H2 ⊂ · · · ⊂ ~Hn−1 ⊂ ~Hn ⊂ · · · , (21) C ⊂ C2 ⊂ · · · ⊂ Cn−1 ⊂ Cn ⊂ · · · . Let emphasize that all previous considerations can be transformed in three- dimensional orthogonal curvilinear coordinate system by introducing of correspond- ing presumptions for functions from the sets F, ~F and Lamé’s coefficients. Finally, let state a few examples where scalar and vector polyharmonic col- lections appear. Example 1. All meaningful products of third-and-higher-order differential opera- tions for vector functions ~f ∈ ~H and scalar functions f ∈ H are anullated. For vector functions ~f ∈ ~H the following equation holds: (22) curl curl ~f = graddiv ~f. Hence, for f ∈ H and ~f ∈ ~H, on the basis of formulas (22) and (10)–(17) the following is true: graddiv gradf = grad (∆f) = ~0, curl curl curl ~f = curl (graddiv) ~f = ~0, div graddiv ~f = div (curl curl) ~f = 0. Thus, all eight meaningful products of third-order differential operations are anul- lated, so that the statement is true. Example 2. If f ∈ Hn−1, then x · f ∈ Hn, n ≥ 2. Let us notice that if f ∈ F, then x · f ∈ F. For an arbitrary scalar function f ∈ F the following equation is directly verified: ∆(x · f) = 2∂f/∂x+ x ·∆(f). Inductive generalization is the following equation: ∆n(x · f) = 2n · ∂ ∆n−1(f) /∂x+ x ·∆n(f). Thus, for (n− 1)-harmonic function f ∈ Hn−1 the conclusion x · f ∈ Hn is true. A note on higher-order differential operations 109 Example 3. If f ∈ Hn−1, then (x 2 + y2 + z2) · f ∈ Hn, n ≥ 2. Let us notice that if f ∈ F, then (x2 + y2 + z2) · f ∈ F. For the arbitrary scalar function f ∈ F the following equations are directly verified: ∆(x2 · f) = 2 · f + 4x · ∂f/∂x+ x2 ·∆(f), ∆2(x2 · f) = 8 · ∂2f/∂x2 + 8x · ∂ /∂x+ 4 ·∆(f) + x2 ·∆2(f). Inductive generalization is the equation as follows: ∆n(x2 · f) = 4n(n− 1) · ∂2 ∆n−2(f) + 4nx · ∂ ∆n−1(f) /∂x+ 2n ·∆n−1(f) + x2 ·∆n(f). Thus, if f ∈ Hn−1, then (x 2 + y2 + z2) · f ∈ Hn. Two previous examples are the generalizations of the corresponding problems con- tained in [4]. Acknowledgement. I wish to express my gratitude to Professors M. Merkle, I. Lazarević and D. Tošić who examined the first version of paper and gave me their suggestions and some very useful remarks. REFERENCES 1. M. L. Krasnov, A. I. Kiselev, G. I. Makarenko: Vector Analysis. Moscow 1981. 2. N. Cutland: Computability. Cambridge University Press, London 1980. 3. D. S. Mitrinović, J. D. Kečkić: Jednačine matematičke fizike. Beograd 1985. 4. D. S. Mitrinović, in association with P. M. Vasić: Diferencijalne jednačine, Novi zbornik problema 4. Beograd 1986. 5. M. J. Crowe: A History of Vector Analysis. University of Notre Dame Press, London 1967. Faculty of Electrical Engineering, (Received May 6, 1996) University of Belgrade, P.O.B 816, 11001 Belgrade, Yugoslavia [email protected]

0704.0748

Experimental Challenges Involved in Searches for Axion-Like Particles and Nonlinear Quantum Electrodynamic Effects by Sensitive Optical Techniques

EXPERIMENTAL CHALLENGES INVOLVED IN MEASURING POLARIZATION CHANGES OF LIGHT PROPAGATING THROUGH A MAGNETIC FIELD IN VACUUM EXPERIMENTAL CHALLENGES INVOLVED IN SEARCHES FOR AXION- LIKE PARTICLES AND NONLINEAR QUANTUM ELECTRODYNAMIC EFFECTS BY SENSITIVE OPTICAL TECHNIQUES Christopher C. Davis1*, Joseph Harris2, Robert W. Gammon3, Igor I. Smolyaninov1 and Kyuman Cho4 1Department of Electrical and Computer Engineering University of Maryland, College Park, MD 20742 2Department of Physics, University of Maryland, College Park, MD 20742 3Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742 4Department of Physics, Sogang University, Seoul, Korea We discuss the experimental techniques used to date for measuring the changes in polarization state of a laser produced by a strong transverse magnetic field acting in a vacuum. We point out the likely artifacts that can arise in such experiments, with particular reference to the recent PVLAS observations and the previous findings of the BFRT collaboration. Our observations are based on studies with a photon-noise limited coherent homodyne interferometer with a polarization sensitivity of 2×10-8 rad Hz 1/2 mW-1/2. P.A.C.S. numbers: 42.15.Eq, 14.80.Mz, 12.20-m, 13.40.-f Introduction The most important magneto-optical interactions that can occur in material media are the Faraday effect, magnetic dichroism, and magnetic birefringence (the Cotton- Mouton effect). Quantum electrodynamics predicts that because of photon-photon interactions even the vacuum becomes birefringent in the presence of a strong magnetic field [1-5]. Further, the interaction with an axion-like particle and two photons via the Primakoff effect will also lend optical properties to the vacuum in the presence of a strong magnetic field [6-10]. The occurrence of an apparent magnetic dichroism of the vacuum would imply the preferential disappearance of left- or right circularly polarized photons from a light beam. To conserve mass and energy this would imply either the production of particles, or photon-splitting. The QED effect and the axion effect are treated in terms of an effective Lagrangian [1-7], in units where 1c= =h and . 2 / 4 1/137eα π= ≈ ( ) ( ) ( ) 2 22 2 2 1 7 1 4 90 4 2 4ae L F F F F F F a a m a F F µν µν µν µ µν µν µν µν µ µν α ⎡ ⎤ = − + + + ∂ ∂ − +⎢ ⎥⎣ ⎦ % %1 a (1) Where the first half of the expression is the Euler-Heisenberg effective Lagrangian, which is appropriate to the QED effect, and the second half is the effective Lagrangian, which is appropriate to the Primakoff effect and accounts for the axion. Here, is the axion field, is the axion mass, and am M is the inverse axion coupling constant. Raffelt and Stodolsky [7] synthesize the results of Adler [4] and solve for the equations of motion. Analysis of the classical wave solutions of the equations of motion produces a picture of mixing between photon and axion modes in a polarized laser experiment with a static transverse magnetic field and an optical cavity to increase path length. In such an experiment, CP arguments predict that the axion will only couple to the parallel components of the beam. Thus, two main effects are predicted. The first effect is a phase difference ∆φ=φ||-φ⊥ between the parallel and perpendicular components of polarized light interacting with the magnetic field. This arises from both QED and the preferential mixing of axion and photon modes. In the mixing part of this picture, a photon mode oscillates into an axion mode before turning back into a photon and gets out of phase. In both cases, this phase difference causes an apparent birefringence. The second main effect, is an apparent linear dichroism which manifests itself as a rotation,ψ ,of the polarization and attenuation. This is caused by the fact that mirrors do not reflect axions and, hence, any axion modes that do not oscillate back to photons before hitting the mirror will appear as lost parallel photon modes. For small axion masses, the theory predicts: φ ω∆ = , ext a ∆ = and lψ = , (2) where l is the length of the cavity, N is the number of passes and L Nl= is the total path length of the beam through the interaction region. The subscripts QED and a, refer to the origin of the effects. In terms of index of refraction, ∆φ=kL(n||-n⊥). Choosing the limit of small axion masses is justified by several experimental results and astrophysical observations [6 -11] which bound the axion mass to and 3 610 10aeV m eV − −> > 1010M GeV> . This result also takes into account Adler’s analysis of the E-H Lagrangian which predicts the following vacuum birefringence: n⊥=1+2ξsin2θ, n||=1+ θξ 2sin 7 with = . (3) Here,θ is the angle between and k .[4,7] extB The expected birefringence, as a function of extB in Tesla, due to the QED effect, is ∆n=n||-n⊥=4×10-23Bext2. The phase shift between the two orthogonal components of a light beam is 2 /L nφ π∆ = ∆ λ . For input light linearly polarized at 45o to the field direction, this translates into an induced ellipticity of the light /L nε π λ= ∆ , for a path length L. For a 1m path and a 1T field the induced ellipticity is expected to be 1.2×10-16. No experiment to date has achieved this sensitivity. The BRFT and PVLAS Experiments Two important experiments have attempted to detect the phenomena that would result from the Primakoff effect. In the BRFT experiment [12] an upper limit of 3.5×10-10 rad was determined for the possible rotation angle for a 2.2km path in a 3.25T field, equivalent to 1.5×10-14 rad m-1T-2, and an ellipticity of 1.6×10-9 was measured on a 299m path in a 3.25T field. The PVLAS experiment [13] claims a rotation of 1.7×10-7rad for a 44km long path in a 5T field, equivalent to 1.55×10-14 rad m-1T-2. The BRFT and PVLAS experiments differ in several important specific ways, although from the standpoint of applying a modulated magnetic field they are similar. BRFT uses a transverse magnetic field modulated at a frequency of 32mHz about a background level of 3.25T. PVLAS uses a transverse magnetic field that rotates around the light propagation axis at 1.89rad/s. This field is equivalent to the simultaneous application of two orthogonal transverse field components oscillating at 0.3Hz, but in quadrature. Neither the BRFT nor PVLAS experiments operated at the photon noise limit. The BRFT experiment used a 200mW argon ion laser and achieved a sensitivity of 4.7×10-7 rad Hz ½ m W -1/2. The PVLAS experiment used a 100mW 1.06µm Nd:YAG laser and achieved a sensitivity of 10-6 rad Hz-1/2 mW-1/2. The photon noise limit at 1.06µm for a detector with a responsivity of 0.4 A/W (a typical value for a Si photodiode at this wavelength) is 2×10-8 rad Hz 1/2 mW 1/2. Discussion We have for several years operated a balanced coherent homodyne polarization interferometer for the study of the Faraday and Cotton-Mouton effects in condensed matter [14], and have achieved a photon noise limited sensitivity of 2×10-8 rad Hz-1/2 mW-1/2 at 632.8nm or 1.06µm. Because we have only a 1kGauss modulated transverse field magnet with 0.1m pole pieces we could not compete with the BRFT and PVLAS experiments in overall sensitivity since we were a factor of 2.3×104 mT2 below BRFT and a factor 109mT2 below PVLAS in terms of path length and field strength. However, our experience with a very sensitive system for measuring elipticity has taught us much about the potential pitfalls of these experiments from an experimental optics standpoint. It is clear to us that the PVLAS experiment suffers from artifacts, as has already been pointed out by Melissinos [15], that the BRFT experiment suffers from artifacts has been acknowledged by its authors, although they do not specify all the sources of these spurious signals. A primary source of spurious signals in sensitive experiments of this kind is motion of optical components caused by a time-varying or a rotating magnetic field. The BRFT experiment acknowledges this and used a feedback system to attempt to minimize its effects. The PVLAS data show clear sideband peaks corresponding to the rotation frequency of their magnet, which should not be present for an effect proportional to B2. Indeed these peaks are approximately 18 times larger than the “real” signal at twice the magnet rotation frequency. They do not explain the origin of the fundamental signal but interpret the second harmonic signal as resulting from an interaction involving a light, neutral, spin-zero particle. In both the BRFT and PVLAS experiments optical components are either close to the magnet or mechanically coupled to the magnet and its cryostat. A primary component of the experiment that is strongly affected by the magnetic field is the evacuated tube passing though the magnet. This tube extends to the cavity end mirrors. All components in the experiment that experience any modulated field or field gradients will experience time-varying diamagnetic or paramagnetic forces. For example, any stainless steel or aluminum optical mounts will experience paramagnetic forces. There are torques acting on induced magnetic dipoles, especially in any components exposed to the field that are not absolutely symmetrically placed with respect to the field direction. A quartz sample tube in the magnet will experience the strongest forces in the regions where it leaves the magnet and experiences the largest field gradients, and will be pulled into the magnet bore. In general time-varying forces all result from any changes in magnetic stored energy that occur as the field is modulated. This generalized force on an object is F= ∫ ⋅∇− .21 dVHB In our sensitive magneto-optical experiments we have verified that significant artifacts can result from any modulated feedback of light into the laser [16]. It has been shown that if a part of its own field is fed back into a laser by an optical component vibrating with small amplitude, then in the weak feedback regime, phase and amplitude of the output beam from the laser are synchronously modulated [17]. This effect is so efficient that when the source laser is influenced by the feedback the modulated light can cause interference in a sensitive measurement even for a balanced homodyne interferometer measuring an extremely small signal. We have performed a rigorous study of the feedback effect for the case of a balanced homodyne polarization interferometer. As a result, we have been able to detect phase and/or amplitude modulation produced in a balanced homodyne polarization interferometer when light from a mirror oscillating with an amplitude of only 9nm is fed back into the laser with 120dB of attenuation. This effect is still present even if the laser is an extremely low phase noise Nd:YAG ring laser [17]. The BRFT experiment is less sensitive to this feedback effect because it uses a multipass, zig-zag Herriott type cavity [18,19] rather than a spherical Fabry-Perot cavity. It is possible for light scattered by any of the optical components in these experiments to cause feedback, even if no specific optical component is used in the normal direction, and this includes scattered light that reflects off the inside walls of the evacuated tube inside the magnet. The BRFT experiment uses a single optical isolator, which probably does not provide sufficient isolation to prevent feedback modulation effects. It appears that, according to the experimental arrangement shown in ref [11], the PVLAS experiment does not use an optical isolator after its laser. In principle, the Fabry-Perot resonator might not reflect significant incident light if the source laser is perfectly frequency locked to the resonator. In practice, however, even for a very high-Q resonator, it is impossible to avoid the feedback due to imperfectness of mirrors and locking electronics. Therefore, in the PVLAS experiment, the feedback modulation effects may cause major interference in measurements. In principle, any correlated intensity noise can be rejected in a balanced homodyne interferometer. However, because of the imperfect performance of real optical and/or electronic components, overall common mode rejection ratio of the interferometer used in our study was approximately 40dB. Synchronous feedback can cause interference in a sensitive experiment even when the signal level is very low. In the case of the PVLAS scheme, by including the feedback effect synchronized at twice the rotating frequency of the magnet, the representation for the light intensity transmitted through the crossed polarizers of the ellipsometer given in Eq (2) of Ref. [11] can be rewritten as 2 20 ,2( ){ [ ( ) ( ) ( )] }mNI I I t t tν σ α η= + + + +Γ where 0I , 2σ , α , η , and Γ have the same meaning as in Ref [11] and ,2 mNI ν is the intensity modulation caused by the feedback. The frequency of this synchronized modulation is given by the vibration frequency of a feedback element, twice the frequency of the rotating magnet. Small misalignment between the polarization components must be included in the quasi-static, uncompensated rotation and ellipticity, , which is much larger than the rotation caused by the Primakoff effect. Thus the term Γ ,22 ( )mN ( )I t tν η Γ in the above equation has not only the same Fourier frequency as 02I αη but also has the same phase relationship when the quarter-wave plate is rotated by 90o. The synchronous interference, thereby cannot be distinguished from the magneto-optical effect being sought. An important, but subtle distinction between the BRFT and PVLAS experiments is that the BRFT uses a mode-matched mirror cavity while the PVLAS apparently does not. Consequently, in the PVLAS experiment as the light beam oscillates between the two cavity mirrors its spot size and radius of curvature both oscillate and the radius of curvature does not match the mirror curvatures. This mismatch in radius leads to local non-normal incidence on the cavity mirrors (except on axis) and causes the local P-and S- polarization components of the beam to suffer different phase shifts, which vary radially on the mirror. A calculation for a typical very high reflectance multilayer mirror shows that this phase difference can be easily 10-11 rad per reflection for an incidence angle of 1.5mrad. The PVLAS cavity is subject to these effects, which would be modulated if the cavity mirrors move, although the BRFT cavity is not. A potential confounder in a search for vacuum magneto optic effects is the Faraday effect resulting from residual axial field components and trace gas. There are residual axial field components in both the BRFT and PVLAS experiments, since the local wave-vector directions in a Gaussian beam are only nominally perpendicular to a transverse field at the beam waist, or on axis. We do not however, believe that these were the sources of sidebands at the magnet oscillation or rotation frequency ωm. Nonetheless, an experiment in which there is no obvious modulation of the effect at frequency ωm is desirable, since an effect proportional to Bext2 only shows up at frequency 2ωm. In an experiment in which the entire field is modulated at frequency ωm a Faraday effect signal, or spurious signal, at ωm is distinguished from the desired signal at frequency 2ωm, which should be further checked by verifying that the desired signal is proportional to Bext2. A complication can arise if the magnet modulation is not a pure harmonic at frequency ωm. Any second harmonic of the magnetic field can produce a spurious signal at 2ωm, but this can be identified since it will be linear in Bext. Features of an Improved Experiment It is our belief that a balanced coherent homodyne interferometer is a better instrument to use than an extinction-based ellipsometer in a search for vacuum magneto- optical effects. Such a system is almost guaranteed to achieve the photon noise limit and provides excellent common mode rejection of laser noise. We also believe that any effect observed should be demonstrated to scale with Bext2 [14]. It will also be desirable to use the largest magnetic field possible, but not to modulate this. An experiment similar to PVLAS can then be performed by rotating the optical train at angular frequency ωm. Conclusions We believe that we have identified the likely causes of artifacts in the PVLAS experiment, and therefore suggest that the case for an interaction involving an axion-like particle has not been made. Furthermore, the PVLAS experiment contradicts the findings of the BRFT experiment, and a series of astrophysical observations that restrict the range of axion particle masses that are possible. An improved experimental arrangement is needed to pursue vacuum magnetic birefringence and polarization rotation effects. With an improved system, detection of the QED- predicted magnetic birefringence [4,5] should be possible, and a more sensitive examination of the existences of any axion-like interactions. * Corresponding author Email address: [email protected] [1] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936). [2] V.F. Weisskopf, K. Dan. Vidensk.Selsk.Mat.Fys.Medd, 14, 6 (1936). [3] J. Schwinger, Phys.Rev. 82, 664 (1951). [4] S.L. Adler, Ann Phys. (N.Y.) 67,599 (1971). [5] S. L. Adler, J. Phys. A 40, F143 (2007). [6] P. Sikivie, Phys. Rev. Lett. 51, 1415 (1983). [7] G. Raffelt and L. Stodolsky, Phys. Rev. D 37, 1237 (1988). [8] S.J. Asztalos et al. Ann.Rev.Nucl.Part.Sci. 56, 293 (2006). [9] P. Sikivie, arXiv:hep-ph/0701198v1 (2007). [10] G. Raffelt, arXiv:hep-ph/0611350 (2006). [11] J. Jaeckel et al, Phys.Rev.D 75, 013004 (2007) [12] R. Cameron et al. Phys Rev. D 47, 3703 (1993) [13] E. Zavattini et al. Phys. Rev. Lett. 96, 110406 (2006) [14] K. Cho, S.P. Bush, D.L. Mazzoni, and C.C. Davis, Phys. Rev. B 43, 965 (1991). [15] A.C. Melissinos, arXiv:hep-ph/0702135v1 13 Feb 2007. [16] K. Cho, Ph.D Thesis, University of Maryland, 1991. [17] M. Sargent III, M.O. Scully and W.E. Lamb, Laser Physics, Addison-Wesley, Reading. Mass, 1974. [18] D. R. Herriott, H. Kogelnik, and R. Kompfner, Appl. Opt. 3, 523 (1964) [19] D. R. Herriott and H.J. Schulte, Appl. Opt. Aug. 4, 883 (1965) mailto:[email protected]

0704.0749

Binary Systems as Test-beds of Gravity Theories

Binary Systems as Test-beds of Gravity Theories∗ Thibault Damour Institut des Hautes Etudes Scientifiques, 35 route de Chartres, F-91440 Bures-sur-Yvette, France We review the general relativistic theory of the motion, and of the timing, of binary systems containing compact objects (neutron stars or black holes). Then we indicate the various ways one can use binary pulsar data to test the strong-field and/or radiative aspects of General Relativity, and of general classes of alternative theories of relativistic gravity. 1 Introduction The discovery of binary pulsars in 1974 [1] opened up a new testing ground for relativistic gravity. Before this discovery, the only available testing ground for relativistic gravity was the solar system. As Einstein’s theory of General Relativity (GR) is one of the basic pillars of modern science, it deserves to be tested, with the highest possible accuracy, in all its aspects. In the solar system, the gravitational field is slowly varying and represents only a very small deformation of a flat spacetime. As a consequence, solar system tests can only probe the quasi-stationary (non radiative) weak-field limit of relativistic gravity. By contrast binary systems containing compact objects (neutron stars or black holes) involve spacetime domains (inside and near the compact objects) where the gravitational field is strong. Indeed, the surface relativistic gravitational field h00 ≃ 2GM/c2R of a neutron star is of order 0.4, which is close to the one of a black hole (2GM/c2R = 1) and much larger than the surface gravitational fields of solar system bodies: (2GM/c2R)Sun ∼ 10−6, (2GM/c2R)Earth ∼ 10−9. In addition, the high stability of “pulsar clocks” has made it possible to monitor the dynamics of its orbital motion down to a precision allowing one to measure the small (∼ (v/c)5) orbital effects linked to the propagation of the gravitational field at the velocity of light between the pulsar and its companion. The recent discovery of the remarkable double binary pulsar PSR J0737− 3039 [2, 3] (see also the contributions of M. Kramer and A. Possenti to these ∗Based on lectures given at the SIGRAV School “A Century from Einstein Relativity: Probing Gravity Theories in Binary Systems”, Villa Olmo (Como Lake, Italy), 17-21 May 2005. To appear in the Proceedings, edited by M. Colpi et al. (to be published by Springer). http://arxiv.org/abs/0704.0749v1 proceedings) has renewed the interest in the use of binary pulsars as test-beds of gravity theories. The aim of these notes is to provide an introduction to the theoretical frameworks needed for interpreting binary pulsar data as tests of GR and alternative gravity theories. 2 Motion of binary pulsars in general relativity The traditional (text book) approach to the problem of motion of N separate bodies in GR consists of solving, by successive approximations, Einstein’s field equations (we use the signature −+++) Rµν − Rgµν = Tµν , (1) together with their consequence ∇ν T µν = 0 . (2) To do so, one assumes some specific matter model, say a perfect fluid, T µν = (ε+ p)uµ uν + p gµν . (3) One expands (say in powers of Newton’s constant) gµν(x λ) = ηµν + h µν + h µν + . . . , (4) together with the use of the simplifications brought by the ‘Post-Newtonian’ approximation (∂0 hµν = c −1 ∂t hµν ≪ ∂i hµν ; v/c ≪ 1, p ≪ ε). Then one integrates the local material equation of motion (2) over the volume of each separate body, labelled say by a = 1, 2, . . . , N . In so doing, one must define some ‘center of mass’ zia of body a, as well as some (approximately conserved) ‘mass’ ma of body a, together with some corresponding ‘spin vector’ S a and, possibly, higher multipole moments. An important feature of this traditional method is to use a unique coor- dinate chart xµ to describe the full N -body system. For instance, the center of mass, shape and spin of each body a are all described within this common coordinate system xµ. This use of a single chart has several inconvenient as- pects, even in the case of weakly self-gravitating bodies (as in the solar system case). Indeed, it means for instance that a body which is, say, spherically sym- metric in its own ‘rest frame’ Xα will appear as deformed into some kind of ellipsoid in the common coordinate chart xµ. Moreover, it is not clear how to construct ‘good definitions’ of the center of mass, spin vector, and higher multipole moments of body a, when described in the common coordinate chart xµ. In addition, as we are interested in the motion of strongly self-gravitating bodies, it is not a priori justified to use a simple expansion of the type (4) be- cause h Gma/(c 2 |x − za|) will not be uniformly small in the common coordinate system xµ. It will be small if one stays far away from each object a, but, as recalled above, it will become of order unity on the surface of a compact body. These two shortcomings of the traditional ‘one-chart’ approach to the rela- tivistic problem of motion can be cured by using a ‘multi-chart’ approach.The multi-chart approach describes the motion of N (possibly, but not necessarily, compact) bodies by using N+1 separate coordinate systems: (i) one global coor- dinate chart xµ (µ = 0, 1, 2, 3) used to describe the spacetime outside N ‘tubes’, each containing one body, and (ii) N local coordinate charts Xαa (α = 0, 1, 2, 3; a = 1, 2, . . . , N) used to describe the spacetime in and around each body a. The multi-chart approach was first used to discuss the motion of black holes and other compact objects [4, 5, 6, 7, 8, 9, 10, 11]. Then it was also found to be very convenient for describing, with the high-accuracy required for dealing with modern technologies such as VLBI, systems of N weakly self-gravitating bodies, such as the solar system [12, 13]. The essential idea of the multi-chart approach is to combine the information contained in several expansions. One uses both a global expansion of the type (4) and several local expansions of the type Gαβ(X a ) = G a ;ma) +H αβ (X a ;ma,mb) + · · · , (5) where G αβ(X ;ma) denotes the (possibly strong-field) metric generated by an isolated body of mass ma (possibly with the additional effect of spin). The separate expansions (4) and (5) are then ‘matched’ in some overlapping domain of common validity of the type Gma/c 2 . Ra ≪ |x−za| ≪ d ∼ |xa−xb| (with b 6= a), where one can relate the different coordinate systems by expansions of the form xµ = zµa (Ta) + e i (Ta)X ij(Ta)X a + · · · (6) The multi-chart approach becomes simplified if one considers compact bodies (of radius Ra comparable to 2Gma/c 2). In this case, it was shown [9], by considering how the ‘internal expansion’ (5) propagates into the ‘external’ one (4) via the matching (6), that, in General Relativity, the internal structure of each compact body was effaced to a very high degree, when seen in the external expansion (4). For instance, for non spinning bodies, the internal structure of each body (notably the way it responds to an external tidal excitation) shows up in the external problem of motion only at the fifth post-Newtonian (5PN) approximation, i.e. in terms of order (v/c)10 in the equations of motion. This ‘effacement of internal structure’ indicates that it should be possible to simplify the rigorous multi-chart approach by skeletonizing each compact body by means of some delta-function source. Mathematically, the use of dis- tributional sources is delicate in a nonlinear theory such as GR. However, it was found that one can reproduce the results of the more rigorous matched- multi-chart approach by treating the divergent integrals generated by the use of delta-function sources by means of (complex) analytic continuation [9]. The most efficient method (especially to high PN orders) has been found to use analytic continuation in the dimension of space d [14]. Finally, the most efficient way to derive the general relativistic equations of motion of N compact bodies consists of solving the equations derived from the action (where g ≡ − det(gµν)) dd+1 x R(g)− −gµν(zλa ) dz a dzνa , (7) formally using the standard weak-field expansion (4), but considering the space dimension d as an arbitrary complex number which is sent to its physical value d = 3 only at the end of the calculation. Using this method1 one has derived the equations of motion of two compact bodies at the 2.5PN (v5/c5) approximation level needed for describing binary pulsars [15, 16, 9]: d2 zia = Aia0(za − zb) + c−2Aia2(za − zb,va,vb) + c−4Aia4(za − zb,va,vb,Sa,Sb) + c−5Aia5(za − zb,va − vb) +O(c−6) . (8) Here Aia0 = −Gmb(zia − zib)/|za − zb|3 denotes the Newtonian acceleration, Aia2 its 1PN modification, Aia4 its 2PN modification (together with the spin-orbit effects), and Aia5 the 2.5PN contribution of order v 5/c5. [See the references above; or the review [17], for more references and the explicit expressions of A2, A4 and A5.] It was verified that the term A a5 has the effect of decreasing the mechanical energy of the system by an amount equal (on average) to the energy lost in the form of gravitational wave flux at infinity. Note, however, that here Aia5 was derived, in the near zone of the system, as a direct consequence of the general relativistic propagation of gravity, at the velocity c, between the two bodies. This highlights the fact that binary pulsar tests of the existence of Aia5 are direct tests of the reality of gravitational radiation. Recently, the equations of motion (8) have been computed to even higher accuracy: 3PN ∼ v6/c6 [18, 19, 20, 21, 22] and 3.5PN ∼ v7/c7 [23, 24, 25] (see also the review [26]). These refinements are, however, not (yet) needed for interpreting binary pulsar data. 3 Timing of binary pulsars in general relativity In order to extract observational effects from the equations of motion (8) one needs to go through two steps: (i) to solve the equations of motion (8) so as to 1Or, more precisely, an essentially equivalent analytic continuation using the so-called ‘Riesz kernels’. get the coordinate positions z1 and z2 as explicit functions of the coordinate time t, and (ii) to relate the coordinate motion za(t) to the pulsar observables, i.e. mainly to the times of arrival of electromagnetic pulses on Earth. The first step has been accomplished, in a form particularly useful for dis- cussing pulsar timing, in Ref. [27]. There (see also [28]) it was shown that, when considering the full (periodic and secular) effects of the A2 ∼ v2/c2 terms in Eq. (8), together with the secular effects of the A4 ∼ v4/c4 and A5 ∼ v5/c5 terms, the relativistic two-body motion could be written in a very simple ‘quasi- Keplerian’ form (in polar coordinates), namely: n dt+ σ = u− et sinu , (9) θ − θ0 = (1 + k) 2 arctan 1 + eθ 1− eθ , (10) R ≡ rab = aR(1 − eR cosu) , (11) ra ≡ |za − zCM | = ar(1− er cosu) , (12) rb ≡ |zb − zCM | = ar′(1− er′ cosu) . (13) Here n ≡ 2π/Pb denotes the orbital frequency, k = ∆θ/2π = 〈ω̇〉/n = 〈ω̇〉Pb/2π the fractional periastron advance per orbit, u an auxiliary angle (‘rel- ativistic eccentric anomaly’), et, eθ, eR, er and er′ various ‘relativistic eccentric- ities’ and aR, ar and ar′ some ‘relativistic semi-major axes’. See [27] for the relations between these quantities, as well as their link to the relativistic energy and angular momentum E, J . A direct study [28] of the dynamical effect of the contribution A5 ∼ v5/c5 in the equations of motion (8) has checked that it led to a secular increase of the orbital frequency n(t) ≃ n(0)+ ṅ(t−t0), and thereby to a quadratic term in the ‘relativistic mean anomaly’ ℓ = n dt+ σ appearing on the left-hand side (L.H.S.) of Eq. (9): ℓ ≃ σ0 + n0(t− t0) + ṅ(t− t0)2 . (14) As for the contribution A4 ∼ v4/c4 it induces several secular effects in the orbital motion: various 2PN contributions to the dimensionless periastron pa- rameter k (δ4 k ∼ v4/c4+ spin-orbit effects), and secular variations in the incli- nation of the orbital plane (due to spin-orbit effects). The second step in relating (8) to pulsar observations has been accomplished through the derivation of a ‘relativistic timing formula’ [29, 30]. The ‘timing formula’ of a binary pulsar is a multi-parameter mathematical function relating the observed time of arrival (at the radio-telescope) of the center of the N th pulse to the integer N . It involves many different physical effects: (i) dispersion effects, (ii) travel time across the solar system, (iii) gravitational delay due to the Sun and the planets, (iv) time dilation effects between the time measured on the Earth and the solar-system-barycenter time, (v) variations in the travel time between the binary pulsar and the solar-system barycenter (due to relative accelerations, parallax and proper motion), (vi) time delays happening within the binary system. We shall focus here on the time delays which take place within the binary system (see the lectures of M. Kramer for a discussion of the other effects). For a proper derivation of the time delays occurring within the binary sys- tem we need to use the multi-chart approach mentionned above. In the ‘rest frame’ (X0a = c Ta, X a) attached to the pulsar a, the pulsar phenomenon can be modelled by the secularly changing rotation of a beam of radio waves: Ωa(Ta) d Ta ≃ Ωa Ta + Ω̇a T Ω̈a T a + · · · , (15) where Φa is the longitude around the spin axis. [Depending on the precise defi- nition of the rest-frame attached to the pulsar, the spin axis can either be fixed, or be slowly evolving, see e.g. [13].] One must then relate the initial direction (Θa,Φa), and proper time Ta, of emission of the pulsar beam to the coordinate direction and coordinate time of the null geodesic representing the electromag- netic beam in the ‘global’ coordinates xµ used to describe the dynamics of the binary system [NB: the explicit orbital motion (9)–(13) refers to such global coordinates x0 = ct, xi]. This is done by using the link (6) in which zia denotes the global coordinates of the ‘center of mass’ of the pulsar, Ta the local (proper) time of the pulsar frame, and where, for instance e0i = c2 rab + · · · + · · · (16) Using the link (6) (with expressions such as (16) for the coefficients e i , . . .) one finds, among other results, that a radio beam emitted in the proper direction N i in the local frame appears to propagate, in the global frame, in the coordinate direction ni where ni = N i + −N i N . (17) This is the well known ‘aberration effect’, which will then contribute to the timing formula. One must also write the link between the pulsar ‘proper time’ Ta and the coordinate time t = x0/c = z0a/c used in the orbital motion (9)–(13). This reads − c2 d T 2a = g̃µν(aλa) dzµa dzνa (18) where the ‘tilde’ denotes the operation consisting (in the matching approach) in discarding in gµν the ‘self contributions’ ∼ (Gma/Ra)n, while keeping the effect of the companion (∼ Gmb/rab, etc. . .). One checks that this is equivalent (in the dimensional-continuation approach) in taking xµ = zµa for sufficiently small values of the real part of the dimension d. To lowest order this yields the link 1− 2Gmb c2 rab 1− Gmb c2 rab which combines the special relativistic and general relativistic time dilation effects. Hence, following [30] we can refer to them as the ‘Einstein time delay’. Then, one must compute the (global) time taken by a light beam emitted by the pulsar, at the proper time Ta (linked to temission by (19)), in the initial global direction ni (see Eq. (17)), to reach the barycenter of the solar system. This is done by writing that this light beam follows a null geodesic: in particular 0 = ds2 = gµν(x λ) dxµ dxν ≃ − 1− 2U c2 dt2 + dx2 (20) where U = Gma/|x−za|+Gmb/|x−zb| is the Newtonian potential within the binary system. This yields (with te ≡ temission, ta ≡ tarrival) ta − te = dt ≃ 1 |dx|+ 2 |x− za| |x− zb| |dx| . (21) The first term on the last RHS of Eq. (21) is the usual ‘light crossing time’ |zbarycenter(ta) − za(te)| between the pulsar and the solar barycenter. It con- tains the ‘Roemer time delay’ due to the fact that za(te) moves on an orbit. The second term on the last RHS of Eq. (21) is the ‘Shapiro time delay’ due to the propagation of the beam in a curved spacetime (only the Gmb piece linked to the companion is variable). When inserting the ‘quasi-Keplerian’ form (9)–(13) of the relativistic motion in the ‘Roemer’ term in (21), together with all other relativistic effects, one finds that the final expression for the relativistic timing formula can be significantly simplified by doing two mathematical transformations. One can redefine the ‘time eccentricity’ et appearing in the ‘Kepler equation’ (9), and one can define a new ‘eccentric anomaly’ angle: u→ unew [we henceforth drop the superscript ‘new’ on u]. After these changes, the binary-system part of the general relativis- tic timing formula [30] takes the form (we suppress the index a on the pulsar proper time Ta) tbarycenter − t0 = D−1[T +∆R(T ) + ∆E(T ) + ∆S(T ) + ∆A(T )] (22) ∆R = x sinω[cosu− e(1 + δr)] + x[1 − e2(1 + δθ)2]1/2 cosω sinu , (23) ∆E = γ sinu , (24) ∆S = −2r ln{1− e cosu− s[sinω(cosu− e) + (1− e2)1/2 cosω sinu]},(25) ∆A = A{sin[ω +Ae(u)] + e sinω}+B{cos[ω +Ae(u)] + e cosω} , (26) where x = x0 + ẋ(T − T0) represents the projected light-crossing time (x = apulsar sin i/c), e = e0 + ė(T − T0) a certain (relativistically-defined) ‘timing eccentricity’, Ae(u) the function Ae(u) ≡ 2 arctan 1 + e , (27) ω = ω0 + k Ae(u) the ‘argument of the periastron’, and where the (relativisti- cally-defined) ‘eccentric anomaly’ u is the function of the ‘pulsar proper time’ T obtained by solving the Kepler equation u− e sinu = 2π T − T0 T − T0 . (28) It is understood here that the pulsar proper time T corresponding to the N th pulse is related to the integer N by an equation of the form N = c0 + νp T + ν̇p T ν̈p T 3 . (29) From these formulas, one sees that δθ (and δr) measure some relativistic distor- tion of the pulsar orbit, γ the amplitude of the ‘Einstein time delay’2 ∆E , and r and s the range and shape of the ‘Shapiro time delay’3 ∆S . Note also that the dimensionless PPK parameter k measures the non-uniform advance of the periastron. It is related to the often quoted secular rate of periastron advance ω̇ ≡ 〈dω/dt〉 by the relation k = ω̇Pb/2π. It has been explicitly checked that binary-pulsar observational data do indeed require to model the relativistic pe- riastron advance by means of the non-uniform (and non-trivial) function of u multiplying k on the R.H.S. of Eq. (27) [31]4. Finally, we see from Eq. (28) that Pb represents the (periastron to periastron) orbital period at the fiducial epoch T0, while the dimensionless parameter Ṗb represents the time derivative of Pb (at T0). Schematically, the structure of the DD timing formula (22) is tbarycenter − t0 = F [TN ; {pK}; {pPK}; {qPK}] , (30) where tbarycenter denotes the solar-system barycentric (infinite frequency) ar- rival time of a pulse, T the pulsar emission proper time (corrected for aberra- tion), {pK} = {Pb, T0, e0, ω0, x0} is the set of Keplerian parameters, {pPK = k, γ, Ṗb, r, s, δθ, ė, ẋ} the set of separately measurable post-Keplerian parameters, 2The post-Keplerian timing parameter γ, first introduced in [29], has the dimension of time, and should not be confused with the dimensionless post-Newtonian Eddington parameter γPPN probed by solar-system experiments (see below). 3The dimensionless parameter s is numerically equal to the sine of the inclination angle i of the orbital plane, but its real definition within the PPK formalism is the timing parameter which determines the ‘shape’ of the logarithmic time delay ∆S(T ). 4Alas this function is theory-independent, so that the non-uniform aspect of the periastron advance cannot be used to yield discriminating tests of relativistic gravity theories. and {qPK} = {δr, A,B,D} the set of not separately measurable post-Keplerian parameters [31]. [The parameter D is a ‘Doppler factor’ which enters as an overall multiplicative factor D−1 on the right-hand side of Eq. (22).] A further simplification of the DD timing formula was found possible. In- deed, the fact that the parameters {qPK} = {δr, A,B,D} are not separately measurable means that they can be absorbed in changes of the other param- eters. The explicit formulas for doing that were given in [30] and [31]: they consist in redefining e, x, Pb, δθ and δr. At the end of the day, it suffices to consider a simplified timing formula where {δr, A,B,D} have been set to some given fiducial values, e.g. {0, 0, 0, 1}, and where one only fits for the remaining parameters {pK} and {pPK}. Finally, let us mention that it is possible to extend the general parametrized timing formula (30) by writing a similar parametrized formula describing the ef- fect of the pulsar orbital motion on the directional spectral luminosity [d(energy) /d(time) d(frequency) d(solid angle)] received by an observer. As discussed in detail in [31] this introduces a new set of ‘pulse-structure post-Keplerian pa- rameters’. 4 Phenomenological approach to testing rela- tivistic gravity with binary pulsar data As said in the Introduction, binary pulsars contain strong gravity domains and should therefore allow one to test the strong-field aspects of relativistic gravity. The question we face is then the following: How can one use binary pulsar data to test strong-field (and radiative) gravity? Two different types of answers can be given to this question: a phenomeno- logical (or theory-independent) one, or various types of theory-dependent ap- proaches. In this Section we shall consider the phenomenological approach. The phenomenological approach to binary-pulsar tests of relativistic gravity is called the parametrized post-Keplerian formalism [32, 31]. This approach is based on the fact that the mathematical form of the multi-parameter DD timing formula (30) was found to be applicable not only in General Relativity, but also in a wide class of alternative theories of gravity. Indeed, any theory in which gravity is mediated not only by a metric field gµν but by a general combination of a metric field and of one or several scalar fields ϕ(a) will induce relativistic timing effects in binary pulsars which can still be parametrized by the formulas (22)–(29). Such general ‘tensor-multi-scalar’ theories of gravity contain arbitrary functions of the scalar fields. They have been studied in full generality in [33]. It was shown that, under certain conditions, such tensor-scalar gravity theories could lead, because of strong-field effects, to very different predictions from those of General Relativity in binary pulsar timing observations [34, 35, 36]. However, the point which is important for this Section, is that even when such strong-field effects develop one can still use the universal DD timing formula (30) to fit the observed pulsar times of arrival. The basic idea of the phenomenological, parametrized post-Keplerian (PPK) approach is then the following: By least-square fitting the observed sequence of pulsar arrival times tN to the parametrized formula (30) (in which TN is defined by Eq. (29) which introduces the further parameters νp, ν̇p, ν̈p) one can phenomenologically extract from raw observational data the (best fit) values of all the parameters entering Eqs. (29) and (30). In particular, one so determines both the set of Keplerian parameters {pK} = {Pb, T0, e0, ω0, x0}, and the set of post-Keplerian (PK) parameters {pPK} = {k, γ, Ṗb, r, s, δθ, ė, ẋ}. In extracting these values, we did not have to assume any theory of gravity. However, each specific theory of gravity will make specific predictions relating the PK param- eters to the Keplerian ones, and to the two (a priori unknown) masses ma and mb of the pulsar and its companion. [For certain PK parameters one must also consider other variables related to the spin vectors of a and b.] In other words, the measurement (in addition of the Keplerian parameters) of each PK param- eter defines, for each given theory, a curve in the (ma,mb) mass plane. For any given theory, the measurement of two PK parameters determines two curves and thereby generically determines the values of the two masses ma and mb (as the point of intersection of these two curves). Therefore, as soon as one mea- sures three PK parameters one obtains a test of the considered gravity theory. The test is passed only if the three curves meet at one point. More generally, the measurement of n PK timing parameters yields n− 2 independent tests of relativistic gravity. Any one of these tests, i.e. any simultaneous measurement of three PK parameters can either confirm or put in doubt any given theory of gravity. As General Relativity is our current most successful theory of gravity, it is clearly the prime target for these tests. We have seen above that the timing data of each binary pulsar provides a maximum of 8 PK parameters: k, γ, Ṗb, r, s, δθ, ė and ẋ. Here, we were talking about a normal ‘single line’ binary pulsar where, among the two compact objects a and b only one of the two, say a is observed as a pulsar. In this case, one binary system can provide up to 8− 2 = 6 tests of GR. In practice, however, it has not yet been possible to measure the parameter δθ (which measures a small relativistic deformation of the elliptical orbit), nor the secular parameters ė and ẋ. The original Hulse-Taylor system PSR 1913+16 has allowed one to measure 3 PK parameters: k ≡ 〈ω̇〉Pb/2π, γ and Ṗb. The two parameters k and γ involve (non radiative) strong-field effects, while, as explained above, the orbital period derivative Ṗb is a direct consequence of the term A5 ∼ v5/c5 in the binary-system equations of motion (5). The term A5 is itself directly linked to the retarded propagation, at the velocity of light, of the gravitational interaction between the two strongly self-gravitating bodies a and b. Therefore, any test involving Ṗb will be a mixed radiative strong-field test. Let us explain on this example what information one needs to implement a phenomenological test such as the (k−γ−Ṗb)1913+16 one. First, we need to know the predictions made by the considered target theory for the PK parameters k, γ and Ṗb as functions of the two masses ma and mb. These predictions have been worked out, for General Relativity, in Refs. [29, 28, 30]. Introducing the notation (where n ≡ 2π/Pb) M ≡ ma +mb (31) Xa ≡ ma/M ; Xb ≡ mb/M ; Xa +Xb ≡ 1 (32) βO(M) ≡ , (33) they read kGR(ma,mb) = 1− e2 β2O , (34) γGR(ma,mb) = Xb(1 +Xb)β O , (35) ṖGRb (ma,mb) = − 1 + 73 e2 + 37 (1− e2)7/2 XaXb β O . (36) However, if we use the three predictions (34)–(36), together with the best current observed values of the PK parameters kobs, γobs, Ṗ obdb [37] we shall find that the three curves kGR(ma,mb) = k obs, γGR(ma,mb) = γ obs, ṖGRb (ma,mb) = Ṗ obsb in the (ma,mb) mass plane fail to meet at about the 13 σ level! Should this put in doubt General Relativity? No, because Ref. [38] has shown that the time variation (notably due to galactic acceleration effects) of the Doppler fac- tor D entering Eq. (22) entailed an extra contribution to the ‘observed’ period derivative Ṗ obsb . We need to subtract this non-GR contribution before drawing the corresponding curve: ṖGRb (ma,mb) = Ṗ b − Ṗ galactic b . Then one finds that the three curves do meet within one σ. This yields a deep confirmation of Gen- eral Relativity, and a direct observational proof of the reality of gravitational radiation. We said several times that this test is also a probe of the strong-field aspects of GR. How can one see this? A look at the GR predictions (34)–(36) does not exhibit explicit strong-field effects. Indeed, the derivation of Eqs. (34)–(36) used in a crucial way the ‘effacement of internal structure’ that occurs in the general relativistic dynamics of compact objects. This non trivial property is rather specific of GR and means that, in this theory, all the strong-field effects can be absorbed in the definition of the masses ma and mb. One can, however, verify that strong-field effects do enter the observable PK parameters k, γ, Ṗb etc. . . by considering how the theoretical predictions (34)–(36) get modified in alternative theories of gravity. The presence of such strong-field effects in PK parameters was first pointed out in Ref. [7] (see also [39]) for the Jordan-Fierz-Brans-Dicke theory of gravity, and in Ref. [8] for Rosen’s bi-metric theory of gravity. A detailed study of such strong-field deviations was then performed in [33, 34, 35] for general tensor-(multi-)scalar theories of gravity. In the following Section we shall exhibit how such strong-field effects enter the various post-Keplerian parameters. Continuing our historical review of phenomenological pulsar tests, let us come to the binary system which was the first one to provide several ‘pure strong-field tests’ of relativistic gravity, without mixing of radiative effects: PSR 1534+12. In this system, it was possible to measure the four (non ra- diative) PK parameters k, γ, r and s. [We see from Eq. (25) that r and s mea- sure, respectively, the range and the shape of the ‘Shapiro time delay’ ∆S .] The measurement of the 4 PK parameters k, γ, r, s define 4 curves in the (ma,mb) mass plane, and thereby yield 2 strong-field tests of GR. It was found in [40] that GR passes these two tests. For instance, the ratio between the measured value sobs of the phenomenological parameter5 s and the value sGR[kobs, γobs] predicted by GR on the basis of the measurements of the two PK parameters k and γ (which determine, via Eqs. (34) , (35), the GR-predicted value of ma and mb) was found to be s obs/sGR[kobs, γobs] = 1.004± 0.007 [40]. The most recent data [41] yield sobs/sGR[kobs, γobs] = 1.000± 0.007. We see that we have here a confirmation of the strong-field regime of GR at the 1% level. Another way to get phenomenological tests of the strong field aspects of gravity concerns the possibility of a violation of the strong equivalence principle. This is parametrized by phenomenologically assuming that the ratio between the gravitational and the inertial mass of the pulsar differs from unity (which is its value in GR): (mgrav/minert)a = 1+∆a. Similarly to what happens in the Earth- Moon-Sun system [42], the three-body system made of a binary pulsar and of the Galaxy exhibits a ‘polarization’ of the orbit which is proportional to ∆ ≡ ∆a − ∆b, and which can be constrained by considering certain quasi-circular neutron- star-white-dwarf binary systems [43]. See [44] for recently published improved limits6 on the phenomenological equivalence-principle violation parameter ∆. The Parkes multibeam survey has recently discovered several new interesting ‘relativistic’ binary pulsars, thereby giving a huge increase in the number of phenomenological tests of relativistic gravity. Among those new binary pulsar systems, two stand out as superb testing grounds for relativistic gravity: (i) PSR J1141−6545 [46, 47], and (ii) the remarkable double binary pulsar PSR J0737−3039A and B [2, 3, 48, 49] (see also the lectures by M. Kramer and A. Possenti). The PSR J1141−6545 timing data have led to the measurement of 3 PK parameters: k, γ, and Ṗb [47]. As in PSR 1913+16 this yields one mixed radiative-strong-field test7. 5As already mentioned the dimensionless parameter s is numerically equal (in all theories) to the sine of the inclination angle i of the orbital plane, but it is better thought, in the PPK formalism, as a phenomenological timing parameter determining the ‘shape’ of the logarithmic time delay ∆S(T ). 6Note, however, that these limits, as well as those previously obtained in [45], assume that the (a priori pulsar-mass dependent) parameter ∆ ≃ ∆a is the same for all the analyzed pulsars. 7In addition, scintillation data have led to an estimate of the sine of the orbital inclination, sin i [50]. As said above, sin i numerically coincides with the PK parameter s measuring the ‘shape’ of the Shapiro time delay. Therefore, one could use the scintillation measurements as an indirect determination of s, thereby obtaining two independent tests from PSR J1141−6545 data. A caveat, however, is that the extraction of sin i from scintillation measurements rests on several simplifying assumptions whose validity is unclear. In fact, in the case of PSR J0737−3039 the direct timing measurement of s disagrees with its estimate via scintillation The timing data of the millisecond binary pulsar PSR J0737−3039A have led to the direct measurement of 5 PK parameters: k, γ, r, s and Ṗb [3, 48, 49]. In addition, the ‘double line’ nature of this binary system (i.e. the fact that one observes both components, A and B, as radio pulsars) allows one to perform new phenomenological tests by using Keplerian parameters. Indeed, the simultaneous measurement of the Keplerian parameters xa and xb representing the projected light crossing times of both pulsars (A and B) gives access to the combined Keplerian parameter Robs ≡ xobsb xobsa . (37) On the other hand, the general derivation of [30] (applicable to any Lorentz- invariant theory of gravity, and notably to any tensor-scalar theory) shows that the theoretical prediction for the the ratio R, considered as a function of the masses ma and mb, is Rtheory = . (38) The absence of any explicit strong-field-gravity effects in the theoretical predic- tion (38) (to be contrasted, for instance, with the predictions for PK parameters in tensor-scalar gravity discussed in the next Section) is mainly due to the con- vention used in [30] and [31] for defining the masses ma and mb. These are always defined so that the Lagrangian for two non interacting compact objects reads L0 = −ma c2(1− v2a/c2)1/2. In other words, ma c2 represents the total energy of body a. This means that one has implicitly lumped in the definition of ma many strong-self-gravity effects. [For instance, in tensor-scalar gravity ma includes not only the usual Einsteinian gravitational binding energy due to the self-gravitational field gµν(x), but also the extra binding energy linked to the scalar field ϕ(x).] Anyway, what is important is that, when performing a phenomenological test from the measurement of a triplet of parameters, e.g. {k, γ,R}, at least one parameter among them be a priori sensitive to strong- field effects. This is enough for guaranteeing that the crossing of the three curves ktheory(ma,mb) = k obs, γtheory(ma,mb) = γ obs, Rtheory(ma,mb) = R is really a probe of strong-field gravity. In conclusion, the two recently discovered binary pulsars PSR J1141−6545 and PSR J0737−3039 have more than doubled the number of phenomenological tests of (radiative and) strong-field gravity. Before their discovery, the ‘canoni- cal’ relativistic binary pulsars PSR 1913+16 and PSR 1534+12 had given us four data [49]. It is therefore safer not to use scintillation estimates of sin i on the same footing as direct timing measurements of the PK parameter s. On the other hand, a safe way of obtaining an s-related gravity test consists in using the necessary mathematical fact that s = sin i ≤ 1. In GR the definition xa = aa sin i/c leads to sin i = nxa/(β0 Xb). Therefore we can write the inequality nxa/(β0(M)Xb) ≤ 1 as a phenomenological test of GR. s ≤ 1 0 0.5 1 1.5 2 2.5 PSR J1141−6545 intersection 0 0.5 1 1.5 2 2.5 PSR B1534+12 intersection 0 0.5 1 1.5 2 2.5 PSR J0737−3039 intersection 0 0.5 1 1.5 2 2.5 2.5 ω s ≤ 1 PSR B1913+16 intersection Figure 1: Phenomenological tests of General Relativity obtained from Keplerian and post-Keplerian timing parameters of four relativistic pulsars. Figure taken from [51]. such tests: one (k−γ−Ṗb) test from PSR 1913+16 and three (k−γ−r−s−Ṗb8) tests from PSR 1534+12. The two new binary systems have given us five9 more phenomenological tests: one (k−γ− Ṗb) (or two, k−γ− Ṗb−s) tests from PSR J1141−6545 and four (k−γ− r−s− Ṗb−R) tests from PSR J0737−303910. As illustrated in Figure 1, these nine phenomenological tests of strong-field (and radiative) gravity are all in beautiful agreement with General Relativity. In addition, let us recall that several quasi-circular wide binaries, made of a neutron star and a white dwarf, have led to high-precision phenomenological confirmations [44] (in strong-field conditions) of one of the deep predictions of General Relativity: the ‘strong’ equivalence principle, i.e. the fact that var- ious bodies fall with the same acceleration in an external gravitational field, independently of the strength of their self-gravity. Finally, let us mention that Ref. [31] has extended the philosophy of the 8The timing measurement of Ṗ obs in PSR 1534+12 is even more strongly affected by kinematic corrections (Ḋ terms) than in the PSR 1913+16 case. In absence of a precise, independent measurement of the distance to PSR 1534+12, the k−γ− Ṗb test yields, at best, a ∼ 15% test of GR. 9Or even six, if we use the scintillation determination of s in PSR J1141−6545. 10The companion pulsar 0737−3039B being non recycled, and being visible only during a small part of its orbit, cannot be timed with sufficient accuracy to allow one to measure any of its post-Keplerian parameters. phenomenological (parametrized post-Keplerian) analysis of timing data, to a similar phenomenological analysis of pulse-structure data. Ref. [31] showed that, in principle, one could extract up to 11 ‘post-Keplerian pulse-structure param- eters’. Together with the 8 post-Keplerian timing parameters of a (single-line) binary pulsar, this makes a total of 19 phenomenological PK parameters. As these parameters depend not only on the two massesma,mb but also on the two angles λ, η determining the direction of the spin axis of the pulsar, the maximum number of tests one might hope to extract from one (single-line) binary pulsar is 19 − 4 = 15. However, the present accuracy with which one can model and measure the pulse structure of the known pulsars has not yet allowed one to measure any of these new pulse-structure parameters in a theory-independent and model-independent way. Nonetheless, it has been possible to confirm the reality (and order of mag- nitude) of the spin-orbit coupling in GR which was pointed out [52, 53] to be observable via a secular change of the intensity profile of a pulsar signal. Confir- mations of general relativistic spin-orbit effects in the evolution of pulsar profiles were obained in several pulsars: PSR 1913+16 [54, 55], PSR B1534+12 [56] and PSR J1141−6545 [57]. In this respect, let us mention that the spin-orbit interac- tion affects also several PK parameters, either by inducing a secular evolution in some of them (see [31]) or by contributing to their value. For instance, the spin- orbit interaction contributes to the observed value of the periastron advance parameter k an amount which is significant for the pulsars (such as 1913+16 and 0737−3039) where k is measured with high-accuracy. It was then pointed out [58] that this gives, in principle, and indirect way of measuring the moment of inertia of neutron stars (a useful quantity for probing the equation of state of nuclear matter [59, 60]). However, this can be done only if one measures, besides k, two other PK parameters with 10−5 accuracy. A rather tall order which will be a challenge to meet. The phenomenological approach to pulsar tests has the advantage that it can confirm or invalidate a specific theory of gravity without making assumptions about other theories. Moreover, as General Relativity has no free parameters, any test of its predictions is a potentially lethal test. From this point of view, it is remarkable that GR has passed with flying colours all the pulsar tests if has been submitted to. [See, notably, Fig. 1.] As argued above, these tests have probed strong-field aspects of gravity which had not been probed by solar-system (or cosmological) tests. On the other hand, a disadvantage of the phenomenological tests is that they do not tell us in any precise way which strong-field structures, have been actually tested. For instance, let us imagine that one day one specific PPK test fails to be satisfied by GR, while the others are OK. This leaves us in a quandary: If we trust the problematic test, we must conclude that GR is wrong. However, the other tests say that GR is OK. This example shows that we would like to have some idea of what physical effects, linked to strong-field gravity, enter in each test, or even better in each PK parameter. The ‘effacement of internal structure’ which takes place in GR does not allow one to discuss this issue. This gives us a motivation for going beyond the phenomenological PPK approach by considering theory-dependent formalisms in which one embeds GR within a space of alternative gravity theories. 5 Theory-space approach to testing relativistic gravity with binary pulsar data A complementary approach to testing gravity with binary pulsar data consists in embedding General Relativity within a multi-parameter space of alternative theories of gravity. In other words, we want to contrast the predictions of GR with the predictions of continuous families of alternative theories. In so doing we hope to learn more about which structures of GR are actually being probed in binary pulsar tests. This is a bit similar to the well-known psycho-physiological fact that the best way to appreciate a nuance of colour is to surround a given patch of colour by other patches with slightly different colours. This makes it much easier to detect subtle differences in colour. In the same way, we hope to learn about the probing power of pulsar tests by seeing how the phenomeno- logical tests summarized in Fig. 1 fail (or continue) to be satisfied when one continuously deform, away from GR, the gravity theory which is being tested. Let us first recall the various ways in which this theory-space approach has been used in the context of the solar-system tests of relativistic gravity. 5.1 Theory-space approaches to solar-system tests of rel- ativistic gravity In the quasi-stationary weak-field context of the solar-system, this theory-space approach has been implemented in two different ways. First, the parametrized post-Newtonian (PPN) formalism [61, 62, 63, 42, 64, 65, 11, 66] describes many ‘directions’ in which generic alternative theories of gravity might dif- fer in their weak-field predictions from GR. In its most general versions the PPN formalism contains 10 ‘post-Einstein’ PPN parameters, γ̄ ≡ γPPN − 111, β̄ ≡ βPPN−1, ξ, α1, α2, α3, ζ1, ζ2, ζ3, ζ4. Each one of these dimensionless quanti- ties parametrizes a certain class of slow-motion, weak-field gravitational effects which deviate from corresponding GR predictions. For instance, γ̄ parametrizes modifications both of the effect of a massive body (say, the Sun) on the light passing near it, and of the terms in the two-body gravitational Lagrangian which are proportional to (Gmamb/rab) · (va − vb)2/c2. A second way of implementing the theory-space philosophy consists in con- sidering some explicit, parameter-dependent family of alternative relativistic theories of gravity. For instance, the simplest tensor-scalar theory of gravity 11The PPN parameter γPPN is usually denoted simply as γ. To distinguish it from the Einstein-time-delay PPK timing parameter γ used above we add the superscript PPN. In addition, as the value of γPPN in GR is 1, we prefer to work with the parameter γ̄ ≡ γPPN−1 which vanishes in GR, and therefore measures a ‘deviation’ from GR in a certain ‘direction’ in theory-space. Similarly with β̄ ≡ βPPN − 1. put forward by Jordan [67], Fierz [68] and Brans and Dicke [69] has a unique free parameter, say α20 = (2ωBD + 3) −1. When α20 → 0, this theory reduces to GR, so that α20 (or 1/ωBD) measures all the deviations from GR. When considering the weak-field limit of the Jordan-Fierz-Brans-Dicke (JFBD) the- ory, one finds that it can be described within the PPN formalism by choosing γ̄ = −2α20(1 + α20)−1, β̄ = 0 and ξ = αi = ζj = 0. Having briefly recalled the two types of theory-space approaches used to discuss solar-system tests, let us now consider the case of binary-pulsar tests. 5.2 Theory-space approaches to binary-pulsar tests of rel- ativistic gravity There exist generalizations of these two different theory-space approaches to the context of strong-field gravity and binary pulsar tests. First, the PPN formalism has been (partially) extended beyond the ‘first post-Newtonian’ (1PN) order deviations from GR (∼ v2/c2+Gm/c2 r) to describe 2PN order deviations from [70]. Remarkably, there appear only two new parameters at the 2PN level12: ǫ and ζ. Also, by expanding in powers of the self-gravity parameters of body a and b the predictions for the PPK timing parameters in generic tensor-multi-scalar theories, one has shown that these predictions depended on several ‘layers’ of new dimensionless parameters [33]. Early among these parameters one finds, the 1PN parameters β̄, γ̄ and then the basic 2PN parameters ǫ and ζ, but one also finds further parameters β3, (ββ ′), β′′, . . . which would not enter usual 2PN effects. The two approaches that we have just mentionned can be viewed as generalizations of the PPN formalism. There exist also useful generalizations to the strong-field context of the idea of considering some explicit parameter-dependent family of alternative theo- ries of relativistic gravity. Early studies [7, 8, 39] focussed either on the one- parameter JFBD tensor-scalar theory, or on some theories which are not con- tinuously connected to GR, such as Rosen’s bimetric theory of gravity. Though the JFBD theory exhibits a marked difference from GR in that it predicts the existence of dipole radiation, it has the disadvantage that the weak field, solar- system constraints on its unique parameter α20 are so strong that they drastically constrain (and essentially forbid) the presence of any non-radiative, strong-field deviations from GR. In view of this, it is useful to consider other ‘mini-spaces’ of alternative theories. A two-parameter mini-space of theories, that we shall denote13 here as ′, β′′), was introduced in [33]. This two-parameter family of tensor-bi-scalar theories was constructed so as to have exactly the same first post-Newtonian limit as GR (i.e. γ̄ = β̄ = · · · = 0), but to differ from GR in its predictions 12When restricting oneself to the general class of tensor-multi-scalar theories. At the 1PN level, this restriction would imply that only the ‘directions’ γ̄ and β̄ are allowed. 13We add here an index 2 to T as a reminder that this is a class of tensor-bi-scalar theories, i.e. that they contain two independent scalar fields ϕ1, ϕ2 besides a dynamical metric gµν . for the various observables that can be extracted from binary pulsar data. Let us give one example of this behaviour of the T2(β ′, β′′) class of theories. For a general theory of gravity we expect to have violations of the strong equiva- lence principle in the sense that the ratio between the gravitational mass of a self-gravitating body to its inertial mass will admit an expansion of the type mgrava minerta ≡ 1 + ∆a = 1− η1 ca + η2 c a + . . . (39) where ca ≡ −2 ∂ lnma∂ lnG measures the ‘gravitational compactness’ (or fractional gravitational binding energy, ca ≃ −2Egrava /ma c2) of body a. The numerical coefficient η1 of the contribution linear in ca is a combination of the first post- Newtonian order PPN parameters, namely η1 = 4 β̄ − γ̄ [42]. The numerical coefficient η2 of the term quadratic in ca is a combination of the 1PN and 2PN parameters. When working in the context of the T2(β ′, β′′) theories, the 1PN parameters vanish exactly (β̄ = 0 = γ̄) and the coefficient of the quadratic term becomes simply proportional to the theory parameter β′ : η2 = Bβ′, where B ≈ 1.026. This example shows explicitly how binary pulsar data (here the data constraining the equivalence principle violation parameter ∆ = ∆a − ∆b, see above) can go beyond solar-system experiments in probing certain strong-self- gravity effects. Indeed, solar-system experiments are totally insensitive to 2PN parameters because of the smallness of ca ∼ Gma/c2Ra and of the structure of 2PN effects [70]. By contrast, the ‘compactness’ of neutron stars is of order ca ∼ 0.21ma/M⊙ ∼ 0.3 [33] so that the pulsar limit |∆| < 5.5×10−3 [44] yields, within the T2(β ′, β′′) framework, a significant limit on the dimensionless (2PN order) parameter β′ : |β′| < 0.12. Ref. [35] introduced a new two-parameter mini-space of gravity theories, denoted here as T1(α0, β0), which, from the point of view of theoretical physics, has several advantages over the T2(β ′, β′′) mini-space mentionned above. First, it is technically simpler in that it contains only one scalar field ϕ besides the metric gµν (hence the index 1 on T1(α0, β0)). Second, it contains only positive- energy excitations (while one combination of the two scalar fields of T2(β ′, β′′) carried negative-energy waves). Third, it is the minimal way to parametrize the huge class of tensor-mono-scalar theories with a ‘coupling function’ a(ϕ) satisfying some very general requirements (see below). Let us now motivate the use of tensor-scalar theories of gravity as alternatives to general relativity. 5.3 Tensor-scalar theories of gravity Let us start by recalling (essentially from [35]) why tensor-(mono)-scalar theories define a natural class of alternatives to GR. First, and foremost, the existence of scalar partners to the graviton is a simple theoretical possibility which has surfaced many times in the development of unified theories, from Kaluza-Klein to superstring theory. Second, they are general enough to describe many inter- esting deviations from GR (both in weak-field and in strong field conditions), but simple enough to allow one to work out their predictions in full detail. Let us therefore consider a general tensor-scalar action involving a metric g̃µν (with signature ‘mostly plus’), a scalar field Φ, and some matter variables ψm (including gauge bosons): 16πG∗ g̃1/2 F (Φ)R̃ − Z(Φ)g̃µν∂µΦ ∂νΦ− U(Φ) + Sm[ψm; g̃µν ] . For simplicity, we assume here that the weak equivalence principle is satisfied, i.e., that the matter variables ψm are all coupled to the same ‘physical metric’ g̃µν . The general model (40) involves three arbitrary functions: a function F (Φ) coupling the scalar Φ to the Ricci scalar of g̃µν , R̃ ≡ R(g̃µν), a function Z(Φ) renormalizing the kinetic term of Φ, and a potential function U(Φ). As we have the freedom of arbitrary redefinitions of the scalar field, Φ → Φ′ = f(Φ), only two functions among F , Z and U are independent. It is often convenient to rewrite (40) in a canonical form, obtained by redefining both Φ and g̃µν according to g∗µν = F (Φ) g̃µν , (41) ϕ = ± F ′2(Φ) F 2(Φ) F (Φ) . (42) This yields 16πG∗ ∗ [R∗ − 2gµν∗ ∂µϕ∂νϕ− V (ϕ)] + Sm 2(ϕ) g∗µν where R∗ ≡ R(g∗µν), where the potential V (ϕ) = F−2(Φ)U(Φ) , (44) and where the conformal coupling function A(ϕ) is given by A(ϕ) = F−1/2(Φ) , (45) with Φ(ϕ) obtained by inverting the integral (42). The two arbitrary functions entering the canonical form (43) are: (i) the con- formal coupling function A(ϕ), and (ii) the potential function V (ϕ). Note that the ‘physical metric’ g̃µν (the one measured by laboratory clocks and rods) is conformally related to the ‘Einstein metric’ g∗µν , being given by g̃µν = A 2(ϕ) g∗µν . The canonical representation is technically useful because it decouples the two 14Actually, most unified models suggest that there are violations of the weak equivalence principle. However, the study of general string-inspired tensor-scalar models [71] has found that the composition-dependent effects would be negligible in the gravitational physics of neutron stars that we consider here. The experimental limits on tests of the equivalence principle would, however, bring a strong additional constraint of order 10−5 α2 ∼ ∆a/a . 10−12. As this constraint is strongly model-dependent, we will not use it in our exclusion plots below. One should, however, keep in mind that a limit on the scalar coupling strength of order α2 . 10−7 [71, 72] is likely to exist in many, physically-motivated, tensor-scalar models. irreducible propagating excitations: the spin-0 excitations are described by ϕ, while the pure spin-2 excitations are described by the Einstein metric g∗µν (with kinetic term the usual Einstein-Hilbert action ∝ R(g∗µν)). In many technical developments it is useful to work with the logarithmic coupling function a(ϕ) such that: a(ϕ) ≡ lnA(ϕ) ; A(ϕ) ≡ ea(ϕ) . (46) In the case of the general model (40) this logarithmic15 coupling function is given by a(ϕ) = − lnF (Φ) , where Φ(ϕ) must be obtained from (42). In the following, we shall assume that the potential V (ϕ) is a slowly varying function of ϕ which, in the domain of variation we shall explore, is roughly equivalent to a very small mass term V (ϕ) ∼ 2m2ϕ(ϕ−ϕ0)2 with m2ϕ of cosmo- logical order of magnitude m2ϕ = O(H20 ), or, at least, with a range λϕ = m−1ϕ much larger than the typical length scales that we shall consider (such as the size of the binary orbit, or the size of the Galaxy when considering violations of the strong equivalence principle). Under this assumption16 the potential func- tion V (ϕ) will only serve the role of fixing the value of ϕ far from the system (to ϕ(r = ∞) = ϕ0), and its effect on the propagation of ϕ within the system will be negligible. In the end, the tensor-scalar phenomenology that we shall explore only depends on one function: the coupling function a(ϕ). Let us consider some examples to see what kind of coupling functions might naturally arise. First, the simplest case is the Jordan-Fierz-Brans-Dicke action, which is of the general type (40) with F (Φ) = Φ (47) Z(Φ) = ωBD Φ −1 , (48) where ωBD is an arbitrary constant. Using Eqs. (42), (45) above, one finds that − 2α0 ϕ = lnΦ and that the (logarithmic) coupling function is simply a(ϕ) = α0 ϕ+ const. , (49) where α0 = ∓(2ωBD + 3)−1/2, depending on the sign chosen in Eq. (42). Inde- pendently of this sign, one has the link α20 = 2ωBD + 3 . (50) 15As we shall mostly work with a(ϕ) below, we shall henceforth drop the adjective ‘loga- rithmic’. 16Note, however, that, as was recently explored in [73, 74, 75], a sufficiently fast varying potential V (ϕ) can change the tensor-scalar phenomenology by endowing ϕ with a mass term m2ϕ = ∂2V/∂ϕ2 which strongly depends on the local value of ϕ and, thereby can get large in sufficiently dense environments. Note that 2ωBD + 3 must be positive for the spin-0 excitations to have the correct (non ghost) sign. Let us now discuss the often considered case of a massive scalar field having a nonminimal coupling to curvature 16πG∗ g̃1/2 R̃− g̃µν∂µΦ ∂νΦ−m2ΦΦ2 + ξR̃Φ2 + Sm[ψm; g̃µν ] . This is of the form (40) with F (Φ) = 1 + ξΦ2 , Z(Φ) = 1 , U(Φ) = m2ΦΦ 2 . (52) The case ξ = − 1 is usually referred to as that of ‘conformal coupling’. With the variables (51) the theory is ghost-free only if 2 (1+ ξΦ2)2 (dϕ/dΦ)2 = 1+ ξ(1 + 6 ξ)Φ2 is everywhere positive. If we do not wish to restrict the initial values of Φ, we must have ξ(1+6 ξ) > 0. Introducing then the notation χ ≡ ξ(1 + 6 ξ), we get the following link between Φ and ϕ: 1 + 2χΦ 1 + χ2 Φ2 + χΦ 1 + χ2 Φ2 − 1 + ξΦ2 . (53) For small values of Φ, this yields ϕ = Φ/ 2 + O(Φ3). The potential and the coupling functions are given by V (ϕ) = 1 + ξΦ2 , (54) a(ϕ) = −1 ln(1 + ξΦ2) . (55) These functions have singularities when 1+ ξΦ2 vanishes. If we do not wish to restrict the initial value of Φ we must assume ξ > 0 (which then implies our previous assumption ξ(1+6 ξ) > 0). Then there is a one-to-one relation between Φ and ϕ over the entire real line. Small values of Φ correspond to small values of ϕ and to a coupling function a(ϕ) = − ξ ϕ2 +O(ϕ4) . (56) On the other hand, large values of |Φ| correspond to large values of |ϕ|, and to a coupling function of the asymptotic form a(ϕ) ≃ − |ϕ|+ const. (57) The potential V (ϕ) has a minimum at ϕ = 0, as well as other minima at ϕ → ±∞. If we assume, for instance, that m2Φ and the cosmological dynamics are such that the cosmological value of ϕ is currently attracted towards zero, the value of ϕ at large distances from the local gravitating systems we shall consider will be ϕ0 ≪ 1. As a final example of a possible tensor-scalar gravity theory, let us discuss the string-motivated dilaton-runaway scenario considered in [76]. The starting action (a functional of ḡµν and Φ) was taken of the general form Bg(Φ) BΦ(Φ) [2 �̄Φ −(∇̄Φ)2]− 1 BF (Φ)F̄ 2 − V (Φ) + · · · and it was assumed that all the functions Bi(Φ) have a regular asymptotic be- havior when Φ → +∞ of the form Bi(Φ) = Ci+O(e−Φ). Under this assumption the early cosmological evolution can push Φ towards +∞ (hence the name ‘run- away dilaton’). In the canonical, ‘Einstein frame’ representation (43), one has, for large values of Φ, Φ ≃ c ϕ, where c is a numerical constant, and the coupling function to hadronic matter is given by ea(ϕ) ∝ ΛQCD(ϕ) ∝ B−1/2g (ϕ) exp[−8π2 b−13 BF (ϕ)] where b3 is the one-loop rational coefficient entering the renormalization-group running of the gauge field coupling g2F . This finally yields a coupling function of the approximate form (for large values of ϕ): a(ϕ) ≃ k e−cϕ + const. , where the dimensionless constants k and c are both expected to be of order unity. [The constant c must be positive, but the sign of k is not a priori restricted.] Summarizing: the JFBD model yields a coupling function which is a linear function of ϕ, Eq. (49), a nonminimally coupled scalar yields a coupling function which interpolates between a quadratic function of ϕ, Eq. (56), and a linear one, Eq. (57), and the dilaton-runaway scenario of Ref. [76] yields a coupling function of a decaying exponential type. 5.4 The role of the coupling function a(ϕ); definition of the two-dimensional space of tensor-scalar gravity theo- ries T1(α0, β0) Let us now discuss how the coupling function a(ϕ) enters the observable pre- dictions of tensor-scalar gravity at the first post-Newtonian (1PN) level, i.e., in the weak-field conditions appropriate to solar-system tests. It was shown in previous work that, if one uses appropriate units in the asymptotic region far from the system, namely units such that the asymptotic value a(ϕ0) of a(ϕ) vanishes17, all observable quantities at the 1PN level depend only on the values 17In these units the Einstein metric g∗µν and the physical metric g̃µν asymptotically coin- cide. of the first two derivatives of the a(ϕ) at ϕ = ϕ0. More precisely, if one defines α(ϕ) ≡ ∂ a(ϕ) ; β(ϕ) ≡ ∂ α(ϕ) ∂2 a(ϕ) , (58) and denotes by α0 ≡ α(ϕ0), β0 ≡ β(ϕ0) their asymptotic values, one finds (see, e.g., [33]) that the effective gravitational constant between two bodies (as measured by a Cavendish experiment) is given by G = G∗(1 + α 0) , (59) while, among the PPN parameters, only the two basic Eddington ones, γ̄ ≡ γPPN − 1, and β̄ ≡ βPPN − 1, do not vanish, and are given by γ̄ ≡ γPPN − 1 = −2 α 1 + α20 , (60) β̄ ≡ βPPN − 1 = 1 α0 β0 α0 (1 + α20) . (61) The structure of the results (60) and (61) can be transparently expressed by means of simple (Feynman-like) diagrams (see, e.g., [77]). Eqs. (59) and (60) correspond to diagrams where the interaction between two worldlines (repre- senting two massive bodies) is mediated by the sum of the exchange of one graviton and one scalar particle. The scalar couples to matter with strength G∗. The exchange of a scalar excitation then leads to a term ∝ α20. On the other hand, Eq. (61) corresponds to a nonlinear interaction between three worldlines involving: (i) the ‘generation’ of a scalar excitation on a first world- line (factor α0), (ii) a nonlinear vertex on a second worldline associated to the quadratic piece of a(ϕ) (aquad(ϕ) = β0(ϕ−ϕ0)2; so that one gets a factor β0), and (iii) the final ‘absorption’ of a scalar excitation on a third worldline (second factor α0). Eqs. (60) and (61) can be summarized by saying that the first two coefficients in the Taylor expansion of the coupling function a(ϕ) around ϕ = ϕ0 (after setting a(ϕ0) = 0) a(ϕ) = α0(ϕ− ϕ0) + β0(ϕ − ϕ0)2 + · · · (62) suffice to determine the quasi-stationary, weak-field (1PN) predictions of any tensor-scalar theory. In other words, the solar-system tests only explore the ‘osculating approximation’ (62) (slope and local curvature) to the function a(ϕ). Note that GR corresponds to a vanishing coupling function a(ϕ) = 0 (so that α0 = β0 = · · · = 0), the JFBD model corresponds to keeping only the first term on the R.H.S. of (62), while, for instance, the nonminimally coupled scalar field (with asymptotic value ϕ0 ≪ 1) does indeed lead to nonzero values for both α0 and β0, namely α0 ≃ − 2 ξ ϕ0 ; β0 ≃ − 2 ξ . (63) Finally the dilaton-runaway scenario considered above leads also to non zero values for both α0 and β0, namely α0 ≃ − k c e−cϕ0 ; β0 ≃ + k c2 e−cϕ0 , (64) for a largish value of ϕ0. Note that the dilaton-runaway model naturally predicts that α0 ≪ 1, and that β0 is of the same order of magnitude as α0 : β0 ≃ − c α0 with c being (positive and) of order unity. The interesting outcome is that such a model is well approximated by the usual JFBD model (with β0 = 0). This shows that a JFBD-like theory could come out from a model which is initially quite different from the usual exact JFBD theory. As we shall discuss in detail below, solar-system tests constrain α20 and α 0 |β0| to be both small. This immediately implies that |α0| must be small, i.e., that the scalar field is linearly weakly coupled to matter. On the other hand, the quadratic coupling parameter β0 is not directly constrained. Both its magnitude and its sign can be more or less arbitrary. Note that there are no a priori sign restrictions on β0. The conformal factor A 2(ϕ) = exp(2 a(ϕ)) entering Eq. (43) had to be positive, but this leads to no restrictions on the sign of a(ϕ) and of its various derivatives18. For instance, in the nonminimally coupled scalar field case, it seemed more natural to require ξ > 0, which leads to a negative β0 in view of Eq. (63). Let us summarize the results above: (i) the most general tensor-scalar the- ory19 is described by one arbitrary function a(ϕ); and (ii) weak-field tests depend only on the first two terms, parametrized by α0 and β0, in the Taylor expansion (62) of a(ϕ) around its asymptotic value ϕ0. From this follows a rather natural way to define a simplemini space of tensor- scalar theories. It suffices to consider the two-dimensional space of theories, say T1(α0, β0), defined by the coupling function which is a quadratic polynomial in ϕ [34, 35], say aα0,β0(ϕ) = α0(ϕ− ϕ0) + β0(ϕ− ϕ0)2 . (65) As indicated, this class of theories depends only on two parameters: α0 and β0. The asymptotic value ϕ0 of ϕ does not count as a third parameter (when using the form (65)) because one can always work with the shifted field ϕ̄ ≡ ϕ−ϕ0, with asymptotic value ϕ̄0 = 0 and coupling function aα0,β0(ϕ̄) = α0 ϕ̄+ β0 ϕ̄ 2. Moreover, as already said, the asymptotic value a(ϕ0) of a(ϕ) has also no physical meaning, because one can always use units such that it vanishes (as done in (65)). 18As explained above, we assume here the presence of a potential term V (ϕ) to fix the asymptotic value ϕ0 of ϕ. If the potential V (ϕ) is absent (or negligible), the ‘attractor mechanism’ of Refs. [78, 71] would attract ϕ to a minimum of the coupling function a(ϕ), thereby favoring a positive value of β0. 19Under the assumption that the potential V (ϕ) is a slowly-varying function of ϕ, which modifies the propagation of ϕ only on very large scales. Note also that an alternative way to represent the same class of theories is to use a coupling function of the very simple form aβ(ϕ) = β ϕ2 , (66) but to keep the asymptotic value ϕ0 as an independent parameter. This class of theories is clearly equivalent to T1(α0, β0), Eq. (65), with the dictionary: α0 = β ϕ0, β0 = β. 5.5 Tensor-scalar gravity, strong-field effects, and binary- pulsar observables Having chosen some mini-space of gravity theories, we now wish to derive what predictions these theories make for the timing observables of binary pulsars. To do this we need to generalize the general relativistic treatment of the motion and timing of binary systems comprising strongly self-gravitating bodies summarized above. Let us recall that this treatment was based on a multi-chart method, using a matching between two separate problems: (i) the ‘internal problem’ considers each strongly self-gravitating body in a suitable approximately freely falling frame where the influence of its companion is small, and (ii) the ‘external problem’ where the two bodies are described as effective point masses which interact via the various fields they are coupled to. Let us first consider the internal problem, i.e., the description of a neutron star in an approximately freely falling frame where the influence of the companion is reduced to imposing some boundary conditions on the tensor and scalar fields with which it interacts [7, 8, 33, 34, 35]. The field equations of a general tensor-scalar theory, as derived from the canonical action (43) (neglecting the effect of V (ϕ)) read R∗µν = 2 ∂µϕ∂νϕ+ 8πG∗ T ∗µν − T ∗g∗µν , (67) �g∗ ϕ = − 4πG∗ α(ϕ)T∗ , (68) where T ∗ ≡ 2 c (g∗)−1/2 δSm/δg∗µν denotes the material stress-energy tensor in ‘Einstein units’, and α(ϕ) the ϕ-derivative of the coupling function, see Eq. (58). All tensorial operations in Eqs. (67) and (68) are performed by using the Einstein metric g∗µν . Explicitly writing the field equations (67) and (68) for a slowly rotating (stationary, axisymmetric) neutron star, labelled20 A, leads to a coupled set of ordinary differential equations constraining the radial dependence of g∗µν and ϕ [35, 79]. Imposing the boundary conditions g∗µν → ηµν , ϕ → ϕa at large radial distances, finally determines the crucial ‘form factors’ (in Einstein units) describing the effective coupling between the neutron star A and the fields to 20We henceforth use the labels A and B for the (recycled) pulsar and its companion, instead of the labels a and b used above. We henceforth use the label a to denote the asymptotic value of some quantity (at large radial distances within the local frame, Xi or Xi , of the considered neutron star A or B). which it is sensitive: total mass mA(ϕa), total scalar charge ωA(ϕa), and inertia moment IA(ϕa). As indicated, these quantities are functions of the asymptotic value ϕa of ϕ felt by the considered neutron star 21. They satisfy the relation ωA(ϕa) = −∂ mA(ϕa)/∂ ϕa. From them, one defines other quantities that play an important role in binary pulsar physics, notably αA(ϕa) ≡ − ≡ ∂ lnmA , (69) βA(ϕa) ≡ , (70) as well as kA(ϕa) ≡ − ∂ ln IA . (71) The quantity αA, Eq. (69), plays a crucial role. It measures the effective coupling strength between the neutron star and the ambient scalar field. If we formally let the self-gravity of the neutron A tend toward zero (i.e., if we consider a weakly self-gravitating object), the function αA(ϕa) becomes replaced by α(ϕa) where α(ϕ) ≡ ∂ a(ϕ)/∂ ϕ is the coupling strength appearing in the R.H.S. of Eq. (68). Roughly speaking, we can think of αA(ϕa) as a (suitable defined) average value of the local coupling strength α(ϕ(r)) over the radial profile of the neutron star 0.5 1 1.5 2 2.5 3 critical maximum maximum mass in GR scalar charge baryonic mass neutron star Figure 2: Dependence upon the baryonic mass m̄A of the coupling parameter αA in the theory T1(α0, β0) with α0 = −0.014, β0 = −6. Figure taken from [80]. It was pointed out in Refs. [34, 35] that the strong self-gravity of a neutron star can cause the effective coupling strength αA(ϕa) to become of order unity, 21This ϕa is a combination of the cosmological background value ϕ0 and of the scalar influence of the companion of the considered neutron star. It varies with the orbital period and is determined as part of the ‘external problem’ discussed below. Note that, strictly speaking, the label a (for asymptotic) should be indexed by the label of the considered neutron star: i.e. one should use a label aA (and a locally asymptotic value ϕaA) when considering the neutron star A, and a label aB (with a corresponding ϕaB ) when considering the neutron star B. even when its weak-field counterpart α0 = α(ϕa) is extremely small (as is im- plied by solar-system tests that put strong constraints on the PPN combination γ̄ = −2α20/(1+α20)). This is illustrated, in the minimal context of the T1(α0, β0) class of theories, in Figure 2. Note that when the baryonic mass m̄A of the neutron star is smaller than the critical mass m̄cr ≃ 1.24M⊙ the effective scalar coupling strength αA of the star is quite small (because it is proportional to its weak-field limit α0 = α(ϕa)). By contrast, when m̄A > m̄cr, |αA| becomes of order unity, nearly independently of the externally imposed α0 = αa = α(ϕa). This interesting non-perturbative behaviour was related in [34, 35] to a mechanism of spontaneous scalarization, akin to the well-known mechanism of spontaneous magnetization of ferromagnets. See also [51] for a simple analytical description of the behaviour of αA. Let us also mention in passing that, in the case where A is a black hole, the effective coupling strength αA actually vanishes [33]. This result is related to the impossibility of having (regular) ‘scalar hair’ on a black hole. We have sketched above the first part of the matching approach to the mo- tion and timing of strongly self-gravitating bodies: the ‘internal problem’. It remains to describe the remaining ‘external problem’. As already mentionned (and emphasized, in the present context, by Eardley [7, 11]), the most efficient way to describe the external problem is, instead of matching in detail the exter- nal fields (g∗µν , ϕ) to the fields generated by each body in its comoving frame, to ‘skeletonize’ the bodies by point masses. Technically this means working with the action 16πG∗ ∗ [R∗ − 2 gµν∗ ∂µϕ∂νϕ] mA(ϕ(zA))(−g∗µν(zA) dz 1/2 , (72) where the function mA(ϕ) in the last term on the R.H.S. is the function mA(ϕa) obtained above by solving the internal problem. Eq. (72) indicates that the ar- gument of this function is taken to be ϕa = ϕ(zA), i.e., the value that the scalar field (as viewed in the external problem) takes at the location z A of the center of mass of body A. However, as body A is described, in the external problem, as a point mass this causes a technical difficulty: the externally determined field ϕ(x) becomes formally singular at the location of the point sources, so that ϕ(zA) is a priori undefined. One can either deal with this problem by coming back to the physically well-defined matching approach (which shows that ϕ(zA) should be replaced by ϕa, the value of ϕ in an intermediate domain RA ≪ r ≪ |zA−zB |), or use the efficient technique of dimensional regularization. This means that the spacetime dimension D in Eq. (72) is first taken to have a complex value such that ϕ(zA) is finite, before being analytically continued to its physical value D = 4. One then derives from the action (72) two important consequences for the motion and timing of binary pulsars. First, one derives the Lagrangian de- scribing the relativistic interaction between N strongly self-gravitating bodies (including orbital ∼ (v/c)2 effects, and neglecting O(v4/c4) ones) [11, 7, 33, 39]. It is the sum of one-body, two-body and three-body terms. The one-body action has the usual form of the sum (over the label A) of the kinetic term of each point mass: one-body A = −mA c 1− v2A/c2 = −mA c2 + (v2A) . (73) Here, we use Einstein units, and the inertial mass mA entering Eq. (73) is mA ≡ mA(ϕ0), where ϕ0 is the asymptotic value of ϕ far away from the considered N -body system. The two-body action is a sum over the pairs A,B of a term L 2-body AB which differs from the GR-predicted 2-body Lagrangian in two ways: (i) the usual gravitational constant G appearing as an overall factor in L 2-body AB must be re- placed by an effective (body-dependent) gravitational constant (in the appro- priate units mentioned above) given by GAB = G∗(1 + αA αB) , (74) and (ii) the relativistic (O(v2/c2)) terms in L2-bodyAB contain, in addition to those predicted by GR, new velocity-dependent terms of the form 2-body AB = (γ̄AB) GAB mAmB (vA − vB)2 , (75) γ̄AB ≡ γAB − 1 = − 2 αA αB 1 + αA αB . (76) In these expressions αA ≡ αA(ϕ0) ≡ ∂ lnmA(ϕ0)/∂ϕ0 (see Eq. (69) with ϕa → Finally, the 3-body action is a sum over the pairs B,C and over A (with A 6= B, A 6= C, but the possibility of having B = C) of 3-body ABC = −(1 + 2 β̄ GAB GAC mAmB mC c2 rAB rAC where β̄ABC ≡ βABC − 1 = αB βA αC (1 + αA αB)(1 + αA αC) , (78) with βA = ∂αA(ϕ0)/∂ϕ0 (see Eq. (70) with ϕa → ϕ0). When comparing the strong-field results (74), (76), (78) to their weak-field counterparts (59), (60), (61) one sees that the body-dependent quantity αA replaces the weak-field coupling strength α0 in all quantities which are linked to a scalar effect generated by body A. Note also that, in keeping with the ‘3-body’ nature of Eq. (77), the quantity βABC −1 is linked to scalar interactions which are generated in bodies B and C and which nonlinearly interact on body A. The notation used above has been chosen to emphasize that γAB and β are strong-field analogs of the usual Eddington parameters γPPN, βPPN, so that γ̄AB and β̄ BC are strong-field analogs of the ‘post-Einstein’ 1PN parameters γ̄ and β̄ (which vanish in GR). Indeed the usual PPN results for the post-Einstein terms in the O(1/c2) 2-body and 3-body Lagrangians are obtained by replacing in Eqs. (75) and (77) γ̄AB → γ̄, β̄ABC → β̄ and GAB → G. The non-perturbative strong-field effects discussed above show that the strong self-gravity of neutron stars can cause γAB and β BC to be significantly different from their GR values γGR = 1, βGR = 1, in some scalar-tensor theories having a small value of the basic coupling parameter α0 (so that γ PPN − 1 ∝ α20 and βPPN − 1 ∝ β0 α20 are both small). For instance, Fig. 2 shows that it is possible to have αA ∼ αB ∼ ± 0.6 which implies γAB − 1 ∼ − 0.53, i.e., a 50% deviation from GR! Even larger effects can arise in βABC − 1 because of the large values that βA = ∂αA/∂ϕ0 can reach near the spontaneous scalarization transition [35]. Those possible strong-field modifications of the effective Eddington param- eters γAB, β BC , which parametrize the ‘first post-Keplerian’ (1PK) effects (i.e., the orbital effects ∼ v2/c2 smaller than those entailed by the Lagrangian A 6=B GAB mAmB/rAB), can then significantly modify the usual GR predictions relating the directly observable parametrized post-Keplerian (PPK) parameters to the values of the masses of the pulsar and its compan- ion. As worked out in Refs. [11, 31, 33, 35] one finds the following modified predictions for the PPK parameters k ≡ 〈ω̇〉/n, r and s: kth(mA,mB) = 1− e2 GAB(mA +mB)n αA αB 1 + αA αB − XA βB α A +XB βA α 6 (1 + αA αB)2 , (79) rth(mA,mB) = G0B mB , (80) sth(mA,mB) = GAB(mA +mB)n ]−1/3 . (81) Here, the label A refers to the object which is timed (‘the pulsar’22), the label B refers to its companion, xA = aA sin i/c denotes the projected semi-major axis of the orbit of A (in light seconds), XA ≡ mA/(mA+mB) andXB ≡ mB/(mA+ mB) = 1 − XA the mass ratios, n ≡ 2π/Pb the orbital frequency and G0B = G∗(1 + α0 αB) the effective gravitational constant measuring the interaction 22In the double binary pulsar, both the first discovered pulsar and its companion are pulsars. However, the companion B is a non recycled, slow pulsar whose motion is well described by Keplerian parameters only. between B and a test object (namely electromagnetic waves on their way from the pulsar toward the Earth). In addition one must replace the unknown bare Newtonian G∗ by its expression in terms of the one measured in Cavendish experiments, i.e., G∗ = G/(1 + α 0) as deduced from Eq. (59). The modified theoretical prediction for the PPK parameter γ entering the ‘Einstein time delay’ ∆E , Eq. (24), is more complicated to derive because one must take into account the modulation of the proper spin period of the pulsar caused by the variation of its moment of inertia IA under the (scalar) influence of its companion [11, 7, 35]. This leads to γth(mA,mB) = 1 + αA αB GAB(mA +mB)n [XB(1 + αA αB) + 1 + kA αB ] , (82) where kA(ϕ0) = −∂ ln IA(ϕ0)/∂ϕ0 (see Eq. (71) with ϕa → ϕ0). Numerical studies [35] show that kA can take quite large values. Actually, the quantity kA αB entering (82) blows up near the scalarization transition when α0 → 0 (keeping β0 < 0 fixed). In other words a theory which is closer to GR in weak- field conditions predicts larger deviations in the strong-field regime. The structure dependence of the effective gravitational constantGAB , Eq. (74), has also the consequence that the object A does not fall in the same way as B in the gravitational field of the Galaxy. As most of the mass of the Galaxy is made of non strongly-self-gravitating bodies, A will fall toward the Galaxy with an acceleration ∝ GA0, while B will fall with an acceleration ∝ GB0. Here, as above, GA0 = G0A = G∗(1 + α0 αA) is the effective gravi- tational constant between A and any weakly self-gravitating body. As pointed out in Ref. [43] this possible violation of the universality of free fall of self- gravitating bodies can be constrained by using observational data on the class of small-eccentricity long-orbital-period binary pulsars. More precisely, the quantity which can be observationally constrained is not exactly the violation ∆AB = (G0A −G0B)/G = (1 +α20)−1(α0 αA −α0 αB) of the strong equivalence principle [which simplifies to ∆A0 = (G0A − G)/G = (1 + α20)−1(α0 αA − α20) in the case of observational relevance where one neglects the self-gravity of the white-dwarf companion] but rather23 [33] ∆effective ≡ 2 γAB − (XA βBAA +XB βABB) + 2 (1 + αA αB) −3/2(1 + α20) −1(α0 αA − α0 αB) . (83) Here, the index B (= white-dwarf companion) can be replaced by 0 (weakly self- gravitating body) so that, for instance, γAB = γA0 = 1− 2αAα0/(1+αA α0) = (1− αA α0)/(1 + αA α0), as deduced from Eq. (76). 23This refinement is given here for pedagogical completeness. However, in practice, the lowest-order result ∆ ≃ (1 + α2 )−1(α0 αA − α ) ≃ α0 αA − α is accurate enough. It remains to discuss the possible strong-field modifications of the theoretical prediction for the orbital period derivative Ṗb = Ṗ b (mA,mB). This is obtained by deriving from the effective action (72) the energy lost by the binary system in the form of fluxes of spin-2 and spin-0 waves at infinity. The needed results in a generic tensor-scalar theory were derived in Refs. [33, 39] (in addition one must take into account the tensor-scalar modification of the additional ‘varying- Doppler’ contribution to the observed Ṗb due to the Galactic acceleration [38]). The final result for Ṗb is of the form Ṗ thb (mA,mB) = Ṗ monopole bϕ + Ṗ dipole bϕ + Ṗ quadrupole bϕ + Ṗ quadrupole galagtic bGR + δ th Ṗ galactic b , (84) where, for instance, Ṗ monopole bϕ is (heuristically 24) related to the monopolar flux of spin-0 waves at infinity. The term Ṗ quadrupole bg∗ corresponds to the usual quadrupolar flux of spin-2 waves at infinity. It reads: quadrupole bg∗ (mA,mB) = − 5(1 + αA αB) (mA +mB)2 GAB(mA +mB)n 1 + 73 e2/24 + 37 e4/96 (1− e2)7/2 with GAB = G∗(1 + αA αB) = G(1 + αA αB)/(1 + α 0), where G∗ is the ‘bare’ gravitational constant appearing in the action, while G is the gravitational con- stant measured in Cavendish experiments. The flux (85) is the only one which survives in GR (although without any αA-related modifications). Among the several other contributions which arise in tensor-scalar theories, let us only write down the explicit expression of the contribution to (84) coming from the dipolar flux of scalar waves. Indeed, this contribution is, in most cases, the dominant one [7] because it scales as (v/c)3, while the monopolar and quadrupolar con- tributions scale as (v/c)5. It reads dipole bϕ (mA,mB) = −2π G∗mAmB n c3(mA +mB) 1 + e2/2 (1 − e2)5/2 (αA − αB)2 . (86) Note that the dipolar effect (86) vanishes when αA = αB . Indeed, a binary system made of two identical objects (A = B) cannot select a preferred direction for a dipole vector, and cannot therefore emit any dipolar radiation. This also implies that double neutron star systems (which tend to have mA ≈ mB ∼ 1.35M⊙) will be rather poor emitters of dipolar radiation (though (86) still tends to dominate over the other terms in (84), because of the remaining difference (mA − mB)/(mA + mB) 6= 0). By contrast, very dissymmetric systems such 24Contrary to the GR case where a lot of effort was spent to show how the observed Ṗb was directly related to the GR predictions for the (v/c)5-accurate orbital equations of motion of a binary system [9], we use here the indirect and less rigorous argument that the energy flux at infinity should be balanced by a corresponding decrease of the mechanical energy of the binary system. as a neutron-star and a white-dwarf (or a neutron-star and a black hole) will be very efficient emitters of dipolar radiation, and will potentially lead to very strong constraints on tensor-scalar theories. See below. 5.6 Theory-space analyses of binary pulsar data Having reviewed the theoretical results needed to discuss the predictions of alternative gravity theories, let us end by summarizing the results of various theory-space analyses of binary pulsar data. Let us first recall what are the best, current solar-system limits on the two 1PN ‘post-Einstein’ parameters γ̄ ≡ γPPN − 1 and β̄ ≡ βPPN − 1. They are: γ̄ = (2.1± 2.3)× 10−5 , (87) from frequency shift measurements made with the Cassini spacecraft [81], which supersedes the constraint γ̄ = (−1.7± 4.5)× 10−4 (88) from VLBI measurements [82], |2 γ̄ − β̄| < 3× 10−3 , (89) from Mercury’s perihelion shift [66, 83], and 4 β̄ − γ̄ = (4.4± 4.5)× 10−4 , (90) from Lunar laser ranging measurements [84]. Concerning binary pulsar data, we can make use of the published measure- ments of various Keplerian and post-Keplerian timing parameters in the binary pulsars: PSR 1913+16 [37], PSR B1534+12 [41], PSR J1141−6545 [47] and PSR J0737−3039A+B [3, 48, 49]. In addition, we can use25 the recently up- dated limit on the parameter ∆ measuring a possible violation of the strong equivalence principle (SEP), namely |∆| < 5.5 × 10−3 at the 95% confidence level [44]. This ensemble of solar-system and binary-pulsar data can then be analyzed within any given parametrized theoretical framework. For instance, one might work within (i) the 4-parameter framework T0(γ̄, β̄; ǫ, ζ) [70] which defines the 2PN exten- sion of the original (Eddington) PPN framework T0(γ̄, β̄); or (ii) the 2-parameter class of tensor-mono-scalar theories T1(α0, β0) [34]; or 25There is, however, a caveat in the theoretical use one can make of the phenomenological limits on ∆. Indeed, in the small-eccentricity long-orbital-period binary pulsar systems used to constrain ∆ one does not have access to enough PK parameters to measure the pulsar mass mA directly. As the theoretical expression of ∆ ≃ α0 αA −α depends on mA (through αA), one needs to assume some fiducial value of mA (say mA ≃ 1.35M⊙). (iii) the 2-parameter class of tensor-bi-scalar theories T2(β ′, β′′) [33]. Here, the index 0 on T0(γ̄, β̄; ǫ, ζ) is a reminder of the fact that this framework is not a family of specific theories (it contains zero explicit dynamical fields), but is a parametrization of 2PN deviations from GR. As a consequence, its use for analyzing binary pulsar data is somewhat ill-defined because one needs to truncate the various timing observables (which are functions of the compactness of the two bodies A and B, say PPK = f(cA, cB)) at the 2PN order (i.e. es- sentially at the quadratic order in cA and/or cB). For some observables (or for product of observables) there might be several ways of defining this truncation. In spite of this slight inconvenience, the use of the T0(γ̄, β̄; ǫ, ζ) framework is conceptually useful because it shows very clearly why and how binary-pulsar data can probe the behaviour of gravitational theories beyond the usual 1PN regime probed by solar-system tests. For instance, the parameter ∆A ≡ mgravA /minertA − 1 measuring the strong equivalence principle (SEP) violation in a neutron star has, within the T0(γ̄, β̄; ǫ, ζ) framework, a 2PN-order expansion of the form [33, 70] ∆A = − (4 β̄ − γ̄) cA + + ζ +O(β̄) bA , (91) where cA = −2 ∂ lnmA∂ lnG ≃ 〈U〉A, bA = 1c4 〈U 2〉A ≃ B c2A, with B ≃ 1.026 and cA ≃ kmA/M⊙ with k ∼ 0.21. The general result (91) is compatible with the result quoted in subsection 5.2 within the context of the theory T2(β ′, β′′) when taking into account the fact that, within T2(β ′, β′′), one has β̄ = γ̄ = 0, ǫ = β′ and ζ = 0 [and that β′′ parametrizes some effects beyond the 2PN level]. On the example of Eq. (91) one sees that, after having used solar-system tests to constrain the first contribution on the RHS to a very small value, one can use binary-pulsar tests of the SEP to set a significant limit on the combination ǫ + ζ of 2PN parameters. Other pulsar data then yield significant limits on other combinations of the two 2PN parameters ǫ and ζ. The final conclusion is that binary-pulsar data allow one to set significant limits (around or better than the 1% level) on the possible 2PN deviations from GR (in contrast to solar-system tests which are unable to yield any limit on ǫ and ζ) [70]. For a recent update of the limits on ǫ and ζ, which makes use of recent pulsar data see [51]. Let us now briefly discuss the use of mini-space of theories, such as T1(α0, β0) or T2(β ′, β′′), for analyzing solar-system and binary-pulsar data. The basic methodology is to compute, for each given theory (e.g. for each given values of α0 and β0 if one chooses to work in the T1(α0, β0) theory space) a goodness- of-fit statistics χ2(α0, β0) measuring the quality of the agreement between the experimental data and the considered theory. For instance, when considering the timing data of a particular pulsar, for which one has measured several PK parameters pi (i = 1, . . . , n) with some standard deviations σ , one defines, for this pulsar χ2(α0, β0) = min mA,mB (σobspi ) theory i (α0, β0;mA,mB)− p 2 , (92) where ‘min’ denotes the result of minimizing over the unknown masses mA,mB and where p theory i (α0, β0;mA,mB) denotes the theoretical prediction (within T1(α0, β0)) for the PK observable pi (given also the observed values of the Keplerian parameters). The goodness-of-fit quantity χ2(α0, β0) will reach its minimum χ min for some values, say αmin0 , β 0 , of α0 and β0. Then, one focusses, for each pulsar, on the level contours of the function ∆χ2(α0, β0) ≡ χ2(α0, β0)− χ2min . (93) Each choice of level contour (e.g. ∆χ2 = 1 or ∆χ2 = 2.3) defines a certain region in theory space, which contains, with a certain corresponding ‘confidence level’, the ‘correct’ theory of gravity (if it belongs to the considered mini-space of theories). When combining together several independent data sets (e.g. solar- system data, and different pulsar data) we can define a total goodness-of-fit statistics χ2tot(α0, β0), by adding together the various individual χ 2(α0, β0). This leads to a corresponding combined contour ∆χ2tot(α0, β0). Let us end by briefly summarizing the results of the theory-space approach to relativistic gravity tests. For detailed discussions the reader should consult Refs. [33, 40, 35, 36, 80], and especially the recent update [51] which uses the latest binary-pulsar data. Regarding the two-parameter class of tensor-bi-scalar theories T2(β ′, β′′) the recent analysis [51] has shown that the ∆χ2(β′, β′′) corresponding to the double binary pulsar PSR J0737−3039 was defining quite a small elliptical allowed region in the (β′, β′′) plane. By contrast the other pulsar data define much wider allowed regions, while the strong equivalence principle tests define (in view of the theoretical result ∆ ≃ 1 + 1 Bβ′(c2A − c2B)) a thin, but infinitely long, strip |β′| < cst. in the (β′, β′′) plane. This highlights the power of the double binary pulsar in probing certain specific strong-field deviations from GR. Contrary to the T2(β ′, β′′) tensor-bi-scalar theories, which were constructed to have exactly the same first post-Newtonian limit as GR26 (so that solar- system tests put no constraints on β′ and β′′), the class of tensor-mono-scalar theories T1(α0, β0) is such that its parameters α0 and β0 parametrize both the weak-field 1PN regime (see Eqs. (60) and (61) above) and the strong-field regime (which plays an important role in compact binaries). This means that each class of solar-system data (see Eqs. (87)–(90) above) will define, via a corresponding goodness-of-fit statistics of the type, say χ2Cassini(α0, β0) = (σ Cassini −2 (γ̄theory(α0, β0)− γ̄Cassini)2 26However, this could be achieved only at the cost of allowing some combination of the two scalar fields to carry a negative energy flux. a certain allowed region27 in the (α0, β0) plane. As a consequence, the analysis in the framework of the T1(α0, β0) space of theories allows one to compare and contrast the probing powers of solar-system tests versus binary-pulsar tests (while comparing also solar-system tests among themselves and binary-pulsar ones among themselves). The result of the recent analysis [51] is shown in Figure 3. general relativity B1534+12 J1141–6545 J0737–3039 B1913+16 −6 −4 −2 0 2 4 6 0.025 matter matter 0.175 0.075 0.125 solar system Figure 3: Solar-system and binary-pulsar constraints on the two-parameter fam- ily of tensor-mono-scalar theories T1(α0, β0). Figure taken from [51]. In Fig. 3, the various solar-system constraints (87)–(90) are concentrated around the horizontal β0 axis. In particular, the high-precision Cassini con- straint is the lower small grey strip. The various pulsar constraints are labelled by the name of the pulsar, except for the strong equivalence principle constraint which is labelled SEP. Note that General Relativity corresponds to the origin of the (α0, β0) plane, and is compatible with all existing tests. The global constraint obtained by combining all the pulsar tests would, to a good accuracy, be obtained by intersecting the various pulsar-allowed regions. One can then see on Fig. 3 that it would be comparable to the pre-Cassini solar-system constraints and that its boundaries would be defined successively (starting from the left) by 1913+16, 1141−6545, 0737−3039, 1913+16 again and 1141−6545 again. A first conclusion is therefore that, at the quantitative level, binary-pulsar tests constrain tensor-scalar gravity theories as strongly as most solar-system 27Actually, in the case of the Cassini data, as it is quite plausible that the positive value of the published central value γ̄Cassini = +2.1× 10−5 is due to unsubtracted systematic effects, we use σCassiniγ = 2.3× 10 −5 but γ̄Cassini = 0. Otherwise, we would get unreasonably strong 1σ limits on α2 because tensor-scalar theories predict that γ̄ must be negative, see Eqs. (60) and (61). tests (excluding the exceptionally accurate Cassini result which constrains α20 to be smaller than 1.15 × 10−5, i.e. |α0| < 3.4 × 10−3). A second conclusion is obtained by comparing the behaviour of the solar-system exclusion plots and of the binary-pulsar ones around the negative β0 axis. One sees that binary- pulsar tests exclude a whole domain of the theory space (located on the left of β0 < −4) which is compatible with all solar-system experiments (even when including the very tight Cassini constraint). This remarkable qualitative feature of pulsar tests is a direct consequence of the existence of (non-perturbative) strong-field effects which start developing when the product −β0 cA (with cA denoting, as above, the compactness of the pulsar) becomes of order unity. 6 Conclusion In conclusion, we hope to have convinced the reader of the superb opportunities that binary pulsar data offer for testing gravity theories. In particular, they have been able to go qualitatively beyond solar-system experiments in probing two physically important regimes of relativistic gravity: the radiative regime and the strong-field one. Up to now, General Relativity has passed with flying colours all the radiative and strong-field tests provided by pulsar data. However, it is important to continue testing General Relativity in all its aspects (weak- field, radiative and strong-field). Indeed, history has taught us that physical theories have a limited range of validity, and that it is quite difficult to predict in which regime a theory will cease to be an accurate description of nature. Let us look forward to new results, and possibly interesting surprises, from binary pulsar data. Acknowledgments It is a pleasure to thank my long-term collaborator Gilles Esposito-Farèse for his useful remarks on the text, and for providing the figures. I wish also to thank the organizers of the 2005 Sigrav School, and notably Monica Colpi and Ugo Moschella, for organizing a warm and intellectually stimulating meeting. 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Lett. 93, 261101 (2004), arXiv:gr-qc/0411113. http://arxiv.org/abs/gr-qc/0402007 http://arxiv.org/abs/gr-qc/0411113 Introduction Motion of binary pulsars in general relativity Timing of binary pulsars in general relativity Phenomenological approach to testing relativistic gravity with binary pulsar data Theory-space approach to testing relativistic gravity with binary pulsar data Theory-space approaches to solar-system tests of relativistic gravity Theory-space approaches to binary-pulsar tests of relativistic gravity Tensor-scalar theories of gravity The role of the coupling function a(); definition of the two-dimensional space of tensor-scalar gravity theories T1 (0 , 0) Tensor-scalar gravity, strong-field effects, and binary-pulsar observables Theory-space analyses of binary pulsar data Conclusion

0704.0750

Some combinatorial aspects of differential operation compositions on space $R^n$

arXiv:0704.0750v1 [math.DG] 5 Apr 2007 Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 9 (1998), 29–33 SOME COMBINATORIAL ASPECTS OF DIFFERENTIAL OPERATION COMPOSITION ON THE SPACE R Branko J. Malešević In this paper we present a recurrent relation for counting meaningful compositions of the higher-order differential operations on the space Rn (n=3,4,...) and extract the non-trivial compositions of order higher than two. 1. DIFFERENTIAL FORMS AND OPERATIONS ON THE SPACE R It is well known that the first-order differential operations grad, curl and div on the space R3 can be introduced using the operator of the exterior differentia- tion d of differential forms [1]: Ω0(R3) −→ Ω1(R3) −→ Ω2(R3) −→ Ω3(R3), where Ωi(R3) is the space of differential forms of degree i = 0, 1, 2, 3 on the space 3 over the ring of functions A = {f : R3 → R | f ∈ C∞(R3)}. In the consideration, which follows, we give definitions of the first-order differential operations. Let us notice that one-dimensional spaces Ω0(R3) and Ω3(R3) are isomorphic toA and let ϕ0 : Ω 0(R3) → A, ϕ3 : Ω 3(R3) → A be the corresponding isomorphisms. Next, the set of vector functionsB = {f =(f1, f2, f3) : R 3 → R3 | f1, f2, f3 ∈ C ∞(R3)}, over the ring A, is three-dimensional. It is isomorphic to Ω1(R3) and Ω2(R3). Let ϕ1 : Ω 1(R3) → B, ϕ2 : Ω 2(R3) → B be the corresponding isomorphisms. In that case, the compositions ϕ−10 ◦ϕ3 : Ω 3(R3) → Ω0(R3) and ϕ−11 ◦ϕ2 : Ω 2(R3) → Ω1(R3) are isomorphisms of the corresponding spaces of differential forms. The first-order differential operations are defined via the operator of the exterior differentiation d of differential forms in the following form: ∇1 = ϕ1◦d◦ϕ 0 : A → B, ∇2 = ϕ2◦d◦ϕ 1 : B → B, ∇3 = ϕ3◦d◦ϕ 2 : B → A. Therefore we obtain explicit expressions for the first order differential operations ∇1, ∇2, ∇3 on the space R 3 in the following form: (1) gradf = ∇1f = e3 : A → B, (2) curlf = ∇2f = e3 : B → B, (3) divf = ∇3f = : B → A. 1991 Mathematics Subject Classification: 26B12, 58A10 http://arxiv.org/abs/0704.0750v1 30 Branko J. Malešević Let us count meaningful compositions of differential operations ∇1,∇2,∇3. Consider the set of functions Θ = {∇1,∇2,∇3}. Let us define a binary relation ρ ”to be in composition” with ∇iρ∇j = ⊤ iff the composition ∇j ◦∇i is meaningful (∇i,∇j ∈ Θ). The Cayley’s table of this relation reads: ρ ∇1 ∇2 ∇3 ∇1 ⊥ ⊤ ⊤ ∇2 ⊥ ⊤ ⊤ ∇3 ⊤ ⊥ ⊥ . We form the graph of relation ρ as follows. If ∇iρ∇j = ⊤ then we put the node ∇j under the node ∇i. Let us mark ∇0 as nowhere-defined function ϑ, with domain and range being the empty set [2]. We shall consider ∇0ρ∇i = ⊤ (i = 1, 2, 3). For the set of functions Θ ∪ {∇0} our graph is the tree with the root in the node ∇0. ∇0 f(0) = 1 ∇2 ❳❳ ∇3 f(1) = 3 ∇1 f(2) = 5 ∇3 f(3) = 8 ✔✔❚❚ ✔✔❚❚ ✔✔❚❚ ✔✔❚❚ ✔✔q∇2 ❚❚q∇3 q∇1 f(4) = 13 ✔✔❚❚ ✔✔❚❚Fig. 1 f(5) = 21 Let fi(k) be a number of meaningful compositions of the k th-order beginning with ∇i. Let f(k) be a number of meaningful composition of the k th-order of operations over Θ. Then f(k) = f1(k) + f2(k) + f3(k). Based on partial self similarity of the tree (Fig. 1), which is formed according to Cayley’s table (4), we get equalities: f1(k) = f2(k − 1) + f3(k − 1) ∧ f2(k) = f2(k − 1) + f3(k − 1) ∧ f3(k) = f1(k − 1). Now, a recurrent relation for f(k) can be derived as follows: f(k) = f1(k) + f2(k) + f3(k) f1(k − 1) + f2(k − 1) + f3(k − 1) f3(k − 1) + f2(k − 1) = f(k − 1) + f1(k − 2) + f2(k − 2) + f3(k − 2) = f(k − 1) + f(k − 2). Based on the initial values: f(1) = 3, f(2) = 5, f(3) = 8 we conclude that f(k) = Fk+3, where is Fibonacci’s number of order k + 3. Let us note that ∇2 ◦∇1 = 0 and ∇3 ◦∇2 = 0, because d 2 = 0. On the other hand, the compositions ∇1 ◦∇3, ∇2 ◦∇2 and ∇3 ◦∇1 are not annihilated, because of ϕ−10 ◦ ϕ3 6= i and ϕ 1 ◦ ϕ2 6= i. Thus, as in the paper [2], we conclude that the non-trivial compositions are of the following form: (∇1◦)∇3 ◦ · · · ◦ ∇1 ◦ ∇3 ◦ ∇1, ∇2 ◦ ∇2 ◦ · · · ◦ ∇2 ◦ ∇2 ◦ ∇2, (∇3◦)∇1 ◦ · · · ◦ ∇3 ◦ ∇1 ◦ ∇3. As non-trivial compositions we consider those which are not identical to the zero function. Terms in parentheses are included in for an odd number of terms and are left out otherwise. Some combinatorial aspects of differential operation compositions ... 31 2. DIFFERENTIAL FORMS AND OPERATIONS ON THE SPACE R Let us present a recurrent relation for counting meaningful compositions of the higher-order differential operations on the space Rn (n = 3, 4, . . .) and extract the non-trivial compositions of order higher than two. Let us form the following sets of functions: Ai = {f : R )|f1, . . . , f(n ) ∈ C for i = 0, 1, . . . ,m where m = [n/2]. Let Ωi(Rn) be a set of differential forms of degree i = 0, 1, . . . , n on the space Rn. Let us notice that Ωi(Rn) and Ωn−i(Rn), over ring A0, are spaces of the same dimension , for i = 0, 1, . . . ,m. They can be identified with Ai, using the corresponding isomorphisms: ϕi : Ω i(Rn) → Ai (0 ≤ i ≤ m) and ϕn−i : Ω n−i(Rn) → Ai (0 ≤ i < n−m). We define the first-order differential operations on the space Rn via the operator of the exterior differentiation d as follows: ∇i = ϕi ◦ d ◦ ϕ i−1 (1 ≤ i ≤ n). (1 ≤ i ≤ m) Therefore, we obtain the first order differential operations on the space Rn, de- pending on pairity of dimension n, in the following form: n = 2m : ∇1 : A0 → A1 ∇2 : A1 → A2 ∇i : Ai → Ai+1 ∇m : Am−1 → Am ∇m+1 : Am → Am−1 ∇n−j : Aj+1 → Aj ∇n−1 : A2 → A1 ∇n : A1 → A0, n = 2m+ 1 : ∇1 : A0 → A1 ∇2 : A1 → A2 ∇i : Ai → Ai+1 ∇m : Am−1 → Am ∇m+1 : Am → Am ∇m+2 : Am → Am−1 ∇n−j : Aj+1 → Aj ∇n−1 : A2 → A1 ∇n : A1 → A0. Consider the set of functions Θ = {∇1,∇2, . . . ,∇n}. Let us define a binary relation ρ ”to be in composition” with ∇iρ∇j = ⊤ iff the composition ∇j ◦∇i is meaningful (∇i,∇j ∈ Θ). It is not difficult to check that Cayley’s table of this relation is determined with: (6) ∇iρ∇j = ⊤ : (j = i+ 1) ∨ (i+ j = n+ 1), ⊥ : (j 6= i+ 1) ∧ (i+ j 6= n+ 1). Let us form an adjacency matrix A = [aij ] ∈ {0, 1} n×n of the graph, determined by relation ρ. Let fi(k) be a number of meaningful compositions of the k th-order 32 Branko J. Malešević beginning with ∇i (notice that fi(1) = 1 for i= 1, . . . , n). Let f(k) be a number of meaningful composition of the kth-order of operations over Θ. Then f(k) = f1(k)+. . .+fn(k). Notice that the following is true: (7) fi(k) = aij · fj(k − 1), for i = 1, . . . , n. Based on (7) we form the system of recurrent equations: f1(k) fn(k) a11 · · · a1n an1 · · · ann f1(k − 1) fn(k − 1) If vn = [ 1 · · · 1 ]1×n then: (9) f(k) = vn · f1(k) fn(k) So, the expression: (10) f(k) = vn · A k−1 · vTn . follows from (8) and (9). Reducing the system of the recurrent equations (8), for any of the functions fi(k) we have: (11) α0fi(k) + α1fi(k − 1) + · · ·+ αnfi(k − n) = 0 (k > n), where α0, . . . , αn are coefficients of the characteristic polynomial Pn(λ) = |A−λI| = n+ . . .+αn. Thus, we conclude that the function f(k) = fi(k) also satisfies: (12) α0f(k) + α1f(k − 1) + · · ·+ αnf(k − n) = 0 (k > n). Hence, the following theorem holds. Theorem 1. The number of meaningful differential operations, on the space R (n = 3, 4, . . .), of the order higher than two, is determined by the formula (10), i.e. by the recurrent formula (12). In n-dimensional space Rn, for dimensions n = 3, 4, 5, . . . , 10, using the pre- vious theorem we form a table of the corresponding recurrent formula: Dimension: Recurrent relations for the number of meaningful compositions: n = 3 f(i+ 2) = f(i + 1) + f(i) n = 4 f(i+ 2) = 2f(i) n = 5 f(i+ 3) = f(i+ 2) + 2f(i+ 1)− f(i) n = 6 f(i+ 4) = 3f(i+ 2)− f(i) n = 7 f(i + 5) = f(i + 3) + 3f(i + 2) − 2f(i + 1)− f(i) n = 8 f(i + 4) = 4f(i + 3) − 3f(i) n = 9 f(i+ 5) = f(i+ 4) + 4f(i+ 3)− 3f(i+ 2)− 3f(i + 1) + f(i) n = 10 f(i + 6) = 5f(i + 4) − 6f(i + 2) + f(i) Some combinatorial aspects of differential operation compositions ... 33 Let us determine non-trivial higher-order meaningful compositions on the space Rn. For isomorphisms ϕk we have: (13) ϕ−1 ◦ ϕn−k 6= i, for k = 1, 2, . . . , n and 2k 6= n. Then, based on (6) and (13), all second-order compositions are given by the formula: (14) ∇j ◦ ∇k = 0 : j = k + 1, gj,k : (k + j = n+ 1) ∧ (2k 6= n), ϑ : (j 6= k + 1) ∧ (k + j 6= n+ 1); where 0 is a trivial composition, gj,k is a non-trivial second-order composition and ϑ is a nowhere-defined function for j, k = 1, . . . , n. Notice that in gj,k = ∇j ◦∇k = ϕn+1−k ◦ d ◦ ϕ ◦ ϕk ◦ d ◦ ϕ k−1 (j=n+1−k ∧ 2k 6=n) and switching the terms is impossible, because in that way we get nowhere-defined function ϑ. Hence, we conclude that the following theorem holds. Theorem 2. All meaningful non-trivial differential operations on the space R (n = 3, 4, . . .), of order higher than, two are given in the form of the following com- positions: (∇k) ◦ ∇j ◦ ∇k ◦ · · · ◦ ∇j ◦ ∇k, (∇j) ◦ ∇k ◦ ∇j ◦ · · · ◦ ∇k ◦ ∇j , with to the condition k+ j = n+ 1 and 2k, 2j 6= n for k, j = 1, 2, . . . , n. Terms in parentheses are included in for an odd number of terms and are left out otherwise. Acknowledgment. I wish to express my gratitude to ProfessorsM. Merkle and M. Prvanović who examined the first version of the paper and gave me their suggestions and some very useful remarks. REFERENCES 1. R.Bott, L.W.Tu: Differential forms in algebraic topology, Springer, New York 1982. 2. B.J.Malešević: A note on higher-order differential operations, Univ. Beograd, Publ. Elektrotehn. Fak.,Ser. Mat. 7 (1996), 105-109. University of Belgrade, (Received September 8, 1997) Faculty of Electrical Engineering, (Revised October 30, 1998) P.O.Box 35-54, 11120 Belgrade, Yugoslavia [email protected]

0704.0751

Hyperbolicity in unbounded convex domains

HYPERBOLICITY IN UNBOUNDED CONVEX DOMAINS FILIPPO BRACCI AND ALBERTO SARACCO ABSTRACT. We provide several equivalent characterizations of Kobayashi hyperbolicity in un- bounded convex domains in terms of peak and anti-peak functions at infinity, affine lines, Bergman metric and iteration theory. 1. INTRODUCTION Despite the fact that linear convexity is not an invariant property in complex analysis, bounded convex domains in CN have been very much studied as prototypes for the general situation. In particular, by Harris’ theorem [6] (see also, [1], [9]) it is known that bounded convex domains are always Kobayashi complete hyperbolic (and thus by Royden’s theorem, they are also taut and hyperbolic). Moreover, by Lempert’s theorem [10], [11], the Kobayashi distance can be realized by means of extremal discs. These are the basic cornerstone for many useful results, especially in pluripotential theory and iteration theory. On the other hand, not much is known about unbounded domains. Clearly, the geometry at infinity must play some important role. In this direction, Gaussier [5] gave some conditions in terms of existence of peak and anti-peak functions at infinity for an unbounded domain to be hyperbolic, taut or complete hyperbolic. Recently, Nikolov and Pflug [14] deeply studied conditions at infinity which guarantee hyperbolicity, up to a characterization of hyperbolicity in terms of the asymptotic behavior of the Lempert function. In these notes we restrict ourselves to the case of unbounded convex domains, where, strange enough, many open questions in the previous directions seem to be still open. In particular an unbounded convex domain needs not to be hyperbolic, as the example of Ck shows. Some esti- mates on the Caratheodory and Bergman metrics in convex domains were obtained by Nikolov and Pflug in [12], [13]. The question is whether one can understand easily hyperbolicity of unbounded convex domains in terms of geometric or analytic properties. A result in this direc- tion was obtained by Barth [3], who proved the equivalence of properties (1), (2) and (6) in the theorem below. 2000 Mathematics Subject Classification. Primary 32Q45 Secondary 32A25; 52A20. Key words and phrases. Kobayashi hyperbolicity; convex domains; taut; peak functions. The second author was partially supported by PRIN project “Proprietà geometriche delle varietà reali e complesse”. http://arxiv.org/abs/0704.0751v3 2 F. BRACCI AND A. SARACCO The aim of the present paper is to show that actually for unbounded convex domains, hyper- bolicity can be characterized in many different ways and can be easily inferred just looking at a single boundary point. The dichotomy we discovered for unbounded convex domains is rather stringent: either the domain behaves like a bounded convex domain or it behaves like Ck. In particular, this pro- vides examples of unbounded domains which admit the Bergman metric and are complete with respect to it. The main result of these notes is the following (notations and terminology are standard and will be recalled in the next section): Theorem 1.1. Let D ⊂ CN be a (possibly unbounded) convex domain. The following are equivalent: (1) D is biholomorphic to a bounded domain; (2) D is (Kobayashi) hyperbolic; (3) D is taut; (4) D is complete (Kobayashi) hyperbolic; (5) D does not contain nonconstant entire curves; (6) D does not contain complex affine lines; (7) D has N linearly independent separating real hyperplanes; (8) D has peak and antipeak functions (in the sense of Gaussier) at infinity; (9) D admits the Bergman metric bD. (10) D is complete with respect to the Bergman metric bD. (11) for any f : D → D holomorphic such that the sequence of its iterates {f ◦k} is not compactly divergent there exists z0 ∈ D such that f(z0) = z0. The first implications of the theorem allow to obtain the following canonical complete hyper- bolic decomposition for unbounded convex domains, which is used in the final part of the proof of the theorem itself. Proposition 1.2. Let D ⊂ CN be a (possibly unbounded) convex domain. Then there exist a unique k (0 ≤ k ≤ N) and a unique complete hyperbolic convex domain D′ ⊂ Ck, such that, up to a linear change of coordinates, D = D′ × CN−k. By using such a canonical complete hyperbolic decomposition, one sees for instance that the “geometry at infinity” of an unbounded convex domain can be inferred from the geometry of any finite point of its boundary (see the last section for precise statements). For example, as an application of Corollary 4.3 and Theorem 1.1, existence of peak and anti-peak functions (in the sense of Gaussier) for an unbounded convex domain equals the absence of complex line in the “CR-part” of the boundary of the domain itself. This answers a question in Gaussier’s paper (see [5, pag. 115]) about geometric conditions for the existence in convex domains of peak and anti-peak plurisubharmonic functions at infinity. The authors want to sincerely thank prof. Nikolov for helpful conversations, and in particular for sharing his idea of constructing antipeak functions. HYPERBOLICITY IN UNBOUNDED CONVEX DOMAINS 3 2. PRELIMINARY A convex domain D ⊂ CN is a domain such that for any couple z0, z1 ∈ D the real segment joining z0 and z1 is contained in D. It is well known that for any point p ∈ ∂D there exists (at least) one real separating hyperplane Hp = {z ∈ C n : ReL(z) = a}, with L a complex linear functional and a ∈ R such that p ∈ Hp and D ∩ Hp = ∅. Such a hyperplane Hp is sometimes also called a tangent hyperplane to D at p. We say that k separating hyperplanes Hj = {ReLj(z) = aj}, j = 1, . . . , k, are linearly independent if L1, . . . , Lk are linearly independent linear functionals. Let D := {ζ ∈ C : |ζ | < 1} be the unit disc. Let D ⊂ CN be a domain. The Kobayashi pseudo-metric for the point z ∈ D and vector v ∈ CN is defined as κD(z; v) := inf{λ > 0|∃ϕ : D −→ D,ϕ(0) = z, ϕ′(0) = v/λ}. If κD(z; v) > 0 for all v 6= 0 then D is said to be (Kobayashi) hyperbolic. The pseudo- distance kD obtained by integrating κD is called the Kobayashi pseudodistance. The domain D is (Kobayashi) complete hyperbolic if kD is complete. The Carathéodory pseudo-distance cD is defined by cD(z, w) = sup{kD(f(z), f(w)) : f : D → D holomorphic}. In general, cD ≤ kD. We refer the reader to the book of Kobayashi [9] for properties of Kobayashi and Carathéodory metrics and distances. Another (pseudo)distance that can be introduced on the domain D is the Bergman (pseudo) distance (see, e.g., [9, Sect. 10, Ch. 4]). Let {ej} be a orthonormal complete basis of the space of square-integrable holomorphic functions on D. Then let lD(z, w) := ej(z)ej(w). If lD(z, z) > 0 one can define a symmetric form bD := 2 hjkdzj ⊗ dzk, with hjk = ∂2 log bD(z,z) ∂zj ,∂zk , which is a positive semi-definite Hermitian form, called the Bergman pseudo-metric of D. If bD is positive definite everywhere, one says that D admits the Bergman metric bD. For instance, Ck, k ≥ 1 does not support square-integrable holomorphic functions, therefore lCk ≡ 0 and C k does not admit the Bergman metric. For the next result, see [9, Corollaries 4.10.19, 4.10.20]: Proposition 2.1. Let D ⊂ CN be a domain. (1) Assume bD(z, z) > 0 for all z ∈ D. If cD is a distance, if it induces the topology of D and if the cD-balls are compact, thenD admits the Bergman metric bD and it is complete with respect to bD. (2) If D is a bounded convex domain then it (admits the Bergman metric and it) is complete with respect to the Bergman metric. 4 F. BRACCI AND A. SARACCO Let G be another domain. We recall that if {ϕk} is a sequence of holomorphic mappings from G to D, then the sequence is said to be compactly divergent if for any two compact sets K1 ⊂ G and K2 ⊂ D it follows that ♯{k ∈ N : ϕk(K1) ∩K2 6= ∅} < +∞. A family F of holomorphic mappings from G to D is said to be normal if each sequence of F admits a subsequence which is either compactly divergent or uniformly convergent on compacta. If the family of all holomorphic mappings from the unit disc D to D is normal, then D is said to be taut. It is known: Theorem 2.2. Let D ⊂ CN be a domain. (1) (Royden) D complete hyperbolic ⇒ D taut ⇒ D hyperbolic. (2) (Kiernan) If D is bounded then D is hyperbolic. (3) (Harris) If D is a bounded convex domain then D is complete hyperbolic. The notion of (complete) hyperbolicity is pretty much related to existence of peak functions at each boundary point. In case D is an unbounded domain, H. Gaussier [5] introduced the following concepts of “peak and antipeak functions” at infinity, which we use in the sequel: Definition 2.3. A function ϕ : D → R ∪ {−∞} is called a global peak plurisubharmonic function at infinity if it is plurisubharmonic on D, continuous up to D (closure in CN ) and ϕ(z) = 0, ϕ(z) < 0 ∀z ∈ D. A function ϕ : D → R∪{−∞} is called a global antipeak plurisubharmonic function at infinity if it is plurisubharmonic on D, continuous up to D and ϕ(z) = −∞, ϕ(z) > −∞ ∀z ∈ D. For short we will simply call them peak and antipeak functions (in the sense of Gaussier) at infinity. Gaussier proved the following result: Theorem 2.4 (Gaussier). Let D ⊂ CN be an unbounded domain. Assume that D is locally taut at each point of ∂D and there exist peak and antipeak functions (in the sense of Gaussier) at infinity. Then D is taut. Obviously a convex domain is locally taut at each boundary point, thus tautness follows from existence of peak and antipeak functions (in the sense of Gaussier) at infinity. Finally, if f : D → D is a holomorphic function, the sequence of its iterates {f ◦k} is defined by induction as f ◦k := f ◦(k−1) ◦ f . If f has a fixed point z0 ∈ D, then {f ◦k} is not compactly divergent. On the other hand, depending on the geometry of D, there exist examples HYPERBOLICITY IN UNBOUNDED CONVEX DOMAINS 5 of holomorphic maps f such that {f ◦k} is not compactly divergent but f has no fixed points in D. It is known (see [2]) that Theorem 2.5 (Abate). Let D ⊂ CN be a taut domain. Assume that Hj(D;Q) = 0 for all j > 0 and let f : X → X holomorphic. Then the sequence of iterates {f ◦k} is compactly divergent if and only if f has no periodic points in D. If D is a bounded convex domain then the sequence of iterates {f ◦k} is compactly divergent if and only if f has no fixed points in D. 3. THE PROOF OF THEOREM 1.1 The proof of Theorem 1.1 is obtained in several steps, which might be of some interest by their own. For a domain D ⊂ CN let us denote by δD the Lempert function given by δD(z, w) = inf{ω(0, t) : t ∈ (0, 1), ∃ϕ ∈ Hol(D, D) : ϕ(0) = z, ϕ(t) = w}. The Lempert function is not a pseudodistance in general because it does not enjoy the triangle inequality. The Kobayashi pseudodistance is the largest minorant of δD which satisfies the triangle inequality. The following lemma (known as Lempert’s theorem in case of bounded convex domains) is probably known, but we provide its simple proof due to the lack of reference. Lemma 3.1. Let D ⊂ CN be a (possibly unbounded) convex domain. Then kD = δD = cD. Proof. The result is due to Lempert [11] in case D is bounded. Assume D is unbounded. Let DR be the intersection of D with a ball of center the origin and radius R > 0. For R >> 1 the set DR is a nonempty convex bounded domain. Therefore kDR = δDR = cDR . Now {DR} is an increasing sequence of domains whose union is D. Hence, limR→∞ kDR = kD, limR→∞ cDR = cD and limR→∞ δDR = δD (see, e.g., [7, Prop. 2.5.1] and [7, Prop. 3.3.5]). Thus kD = δD = cD. � Proposition 3.2. Let D ⊂ CN be a (possibly unbounded) convex domain. Then the Kobayashi balls in D are convex. Proof. For the bounded case, see [1, Proposition 2.3.46]. For the unbounded case, let Bǫ be the Kobayashi ball of radius ǫ and center z0 ∈ D, let DR be the intersection of D with an Euclidean ball of center the origin and radius R > 0, and let BRǫ be the Kobayashi ball in DR of radius ǫ and center z0. Then the convex sets B ǫ ⊂ B ǫ ⊂ Bǫ for all R >> 1, δ > 0, and their convex increasing union ∪RB ǫ = Bǫ, since limR→∞ kDR = kD. � Lemma 3.3. Let D ⊂ CN be a (possibly unbounded) taut convex domain. Then for any couple z, w ∈ D there exists ϕ ∈ Hol(D, D) such that ϕ(0) = z, ϕ(t) = w t ∈ [0, 1) and kD(z, w) = ω(0, t). 6 F. BRACCI AND A. SARACCO Proof. By Lemma 3.1, kD = δD, so there exists a sequence {ϕk} of holomorphic discs and tk ∈ (0, 1) such that ϕk(0) = z and ϕk(tk) = w and kD(z, w) = lim ω(0, tk). Since D is taut and ϕk(0) = z for all k, we can assume that {ϕk} converges uniformly on compacta to a (holomorphic) map ϕ : D → D. Then ϕ(0) = z. Moreover, since kD(z, w) <∞, there exists t0 < 1 such that tk ≤ t0 for all k. We can assume (up to subsequences) that tk → t ≤ t0. Then kD(z, w) = lim ω(0, tk) = ω(0, t). Moreover, ϕ(t) = limk→∞ ϕk(tk) = w and we are done. � Proposition 3.4. Let D ⊂ CN be a (possibly unbounded) convex domain. Then D is taut if and only if it is complete hyperbolic. Proof. One direction is contained in Royden’s theorem. Conversely, assume that D is taut. We are going to prove that every closed Kobayashi balls is compact (which is equivalent to be complete hyperbolic, see [9] or [1, Proposition 2.3.17]). Let R > 0, z ∈ D and let B(z, R) = {w ∈ D : kD(z, w) ≤ R}. If B(z, R) is not compact then there exists a sequence {wk} such that wk → p ∈ ∂D ∪ {∞} and kD(z, wk) ≤ R. For any k, let ϕk ∈ Hol(D, D) be the extremal disc given by Lemma 3.3 such that ϕk(0) = z, ϕk(tk) = wk for some tk ∈ (0, 1) and kD(z, wk) = ω(0, tk). Notice that, since kD(z, wk) ≤ R, then there exists t0 < 1 such that tk ≤ t0 for all k. We can assume up to subsequences that tk → t with t < 1. Since D is taut and ϕk(0) = z, up to extracting subsequences, the sequence {ϕk} is converging uniformly on compacta to a holomorphic disc ϕ : D → D such that ϕ(0) = z. However, ϕ(t) = lim ϕk(tk) = lim wk = p, a contradiction. Therefore B(z, R) is compact and D is complete hyperbolic. � For the next proposition, cfr. [4, Lemma 3]. Proposition 3.5. Let D ⊂ CN be a convex domain, which does not contain complex affine lines. Then there exist {L1 = 0}, . . . , {LN = 0} linearly independent hyperplanes containing the origin and a1, . . . , aN ∈ R such that D ⊂ {Re L1 > a1, . . . ,Re LN > aN}. Proof. Without loss of generality we can assume thatO ∈ D. SinceD does not contain complex affine lines, ∂D is not empty. Take a point p1 ∈ ∂D and a tangent real hyperplane through p1, given by {Re L1 = a1} (if the boundary is smooth there is only one tangent hyperplane), where L1 is defined so that D ⊂ {Re L1 > a1}. Suppose that L1, . . . , Lk, k < N , are already defined, they are linearly independent and D ⊂ {Re L1 > a1, . . . ,Re Lk > ak}. HYPERBOLICITY IN UNBOUNDED CONVEX DOMAINS 7 The intersection hk = ∩ 1{Li = 0} is a complex (N − k)-dimensional plane through the origin O (which is also contained in D by hypothesis). Since D does not contain complex affine lines, ∂D ∩ hk is not empty. Take a point pk+1 ∈ ∂D ∩ hk and consider a tangent real hyperplane through pk+1, {Re Lk+1 = ak+1}, where Lk+1 is defined so that D ⊂ {Re Lk+1 > ak+1}. By construction Lk+1 is linearly independent from L1, . . . , Lk and D ⊂ {Re L1 > a1, . . . ,Re Lk+1 > ak+1}. Continuing this way, the proof is concluded. � Now we are in a good shape to prove part of Theorem 1.1: Proposition 3.6. Let D ⊂ CN be a convex domain. The following are equivalent: (1) D is biholomorphic to a bounded domain; (2) D is (Kobayashi) hyperbolic; (3) D is taut; (4) D is complete (Kobayashi) hyperbolic; (5) D does not contain nonconstant entire curves; (6) D does not contain complex affine lines; (7) D has N linearly independent separating real hyperplanes; (8) D has peak and antipeak functions (in the sense of Gaussier) at infinity; Proof. (1) ⇒ (2): every bounded domain in CN is hyperbolic by [8] (see, also, [1, Thm. 2.3.14]) (2) ⇒ (5) ⇒ (6): obvious. (6) ⇒ (7): it is Proposition 3.5. (7) ⇒ (1): letL1, . . . , LN be linearly independent complex linear functionals and let a1, . . . , aN ∈ R be such that {ReLj = aj} for j = 1, . . . , N are real separating hyperplanes for D. Up to sign changes, we can assume that D ⊂ {ReLj > aj}. Then the map F (z1, . . . , zN) := L1(z)− a1 + 1 , . . . , LN(z)− aN + 1 maps D biholomorphically on a bounded convex domain of CN . (6) ⇒ (8): let L1, . . . , LN be as in Proposition 3.5. Up to a linear change of coordinates, we can suppose that zj = Lj for all 1 ≤ j ≤ N . A peak function is given by zj − aj + 1 Let Dj := {ReLj > aj}. Then D ⊂ j=1Dj , and Dj is biholomorphic to D for each j. In particular C \ Dj is not a polar set. We may assume that 0 6∈ Dj. Let Gj be the image of Dj under the transformation z → 1/z. Since C \ Gj is not a polar set, there exists ε > 0 such that C \ Gεj is not polar, too, where G j = Gj ∪ εD. Denote by g j the Green function of Gεj . Then hj = g j (0; ·) is a negative harmonic function on Gj with limz→0 hj(z) = −∞ and 8 F. BRACCI AND A. SARACCO infGj\rD hj > −∞ for any r > 0. Then ψj(z) = hj(1/z) is an antipeak function of Dj at ∞ and hence ψ = j=1 ψj is an antipeak function for D at ∞. (8) ⇒ (3): it is Gaussier’s theorem [5, Prop. 2]. (3) ⇒ (4): it is Proposition 3.4. (4) ⇒ (3) ⇒ (2): it is Royden’s theorem [15, Prop. 5, pag. 135 and Corollary p.136]. � As a consequence we have Proposition 1.2, which gives a canonical complete hyperbolic decomposition of a convex domain as the product of a complete hyperbolic domain and a copy of Ck. Proof of Proposition 1.2. We prove the result by induction on N . If N = 1 then either D = C or D is biholomorphic to the disc and hence (complete) hyperbolic. Assume the result is true for N , we prove it holds for N + 1. Let D ⊂ CN+1 be a convex domain. Then, by Proposition 3.6, either D is complete hyperbolic or D contains an affine line, say, up to a linear change of coordinates lN+1 = {z1, . . . , zN = 0} ⊂ D. Clearly, there exists c ∈ C such that D ∩ {zN+1 = c} 6= ∅. Up to translation we can assume c = 0. Let us define DN = D ∩ {zN+1 = 0}. DN ⊂ C N is convex. We claim that D = DN × C. Induction will then conclude the proof. Let z0 ∈ DN . We want to show that (z0, ζ) ∈ D for all ζ ∈ C. Since lN+1 ⊂ D then (0, ζ) ∈ D for all ζ ∈ C. Assume z0 6= 0. Fix ζ ∈ C. Since DN is open, there exists ε0 > 0 such that z1 := (1 + ε0)z0 ∈ DN . Since D is convex, for any t ∈ [0, 1] it follows t(z1, 0)+ (1− t)(0, ξ) ∈ D for all ξ ∈ C. Setting ξ0 := ζ ∈ C and t0 = (1+ ε0) −1 ∈ (0, 1) we obtain (z0, ζ) = t0 (z1, 0) + (1− t0) (0, ξ0) ∈ D, completing the proof. � In order to finish the proof of Theorem 1.1 we need to show that the first eight conditions, which are all and the same thanks to Proposition 3.6, are equivalent to (9), (10), (11). Proof of Theorem 1.1. Conditions (1) to (8) are all equivalent by Proposition 3.6. (10)⇒(9): obvious. (4)⇒(10): By Lemma 3.1, the Caratheodory distance cD equals the Kobayashi distance kD, thus, since D is (Kobayashi) complete hyperbolic, cD is a distance which induces the topology on D and the cD-balls are compact. By Proposition 2.1 then D admits the Bergman metric and it is complete with respect to it. (9)⇒(4): Assume D is not complete hyperbolic. Then by Proposition 1.2, up to a linear change of coordinates, D = D′ × Ck for some complete hyperbolic domain D′ and k ≥ 1. By the product formula (see [9, Prop. 4.10.17]) lD = lD′ · lCk ≡ 0 and thus D does not admit the Bergman metric. HYPERBOLICITY IN UNBOUNDED CONVEX DOMAINS 9 (11) ⇒ (4): Assume D is not complete hyperbolic. We have to exhibit a holomorphic self- map f : D → D such that {f ◦k} is not compactly divergent but there exists no z0 ∈ D such that f(z0) = z0. By Proposition 1.2, up to a linear change of coordinates, D = D′ × Ck for some complete hyperbolic domain D′ and k ≥ 1. Let f : D′ × Ck−1 × C ∋ (z, w′, w) 7→ (z, w′, ew + w) ∈ D′ × Ck−1 × C. Then clearly f has no fixed points in D. However, if w0 = log(iπ), then f ◦2(z, w′, w0) = (z, w′, w0), and therefore the sequence {f ◦k} is not compactly divergent. (4) ⇒ (11): According to the theory developed so far, if D is complete hyperbolic, then it is taut and its Kobayashi balls are convex and compact. With these ingredients, the proof for bounded convex domains go through also in the unbounded case (see [1, Thm. 2.4.20]). � 4. APPLICATIONS Corollary 4.1. Let D ⊂ CN be a convex domain. If there exists a point p ∈ ∂D such that ∂D is strongly convex at p then D is complete hyperbolic. Proof. By Proposition 1.2, if D were not complete hyperbolic, up to linear changes of coor- dinates, D = D′ × Ck for some complete hyperbolic convex domain D′ and k ≥ 1. Then ∂D = ∂D′ × Ck could not be strongly convex anywhere. � Note that the converse to the previous corollary is false: the half-plane {ζ ∈ C : Re ζ > 0} is a complete hyperbolic convex domain in C with boundary which is nowhere strongly convex. Proposition 4.2. Let D ⊂ CN be an (unbounded) convex domain. Then D has canonical complete hyperbolic decomposition (up to a linear change of coordinates) D = D′ × Ck with D′ complete hyperbolic, if and only if for every p ∈ ∂D and every separating hyperplane Hp, (Hp ∩ ∂D) ∩ i(Hp ∩ ∂D) contains a copy of C k but contains no copies of Ck+1. Proof. (⇒) Since D = D′ × Ck, for every p ∈ ∂D and every separating hyperplane Hp, (Hp ∩ ∂D) ∩ i(Hp ∩ ∂D) = [(Hp ∩ ∂D ′) ∩ i(Hp ∩ ∂D ′)]× Ck. Since D′ is complete hyperbolic, its boundary does not contain complex lines. (⇐) Since D is convex, D = D′ × Ck , by Proposition 1.2. By the first part of the present proof, for every p ∈ ∂D and every separating hyperplane Hp, (Hp ∩ ∂D) ∩ i(Hp ∩ ∂D) = C Hence k′ = k. � Corollary 4.3. Let D ⊂ CN be an (unbounded) convex domain. If there exist p ∈ ∂D and a separating hyperplane Hp such that (Hp ∩ ∂D) ∩ i(Hp ∩ ∂D) does not contain any complex affine line then D is complete hyperbolic. Conversely, if D is complete hyperbolic, then for any point p ∈ ∂D and any separating hyperplane Hp, it follows that (Hp ∩ ∂D)∩ i(Hp ∩ ∂D) does not contain any complex affine line. 10 F. BRACCI AND A. SARACCO As a final remark, we notice that, as Abate’s theorem 2.5 is the cornerstone to the study of iteration theory in bounded convex domains, our Theorem 1.1 and Proposition 1.2 can be used effectively well to the same aim for unbounded convex domains. In fact, if D = D′ ×Ck is the canonical complete hyperbolic decomposition of D, then a holomorphic self map f : D → D can be written in the coordinates (z, w) ∈ D′ × Ck as f(z, w) = (ϕ(z, w), ψ(z, w)), where ϕ : D′ ×Ck → D′ and ψ : D′ ×Ck → Ck. In particular, since D′ is complete hyperbolic, then ϕ depends only on z, namely, f(z, w) = (ϕ(z), ψ(z, w)). The map ψ(z, w) can be as worse as entire functions in Ck are, but the map ϕ is a holomorphic self-map of a complete hyperbolic convex domains and its dynamics goes similarly to that of holomorphic self-maps of bounded convex domains. For instance, if the sequence {f ◦k} is non-compactly divergent, then f might have no fixed points, but the sequence {ϕ◦k} must have at least one. REFERENCES [1] M. Abate, Iteration Theory of Holomorphic Maps on Taut Manifolds, Mediterranean Press, Rende, Cosenza, 1989. [2] M. Abate, Iteration theory, compactly divergent sequences and commuting holomorphic maps. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1991), 2, 167–191. [3] T. J. Barth, Convex domains and Kobayashi hyperbolicity. Proc. Amer. Math. Soc. 79 (1980), no. 4, 556–558. [4] B. Drinovec Drnovsek, Proper holomorphic discs avoiding closed convex sets. Math. Z. 241 (2002), 3, 593– [5] H. Gaussier, Tautness and complete hyperbolicity of domains in Cn, Proc. Amer. Math. Soc. 127, 1 (1999), 105–116. [6] L. A. Harris, Schwarz-Pick systems of pseudometric for domains in normed linear spaces. In Advances in Holomorphy, Notase de Matematica 65, North-Holland, Amsterdam, 1979, 345–406. [7] M. Jarnicki, P. Pflug, Invariant distances and metrics in complex analysis W. de Gruyter, Berlin, New York 1993. [8] P. Kiernan, On the relations between tight, taut and hyperbolic manifolds. Bull. Amer. Math. Soc. 76 (1970), 49–51. [9] S. Kobayashi, Hyperbolic Complex Spaces, Springer-Verlag, Grundlehren der mathematischen Wis- senschaften 318, 1998. [10] L. Lempert, La métrique de Kobayashi et la representation des domaines sur la boule. Bull. Soc. Math. Fr. 109 (1981), 427–474. [11] L. Lempert, Holomorphic retracts and intrinsic metrics in convex domains. Analysis Math. 8 (1982), 257– [12] N. Nikolov, P. Pflug, Behavior of the Bergman kernel and metric near convex boundary points. Proc. Amer. Math. Soc. 131 (2003), no. 7, 2097–2102 (electronic). [13] N. Nikolov, P. Pflug, Estimates for the Bergman kernel and metric of convex domains in Cn. Ann. Polon. Math. 81 (2003), no. 1, 73–78. [14] N. Nikolov, P. Pflug, Local vs. global hyperconvexity, tautness or k-completeness for unbounded open sets in Cn. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), no. 4, 601–618. [15] H. L. Royden, Remarks on the Kobayashi metric, Lect. Notes in Math 185, Springer, Berlin (1971), 125–137. HYPERBOLICITY IN UNBOUNDED CONVEX DOMAINS 11 F. BRACCI: DIPARTIMENTO DI MATEMATICA, UNIVERSITÀ DI ROMA “TOR VERGATA”, VIA DELLA RICERCA SCIENTIFICA 1, 00133, ROMA, ITALY E-mail address: [email protected] A. SARACCO: SCUOLA NORMALE SUPERIORE, PIAZZA DEI CAVALIERI 7, 56126, PISA, ITALY E-mail address: [email protected] 1. Introduction 2. Preliminary 3. The proof of Theorem ?? 4. Applications References

0704.0752

Actions for the Bosonic String with the Curved Worldsheet

arXiv:0704.0752v3 [hep-th] 18 Apr 2008 Actions for the Bosonic String with the Curved Worldsheet Davoud Kamani Faculty of Physics, Amirkabir University of Technology (Tehran Polytechnic) P.O.Box: 15875-4413, Tehran, Iran e-mail: [email protected] Abstract At first we introduce an action for the string, which leads to a worldsheet that always is curved. For this action we study the Poincaré symmetry and the associated conserved currents. Then, a generalization of the above action, which contains an arbitrary function of the two-dimensional scalar curvature, will be introduced. An extra scalar field enables us to modify these actions to Weyl invariant models. PACS: 11.25.-w; 11.30.Cp Keywords: Curved worldsheet; 2d scalar curvature; Poincaré symmetry. http://arxiv.org/abs/0704.0752v3 1 Introduction The two-dimensional models have widely been used in the context of the two-dimensional gravity (e.g. see [1, 2, 3, 4] and references therein) and string theory. From the 2d-gravity point of view, higher-dimensional gravity models, by dimensional reduction reduce to the 2d-gravity [1, 2, 3]. From the string theory point of view, the (1+1)-dimensional actions are fundamental tools of the theory. However, 2d-gravity and 2d- string theory are closely related to each other. The known sigma models for string, in the presence of the dilaton field Φ(X), contain the two-dimensional scalar curvature R(hab), hRΦ(X). (1) In two dimensions the combination hR is total derivative. Thus, in the absence of the dilaton field, this action is a topological invariant that gives no dynamics to the worldsheet metric hab. In fact, in the action (1), the dilaton is not the only choice. For example, replacing the dilaton field with the scalar curvature R, leads to the R2-gravity [1, 4, 5]. In particular the Polyakov action is replaced by a special combination of the worldsheet fields, which include an overall factor R−1. Removing the dilaton and replacing it with another quantities motivated us to study a class of two-dimensional actions. They are useful in the context of the non-critical strings with curved worldsheet, and the 2-dimensional gravity. Instead of the dilaton field, we introduce some combinations of hab, R and the induced metric on the worldsheet, i.e. γab, which give dynamics to hab. These non-linear combinations can contain an arbitrary function f(R) of the scalar curvature R. We observe that these dynamics lead to the constraint equation for hab, extracted from the Polyakov action. For the flat spacetime, these models have the Poincaré symmetry. In addition, they are reparametrization invariant. However, for any function f(R), they do not have the Weyl symmetry. Therefore, the string worldsheet at most is conformally flat. By introducing an extra scalar field in these actions, they also find the Weyl symmetry. Note that a Weyl non-invariant string theory has noncritical dimension, e.g. see [6]. This paper is organized as follows. In section 2, we introduce a new action for the string in which the corresponding worldsheet always is curved. In section 3, the Poincaré symmetry of this string model will be studied. In section 4, the generalized form of the above action will be introduced and it will be analyzed. 2 Curved worldsheet in the curved spacetime We consider the following action for the string, which propagates in the curved spacetime S = −T habγab , (2) where h = − det hab, and T is a dimensionless constant. In addition, R denotes the two- dimensional scalar curvature which is made from hab. The string coordinates are {Xµ(σ, τ)}. The induced metric on the worldsheet, i.e. γab, is also given by γab = gµν(X)∂aX µ(σ, τ)∂bX ν(σ, τ), (3) where gµν(X) is the spacetime metric. In two dimensions, the symmetries of the curvature tensor imply the identity Rab − habR = 0. (4) Therefore, the variation of the action (2) leads to the following equation of motion for hab, Rab − γab = 0. (5) This implies that the energy-momentum tensor, extracted from the action (2), vanishes. Contraction of this equation by hab gives R = 1 habγab. Introducing this equation and the equation (5) into (4) leads to (Polyakov) ab ≡ γab − hab(h a′b′γa′b′) = 0. (6) This is the constraint equation, extracted from the Polyakov action. Note that the energy- momentum tensor, due to the action (2), is proportional to the left-hand-side of the equation (5). Thus, it is different from (6). The equation of motion of the string coordinate Xµ(σ, τ) also is hRhab∂bX hRhabΓ νλ∂aX λ = 0. (7) Presence of the scalar curvature R distinguishes this equation from its analog, extracted from the Polyakov action. Now consider those solutions of the equations of motion (5) and (7), which admit constant scalar curvature R. For these solutions, the equation (7) reduces to the equation of motion of the string coordinates, extracted from the Polyakov action with the curved background. However, for general solutions the scalar curvature R depends on the worldsheet coordinates σ and τ , and hence this coincidence does not occur. 2.1 The model in the conformal gauge Under reparametrization of σ and τ , the action (2) is invariant. That is, in two dimensions the general coordinate transformations σ → σ′(σ, τ) and τ → τ ′(σ, τ), depend on two free functions, namely the new coordinates σ′ and τ ′. By means of such transformations any two of the three independent components of hab can be eliminated. A standard choice is a parametrization of the worldsheet such that hab = e φ(σ,τ)ηab, (8) where ηab = diag(−1, 1), and eφ(σ,τ) is an unknown conformal factor. The choice (8) is called the conformal gauge. Since the action (2) does not have the Weyl symmetry (a local rescaling of the worldsheet metric hab) we cannot choose the gauge hab = ηab. The scalar curvature corresponding to the metric (8) is R = −e−φ∂2φ, (9) where ∂2 = ηab∂a∂b. Thus, the action (2) reduces to S ′ = −T d2σe−φ∂2φ ηabγab . (10) According to the gauge (8), this action describes a conformally flat worldsheet. 3 Poincaré symmetry of the model In this section we consider flat Minkowski space, i.e. gµν(X) = ηµν . Therefore, the equations of motion are simplified to Rab − ηµν∂aX ν = 0, (11) hRhab∂bX µ) = 0. (12) The Poincaré symmetry reflects the symmetry of the background in which the string is propagating. It is described by the transformations δXµ = aµνX ν + bµ, δhab = 0, (13) where aµν and b µ are independent of the worldsheet coordinates σ and τ , and aµν = ηµλa is antisymmetric. Thus, from the worldsheet point of view, these transformations are global symmetries. Under these transformations the action (2) is invariant. 3.1 The conserved currents The Poincaré invariance of the action (2) is associated to the following Noether currents J µνa = T hRhab(Xµ∂bX ν −Xν∂bXµ), Pµa = T hRhab∂bX µ, (14) where the current Pµa is corresponding to the translation invariance and J µνa is the current associated to the Lorentz symmetry. According to the equation of motion (12) these are conserved currents ∂aJ µνa = 0, ∂aPµa = 0. (15) 3.2 The covariantly conserved currents It is possible to construct two other currents from (14), in which they be covariantly con- served. For this, there is the useful formula ∇aKa = a), (16) where Ka is a worldsheet vector. Therefore, we define the currents Jµνa and P µa as in the following Jµνa = J µνa, P µa = Pµa. (17) According to the equations (15) and (16), these are covariantly conserved currents, i.e., ∇aJµνa = ∇aP µa = 0. (18) The currents (17) can also be written as Jµνa = R(Xµ∂aX ν −Xν∂aXµ), P µa = µ. (19) Since there is ∇ahbc = 0, the conservation laws (18) also imply the covariantly conservation of the currents (19). 4 Generalization of the model The generalized form of the action (2) is I = −T f(R)− , (20) where f(R) is an arbitrary differentiable function of the scalar curvature R. The set {Xµ(σ, τ)} describes a string worldsheet in the spacetime. These string coordinates ap- peared in the induced metric γab through the equation (3). Thus, (20) is a model for the string action. The equation of motion of Xµ is as previous, i.e. (7). Vanishing the variation of this action with respect to the worldsheet metric hab, gives the equation of motion of hab, df(R) γab = 0. (21) The trace of this equation is df(R) habγab = 0. (22) Combining the equations (4), (21) and (22) again leads to the equation (6). As an example, consider the function f(R) = α lnR + β. Thus, the field equation (21) implies that the intrinsic metric hab becomes proportional to the induced metric γab, that is hab = Since the Poincaré transformations contain δhab = 0, the generalized action (20) for the flat background metric gµν = ηµν , also has the Poincaré invariance. This leads to the previous conserved currents, i.e. (14) and (19). 4.1 Weyl invariance in the presence of a new scalar field The action (20) under the reparametrization transformations is symmetric. The Weyl trans- formation is also defined by hab −→ h′ab = eρ(σ,τ)hab. (23) Thus, the scalar curvature transforms as R −→ R′ = e−ρ(R−∇2ρ), (24) where ∇2ρ = 1√ hhab∂bρ). The equations (23) and (24) imply that the action (20), for any function f(R), is Weyl non-invariant. Introducing (23) and (24) into the action (20) gives a new action which contains the field ρ(σ, τ), I ′ = −T h(R−∇2ρ) f [e−ρ(R−∇2ρ)]− 1 e−ρhabγab . (25) We can ignore the origin of this action. In other words, it is another model for string. However, under the Weyl transformations hab −→ eu(σ,τ)hab, ρ −→ ρ− u, (26) the action I ′, for any function f , is symmetric. Note that according to the definition of ∇2 there is the transformation ∇2 → e−u∇2. 5 Conclusions We considered some string actions which give dynamics to the worldsheet metric hab. Due to the absence of the Weyl invariance, these models admit at most conformally flat (but not flat) worldsheet. We observed that the constraint equation on the metric, extracted from the Polyakov action, is a special result of the field equations of our string models. Obtaining this constraint equation admits us to introduce an arbitrary function of the scalar curvature to the action. For the case f(R) = α lnR + β, the metric hab becomes proportional to the induced metric of the worldsheet. By introducing a new degree of freedom we obtained a string action, in which for any function f is Weyl invariant. Our string models with arbitrary f(R), in the flat background have the Poincaré sym- metry. The associated conserved currents are proportional to the scalar curvature R. We also constructed the covariantly conserved currents from the Poincaré currents. References [1] H.J. Schmidt, Int. J. Mod. Phys. D7 (1998) 215, gr-qc/9712034. [2] D. Park and Y. Kiem, Phys. Rev. D53 (1996) 5513; Phys. Rev. D53 (1996) 747. [3] A. Achucarro and M. Ortiz, Phys. Rev. D48 (1993) 3600. [4] D. Grumiller, W. Kummer and D.V. Vassilevich, Phys. Rept. 369 (2002) 327-429, hep- th/0204253. [5] M.O. Katanaev and I.V. Volovich, Phys. Lett. B175 (1986) 413, hep-th/0209014. [6] F. David, Mod. Phys. Lett. A3 (1988) 1651; J. Distler, H. Kawai, Nucl. Phys. B321 (1989) 509; A. A. Tseytlin, Int. Jour. Mod. Phys. A4 (1989) 1257; J. Polchinski, Nucl. Phys. B324 (1989) 123.

0704.0753

Bianchi Type I Massive String Magnetized Barotropic Perfect Fluid Cosmological Model in General Relativity

Bianchi Type I Massive String Magnetized Barotropic Perfect Fluid Cosmological Model in General Relativity Raj Bali 1, Umesh Kumar Pareek2 and Anirudh Pradhan3 1Department of Mathematics, University of Rajasthan, Jaipur-302 004, India E-mail : [email protected] 2Department of Mathematics, Jaipur Engineering College and Research Centre, Jaipur-303 905, India E-mail : [email protected] 3Department of Mathematics, Hindu Post-graduate College, Zamania-232 331, Ghazipur, India E-mail : [email protected], [email protected] Abstract Bianchi type I massive string cosmological model with magnetic field of barotropic perfect fluid distribution through the techniques used by Latelier and Stachel, is investigated. To get the deterministic model of the universe, it is assumed that the universe is filled with barotropic per- fect fluid distribution. The magnetic field is due to electric current pro- duced along x-axis with infinite electrical conductivity. The behaviour of the model in presence and absence of magnetic field together with other physical aspects is further discussed. Keywords: Massive string, magnetic field, Bianchi type I model, perfect fluid PACS: 98.80.Cq, 04.20.-q 1 Introduction The cosmic strings play an important role in the study of the early universe. These strings arise during the phase transition after the big bang explosion as the temperature drops down below some critical temperature as predicted by grand unified theories [1-5]. It is thought that cosmic strings cause density perturbations leading to the formation of galaxies [6]. These cosmic strings have stress-energy and couple with the gravitational field. Therefore, it is interesting to study the gravitational effects that arise from strings. The general relativistic treatment of strings was started by Letelier [7, 8] and Stachel [9]. Exact solutions of string cosmology in various space-times have been studied by several authors [10-23]. http://arxiv.org/abs/0704.0753v2 On the other hand, the magnetic field has an important role at the cosmo- logical scale and is present in galactic and intergalactic spaces. The importance of the magnetic field for various astrophysical phenomena has been studied in many papers. Melvin [24] has pointed out that during the evolution of the uni- verse, the matter was in a highly ionized state and is smoothly coupled with the field and forms a neutral matter as a result of universe expansion. FRW models are approximately valid as present day magnetic field strength is very small. In the early universe, the strength might have been appreciable. The break-down of isotropy is due to the magnetic field. Therefore the possibility of the presence of magnetic field in the cloud string universe is not unrealistic and has been investigated by many authors [25-28]. In this paper, we have investigated Bianchi type I massive string magnetized barotropic perfect fluid cosmological model in General Relativity. The magnetic field is due to an electric current produced along x-axis with infinite electrical conductivity. Also the behaviour of the model in the presence and absence of magnetic field together with other physical aspects is discussed. 2 The Metric and Field Equations We consider the space-time of Bianchi type-I in the form ds2 = −dt2 +A2(t)dx2 +B2(t)dy2 + C2(t)dz2. (1) The energy momentum tensor for a cloud of massive string and perfect fluid distribution with electromagnetic field is taken as i = (ρ+ p)viv j + pg i − λxix j + E i , (2) where vi and xi satisfy condition vivi = −xixi = −1, vixi = 0, (3) p is the isotropic pressure, ρ is the proper energy density for a cloud string with particles attached to them, λ is the string tension density, vi the four-velocity of the particles, and xi is a unit space-like vector representing the direction of string. In a co-moving co-ordinate system, we have vi = (0, 0, 0, 1), xi = , 0, 0, 0 . (4) The electromagnetic field E i given by Lichnerowicz [29] as i = µ̄ | h |2 − hihj . (5) Here the flow-vector vi satisfies ivj = −1, (6) and µ̄ is the magnetic permeability, hi the magnetic flux vector defined by ǫijklF klvj , (7) where Fkl is the electromagnetic field tensor and ǫijkl is the Levi Civita tensor density. The incidental magnetic field is taken along x-axis, so that h1 6= 0, h2 = h3 = h4 = 0. We assume that F23 is the only non-vanishing component of Fij . The Maxwell’s equations Fij;k + Fjk;i + Fki;j = 0, ;j = 0, (8) are satisfied by F23 = constant = H(say). Here F14 = 0 = F24 = F34, due to the assumption of infinite electrical conduc- tivity [30]. Hence . (9) Since | h |2= hlhl = h1h1 = g11(h1)2, therefore | h |2= µ̄2B2C2 . (10) Using Eqs. (9) and (10) in (5), we have E11 = − 2µ̄B2C2 = −E22 = −E33 = E44 . (11) If the particle density of the configuration is denoted by ρp, then we have ρ = ρp + λ. (12) The Einstein’s field equations (in gravitational units c = 1, G = 1) read as i = −T i , (13) where R i is the Ricci tensor; R = g ijRij is the Ricci scalar. The field equations (13) with (2) subsequently lead to the following system of equations: = −p+ λ+ H 2µ̄B2C2 , (14) 2µ̄B2C2 , (15) 2µ̄B2C2 , (16) 2µ̄B2C2 , (17) where the suffix 4 at the symbols A, B and C denotes ordinary differentiation with respect to t. 3 Solution of Field Equations The field Eqs. (14)-(17) are a system of four equations with six unknown param- eters A, B, C, p, λ and ρ. Two additional constraints relating these parameters are required to obtain explicit solutions of the system. From Eq. (16), we have p = −A44 − B44 − A4B4 , (18) where K = H . Now from Eq. (17), we have . (19) To get deterministic solution, we first assume that the universe is filled with barotropic perfect fluid which leads to p = γρ, (20) where γ(0 ≤ γ ≤ 1) is a constant. Putting the values of p and ρ from Eqs. (18) and (19) in (20), we obtain + (1 + γ) + (1 − γ) K = 0. (21) Equations (15) and (16) lead to (CB4 −BC4)4 (CB4 −BC4) = −A4 , (22) which again leads to , (23) where L is an integrating constant and BC = µ, = ν. (24) Thus from Eqs. (23) and (24), we have . (25) For deterministic solution, we secondly assume A = constant = α(say). (26) Thus Eq. (25) leads to . (27) From Eqs. (21) and (26), we have (1− γ)K = 0. (28) Using (24) in Eq. (28), we obtain + (γ − 1) µ + (1− γ) L 4α2µ2 + (1− γ)K = 0, (29) which again leads to 2µ44 + (γ − 1) , (30) where a = (γ − 1)L + 4(γ − 1)K. (31) Let us assume that µ4 = f(µ). Thus µ44 = ff ′, where f ′ = df . Accordingly Eq. (30) leads to (f2) + (γ − 1) 1 , (32) which again reduces to γ − 1 + bµ1−γ . (33) Now from Eq. (27), we have dµ (34) Using Eq. (33) in Eq. (34), we have Lµ̄γdµ bµ1−γ ℓ2 + µ1−γ , (35) where ℓ2 = a (γ−1)b . Eq. (35), after integration, leads to ν = S ℓ2 + µ1−γ − ℓ ℓ2 + µ1−γ + ℓ α(1−γ)ℓ , (36) where S is the constant of integration. Thus the metric (1) reduces to the form ds2 = − dµ2 + α2dx2 + µ ℓ2 + µ1−γ − ℓ ℓ2 + µ1−γ + ℓ αℓ(1−γ) ℓ2 + µ1−γ − ℓ ℓ2 + µ1−γ + ℓ αℓ(1−γ) , (37) which after suitable transformation of coordinates, leads to ds2 = − dT b(ℓ2 + T 1−γ) + dX2 + T ℓ2 + µ1−γ − ℓ ℓ2 + µ1−γ + ℓ αℓ(1−γ) ℓ2 + µ1−γ − ℓ ℓ2 + µ1−γ + ℓ αℓ(1−γ) , (38) where αx = X, Sy = Y, 1√ z = z, µ = T . In the absence of the magnetic field, i.e. when K → 0, then the metric (37) reduces to 2 = − + T 1−γ ) + dX2 + T + T 1−γ − L + T 1−γ + L + T 1−γ − L + T 1−γ + L . (39) 4 The Geometric and Physical Significance of Model The energy density (ρ), the string tension density (λ), the particle density (ρp), the isotropic pressure (p), the scalar of expansion (θ), and shear tensor (σ) for the model (38) are given by b(ℓ2 + T 1−γ)− L , (40) b(1− γ)T 1−γ , (41) b(ℓ2 + γT 1−γ)− L , (42) b(ℓ2 + T 1−γ)− L . (43) ℓ2 + T 1−γ , (44) 6b(ℓ2 + T 1−γ) + . (45) ρ+ p = b{2ℓ2 − (1− γ)T 1−γ} − 2L , (46) ρ+ 3p = b{4ℓ2 + (1 + 3γ)T 1−γ − 4L − 16K . (47) The reality conditions given by Ellis [31] as (i)ρ+ p > 0, (ii)ρ+ 3p > 0, are satisfied when T 1−γ < ℓ2 − L The energy conditions ρ ≥ 0 and ρp ≥ 0 are satisfied in the presence of magnetic field for the model (38). The condition ρ ≥ 0 leads to b(ℓ2 + T 1−γ) ≥ + 4K. The condition ρp ≥ 0 leads to b(ℓ2 + T 1−γ) ≥ L − 4K. From Eq. (42), we observe that the string tension density λ ≥ 0 provided b(1− γ)T 1−γ ≥ 8K. The model (38) starts with a big bang at T = 0 and the expansion in the model decreases as time increases. When T → 0 then ρ → ∞, λ → ∞. When T → ∞ then ρ → 0, λ → 0. Also p → ∞ when T → 0 and p → 0 when T → ∞. Since limT→∞ 6= 0, hence the model does not isotropize in general. However, if L = 0 then the model (38) isotropizes for large values of T . There is a point type singularity [32] in the model (38) at T = 0. The ratio of magnetic energy to material energy is given by b(ℓ2 + T (1−γ))− L2 , (48) where 0 ≤ γ ≤ 1. The ratio E is non-zero finite quantity initially and tends to zero as T → ∞. The scale factor (R) is given by R3 = ABC = αµ = αT. (49) Thus R increases as T increases. The deceleration parameter (q) in presence of magnetic field is given by q = −RR44 a(3a− 2) + 1 b(1 + 3γ)T (1−γ) γ−1 + bT (1−γ) ) . (50) The deceleration parameter (q) approaches the value (−1) as in the case of de-Sitter universe if T (1−γ) = 2a(3a− 2)(γ − 1)− a b(1− γ)(1− 3γ) In the absence of magnetic field, i.e. K → 0, the above mentioned quantities are given by 4T 1+γ , (51) b(1− γ) 4T 1+γ , (52) 4T 1+γ , (53) 4T 1+γ , (54) In the absence of magnetic field when γ = 1, then ρ = b and also the string tension density becomes zero. The energy conditions ρ ≥ 0 and ρp ≥ 0 are satisfied for the model (38) when b ≥ 0. The reality conditions given by Ellis [31] as (i) ρ+ p > 0, (ii) ρ+ 3p > 0, are satisfied when b > 0. + bT 1−γ , (55) + 6bT 1−γ . (56) In the absence of magnetic field, the model (39) starts with a big bang at T = 0 and the expansion in the model decreases as time increases. When T → 0 then ρ → ∞, λ → ∞ and p → ∞. When T → ∞ then ρ → 0, λ → 0 and p → 0. In the absence of magnetic field, the particle density (ρp) and the isotropic pressure (p) are equal. Since limT→∞ 6= 0, therefore the model does not isotropize in general. However, if L = 0 then the model (39) isotropizes for large values of T . There is a point type singularity [32] in the model (39) at T = 0. In absence of magnetic field, the scale factor (R) is given by R3 = αT. (57) The R increases as T increases in this case also. The deceleration parameter (q) is given by q = − b(1 + 3γ)T (1−γ) − L {3L2(1− γ) + α2} + bT (1−γ) We observe that q < 0 if T (1−γ) > 2{3L2(1−γ)α2} b(1+3γ)α4 . The deceleration parameter (q) approaches the value (−1) as in the case of de-Sitter universe if T (1−γ) = L2{3L2(1− γ) + α2} 3bγα4 Acknowledgments Authors would like to thank the Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, India for providing facility and support where this work was carried out. Authors also thank to the referee for their fruitful comments. References [1] Kibble T W B 1976 J. Phys. A: Math. Gen. 9 1387 [2] Zel’dovich Ya B, Kobzarev, I Yu, and Okun, L B 1975 Zh. Eksp. Teor. Fiz. 67 3 Zel’dovich Ya B, Kobzarev, I Yu, and Okun, L B 1975 Sov. Phys.-JETP [3] Kibble T W B 1980 Phys. Rep. 67 183 [4] Everett A E 1981 Phys. Rev. 24 858 [5] Vilenkin A 1981 Phys. Rev. D 24 2082 [6] Zel’dovich Ya B 1980 Mon. Not. R. Astron. Soc. 192 663 [7] Letelier P S 1979 Phys. Rev. D 20 1249 [8] Letelier P S 1983 Phys. Rev. D 28 2414 [9] Stachel J 1980 Phys. Rev. D 21 2171 [10] Banerjee A, Sanyal A K and Chakraborty S 1990 Pramana-J. Phys. 34 1 [11] Chakraborty S 1991 Ind. J. Pure Appl. Phys. 29 31 [12] Tikekar R and Patel L K 1992 Gen. Rel. Grav. 24 397 [13] Tikekar R and Patel L K 1994 Pramana-J. Phys. 42 483 [14] Patel, L K and Maharaj S D 1996 Pramana-J. Phys. 47 33 [15] Ram, S and Singh, T K 1995 Gen. Rel. Grav. 27 1207 [16] Carminati J and McIntosh C B G 1980 J. Phys. A: Math. Gen. 13 953 [17] Krori K D, Chaudhury T, Mahanta C R and Mazumdar A 1990 Gen. Rel. Grav. 22 123 [18] Wang X X 2003 Chin. Phys. Lett. 20 615 [19] Singh G P and Singh T 1999 Gen. Relativ. Gravit. 31 371 [20] Bali R and Upadhaya R D 2003 Astrophys. Space Sci. 283 97 [21] Bali R and Pradhan A 2007 Chin. Phys. Lett. 24 585 [22] Bali R and Anjali 2006 Astrophys. Space Sci. 302 201 [23] Yadav M K, Rai A and Pradhan A 2007 Int. J. Theor. Phys. to appear (gr-qc/0611032). [24] Melvin M A 1975 Ann. New York Acad. Sci. 262 253 [25] Wang X X 2006 Chin. Phys. Lett. 23 1702 [26] Wang X X 2004 Astrophys. Space Sci. 293 933 [27] Chakraborty N C and Chakraborty 2001 Int. J. Mod. Phys. D 10 723 [28] Singh G P and Singh T 1999 Gen. Rel. Gravit. 31 371 [29] Lichnerowicz A 1967 Relativistic Hydrodynamics and Magnetohydrody- namics Benjamin New York p. 13 [30] Roy Maartens 2000 Pramana-J. Phys. 55 576 [31] Ellis G F R 1971 General Relativity and Cosmology ed. Sachs R K Claren- don Press p. 117 [32] MacCallum M A H 1971 Comm. Math. Phys. 20 57 http://arxiv.org/abs/gr-qc/0611032 Introduction The Metric and Field Equations Solution of Field Equations The Geometric and Physical Significance of Model

0704.0754

General Relativity Today

General Relativity Today∗† Thibault Damour Institut des Hautes Etudes Scientifiques 35 route de Chartres, 91440 Bures-sur-Yvette, France Abstract: After recalling the conceptual foundations and the basic struc- ture of general relativity, we review some of its main modern developments (apart from cosmology) : (i) the post-Newtonian limit and weak-field tests in the solar system, (ii) strong gravitational fields and black holes, (iii) strong-field and radiative tests in binary pulsar observations, (iv) gravitational waves, (v) general relativity and quantum theory. 1 Introduction The theory of general relativity was developed by Einstein in work that extended from 1907 to 1915. The starting point for Einstein’s thinking was the compo- sition of a review article in 1907 on what we today call the theory of special relativity. Recall that the latter theory sprang from a new kinematics governing length and time measurements that was proposed by Einstein in June of 1905 [1], [2], following important pioneering work by Lorentz and Poincaré. The theory of special relativity essentially poses a new fundamental framework (in place of the one posed by Galileo, Descartes, and Newton) for the formulation of physical laws: this framework being the chrono-geometric space-time structure of Poincaré and Minkowski. After 1905, it therefore seemed a natural task to formulate, reformulate, or modify the then known physical laws so that they fit within the framework of special relativity. For Newton’s law of gravitation, this task was begun (before Einstein had even supplied his conceptual crystallization in 1905) by Lorentz (1900) and Poincaré (1905), and was pursued in the period from 1910 to 1915 by Max Abraham, Gunnar Nordström and Gustav Mie (with these latter researchers developing scalar relativistic theories of gravitation). Meanwhile, in 1907, Einstein became aware that gravitational interactions possessed particular characteristics that suggested the necessity of generalizing the framework and structure of the 1905 theory of relativity. After many years of intense intellectual effort, Einstein succeeded in constructing a generalized ∗Talk given at the Poincaré Seminar “Gravitation et Expérience” (28 October 2006, Paris); to appear in the proceedings to be published by Birkhäuser. †Translated from the French by Eric Novak. http://arxiv.org/abs/0704.0754v1 theory of relativity (or general relativity) that proposed a profound modification of the chrono-geometric structure of the space-time of special relativity. In 1915, in place of a simple, neutral arena, given a priori, independently of all material content, space-time became a physical “field” (identified with the gravitational field). In other words, it was now a dynamical entity, both influencing and influenced by the distribution of mass-energy that it contains. This radically new conception of the structure of space-time remained for a long while on the margins of the development of physics. Twentieth century physics discovered a great number of new physical laws and phenomena while working with the space-time of special relativity as its fundamental framework, as well as imposing the respect of its symmetries (namely the Lorentz-Poincaré group). On the other hand, the theory of general relativity seemed for a long time to be a theory that was both poorly confirmed by experiment and without connection to the extraordinary progress springing from application of quantum theory (along with special relativity) to high-energy physics. This marginaliza- tion of general relativity no longer obtains. Today, general relativity has become one of the essential players in cutting-edge science. Numerous high-precision ex- perimental tests have confirmed, in detail, the pertinence of this theory. General relativity has become the favored tool for the description of the macroscopic uni- verse, covering everything from the big bang to black holes, including the solar system, neutron stars, pulsars, and gravitational waves. Moreover, the search for a consistent description of fundamental physics in its entirety has led to the exploration of theories that unify, within a general quantum framework, the description of matter and all its interactions (including gravity). These theo- ries, which are still under construction and are provisionally known as string theories, contain general relativity in a central way but suggest that the funda- mental structure of space-time-matter is even richer than is suggested separately by quantum theory and general relativity. 2 Special Relativity We begin our exposition of the theory of general relativity by recalling the chrono-geometric structure of space-time in the theory of special relativity. The structure of Poincaré-Minkowski space-time is given by a generalization of the Euclidean geometric structure of ordinary space. The latter structure is sum- marized by the formula L2 = (∆x)2 + (∆y)2 + (∆z)2 (a consequence of the Pythagorean theorem), expressing the square of the distance L between two points in space as a sum of the squares of the differences of the (orthonormal) coordinates x, y, z that label the points. The symmetry group of Euclidean ge- ometry is the group of coordinate transformations (x, y, z) → (x′, y′, z′) that leave the quadratic form L2 = (∆x)2 + (∆y)2 +(∆z)2 invariant. (This group is generated by translations, rotations, and “reversals” such as the transformation given by reflection in a mirror, for example: x′ = −x, y′ = y, z′ = z.) The Poincaré-Minkowski space-time is defined as the ensemble of events (ide- alizations of what happens at a particular point in space, at a particular moment in time), together with the notion of a (squared) interval S2 defined between any two events. An event is fixed by four coordinates, x, y, z, and t, where (x, y, z) are the spatial coordinates of the point in space where the event in question “occurs,” and where t fixes the instant when this event “occurs.” An- other event will be described (within the same reference frame) by four different coordinates, let us say x+∆x, y+∆y, z+∆z, and t+∆t. The points in space where these two events occur are separated by a distance L given by the for- mula above, L2 = (∆x)2+(∆y)2+(∆z)2. The moments in time when these two events occur are separated by a time interval T given by T = ∆t. The squared interval S2 between these two events is given as a function of these quantities, by definition, through the following generalization of the Pythagorean theorem: S2 = L2 − c2 T 2 = (∆x)2 + (∆y)2 + (∆z)2 − c2(∆t)2 , (1) where c denotes the speed of light (or, more precisely, the maximum speed of signal propagation). Equation (1) defines the chrono-geometry of Poincaré-Minkowski space-time. The symmetry group of this chrono-geometry is the group of coordinate trans- formations (x, y, z, t) → (x′, y′, z′, t′) that leave the quadratic form (1) of the interval S invariant. We will show that this group is made up of linear trans- formations and that it is generated by translations in space and time, spatial rotations, “boosts” (meaning special Lorentz transformations), and reversals of space and time. It is useful to replace the time coordinate t by the “light-time” x0 ≡ ct, and to collectively denote the coordinates as xµ ≡ (x0, xi) where the Greek indices µ, ν, . . . = 0, 1, 2, 3, and the Roman indices i, j, . . . = 1, 2, 3 (with x1 = x, x2 = y, and x3 = z). Equation (1) is then written S2 = ηµν ∆x µ ∆xν , (2) where we have used the Einstein summation convention1 and where ηµν is a diagonal matrix whose only non-zero elements are η00 = −1 and η11 = η22 = η33 = +1. The symmetry group of Poincaré-Minkowski space-time is therefore the ensemble of Lorentz-Poincaré transformations, x′µ = Λµν x ν + aµ , (3) where ηαβ Λ ν = ηµν . The chrono-geometry of Poincaré-Minkowski space-time can be visualized by representing, around each point x in space-time, the locus of points that are separated from the point x by a unit (squared) interval, in other words the ensemble of points x′ such that S2xx′ = ηµν(x ′µ−xµ)(x′ν −xν) = +1. This locus is a one-sheeted (unit) hyperboloid. If we were within an ordinary Euclidean space, the ensemble of points x′ would trace out a (unit) sphere centered on x, and the “field” of these spheres 1Every repeated index is supposed to be summed over all of its possible values. centered on each point x would allow one to completely characterize the Eu- clidean geometry of the space. Similarly, in the case of Poincaré-Minkowski space-time, the “field” of unit hyperboloids centered on each point x is a visual characterization of the geometry of this space-time. See Figure 1. This figure gives an idea of the symmetry group of Poincaré-Minkowski space-time, and renders the rigid and homogeneous nature of its geometry particularly clear. Figure 1: Geometry of the “rigid” space-time of the theory of special relativity. This geometry is visualized by representing, around each point x in space-time, the locus of points separated from the point x by a unit (squared) interval. The space-time shown here has only three dimensions: one time dimension (repre- sented vertically), x0 = ct, and two spatial dimensions (represented horizon- tally), x, y. We have also shown the ‘space-time line’, or ‘world-line’, (moving from the bottom to the top of the “space-time block,” or from the past towards the future) representing the history of a particle’s motion. The essential idea in Einstein’s article of June 1905 was to impose the group of transformations (3) as a symmetry group of the fundamental laws of physics (“the principle of relativity”). This point of view proved to be extraordinarily fruitful, since it led to the discovery of new laws and the prediction of new phe- nomena. Let us mention some of these for the record: the relativistic dynamics of classical particles, the dilation of lifetimes for relativistic particles, the re- lation E = mc2 between energy and inertial mass, Dirac’s relativistic theory of quantum spin 1 particles, the prediction of antimatter, the classification of particles by rest mass and spin, the relation between spin and statistics, and the CPT theorem. After these recollections on special relativity, let us discuss the special fea- ture of gravity which, in 1907, suggested to Einstein the need for a profound generalization of the chrono-geometric structure of space-time. 3 The Principle of Equivalence Einstein’s point of departure was a striking experimental fact: all bodies in an external gravitational field fall with the same acceleration. This fact was pointed out by Galileo in 1638. Through a remarkable combination of logical reason- ing, thought experiments, and real experiments performed on inclined planes,2 Galileo was in fact the first to conceive of what we today call the “universality of free-fall” or the “weak principle of equivalence.” Let us cite the conclusion that Galileo drew from a hypothetical argument where he varied the ratio between the densities of the freely falling bodies under consideration and the resistance of the medium through which they fall: “Having observed this I came to the con- clusion that in a medium totally devoid of resistance all bodies would fall with the same speed” [3]. This universality of free-fall was verified with more pre- cision by Newton’s experiments with pendulums, and was incorporated by him into his theory of gravitation (1687) in the form of the identification of the iner- tial massmi (appearing in the fundamental law of dynamics F = mi a) with the gravitational mass mg (appearing in the gravitational force, Fg = Gmgm mi = mg . (4) At the end of the nineteenth century, Baron Roland von Eötvös verified the equivalence (4) between mi and mg with a precision on the order of 10 and Einstein was aware of this high-precision verification. (At present, the equivalence between mi and mg has been verified at the level of 10 −12 [4].) The point that struck Einstein was that, given the precision with whichmi = mg was verified, and given the equivalence between inertial mass and energy discovered by Einstein in September of 1905 [2] (E = mi c 2), one must conclude that all of the various forms of energy that contribute to the inertial mass of a body (rest mass of the elementary constituents, various binding energies, internal kinetic energy, etc.) do contribute in a strictly identical way to the gravitational mass of this body, meaning both to its capacity for reacting to an external gravitational field and to its capacity to create a gravitational field. In 1907, Einstein realized that the equivalence betweenmi andmg implicitly contained a deeper equivalence between inertia and gravitation that had impor- tant consequences for the notion of an inertial reference frame (which was a fun- damental concept in the theory of special relativity). In an ingenious thought experiment, Einstein imagined the behavior of rigid bodies and reference clocks within a freely falling elevator. Because of the universality of free-fall, all of the objects in such a “freely falling local reference frame” would appear not to be accelerating with respect to it. Thus, with respect to such a reference frame, the exterior gravitational field is “erased” (or “effaced”). Einstein therefore pos- tulated what he called the “principle of equivalence” between gravitation and inertia. This principle has two parts, that Einstein used in turns. The first part says that, for any external gravitational field whatsoever, it is possible to 2The experiment with falling bodies said to be performed from atop the Leaning Tower of Pisa is a myth, although it aptly summarizes the essence of Galilean innovation. locally “erase” the gravitational field by using an appropriate freely falling local reference frame and that, because of this, the non-gravitational physical laws apply within this local reference frame just as they would in an inertial reference frame (free of gravity) in special relativity. The second part of Einstein’s equiv- alence principle says that, by starting from an inertial reference frame in special relativity (in the absence of any “true” gravitational field), one can create an apparent gravitational field in a local reference frame, if this reference frame is accelerated (be it in a straight line or through a rotation). 4 Gravitation and Space-Time Chrono-Geometry Einstein was able (through an extraordinary intellectual journey that lasted eight years) to construct a new theory of gravitation, based on a rich general- ization of the 1905 theory of relativity, starting just from the equivalence prin- ciple described above. The first step in this journey consisted in understanding that the principle of equivalence would suggest a profound modification of the chrono-geometric structure of Poincaré-Minkowski space-time recalled in Equa- tion (1) above. To illustrate, let Xα, α = 0, 1, 2, 3, be the space-time coordinates in a lo- cal, freely-falling reference frame (or locally inertial reference frame). In such a reference frame, the laws of special relativity apply. In particular, the infinites- imal space-time interval ds2 = dL2 − c2 dT 2 between two neighboring events within such a reference frame Xα, X ′α = Xα + dXα (close to the center of this reference frame) takes the form ds2 = dL2 − c2 dT 2 = ηαβ dXα dXβ , (5) where we recall that the repeated indices α and β are summed over all of their values (α, β = 0, 1, 2, 3). We also know that in special relativity the local energy and momentum densities and fluxes are collected into the ten components of the energy-momentum tensor Tαβ. (For example, the energy density per unit volume is equal to T 00, in the reference frame described by coordinates Xα = (X0, X i), i = 1, 2, 3.) The conservation of energy and momentum translates into the equation ∂β T αβ = 0, where ∂β = ∂/∂ X The theory of special relativity tells us that we can change our locally in- ertial reference frame (while remaining in the neighborhood of a space-time point where one has “erased” gravity) through a Lorentz transformation, X ′α = Λαβ X β. Under such a transformation, the infinitesimal interval ds2, Equation (5), remains invariant and the ten components of the (symmetric) tensor Tαβ are transformed according to T ′αβ = Λαγ Λ γδ. On the other hand, when we pass from a locally inertial reference frame (with coordinates Xα) to an extended non-inertial reference frame (with coordinates xµ; µ = 0, 1, 2, 3), the transformation connecting the Xα to the xµ is no longer a linear transforma- tion (like the Lorentz transformation) but becomes a non-linear transformation Xα = Xα(xµ) that can take any form whatsoever. Because of this, the value of the infinitesimal interval ds2, when expressed in a general, extended reference frame, will take a more complicated form than the very simple one given by Equation (5) that it had in a reference frame that was locally in free-fall. In fact, by differentiating the non-linear functions Xα = Xα(xµ) we obtain the relation dXα = ∂Xα/∂xµ dxµ. By substituting this relation into (5) we then obtain ds2 = gµν(x λ) dxµ dxν , (6) where the indices µ, ν are summed over 0, 1, 2, 3 and where the ten functions gµν(x) (symmetric over the indices µ and ν) of the four variables x λ are de- fined, point by point (meaning that for each point xλ we consider a refer- ence frame that is locally freely falling at x, with local coordinates Xαx ) by gµν(x) = ηαβ ∂X x (x)/∂x µ ∂Xβx (x)/∂x ν . Because of the nonlinearity of the functions Xα(x), the functions gµν(x) generally depend in a nontrivial way on the coordinates xλ. The local chrono-geometry of space-time thus appears to be given, not by the simple Minkowskian metric (2), with constant coefficients ηµν , but by a quadratic metric of a much more general type, Equation (6), with coefficients gµν(x) that vary from point to point. Such general metric spaces had been introduced and studied by Gauss and Riemann in the nineteenth century (in the case where the quadratic form (6) is positive definite). They carry the name Riemannian spaces or curved spaces. (In the case of interest for Einstein’s theory, where the quadratic form (6) is not positive definite, one speaks of a pseudo-Riemannian metric.) We do not have the space here to explain in detail the various geometric structures in a Riemannian space that are derivable from the data of the in- finitesimal interval (6). Let us note simply that given Equation (6), which gives the distance ds between two infinitesimally separated points, we are able, through integration along a curve, to define the length of an arbitrary curve connecting two widely separated points A and B: LAB = ds. One can then define the “straightest possible line” between two given points A and B to be the shortest line, in other words the curve that minimizes (or, more generally, extremizes) the integrated distance LAB. These straightest possible lines are called geodesic curves. To give a simple example, the geodesics of a spherical surface (like the surface of the Earth) are the great circles (with radius equal to the radius of the sphere). If one mathematically writes the condition for a curve, as given by its parametric representation xµ = xµ(s), where s is the length along the curve, to extremize the total length LAB one finds that x must satisfy the following second-order differential equation: d2 xλ + Γλµν(x) = 0 , (7) where the quantities Γλµν , known as the Christoffel coefficients or connection coefficients, are calculated, at each point x, from the metric components gµν(x) by the equation Γλµν ≡ gλσ(∂µ gνσ + ∂ν gµσ − ∂σ gµν) , (8) where gµν denotes the matrix inverse to gµν (g µσ gσν = δ ν where the Kronecker symbol δµν is equal to 1 when µ = ν and 0 otherwise) and where ∂µ ≡ ∂/∂xµ denotes the partial derivative with respect to the coordinate xµ. To give a very simple example: in the Poincaré-Minkowski space-time the components of the metric are constant, gµν = ηµν (when we use an inertial reference frame). Because of this, the connection coefficients (8) vanish in an inertial reference frame, and the differential equation for geodesics reduces to d2 xλ/ds2 = 0, whose solutions are ordinary straight lines: xλ(s) = aλ s + bλ. On the other hand, in a general “curved” space-time (meaning one with components gµν that depend in an arbitrary way on the point x) the geodesics cannot be globally represented by straight lines. One can nevertheless show that it always remains possible, for any gµν(x) whatsoever, to change coordinates x µ → Xα(x) in such a way that the connection coefficients Γαβγ , in the new system of coordinates Xα, vanish locally, at a given point Xα0 (or even along an arbitrary curve). Such locally geodesic coordinate systems realize Einstein’s equivalence principle mathematically: up to terms of second order, the components gαβ(X) of a “curved” metric in locally geodesic coordinates Xα (ds2 = gαβ(X) dX α dXβ) can be identified with the components of a “flat” Poincaré-Minkowski metric: gαβ(X) = ηαβ +O((X−X0)2), where X0 is the point around which we expand. 5 Einstein’s Equations: Elastic Space-Time Having postulated that a consistent relativistic theory of the gravitational field should include the consideration of a far-reaching generalization of the Poincaré- Minkowski space-time, Equation (6), Einstein concluded that the same ten functions gµν(x) should describe both the geometry of space-time as well as gravitation. He therefore got down to the task of finding which equations must be satisfied by the “geometric-gravitational field” gµν(x). He was guided in this search by three principles. The first was the principle of general relativity, which asserts that in the presence of a gravitational field one should be able to write the fundamental laws of physics (including those governing the gravitational field itself) in the same way in any coordinate system whatsoever. The second was that the “source” of the gravitational field should be the energy-momentum tensor T µν . The third was a principle of correspondence with earlier physics: in the limit where one neglects gravitational effects, gµν(x) = ηµν should be a solution of the equations being sought, and there should also be a so-called Newtonian limit where the new theory reduces to Newton’s theory of gravity. Note that the principle of general relativity (contrary to appearances and contrary to what Einstein believed for several years) has a different physical status than the principle of special relativity. The principle of special relativity was a symmetry principle for the structure of space-time that asserted that physics is the same in a particular class of reference frames, and therefore that certain “corresponding” phenomena occur in exactly the same way in different reference frames (“active” transformations). On the other hand, the principle of general relativity is a principle of indifference: the phenomena do not (in general) take place in the same way in different coordinate systems. However, none of these (extended) coordinate systems enjoys any privileged status with respect to the others. The principle asserting that the energy-momentum tensor T µν should be the source of the gravitational field is founded on two ideas: the relations E = mi c and the weak principle of equivalence mi = mg show that, in the Newtonian limit, the source of gravitation, the gravitational mass mg, is equal to the total energy of the body considered, or in other words the integral over space of the energy density T 00, up to the factor c−2. Therefore at least one of the components of the tensor T µν must play the role of source for the gravitational field. However, since the gravitational field is encoded, according to Einstein, by the ten components of the metric gµν , it is natural to suppose that the source for gµν must also have ten components, which is precisely the case for the (symmetric) tensor T µν . In November of 1915, after many years of conceptually arduous work, Ein- stein wrote the final form of the theory of general relativity [6]. Einstein’s equa- tions are non-linear, second-order partial differential equations for the geometric- gravitational field gµν , containing the energy-momentum tensor Tµν ≡ gµκ gνλ T κλ on the right-hand side. They are written as follows: Rµν − Rgµν = Tµν (9) where G is the (Newtonian) gravitational constant, c is the speed of light, and R ≡ gµν Rµν and the Ricci tensor Rµν are calculated as a function of the connection coefficients Γλµν (8) in the following way: Rµν ≡ ∂α Γαµν − ∂ν Γαµα + Γαβα Γβµν − Γαβν Γβµα . (10) One can show that, in a four-dimensional space-time, the three principles we have described previously uniquely determine Einstein’s equations (9). It is nevertheless remarkable that these equations may also be developed from points of view that are completely different from the one taken by Einstein. For example, in the 1960s various authors (in particular Feynman, Weinberg and Deser; see references in [4]) showed that Einstein’s equations could be obtained from a purely dynamical approach, founded on the consistency of interactions of a long-range spin 2 field, without making any appeal, as Einstein had, to the geometric notions coming from mathematical work on Riemannian spaces. Let us also note that if we relax one of the principles described previously (as Einstein did in 1917) we can find a generalization of Equation (9) in which one adds the term +Λ gµν to the left-hand side, where Λ is the so-called cosmological constant. Such a modification was proposed by Einstein in 1917 in order to be able to write down a globally homogeneous and stationary cosmological solution. Einstein rejected this additional term after work by Friedmann (1922) showed the existence of expanding cosmological solutions of general relativity and after the observational discovery (by Hubble in 1929) of the expanding motion of galaxies within the universe. However, recent cosmological data have once again made this possibility fashionable, although in the fundamental physics of today one tends to believe that a term of the type Λ gµν should be considered as a particular physical contribution to the right-hand side of Einstein’s equations (more precisely, as the stress-energy tensor of the vacuum, T Vµν = − c Λ gµν), rather than as a universal geometric modification of the left-hand side. Let us now comment on the physical meaning of Einstein’s equations (9). The essential new idea is that the chrono-geometric structure of space-time, Equation (6), in other words the structure that underlies all of the measurements that one could locally make of duration, dT , and of distance, dL, (we recall that, locally, ds2 = dL2 − c2 dT 2) is no longer a rigid structure that is given a priori, once and for all (as was the case for the structure of Poincaré-Minkowski space-time), but instead has become a field, a dynamical or elastic structure, which is created and/or deformed by the presence of an energy-momentum distribution. See Figure 2, which visualizes the “elastic” geometry of space- time in the theory of general relativity by representing, around each point x, the locus of points (assumed to be infinitesimally close to x) separated from x by a constant (squared) interval: ds2 = ε2. As in the case of Poincaré-Minkowski space-time (Figure 1), one arrives at a “field” of hyperboloids. However, this field of hyperboloids no longer has a “rigid” and homogeneous structure. Figure 2: “Elastic” space-time geometry in the theory of general relativity. This geometry is visualized by representing, around each space-time point x, the locus of points separated from x by a given small positive (squared) interval. The space-time field gµν(x) describes the variation from point to point of the chrono-geometry as well as all gravitational effects. The simplest example of space-time chrono-geometric elasticity is the effect that the proximity of a mass has on the “local rate of flow for time.” In concrete terms, if you separate two twins at birth, with one staying on the surface of the Earth and the other going to live on the peak of a very tall mountain (in other words farther from the Earth’s center), and then reunite them after 100 years, the “highlander” will be older (will have lived longer) than the twin who stayed on the valley floor. Everything takes place as if time flows more slowly the closer one is to a given distribution of mass-energy. In mathematical terms this effect is due to the fact that the coefficient g00(x) of (dx 0)2 in Equation (6) is deformed with respect to its value in special relativity, gMinkowski00 = η00 = −1, to become gEinstein00 (x) ≃ −1 + 2GM/c2r, where M is the Earth’s mass (in our example) and r the distance to the center of the Earth. In the example considered above of terrestrial twins the effect is extremely small (a difference in the amount of time lived of about one second over 100 years), but the effect is real and has been verified many times using atomic clocks (see the references in [4]). Let us mention that today this “Einstein effect” has important practical repercussions, for example in aerial or maritime navigation, for the piloting of automobiles, or even farm machinery, etc. In fact, the GPS (Global Positioning System), which uses the data transmitted by a constellation of atomic clocks on board satellites, incorporates the Einsteinian deformation of space-time chrono-geometry into its software. The effect is only on the order of one part in a billion, but if it were not taken into account, it would introduce an unacceptably large error into the GPS, which would continually grow over time. Indeed, GPS performance relies on the high stability of the orbiting atomic clocks, a stability better than 10−13, or in other words 10,000 times greater than the apparent change in frequency(∼ 10−9) due to the Einsteinian deformation of the chrono-geometry. 6 TheWeak-Field Limit and the Newtonian Limit To understand the physical consequences of Einstein’s equations (9), it is useful to begin by considering the limiting case of weak geometric-gravitational fields, namely the case where gµν(x) = ηµν + hµν(x), with perturbations hµν(x) that are very small with respect to unity: |hµν(x)| ≪ 1. In this case, a simple calculation (that we encourage the reader to perform) starting from Definitions (8) and (10) above, leads to the following explicit form of Einstein’s equations (where we ignore terms of order h2 and hT ): � hµν − ∂µ ∂α hαν − ∂ν ∂α hαµ + ∂µν hαα = − 16 πG T̃µν , (11) where � = ηµν ∂µν = ∆−∂20 = ∂2/∂x2+∂2/∂y2+∂2/∂z2− c−2 ∂2/∂t2 denotes the “flat” d’Alembertian (or wave operator; xµ = (ct, x, y, z)), and where indices in the upper position have been raised by the inverse ηµν of the flat metric ηµν (numerically ηµν = ηµν , meaning that −η00 = η11 = η22 = η33 = +1). For example ∂α hαν denotes η αβ ∂α hβν and h α ≡ ηαβ hαβ = −h00 + h11 + h22 + h33. The “source” T̃µν appearing on the right-hand side of (11) denotes the combination T̃µν ≡ Tµν − 12 T α ηµν (when space-time is four-dimensional). The “linearized” approximation (11) of Einstein’s equations is analogous to Maxwell’s equations �Aµ − ∂µ ∂αAα = −4π Jµ , (12) connecting the electromagnetic four-potential Aµ ≡ ηµν Aν (where A0 = V , Ai = A, i = 1, 2, 3) to the four-current density Jµ ≡ ηµν Jν (where J0 = ρ is the charge density and J i = J is the current density). Another analogy is that the structure of the left-hand side of Maxwell’s equations implies that the “source” Jµ appearing on the right-hand side must satisfy ∂ µ Jµ = 0 (∂ µ ≡ ηµν ∂ν), which expresses the conservation of electric charge. Likewise, the structure of the left-hand side of the linearized form of Einstein’s equations (11) implies that the “source” Tµν = T̃µν − 12 T̃ α ηµν must satisfy ∂ µ Tµν = 0, which expresses the conservation of energy and momentum of matter. (The structure of the left- hand side of the exact form of Einstein’s equations (9) implies that the source Tµν must satisfy the more complicated equation ∂µ T µν+Γµσµ T σν+Γνσµ T µσ = 0, where the terms in ΓT can be interpreted as describing an exchange of energy and momentum between matter and the gravitational field.) The major dif- ference is that, in the case of electromagnetism, the field Aµ and its source Jµ have a single space-time index, while in the gravitational case the field hµν and its source T̃µν have two space-time indices. We shall return later to this anal- ogy/difference between Aµ and hµν which suggests the existence of a certain relation between gravitation and electromagnetism. We recover the Newtonian theory of gravitation as the limiting case of Ein- stein’s theory by assuming not only that the gravitational field is a weak defor- mation of the flat Minkowski space-time (hµν ≪ 1), but also that the field hµν is slowly varying (∂0 hµν ≪ ∂i hµν) and that its source Tµν is non-relativistic (Tij ≪ T0i ≪ T00). Under these conditions Equation (11) leads to a Poisson- type equation for the purely temporal component, h00, of the space-time field, ∆h00 = − 16 πG T̃00 = − (T00 + Tii) ≃ − T00 , (13) where ∆ = ∂2x + ∂ y + ∂ z is the Laplacian. Recall that, according to Laplace and Poisson, Newton’s theory of gravity is summarized by saying that the grav- itational field is described by a single potential U(x), produced by the mass density ρ(x) according to the Poisson equation ∆U = −4 πGρ, that deter- mines the acceleration of a test particle placed in the exterior field U(x) ac- cording to the equation d2 xi/dt2 = ∂i U(x) ≡ ∂U/∂xi. Because of the relation mi = mg = E/c 2 one can identify ρ = T 00/c2. We therefore find that (13) reproduces the Poisson equation if h00 = +2U/c 2. It therefore remains to ver- ify that Einstein’s theory indeed predicts that a non-relativistic test particle is accelerated by a space-time field according to d2 xi/dt2 ≃ 1 c2 ∂i h00. Einstein understood that this was a consequence of the equivalence principle. In fact, as we discussed in Section 4 above, the principle of equivalence states that the gravitational field is (locally) erased in a locally inertial reference frame Xα (such that gαβ(X) = ηαβ +O((X −X0)2)). In such a reference frame, the laws of special relativity apply at the point X0. In particular an isolated (and elec- trically neutral) body must satisfy a principle of inertia in this frame: its center of mass moves in a straight line at constant speed. In other words it satisfies the equation of motion d2Xα/ds2 = 0. By passing back to an arbitrary (ex- tended) coordinate system xµ, one verifies that this equation for inertial motion transforms into the geodesic equation (7). Therefore (7) describes falling bodies, such as they are observed in arbitrary extended reference frames (for example a reference frame at rest with respect to the Earth or at rest with respect to the center of mass of the solar system). From this one concludes that the relativis- tic analog of the Newtonian field of gravitational acceleration, g(x) = ∇U(x), is gλ(x) ≡ −c2 Γλµν dxµ/ds dxν/ds. By considering a particle whose motion is slow with respect to the speed of light (dxi/ds ≪ dx0/ds ≃ 1) one can easily verify that gi(x) ≃ −c2 Γi00. Finally, by using the definition (8) of Γαµν , and the hypothesis of weak fields, one indeed verifies that gi(x) ≃ 1 c2 ∂i h00, in perfect agreement with the identification h00 = 2U/c 2 anticipated above. We encour- age the reader to personally verify this result, which contains the very essence of Einstein’s theory: gravitational motion is no longer described as being due to a force, but is identified with motion that is “as inertial as possible” within a space-time whose chrono-geometry is deformed in the presence of a mass-energy distribution. Finding the Newtonian theory as a limiting case of Einstein’s theory is ob- viously a necessity for seriously considering this new theory. But of course, from the very beginning Einstein explored the observational consequences of general relativity that go beyond the Newtonian description of gravitation. We have already mentioned one of these above: the fact that g00 = η00 + h00 ≃ −1 + 2U(x)/c2 implies a distortion in the relative measurement of time in the neighborhood of massive bodies. In 1907 (as soon as he had developed the principle of equivalence, and long before he had obtained the field equations of general relativity) Einstein had predicted the existence of such a distortion for measurements of time and frequency in the presence of an external gravita- tional field. He realized that this should have observable consequences for the frequency, as observed on Earth, of the spectral rays emitted from the surface of the Sun. Specifically, a spectral ray of (proper local) frequency ν0 emitted from a point x0 where the (stationary) gravitational potential takes the value U(x0) and observed (via electromagnetic signals) at a point x where the potential is U(x) should appear to have a frequency ν such that g00(x0) g00(x) ≃ 1 + 1 [U(x)− U(x0)] . (14) In the case where the point of emission x0 is in a gravitational potential well deeper than the point of observation x (meaning that U(x0) > U(x)) one has ν < ν0, in other words a reddening effect on frequencies. This effect, which was predicted by Einstein in 1907, was unambiguously verified only in the 1960s, in experiments by Pound and collaborators over a height of about twenty me- ters. The most precise verification (at the level of ∼ 10−4) is due to Vessot and collaborators, who compared a hydrogen maser, launched aboard a rocket that reached about 10,000 km in altitude, to a clock of similar construction on the ground. Other experiments compared the times shown on clocks placed aboard airplanes to clocks remaining on the ground. (For references to these experiments see [4].) As we have already mentioned, the “Einstein effect” (14) must be incorporated in a crucial way into the software of satellite positioning systems such as the GPS. In 1907, Einstein also pointed out that the equivalence principle would sug- gest that light rays should be deflected by a gravitational field. Indeed, a gener- alization of the reasoning given above for the motion of particles in an external gravitational field, based on the principle of equivalence, shows that light must itself follow a trajectory that is “as inertial as possible,” meaning a geodesic of the curved space-time. Light rays must therefore satisfy the geodesic equa- tion (7). (The only difference from the geodesics followed by material particles is that the parameter s in Equation (7) can no longer be taken equal to the “length” along the geodesic, since a “light” geodesic must also satisfy the con- straint gµν(x) dx µ dxν = 0, ensuring that its speed is equal to c, when it is measured in a locally inertial reference frame.) Starting from Equation (7) one can therefore calculate to what extent light is deflected when it passes through the neighborhood of a large mass (such as the Sun). One nevertheless soon realizes that in order to perform this calculation one must know more than the component h00 of the gravitational field. The other components of hµν , and in particular the spatial components hij , come into play in a crucial way in this calculation. This is why it was only in November of 1915, after having obtained the (essentially) final form of his theory, that Einstein could predict the total value of the deflection of light by the Sun. Starting from the linearized form of Einstein’s equations (11) and continuing by making the “non-relativistic” sim- plifications indicated above (Tij ≪ T0i ≪ T00, ∂0 h≪ ∂i h) it is easy to see that the spatial component hij , like h00, can be written (after a helpful choice of coordinates) in terms of the Newtonian potential U as hij(x) ≃ +2U(x) δij/c2, where δij takes the value 1 if i = j and 0 otherwise (i, j = 1, 2, 3). By inserting this result, as well as the preceding result h00 = +2U/c 2, into the geodesic equation (7) for the motion of light, one finds (as Einstein did in 1915) that general relativity predicts that the Sun should deflect a ray of light by an angle θ = 4GM/(c2b) where b is the impact parameter of the ray (meaning its mini- mum distance from the Sun). As is well known, the confirmation of this effect in 1919 (with rather weak precision) made the theory of general relativity and its creator famous. 7 The Post-Newtonian Approximation and Ex- perimental Confirmations in the Regime of Weak and Quasi-Stationary Gravitational Fields We have already pointed out some of the experimental confirmations of the theory of general relativity. At present, the extreme precision of certain mea- surements of time or frequency in the solar system necessitates a very careful account of the modifications brought by general relativity to the Newtonian de- scription of space-time. As a consequence, general relativity is used in a great number of situations, from astronomical or geophysical research (such as very long range radio interferometry, radar tracking of the planets, and laser tracking of the Moon or artificial satellites) to metrological, geodesic or other applica- tions (such as the definition of international atomic time, precision cartography, and the G.P.S.). To do this, the so-called post-Newtonian approximation has been developed. This method involves working in the Newtonian limit sketched above while keeping the terms of higher order in the small parameter ε ∼ v ∼ |hµν | ∼ |∂0 h/∂i h|2 ∼ |T 0i/T 00|2 ∼ |T ij/T 00| , where v denotes a characteristic speed for the elements in the system considered. For all present applications of general relativity to the solar system it suffices to include the first post-Newtonian approximation, in other words to keep the relative corrections of order ε to the Newtonian predictions. Since the theory of general relativity was poorly verified for a long time, one found it useful (as in the pioneering work of A. Eddington, generalized in the 1960s by K. Nordtvedt and C.M. Will) to study not only the precise predictions of the equations (9) defining Einstein’s theory, but to also consider possible deviations from these predictions. These possible deviations were parameterized by means of several non-dimensional “post-Newtonian” parameters. Among these parameters, two play a key role: γ and β. The parameter γ describes a possible deviation from general relativity that comes into play starting at the linearized level, in other words one that modifies the linearized approximation given above. More pre- cisely, it is defined by writing that the difference hij ≡ gij − δij between the spatial metric and the Euclidean metric can take the value hij = 2γ U δij/c 2 (in a suitable coordinate system), rather than the value hGRij = 2U δij/c 2 that it takes in general relativity, thus differing by a factor γ. Therefore, by definition γ takes the value 1 in general relativity, and γ− 1 measures the possible deviation with respect to this theory. As for the parameter β (or rather β−1), it measures a pos- sible deviation (with respect to general relativity) in the value of h00 ≡ g00−η00. The value of h00 in general relativity is h 00 = 2U/c 2 − 2U2/c4, where the first term (discussed above) reproduces the Newtonian approximation (and cannot therefore be modified, as the idea is to parameterize gravitational physics be- yond Newtonian predictions) and where the second term is obtained by solving Einstein’s equations (9) at the second order of approximation. One then writes an h00 of a more general parameterized type, h00 = 2U/c 2 − 2 β U2/c4, where, by definition, β takes the value 1 in general relativity. Let us finally point out that the parameters γ−1 and β−1 completely parameterize the post-Newtonian regime of the simplest theoretical alternatives to general relativity, namely the tensor-scalar theories of gravitation. In these theories, the gravitational inter- action is carried by two fields at the same time: a massless tensor (spin 2) field coupled to T µν , and a massless scalar (spin 0) field ϕ coupled to the trace Tαα . In this case the parameter −(γ − 1) plays the key role of measuring the ratio between the scalar coupling and the tensor coupling. All of the experiments performed to date within the solar system are com- patible with the predictions of general relativity. When they are interpreted in terms of the post-Newtonian (and “post-Einsteinian”) parameters γ − 1 and β−1, they lead to strong constraints on possible deviations from Einstein’s the- ory. We make note of the following among tests performed in the solar system: the deflection of electromagnetic waves in the neighborhood of the Sun, the grav- itational delay (‘Shapiro effect’) of radar signals bounced from the Viking lander on Mars, the global analysis of solar system dynamics (including the advance of planetary perihelia), the sub-centimeter measurement of the Earth-Moon dis- tance obtained from laser signals bounced off of reflectors on the Moon’s surface, etc. At present (October of 2006) the most precise test (that has been published) of general relativity was obtained in 2003 by measuring the ratio 1 + y ≡ f/f0 between the frequency f0 of radio waves sent from Earth to the Cassini space probe and the frequency f of coherent radio waves sent back (with the same local frequency) from Cassini to Earth and compared (on Earth) to the emitted frequency f0. The main contribution to the small quantity y is an effect equal, in general relativity, to yGR = 8(GM/c 3 b) db/dt (where b is, as before, the impact parameter) due to the propagation of radio waves in the geometry of a space-time deformed by the Sun: ds2 ≃ −(1−2U/c2) c2 dt2+(1+2U/c2)(dx2+dy2+dz2), where U = GM/r. The maximum value of the frequency change predicted by general relativity was only |yGR| . 2 × 10−10 for the best observations, but thanks to an excellent frequency stability ∼ 10−14 (after correction for the perturbations caused by the solar corona) and to a relatively large number of individual measurements spread over 18 days, this experiment was able to verify Einstein’s theory at the remarkable level of ∼ 10−5 [7]. More precisely, when this experiment is interpreted in terms of the post-Newtonian parameters γ− 1 and β − 1, it gives the following limit for the parameter γ − 1 [7] γ − 1 = (2.1± 2.3)× 10−5 . (15) As for the best present-day limit on the parameter β−1, it is smaller than 10−3 and comes from the non-observation, in the data from lasers bounced off of the Moon, of any eventual polarization of the Moon’s orbit in the direction of the Sun (‘Nordtvedt effect’; see [4] for references) 4(β − 1)− (γ − 1) = −0.0007± 0.0010 . (16) Although the theory of general relativity is one of the best verified theories in physics, scientists continue to design and plan new or increasingly precise tests of the theory. This is the case in particular for the space mission Gravity Probe B (launched by NASA in April of 2004) whose principal aim is to directly observe a prediction of general relativity that states (intuitively speaking) that space is not only “elastic,” but also “fluid.” In the nineteenth century Foucalt invented both the gyroscope and his famous pendulum in order to render Newton’s absolute (and rigid) space directly observable. His experiments in fact showed that, for example, a gyroscope on the surface of the Earth continued, despite the Earth’s rotation, to align itself in a direction that is “fixed” with respect to the distant stars. However, in 1918, when Lense and Thirring analyzed some of the consequences of the (linearized) Einstein equations (11), they found that general relativity predicts, among other things, the following phenomenon: the rotation of the Earth (or any other ball of matter) creates a particular deformation of the chrono-geometry of space-time. This deformation is described by the “gravito- magnetic” components h0i of the metric, and induces an effect analogous to the “rotation drag” effect caused by a ball of matter turning in a fluid: the rotation of the Earth (minimally) drags all of the space around it, causing it to continually “turn,” as a fluid would.3 This “rotation of space” translates, in an observable way, into a violation of the effects predicted by Newton and confirmed by Foucault’s experiments: in particular, a gyroscope no longer aligns itself in a direction that is “fixed in absolute space,” rather its axis of rotation is “dragged” by the rotating motion of the local space where it is located. This effect is much too small to be visible in Foucalt’s experiments. Its observation by Gravity Probe B (see [8] and the contribution of John Mester to this Poincaré seminar) is important for making Einstein’s revolutionary notion of a fluid space-time tangible to the general public. Up till now we have only discussed the regime of weak and slowly varying gravitational fields. The theory of general relativity predicts the appearance of new phenomena when the gravitational field becomes strong and/or rapidly varying. (We shall not here discuss the cosmological aspects of relativistic grav- itation.) 8 Strong Gravitational Fields and Black Holes The regime of strong gravitational fields is encountered in the physics of grav- itationally condensed bodies. This term designates the final states of stellar evolution, and in particular neutron stars and black holes. Recall that most of the life of a star is spent slowly burning its nuclear fuel. This process causes the star to be structured as a series of layers of differentiated nuclear structure, surrounding a progressively denser core (an “onion-like” structure). When the initial mass of the star is sufficiently large, this process ends into a catastrophic phenomenon: the core, already much denser than ordinary matter, collapses in on itself under the influence of its gravitational self-attraction. (This implosion of the central part of the star is, in many cases, accompanied by an explosion of the outer layers of the star—a supernova.) Depending on the quantity of mass that collapses with the core of a star, this collapse can give rise to either a neutron star or a black hole. A neutron star condenses a mass on the order of the mass of the Sun inside a radius on the order of 10 km. The density in the interior of a neutron star (named thus because neutrons dominate its nuclear composition) is more than 100 million tons per cubic centimeter (1014 g/cm3)! It is about the same as the density in the interior of atomic nuclei. What is important for our discussion is that the deformation away from the Minkowski metric in the immediate neigh- borhood of a neutron star, measured by h00 ∼ hii ∼ 2GM/c2R, where R is the radius of the star, is no longer a small quantity, as it was in the solar system. In fact, while h ∼ 2GM/c2R is on the order of 10−9 for the Earth and 10−6 for the Sun, one finds that h ∼ 0.4 for a typical neutron star (M ≃ 1.4M⊙, R ∼ 10 3Recent historical work (by Herbert Pfister) has in fact shown that this effect had already been derived by Einstein within the framework of the provisory relativistic theory of gravity that he started to develop in 1912 in collaboration with Marcel Grossmann. km). One thus concludes that it is no longer possible, as was the case in the solar system, to study the structure and physics of neutron stars by using the post-Newtonian approximation outlined above. One must consider the exact form of Einstein’s equations (9), with all of their non-linear structure. Because of this, we expect that observations concerning neutron stars will allow us to confirm (or refute) the theory of general relativity in its strongly non-linear regime. We shall discuss such tests below in relation to observations of binary pulsars. A black hole is the result of a continued collapse, meaning that it does not stop with the formation of an ultra-dense star (such as a neutron star). (The physical concept of a black hole was introduced by J.R. Oppenheimer and H. Snyder in 1939. The global geometric structure of black holes was not un- derstood until some years later, thanks notably to the work of R. Penrose. For a historical review of the idea of black holes see [9].) It is a particular structure of curved space-time characterized by the existence of a boundary (called the “black hole surface” or “horizon”) between an exterior region, from which it is possible to emit signals to infinity, and an interior region (of space-time), within which any emitted signal remains trapped. See Figure 3. r = 0 SINGULARITY r = 2M HORIZON FLASH OF LIGHT EMITTED FROM CENTER COLLAPSING space Figure 3: Schematic representation of the space-time for a black hole created from the collapse of a spherical star. Each cone represents the space-time history of a flash of light emitted from a point at a particular instant. (Such a “cone field” is obtained by taking the limit ε2 = 0 from Figure 2, and keeping only the upper part, in other words the part directed towards the future, of the double cones obtained as limits of the hyperboloids of Figure 2.) The interior of the black hole is shaded, its outer boundary being the “black hole surface” or “horizon.” The “inner boundary” (shown in dark grey) of the interior region of the black hole is a space-time singularity of the big-crunch type. The cones shown in this figure are called “light cones.” They are defined as the locus of points (infinitesimally close to x) such that ds2 = 0, with dx0 = cdt ≥ 0. Each represents the beginning of the space-time history of a flash of light emitted from a certain point in space-time. The cones whose vertices are located outside of the horizon (the shaded zone) will evolve by spreading out to infinity, thus representing the possibility for electromagnetic signals to reach infinity. On the other hand, the cones whose vertices are located inside the horizon (the grey zone) will evolve without ever succeeding in escaping the grey zone. It is therefore impossible to emit an electromagnetic signal that reaches infinity from the grey zone. The horizon, namely the boundary between the shaded zone and the unshaded zone, is itself the history of a particular flash of light, emitted from the center of the star over the course of its collapse, such that it asymptotically stabilizes as a space-time cylinder. This space-time cylinder (the asymptotic horizon) therefore represents the space-time history of a bubble of light that, viewed locally, moves outward at the speed c, but which globally “runs in place.” This remarkable behavior is a striking illustration of the “fluid” character of space-time in Einstein’s theory. Indeed, one can compare the pre- ceding situation with what may take place around the open drain of an emptying sink: a wave may move along the water, away from the hole, all the while run- ning in place with respect to the sink because of the falling motion of the water in the direction of the drain. Note that the temporal development of the interior region is limited, ter- minating in a singularity (the dark gray surface) where the curvature becomes infinite and where the classical description of space and time loses its meaning. This singularity is locally similar to the temporal inverse of a cosmological sin- gularity of the big bang type. This is called a big crunch. It is a space-time frontier, beyond which space-time ceases to exist. The appearance of singulari- ties associated with regions of strong gravitational fields is a generic phenomenon in general relativity, as shown by theorems of R. Penrose and S.W. Hawking. Black holes have some remarkable properties. First, a uniqueness theorem (due to W. Israel, B. Carter, D.C. Robinson, G. Bunting, and P.O. Mazur) asserts that an isolated, stationary black hole (in Einstein-Maxwell theory) is completely described by three parameters: its mass M , its angular momen- tum J , and its electric charge Q. The exact solution (called the Kerr-Newman solution) of Einstein’s equations (11) describing a black hole with parameters M,J,Q is explicitly known. We shall here content ourselves with writing the space-time geometry in the simplest case of a black hole: the one in which J = Q = 0 and the black hole is described only by its mass (a solution discov- ered by K. Schwarzschild in January of 1916): ds2 = − 1− 2GM c2 dt2 + 1− 2GM + r2(dθ2 + sin2 θ dϕ2) . (17) We see that the purely temporal component of the metric, g00 = −(1−2GM/c2r), vanishes when the radial coordinate r takes the value r = rH ≡ 2GM/c2. Ac- cording to the earlier equation (14), it would therefore seem that light emitted from an arbitrary point on the sphere r0 = rH , when it is viewed by an observer located anywhere in the exterior (in r > rH), would experience an infinite red- dening of its emission frequency (ν/ν0 = 0). In fact, the sphere rH = 2GM/c is the horizon of the Schwarzschild black hole, and no particle (that is capable of emitting light) can remain at rest when r = rH (nor, a fortiori, when r < rH). To study what happens at the horizon (r = rH) or in the interior (r < rH) of a Schwarzschild black hole, one must use other space-time coordinates than the coordinates (t, r, θ, ϕ) used in Equation (17). The “big crunch” singularity in the interior of a Schwarzschild black hole, in the coordinates of (17), is located at r = 0 (which does not describe, as one might believe, a point in space, but rather an instant in time). The space-time metric of a black hole space-time, such as Equation (17) in the simple case J = Q = 0, allows one to study the influence of a black hole on particles and fields in its neighborhood. One finds that a black hole is a gravitational potential well that is so deep that any particle or wave that penetrates the interior of the black hole (the region r < rH) will never be able to come out again, and that the total energy of the particle or wave that “falls” into the black hole ends up augmenting the total mass-energy M of the black hole. By studying such black hole “accretion” processes with falling particles (following R. Penrose), D. Christodoulou and R. Ruffini showed that a black hole is not only a potential well, but also a physical object possessing a significant free energy that it is possible, in principle, to extract. Such black hole energetics is encapsulated in the “mass formula” of Christodoulou and Ruffini (in units where c = 1) Mirr + 4GMirr 4G2M2irr , (18) where Mirr denotes the irreducible mass of the black hole, a quantity that can only grow, irreversibly. One deduces from (18) that a rotating (J 6= 0) and/or charged (Q 6= 0) black hole possesses a free energy M − Mirr > 0 that can, in principle, be extracted through processes that reduce its angular momentum and/or its electric charge. Such black hole energy-extraction processes may lie at the origin of certain ultra-energetic astrophysical phenomena (such as quasars or gamma ray bursts). Let us note that, according to Equation (18), (rotating or charged) black holes are the largest reservoirs of free energy in the Universe: in fact, 29% of their mass energy can be stored in the form of rotational energy, and up to 50% can be stored in the form of electric energy. These percentages are much higher than the few percent of nuclear binding energy that is at the origin of all the light emitted by stars over their lifetimes. Even though there is not, at present, irrefutable proof of the existence of black holes in the universe, an entire range of very strong presumptive evidence lends credence to their existence. In particular, more than a dozen X-ray emitting binary systems in our galaxy are most likely made up of a black hole and an ordinary star. Moreover, the center of our galaxy seems to contain a very compact concentration of mass ∼ 3 × 106M⊙ that is probably a black hole. (For a review of the observational data leading to these conclusions see, for example, Section 7.6 of the recent book by N. Straumann [6].) The fact that a quantity associated with a black hole, here the irreducible mass Mirr, or, according to a more general result due to S.W. Hawking, the total area A of the surface of a black hole (A = 16 πG2M2irr), can evolve only by irreversibly growing is reminiscent of the second law of thermodynamics. This result led J.D. Bekenstein to interpret the horizon area, A, as being propor- tional to the entropy of the black hole. Such a thermodynamic interpretation is reinforced by the study of the growth of A under the influence of external perturbations, a growth that one can in fact attribute to some local dissipative properties of the black hole surface, notably a surface viscosity and an electrical resistivity equal to 377 ohm (as shown in work by T. Damour and R.L. Zna- jek). These “thermodynamic” interpretations of black hole properties are based on simple analogies at the level of classical physics, but a remarkable result by Hawking showed that they have real content at the level of quantum physics. In 1974, Hawking discovered that the presence of a horizon in a black hole space-time affected the definition of a quantum particle, and caused a black hole to continuously emit a flux of particles having the characteristic spectrum (Planck spectrum) of thermal emission at the temperature T = 4 ~G∂M/∂A, where ~ is the reduced Planck constant. By using the general thermodynamic relation connecting the temperature to the energy E = M and the entropy S, T = ∂M/∂S, we see from Hawking’s result (in conformity with Bekenstein’s ideas) that a black hole possesses an entropy S equal (again with c = 1) to . (19) The Bekenstein-Hawking formula (19) suggests an unexpected, and perhaps pro- found, connection between gravitation, thermodynamics, and quantum theory. See Section 11 below. 9 Binary Pulsars and Experimental Confirma- tions in the Regime of Strong and Radiating Gravitational Fields Binary pulsars are binary systems made up of a pulsar (a rapidly spinning neutron star) and a very dense companion star (either a neutron star or a white dwarf). The first system of this type (called PSR B1913+16) was discovered by R.A. Hulse and J.H. Taylor in 1974 [10]. Today, a dozen are known. Some of these (including the first-discovered PSR B1913+16) have revealed themselves to be remarkable probes of relativistic gravitation and, in particular, of the regime of strong and/or radiating gravitational fields. The reason for which a binary pulsar allows for the probing of strong gravitational fields is that, as we have already indicated above, the deformation of the space-time geometry in the neighborhood of a neutron star is no longer a small quantity, as it is in the solar system. Rather, it is on the order of unity: hµν ≡ gµν − ηµν ∼ 2GM/c2R ∼ 0.4. (We note that this value is only 2.5 times smaller than in the extreme case of a black hole, for which 2GM/c2R = 1.) Moreover, the fact that the gravitational interaction propagates at the speed of light (as indicated by the presence of the wave operator, � = ∆ − c−2∂2/∂t2 in (11)) between the pulsar and its companion is found to play an observationally significant role for certain binary pulsars. Let us outline how the observational data from binary pulsars are used to probe the regime of strong (hµν on the order of unity) and/or radiative (effects propagating at the speed c) gravitational fields. (For more details on the obser- vational data from binary pulsars and their use in probing relativistic gravita- tion, see Michael Kramer’s contribution to this Poincaré seminar.) Essentially, a pulsar plays the role of an extremely stable clock. Indeed, the “pulsar phe- nomenon” is due to the rotation of a bundle of electromagnetic waves, created in the neighborhood of the two magnetic poles of a strongly magnetized neutron star (with a magnetic field on the order of 1012 Gauss, 1012 times the size of the terrestrial magnetic field). Since the magnetic axis of a pulsar is not aligned with its axis of rotation, the rapid rotation of the pulsar causes the (inner) magnetosphere as a whole to rotate, and likewise the bundle of electromagnetic waves created near the magnetic poles. The pulsar is therefore analogous to a lighthouse that sweeps out space with two bundles (one per pole) of electromag- netic waves. Just as for a lighthouse, one does not see the pulsar from Earth except when the bundle sweeps the Earth, thus causing a flash of electromag- netic noise with each turn of the pulsar around itself (in some cases, one even sees a secondary flash, due to emission from the second pole, after each half- turn). One can then measure the time of arrival at Earth of (the center of) each flash of electromagnetic noise. The basic observational data of a pulsar are thus made up of a regular, discrete sequence of the arrival times at Earth of these flashes or “pulses.” This sequence is analogous to the signal from a clock: tick, tick, tick, . . .. Observationally, one finds that some pulsars (and in particular those that belong to binary systems) thus define clocks of a stability comparable to the best atomic clocks [11]. In the case of a solitary pulsar, the sequence of its arrival times is (in essence) a regular “arithmetic sequence,” TN = aN + b, where N is an integer labelling the pulse considered, and where a is equal to the period of rotation of the pulsar around itself. In the case of a binary pulsar, the sequence of arrival times is a much richer signal, say TN = aN + b+∆N , where ∆N measures the deviation with respect to a regular arithmetic sequence. This deviation (after the subtraction of effects not connected to the orbital period of the pulsar) is due to a whole ensemble of physical effects connected to the orbital motion of the pulsar around its companion or, more precisely, around the center of mass of the binary system. Some of these effects could be pre- dicted by a purely Keplerian description of the motion of the pulsar in space, and are analogous to the “Rœmer effect” that allowed Rœmer to determine, for the first time, the speed of light from the arrival times at Earth of light signals coming from Jupiter’s satellites (the light signals coming from a body moving in orbit are “delayed” by the time taken by light to cross this orbit and arrive at Earth). Other effects can only be predicted and calculated by using a relativistic description, either of the orbital motion of the pulsar, or of the prop- agation of electromagnetic signals between the pulsar and Earth. For example, the following facts must be accounted for: (i) the “pulsar clock” moves at a large speed (on the order of 300 km/s ∼ 10−3c) and is embedded in the varying gravitational potential of the companion; (ii) the orbit of the pulsar is not a simple Keplerian ellipse, but (in general relativity) a more complicated orbit that traces out a “rosette” around the center of mass; (iii) the propagation of electromagnetic signals between the pulsar and Earth takes place in a space-time that is curved by both the pulsar and its companion, which leads to particular effects of relativistic delay; etc. Taking relativistic effects in the theoretical de- scription of arrival times for signals emitted by binary pulsars into account thus leads one to write what is called a timing formula. This timing formula (due to T. Damour and N. Deruelle) in essence allows one to parameterize the sequence of arrival times, TN = aN + b +∆N , in other words to parameterize ∆N , as a function of a set of “phenomenological parameters” that include not only the so-called “Keplerian” parameters (such as the orbital period P , the projection of the semi-major axis of the pulsar’s orbit along the line of sight xA = aA sin i, and the eccentricity e), but also the post-Keplerian parameters associated with the relativistic effects mentioned above. For example, effect (i) discussed above is parameterized by a quantity denoted γT ; effect (ii) by (among others) the quantities ω̇, Ṗ ; effect (iii) by the quantities r, s; etc. The way in which observations of binary pulsars allow one to test rela- tivistic theories of gravity is therefore the following. A (least-squares) fit be- tween the observational timing data, ∆obsN , and the parameterized theoreti- cal timing formula, ∆thN (P, xA, e; γT , ω̇, Ṗ , r, s), allows for the determination of the observational values of the Keplerian (P obs, xobsA , e obs) and post-Keplerian (γobsT , ω̇ obs, Ṗ obs, robs, sobs) parameters. The theory of general relativity pre- dicts the value of each post-Keplerian parameter as a function of the Keple- rian parameters and the two masses of the binary system (the mass mA of the pulsar and the mass mB of the companion). For example, the theoretical value predicted by general relativity for the parameter γT is γ T (mA,mB) = en−1(GMn/c3)2/3mB(mA + 2mB)/M 2, where e is the eccentricity, n = 2π/P the orbital frequency, andM ≡ mA+mB. We thus see that, if one assumes that general relativity is correct, the observational measurement of a post-Keplerian parameter, for example γobsT , determines a curve in the plane (mA,mB) of the two masses: γGRT (mA,mB) = γ T , in our example. The measurement of two post-Keplerian parameters thus gives two curves in the (mA,mB) plane and generically allows one to determine the values of the two masses mA and mB, by considering the intersection of the two curves. We obtain a test of general relativity as soon as one observationally measures three or more post-Keplerian parameters: if the three (or more) curves all intersect at one point in the plane of the two masses, the theory of general relativity is confirmed, but if this is not the case the theory is refuted. At present, four distinct binary pulsars have allowed one to test general relativity. These four “relativistic” binary pulsars are: the first binary pulsar PSR B1913+16, the pulsar PSR B1534+12 (dis- covered by A. Wolszczan in 1991), and two recently discovered pulsars: PSR J1141−6545 (discovered in 1999 by V.M. Kaspi et al., whose first timing results are due to M. Bailes et al. in 2003), and PSR J0737−3039 (discovered in 2003 by M. Burgay et al., whose first timing results are due to A.G. Lyne et al. and M. Kramer et al.). With the exception of PSR J1141−6545, whose companion is a white dwarf, the companions of the pulsars are neutron stars. In the case of PSR J0737−3039 the companion turns out to also be a pulsar that is visible from Earth. In the system PSR B1913+16, three post-Keplerian parameters have been measured (ω̇, γT , Ṗ ), which gives one test of the theory. In the system PSR J1141−65, three post-Keplerian parameters have been measured (ω̇, γT , Ṗ ), which gives one test of the theory. (The parameter s is also measured through scin- tillation phenomena, but the use of this measurement for testing gravitation is more problematic.) In the system PSR B1534+12, five post-Keplerian param- eters have been measured, which gives three tests of the theory. In the system PSR J0737−3039,six post-Keplerian parameters,4 which gives four tests of the theory. It is remarkable that all of these tests have confirmed general relativ- ity. See Figure 4 and, for references and details, [4, 11, 12, 13], as well as the contribution by Michael Kramer. Note that, in Figure 4, some post-Keplerian parameters are measured with such great precision that they in fact define very thin curves in the mA,mB plane. On the other hand, some of them are only measured with a rough fractional precision and thus define “thick curves,” or “strips” in the plane of the masses (see, for example, the strips associated with Ṗ , r and s in the case of PSR B1534+12). In any case, the theory is confirmed when all of the strips (thick or thin) have a non-empty common intersection. (One should also note that the strips represented in Figure 4 only use the “one sigma” error bars, in other words a 68% level of confidence. Therefore, the fact that the Ṗ strip for PSR B1534+12 is a little bit disjoint from the intersection of the other strips is not significant: a “two sigma” figure would show excellent agreement between observation and general relativity.) In view of the arguments presented above, all of the tests shown in Figure 4 confirm the validity of general relativity in the regime of strong gravitational fields (hµν ∼ 1). Moreover, the four tests that use measurements of the pa- rameter Ṗ (in the four corresponding systems) are direct experimental confir- mations of the fact that the gravitational interaction propagates at the speed c between the companion and the pulsar. In fact, Ṗ denotes the long-term variation 〈dP/dt〉 of the orbital period. Detailed theoretical calculations of the motion of two gravitationally condensed objects in general relativity, that take into account the effects connected to the propagation of the gravitational inter- action at finite speed[14], have shown that one of the observable effects of this propagation is a long-term decrease in the orbital period given by the formula ṖGR(mA,mB) = − 192 π 1 + 73 e2 + 37 (1− e2)7/2 4In the case of PSR J0737−3039, one of the six measured parameters is the ratio xA/xB between a Keplerian parameter of the pulsar and its analog for the companion, which turns out to also be a pulsar. s ≤ 1 0 0.5 1 1.5 2 2.5 PSR J1141−6545 intersection 0 0.5 1 1.5 2 2.5 PSR B1534+12 intersection 0 0.5 1 1.5 2 2.5 PSR J0737−3039 intersection 0 0.5 1 1.5 2 2.5 2.5 ω s ≤ 1 PSR B1913+16 intersection Figure 4: Tests of general relativity obtained from observations of four binary pulsars. For each binary pulsar one has traced the “curves,” in the plane of the two masses (mA = mass of the pulsar, mB = mass of the companion), defined by equating the theoretical expressions for the various post-Keplerian parameters, as predicted by general relativity, to their observational value, de- termined through a least-squares fit to the parameterized theoretical timing formula. Each “curve” is in fact a “strip,” whose thickness is given by the (one sigma) precision with which the corresponding post-Keplerian parameter is measured. For some parameters, these strips are too thin to be visible. The grey zones would correspond to a sine for the angle of inclination of the or- bital plane with respect to the plane of the sky that is greater than 1, and are therefore physically excluded. The direct physical origin of this decrease in the orbital period lies in the mod- ification, produced by general relativity, of the usual Newtonian law of gravi- tational attraction between two bodies, FNewton = GmAmB/r AB. In place of such a simple law, general relativity predicts a more complicated force law that can be expanded in the symbolic form FEinstein = GmAmB + · · · , (20) where, for example, “v2/c2” represents a whole set of terms of order v2A/c v2B/c 2, vA vB/c 2, or even GmA/c 2 r or GmB/c 2 r. Here vA denotes the speed of body A, vB that of body B, and rAB the distance between the two bod- ies. The term of order v5/c5 in Equation (20) is particularly important. This term is a direct consequence of the finite-speed propagation of the gravitational interaction between A and B, and its calculation shows that it contains a com- ponent that is opposed to the relative speed vA − vB of the two bodies and that, consequently, slows down the orbital motion of each body, causing it to evolve towards an orbit that lies closer to its companion (and therefore has a shorter orbital period). This “braking” term (which is correlated with the emis- sion of gravitational waves), δFEinstein ∼ v5/c5 FNewton, leads to a long-term decrease in the orbital period ṖGR ∼ −(v/c)5 ∼ −10−12 that is very small, but whose reality has been verified with a fractional precision of order 10−3 in PSR B1913+16 and of order 20% in PSR B1534+12 and PSR J1141−6545 [4, 11, 13]. To conclude this brief outline of the tests of relativistic gravitation by binary pulsars, let us note that there is an analog, for the regime of strong gravitational fields, of the formalism of parametrization for possible deviations from general relativity mentioned in Section 6 in the framework of weak gravitational fields (using the post-Newtonian parameters γ−1 and β−1). This analog is obtained by considering a two-parameter family of relativistic theories of gravitation, assuming that the gravitational interaction is propagated not only by a tensor field gµν but also by a scalar field ϕ. Such a class of tensor-scalar theories of gravitation allows for a description of possible deviations in both the solar system and in binary pulsars. It also allows one to explicitly demonstrate that binary pulsars indeed test the effects of strong fields that go beyond the tests of the weak fields of the solar system by exhibiting classes of theories that are compatible with all of the observations in the solar system but that are incompatible with the observations of binary pulsars, see [4, 13]. 10 Gravitational Waves: Propagation, Genera- tion, and Detection As soon as he had finished constructing the theory of general relativity, Ein- stein realized that it implied the existence of waves of geometric deformations of space-time, or “gravitational waves” [15, 2]. Mathematically, these waves are analogs (with the replacement Aµ → hµν) of electromagnetic waves, but concep- tually they signify something remarkable: they exemplify, in the purest possible way, the “elastic” nature of space-time in general relativity. Before Einstein space-time was a rigid structure, given a priori, which was not influenced by the material content of the Universe. After Einstein, a distribution of matter (or more generally of mass-energy) that changes over the course of time, let us say for concreteness a binary system of two neutron stars or two black holes, will not only deform the chrono-geometry of the space-time in its immediate neighborhood, but this deformation will propagate in every possible direction away from the system considered, and will travel out to infinity in the form of a wave whose oscillations will reflect the temporal variations of the matter distri- bution. We therefore see that the study of these gravitational waves poses three separate problems: that of generation, that of propagation, and, finally, that of detection of such gravitational radiation. These three problems are at present being actively studied, since it is hoped that we will soon detect gravitational waves, and thus will be able to obtain new information about the Universe [16]. We shall here content ourselves with an elementary introduction to this field of research. For a more detailed introduction to the detection of gravitational waves see the contribution by Jean-Yves Vinet to this Poincaré seminar. Let us first consider the simplest case of very weak gravitational waves, outside of their material sources. The geometry of such a space-time can be written, as in Section 6, as gµν(x) = ηµν+hµν(x), where hµν ≪ 1. At first order in h, and outside of the source (namely in the domain where Tµν(x) = 0), the perturbation of the geometry, hµν(x), satisfies a homogeneous equation obtained by replacing the right-hand side of Equation (11) with zero. It can be shown that one can simplify this equation through a suitable choice of coordinate system. In a transverse traceless (TT) coordinate system the only non-zero components of a general gravitational wave are the spatial components hTTij , i, j = 1, 2, 3 (in other words hTT00 = 0 = h 0i ), and these components satisfy � hTTij = 0 , ∂j h ij = 0 , h jj = 0 . (21) The first equation in (21), where the wave operator � = ∆ − c−2 ∂2t appears, shows that gravitational waves (like electromagnetic waves) propagate at the speed c. If we consider for simplicity a monochromatic plane wave (hTTij = ζij exp(ik ·x− i ω t)+ complex conjugate, with ω = c |k|), the second equation in (21) shows that the (complex) tensor ζij measuring the polarization of a gravitational wave only has non-zero components in the plane orthogonal to the wave’s direction of propagation: ζij k j = 0. Finally, the third equation in (21) shows that the polarization tensor ζij has vanishing trace: ζjj = 0. More concretely, this means that if a gravitational wave propagates in the z- direction, its polarization is described by a 2 × 2 matrix, ζxx ζxy ζyx ζyy , which is symmetric and traceless. Such a polarization matrix therefore only contains two independent (complex) components: ζ+ ≡ ζxx = −ζyy, and ζ× ≡ ζxy = ζyx. This is the same number of independent (complex) components that an electromagnetic wave has. Indeed, in a transverse gauge, an electromagnetic wave only has spatial components ATi that satisfy �ATi = 0 , ∂j A j = 0 . (22) As in the case above, the first equation (22) means that an electromagnetic wave propagates at the speed c, and the second equation shows that a monochromatic plane electromagnetic wave (ATi = ζi exp(ik · x− i ω t)+ c.c., ω = c |k|) is de- scribed by a (complex) polarization vector ζi that is orthogonal to the direction of propagation: ζj k j = 0. For a wave propagating in the z-direction such a vector only has two independent (complex) components, ζx and ζy. It is in- deed the same number of components that a gravitational wave has, but we see that the two quantities measuring the polarization of a gravitational wave, ζ+ = ζxx = −ζyy, ζ× = ζxy = ζyx are mathematically quite different from the two quantities ζx, ζy measuring the polarization of an electromagnetic wave. However, see Section 11 below. We have here discussed the propagation of a gravitational wave in a back- ground space-time described by the Minkowski metric ηµν . One can also con- sider the propagation of a wave in a curved background space-time, namely by studying solutions of Einstein’s equations (9) of the form gµν(x) = g µν(x) + hµν(x) where hµν is not only small, but varies on temporal and spatial scales much shorter than those of the background metric gBµν(x). Such a study is nec- essary, for example, for understanding the propagation of gravitational waves in the cosmological Universe. The problem of generation consists in searching for the connection between the tensorial amplitude hTTij of the gravitational radiation in the radiation zone and the motion and structure of the source. If one considers the simplest case of a source that is sufficiently diffuse that it only creates waves that are everywhere weak (gµν − ηµν = hµν ≪ 1), one can use the linearized approximation to Ein- stein’s equations (9), namely Equations (11). One can solve Equations (11) by the same technique that is used to solve Maxwell’s equations (12): one fixes the coordinate system by imposing ∂α hαµ − 12 ∂µ h α = 0 (analogous to the Lorentz gauge condition ∂αAα = 0), then one inverts the wave operator by using re- tarded potentials. Finally, one must study the asymptotic form, at infinity, of the emitted wave, and write it in the reduced form of a transverse and traceless amplitude hTTij satisfying Equations (21) (analogous to a transverse electromag- netic wave ATi satisfying (22)). One then finds that, just as charge conservation implies that there is no monopole type electro-magnetic radiation, but only dipole or higher orders of polarity, the conservation of energy-momentum im- plies the absence of monopole and dipole gravitational radiation. For a slowly varying source (v/c≪ 1), the dominant gravitational radiation is of quadrupole type. It is given, in the radiation zone, by an expression of the form hTTij (t, r,n) ≃ [Iij(t− r/c)]TT . (23) Here r denotes the distance to the center of mass of the source, Iij(t) ≡ d3x c−2 T 00(t,x) xixj − 1 x2δij is the quadrupole moment of the mass-energy distri- bution, and the upper index TT denotes an algebraic projection operation for the quadrupole tensor Iij (which is a 3 × 3 matrix) that only retains the part orthogonal to the local direction of wave propagation ni ≡ xi/r with vanish- ing trace (ITTij is therefore locally a (real) 2× 2 symmetric, traceless matrix of the same type as ζij above). Formula (23) (which was in essence obtained by Einstein in 1918 [15]) is only the first approximation to an expansion in powers of v/c, where v designates an internal speed characteristic of the source. The prospect of soon being able to detect gravitational waves has motivated theo- rists to improve Formula (23): (i) by describing the terms of higher order in v/c, up to a very high order, and (ii) by using new approximation methods that allow one to treat sources containing regions of strong gravitational fields (such as, for example, a binary system of two black holes or two neutron stars). See below for the most recent results. Finally, the problem of detection, of which the pioneer was Joseph Weber in the 1960s, is at present giving rise to very active experimental research. The principle behind any detector is that a gravitational wave of amplitude hTTij induces a change in the distance L between two bodies on the order of δL ∼ hL during its passage. One way of seeing this is to consider the action of a wave hTTij on two free particles, at rest before the arrival of the wave at the positions xi1 and x 2 respectively. As we have seen, each particle, in the presence of the wave, will follow a geodesic motion in the geometry gµν = ηµν + hµν (with h00 = h0i = 0 and hij = h ij ). By writing out the geodesic equation, Equation (7), one finds that it simply reduces (at first order in h) to d2xi/ds2 = 0. Therefore, particles that are initially at rest (xi = const.) remain at rest in a transverse and traceless system of coordinates! This does not however mean that the gravitational wave has no observable effect. In fact, since the spatial geometry is perturbed by the passage of the wave, gij(t,x) = δij + h ij (t,x), one finds that the physical distance between the two particles xi1, x 2 (which is observable, for example, by measuring the time taken for light to make a round trip between the two particles) varies, during the passage of the wave, according to L2 = (δij + h ij )(x 2 − xi1)(x 2 − x The problem of detecting a gravitational wave thus leads to the problem of detecting a small relative displacement δL/L ∼ h. By using Formula (23), one finds that the order of magnitude of h, for known or hoped for astrophysical sources (for example,a very close system of two neutron stars or two black holes), situated at distances such that one may hope to see several events per year (r & 600 million light-years), is in fact extremely small: h . 10−22 for signals whose characteristic frequency is around 100 Hertz. Several types of detectors have been developed since the pioneering work of J. Weber [16]. At present, the detectors that should succeed in the near future at detecting amplitudes h ∼ δL/L ∼ 10−22 are large interferometers, of the Michelson or Fabry-Pérot type, having arms that are many kilometers in length into which a very powerful monochromatic laser beam is injected. Such terrestrial interferometric detectors presently exist in the U.S.A. (the LIGO detectors [17]), in Europe (the VIRGO [18] and GEO 600 [19] detectors) and elsewhere (such as the TAMA detector in Japan). Moreover, the international space project LISA [20], made up of an interferometer between satellites that are several million kilometers apart, should allow one to detect low frequency (∼ one hundredth or one thousandth of a Hertz) gravitational waves in a dozen years or so. This collection of gravi- tational wave detectors promises to bring invaluable information for astronomy by opening a new “window” on the Universe that is much more transparent than the various electromagnetic (or neutrino) windows that have so greatly expanded our knowledge of the Universe in the twentieth century. The extreme smallness of the expected gravitational signals has led a num- ber of experimentalists to contribute, over many years, a wealth of ingenuity and know-how in order to develop technology that is sufficiently precise and trustworthy (see [17, 18, 19, 20]). To conclude, let us also mention how much concerted theoretical effort has been made, both in calculating the general rel- ativistic predictions for gravitational waves emitted by certain sources, and in developing methods adapted to the extraction of the gravitational signal from the background noise in the detectors. For example, one of the most promising sources for terrestrial detectors is the wave train for gravitational waves emitted by a system of two black holes, and in particular the final (most intense) portion of this wave train, which is emitted during the last few orbits of the system and the final coalescence of the two black holes into a single, more massive black hole. We have seen above (see Section 9) that the finite speed of propagation of the gravitational interaction between the two bodies of a binary system gives rise to a progressive acceleration of the orbital frequency, connected to the pro- gressive approach of the two bodies towards each other. Here we are speaking of the final stages in such a process, where the two bodies are so close that they orbit around each other in a spiral pattern that accelerates until they attain (for the final “stable” orbits) speeds that become comparable to the speed of light, all the while remaining slightly slower. In order to be able to determine, with a precision that is acceptable for the needs of detection, the dynamics of such a binary black hole system in such a situation, as well as the gravitational amplitude hTTij that it emits, it was necessary to develop a whole ensemble of analytic techniques to a very high level of precision. For example, it was neces- sary to calculate the expansion (20) of the force determining the motion of the two bodies to a very high order and also to calculate the amplitude hTTij of the gravitational radiation emitted to infinity with a precision going well beyond the quadrupole approximation (23). These calculations are comparable in complex- ity to high-order calculations in quantum field theory. Some of the techniques developed for quantum field theory indeed proved to be extremely useful for these calculations in the (classical) theory of general relativity (such as certain resummation methods and the mathematical use of analytic continuation in the number of space-time dimensions). For an entryway into the literature of these modern analytic methods, see [21], and for an early example of a result obtained by such methods of direct interest for the physics of detection see Figure 5 [22], which shows a component of the gravitational amplitude hTTij (t) emitted during the final stages of evolution of a system of two black holes of equal mass. The first oscillations shown in Figure 5 are emitted during the last quasi-circular orbits (accelerated motion in a spiral of decreasing radius). The middle part of the signal corresponds to a phase where, having moved past the last stable orbit, the two black holes “fall” toward each other while spiraling rapidly. In fact, contrarily to Newton’s theory, which predicts that two condensed bodies would be able to orbit around each other with an orbit of arbitrarily small ra- dius (basically up until the point that the two bodies touch), Einstein’s theory predicts a modified law for the force between the two bodies, Equation (20), whose analysis shows that it is so attractive that it no longer allows for sta- ble circular orbits when the distance between the two bodies becomes smaller than around 6G(mA +mB)/c 2. In the case of two black holes, this distance is sufficiently larger than the black hole “radii” (2GmA/c 2 and 2GmB/c 2) that one is still able to analytically treat the beginning of the “spiralling plunge” of the two black holes towards each other. The final oscillations in Figure 5 are emitted by the rotating (and initially highly deformed) black hole formed from the merger of the two initial, separate black holes. −200 −100 0 100 −0.48 −0.38 −0.28 −0.18 −0.08 inspiral + plunge merger + ring−down Figure 5: The gravitational amplitude h(t) emitted during the final stages of evolution of a system of two equal-mass black holes. The beginning of the signal (the left side of the figure), which is sinusoidal, corresponds to an inspiral motion of two separate black holes (with decreasing distance); the middle corresponds to a rapid “inspiralling plunge” of the two black holes towards each other; the end (at right) corresponds to the oscillations of the final, rotating black hole formed from the merger of the two initial black holes. Up until quite recently the analytic predictions illustrated in Figure 5 con- cerning the gravitational signal h(t) emitted by the spiralling plunge and merger of two black holes remained conjectural, since they could be compared to neither other theoretical predictions nor to observational data. Recently, worldwide ef- forts made over three decades to attack the problem of the coalescence of two black holes by numerically solving Einstein’s equations (9) have spectacularly begun to bear fruit. Several groups have been able to numerically calculate the signal h(t) emitted during the final orbits and merger of two black holes [23]. In essence, there is good agreement between the analytical and numerical predictions. In order to be able to detect the gravitational waves emitted by the coalescence of two black holes, it will most likely be necessary to properly combine the information on the structure of the signal h(t) obtained by the two types of methods, which are in fact complementary. 11 General Relativity and Quantum Theory: From Supergravity to String Theory Up until now, we have discussed the classical theory of general relativity, ne- glecting any quantum effects. What becomes of the theory in the quantum regime? This apparently innocent question in fact opens up vast new prospects that are still under construction. We will do nothing more here than to touch upon the subject, by pointing out to the reader some of the paths along which contemporary physics has been led by the challenge of unifying general relativity and quantum theory. For a more complete introduction to the various possi- bilities “beyond” general relativity suggested within the framework of string theory (which is still under construction) one should consult the contribution of Ignatios Antoniadis to this Poincaré Seminar. Let us recall that, from the very beginning of the quasi-definitive formula- tion of quantum theory (1925–1930), the creators of quantum mechanics (Born, Heisenberg, Jordan; Dirac; Pauli; etc.) showed how to “quantize” not only systems with several particles (such as an atom), but also fields, continuous dy- namical systems whose classical description implies a continuous distribution of energy and momentum in space. In particular, they showed how to quantize (or in other words how to formulate within a framework compatible with quantum theory) the electromagnetic field Aµ, which, as we have recalled above, satisfies the Maxwell equations (12) at the classical level. They nevertheless ran into dif- ficulty due to the following fact. In quantum theory, the physics of a system’s evolution is essentially contained in the transition amplitudes A(f, i) between an initial state labelled by i and a final state labelled by f . These amplitudes A(f, i) are complex numbers. They satisfy a “transitivity” property of the type A(f, i) = A(f, n)A(n, i) , (24) which contains a sum over all possible intermediate states, labelled by n (with this sum becoming an integral when there is a continuum of intermediate pos- sible states). R. Feynman used Equation (24) as a point of departure for a new formulation of quantum theory, by interpreting it as an analog of Huy- gens’ Principle: if one thinks of A(f, i) as the amplitude, “at the point f ,” of a “wave” emitted “from the point i,” Equation (24) states that this amplitude can be calculated by considering the “wave” emitted from i as passing through all possible intermediate “points” n (A(n, i)), while reemitting “wavelets” start- ing from these intermediate points (A(f, n)), which then superpose to form the total wave arriving at the “final point f .” Property (24) does not pose any problem in the quantum mechanics of dis- crete systems (particle systems). It simply shows that the amplitude A(f, i) behaves like a wave, and therefore must satisfy a “wave equation” (which is in- deed the case for the Schrödinger equation describing the dependence of A(f, i) on the parameters determining the final configuration f). On the other hand, Property (24) poses formidable problems when one applies it to the quantiza- tion of continuous dynamical systems (fields). In fact, for such systems the “space” of intermediate possible states is infinitely larger than in the case of the mechanics of discrete systems. Roughly speaking, the intermediate possible states for a field can be described as containing ℓ = 1, 2, 3, . . . quantum excita- tions of the field, with each quantum excitation (or pair of “virtual particles”) being described essentially by a plane wave, ζ exp(i kµ x µ), where ζ measures the polarization of these virtual particles and kµ = ηµν kν , with k 0 = ω and ki = k, their angular frequency and wave vector, or (using the Planck-Einstein- de Broglie relations E = ~ω, p = ~k) their energy-momentum pµ = ~ kµ. The quantum theory shows (basically because of the uncertainty principle) that the four-frequencies (and four-momenta) pµ = ~ kµ of the intermediate states cannot be constrained to satisfy the classical equation ηµν p µ pν = −m2 (or in other words E2 = p2 +m2 ; we use c = 1 in this section). As a consequence, the sum over intermediate states for a quantum field theory has the following properties (among others): (i) when ℓ = 1 (an intermediate state containing only one pair of virtual particles, called a one-loop contribution), there is an in- tegral over a four-momentum pµ, d4p = dp; (ii) when ℓ = 2 (two pairs of virtual particles; a two-loop contribution), there is an integral over two four- momenta p 1 , p d4p1 d 4p2; etc. The delicate point comes from the fact that the energy-momentum of an intermediate state can take arbitrarily high values. This possibility is directly connected (through a Fourier transform) to the fact that a field possesses an infinite number of degrees of freedom, corresponding to configurations that vary over arbitrarily small time and length scales. The problems posed by the necessity of integrating over the infinite domain of four-momenta of intermediate virtual particles (or in other words of account- ing for the fact that field configurations can vary over arbitrarily small scales) appeared in the 1930s when the quantum theory of the electromagnetic field Aµ (called quantum electrodynamics, or QED) was studied in detail. These problems imposed themselves in the following form: when one calculates the transition amplitude for given initial and final states (for example the collision of two light quanta, with two photons entering and two photons leaving) by using (24), one finds a result given in the form of a divergent integral, because of the integral (in the one-loop approximation, ℓ = 1) over the arbitrarily large energy-momentum describing virtual electron-positron pairs appearing as pos- sible intermediate states. Little by little, theoretical physicists understood that the types of divergent integrals appearing in QED were relatively benign and, after the second world war, they developed a method (renormalization theory) that allowed one to unambiguously isolate the infinite part of these integrals, and to subtract them by expressing the amplitudes A(f, i) solely as a function of observable quantities [24] (work by J. Schwinger, R. Feynman, F. Dyson etc.). The preceding work led to the development of consistent quantum theories not only for the electromagnetic field Aµ (QED), but also for generalizations of electromagnetism (Yang-Mills theory or non-abelian gauge theory) that turned out to provide excellent descriptions of the new interactions between elementary particles discovered in the twentieth century (the electroweak theory, partially unifying electromagnetism and weak nuclear interactions, and quantum chro- modynamics, describing the strong nuclear interactions). All of these theories give rise to only relatively benign divergences that can be “renormalized” and thus allowed one to compute amplitudes A(f, i) corresponding to observable physical processes [24] (notably, work by G. ’t Hooft and M. Veltman). What happens when we use (24) to construct a “perturbative” quantum theory of general relativity (namely one obtained by expanding in the number ℓ of virtual particle pairs appearing in the intermediate states)? The answer is that the integrals over the four-momenta of intermediate virtual particles are not at all of the benign type that allowed them to be renormalized in the simpler case of electromagnetism. The source of this difference is not accidental, but is rather connected with the basic physics of relativistic gravitation. Indeed, as we have mentioned, the virtual particles have arbitrarily large energies E. Because of the basic relations that led Einstein to develop general relativity, namely E = mi and mi = mg, one deduces that these virtual particles correspond to arbitrarily large gravitational masses mg. They will therefore end up creating intense gravitational effects that become more and more intense as the number ℓ of virtual particle pairs grows. These gravitational interactions that grow without limit with energy and momentum correspond (by Fourier transform) to field configurations concentrated in arbitrarily small space and time scales. One way of seeing why the quantum gravitational field creates much more violent problems than the quantum electromagnetic field is, quite simply, to go back to dimensional analysis. Simple considerations in fact show that the relative (non- dimensional) one-loop amplitude A1 must be proportional to the product ~G and must contain an integral d4k. However, in 1900 Planck had noticed that (in units where c = 1) the dimensions of ~ and G were such that the product ~G had the dimensions of length (or time) squared: ≃ 1.6× 10−33 cm, tP ≡ ≃ 5.4× 10−44 s . (25) One thus deduces that the integral d4k f(k) must have the dimensions of a squared frequency, and therefore that A1 must (when k → ∞) be of the type, A1 ∼ ~G d4k/k2. Such an integral diverges quadratically with the upper limit Λ of the integral (the cutoff frequency, such that |k| ≤ Λ), so that A1 ∼ ~GΛ2 ∼ t2P Λ2. The extension of this dimensional analysis to the intermediate states with several loops (ℓ > 1) causes even more severe polynomial divergences to appear, of a type such that the power of Λ that appears grows without limit with ℓ. In summary, the essential physical characteristics of gravitation (E = mi = mg and the dimension of Newton’s constant G) imply the impossibility of gener- alizing to the gravitational case the methods that allowed a satisfactory quantum treatment of the other interactions (electromagnetic, weak, and strong). Several paths have been explored to get out of this impasse. Some researchers tried to quantize general relativity non-perturbatively, without using an expansion in intermediate states (24) (work by A. Ashtekar, L. Smolin, and others). others have tried to generalize general relativity by adding a fermionic field to Einstein’s (bosonic) gravitational field gµν(x), the gravitino field ψµ(x). It is indeed re- markable that it is possible to define a theory, known as supergravity, that gener- alizes the geometric invariance of general relativity in a profound way. After the 1974 discovery (by J. Wess and B. Zumino) of a possible new global symmetry for interacting bosonic and fermionic fields, supersymmetry (which is a sort of global rotation transforming bosons to fermions and vice versa), D.Z. Freedman, P. van Nieuwenhuizen, and S. Ferrara; and S. Deser and B. Zumino; showed that one could generalize global supersymmetry to a local supersymmetry, meaning that it varies from point to point in space-time. Local supersymmetry is a sort of fermionic generalization (with anti-commuting parameters) of the geometric invariance at the base of general relativity (the invariance under any change in coordinates). The generalization of Einstein’s theory of gravitation that admits such a local supersymmetry is called supergravity theory. As we have mentioned, in four dimensions this theory contains, in addition to the (commuting) bosonic field gµν(x), an (anti-commuting) fermionic field ψµ(x) that is both a space- time vector (with index µ) and a spinor. (It is a massless field of spin 3/2, intermediate between a massless spin 1 field like Aµ and a massless spin 2 field like hµν = gµν − ηµν .) Supergravity was extended to richer fermionic struc- tures (with many gravitinos), and was formulated in space-times having more than four dimensions. It is nevertheless remarkable that there is a maximal dimension, equal to D = 11, admitting a theory of supergravity (the maximal supergravity constructed by E. Cremmer, B. Julia, and J. Scherk). The initial hope underlying the construction of these supergravity theories was that they would perhaps allow one to give meaning to the perturbative calculation (24) of quantum amplitudes. Indeed, one finds for example that at one loop, ℓ = 1, the contributions coming from intermediate fermionic states have a sign oppo- site to the bosonic contributions and (because of the supersymmetry, bosons ↔ fermions) exactly cancel them. Unfortunately, although such cancellations exist for the lowest orders of approximation, it appeared that this was probably not going to be the case at all orders5. The fact that the gravitational interaction constant G has “a bad dimension” remains true and creates non-renormalizable divergences starting at a certain number of loops ℓ. Meanwhile, a third way of defining a consistent quantum theory of gravity was developed, under the name of string theory. Initially formulated as models for the strong interactions (in particular by G. Veneziano, M. Virasoro, P. Ra- mond, A. Neveu, and J.H. Schwarz), the string theories were founded upon the quantization of the relativistic dynamics of an extended object of one spatial di- mension: a “string.” This string could be closed in on itself, like a small rubber band (a closed string), or it could have two ends (an open string). Note that the point of departure of string theory only includes the Poincaré-Minkowski space-time, in other words the metric ηµν of Equation (2), and quantum theory (with the constant ~ = h/2π). In particular, the only symmetry manifest in the classical dynamics of a string is the Poincaré group (3). It is, however, remark- 5Recent work by Z. Bern et al. and M. Green et al., has, however, suggested that such cancellations take place at all orders for the case of maximal supergravity, dimensionally reduced to D = 4 dimensions. able that (as shown by T. Yoneya, and J. Scherk and J.H. Schwarz, in 1974) one of the quantum excitations of a closed string reproduces, in a certain limit, all of the non-linear structure of general relativity (see below). Among the other remarkable properties of string theory [25], let us point out that it is the first physical theory to determine the space-time dimension D. In fact, this theory is only consistent if D = 10, for the versions allowing fermionic excitations (the purely bosonic string theory selects D = 26). The fact that 10 > 4 does not mean that this theory has no relevance to the real world. Indeed, it has been known since the 1930s (from work of T. Kaluza and O. Klein) that a space- time of dimension D > 4 is compatible with experiment if the supplementary (spatial) dimensions close in on themselves (meaning they are compactified) on very small distance scales. The low-energy physics of such a theory seems to take place in a four-dimensional space-time, but it contains new (a priori mass- less) fields connected to the geometry of the additional compactified dimensions. Moreover, recent work (due in particular to I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali) has suggested the possibility that the additional dimensions are compactified on scales that are small with respect to everyday life, but very large with respect to the Planck length. This possibility opens up an entire phenomenological field dealing with the eventual observation of signals coming from string theory (see the contribution of I. Antoniadis to this Poincaré seminar). However, string theory’s most remarkable property is that it seems to avoid, in a radical way, the problems of divergent (non-renormalizable) integrals that have weighed down every direct attempt at perturbatively quantizing gravity. In order to explain how string theory arrives at such a result, we must discuss some elements of its formalism. Recall that the classical dynamics of any system is obtained by minimizing a functional of the time evolution of the system’s configuration, called the action (the principle of least action). For example, the action for a particle of mass m, moving in a Riemannian space-time (6), is proportional to the length of the line that it traces in space-time: S = −m ds. This action is minimized when the particle follows a geodesic, in other words when its equation of motion is given by (7). According to Y. Nambu and T. Goto, the action for a string is S = −T dA, where the parameter T (analogous to m for the particle) is called the string tension, and where dA is the area of the two-dimensional surface traced out by the evolution of the string in the (D-dimensional) space- time in which it lives. In quantum theory, the action functional serves (as shown by R. Feynman) to define the transition amplitude (24). Basically, when one considers two intermediate configurations m and n (in the sense of the right-hand side of (24)) that are close to each other, the amplitude A(n,m) is proportional to exp(i S(n,m)/~), where S(n,m) is the minimal classical action such that the system considered evolves from the configuration labelled by n to that labelled by m. Generalizing the decomposition in (24) by introducing an infinite number of intermediate configurations that lie close to each other, one ends up (in a generalization of Huygens’ principle) expressing the amplitude A(f, i) as a multiple sum over all of the “paths” (in the configuration space of the system studied) connecting the initial state i to the final state f . Each path contributes a term eiφ where the phase φ = S/~ is proportional to the action S corresponding to this “path,” or in other words to this possible evolution of the system. In string theory, φ = −(T/~) dA. Since the phase is a non- dimensional quantity, and dA has the dimension of an area, we see that the quantum theory of strings brings in the quantity ~/T , having the dimensions of a length squared, at a fundamental level. More precisely, the fundamental length of string theory, ℓs, is defined by ℓ2s ≡ α′ ≡ 2 π T . (26) This fundamental length plays a central role in string theory. Roughly speak- ing, it defines the characteristic “size” of the quantum states of a string. If ℓs is much smaller than the observational resolution with which one studies the string, the string will look like a point-like particle, and its interactions will be described by a quantum theory of relativistic particles, which is equivalent to a theory of relativistic fields. It is precisely in this sense that general relativity emerges as a limit of string theory. Since this is an important conceptual point for our story, let us give some details about the emergence of general relativity from string theory. The action functional that is used in practice to quantize a string is not really −T dA, but rather (as emphasized by A. Polyakov) = − 1 4 π ℓ2s −γ γab ∂aXµ ∂bXν ηµν + · · · , (27) where σa, a = 0, 1 are two coordinates that allow an event to be located on the space-time surface (or ‘world-sheet’) traced out by the string within the ambient space-time; γab is an auxiliary metric (dΣ 2 = γab(σ) dσ a dσb) defined on this surface (with γab being its inverse, and γ its determinant); and Xµ(σa) defines the embedding of the string in the ambient (flat) space-time. The dots indicate additional terms, and in particular terms of fermionic type that were introduced by P. Ramond, by A. Neveu and J.H. Schwarz, and by others. If one separates the two coordinates σa = (σ0, σ1) into a temporal coordinate, τ ≡ σ0, and a spatial coordinate, σ ≡ σ1, the configuration “at time τ” of the string is described by the functions Xµ(τ, σ), where one can interpret σ as a curvilinear abscissa describing the spatial extent of the string. If we consider a closed string, one that is topologically equivalent to a circle, the function Xµ(τ, σ) must be periodic in σ. One can show that (modulo the imposition of certain constraints) one can choose the coordinates τ and σ on the string such that dΣ2 = −dτ2+dσ2. Then, the dynamical equations for the string (obtained by minimizing the action (27)) reduce to the standard equation for waves on a string: −∂2Xµ/∂τ2 + ∂2Xµ/∂σ2 = 0. The general solution to this equation describes a superposition of waves travelling along the string in both possible directions: Xµ = X L(τ+σ)+X R(τ−σ). If we consider a closed string (one that is topologically equivalent to a circle), these two types of wave are independent of each other. For an open string (with certain reflection conditions at the endpoints of the string) these two types of waves are connected to each other. Moreover, since the string has a finite length in both cases, one can decompose the left- or right-moving waves X L(τ + σ) or X R(τ − σ) as a Fourier series. For example, for a closed string one may write Xµ(τ, σ) = X 0 (τ) + e−2in(τ−σ) + ãµn√ e−2in(τ+σ) + h.c. (28) Here X 0 (τ) = x µ + 2 ℓ2s p µτ describes the motion of the string’s center of mass, and the remainder describes the decomposition of the motion around the center of mass into a discrete set of oscillatory modes. Like any vibrating string, a rel- ativistic string can vibrate in its fundamental mode (n = 1) or in a “harmonic” of the fundamental mode (for an integer n > 1). In the classical case the com- plex coefficients aµn, ã n represent the (complex) amplitudes of vibration for the modes of oscillation at frequency n times the fundamental frequency. (with aµn corresponding to a wave travelling to the right, while ãµn corresponds to a wave travelling to the left.) When one quantizes the string dynamics the position of the string Xµ(τ, σ) becomes an operator (acting in the space of quantum states of the system), and because of this the quantities xµ, pµ, aµn and ã n in (28) be- come operators. The notation h.c. signifies that one must add the hermitian conjugates of the oscillation terms, which will contain the operators (aµn) † and (ãµn) †. (The notation † indicates hermitian conjugation, in other words the oper- ator analog of complex conjugation.) One then finds that the operators xµ and pµ describing the motion of the center of mass satisfy the usual commutation re- lations of a relativistic particle, [xµ, pµ] = i ~ ηµν , and that the operators aµn and ãµn become annihilation operators, like those that appear in the quantum theory of any vibrating system: [aµn, (a †] = ~ ηµν δnm, [ã n, (ã †] = ~ ηµν δmn. In the case of an open string, one only has one set of oscillators, let us say aµn. The discussion up until now has neglected to mention that the oscillation am- plitudes aµn, ã n must satisfy an infinite number of constraints (connected with the equation obtained by minimizing (27) with respect to the auxiliary metric γab). One can satisfy these by expressing two of the space-time components of the oscillators aµn, ã n (for each n) as a function of the other. Because of this, the physical states of the string are described by oscillators ain, ã n where the index i only takes D−2 values in a space-time of dimension D. Forgetting this subtlety for the moment (which is nevertheless crucial physically), let us conclude this discussion by summarizing the spectrum of a quantum string, or in other words the ensemble of quantum states of motion for a string. For an open string, the ensemble of quantum states describes the states of motion (the momenta pµ) of an infinite collection of relativistic particles, having squared massesM2 = −ηµν pµ pν equal to (N−1) m2s, whereN is a non-negative integer andms ≡ ~/ℓs is the fundamental mass of string theory associated to the fundamental length ℓs. For a closed string, one finds another “infinite tower” of more and more massive particles, this time with M2 = 4(N − 1)m2s. In both cases the integer N is given, as a function of the string’s oscillation amplitudes (travelling to the right), by n ηµν(a † aνn . (29) In the case of a closed string one must also satisfy the constraint N = Ñ where Ñ is the operator obtained by replacing aµn by ã n in (29). The preceding result essentially states that the (quantized) internal energy of an oscillating string defines the squared mass of the associated particle. The presence of the additional term −1 in the formulae given above for M2 means that the quantum state of minimum internal energy for a string, that is, the “vacuum” state |0〉 where all oscillators are in their ground state, aµn | 0〉 = 0, corresponds to a negative squared mass (M2 = −m2s for the open string and M2 = −4m2s for the closed string). This unusual quantum state (a tachyon) cor- responds to an instability of the theory of bosonic strings. It is absent from the more sophisticated versions of string theory (“superstrings”) due to F. Gliozzi, J. Scherk, and D. Olive, to M. Green and J.H. Schwarz, and to D. Gross and collaborators. Let us concentrate on the other states (which are the only ones that have corresponding states in superstring theory). One then finds that the first possible physical quantum states (such that N = 1) describe some massless particles. In relativistic quantum theory it is known that any particle is the quantized excitation of a corresponding field. Therefore the massless particles that appear in string theory must correspond to long-range fields. To know which fields appear in this way one must more closely examine which possible combinations of oscillator excitations a 1 , a 2 , a 3 , . . ., appearing in Formula (29), can lead to N = 1. Because of the factor n in (29) multiplying the harmonic contribution of order n to the mass squared, only the oscillators of the fun- damental mode n = 1 can give N = 1. One then deduces that the internal quantum states of massless particles appearing in the theory of open strings are described by a string oscillation state of the form † | 0〉 . (30) On the other hand, because of the constraint N = Ñ = 1, the internal quantum states of the massless particles appearing in the theory of closed strings are described by a state of excitation containing both a left-moving oscillation and a right-moving oscillation: ζµν(a † (ãν1) † | 0〉 . (31) In Equations (30) and (31) the state |0〉 denotes the ground state of all oscillators (aµn | 0〉 = ãµn | 0〉 = 0). The state (30) therefore describes a massless particle (with momentum sat- isfying ηµν p µ pν = 0), possessing an “internal structure” described by a vector polarization ζµ. Here we recognize exactly the definition of a photon, the quan- tum state associated with a wave Aµ(x) = ζµ exp(i kλ x λ), where pµ = ~ kµ. The theory of open strings therefore contains Maxwell’s theory. (One can also show that, because of the constraints briefly mentioned above, the polarization ζµ must be transverse, k µ ζµ = 0, and that it is only defined up to a gauge transformation: ζ′µ = ζµ + a kµ.) As for the state (31), this describes a massless particle (ηµν p µ pν = 0), possessing an “internal structure” described by a tensor polarization ζµν . The plane wave associated with such a particle is therefore of the form h̄µν(x) = ζµν exp(i kλ x λ), where pµ = ~ kµ. As in the case of the open string, one can show that ζµν must be transverse, ζµν k ν = 0 and that it is only defined up to a gauge transformation, ζ′µν = ζµν+kµ aν+kν bµ. We here see the same type of structure appear that we had in general relativity for plane waves. However, here we have a structure that is richer than that of general relativity. Indeed, since the state (31) is obtained by combining two independent states of oscillation, (a † and (ã †, the polarization tensor ζµν is not constrained to be symmetric. Moreover it is not constrained to have vanishing trace. Therefore, if we decompose ζµν into its possible irreducible parts (a symmetric traceless part, a symmetric part with trace, and an antisymmetric part) we find that the field h̄µν(x) associated with the massless states of a closed string decomposes into: (i) a field hµν(x) (the graviton) representing a weak gravitational wave in general relativity, (ii) a scalar field Φ(x) (called the dilaton), and (iii) an antisymmetric tensor field Bµν(x) = −Bνµ(x) subject to the gauge invariance B′µν(x) = Bµν(x) + ∂µ aν(x) − ∂ν aµ(x). Moreover, when one studies the non- linear interactions between these various fields, as described by the transition amplitudes A(f, i) in string theory, one can show that the field hµν(x) truly represents a deformation of the flat geometry of the background space-time in which the theory was initially formulated. Let us emphasize this remarkable result. We started from a theory that studied the quantum dynamics of a string in a rigid background space-time. This theory predicts that certain quantum excitations of a string (that propagate at the speed of light) in fact represent waves of deformation of the space-time geometry. In intuitive terms, the “elas- ticity” of space-time postulated by the theory of general relativity appears here as being due to certain internal vibrations of an elastic object extended in one spatial dimension. Another suggestive consequence of string theory is the link suggested by the comparison between (30) and (31). Roughly, Equation (31) states that the internal state of a closed string corresponding to a graviton is constructed by taking the (tensor) product of the states corresponding to photons in the theory of open strings. This unexpected link between Einstein’s gravitation (gµν) and Maxwell’s theory (Aµ) translates, when we look at interactions in string theory, into remarkable identities (due to H. Kawai, D.C. Lewellen, and S.-H.H. Tye) between the transition amplitudes of open strings and those of closed strings. This affinity between electromagnetism, or rather Yang-Mills theory, and gravitation has recently given rise to fascinating conjectures (due to A. Polyakov and J. Maldacena) connecting quantum Yang-Mills theory in flat space-time to quasi-classical limits of string theory and gravitation in curved space-time. Einstein would certainly have been interested to see how classical general relativity is used here to clarify the limit of a quantum Yang-Mills theory. Having explained the starting point of string theory, we can outline the in- tuitive reason for which this theory avoids the problems with divergent integrals that appeared when one tried to directly quantize gravitation. We have seen that string theory contains an infinite tower of particles whose masses grow with the degree of excitation of the string’s internal oscillators. The gravita- tional field appears in the limit that one considers the low energy interactions (E ≪ ms) between the massless states of the theory. In this limit the gravi- ton (meaning the particle associated with the gravitational field) is treated as a “point-like” particle. When we consider more complicated processes (at one loop, ℓ = 1, see above), virtual elementary gravitons could appear with arbitrar- ily high energy. It is these virtual high-energy gravitons that are responsible for the divergences. However, in string theory, when we consider any intermediate process whatsoever where high energies appear, it must be remembered that this high intermediate energy can also be used to excite the internal state of the virtual gravitons, and thus reveal that they are “made” from an extended string. An analysis of this fact shows that string theory introduces an effective truncation of the type E . ms on the energies of exchanged virtual particles. In other words, the fact that there are no truly “point-like” particles in string theory, but only string excitations having a characteristic length ∼ ℓs, elimi- nates the problem of infinities connected to arbitrarily small length and time scales. Because of this, in string theory one can calculate the transition ampli- tudes corresponding to a collision between two gravitons, and one finds that the result is given by a finite integral [25]. Up until now we have only considered the starting point of string theory. This is a complex theory that is still in a stage of rapid development. Let us briefly sketch some other aspects of this theory that are relevant for this exposé centered around relativistic gravitation. Let us first state that the more sophis- ticated versions of string theory (superstrings) require the inclusion of fermionic oscillators bµn, b̃ n, in addition to the bosonic oscillators a n, ã n introduced above. One then finds that there are no particles of negative mass-squared, and that the space-time dimension D must be equal to 10. One also finds that the mass- less states contain more states than those indicated above. In fact, one finds that the fields corresponding to these states describe the various possible theo- ries of supergravity in D = 10. Recently (in work by J. Polchinski) it has also been understood that string theory contains not only the states of excitation of strings (in other words of objects extended in one spatial direction), but also the states of excitation of objects extended in p spatial directions, where the integer p can take other values than 1. For example, p = 2 corresponds to a membrane. It even seems (according to C. Hull and P. Townsend) that one should recognize that there is a sort of “democracy” between several different values for p. An object extended in p spatial directions is called a p-brane. In general, the masses of the quantum states of these p-branes are very large, be- ing parametrically higher than the characteristic mass ms. However, one may also consider a limit where the mass of certain p-branes tends towards zero. In this limit, the fields associated with these p-branes become long-range fields. A surprising result (by E. Witten) is that, in this limit, the infinite tower of states of certain p-branes (in particular for p = 0) corresponds exactly to the infinite tower of states that appear when one considers the maximal supergravity in D = 11 dimensions, with the eleventh (spatial) dimension compactified on a circle (that is to say with a periodicity condition on x11). In other words, in a certain limit, a theory of superstrings in D = 10 transforms into a theory that lives in D = 11 dimensions! Because of this, many experts in string theory believe that the true definition of string theory (which is still to be found) must start from a theory (to be defined) in 11 dimensions (known as “M -theory”). We have seen in Section 8 that one point of contact between relativistic grav- itation and quantum theory is the phenomenon of thermal emission from black holes discovered by S.W. Hawking. String theory has shed new light upon this phenomenon, as well as on the concept of black hole “entropy.” The essential question that the calculation of S.W. Hawking left in the shadows is: what is the physical meaning of the quantity S defined by Equation (19)? In the ther- modynamic theory of ordinary bodies, the entropy of a system is interpreted, since Boltzmann’s work, as the (natural) logarithm of the number of micro- scopic states N having the same macroscopic characteristics (energy, volume, etc.) as the state of the system under consideration: S = logN . Bekenstein had attempted to estimate the number of microscopic internal states of a macroscop- ically defined black hole, and had argued for a result such that logN was on the order of magnitude of A/~G, but his arguments remained indirect and did not allow a clear meaning to be attributed to this counting of microscopic states. Work by A. Sen and by A. Strominger and C. Vafa, as well as by C.G. Callan and J.M. Maldacena has, for the first time, given examples of black holes whose microscopic description in string theory is sufficiently precise to allow for the calculation (in certain limits) of the number of internal quantum states, N . It is therefore quite satisfying to find a final result for N whose logarithm is pre- cisely equal to the expression (19). However, there do remain dark areas in the understanding of the quantum structure of black holes. In particular, the string theory calculations allowing one to give a precise statistical meaning to the en- tropy (19) deal with very special black holes (known as extremal black holes, which have the maximal electric charge that a black hole with a regular horizon can support). These black holes have a Hawking temperature equal to zero, and therefore do not emit thermal radiation. They correspond to stable states in the quantum theory. One would nevertheless also like to understand the detailed internal quantum structure of unstable black holes, such as the Schwarzschild black hole (17), which has a non-zero temperature, and which therefore loses its mass little by little in the form of thermal radiation. What is the final state to which this gradual process of black hole “evaporation” leads? Is it the case that an initial pure quantum state radiates all of its initial mass to transform itself entirely into incoherent thermal radiation? Or does a Schwarzschild black hole transform itself, after having obtained a minimum size, into something else? The answers to these questions remain open to a large extent, although it has been argued that a Schwarzschild black hole transforms itself into a highly massive quantum string state when its radius becomes on the order of ℓs [26]. We have seen previously that string theory contains general relativity in a certain limit. At the same time, string theory is, strictly speaking, infinitely richer than Einstein’s gravitation, for the graviton is nothing more than a partic- ular quantum excitation of a string, among an infinite number of others. What deviations from Einstein’s gravity are predicted by string theory? This question remains open today because of our lack of comprehension about the connection between string theory and the reality observed in our everyday environment (4-dimensional space-time; electromagnetic, weak, and strong interactions; the spectrum of observed particles; . . .). We shall content ourselves here with out- lining a few possibilities. (See the contribution by I. Antoniadis for a discussion of other possibilities.) First, let us state that if one considers collisions between gravitons with energy-momentum k smaller than, but not negligible with respect to, the characteristic string mass ms, the calculations of transition amplitudes in string theory show that the usual Einstein equations (in the absence of mat- ter) Rµν = 0 must be modified, by including corrections of order (k/ms) 2. One finds that these modified Einstein equations have the form (for bosonic string theory) Rµν + ℓ2s Rµαβγ R ν + · · · = 0 , (32) where �ναβ ≡ ∂α Γ νβ + Γ νβ − ∂β Γµνα − Γ να , (33) denotes the “curvature tensor” of the metric gµν . (the quantity Rµν defined in Section 5 that appears in Einstein’s equations in an essential way is a “trace” of this tensor: Rµν = R �µσν .) As indicated by the dots in (32), the terms written are no more than the two first terms of an infinite series in growing powers of ℓ2s ≡ α′. Equation (32) shows how the fact that the string is not a point, but is rather extended over a characteristic length ∼ ℓs, modifies the Einsteinian description of gravity. The corrections to Einstein’s equation shown in (32) are nevertheless completely negligible in most applications of general relativity. In fact, it is expected that ℓs is on the order of the Planck scale ℓp, Equation (25). More precisely, one expects that ℓs is on the order of magnitude of 10 −32 cm. (Nevertheless, this question remains open, and it has been recently suggested that ℓs is much larger, and perhaps on the order of 10 −17 cm.) If one assumes that ℓs is on the order of magnitude of 10 −32 cm (and that the extra dimensions are compactified on distances scales on the order of ℓs), the only area of general relativistic applications where the modifications shown in (32) should play an important role is in primordial cosmology. Indeed, close to the initial singularity of the Big Bang (if it exists), the “curvature” Rµναβ becomes extremely large. When it reaches values comparable to ℓ−2s the infinite series of corrections in (32) begins to play a role comparable to the first term, discovered by Einstein. Such a situation is also found in the interior of a black hole, when one gets very close to the singularity (see Figure 3). Unfortunately, in such situations, one must take the infinite series of terms in (32) into account, or in other words replace Einstein’s description of gravitation in terms of a field (which corresponds to a point-like (quantum) particle) by its exact stringy description. This is a difficult problem that no one really knows how to attack today. However, a priori string theory predicts more drastic low energy (k ≪ ms) modifications to general relativity than the corrections shown in (32). In fact, we have seen in Equation (31) above that Einsteinian gravity does not appear alone in string theory. It is always necessarily accompanied by other long-range fields, in particular a scalar field Φ(x), the dilaton, and an antisymmetric ten- sor Bµν(x). What role do these “partners” of the graviton play in observable reality? This question does not yet have a clear answer. Moreover, if one recalls that (super)string theory must live in a space-time of dimension D = 10, and that it includes the D = 10 (and eventually the D = 11) theory of supergravity, there are many other supplementary fields that add themselves to the ten com- ponents of the usual metric tensor gµν (in D = 4). It is conceivable that all of these supplementary fields (which are massless to first approximation in string theory) acquire masses in our local universe that are large enough that they no longer propagate observable effects over macroscopic scales. It remains possible, however, that one or several of these fields remain (essentially) massless, and therefore can propagate physical effects over distances that are large enough to be observable. It is therefore of interest to understand what physical effects are implied, for example, by the dilaton Φ(x) or by Bµν(x). Concerning the latter, it is interesting to note that (as emphasized by A. Connes, M. Douglas, and A. Schwartz), in a certain limit, the presence of a background Bµν(x) has the effect of deforming the space-time geometry in a “non-commutative” way. This means that, in a certain sense, the space-time coordinates xµ cease to be sim- ple real (commuting) numbers in order to become non-commuting quantities: xµxν − xνxµ = εµν where εµν = −ενµ is connected to a (uniform) background Bµν . To conclude, let us consider the other obligatory partner of the graviton gµν(x), the dilaton Φ(x). This field plays a central role in string theory. In fact, the average value of the dilaton (in the vacuum) determines the string theory coupling constant, gs = e Φ. The value of gs in turn determines (along with other fields) the physical coupling constants. For example, the gravitational coupling constant is given by a formula of the type ~G = ℓ2s(g s + · · · ) where the dots denote correction terms (which can become quite important if gs is not very small). Similarly, the fine structure constant, α = e2/~c ≃ 1/137, which deter- mines the intensity of electromagnetic interactions is a function of g2s . Because of these relations between the physical coupling constants and gs (and therefore the value of the dilaton; gs = e Φ), we see that if the dilaton is massless (or in other words is long-range), its value Φ(x) at a space-time point x will depend on the distribution of matter in the universe. For example, as is the case with the gravitational field (for example g00(x) ≃ −1 + 2GM/c2r), we expect that the value of Φ(x) depends on the masses present around the point x, and should be different at the Earth’s surface than it is at a higher altitude. One may also expect that Φ(x) would be sensitive to the expansion of the universe and would vary over a time scale comparable to the age of the universe. However, if Φ(x) varies over space and/or time, one concludes from the relations shown above between gs = e Φ and the physical coupling constants that the latter must also vary over space and/or time. Therefore, for example, the value, here and now, of the fine structure constant α could be slightly different from the value it had, long ago, in a very distant galaxy. Such effects are accessible to detailed astronomical observations and, in fact, some recent observations have suggested that the interaction constants were different in distant galaxies. However, other experimental data (such as the fossil nuclear reactor at Oklo and the isotopic composition of ancient terrestrial meteorites) put very severe limits on any vari- ability of the coupling “constants.” Let us finally note that if the fine structure “constant” α, as well as other coupling “constants,” varies with a massless field such as the dilaton Φ(x), then this implies a violation of the basic postulate of general relativity: the principle of equivalence. In particular, one can show that the universality of free fall is necessarily violated, meaning that bodies with dif- ferent nuclear composition would fall with different accelerations in an external gravitational field. This gives an important motivation for testing the principle of equivalence with greater precision. For example, the MICROSCOPE space mission [27] (of the CNES) should soon test the universality of free fall to the level of 10−15, and the STEP space project (Satellite Test of the Equivalence Principle) [28] could reach the level 10−18. Another interesting phenomenological possibility is that the dilaton (and/or other scalar fields of the same type, called moduli) acquires a non-zero mass that is however very small with respect to the string mass scale ms. One could then observe a modification of Newtonian gravitation over small distances (smaller than a tenth of a millimeter). For a discussion of this theoretical possibility and of its recent experimental tests see, respectively, the contributions by I. Anto- niadis and J. Mester to this Poincaré seminar. 12 Conclusion For a long time general relativity was admired as a marvellous intellectual con- struction, but it only played a marginal role in physics. Typical of the appraisal of this theory is the comment by Max Born [29] made upon the fiftieth an- niversary of the annus mirabilis: “The foundations of general relativity seemed to me then, and they still do today, to be the greatest feat of human thought concerning Nature, the most astounding association of philosophical penetra- tion, physical intuition, and mathematical ability. However its connections to experiment were tenuous. It seduced me like a great work of art that should be appreciated and admired from a distance.” Today, one century after the annus mirabilis, the situation is quite different. General relativity plays a central role in a large domain of physics, including everything from primordial cosmology and the physics of black holes to the observation of binary pulsars and the definition of international atomic time. It even has everyday practical applications, via the satellite positioning sys- tems (such as the GPS and, soon, its European counterpart Galileo). Many ambitious (and costly) experimental projects aim to test it (G.P.B., MICRO- SCOPE, STEP, . . .), or use it as a tool for deciphering the distant universe (LIGO/VIRGO/GEO, LISA, . . .). The time is therefore long-gone that its con- nection with experiment was tenuous. Nevertheless, it is worth noting that the fascination with the structure and physical implications of the theory evoked by Born remains intact. One of the motivations for thinking that the theory of strings (and other extended objects) holds the key to the problem of the unification of physics is its deep affinity with general relativity. Indeed, while the attempts at “Grand Unification” made in the 1970s completely ignored the gravitational interaction, string theory necessarily leads to Einstein’s fundamen- tal concept of a dynamical space-time. At any rate, it seems that one must more deeply understand the “generalized quantum geometry” created through the in- teraction of strings and p-branes in order to completely formulate this theory and to understand its hidden symmetries and physical implications. Einstein would no doubt appreciate seeing the key role played by symmetry principles and gravity within modern physics. References [1] A. Einstein, Zur Elektrodynamik bewegter Körper, Annalen der Physik 17, 891 (1905). [2] See http://www.einstein.caltech.edu for an entry into the Einstein Col- lected Papers Project. The French reader will have access to Einstein’s main papers in Albert Einstein, Œuvres choisies, Paris, Le Seuil/CNRS, 1993, under the direction of F. Balibar. See in particular Volumes 2 (Rel- ativités I) and 3 (Relativités II). One can also consult the 2005 Poincaré seminar dedicated to Einstein (http://www.lpthe.jussieu.fr/poincare): Einstein, 1905-2005, Poincaré Seminar 2005, edited by T. Damour, O. Darrigol, B. Duplantier and V. Rivasseau (Birkhäuser Verlag, Basel, Suisse, 2006). See also the excellent summary article by D. Giulini and N. Straumann, “Einstein’s impact on the physics of the twentieth cen- tury,” Studies in History and Philosophy of Modern Physics 37, 115- 173 (2006). For online access to many of Einstein’s original articles and to documents about him, see http://www.alberteinstein.info/. We also note that most of the work in progress on general relativity can be consulted on various archives at http://xxx.lanl.gov, in particular the archive gr-qc. Review articles on certain sub-fields of general relativ- ity are accessible at http://relativity.livingreviews.org. Finally, see T. Damour, Once Upon Einstein, A K Peters Ltd, Wellesley, 2006, for a recent non-technical account of the formation of Einstein’s ideas. [3] Galileo, Dialogues Concerning Two New Sciences, translated by Henry Crew and Alfonso di Salvio, Macmillan, New York, 1914. [4] The reader interested in learning about recent experimental tests of gravitational theories may consult, on the internet, ei- ther the highly detailed review by C.M. Will in Living Re- views (http://relativity.livingreviews.org/Articles/lrr-2001-4) or the brief review by T. Damour in the Review of Particle Physics http://www.einstein.caltech.edu http://www.lpthe.jussieu.fr/poincare http://www.alberteinstein.info/ http://xxx.lanl.gov http://relativity.livingreviews.org http://relativity.livingreviews.org/Articles/lrr-2001-4 (http://pdg.lbl.gov/). See also John Mester’s contribution to this Poincaré seminar. [5] A. Einstein, Die Feldgleichungen der Gravitation, Sitz. Preuss. Akad. Wiss., 1915, p. 844. [6] The reader wishing to study the formalism and applications of general relativity in detail can consult, for example, the following works: L. Landau and E. Lifshitz, The Classical Theory of Fields, Butterworth-Heinemann, 1995; S. Weinberg, Gravitation and Cos- mology, Wiley, New York, 1972; H.C. Ohanian and R. Ruffini, Gravitation and Spacetime, Second Edition, Norton, New York, 1994; N. Straumann, General Relativity, With Applications to Astro- physics, Springer Verlag, 2004. Let us also mention detailed course notes on general relativity by S.M. Carroll, available on the in- ternet: http://pancake.uchicago.edu/∼carroll/notes/∼; as well as at gr-qc/9712019. Finally, let us mention the recent book (in French) on the history of the discovery and reception of general relativity: J. Eisen- staedt, Einstein et la relativité générale, CNRS, Paris, 2002. [7] B. Bertotti, L. Iess, and P. Tortora, A Test of General Relativity Using Radio Links with the Cassini Spacecraft, Nature 425, 374 (2003). [8] http://einstein.stanford.edu [9] W. Israel, Dark stars: the evolution of an idea, in 300 Years of Grav- itation, edited by S.W. Hawking and W. Israel, Cambridge University Press, Cambridge, 1987, Chapter 7, pp. 199-276. [10] The discovery of binary pulsars is related in Hulse’s Nobel Lecture: R.A. Hulse, Reviews of Modern Physics 66, 699 (1994). [11] For an introduction to the observational characteristics of pulsars, and their use in testing relativistic gravitation, see Taylor’s Nobel Lecture: J.H. Taylor, Reviews of Modern Physics 66, 711 (1994). See also Michael Kramer’s contribution to this Poincaré seminar. [12] For an update on the observational characteristics of pulsars, and their use in testing general relativity, see the Living Review by I.H. Stairs, available at http://relativity.livingreviews.org/Articles/lrr-2003-5/ and the contribution by Michael Kramer to this Poincaré seminar. [13] For a recent update on tests of relativistic gravitation (and of tensor- scalar theories) obtained through the chronometry of binary pulsars, see G. Esposito-Farèse, gr-qc/0402007 (available on the general relativity and quantum cosmology archive at the address http://xxx.lanl.gov), and T. Damour and G. Esposito-Farèse, in preparation. Figure 4 is adapted from these references. http://pdg.lbl.gov/ http://pancake.uchicago.edu/~carroll/notes/~ http://arxiv.org/abs/gr-qc/9712019 http://einstein.stanford.edu http://relativity.livingreviews.org/Articles/lrr-2003-5/ http://arxiv.org/abs/gr-qc/0402007 http://xxx.lanl.gov [14] For a review of the problem of the motion of two gravitationally con- densed bodies in general relativity, up to the level where the effects con- nected to the finite speed of propagation of the gravitational interaction appear, see T. Damour, The problem of motion in Newtonian and Ein- steinian gravity, in 300 Years of Gravitation, edited by S.W. Hawking and W. Israel, Cambridge University Press, Cambridge, 1987, Chapter 6, pp. 128-198. [15] A. Einstein, Näherungsweise Integration der Feldgleichungen der Gravitation, Sitz. Preuss. Akad. Wiss., 1916, p. 688 ; ibidem, Über Grav- itationswellen, 1918, p. 154. [16] For a highly detailed introduction to these three problems, see K.S. Thorne Gravitational radiation, in 300 Years of Gravitation, edited by S.W. Hawking and W. Israel, Cambridge University Press, Cam- bridge, 1987, Chapter 9, pp. 330-458. [17] http://www.ligo.caltech.edu/ [18] http://www.virgo.infn.it/ [19] http://www/geo600.uni-hanover.de/ [20] http://lisa.jpl.nasa.gov/ [21] L. Blanchet et al., gr-qc/0406012 ; see also the Living Review by L. Blanchet, available at http://relativity.livingreviews.org/Articles. [22] Figure 5 is adapted from work by A. Buonanno and T. Damour, gr-qc/0001013. [23] F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005), gr-qc/0507014; M. Campanelli et al., Phys. Rev. Lett. 96, 111101 (2006), gr-qc/0511048 J. Baker et al., Phys. Rev. D 73, 104002 (2006), gr-qc/0602026. [24] For a particularly clear exposé of the development of the quantum the- ory of fields, see, for example, the first chapter of S. Weinberg, The Quantum Theory of Fields, volume 1, Foundations, Cambridge Univer- sity Press, Cambridge, 1995. [25] For an introduction to the theory of (super)strings see http://superstringtheory.com/. For a detailed (and technical) in- troduction to the theory see the books: K. Becker, M. Becker, and J.H. Schwarz, String Theory and M-theory: An Introduction, Cambridge University Press, Cambridge, 2006; B. Zwiebach, A First Course in String Theory, Cambridge University Press, Cambridge, 2004; M.B. Green, J.H. Schwarz et E. Witten, Superstring theory, 2 vol- umes, Cambridge University Press, Cambridge, 1987 ; and J. Polchinski, String Theory, 2 volumes, Cambridge University Press, Cambridge, http://www.ligo.caltech.edu/ http://www.virgo.infn.it/ http://www/geo600.uni-hanover.de/ http://lisa.jpl.nasa.gov/ http://arxiv.org/abs/gr-qc/0406012 http://relativity.livingreviews.org/Articles http://arxiv.org/abs/gr-qc/0001013 http://arxiv.org/abs/gr-qc/0507014 http://arxiv.org/abs/gr-qc/0511048 http://arxiv.org/abs/gr-qc/0602026 http://superstringtheory.com/ 1998. To read review articles or to research this theory as it develops see the hep-th archive at http://xxx.lanl.gov. To search for information on the string theory literature (and more generally that of high-energy physics) see also the site http://www.slac.stanford.edu/spires/find/hep. [26] For a detailed introduction to black hole physics see P.K. Townsend, gr-qc/9707012; for an entry into the vast literature on black hole en- tropy, see, for example, T. Damour, hep-th/0401160 in Poincaré Sem- inar 2003, edited by Jean Dalibard, Bertrand Duplantier, and Vincent Rivasseau (Birkhäuser Verlag, Basel, 2004), pp. 227-264. [27] http://www.onera.fr/microscope/ [28] http://www.sstd.rl.ac.uk/fundphys/step/. [29] M. Born, Physics and Relativity, in Fünfzig Jahre Relativitätstheorie, Bern, 11-16 Juli 1955, Verhandlungen, edited by A. Mercier and M. Kervaire, Helvetica Physica Acta, Supplement 4, 244-260 (1956). http://xxx.lanl.gov http://www.slac.stanford.edu/spires/find/hep http://arxiv.org/abs/gr-qc/9707012 http://arxiv.org/abs/hep-th/0401160 http://www.onera.fr/microscope/ http://www.sstd.rl.ac.uk/fundphys/step/ Introduction Special Relativity The Principle of Equivalence Gravitation and Space-Time Chrono-Geometry Einstein's Equations: Elastic Space-Time The Weak-Field Limit and the Newtonian Limit The Post-Newtonian Approximation and Experimental Confirmations in the Regime of Weak and Quasi-Stationary Gravitational Fields Strong Gravitational Fields and Black Holes Binary Pulsars and Experimental Confirmations in the Regime of Strong and Radiating Gravitational Fields Gravitational Waves: Propagation, Generation, and Detection General Relativity and Quantum Theory: From Supergravity to String Theory Conclusion

0704.0755

A procedure for finding the k-th power of a matrix

published elec. at www.maplesoft.com A PROCEDURE FOR FINDING THE kTH POWER OF A MATRIX Branko Malešević Faculty of Electrical Engineering, University of Belgrade, Serbia [email protected] Ivana Jovović PhD student, Faculty of Mathematics, University of Belgrade, Serbia [email protected] 1 Introduction This worksheet demonstrates the use of Maple in Linear Algebra. We give a new procedure (PowerMatrix) in Maple for finding the kth power of n-by-n square matrix A, in a symbolic form, for any positive integer k, k ≥ n. The algorithm is based on an application of Cayley-Hamilton theorem. We used the fact that the entries of the matrix Ak satisfy the same recurrence relation which is determined by the characteristic polynomial of the matrix A (see [1]). The order of these recurrences is n− d, where d is the lowest degree of the characteristic polynomial of the matrix A. For non-singular matrices the procedure can be extended for k not only a positive integer. 2 Initialization > restart: with(LinearAlgebra): 2.1 Procedure Definition 2.1.1 PowerMatrix Input data are a square matrix A and a parameter k. Elements of the matrix A can be numbers and/or parameters. The parameter k can take numeric value or be a symbol. The output data is the kth power of the matrix. The procedure PowerMatrix is as powerful as the procedure rsolve. > PowerMatrix := proc(A::Matrix,k) local i,j,m,r,q,n,d,f,P,F,C; P := x->CharacteristicPolynomial(A,x); n := degree(P(x),x); d := ldegree(P(x),x); http://arxiv.org/abs/0704.0755v2 http://www.maplesoft.com/ mailto:[email protected] mailto:[email protected] F := (i,j)->rsolve(sum(coeff(P(x),x,m)*f(m+q),m=0..n)=0,seq(f(r)=(A^r)[i,j], r=d+1..n),f); C := q->Matrix(n,n,F); if (type(k,integer)) then return(simplify(A^k)) elif (Determinant(A)=0 and not type(k,numeric)) then printf("The %ath power of the matrix for %a>=%d:", k,k,n) elif (Determinant(A)=0 and type(k,numeric)) then return(simplify(A^k)) fi; return(simplify(subs(q=k,C(q)))); 3 Examples 3.1 Example 1. > A := Matrix([[4,-2,2],[-5,7,-5],[-6,6,-4]]); 4 −2 2 −5 7 −5 −6 6 −4 > PowerMatrix(A,k); −2k + 2 · 3k 2(1+k) − 2 · 3k −2(1+k) + 2 · 3k −5 · 3k + 5 · 2k 5 · 3k − 4 · 2k −5 · 3k + 5 · 2k 6 · 2k − 6 · 3k −6 · 2k + 6 · 3k −6 · 3k + 7 · 2k > Determinant(A); > B := A^(-1); > PowerMatrix(B,k); −2(−k) + 2 · 3(−k) 2(1−k) − 2 · 3(−k) −2(1−k) + 2 · 3(−k) −5 · 3(−k) + 5 · 2(−k) 5 · 3(−k) − 4 · 2(−k) −5 · 3(−k) + 5 · 2(−k) −6 · 3(−k) + 6 · 2(−k) −6 · 2(−k) + 6 · 3(−k) −6 · 3(−k) + 7 · 2(−k) 3.2 Example 2. > A := Matrix([[1-p,p],[p,1-p]]); 1− p p p 1− p > PowerMatrix(A,k); (1− 2 p)k (1− 2 p)k (1− 2 p)k (1− 2 p)k The example is from [4], page 272, exercise 19. 3.3 Example 3. > A := Matrix([[a,b,c],[d,e,f],[g,h,i]]); a b c d e f g h i > PowerMatrix(A,k)[1,1]; R = RootOf( (gbf + hdc + iea − gce − hfa− idb) Z3 + (gc + hf + db − ie − ia − ea) Z2 + (i+ e + a) Z − 1 ) R2ie− R2hf − Re− Ri+ 1 (3 R2gbf + 3 R2hdc+ 3 R2iea −3 R2gce− 3 R2hfa− 3 R2idb+ 2 Rgc+ 2 Rhf + 2 Rdb− 2 Rie− 2 Ria− 2 Rea+ i+ e+ a) R # Warning! In this example MatrixPower and MatrixFuction procedures cannot be done in real-time. # MatrixPower(A,k)[1,1]; # MatrixFunction(A,v^k,v)[1,1]; 3.4 Example 4. > A := Matrix([[0,0,1,0,1],[1,0,0,0,1],[0,0,0,1,1],[0,1,0,0,1],[1,1,1,1,0]]); 0 0 1 0 1 1 0 0 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 1 0 > PowerMatrix(A,k)[1,5]; Replace ’:’ with ’;’ and see result! > MatrixPower(A,k)[1,5]: > assume(m::integer):simplify(MatrixPower(A,k)[1,5]): The example is from [3], page 101. 3.5 Example 5. and Example 6. Pay attention what happens for singular matrices. 3.5.1 Example 5. > A := Matrix([[0,2,1,3],[0,0,-2,4],[0,0,0,5],[0,0,0,0]]); 0 2 1 3 0 0 −2 4 0 0 0 5 0 0 0 0 > PowerMatrix(A,2); 0 0 −4 13 0 0 0 −10 0 0 0 0 0 0 0 0 > PowerMatrix(A,3); 0 0 0 −20 0 0 0 0 0 0 0 0 0 0 0 0 > PowerMatrix(A,k); The kth power of the matrix A for k ≥ 4: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 > MatrixPower(A,k); Error, (in LinearAlgebra:-LA_Main:-MatrixPower) power k is not defined for this Matrix > MatrixFunction(A,v^k,v); Error, (in LinearAlgebra:-LA_Main:-MatrixFunction) Matrix function vk is not defined for this Matrix The example is from [2], page 151, exercise 23. 3.5.2 Example 6. > A := Matrix([[1,1,1,0],[1,1,1,-1],[0,0,-1,1],[0,0,1,-1]]); 1 1 1 0 1 1 1 −1 0 0 −1 1 0 0 1 −1 > PowerMatrix(A,k); > The kth power of the matrix for k ≥ 4: 2(−1+k) 2(−1+k) (−1)(1+k) · 2k 5 · 2k (−1)k · 2k 2(−1+k) 2(−1+k) 5 · 2k 5 · (−1)(1+k) · 2k 5 · (−1)k · 2k 0 0 (−1)k · 2(−1+k) (−1)(1+k) · 2(−1+k) 0 0 (−1)(1+k) · 2(−1+k) (−1)k · 2(−1+k) > MatrixPower(A,k); Error, (in LinearAlgebra:-LA_Main:-MatrixPower) power k is not defined for this Matrix > MatrixFunction(A,v^k,v); Error, (in LinearAlgebra:-LA_Main:-MatrixFunction) Matrix function vk is not defined for this Matrix 4 References [1] BrankoMalešević: Some combinatorial aspects of the composition of a set of functions, NSJOM 2006 (36), 3-9, URLs: http://www.im.ns.ac.yu/NSJOM/Papers/36 1/NSJOM 36 1 003 009.pdf, http://arxiv.org/abs/math.CO/0409287. [2] JohnB. Johnston, G.BaleyPrice, Fred S.Van Vleck: Linear Equations and Matrices, Addi- son-Wesley, 1966. [3] Carl D.Meyer: Matrix Analysis and Applied Linear Algebra Book and Solutions Manual SIAM, 2001. [4] Robert Messer: Linear Algebra Gateway to Mathematics, New York, Harper-Collins College Publisher, 1993. 5 Conclusions This procedure has an educational character. It is an interesting demonstration for finding the kth power of a matrix in a symbolic form. Sometimes, it gives solutions in the better form than the existing procedure MatrixPower (see example 4.). See also example 5. and example 6., where we consider singular matrices. In these cases the procedure MatrixPower does not give a solution. The procedure PowerMatrix calculates the kth power of any singular matrices. In some examples it is possible to get a solution in the better form with using the procedure allvalues (see example 3.). Legal Notice: The copyright for this application is owned by the authors. Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities. http://www.im.ns.ac.yu/NSJOM/Papers/36_1/NSJOM_36_1_003_009.pdf http://arxiv.org/abs/math.CO/0409287 Introduction Initialization Procedure Definition PowerMatrix Examples Example 1. Example 2. Example 3. Example 4. Example 5. and Example 6. Example 5. Example 6. References Conclusions

0704.0756

The dynamical Casimir effect in braneworlds

The dynamical Casimir effect in braneworlds Ruth Durrer∗ and Marcus Ruser† Département de Physique Théorique, Université de Genève, 24 quai Ernest Ansermet, CH–1211 Genève 4, Switzerland In braneworld cosmology the expanding Universe is realized as a brane moving through a warped higher-dimensional spacetime. Like a moving mirror causes the creation of photons out of vacuum fluctuations, a moving brane leads to graviton production. We show that, very generically, KK- particles scale like stiff matter with the expansion of the Universe and can therefore not represent the dark matter in a warped braneworld. We present results for the production of massless and Kaluza-Klein (KK) gravitons for bouncing branes in five-dimensional Anti de Sitter space. We find that for a realistic bounce the back reaction from the generated gravitons will be most likely relevant. This Letter summarizes the main results and conclusions from numerical simulations which are presented in detail in a long paper [M. Ruser and R. Durrer, Phys. Rev. D 76, 104014 (2007), arXiv:0704.0790]. PACS numbers: 04.50.+h, 11.10.Kk, 98.80.Cq Introduction: String theory, the most serious can- didate for a quantum theory of gravity, predicts the ex- istence of ’branes’, i.e. hypersurfaces in the 10- (or 11- ) dimensional spacetime on which ordinary matter, e.g. gauge particles and fermions, are confined. Gravitons can move freely in the ’bulk’, the full higher dimensional spacetime [1]. The scenario, where our Universe moves through a five- dimensional Anti de Sitter (AdS) spacetime has been especially successful in reproducing the observed four- dimensional behavior of gravity. It has been shown that at sufficiently low energies and large scales, not only grav- ity on the brane looks four dimensional [2], but also cos- mological expansion can be reproduced [3]. We shall con- centrate here on this example and comment on behavior which may survive in other warped braneworlds. We consider the following situation: A fixed ’static brane’ is sitting in the bulk. The ’physical brane’, our Universe, is first moving away from the AdS Cauchy hori- zon, approaching the second brane. This motion corre- sponds to a contracting Universe. After a closest en- counter the physical brane turns around and moves away from the static brane. This motion mimics the observed expanding Universe. The moving brane acts as a time-dependent boundary for the 5D bulk leading to production of gravitons from vac- uum fluctuations in the same way a moving mirror causes photon creation from vacuum in dynamical cavities [4]. Apart from massless gravitons, braneworlds allow for a tower of Kaluza-Klein (KK) gravitons which appear as massive particles on the brane leading possibly to phe- nomenological consequences. We postulate, that high energy stringy physics will lead to a turnaround of the brane motion, i.e., provoke a re- pulsion of the physical brane from the static one. This ∗Electronic address: [email protected] †Electronic address: [email protected] motion is modeled by a kink where the brane velocity changes sign. As we shall see, a perfect kink leads to divergent particle production due to its infinite acceler- ation. We therefore assume that the kink is rounded off at the string scale Ls. Then particles with energies E > Es = 1/Ls are not generated. This setup represents a regular ’bouncing Universe’ as, for example the ’ekpy- rotic Universe’ [5]. Four-dimensional bouncing Universes have also been studied in Ref. [6]. Moving brane in AdS5: Our starting point is the met- ric of AdS5 in Poincaré coordinates: ds2 = gABdx AdxB = −dt2 + δijdxidxj + dy2 The physical brane (our Universe) is located at some time-dependent position y = yb(t), while the static brane is at a fixed position y = ys > yb(t). The scale factor on the brane is a(η) = yb(t) , dη = 1− v2dt = γ−1dt , v = dyb where we have introduced the brane velocity v and the conformal time η on the brane. If v ≪ 1, the junction conditions lead to the Friedmann equations on the brane. For reviews see [7, 8]. Defining the string and Planck scales by κ5 ≡ L3s and κ4 ≡ L2Pl the Randall-Sundrum (RS) fine tuning condition [2] implies . (2) We assume that the brane energy density is dominated by a radiation component. The contracting (t < 0) and expanding (t > 0) phases are then described by a(t) = |t|+ tb , yb(t) = |t|+ tb , (3) v(t) = − sign(t)L (|t|+ tb)2 ≃ −HL (4) http://arxiv.org/abs/0704.0756v3 http://arxiv.org/abs/0704.0790 mailto:[email protected] mailto:[email protected] where H = (da/dη)/a2 is the Hubble parameter and we have used that η ≃ t if v ≪ 1. A small velocity also requires yb(t) ≪ L. The transition from contraction to expansion is approximated by a kink at t = 0, such that at the moment of the bounce |v(0)| ≡ vb = , ab = a(0) = , H2b = . (5) Tensor perturbations: We now consider tensor per- turbations hij on this background, ds2 = −dt2 + (δij + 2hij)dxidxj + dy2 . (6) For each polarization, their amplitude h satisfies the Klein-Gordon equation in AdS5 [8] ∂2t + k 2 − ∂2y + h(t, y;k) = 0 (7) where k = |k| is the momentum parallel to the brane and h is subject to the boundary (2nd junction) conditions (v∂t + ∂y)h|yb(t) = 0 → ∂yh|yb(t) = 0 and ∂yh|ys = 0 . Being interested in late-time (low energy) effects, we have approximated the first of those conditions by a Neumann condition (v ≪ 1). Then, the spatial part of Eq. (7) together with (8) forms a Strum-Liouville problem at any given time and therefore has a complete orthonormal set of eigenfunctions {φα(t, y)}∞α=0. These ’instantaneous’ mode functions are given by φ0(t) = ysyb(t) y2s − y2b (t) . (9) φn(t, y) = Nn(t)y 2C2(mn(t), yb(t), y) with Cν(m,x, y) = Y1(mx)Jν(my)−J1(mx)Yν (my) (10) and satisfy [−∂2y + (3/y)∂y]φα(y) = m2αφα(y) as well as (8). Nn is a time-dependent normalization condition. More details can be found in [9]. The massless mode φ0 represents the ordinary four-dimensional graviton on the brane, while the massive modes are KK gravitons. Their masses are quantized by the boundary condition at the static brane which requires C1(mn, yb, ys) = 0. At late times and for large n the KK masses are roughly given by mn ≃ nπ/ys. The gravity wave amplitude h may now be decomposed as [9] h(t, y;k) = qα,k(t)φα(t, y) (11) where the prefactor assures that the variables qα,k are canonically normalized. Their time evolution is deter- mined by the brane motion [cf. Eq. (14)]. Localization of gravity: From the above expressions and using L/yb(t) = a(t), we can determine the late- time behavior of the mode functions φα on the brane (yb ≪ L ≪ ys) φ0(t, yb) → , φn(t, yb) → . (12) At this point we can already make two crucial ob- servations: First, the mass mn is a comoving mass. The instantaneous energy of a KK graviton is ωn,k = k2 +m2n, where k denotes comoving wave number. The ’physical mass’ of a KK mode measured by an ob- server on the brane with cosmic time dτ = adt is therefore mn/a, i.e. the KK masses are redshifted with the expan- sion of the Universe. This comes from the fact that mn is the wave number corresponding to the y direction with respect to the bulk time t which corresponds to conformal time η on the brane and not to physical time. It implies that the energy of KK particles on a moving AdS brane is redshifted like that of massless particles. From this alone we would expect the energy density of KK modes on the brane decays like 1/a4. But this is not all. In contrast to the zero mode which behaves as φ0(t, yb) ∝ 1/a the KK-mode functions φn(t, yb) decay as 1/a 2 with the expansion of the Uni- verse and scale like 1/ ys. Consequently the amplitude of the KK modes on the brane dilutes rapidly with the expansion of the Universe and is in general smaller the larger ys. This can be understood by studying the prob- ability of finding a KK-graviton at position y in the bulk which turns out to be much larger in regions of less warp- ing than in the vicinity of the physical brane[9]. If KK gravitons are present on the brane, they escape rapidly into the bulk, i.e., the moving brane looses them, since their wave function is repulsed away from the brane. This causes the additional 1/a-dependence of φn(t, yb) com- pared to φ0(t, yb). The 1/ ys-dependence expresses the fact that the larger the bulk the smaller the probabil- ity to find a KK-graviton at the position of the moving brane. This behavior reflects the localization of gravity: traces of the five-dimensional nature of gravity like KK gravitons become less and less ’visible’ on the brane as time evolves. As a consequence, the energy density of KK gravitons at late times on the brane behaves as ρKK ∝ 1/a6 . (13) It means that KK gravitons redshift like stiff matter and cannot be the dark matter in an AdS braneworld since their energy density does not have the required 1/a3 be- havior. They also do not behave like dark radiation [7, 8] as one might naively expect. This new result is derived in detail in Ref. [9]. It is based on the calculation of 〈ḣ2(t, yb,k)〉 ∝ 〈q̇2α,k(t)〉φ2α(t, yb) where the bracket in- corporates a quantum expectation value with respect to a well-defined initial vacuum state and averaging over several oscillations of the field [9]. An overdot denotes the derivative with respect to t. The scaling behaviour (13) is due to φ2α(t, yb) only, since graviton production from vacuum fluctuations has ceased at late times (like in radiation domination) which is necessary for a mean- ingful particle definition. Then, 〈q̇2α,k(t)〉 is related to the number of produced gravitons and is constant in time. In case that amplification of tensor perturbation is still on- going, e.g., during a de Sitter phase, the energy density related to the massive modes might scale differently. The scaling behavior (13) remains valid also when the fixed brane is sent off to infinity and we end up with a single braneworld in AdS5, like in the Randall-Sundrum II sce- nario [2]. The situation is not altered if we replace the graviton by a scalar or vector degree of freedom in the bulk. Since every bulk degree of freedom must satisfy the five-dimensional Klein-Gordon equation, the mode func- tions will always be the functions φα, and the energy density of the KK-modes decays like 1/a6. KK particles on a brane moving through an AdS bulk cannot play the role of dark matter. It is important here that we consider a static bulk and the time depencence of the brane comes solely from its motion through the bulk. In Ref. [10] the situation of a fixed brane in a time-dependent bulk is discussed. There it is shown that under certain assumption (separability of the y and t dependence of fluctuations), the energy density of KK modes on a low energy cosmological brane does scale like 1/a3 which seems to be in contradiction with our result. However, the approximations used in [10] lead to a system of equations governing the expansion of the Universe but neglecting the time dependence of the bulk. The situation is then effectively four dimensional even for the KK modes; effects of the fifth dimension like the possibility of KK gravitons escaping into the bulk seem to be lost in this approach. In our case we would have a similar situation if we keep the expansion on the brane a(t) but take the position of the brane in the bulk as static yb(t) = const, which is not consistent with the general relation yb(t) = L/a(t). [For a fixed physical mass M = m/a, if we neglect the time dependence of φn(yb(t)) ∝ 1/a2 we also obtain an energy density for this mass proportional to 1/a3.] Particle production: The equation of motion for the canonical variables qα,k is of the form, see Ref. [9], q̈α,k + ω α,kqα,k = β 6=α Mαβ q̇β,k + Nαβqβ,k . (14) Here ωα,k = k2 +m2α is the frequency of the mode and M andN are coupling matrices. When we quantize these variables, gravitons can be created by two effects: First, the time dependence of the effective frequency (ωeffα,k) ω2α,k−Nαα and second, the time dependence of the mode couplings described by the antisymmetric matrix M and the off-diagonal part of N . Note that Equation (14) is derived from the corre- sponding action for the variables qα,k rather than from the wave equation (7) itself. In this way the approxi- mated boundary conditions (8) can be implemented con- sistently [9, 11]. In the technical paper [9] we have studied graviton pro- duction provoked by a brane moving according to (3) in great detail numerically. We have found that for long wavelengths, kL ≪ 1, the zero mode is mainly generated by its self-coupling, i.e. the time dependence of its ef- fective frequency. One actually finds that N00 ∝ δ(t), so that there is an instability at the moment of the kink which leads to particle creation, and the number of 4D- gravitons is given by 2vb/(kL) 2. This is specific to ra- diation dominated expansion where H2a2 = −∂η(Ha). For another expansion law we would also obtain particle creation during the contraction and expansion phases. Light KK gravitons are produced mainly via their cou- pling to the zero mode. This behavior changes drastically for short wavelengths kL ≫ 1. Then the evolution of the zero mode couples strongly to the KK modes and pro- duction of 4D gravitons via the decay of KK modes takes place. In this case the number of produced 4D gravitons decays only like ∝ 1/(kL). Results and discussion: The numerical simulations have revealed a multitude of interesting effects. In the following we summarize the main findings. We refer the interested reader to Ref. [9] for an extensive discussion. For the zero-mode power spectrum we find on scales kL ≪ 1 on which we observe cosmological fluctuations (Mpc or larger) P0(k) = k2 if kt ≪ 1 (La)−2 if kt ≫ 1 . (15) The spectrum of tensor perturbations is blue on super- horizon scales as one would expect for an ekpyrotic sce- nario. On cosmic microwave background scales the am- plitude of perturbations is of the order of (H0/mPl) 2 and hence unobservably small. Calculating the energy density of the produced massless gravitons one obtains [9] ρh0 ≃ . (16) Comparing this with the radiation energy density, ρrad = (3/(κ4L 2))a−4, the RS fine-tuning condition leads to the simple relation ρh0/ρrad ≃ vb/2. (17) The nucleosynthesis bound [12] requests ρh0 <∼ 0.1ρrad, which implies vb ≤ 0.2, justifying our low energy ap- proach. The model is not severely constrained by the zero-mode. More stringent bounds come from the KK modes. Their energy density on the brane is found to be ρKK ≃ . (18) This result is dominated by high energy KK gravitons which are produced due to the kink. It is reasonable to require that the KK-energy density on the brane be (much) smaller than the radiation density at all times, and in particular, right after the bounce where ρKK is greatest. If this is not satisfied, back reaction cannot be neglected. We obtain with ρrad(0) = 3H b /κ4 a=a(0)=1/ ≃ 100 v3b . (19) If we use the largest value for the brane velocity vb ad- mitted by the nucleosynthesis bound vb ≃ 0.2 and re- quire that ρKK/ρrad be (much) smaller than one for back- reaction effects to be negligible, we obtain the very strin- gent condition . (20) Taking the largest allowed value for L ≃ 0.1mm, the RS fine-tuning condition Eq. (2) determines Ls = (LL 1/3 ≃ 10−22mm ≃ 1/(106TeV) and (L/Ls) 2 ≃ 1042 so that ys > L(L/Ls)2 ≃ 1041mm ∼ 1016Mpc. This is about 12 orders of magnitude larger than the present Hubble scale. Also, since yb(t) ≪ L in the low energy regime, and ys ≫ L according to the inequality (20), the physical brane and the static brane need to be far apart at all times otherwise back reaction is not negligible. This situation is probably not very realistic. We need some high energy, stringy effects to provoke the bounce and these may well be relevant only when the branes are sufficiently close, i.e. at a distance of order Ls. But in this case the constraint (20) will be violated which implies that back reaction will be relevant. On the other hand, if we want that ys ≃ L and back reaction to be unimportant, then Eq. (19) implies that the bounce velocity has to be exceedingly small, vb <∼ 10−15. One might first hope to find a way out of these conclusions by allowing the bounce to happen in the high energy regime. But then vb ≃ 1 and the nucleosynthesis bound is violated since too many zero-mode gravitons are being produced. Clearly our low energy approach looses its justification if vb ≃ 1, but it seems unlikely that modifications coming from the high energy regime alleviate the bounds. Conclusions: Studying graviton production in an AdS braneworld we have found the following. First, the energy density of KK gravitons on the brane behaves as ∝ 1/a6, i.e. it scales like stiff matter with the expansion of the Universe and can therefore not serve as a can- didate for dark matter. Furthermore, if gravity looks four dimensional on the brane, its higher-dimensional aspects, like the KK modes, are repelled from the brane. Even if KK gravitons are produced on the brane they rapidly escape into the bulk as time evolves, leaving no traces of the underlying higher-dimensional nature of gravity. This is likely to survive also in other warped braneworlds when expansion can be mimicked by brane motion. Secondly, a braneworld bouncing at low energies is not constrained by massless 4D gravitons and satisfies the nucleosynthesis bound as long as vb <∼ 0.2. However, for interesting values of the string and AdS scales and the largest admitted bounce velocity the back reaction of the KK modes is only negligible if the two branes are far apart from each other at all times, which seems rather unrealistic. For a realistic bounce the back reaction from KK modes can most likely not be neglected. Even if the energy density of the KK gravitons on the brane dilutes rapidly after the bounce, the corresponding energy density in the bulk could even lead to important changes of the bulk geometry. The present model seems to be adequate to address the back reaction issue since the creation of KK gravitons happens exclusively at the bounce. This and the treatment of the high energy regime vb ≃ 1 is reserved for future work. We thank Kazuya Koyama for discussions. This work is supported by the Swiss National Science Foundation. [1] J. Polchinski, String theory. An introduction to the bosonic string, Vol. I, and String theory. Superstring the- ory and beyond, Vol. II , Cambridge University Press (1998). J. Polchinski, Phys. Rev. Lett. 75, 4724 (1995), hep- th/9910219. [2] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999), hep-th/9905221; 83, 4690, hep-th/9906064 [3] P. Binetruy, C. Deffayet, U. Ellwanger, and D. Langlois, Phys. Lett. B477, 285 (2000), hep-th/9910219. [4] M. Ruser, Phys. Rev. A73, 043811 (2006); J. Phys. A39, 6711 (2006), and references therein. [5] J. Khoury, P. Steinhardt and N. Turok, Phys. Rev. Lett. 92, 031302 (2004), hep-th/0307132; Phys. Rev. Lett. 91 161301 (2003), astro-ph/0302012. [6] R. Durrer and F. Vernizzi, Phys.Rev.D66 083503 (2002), hep-ph/0203275; C. Cartier, E. Copeland and R. Durrer, Phys. Rev. D67, 103517 (2003), hep-th/0301198. [7] R. Maartens, Living Rev. Rel. 7, 7 (2004), gr-qc/0312059. [8] R. Durrer, Braneworlds, at the XI Brazilian School of Cosmology and Gravitation, Edt. M. Novello and S.E. Perez Bergliaffa, AIP Conference Proceedings, 782 (2005), hep-th/0507006. [9] M. Ruser and R. Durrer, Phys. Rev. D 76, 104014 (2007), arXiv:0704.0790. [10] M. Minamitsuji, M. Sasaki and D. Langlois, Phys. Rev. D71, 084019 (2005). [11] C. Cartier, R. Durrer and M. Ruser, Phys. Rev. D72, 104018 (2005). [12] M. Maggiore. Phys. Rept. 331, 283 (2000).

0704.0757

Bounds on Negativity of Superpositions

Bounds on Negativity of Superpositions Yong-Cheng Ou and Heng Fan Institute of Physics, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China The entanglement quantified by negativity of pure bipartite superposed states is studied. We show that if the entanglement is quantified by the concurrence two pure states of high fidelity to one another still have nearly the same entanglement. Furthermore this conclusion can be guaranteed by our obtained inequality, and the concurrence is shown to be a continuous function even in infinite dimensions. The bounds on the negativity of superposed states in terms of those of the states being superposed are obtained. These bounds can find useful applications in estimating the amount of the entanglement of a given pure state. PACS numbers: 03.67.Mn, 03.65.Ta, 03.65.Ud Quantum entanglement plays an important role both in many aspects of quantum information theory[1] and in describing quantum phase transition in quantum many- body systems[2, 3]. As such characterization quantifi- cation of quantum entanglement is a fundamental issue. Consequently the legitimate measures of entanglement are desirable as a first step. The existing well-known bi- partite measure of entanglement with an elegant formula is the concurrence derived analytically by Wootters[4] and the entanglement of formation[5, 6] is a monoton- ically increasing function of the concurrence. In general for a multipartite or higher-dimensional system it is a formidable task of quantifying its entanglement since it needs complicate convex-roof extension. In the last 10 years some important properties of quantum entangle- ment were found, one of which is the monogamy prop- erty described by Coffman-Kundu-Wootters inequality in terms of concurrence [7]. In our previous work we have shown that the monogamy inequality can not general- ize to higher-dimensional systems[8] and established a monogamy inequality in terms of negativity giving a dif- ferent residual entanglement[9]. On the other hand, quantum entanglement is a direct consequence of the superposition principle. It is an in- teresting physical phenomenon that the superposition of two separable states may give birth to an entangled state, on the contrary, the superposition of two entangled states may give birth to a separable state. The relation between the entanglement of the state and the entanglement of the individual terms that by superposition yield the state has been studied, where the entanglement is quantified by the von Neumann entropy[10] and the concurrence[12]. Re- cently it was generalized to the superposition of more than two components[13]. If the entanglement is quan- tified by negativity, it would be interesting to establish the analogous relation and obtain the bound of entangle- ment for the superposition state. In this paper, we first show that, by contrast to the von Neumann entropy, the concurrence is a continuous function even in infinite di- mensions. We deduce an inequality to guarantee this property. Next we give the bounds of the negativity of the superposition state. The discussion and conclusion are presented in the end. The authors in[10] have shown that two states of high fidelity to one another may not have the same entan- glement, i.e., |〈ψ|φ〉|2 → 1 may not generally result in E(ψ) → E(φ), where E is the von Neumann entropy. For a bipartite pure state |Φ〉AB the von Neumann en- tropy is defined as E(ΦAB) ≡ S(TrB|Φ〉AB〈Φ|) = S(TrA|Φ〉AB〈Φ|), (1) where S(ρ) = −Tr(ρ log ρ), and the concurrence is de- fined as C(ΦAB) ≡ 2 (1− Trρ2 , (2) where ρA = TrB|Φ〉AB〈Φ| with the eigenvalues µi. How- ever, if we employ the concurrence to quantify the entan- glement, |〈ψ|φ〉|2 → 1 must result in C(ψ) → C(φ). Let us see their example letting |φ〉AB = |00〉, (3) |ψ〉AB = 1− ǫ|φ〉AB + [|11〉+ |22〉+ · · ·+ |dd〉]. (4) It is obviously true that E(φAB) = C(φAB) = 0, while according to [10] the von Neumann entropy of the state |ψ〉AB is E(ψAB) ≈ ǫ log2 d→ ∞, (5) specially when d is as large as we expect. It follows from Eq.(2) that the concurrence of the state |ψ〉AB give us the result C2(ψAB) = 2 2ǫ− ǫ2 − ǫ → 0, (6) when ǫ is adequately small. By contrast to E(ψAB) in Eq.(5), C2(ψAB) in Eq.(6) is independent of d. Note that when ǫ is small the two states have high fidelity |〈ψ|φ〉|2 = 1 − ǫ → 1. Comparing Eq.(5) to Eq.(6), we can draw a http://arxiv.org/abs/0704.0757v1 conclusion that if the entanglement is quantified by the concurrence two states of high fidelity to one another still have nearly the same entanglement. It is indeed that the difference of the von Neumann en- tropy between two pure states of fixed dimension can be bounded using Fannes’ inequality[11], while the von Neu- mann entropy is not a continuous function and no such bound applies in infinite dimensions. However, as we will show here, a similar bound still works if the entanglement is quantified by the concurrence and the concurrence is a continuous function even in infinite dimensions. In or- der to explain our above viewpoint we present the fol- lowing Theorem which is similar to the original Fannes’ inequality except that the entanglement is quantified by the concurrence. Theorem 1. Suppose ρAB and σAB are density matri- ces of two bipartite pure states in arbitrary dimensions. For the trace distance T (ρA, σA) ≡ Tr|ρA − σB| between ρA = TrBρAB and σA = TrBσAB we have |C2(ρAB)− C2(σAB)| ≤ 4T (ρA, σA). (7) Proof. Let r1 ≥ r2 ≥ · · · ≥ rd be the eigenvalues of ρA, in decreasing order, and s1 ≥ s2 ≥ · · · ≥ sd be the eigenvalues of σA, also in decreasing order. According to[1], it follows that |ri − si| ≤ T (ρA, σA). (8) From the observation of the definition of the concurrence in Eq.(2), we can rewrite the left-hand-side of Eq.(7) as ∣∣C2(ρAB)− C2(σAB) ∣∣ = 2 ∣∣∣∣∣ (r2i − s2i ) ∣∣∣∣∣ ∣∣r2i − s2i |ri + si||ri − si| |ri − si|. (9) The second formula is obtained from the observation that |a+ b+ · · ·+ k| ≤ |a| + |b| + · · · + |k| for any complex quantities a, b, · · ·, k. In the derivation of the last formula we have taken into account the fact that ri + si ≤ 2 since each eigenvalue of ri and si is not greater than one. Combining Eqs.(8) and (9) can give Eq.(7). Thus the proof is completed. From the Theorem 1 it can be seen that the difference of the concurrences of two pure states is a function of fidelity and can be bounded by Eq.(7). What’s more, by contrast to the von Neumann entropy[10] the concurrence is a continuous function and such a bound still works in infinite dimensions. Note that whether a similar bound in Eq.(7) holds for the negativity is still open. In the next paragraphs we are devoted to deducing the bounds on the negativity of any bipartite pure state as a superposition of two terms |Γ〉AB = α|Ψ〉+ β|Φ〉. Before embarking on this study, we first recall some basic definitions of the negativity. As for detecting en- tangled state in higher-dimensional Hilbert space, Peres- Horodecki criterion based on partial transpose[15, 16] is a convenient method. Given a density matrix ρ in a bi- partite pure system of A and B, the partial transpose with respect to A subsystem is described by (ρTA)ij,kl = (ρ)kj,il and the negativity is defined as N = 1 (‖ρTA‖ − 1). (10) The trace norm ‖R‖ is given by ‖R‖ = Tr RR†. Note that N > 0 is the necessary and sufficient condition for entangled bipartite pure states. There are two key ingredients to obtain the bounds of the negativity for bipartite superposition pure states. One is that the negativity can be expressed by means of Schmidt coefficients of a pure state. Suppose that a pure m⊗ n(m ≤ n) quantum state has the standard Schmidt form |ψ〉AB = µi|aibi〉, where µi(i = 1, · · · ,m) are the Schmidt coefficients, ai and bi are the orthogonal basis in HA and HB, respectively. For the pure bipartite state we can derive ‖ρTA‖ = [18], and there- fore Eq.(10) can be reexpressed as N = 1  . (11) In order for the later use we can transform Eq.(11) into = 2N + 1. (12) The other is the Theorem[17], which states that for any two Hermitian matrix H and K defined in Cn×n, µi(H) + µ1(K) ≤ µi(H +K) ≤ µi(H) + µn(K), (13) holds, where µi(·) are the eigenvalues in increasing order. If µ1(K) ≥ 0, from Eq.(13) it is easy to check that µi(H) ≤ µi(H +K) ≤ µi(H) + µn(K), (14) holds also. Then Eq.(14) will be used repeatedly in what follows. For the negativity of the arbitrary superposition state let us first see the simplest case in which two bi- partite states we are superposing, Φ1 and Ψ1, are biorthogonal[10], i.e., Φ1Ψ = Ψ1Φ = 0[12]. Since the matrix representation of a reduced density matrix will be used, we explain the corresponding notations in the following. For the pure state |Φ〉AB defined in m ⊗ n dimensions, generally it can be considered as a vector: |Φ〉AB = [a00, a01, · · · , a0m, a10, a11, · · · , amn]T with the superscript T denoting transpose operation. With the matrix notation, the reduced density matrix reads ρA = ΦΦ †, (15) whose eigenvalues are µi appearing in Eq.(11). Theorem 2. Suppose that two biorthogonal pure states Φ1 and Ψ1, which are defined in m ⊗ n(n ≤ m) di- mensions. The negativity of their superposed states Γ1 = αΦ1 + βΨ1 with |α2|+ |β|2 = 1 satisfies 2|α|2N (Φ1) + 2|β|2N (Ψ1)− 1 ≤ N (αΦ1 + βΨ1) ≤ 2|α|2Ñ (Φ1) + 2|β|2Ñ (Ψ1)− 1 , (16) where Ñ (Φ1) = N (Φ1) + µn(Ψ1)[2N (Φ1) + 1] |α| + n2|β|2µn(Ψ1) 2|α|2 , Ñ (Ψ1) = N (Ψ1) + µn(Φ1)[2N (Ψ1) + 1] |β| + n2|α|2µn(Φ1) 2|β|2 . Proof. From Eq.(15) the reduced density matrix of the state Γ1 can read = |α|2Φ1Φ†1 + |β|2Ψ1Ψ + αβ∗Φ1Ψ + α∗βΨ1Φ . (17) The biorthogonal condition with Φ1Ψ = 0 and Ψ1Φ 0 makes Eq.(17) reduce to = |α|2Φ1Φ†1 + |β|2Ψ1Ψ . (18) Substituting Eq.(18) into the left inequality of Eq.(13) we have |α|2µi(Φ1Φ†1) + |β|2µ1(Ψ1Ψ ) ≤ µi(Γ1Γ†1). (19) Since Ψ1Ψ is positive semidefinite, µ1(Ψ1Ψ ) ≥ 0. Thus Eq.(19) becomes |α|2µi(Φ1Φ†1) ≤ µi(Γ1Γ ). (20) Taking the square root of both sides in Eq.(20) and the sum of µi(·) over all index i, we have µi(Φ1Φ µi(Γ1Γ ). (21) In a similar way, substituting Eq.(18) into the right in- equality of Eq.(14) and taking the sum of µi(·) over all index i, we have µi(Γ1Γ ) ≤ |α| µi(Φ1Φ ) + n|β| µn(Ψ1Ψ Substituting Eqs.(21) and (22) into Eq.(12), respectively, we can obtain |α|2N (Φ1) + |α|2 − 1 ≤ N (αΦ1 + βΨ1) ≤ |α|2Ñ (Φ1) + |α|2 − 1 .(23) If we replace the matrix |α|2Φ1Φ†1 with |β|2Ψ1Ψ Eqs.(20) and (21), i.e., equivalently exchange the ma- trixes H and K in Eq.(14), finally we can also obtain |β|2N (Ψ1) + |β|2 − 1 ≤ N (αΦ1 + βΨ1) ≤ |β|2Ñ (Ψ1) + |β|2 − 1 .(24) Then combining Eqs.(23) and (24) gives Eq.(16). Thus the proof is completed. Note that the lower bound in Eq.(16) can provide a nonzero value only when 2|α|2N (Φ1) + 2|β|2N (Ψ1) > 1. Next we show an example to illustrate the validity of our bound. Consider the state |φ〉AB = α|ϕ〉AB + β|ψ〉AB , (25) |ϕ〉AB = |00〉+ 1√ |11〉, (26) |ψ〉AB = |22〉+ 1√ |33〉, (27) where α = β = 1/ 2. It is easy to check that |ϕ〉AB and |ψ〉AB are biorthogonal, N (|φ〉AB) = 3/2, N (|ϕ〉AB) = N (|ψ〉AB) = 1/2, and µ4(|ϕ〉AB) = µ4(|ψ〉AB) = 1/2. Accordingly from Eq.(16) we obtain the lower and upper bounds 0 < N (|φ〉AB) = < 4, (28) which work well. Finally we directly present the main Theorem of this paper, in which the two states being superposed can be biorthoganal, orthogonal, or nonorthogonal. Theorem 3. Suppose that two arbitrary normalized pure states Φ2 with rank r1 and Ψ2 with rank r2, which are defined in any dimensions. The negativity of their superposed states Γ2 = αΦ2 + βΨ2 with rank r3 and |α2|+ |β|2 = 1 satisfies 2‖α|Φ2〉+ β|Ψ2〉‖2N (αΦ2 + βΨ2) ≤ 2|α|2Ñ (Φ2) + 2|β|2Ñ (Ψ2)− ‖α|Φ2〉+ β|Ψ2〉‖2 + 1, (29) where Ñ (Φ2) = N (Φ2) + µn(Ψ2)[2N (Φ2) + 1] |α| + r2|β|2µn(Ψ2) 2|α|2 , Ñ (Ψ2) = N (Ψ2) + µn(Φ2)[2N (Ψ2) + 1] |β| + r2|α|2µn(Φ2) 2|β|2 , where r = max{r1, r2, r3}. Proof. Consider the matrix M = |α|2Φ2Φ†2 + |β|2Ψ2Ψ , (30) which can be rewritten as ‖Γ2‖2 Γ̂2(Γ̂2) )†, (31) where Γ− = αΦ2 − βΨ2, Γ̂2 = Γ2/‖Γ2‖, and Γ̂−2 = ‖. Thus Eqs.(13) shows that |α|2µi(Φ2Φ†2) + |β|2µ1(Ψ2Ψ ≤ µi(M) ≤ |α|2µi(Φ2Φ†2) + |β|2µn(Ψ2Ψ ), (32) ‖Γ2‖2 Γ̂2Γ̂ ≤ µi(M) ≤ ‖Γ2‖2 Γ̂2Γ̂ .(33) Since µ1(Ψ2Ψ ) ≥ 0 and µ1 ≥ 0, observing the left inequality of Eq.(33) and the right inequality in Eq.(32) we have ‖Γ2‖√ Γ̂2Γ̂ ≤ |α| µi(Φ2Φ ) + |β| µn(Ψ2Ψ Substituting Eqs.(34) into Eq.(12) we have ‖α|Φ2〉+ β|Ψ2〉‖2N (αΦ2 + βΨ2) ≤ 2|α|2Ñ (Φ2)− ‖α|Φ2〉+ β|Ψ2〉‖2 + |α|2. (35) Likewise, if we replace the two matrixes |α|2Φ2Φ†2 with |β|2Ψ2Ψ†2 in Eq.(32), we can obtain ‖α|Φ2〉+ β|Ψ2〉‖2N (αΦ2 + βΨ2) ≤ 2|β|2Ñ (Ψ2)− ‖α|Φ2〉+ β|Ψ2〉‖2 + |β|2. (36) Combining Eqs.(35) and (36) gives Eq.(29). Thus the proof is completed. Since there exists a extra term of the maximal eigen- value in the second inequality in Eq.(33), generally it is difficult to achieve the universal formula for the lower bound of the negativity in this case. But it is our interest in the future work. In conclusion, we have shown that if the entanglement is quantified by the concurrence two pure states of high fidelity to one another still have nearly the same en- tanglement and obtained an inequality that can guaran- tee that the concurrence is a continuous function even in infinite dimensions. However, whether the similar property can apply to the negativity is still open. The bounds on the negativity of superposed states in terms of those of the states being superposed were obtained. So far some bounds of the wildly-studied measures of entanglement like the von Neumann entropy[10], the concurrence[12] and the negativity in this paper for the superposition states have been provided. In view of that the concurrence can be directly accessible in laboratory experiment[19], these bounds can find useful applications in estimating the amount of the entanglement of a given pure state. The author Y.C.O. was supported from China Post- doctoral Science Foundation and the author H.F. was supported by ’Bairen’ program NSFC grant and ’973’ program (2006CB921107). [1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000). [2] A. Osterloh, L. Amico, G. Falci, and R. Fazio Na- ture(London) 416, 608(2002). [3] L. A. Wu, M. S. Sarandy, and D. A. Lidar, Phys. Rev. Lett. 93, 250404(2004). [4] W. K. Wootters, Phys. Rev. Lett. 80, 2245(1998). [5] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824(1996). [6] S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022(1997). [7] V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A 61, 052306(2000). [8] Y. C. Ou, Phys. Rev. A 75, 034305(2007). [9] Y. C. Ou and H. Fan, quant-ph/0702127. [10] N. Linden, S. Popescu, and J. A. Smolin, Phys. Rev. Lett. 97, 100502(2006). [11] M. Ohya and D. Petz, Quantum Entropy and Its Use (Springer-Verlag, Berlin 1983); see also Ref.(1). [12] C. S. Yu, X. X. Yi, and H. S. Song, Phys. Rev. A 75, 022332(2007). [13] Y. Xiang, S. J. Xiong, and F. Y. Hong, quant-ph/0701188. [14] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314(2002). [15] A. Peres, Phys. Rev. Lett. 77, 1413(1996). [16] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223, 1(1996). [17] R. A. Horn and C. R. Johnson, Matrix Analysis (Cam- bridge University Press, New York, 1985), see Theorem 4.3.1. [18] K. Chen, S. Albeverio, and S. M. Fei, Phys. Rev. Lett. 95, 040504(2005). [19] S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich, F. Mintert, and A. Buchleitner, Nature(London) 440, 1022(2006). http://arxiv.org/abs/quant-ph/0702127 http://arxiv.org/abs/quant-ph/0701188

0704.0758

Entangling Independent Photons by Time Measurement

Halder_articleFigureArxiv Entangling Independent Photons by Time Measurement Matthäus Halder, Alexios Beveratos, Nicolas Gisin, Valerio Scarani, Christoph Simon & Hugo Zbinden Group of Applied Physics, University of Geneva, 20, rue de l'Ecole-de-Médecine, 1211 Geneva 4, Switzerland A quantum system composed of two or more subsystems can be in an entangled state, i.e. a state in which the properties of the global system are well defined but the properties of each subsystem are not. Entanglement is at the heart of quantum physics, both for its conceptual foundations and for applications in information processing and quantum communication. Remarkably, entanglement can be “swapped”: if one prepares two independent entangled pairs A1-A2 and B1-B2, a joint measurement on A1 and B1 (called a “Bell-State Measurement”, BSM) has the effect of projecting A2 and B2 onto an entangled state, although these two particles have never interacted or shared any common past1,2. Experiments using twin photons produced by spontaneous parametric down-conversion (SPDC) have already demonstrated entanglement swapping3-6, but here we present its first realization using continuous wave (CW) sources, as originally proposed2. The challenge was to achieve sufficiently sharp synchronization of the photons in the BSM. Using narrow-band filters, the coherence time of the photons that undergo the BSM is significantly increased, exceeding the temporal resolution of the detectors. Hence pulsed sources can be replaced by CW sources, which do not require any synchronization6,7, allowing for the first time the use of completely autonomous sources. Our experiment exploits recent progress in the time precision of photon detectors, in the efficiency of photon pair production by SPDC with waveguides in nonlinear crystals8, and in the stability of narrow-band filters. This approach is independent of the form of entanglement; we employed time-bin entangled photons9 at telecom wavelengths. In addition to entangling photons from autonomous sources, a fundamental quantum phenomenon, our setup is robust against thermal or mechanical fluctuations in optical fibres thanks to cm-long coherence lengths. The present experiment is thus an important step towards real- world quantum networks with truly independent and distant nodes. The BSM is the essential element in an entanglement-swapping experiment. Linear optics allows the realization of only a partial BSM10 by coupling the two incoming modes on a beam-splitter (BS) and observing a suitable detection pattern in the outgoing modes. Such a measurement is successful in at most 50% of the cases. Still, a successful partial BSM entangles two photons that were, up to then, independent. The physics behind this realization is the bosonic character of photons, it is therefore crucial that the two incoming photons are indistinguishable: they must be identical in their spectral, spatial, polarization and temporal modes at the BS: Spectral overlap is achieved by the use of similar filters, spatial overlap by the use of single-mode optical fibres and polarization is matched by a polarization controller. In addition, the temporal resolution must be unambiguous: detection at a time t ± ∆td, with ∆td the temporal resolution of the detector, must single out a unique time mode. In previous experiments, synchronised pulsed sources created both the photons at the same time and path lengths had to be matched to obtain the required temporal overlap. The pulse length, i.e. the coherence length of the photons, was τc << ∆td (typically τc <1ps), but two subsequent pulses were separated by more than ∆td11. The drawback of such a realization is that the two sources cannot be totally autonomous, because of the indispensable synchronization. Here, by using stable narrow-band filters and detectors with low jitter, we reach the regime where τc > ∆td12. In this case, the detectors always single out a unique time mode. As a benefit, we can give up the pulsed character of the sources and the synchronization between them, realizing for the first time the entanglement swapping scheme as originally proposed in Ref.2. The experimental scheme is sketched in Fig.1. Each of the two non-linear crystals emits pairs of energy-time entangled photons13 produced by SPDC of a photon originating from a CW laser. A pair can be created at any time t, and all these processes are coherent within the km-long coherence length of the laser: tt,ψ describes a pair of signal and idler photons emitted by source A. Thus, the state produced by two independent sources can conveniently be represented as ( )∑ ∑  ++++++∝=Ψ BABABABAprep tttttttttttt ,,,,,, ττττψψ . The first term in the above sum describes 4 photons all arriving at the same time t at a BS. Since for this case two identical photons bunch in the same mode, due to their bosonic nature, this term leads to a Hong-Ou-Mandel (HOM) dip14. The second term describes two photon pairs arriving with a time difference τ>0. The two photons A1 and B1 are sent through a 50/50 BS. This fibre-coupler and the two detectors behind it realize a partial BSM10: in particular, when one of the detectors fires at time t and the other one at time t+τ, this corresponds to a measurement of the Bell-state −Ψ for A1 and B19. In consequence the remaining two photons A2 and B2 are projected in the state 22222222 BABABABA tttt ττψ +−+∝Ψ≡ − , which is a singlet state for time-bin entanglement. Hence entanglement has been swapped. This process can be seen as teleportation15-19 of entanglement. It can be tested by sending the photons in unbalanced interferometers such that the path difference between the two arms corresponds to τ. Interference between temporally distinguishable events (at t and t+τ, respectively) is obtained by erasing the time information via unbalanced interferometers9,12,20 as shown in Fig.1. Note that the value of τ varies from one successful entanglement swapping event to another. As in our experiment the path differences of the analysing interferometers are fixed, we test the entanglement of the swapped pairs produced with one fixed τ. We now describe our experiment in more detail. Above we have assumed that the detection times t and t+τ of the BSM are sharply defined. In physical terms, this requirement means that the detection times have to be determined with sub-coherence- time precision: this is the key ingredient that makes it possible to achieve synchronisation of photons A1 and B1 by detection, thus to use CW sources. Since single-photon detectors have a certain intrinsic minimal jitter, the coherence length of the photons has to be increased to exceed this value by narrow filtering. Consider the case where each of the two sources emits one entangled pair of photons, and where A1 and B1 take different exits of the BS. The photon that takes output port 1 is detected by a NbN superconducting single-photon detector (SSPD)21 with a time resolution ∆td = 74ps. The photon in output port 2 is detected by an InGaAs single photon avalanche diode (APD, ∆td = 105ps) triggered by the detection in the SSPD. The time resolution of these detectors is several times smaller than that of commercial telecommunication photon detection modules. To enable synchronization of the photons at the BS by post-selection, the coherence length of the photons has to exceed ∆td. This is achieved by using filters of 10pm bandwidth, corresponding to a coherence time τc of 350ps. We are able to tolerate the losses due to filtering because we use cm-long wave guides in PPLN crystals with a high down-conversion efficiency of 5*10-7 per pump photon and per nm of the created spectrum. For 2mW of laser power, an emission flux q of 2*10-2 pairs per coherence time is obtained. This q is independent of the filtered bandwidth: in fact, narrower filtering decreases the number of photons per second but increases their coherence time by the same factor, hence keeping q constant. Any two-detector click in the BSM prepares the two remaining photons in a time- bin entangled state. In our experiment the creation rate for such entangled photon pairs is ≈104 per second, with time delays τ ranging up to 10ns. This is two orders of magnitude larger than in previous experiments at shorter and similar wavelengths3-6. As the probability of both the pairs originating from different sources equals the probability of creating them in the same source, the first cases have to be post selected by considering only 4-fold-events. Furthermore only one fixed τ is tested. The resulting rate is smaller by two orders of magnitude compared to the creation rate. To verify their entanglement, the two photons are sent through unbalanced interferometers (a and b) in Michelson configuration. The path length differences of the interferometers must be identical only within the coherence length of the analyzed photons (7cm), but stable in phase (α and β): this is achieved by active stabilization22. On each side, both output ports of the interferometer are connected to InGaAs APD, triggered by the detection of both the photons in the BSM. Four-fold coincidences, between one click in each BSM detector and one behind each interferometer, are registered by a multistop time to digital converter (TDC) and the arrival times (t, t+τ) are stored in a table. For τ = 0, we observe a decrease in this coincidence count rate (see Fig.2). The visibility of this HOM dip of 77% indicates the degree of indistinguishability of the two photons A1 and B1 and could be further improved by increasing τc/∆td. The width of the dip corresponds to the convolution of τc for the two photons with the jitter of the detectors. Note that photons which are detected after the BS at measurable different times, but within τc, do still partially bunch, which confirms that the relevant time precision is set by the coherence time of the photons. To test for successful entanglement swapping, the relative phase α-β between the interferometers is changed by keeping α fixed and scanning β. As usual for the analysis of time-bin entanglement9, interference is observable in the case where, at the output of the interferometers, both photons are detected at the same time. We measured the four possible 4-fold coincidence count rates ),( βαijR (clicks in two outer detectors conditioned on a successful BSM) with { }−+∈ ,, ji the different detectors behind interferometer a and b, respectively. Thus the two-photon spin- correlation coefficient ),(),(),(),( ),(),(),(),( βαβαβαβα βαβαβαβα −−+−−+++ −−+−−+++ E is obtained as a function of the phase settings α and β and plotted in Fig.3 for α fixed. A fit of the form )cos(),( βαβα −= VE to our experimental data gives a visibility V=0.63. If one assumes that the two photons are in a Werner state (which corresponds to white noise), one can show that 31>V is sufficient to demonstrate entanglement 5,23. Our experimental visibility clearly exceeds this bound24. The plain squares show that the 3- fold coincidence count rate between a successful BSM and only one of the outside detectors is independent of the phase setting, as expected for a −Ψ -state. V is limited by imperfections in the matching of wavelengths, polarisations and temporal synchronisation. In our setup, the latter is the main source of errors. The integration time of this measurement was 1 hour for each of the 13 phase settings and the experiment was run 8 times, hence took 104 hours, which demonstrates the stability of our setup. Such long integration times are necessary because of low count rates (5 four- fold coincidences per hour), which are mainly due to poor coupling efficiencies of the photons into optical fibres, losses in optical components like filters and interferometers, as well as the limited detectors efficiencies. All these factors decrease the probability of detecting all four photons of a two-pair event. Exploiting all the produced entangled pairs with different delays τ is possible in principle using rapidly adjustable delays in the interferometers or quantum memories. This would be an important step towards the realization of recent proposals for long- distance quantum communication25. Time-bin entanglement is particularly stable and well suited for fibre optic communications26, and the coherence length of 7cm allows tolerating significant fiber length fluctuations as expected in field experiments. If additionally, count rates are further improved, long distance applications like quantum relays27,28 become realistic. In conclusion, we realized an entanglement swapping experiment with completely autonomous CW sources. This is possible thanks to the low jitter of new NbN superconducting and InGaAs avalanche single-photon detectors and to the long coherence length of the created photon pairs after narrow-band filtering. The setup does not require any synchronization between the sources and is highly stable against length fluctuations of the quantum channels. Methods Schematic description of the setup. Both sources consist of an external cavity diode laser in CW mode at 780.027nm (Toptica DL100), stabilized against a Rubidium transition (85Rb F = 3), pumping a nonlinear periodically poled Lithium Niobate waveguide8 (PPLN, HC photonics Corp) at a power of 2mW. The process of SPDC creates 4*1011 pairs of photons per second with a spectral width of 80nm FWHM centered at 1560nm. The photons are emitted collinearly and coupled into a single-mode fiber with 25% efficiency and the remaining laser light is blocked with a silicon high- pass filter (Si). Signal and idler photons are separated and filtered down to a bandwidth of 10pm by custom-made tunable phase-shifted Bragg gratings (AOS GmbH). These filters have a rejection of >40dB, 3dB insertion losses, and can be tuned independently over a range of 400pm. Once a signal photon has been filtered to ωs, the corresponding idler photon has a well-defined frequency ωi, due to stabilized pump wavelength and energy conservation in the process of SPDC (ωs + ωi = ωlaser). After filtering, the effective conversion efficiency for creating a photon pair within these 10pm is 5*10-9 per pump photon. In principle, the available pump power permits us to produce narrow band entangled photon pairs at rates up to 3*108 pairs per second, which translates to an emission flux of more than 0.1 photons per coherence time. In this experiment, we limited the laser to 2mW, in order to reduce the probability of multiple pair creation which would decrease the interference visibility29. After the beam splitter (BS), the first photon is detected by a NbN superconducting single-photon detector (SSPD, Scontel) operated in free running mode21, with a total detection efficiency of 4.5%, 300 dark counts/sec and a timing resolution of 74ps, including the time jitter of both the detector and the amplification and discrimination electronics. The second photon is detected by an InGaAs single- photon avalanche diode operated in Geiger mode and actively triggered by the detection in the SSPD. With home-made electronics this detector has a time jitter of 105ps. The observed HOM-dip with a visibility of 77% was obtained with two SSPD detectors, which were used because of their smaller time jitter. For the entanglement swapping, we used an APD, because of its higher efficiency, in order to shorten the integration time. This means that the visibility of the interference fringe in Fig.3 could further be increased by the use of two SSPDs, but with the drawback of longer measurement times. Photons A2 and B2 are also detected by InGaAs APDs (ID200, idQuantique). All the APDs have quantum efficiencies of 30% and dark count probabilities of 10-4 per ns. The interferometers are actively stabilized against a laser locked on an atomic transition, have a path length difference of 1.2ns and insertion losses of 4dB each. 1. Yurke, B. & Stoler, D. Bell’s-inequality experiments using independent-particle sources. Phys. Rev. A 46 2229 (1992). 2. Żukowski M., Zeilinger A., Horne M. A. & Ekert A. K. ”Event-ready-detectors” Bell experiment via entanglement swapping. Phys. Rev. Lett. 71, 4287-4290 (1993). 3. Pan J.-W., Bouwmeester D., Weinfurter H. & Zeilinger A. Experimental Entanglement Swapping: Entangling Photons That Never Interacted. Phys. Rev. Lett. 80, 3891-3894 (1998). 4. Jennewein T., Weihs G., Pan J.-W. & Zeilinger A. Experimental Nonlocality Proof of Quantum Teleportation and Entanglement Swapping. Phys. Rev. Lett. 88, 017903 (2002). 5. de Riedmatten, H., Marcikic, I., Tittel, W., Zbinden, H. & Gisin, N. Long-distance entanglement swapping with photons from separated sources. Phys. Rev. A 71, 050302 (2005). 6. Yang, T. et al. Experimental Synchronization of Independent Entangled Photon Sources. Phys. Rev. Lett. 96, 110501 (2006). 7. Kaltenbaek, R., Blauensteiner, B., Żukowski, M., Aspelmeyer, M., & Zeilinger A. Experimental interference of independent photons. Phys. Rev. Lett. 96, 240502 (2006). 8. Tanzilli S. et al. PPLN waveguide for quantum communication. Eur. Phys. .J D 18, 155 (2002). 9. Brendel, J., Gisin, N., Tittel, W. & Zbinden, H. Pulsed Energy-Time Entangled Twin- Photon Source for Quantum Communication. Phys. Rev. Lett. 82, 2597 (1999). 10. Weinfurter, H. Experimental Bell-State Analysis. Europhys Lett. 25, 559 (1994). 11. Bouwmeester, D. et al. Experimental quantum teleportation. Nature 390, 575 (1997). 12. Legero, T., Wilk, T., Henrich, M., Rempe, G. & Kuhn, A. Quantum Beat of Two Single Photons. Phys. Rev. Lett. 93, 070503 (2004). 13. Franson, J.D. Bell inequality for position and time. Phys. Rev. Lett. 62, 2205 (1989). 14. Hong, C.K., Ou, Z.Y. & Mandel, L. Measurement of Subpicosecond Time Intervals between Two Photons by Interference. Phys. Rev. Lett. 59, 2044 (1987). 15. Furusawa, A. Unconditional Quantum Teleportation. Science 282, 706 (1998) 16. Boschi, D.,Branca, S., De Martini, F., Hardy, L. & Popescu, S. Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels. Phys. Rev. Lett. 80, 1121 (1998). 17. Marcikic, I., de Riedmatten, H., Tittel, W., Zbinden, H. & Gisin, N. Long-distance teleportation of qubits at telecommunication wavelengths. Nature 421, 509 (2003). 18. Riebe, M. et al. Deterministic quantum teleportation with atoms. Nature 429, 734 (2004). 19. Barrett, M. D. et al. Deterministic quantum teleportation of atomic qubits. Nature 429, 737 (2004). 20. Pittman, T. B. et al. Can Two-Photon Interference be Considered the Interference of Two Photons? Phys. Rev. Lett. 77, 1917 (1996). 21. Milostnaya, I. et al. Superconducting single-photon detectors designed for operation at 1.55-µm telecommunication wavelength. J. Phys. Conference Series 43, 1334 (2006). 22. Marcikic, I. et al. Distribution of Time-Bin Entangled Qubits over 50 km of Optical Fiber. Phys. Rev. Lett. 93, 180502 (2004). 23. Peres, A. Separability Criterion for Density Matrices. Phys. Rev. Lett. 77, 1413 (1996). 24. In fact, in our experimental situation entanglement is present even for values of V that are smaller than 0.33 because the noise is dominated by phase errors due to the partial distinguishability of the two photons involved in the BSM. 25. Duan, L.-M., Lukin, M.D., Cirac, J.I. & Zoller, P. Long-distance quantum communication with atomic ensembles and linear optics. Nature 414, 413 (2001). 26. Thew R. T., Tanzilli S., Tittel W., Zbinden H., & Gisin N. Experimental investigation of the robustness of partially entangled qubits over 11 km. Phys. Rev. A 66, 062304 (2002). 27. Jacobs B. C., Pittman T. B. & Franson J. D. Quantum relays and noise suppression using linear optics. Phys. Rev. A 66, 052307 (2002). 28. Collins, D., Gisin, N. & de Riedmatten, H. Quantum Relay for Long Distance Quantum Cryptography. J. Mod. Opt. 522, 735 (2005). 29. Scarani, V., de Riedmatten, H., Marcikic, I., Zbinden, H. & Gisin, N. Four-photon correction in two-photon Bell experiments. Eur. Phys. J. D 32, 129 (2005). Acknowledgements: We thank C. Barreiro, J.-D. Gautier, G. Gol’tsman, C. Jorel, S Tanzilli and J. van Houwelingen for technical support, and H. de Riedmatten, S. Iblisdir and R. Thew for helpful discussions. Financial support by the EU projects QAP and SINPHONIA and by the Swiss NCCR Quantum Photonics is acknowledged. Author Information: Reprints and permissions information is available at npg.nature.com/reprintsandpermissions. The authors declare that they have no competing financial interests Correspondence and requests for materials should be addressed to M.H. ([email protected]). Figure 1: Experimental Setup. Two pairs of entangled photons (A1- A2 and B1-B2) are produced, one by each source (A and B), and all the photons are narrowly filtered (10pm). One photon of each pair is sent into a 50/50 beam splitter (BS) and both undergo a partial Bell-State measurement (BSM). By detection in different output ports of BS and with a certain time delay τ the two photons A1 and B1 are projected on the −Ψ -state for time bin qubits, projecting the two remaining photons on the −Ψ -state as well. The entanglement is swapped onto the photons A2 and B2 and can be tested by passing them through interferometers with phases α and β, and detecting them by single photon avalanche detectors (APD) in both outputs (+,-) of each interferometer. Figure 2: 4-fold coincidence count rate as a function of the temporal delay τ. It can be seen, that the detection probability decreases if the two photons A1 and B1 arrive simultaneously (τ=0) at the beam splitter due to photon bunching, leading to a Hong-Ou-Mandel dip with 77% visibility. Figure3 Figure 3: The correlation coefficient E(α,β) between photons A2 and B2, conditioned on a BSM of photons A1 and B1, as a function of the relative phase α–β of the interferometers (open points). A fit of the form )cos(),( βαβα −= VE gives a visibility V=0.63. This proves successful entanglement swapping (see text). The coincidence count rate of only one detector conditioned on a successful BSM (3-fold coincidence) is independent of the phase setting as expected for a −Ψ -state (squares).

0704.0759

Energy conservation and Onsager's conjecture for the Euler equations

arXiv:0704.0759v1 [math.AP] 5 Apr 2007 ENERGY CONSERVATION AND ONSAGER’S CONJECTURE FOR THE EULER EQUATIONS A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY ABSTRACT. Onsager conjectured that weak solutions of the Euler equa- tions for incompressible fluids in R3 conserve energy only if they have a certain minimal smoothness, (of order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space B 3,c(N) show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood-Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to B 3,c(N) conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range. 1. INTRODUCTION The Euler equations for the motion of an incompressible inviscid fluid + (u · ∇)u = −∇p, (2) ∇ · u = 0, where u(x, t) denotes the d-dimensional velocity, p(x, t) denotes the pres- sure, and x ∈ Rd. We mainly consider the case d = 3. When u(x, t) is a classical solution, it follows directly that the total energyE(t) = 1 |u|2 dx is conserved. However, conservation of energy may fail for weak solutions (see Scheffer [25], Shnirelman [24]). This possibility has given rise to a considerable body of literature and it is closely connected with statistical theories of turbulence envisioned 60 years ago by Kolmogorov and On- sager. For reviews see, for example, Eyink and Sreenivasan [14], Robert [23], and Frisch [15]. Date: April 5, 2007. 2000 Mathematics Subject Classification. Primary: 76B03; Secondary: 76F02. Key words and phrases. Euler equations, anomalous dissipation, energy flux, Onsager conjecture, turbulence, Littlewood-Paley spectrum. http://arxiv.org/abs/0704.0759v1 2 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY Onsager [22] conjectured that in 3-dimensional turbulent flows, energy dissipation might exist even in the limit of vanishing viscosity. He sug- gested that an appropriate mathematical description of turbulent flows (in the inviscid limit) might be given by weak solutions of the Euler equations that are not regular enough to conserve energy. According to this view, non- conservation of energy in a turbulent flow might occur not only from vis- cous dissipation, but also from lack of smoothness of the velocity. Specif- ically, Onsager conjectured that weak solutions of the Euler equation with Hölder continuity exponent h > 1/3 do conserve energy and that turbulent or anomalous dissipation occurs when h ≤ 1/3. Eyink [12] proved energy conservation under a stronger assumption. Subsequently, Constantin, E and Titi [7] proved energy conservation for u in the Besov spaceBα3,∞, α > 1/3. More recently the result was proved under a slightly weaker assumption by Duchon and Robert [11]. In this paper we sharpen the result of [7]: we prove that energy is con- served for velocities in the Besov space of tempered distributions B 3,p . In fact we prove the result for velocities in the slightly larger spaceB 3,c(N) Section 3). This is a space in which the “Hölder exponent” is exactly 1/3, but the slightly better regularity is encoded in the summability condition. The method of proof combines the approach of [7] in bounding the trilinear term in (3) with a suitable choice of the test function for weak solutions in terms of a Littlewood-Paley decomposition. Certain cancelations in the tri- linear term become apparent using this decomposition. We observe that the space B 3,c(N) is sharp in the context of no anomalous dissipation. We give an example of a divergence free vector field in B 3,∞ for which the energy flux due to the trilinear term is bounded from below by a positive constant. This construction follows ideas in [12]. However, because it is not a solu- tion of the unforced Euler equation, the example does not prove that indeed there exist unforced solutions to the Euler equation that live in B 3,∞ and dissipate energy. Experiments and numerical simulations indicate that for many turbu- lent flows the energy dissipation rate appears to remain positive at large Reynolds numbers. However, there are no known rigorous lower bounds for slightly viscous Navier-Stokes equations. The existence of a weak so- lution of Euler’s equation, with positive smoothness and that does not con- serve energy remains an open question. For a discussion see, for example, Duchon and Robert [11], Eyink [12], Shnirelman [25], Scheffer [24], de Lellis and Szekelyhidi [10]. We note that the proof in Section 3 applied to Burger’s equation for 1- dimensional compressible flow gives conservation of energy in B1/3 3,c(N) ENERGY CONSERVATION 3 this case it is easy to show that conservation of energy can fail in B which is the sharp space for shocks. The Littlewood-Paley approach to the issue of energy conservation ver- sus turbulent dissipation is mirrored in a study of a discrete dyadic model for the forced Euler equations [4, 5]. By construction, all the interactions in that model system are local and energy cascades strictly to higher wave numbers. There is a unique fixed point which is an exponential global at- tractor. Onsager’s conjecture is confirmed for the model in both directions, i.e. solutions with bounded H5/6 norm satisfy the energy balance condition and turbulent dissipation occurs for all solutions when the H5/6 norm be- comes unbounded, which happens in finite time. The absence of anomalous dissipation for inviscid shell models has been obtained in [8] in a space with regularity logarithmically higher than 1/3. In Section 3.2 we present the definition of the energy flux employed in the paper. This is the flux of the Littlewood-Paley spectrum, ([6]) which is a mathematically convenient variant of the physical concept of flux from the turbulence literature. Our estimates employing the Littlewood-Paley de- composition produce not only a sharpening of the conditions under which there is no anomalous dissipation, but also provide detailed information concerning the cascade of energy flux through frequency space. In sec- tion 3.3. we prove that the energy flux through the sphere of radius κ is controlled primarily by scales of order κ. Thus we give a mathematical justification for the physical intuition underlying much of turbulence the- ory, namely that the flux is controlled by local interactions (see, for exam- ple, Kolmogorov [16] and also [13], where sufficient conditions for locality were described). Our analysis makes precise an exponential decay of non- local contributions to the flux that was conjectured by Kraichnan [17]. The energy is not the only scalar quantity that is conserved under evolu- tion by classical solutions of the Euler equations. For 3-dimensional flows the helicity is an important quantity related to the topological configura- tions of vortex tubes (see, for example, Moffatt and Tsinober [21]). The total helicity is conserved for smooth ideal flows. In Section 4 we observe that the techniques used in Section 3 carry over exactly to considerations of the helicity flux, i.e., there is locality for turbulent cascades of helicity and every weak solution of the Euler equation that belongs to B 3,c(N) conserves helicity. This strengthens a recent result of Chae [2]. Once again our argu- ment is sharp in the sense that a divergence free vector field in B 3,∞ can be constructed to produce an example for which the helicity flux is bounded from below by a positive constant. 4 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY An important property of smooth flows of an ideal fluid in two dimen- sions is conservation of enstrophy (i.e. the L2 norm of the curl of the ve- locity). In section 4.2 we apply the techniques of Section 3 to the weak formulation of the Euler equations for velocity using a test function that permits estimation of the enstrophy. We obtain the result that, unlike the cases of the energy and the helicity, the locality in the enstrophy cascade is strong only in the ultraviolet range. In the infrared range there are nonlo- cal effects. Such ultraviolet locality was predicted by Kraichnan [18] and agrees with numerical and experimental evidence. Furthermore, there are arguments in the physical literature that hold that the enstrophy cascade is not local in the infrared range. We present a concrete example that exhibits this behavior. In the final section of this paper, we study the bilinear term B(u, v). We show that the trilinear map (u, v, w) → 〈B(u, v), w)〉 defined for smooth vector fields in L3 has a unique continuous extension to {B1/2 18/7,2 }3 (and a fortiori to {H5/6}3, which is the relevant space for the dyadic model prob- lem referred to above). We present an example to show that this result is optimal. We stress that the borderline space for energy conservation is much rougher than the space of continuity for 〈B(u, v), w〉. 2. PRELIMINARIES We will use the notation λq = 2 q (in some inverse length units). Let B(0, r) denote the ball centered at 0 of radius r in Rd. We fix a nonnegative radial function χ belonging to C∞0 (B(0, 1)) such that χ(ξ) = 1 for |ξ| ≤ 1/2. We further define (3) ϕ(ξ) = χ(λ−11 ξ)− χ(ξ). Then the following is true (4) χ(ξ) + ϕ(λ−1q ξ) = 1, (5) |p− q| ≥ 2 ⇒ Supp ϕ(λ−1q ·) ∩ Supp ϕ(λ−1p ·) = ∅. We define a Littlewood-Paley decomposition. Let us denote by F the Fourier transform on Rd. Let h, h̃, ∆q (q ≥ −1) be defined as follows: ENERGY CONSERVATION 5 h = F−1ϕ and h̃ = F−1χ, ∆qu = F−1(ϕ(λ−1q ξ)Fu) = λdq h(λqy)u(x− y)dy, q ≥ 0 ∆−1u = F−1(χ(ξ)Fu) = h̃(y)u(x− y)dy. For Q ∈ N we define (6) SQ = Due to (3) we have (7) SQu = F−1(χ(λ−1Q+1ξ)Fu). Let us now recall the definition of inhomogeneous Besov spaces. Definition 2.1. Let s be a real number, p and r two real numbers greater than 1. Then ‖u‖Bsp,r = ‖∆−1u‖Lp + λsq‖∆qu‖Lp ℓr(N) is the inhomogeneous Besov norm. Definition 2.2. Let s be a real number, p and r two real numbers greater than 1. The inhomogeneous Besov space Bsp,r is the space of tempered distributions u such that the norm ‖u‖Bsp,r is finite. We refer to [3] and [19] for background on harmonic analysis in the con- text of fluids. We will use the Bernstein inequalities Lemma 2.3. ‖∆qu‖Lb ≤ λ q ‖∆qu‖La for b ≥ a ≥ 1. As a consiquence we have the following inclusions. Corollary 2.4. If b ≥ a ≥ 1, then we have the following continuous em- beddings Bsa,r ⊂ B B0a,2 ⊂ La 6 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY In particular, the following chain of inclusions will be used throughout the text. (8) H 6 (R3) ⊂ B (R3) ⊂ B (R3) ⊂ B 3,2(R 3. ENERGY FLUX AND LOCALITY 3.1. Weak solutions. Definition 3.1. A function u is a weak solution of the Euler equations with initial data u0 ∈ L2(Rd) if u ∈ Cw([0, T ];L2(Rd)), (the space of weakly continuous functions) and for every ψ ∈ C1([0, T ];S(Rd)) with S(Rd) the space of rapidly decaying functions, with ∇x · ψ = 0 and 0 ≤ t ≤ T , we (u(t), ψ(t))− (u(0), ψ(0))− (u(s), ∂sψ(s))ds = b(u, ψ, u)(s)ds, where (u, v) = u · vdx, b(u, v, w) = u · ∇v · w dx, and ∇x · u(t) = 0 in the sense of distributions for every t ∈ [0, T ]. Clearly, (9) implies Lipschitz continuity of the maps t → (u(t), ψ) for fixed test functions. By an approximation argument one can show that for any weak solution u of the Euler equation, the relationship (9) holds for all ψ that are smooth and localized in space, but only weakly Lipschitz in time. This justifies the use of physical space mollifications of u as test functions ψ. Because we do not have an existence theory of weak solutions, this is a rather academic point. 3.2. Energy flux. For a divergence-free vector field u ∈ L2 we introduce the Littlewood-Paley energy flux at wave number λQ by (10) ΠQ = Tr[SQ(u⊗ u) · ∇SQu]dx. If u(t) is a weak solution to the Euler equation, then substituting the test function ψ = S2Qu into the weak formulation of the Euler equation (9) we obtain (11) ΠQ(t) = ‖SQu(t)‖22. ENERGY CONSERVATION 7 Let us introduce the following localization kernel (12) K(q) = q , q ≤ 0; q , q > 0, For a tempered distribution u in R3 we denote dq = λ q ‖∆qu‖3,(13) d2 = {d2q}q≥−1.(14) Proposition 3.2. The energy flux of a divergence-free vector field u ∈ L2 satisfies the following estimate (15) |ΠQ| ≤ C(K ∗ d2)3/2(Q). From (15) we immediately obtain (16) lim sup |ΠQ| ≤ lim sup We define B 3,c(N) to be the class of all tempered distributions u in R3 for which (17) lim λ1/3q ‖∆qu‖3 = 0, and hence dq → 0. We endow B1/33,c(N) with the norm inherited from B Notice that the Besov spaces B 3,p for 1 ≤ p < ∞, and in particular B are included in B 3,c(N) As a consequence of (11) and (16) we obtain the following theorem. Theorem 3.3. The total energy flux of any divergence-free vector field in the class B 3,c(N) ∩ L2 vanishes. In particular, every weak solution to the Euler equation that belongs to the class L3([0, T ];B 3,c(N) ) ∩Cw([0, T ];L2) conserves energy. Proof of Proposition 3.2. In the argument below all the inequalities should be understood up to a constant multiple. Following [7] we write (18) SQ(u⊗ u) = rQ(u, u)− (u− SQ)⊗ (u− SQ) + SQu⊗ SQu, where rQ(u, u) = hQ(y)(u(x− y)− u(x))⊗ (u(x− y)− u(x))dy, h̃Q(y) = λ Q+1h̃(λQ+1y). 8 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY After substituting (18) into (10) we find Tr[rQ(u, u) · ∇SQu]dx(19) Tr[(u− SQ)⊗ (u− SQ) · ∇SQu]dx.(20) We can estimate the term in (19) using the Hölder inequality by ‖rQ(u, u)‖3/2‖∇SQu‖3, whereas (21) ‖rQ(u, u)‖3/2 ≤ ∣∣∣h̃Q(y) ∣∣∣ ‖u(· − y)− u(·)‖23dy. Let us now use Bernstein’s inequalities and Corollary 2.4 to estimate ‖u(· − y)− u(·)‖23 ≤ |y|2λ2q‖∆qu‖23 + ‖∆qu‖23(22) Q |y|2 Q−q d q + λ q(23) ≤ (λ4/3Q |y|2 + λ Q )(K ∗ d2)(Q).(24) Collecting the obtained estimates we find Tr[rQ(u, u) · ∇SQu]dx ≤ (K ∗ d2)(Q) ∣∣∣h̃Q(y) ∣∣∣λ4/3Q |y|2dy + λ λ2q‖∆qu‖23 ≤ (K ∗ d2)(Q)λ−2/3Q λ4/3q d ≤ (K ∗ d2)3/2(Q) Analogously we estimate the term in (20) Tr[(u− SQ)⊗ (u− SQ) ·∆SQu]dx ≤ ‖u− SQu‖23‖∆SQu‖3 ‖∆qu‖23 λ2q‖∆qu‖23 ≤ (K ∗ d2)3/2(Q). This finishes the proof. ENERGY CONSERVATION 9 3.3. Energy flux through dyadic shells. Let us introduce the energy flux through a sequence of dyadic shells between scales −1 ≤ Q0 < Q1 < ∞ as follows (25) ΠQ0Q1 = Tr[SQ0Q1(u⊗ u) · ∇SQ0Q1u] dx, where (26) SQ0Q1 = Q0≤q≤Q1 ∆q = SQ1 − SQ0. We will show that similar to formula (15) the flux through dyadic shells is essentially controlled by scales near the inner and outer radii. In fact it almost follows from (15) in view of the following decomposition S2Q0Q1 = (SQ1 − SQ0−1) = S2Q1 + S Q0−1 − 2SQ0−1SQ1 = S2Q1 + S Q0−1 − 2SQ0−1 = S2Q1 − S Q0−1 − 2SQ0−1(1− SQ0−1) = S2Q1 − S Q0−1 − 2∆Q0−1∆Q0 . Therefore (28) ΠQ0Q1 = ΠQ1 − ΠQ0−1 − 2 Tr[∆̄Q0(u⊗ u) · ∇∆̄Q0u] dx, where (29) ∆̄Q0(u) = h̄Q0(y)u(x− y) dy, and h̄Q0(x) = F−1 ϕ(λ−1Q0−1ξ)ϕ(λ Note that the flux through a sequence of dyadic shells is equal to the difference between the fluxes across the dyadic spheres on the boundary plus a small error term that can be easily estimated. Indeed, let us rewrite the tensor product term as follows (30) ∆̄Q0(u⊗ u) = r̄Q0(u, u) + ∆̄Q0u⊗ u+ u⊗ ∆̄Q0u, where r̄Q(u, u) = h̄Q(y)(u(x− y)− u(x))⊗ (u(x− y)− u(x)) dy. 10 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY Thus we have Tr[∆̄Q0(u⊗ u) · ∇∆̄Q0u] dx = Tr[r̄Q(u, u) · ∇∆̄Q0u] dx ∆̄Q0u · ∇u · ∆̄Q0u dx We estimate the first integral as previously to obtain Tr[r̄Q0(u, u) · ∇∆̄Q0u] dx ∣∣∣∣ ≤ dQ0(K ∗ d 2)(Q0). As to the second integral we have ∆̄Q0u · ∇u · ∆̄Q0u dx ∣∣∣∣ = ∆̄Q0u · ∇SQ0u · ∆̄Q0u dx ≤ d2Q0(K ∗ d 2)1/2(Q0). Applying these estimates to the flux (28) we arrive at the following con- clusion. Theorem 3.4. The energy flux through dyadic shells between wavenumbers λQ0 and λQ1 is controlled primarily by the end-point scales. More precisely, the following estimate holds (33) |ΠQ0Q1| ≤ C(K ∗ d2)3/2(Q0) + C(K ∗ d2)3/2(Q1). 3.4. Construction of a divergence free vector field with non-vanishing energy flux. In this section we give a construction of a divergence free vector field in B 3,∞(R 3) for which the energy flux is bounded from below by a positive constant. This suggests the sharpness ofB 3,c(N) (R3) for energy conservation. Our construction is based on Eyink’s example on a torus [12], which we transform to R3 using a method described below. Let χQ(ξ) = χ(λ Q+1ξ). We define P ξ for vectors ξ ∈ R3, ξ 6= 0 by P⊥ξ v = v − |ξ|−2(v · ξ)ξ = I− |ξ|−2(ξ ⊗ ξ) for v ∈ C3 and we use v · w = vjwj for v, w ∈ C3. Lemma 3.5. Let Φk(x) be R3 – valued functions, such that Ik := |ξ||FΦk(ξ)| dξ <∞. ENERGY CONSERVATION 11 Let also Ψk(x) = P(e ik·xΦk(x)) where P is the Leray projector onto the space of divergence free vectors. Then (34) sup ∣∣Ψk(x)− eik·x(P⊥k Φk)(x) ∣∣ ≤ 1 |k| , (35) sup ∣∣(S2QΨk)(x)− χ2Q(k)Ψk(x) ∣∣ ≤ c (2π)3 where c is the the Lipschitz constant of χ(ξ)2. Proof. First, note that for any k, ξ ∈ R3 and v ∈ C3 we have (v · ξ)ξ |ξ|2 + (v · ξ)k ∣∣∣∣ ≤ |ξ| ξ + |v||ξ + k| |k| . In addition, it follows that (v · k)k |k|2 + (v · ξ)k ∣∣∣∣ = |(v · (k + ξ))k| ≤ |v||ξ + k||k| . Adding (36) and (37) we obtain |P⊥ξ v − P⊥k v| = (v · ξ)ξ |ξ|2 − (v · k)k (v · ξ)ξ |ξ|2 + (v · ξ)k ∣∣∣∣ + (v · k)k |k|2 + (v · ξ)k ≤ 2 |v||ξ + k||k| . Using this inequality we can now derive the following estimate: |Ψk(x)− eik·x(P⊥k Φk)(x)| = |F−1[P⊥ξ (FΦk)(ξ + k)− P⊥k (FΦk)(ξ + k)]| (2π)3 |ξ + k| |k| |(FΦk)(ξ + k)| dξ = |k|−1 1 |ξ||(FΦk(ξ))| dξ. 12 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY Finally, we have |(S2QΨk)(x)− χQ(k)2Ψk(x)| = |F−1[(χQ(ξ)2 − χQ(k)2)(FΨk)(ξ)]| (2π)3 c|ξ + k| |(FΦk)(ξ + k)| dξ = λ−1Q+1 (2π)3 |ξ||(FΦk)(ξ)| dξ, where c is the the Lipschitz constant of χ(ξ)2. This concludes the proof. � Example illustrating the sharpness of Theorem 3.3. Now we proceed to construct a divergence free vector field in B 3,∞(R 3) with non-vanishing energy flux. Let U(k) be a vector field U : Z3 → C3 as in Eyink’s example [12] with U(λq, 0, 0) = iλ q (0, 0,−1), U(−λq, 0, 0) = iλ−1/3q (0, 0, 1), U(0, λq, 0) = iλ q (1, 0, 1), U(0,−λq, 0) = iλ−1/3q (−1, 0,−1), U(λq, λq, 0) = iλ q (0, 0, 1), U(−λq,−λq, 0) = iλ−1/3q (0, 0,−1), U(λq,−λq, 0) = iλ−1/3q (1, 1,−1), U(−λq, λq, 0) = iλ−1/3q (−1,−1, 1), for all q ∈ N and zero otherwise. Denote ρ(x) = F−1χ(4ξ) and A =∫ ρ(x)3 dx. Since χ(ξ) is radial, ρ(x) is real. Moreover, ρ(x)3 dx = (2π)3 F(ρ2)Fρ dξ (2π)6 χ(4η)χ(4(ξ − η))χ(4ξ) dηdξ > 0. Now let u(x) = P U(k)eik·xρ(x). Note that u ∈ B1/33,∞(R3). Our goal is to estimate the flux ΠQ for the vector field u. Define Φk = |k|1/3U(k)ρ(x) and Ψk(x) = P(eik·xΦk(x)). Then clearly Φk(x) and Ψk(x) satisfy the conditions of Lemma 3.5, and we (41) u(x) = k∈Z3\{0} |k|−1/3Ψk(x). ENERGY CONSERVATION 13 Now note that Ψk1 · ∇S2QΨk2 = Ψk1 · S2QP[∇(eik·xΦk2)] = i(Ψk1 · k2)S2QΨk2 +Ψk1 · S2QP(eik2·x∇Φk2). In addition, the following equality holds by construction: (43) P⊥k Φk = Φk, ∀k ∈ Z3. Define the annulusAQ = Z 3∩B(0, λQ+2)\B(0, λQ−1). Thanks to Lemma 3.5, for any sequences k1(Q), k2(Q), k3(Q) ∈ AQ with k1 + k2 = k3, we have (Ψk1 · ∇S2QΨk2) ·Ψ∗k3 dx = i (Ψk1 · k2)S2QΨk2 ·Ψ∗k3 dx+O(λ (eik1·xΦk1 · k2)χQ(k2)2eik2·xΦk2 · e−ik3·xΦ∗k3 dx+O(λ = i(|k1||k2||k3|)1/3A(U(k1) · k2)χQ(k2)2U(k2) · U(k3)∗ +O(λ0Q). On the other hand, since the Fourier transform of Ψk is supported in B(k, 1/4), we have (Ψk1 · ∇S2QΨk2) ·Ψ∗k3 dx = 0, whenever k1 + k2 6= k3. In addition, due to locality of interactions in this example, (44) also holds if Aq \ {k1, k2, k3} 6= ∅ for all q ∈ N. Finally, (Ψk1 · ∇S2QΨk2) ·Ψ∗k3 dx+ (Ψk1 · ∇S2QΨk3) ·Ψ∗k2 dx = 0, whenever k2 /∈ AQ and k3 /∈ AQ. Hence, the flux for u can be written as (46) ΠQ = − k1,k2,k3∈AQ k1+k2+k3=0 (|k1||k2||k3|)−1/3 (Ψk1 · ∇S2QΨk2) ·Ψk3 dx. Since the number of nonzero terms in the above sum is independent of Q, we obtain (47) ΠQ = AΠ̃Q +O(λ where Π̃ is the flux for the vector field U , i.e., (48) Π̃Q := − k1,k2,k3∈AQ k1+k2+k3=0 i(U(k1) · k2)χQ(k2)2U(k2) · U(k3). 14 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY The flux Π̃Q has only the following non-zero terms (see [12] for details): |k2|=λQ |k3|= i(U1(−k2 − k3) · k2)U2(k2) · U3(k3)(χQ(k2)2 − χQ(k3)2) ≥ 4(χ(1/2)2 − χ(1/ 2)2), |k2|= |k3|=2λQ i(U1(−k2 − k3) · k2)U2(k2) · U3(k3)(χQ(k2)2 − χQ(k3)2) ≥ 4(χ(1/ 2)2 − χ(1)2). Hence Π̃Q ≥ 4(χ(1/2)2 − χ(1/ 2)2 + χ(1/ 2)2 − χ(1)2) = 4. This together with (47) implies that lim inf ΠQ ≥ 4A. 4. OTHER CONSERVATION LAWS In this section we apply similar techniques to derive optimal results con- cerning the conservation of helicity in 3D and that of enstrophy in 2D for weak solutions of the Euler equation. In the case of the helicity flux we prove that simultaneous infrared and ultraviolet localization occurs, as for the energy flux. However, the enstrophy flux exhibits strong localization only in the ultraviolet region, and a partial localization in the infrared re- gion. A possibility of such a type of localization was discussed in [18]. 4.1. Helicity. For a divergence-free vector field u ∈ H1/2 with vorticity ω = ∇ × u ∈ H−1/2 we define the helicity and truncated helicity flux as follows u · ω dx(49) Tr [SQ(u⊗ u) · ∇SQω + SQ(u ∧ ω) · ∇SQu] dx,(50) where u ∧ ω = u ⊗ ω − ω ⊗ u. Thus, if u was a solution to the Euler equation, then HQ would be the time derivative of the Littlewood-Paley helicity at frequency λQ, SQu · SQω dx. ENERGY CONSERVATION 15 Let us denote bq = λ q ‖∆qu‖3,(51) b2 = {b2q}∞q=−1,(52) T (q) = q , q ≤ 0; q , q > 0, Proposition 4.1. The helicity flux of a divergence-free vector field u ∈ H1/2 satisfies the following estimate (54) |HQ| ≤ C(T ∗ b2)3/2(Q). Theorem 4.2. The total helicity flux of any divergence-free vector field in the class B 3,c(N) ∩H1/2 vanishes, i.e. (55) lim HQ = 0. Consequently, every weak solution to the Euler equation that belongs to the class L3([0, T ];B 3,c(N) ) ∩ L∞([0, T ];H1/2) conserves helicity. Proposition 4.1 and Theorem 4.2 are proved by direct analogy with the proofs of Proposition 3.2 and Theorem 3.3. Example illustrating the sharpness of Theorem 4.2. We can also construct an example of a vector field in B 3) for which the helicity flux is bounded from below by a positive constant. Indeed, let U(k) be a vector field U : Z3 → C3 with U(±λq, 0, 0) = λ−2/3q (0, 0,−1), U(0,±λq, 0) = λ−2/3q (1, 0, 1), U(±λq,±λq, 0) = λ−2/3q (0, 0, 1), U(±λq,∓λq, 0) = λ−2/3q (1, 1,−1), for all q ∈ N and zero otherwise. Denote ρ(x) = F−1χ(4ξ),A = ρ(x)3 dx, and let (56) u(x) = P U(k)eik·xρ(x). Note that u ∈ B2/33,∞(R3). On the other hand, a computation similar to the one in Section 3.4 yields (57) lim inf |HQ| ≥ 4A. 16 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY 4.2. Enstrophy. We work with the case of a two dimensional fluid in this section. In order to obtain an expression for the enstrophy flux one can use the original weak formulation of the Euler equation for velocities (9) with the test function chosen to be (58) ψ = ∇⊥S2Qω. Let us denote by ΩQ the expression resulting on the right hand side of (9): (59) ΩQ = SQ(u⊗ u) · ∇∇⊥SQω Thus, d‖SQω‖22 = ΩQ. As before we write rQ(u, u) · ∇∇⊥SQω (u− SQu)⊗ (u− SQu) · ∇∇⊥SQω Let us denote cq = ‖∆qω‖3,(61) c2 = {c2q}∞q=−1,(62) W (q) = λ2q, q ≤ 0; λ−4q , q > 0, We have the following estimate (absolute constants are omitted) |ΩQ| ≤ ∣∣∣h̃Q(y) ∣∣∣ (‖∇SQu‖23|y|2 + ‖(I − SQ)u‖23)‖∇2SQω‖3dy + ‖(I − SQ)u‖23‖∇2SQω‖3 λ−2Q ‖SQω‖23 + λ−2q c λ−2q c ≤ ‖SQω‖23 λ−4Q−qc λ2Q−qc λ−4Q−qc ≤ ‖SQω‖23(W ∗ c2)1/2(Q) + (W ∗ c2)3/2(Q) Thus, we have proved the following proposition. ENERGY CONSERVATION 17 FIGURE 1. Construction of the vector field illustrating in- frared nonlocality. Proposition 4.3. The enstrophy flux of a divergence-free vector field satis- fies the following estimate up to multiplication by an absolute constant (64) |ΩQ| ≤ ‖SQω‖23(W ∗ c2)1/2(Q) + (W ∗ c2)3/2(Q). Consequently, every weak solution to the 2D Euler equation with ω ∈ L3([0, T ];L3) conserves enstrophy. Much stronger results concerning conservation of enstrophy are available for the Euler equations ([13], [20]) and for the long time zero-viscosity limit for damped and driven Navier-Stokes equations ([9]). Example illustrating infrared nonlocality. We conclude this section with a construction of a vector field for which the enstrophy cascade is nonlocal in the infrared range. Let θq = arcsin(λq−Q−2) and (65) U lq = (cos(θq),− sin(θq)), Uhq = (sin(θq), cos(θq)), klq = λq(sin(θq), cos(θq)), k λ2Q+2 − λ2q(cos(θq),− sin(θq)), see Fig. 4.2 for the case q = Q. Denote ρ(x) = δh̃(δx),A = ρ(x)3 dx =∫ h̃(x)3 dx. Note that A > 0 and is independent of δ. Now let (67) ulq(x) = P[U q sin(k q · x)ρ(x)], uhq (x) = P[Uhq sin(khq · x)ρ(x)]. (68) uq(x) = u q(x) + u q (x) 18 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY for q = 0, . . . , Q, and (69) uQ+1(x) = P[V sin(λQ+2x1)ρ(x)], where V = (0, 1). Now define (70) u(x) = uq(x). Our goal is to estimate the enstrophy flux for u. Since Fu is compactly supported, the expression (59) is equivalent to (71) ΩQ = (u · ∇)S2Qω · ω dx. It is easy to see that (72) ΩQ ≥ (uhq · ∇)S2Q(∇⊥ · ulq)(∇⊥ · uQ+1) dx. Using Lemma 3.5 we obtain ΩQ ≥ A |Uhq |λ2q|U lq|λQ+2|V |+O(δ) = λQ+2‖∆Q+2u‖3 λ2q‖∆qu‖23 +O(δ), which shows sharpness of (64) in the infrared range. 5. INEQUALITIES FOR THE NONLINEAR TERM We take d = 3 and consider u, v ∈ B 3,2 with ∇ · u = 0 and wish to examine the advective term (74) B(u, v) = P(u · ∇v) = ΛH(u⊗ v) where (75) [H(u⊗ v)]i = Rj(ujvi) +Ri(RkRl(ukvl)) and P is the Leray-Hodge projector, Λ = (−∆) 12 is the Zygmund operator and Rk = ∂kΛ −1 are Riesz transforms. Proposition 5.1. The bilinear advective term B(u, v) maps continuously the space B 3,2 × B 3,2 to the space B . More precisely, there exist ENERGY CONSERVATION 19 bilinear continuous maps C(u, v), I(u, v) so that B(u, v) = C(u, v) + I(u, v) and constants C such that, for all u, v ∈ B 3,2 with ∇ · u = 0, (76) ‖C(u, v)‖ ≤ C‖u‖ (77) ‖I(u, v)‖ ≤ C‖u‖ hold. If u, v, w ∈ B (78) |〈B(u, v), w〉| ≤ C‖u‖ holds. So the trilinear map (u, v, w) 7→ 〈B(u, v), w〉 defined for smooth vector fields in L3 has a unique continuous extension to and a fortiori to Proof. We use duality. We take w smooth (w ∈ B ) and take the scalar product 〈B(u, v), w〉 = B(u, v) · wdx We write, in the spirit of the paraproduct of Bony ([1]) (79) ∆q(B(u, v)) = Cq(u, v) + Iq(u, v) (80) Cq(u, v) = p≥q−2, |p−p′|≤2 ∆q(ΛH(∆pu,∆p′v)) Iq(u, v) = [∆qΛH(Sq+j−2u,∆q+jv) + ∆qΛH(Sq+j−2v,∆q+ju)] 20 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY We estimate the contribution coming from the Cq(u, v): |〈Cq(u, v), w〉| |q−q′|≤1 p≥q−2, |p−p′|≤2 3∆pu‖L3‖Λ 3∆p′v‖L3‖∆q′w‖L3 |p−p′|≤2 3∆pu‖L3‖Λ 3∆p′v‖L3 q≤p+2,|q−q′|≤1 3∆q′w‖L3 |p−p′|≤2 ‖Λ 13∆pu‖ L3‖Λ 3∆p′v‖L3  ‖w‖ ≤ C‖u‖ This shows that the bilinear map C(u, v) = q≥−1Cq(u, v) maps continu- ously (82) |〈C(u, v), w〉| ≤ C‖u‖ The terms Iq(u, v) contribute |〈Iq(u, v), w〉| |j|≤2, |q−q′|≤1 λq‖Sq+j−2u‖ ‖∆q+jv‖L3‖∆q′w‖ |j|≤2, |q−q′|≤1 λq‖Sq+j−2v‖ ‖∆q+ju‖L3‖∆q′w‖ ≤ C‖u‖ |j|≤2,|q−q′|≤1 q ‖∆q+jv‖L3λ q ‖∆q′w‖ +C‖v‖ |j|≤2,|q−q′|≤1 q ‖∆q+ju‖L3λ q ‖∆q′w‖ ≤ C‖u‖ Here we used the fact that ‖Squ‖ ≤ C‖u‖ ENERGY CONSERVATION 21 This last fact is proved easily: ‖Sq(u)‖ ∥∥∥∥∥∥ |∆ju|2 ∥∥∥∥∥∥ ‖∆ju‖2 ≤ C‖u‖ We used Minkowski’s inequality in L 4 in the penultimate inequality and Bernstein’s inequality in the last. This proves that I maps continuously 3,2 × B 3,2 to B The proof of (78) follows along the same lines. Because of Bernstein’s inequalities, the inequality (82) for the trilinear term 〈C(u, v), w〉 is stronger than (78). The estimate of I follows: |〈Iq(u, v), w〉| |j|≤2, |q−q′|≤1 λq‖Sq+j−2u‖ ‖∆q+jv‖ ‖∆q′w‖ |j|≤2, |q−q′|≤1 λq‖Sq+j−2v‖ ‖∆q+ju‖ ‖∆q′w‖ ≤ C‖u‖ |j|≤2,|q−q′|≤1 q ‖∆q+jv‖ q ‖∆q′w‖ +C‖v‖ |j|≤2,|q−q′|≤1 q ‖∆q+ju‖ q ‖∆q′w‖ + ‖v‖ This concludes the proof. ✷ The inequality (82) is not true for 〈B(u, v), w〉 and (78) is close to being optimal: Proposition 5.2. For any 0 ≤ s ≤ 1 , 1 < p < ∞, 2 < r ≤ ∞ there exist functions u, v, w ∈ Bsp,r and smooth, rapidly decaying functions un, vn, wn, such that limn→∞ un = u, limn→∞ vn = v, limn→∞wn = w hold in the norm of Bsp,r and such that 〈B(un, vn), wn〉 = ∞ Proof. We start the construction with a divergence-free, smooth function u such that Fu ∈ C∞0 (B(0, 14)) and u31dx > 0. We select a direction 22 A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY e = (1, 0, 0) and set Φ = (0, u1, 0). Then (83) A := (u(x) · e) ∣∣P⊥e Φ(x) ∣∣2 dx > 0. Next we consider the sequence aq = so that (aq) ∈ ℓr(N) for r > 2, but not for r = 2, and the functions (84) vn = q aqP [sin(λqe · x)Φ(x)] (85) wn = q aqP [cos(λqe · x)Φ(x)] . Clearly, the limits v = limn→∞ vn and w = limn→∞wn exist in norm in every Bsp,r with 0 ≤ s ≤ 12 , 1 < p < ∞ and r > 2. Manifestly, by construction, u, vn and wn are divergence-free, and because their Fourier transforms are in C∞0 , they are rapidly decaying functions. Clearly also 〈B(u, vn), wn〉 = P(u · ∇vn)wndx = (u · ∇vn) · wndx. The terms corresponding to each q in u · ∇vn = (u(x) · e)aqλ q P [cos(λqe · x)Φ(x)] q u(x) · P [sin(λqe · x)∇Φ(x)] and in (85) have Fourier transforms supported B(λqe, ) ∪ B(−λq, 12) and respectively B(λqe, ) ∪ B(−λqe, 14). These are mutually disjoint sets for distinct q and, consequently, the terms corresponding to different indices q do not contribute to the integral (u · ∇vn) · wndx. The terms from the second sum in (86) form a convergent series. 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Constantin) DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CHICAGO, CHICAGO, IL 60637 E-mail address: [email protected] (S. Friedlander and R. Shvydkoy) DEPARTMENT OF MATHEMATICS, STAT. AND COMP. SCI., UNIVERSITY OF ILLINOIS, CHICAGO, IL 60607 E-mail address: [email protected] E-mail address: [email protected]

0704.0760

Search for Heavy, Long-Lived Particles that Decay to Photons at CDF II

Search for Heavy, Long-Lived Particles that Decay to Photons at CDF II A. Abulencia,24 J. Adelman,13 T. Affolder,10 T. Akimoto,55 M.G. Albrow,17 S. Amerio,43 D. Amidei,35 A. Anastassov,52 K. Anikeev,17 A. Annovi,19 J. Antos,14 M. Aoki,55 G. Apollinari,17 T. Arisawa,57 A. Artikov,15 W. Ashmanskas,17 A. Attal,3 A. Aurisano,42 F. Azfar,42 P. Azzi-Bacchetta,43 P. Azzurri,46 N. Bacchetta,43 W. Badgett,17 A. Barbaro-Galtieri,29 V.E. Barnes,48 B.A. Barnett,25 S. Baroiant,7 V. Bartsch,31 G. Bauer,33 P.-H. Beauchemin,34 F. Bedeschi,46 S. Behari,25 G. Bellettini,46 J. Bellinger,59 A. Belloni,33 D. Benjamin,16 A. Beretvas,17 J. Beringer,29 T. Berry,30 A. Bhatti,50 M. Binkley,17 D. Bisello,43 I. Bizjak,31 R.E. Blair,2 C. Blocker,6 B. Blumenfeld,25 A. Bocci,16 A. Bodek,49 V. Boisvert,49 G. Bolla,48 A. Bolshov,33 D. Bortoletto,48 J. Boudreau,47 A. Boveia,10 B. Brau,10 L. Brigliadori,5 C. Bromberg,36 E. Brubaker,13 J. Budagov,15 H.S. Budd,49 S. Budd,24 K. Burkett,17 G. Busetto,43 P. Bussey,21 A. Buzatu,34 K. L. Byrum,2 S. Cabreraq,16 M. Campanelli,20 M. Campbell,35 F. Canelli,17 A. Canepa,45 S. Carilloi,18 D. Carlsmith,59 R. Carosi,46 S. Carron,34 B. Casal,11 M. Casarsa,54 A. Castro,5 P. Catastini,46 D. Cauz,54 M. Cavalli-Sforza,3 A. Cerri,29 L. Cerritom,31 S.H. Chang,28 Y.C. Chen,1 M. Chertok,7 G. Chiarelli,46 G. Chlachidze,17 F. Chlebana,17 I. Cho,28 K. Cho,28 D. Chokheli,15 J.P. Chou,22 G. Choudalakis,33 S.H. Chuang,52 K. Chung,12 W.H. Chung,59 Y.S. Chung,49 M. Cilijak,46 C.I. Ciobanu,24 M.A. Ciocci,46 A. Clark,20 D. Clark,6 M. Coca,16 G. Compostella,43 M.E. Convery,50 J. Conway,7 B. Cooper,31 K. Copic,35 M. Cordelli,19 G. Cortiana,43 F. Crescioli,46 C. Cuenca Almenarq,7 J. Cuevasl,11 R. Culbertson,17 J.C. Cully,35 S. DaRonco,43 M. Datta,17 S. D’Auria,21 T. Davies,21 D. Dagenhart,17 P. de Barbaro,49 S. De Cecco,51 A. Deisher,29 G. De Lentdeckerc,49 G. De Lorenzo,3 M. Dell’Orso,46 F. Delli Paoli,43 L. Demortier,50 J. Deng,16 M. Deninno,5 D. De Pedis,51 P.F. Derwent,17 G.P. Di Giovanni,44 C. Dionisi,51 B. Di Ruzza,54 J.R. Dittmann,4 M. D’Onofrio,3 C. Dörr,26 S. Donati,46 P. Dong,8 J. Donini,43 T. Dorigo,43 S. Dube,52 J. Efron,39 R. Erbacher,7 D. Errede,24 S. Errede,24 R. Eusebi,17 H.C. Fang,29 S. Farrington,30 I. Fedorko,46 W.T. Fedorko,13 R.G. Feild,60 M. Feindt,26 J.P. Fernandez,32 R. Field,18 G. Flanagan,48 R. Forrest,7 S. Forrester,7 M. Franklin,22 J.C. Freeman,29 I. Furic,13 M. Gallinaro,50 J. Galyardt,12 J.E. Garcia,46 F. Garberson,10 A.F. Garfinkel,48 C. Gay,60 H. Gerberich,24 D. Gerdes,35 S. Giagu,51 P. Giannetti,46 K. Gibson,47 J.L. Gimmell,49 C. Ginsburg,17 N. Giokarisa,15 M. Giordani,54 P. Giromini,19 M. Giunta,46 G. Giurgiu,25 V. Glagolev,15 D. Glenzinski,17 M. Gold,37 N. Goldschmidt,18 J. Goldsteinb,42 A. Golossanov,17 G. Gomez,11 G. Gomez-Ceballos,33 M. Goncharov,53 O. González,32 I. Gorelov,37 A.T. Goshaw,16 K. Goulianos,50 A. Gresele,43 S. Grinstein,22 C. Grosso-Pilcher,13 R.C. Group,17 U. Grundler,24 J. Guimaraes da Costa,22 Z. Gunay-Unalan,36 C. Haber,29 K. Hahn,33 S.R. Hahn,17 E. Halkiadakis,52 A. Hamilton,20 B.-Y. Han,49 J.Y. Han,49 R. Handler,59 F. Happacher,19 K. Hara,55 D. Hare,52 M. Hare,56 S. Harper,42 R.F. Harr,58 R.M. Harris,17 M. Hartz,47 K. Hatakeyama,50 J. Hauser,8 C. Hays,42 M. Heck,26 A. Heijboer,45 B. Heinemann,29 J. Heinrich,45 C. Henderson,33 M. Herndon,59 J. Heuser,26 D. Hidas,16 C.S. Hillb,10 D. Hirschbuehl,26 A. Hocker,17 A. Holloway,22 S. Hou,1 M. Houlden,30 S.-C. Hsu,9 B.T. Huffman,42 R.E. Hughes,39 U. Husemann,60 J. Huston,36 J. Incandela,10 G. Introzzi,46 M. Iori,51 A. Ivanov,7 B. Iyutin,33 E. James,17 D. Jang,52 B. Jayatilaka,16 D. Jeans,51 E.J. Jeon,28 S. Jindariani,18 W. Johnson,7 M. Jones,48 K.K. Joo,28 S.Y. Jun,12 J.E. Jung,28 T.R. Junk,24 T. Kamon,53 P.E. Karchin,58 Y. Kato,41 Y. Kemp,26 R. Kephart,17 U. Kerzel,26 V. Khotilovich,53 B. Kilminster,39 D.H. Kim,28 H.S. Kim,28 J.E. Kim,28 M.J. Kim,17 S.B. Kim,28 S.H. Kim,55 Y.K. Kim,13 N. Kimura,55 L. Kirsch,6 S. Klimenko,18 M. Klute,33 B. Knuteson,33 B.R. Ko,16 K. Kondo,57 D.J. Kong,28 J. Konigsberg,18 A. Korytov,18 A.V. Kotwal,16 A.C. Kraan,45 J. Kraus,24 M. Kreps,26 J. Kroll,45 N. Krumnack,4 M. Kruse,16 V. Krutelyov,10 T. Kubo,55 S. E. Kuhlmann,2 T. Kuhr,26 N.P. Kulkarni,58 Y. Kusakabe,57 S. Kwang,13 A.T. Laasanen,48 S. Lai,34 S. Lami,46 S. Lammel,17 M. Lancaster,31 R.L. Lander,7 K. Lannon,39 A. Lath,52 G. Latino,46 I. Lazzizzera,43 T. LeCompte,2 E. Lee,53 J. Lee,49 J. Lee,28 Y.J. Lee,28 S.W. Leeo,53 R. Lefèvre,20 N. Leonardo,33 S. Leone,46 S. Levy,13 J.D. Lewis,17 C. Lin,60 C.S. Lin,17 M. Lindgren,17 E. Lipeles,9 A. Lister,7 D.O. Litvintsev,17 T. Liu,17 N.S. Lockyer,45 A. Loginov,60 M. Loreti,43 R.-S. Lu,1 D. Lucchesi,43 P. Lujan,29 P. Lukens,17 G. Lungu,18 L. Lyons,42 J. Lys,29 R. Lysak,14 E. Lytken,48 P. Mack,26 D. MacQueen,34 R. Madrak,17 K. Maeshima,17 K. Makhoul,33 T. Maki,23 P. Maksimovic,25 S. Malde,42 S. Malik,31 G. Manca,30 F. Margaroli,5 R. Marginean,17 C. Marino,26 C.P. Marino,24 A. Martin,60 M. Martin,25 V. Marting,21 M. Mart́ınez,3 R. Mart́ınez-Ballaŕın,32 T. Maruyama,55 P. Mastrandrea,51 T. Masubuchi,55 H. Matsunaga,55 M.E. Mattson,58 R. Mazini,34 P. Mazzanti,5 K.S. McFarland,49 P. McIntyre,53 R. McNultyf ,30 A. Mehta,30 P. Mehtala,23 S. Menzemerh,11 A. Menzione,46 P. Merkel,48 C. Mesropian,50 A. Messina,36 T. Miao,17 N. Miladinovic,6 J. Miles,33 R. Miller,36 C. Mills,10 M. Milnik,26 A. Mitra,1 G. Mitselmakher,18 A. Miyamoto,27 S. Moed,20 N. Moggi,5 B. Mohr,8 C.S. Moon,28 http://arxiv.org/abs/0704.0760v1 R. Moore,17 M. Morello,46 P. Movilla Fernandez,29 J. Mülmenstädt,29 A. Mukherjee,17 Th. Muller,26 R. Mumford,25 P. Murat,17 M. Mussini,5 J. Nachtman,17 A. Nagano,55 J. Naganoma,57 K. Nakamura,55 I. Nakano,40 A. Napier,56 V. Necula,16 C. Neu,45 M.S. Neubauer,9 J. Nielsenn,29 L. Nodulman,2 O. Norniella,3 E. Nurse,31 S.H. Oh,16 Y.D. Oh,28 I. Oksuzian,18 T. Okusawa,41 R. Oldeman,30 R. Orava,23 K. Osterberg,23 C. Pagliarone,46 E. Palencia,11 V. Papadimitriou,17 A. Papaikonomou,26 A.A. Paramonov,13 B. Parks,39 S. Pashapour,34 J. Patrick,17 G. Pauletta,54 M. Paulini,12 C. Paus,33 D.E. Pellett,7 A. Penzo,54 T.J. Phillips,16 G. Piacentino,46 J. Piedra,44 L. Pinera,18 K. Pitts,24 C. Plager,8 L. Pondrom,59 X. Portell,3 O. Poukhov,15 N. Pounder,42 F. Prakoshyn,15 A. Pronko,17 J. Proudfoot,2 F. Ptohose,19 G. Punzi,46 J. Pursley,25 J. Rademackerb,42 A. Rahaman,47 V. Ramakrishnan,59 N. Ranjan,48 I. Redondo,32 B. Reisert,17 V. Rekovic,37 P. Renton,42 M. Rescigno,51 S. Richter,26 F. Rimondi,5 L. Ristori,46 A. Robson,21 T. Rodrigo,11 E. Rogers,24 S. Rolli,56 R. Roser,17 M. Rossi,54 R. Rossin,10 P. Roy,34 A. Ruiz,11 J. Russ,12 V. Rusu,13 H. Saarikko,23 A. Safonov,53 W.K. Sakumoto,49 G. Salamanna,51 O. Saltó,3 L. Santi,54 S. Sarkar,51 L. Sartori,46 K. Sato,17 P. Savard,34 A. Savoy-Navarro,44 T. Scheidle,26 P. Schlabach,17 E.E. Schmidt,17 M.P. Schmidt,60 M. Schmitt,38 T. Schwarz,7 L. Scodellaro,11 A.L. Scott,10 A. Scribano,46 F. Scuri,46 A. Sedov,48 S. Seidel,37 Y. Seiya,41 A. Semenov,15 L. Sexton-Kennedy,17 A. Sfyrla,20 S.Z. Shalhout,58 M.D. Shapiro,29 T. Shears,30 P.F. Shepard,47 D. Sherman,22 M. Shimojimak,55 M. Shochet,13 Y. Shon,59 I. Shreyber,20 A. Sidoti,46 P. Sinervo,34 A. Sisakyan,15 A.J. Slaughter,17 J. Slaunwhite,39 K. Sliwa,56 J.R. Smith,7 F.D. Snider,17 R. Snihur,34 M. Soderberg,35 A. Soha,7 S. Somalwar,52 V. Sorin,36 J. Spalding,17 F. Spinella,46 T. Spreitzer,34 P. Squillacioti,46 M. Stanitzki,60 A. Staveris-Polykalas,46 R. St. Denis,21 B. Stelzer,8 O. Stelzer-Chilton,42 D. Stentz,38 J. Strologas,37 D. Stuart,10 J.S. Suh,28 A. Sukhanov,18 H. Sun,56 I. Suslov,15 T. Suzuki,55 A. Taffardp,24 R. Takashima,40 Y. Takeuchi,55 R. Tanaka,40 M. Tecchio,35 P.K. Teng,1 K. Terashi,50 J. Thomd,17 A.S. Thompson,21 E. Thomson,45 P. Tipton,60 V. Tiwari,12 S. Tkaczyk,17 D. Toback,53 S. Tokar,14 K. Tollefson,36 T. Tomura,55 D. Tonelli,46 S. Torre,19 D. Torretta,17 S. Tourneur,44 W. Trischuk,34 S. Tsuno,40 Y. Tu,45 N. Turini,46 F. Ukegawa,55 S. Uozumi,55 S. Vallecorsa,20 N. van Remortel,23 A. Varganov,35 E. Vataga,37 F. Vazquezi,18 G. Velev,17 G. Veramendi,24 V. Veszpremi,48 M. Vidal,32 R. Vidal,17 I. Vila,11 R. Vilar,11 T. Vine,31 I. Vollrath,34 I. Volobouevo,29 G. Volpi,46 F. Würthwein,9 P. Wagner,53 R.G. Wagner,2 R.L. Wagner,17 J. Wagner,26 W. Wagner,26 R. Wallny,8 S.M. Wang,1 A. Warburton,34 D. Waters,31 M. Weinberger,53 W.C. Wester III,17 B. Whitehouse,56 D. Whiteson,45 A.B. Wicklund,2 E. Wicklund,17 G. Williams,34 H.H. Williams,45 P. Wilson,17 B.L. Winer,39 P. Wittichd,17 S. Wolbers,17 C. Wolfe,13 T. Wright,35 X. Wu,20 S.M. Wynne,30 A. Yagil,9 K. Yamamoto,41 J. Yamaoka,52 T. Yamashita,40 C. Yang,60 U.K. Yangj,13 Y.C. Yang,28 W.M. Yao,29 G.P. Yeh,17 J. Yoh,17 K. Yorita,13 T. Yoshida,41 G.B. Yu,49 I. Yu,28 S.S. Yu,17 J.C. Yun,17 L. Zanello,51 A. Zanetti,54 I. Zaw,22 X. Zhang,24 J. Zhou,52 and S. Zucchelli5 (CDF Collaboration∗) 1Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China 2Argonne National Laboratory, Argonne, Illinois 60439 3Institut de Fisica d’Altes Energies, Universitat Autonoma de Barcelona, E-08193, Bellaterra (Barcelona), Spain 4Baylor University, Waco, Texas 76798 5Istituto Nazionale di Fisica Nucleare, University of Bologna, I-40127 Bologna, Italy 6Brandeis University, Waltham, Massachusetts 02254 7University of California, Davis, Davis, California 95616 8University of California, Los Angeles, Los Angeles, California 90024 9University of California, San Diego, La Jolla, California 92093 10University of California, Santa Barbara, Santa Barbara, California 93106 11Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain 12Carnegie Mellon University, Pittsburgh, PA 15213 13Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637 14Comenius University, 842 48 Bratislava, Slovakia; Institute of Experimental Physics, 040 01 Kosice, Slovakia 15Joint Institute for Nuclear Research, RU-141980 Dubna, Russia 16Duke University, Durham, North Carolina 27708 17Fermi National Accelerator Laboratory, Batavia, Illinois 60510 18University of Florida, Gainesville, Florida 32611 19Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, I-00044 Frascati, Italy 20University of Geneva, CH-1211 Geneva 4, Switzerland 21Glasgow University, Glasgow G12 8QQ, United Kingdom 22Harvard University, Cambridge, Massachusetts 02138 23Division of High Energy Physics, Department of Physics, University of Helsinki and Helsinki Institute of Physics, FIN-00014, Helsinki, Finland 24University of Illinois, Urbana, Illinois 61801 25The Johns Hopkins University, Baltimore, Maryland 21218 26Institut für Experimentelle Kernphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany 27High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305, Japan 28Center for High Energy Physics: Kyungpook National University, Taegu 702-701, Korea; Seoul National University, Seoul 151-742, Korea; SungKyunKwan University, Suwon 440-746, Korea 29Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, California 94720 30University of Liverpool, Liverpool L69 7ZE, United Kingdom 31University College London, London WC1E 6BT, United Kingdom 32Centro de Investigaciones Energeticas Medioambientales y Tecnologicas, E-28040 Madrid, Spain 33Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 34Institute of Particle Physics: McGill University, Montréal, Canada H3A 2T8; and University of Toronto, Toronto, Canada M5S 1A7 35University of Michigan, Ann Arbor, Michigan 48109 36Michigan State University, East Lansing, Michigan 48824 37University of New Mexico, Albuquerque, New Mexico 87131 38Northwestern University, Evanston, Illinois 60208 39The Ohio State University, Columbus, Ohio 43210 40Okayama University, Okayama 700-8530, Japan 41Osaka City University, Osaka 588, Japan 42University of Oxford, Oxford OX1 3RH, United Kingdom 43University of Padova, Istituto Nazionale di Fisica Nucleare, Sezione di Padova-Trento, I-35131 Padova, Italy 44LPNHE, Universite Pierre et Marie Curie/IN2P3-CNRS, UMR7585, Paris, F-75252 France 45University of Pennsylvania, Philadelphia, Pennsylvania 19104 46Istituto Nazionale di Fisica Nucleare Pisa, Universities of Pisa, Siena and Scuola Normale Superiore, I-56127 Pisa, Italy 47University of Pittsburgh, Pittsburgh, Pennsylvania 15260 48Purdue University, West Lafayette, Indiana 47907 49University of Rochester, Rochester, New York 14627 50The Rockefeller University, New York, New York 10021 51Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1, University of Rome “La Sapienza,” I-00185 Roma, Italy 52Rutgers University, Piscataway, New Jersey 08855 53Texas A&M University, College Station, Texas 77843 54Istituto Nazionale di Fisica Nucleare, University of Trieste/ Udine, Italy 55University of Tsukuba, Tsukuba, Ibaraki 305, Japan 56Tufts University, Medford, Massachusetts 02155 57Waseda University, Tokyo 169, Japan 58Wayne State University, Detroit, Michigan 48201 59University of Wisconsin, Madison, Wisconsin 53706 60Yale University, New Haven, Connecticut 06520 (Dated: November 3, 2018; Version 5.1) We present the first search for heavy, long-lived particles that decay to photons at a hadron collider. We use a sample of γ+jet+missing transverse energy events in pp̄ collisions at 1.96 TeV taken with the CDF II detector. Candidate events are selected based on the arrival time of the photon at the detector. Using an integrated luminosity of 570 pb−1 of collision data, we observe 2 events, consistent with the background estimate of 1.3±0.7 events. While our search strategy does not rely on model-specific dynamics, we set cross section limits in a supersymmetric model with eχ01 → γ eG and place the world-best 95% C.L. lower limit on the eχ01 mass of 101 GeV/c2 at τχ̃0 = 5 ns. PACS numbers: 13.85.Rm, 12.60.Jv, 13.85.Qk, 14.80.Ly ∗With visitors from aUniversity of Athens, bUniversity of Bristol, cUniversity Libre de Bruxelles, dCornell University, eUniversity of Cyprus, fUniversity of Dublin, gUniversity of Ed- inburgh, hUniversity of Heidelberg, iUniversidad Iberoamericana, jUniversity of Manchester, kNagasaki Institute of Applied Science, lUniversity de Oviedo, mUniversity of London, Queen Mary Col- lege, nUniversity of California Santa Cruz, oTexas Tech University, Searches for events with final state photons and miss- ing transverse energy (E/T ) [1] at collider experiments are sensitive to new physics from a wide variety of mod- els [2] including gauge mediated supersymmetry breaking (GMSB) [3]. In these models the lightest neutralino (χ̃01) decays into a photon (γ) and a weakly interacting, stable gravitino (G̃) that gives rise to E/T by leaving the detec- tor without depositing any energy. The observation of an eeγγE/T candidate event by the CDF experiment during Run I at the Fermilab Tevatron [4] has increased the in- terest in experimental tests of this class of theories. Most subsequent searches have focused on promptly produced photons [5, 6], however the χ̃01 can have a lifetime on the order of nanoseconds or more. This is the first search for heavy, long-lived particles that decay to photons at a hadron collider. We optimize our selection requirements using a GMSB model with a standard choice of parameters [7] and vary the values of the χ̃01 mass and lifetime. However, the final search strategy is chosen to be sufficiently general and independent of the specific GMSB model dynamics to yield results that are approximately valid for any model producing the same reconstructed final state topology and kinematics [8]. In pp̄ collisions at the Tevatron the inclusive GMSB production cross section is dominated by pair production of gauginos. The gauginos decay promptly, resulting in a pair of long-lived χ̃01’s in asso- ciation with other final state particles that can be identi- fied as jets. For a heavy χ̃01 decaying inside the detector, the photon can arrive at the face of the detector with a time delay relative to promptly produced photons. To have good sensitivity for nanosecond-lifetime χ̃01’s [8], we search for events that contain a time-delayed photon, E/T , and ≥ 1 jet. This is equivalent to requiring that at least one of the long-lived χ̃01’s decays inside the detector. This Letter summarizes [9] the first search for heavy, long-lived particles that decay to photons at a hadron collider. The data comprise 570±34 pb−1 of pp̄ collisions collected with the CDF II detector [10] at s = 1.96 TeV. Previous searches for nanosecond-lifetime particles using non-timing techniques yielded null results [11]. A full description of the CDF II detector can be found elsewhere [10]. Here we briefly describe the aspects of the detector relevant to this analysis. The magnetic spec- trometer consists of tracking devices inside a 3-m diame- ter, 5-m long superconducting solenoid magnet that op- erates at 1.4 T. An eight-layer silicon microstrip detector array and a 3.1-m long drift chamber with 96 layers of sense wires measure the position (~xi) and time (ti) of the pp̄ interaction [12] and the momenta of charged particles. Muons from collisions or cosmic rays are identified by a pUniversity of California Irvine, qIFIC(CSIC-Universitat de Valen- cia), system of drift chambers situated outside the calorime- ters in the region with pseudorapidity |η| < 1.1 [1]. The calorimeter consists of projective towers with elec- tromagnetic and hadronic compartments. It is divided into a central barrel that surrounds the solenoid coil (|η| < 1.1) and a pair of end-plugs that cover the region 1.1 < |η| < 3.6. Both calorimeters are used to identify and measure the energy and position of photons, elec- trons, jets, and E/T . The electromagnetic calorimeters were recently instrumented with a new system, the EM- Timing system (completed in Fall 2004) [13], that mea- sures the arrival time of electrons and photons in each tower with |η| < 2.1 for all energies above ∼5 GeV. The time and position of arrival of the photon at the calorimeter, tf and ~xf , are used to separate the photons from the decays of heavy, long-lived χ̃01’s from promptly produced photons or photons from non-collision sources. We define the corrected arrival time of the photon as tγc ≡ tf − ti − |~xf − ~xi| The tγc distribution for promptly produced, high energy photons is Gaussian with a mean of zero by construction and with a standard deviation that depends only on the measurement resolution assuming that the pp̄ production vertex has been correctly identified. Photons from heavy, long-lived particles can have arrival times that are many standard deviations larger than zero. The analysis preselection is summarized in Table I. It begins with events passing an online, three-level trigger by having a photon candidate in the region |η| < 1.1 with ET> 25 GeV and E/T> 25 GeV. Offline, the high- est ET photon candidate in the fiducial region of the calorimeter is required to have ET > 30 GeV and to pass the standard photon identification requirements [5] with a minor modification [14]. We require the event to have E/T > 30 GeV where the trigger is 100% effi- cient. We require at least one jet with |ηjet| < 2.0 and > 30 GeV [15]. Since a second photon can be identi- fied as a jet, the analysis is sensitive to signatures where one or both χ̃01’s decay inside the detector. To ensure a high quality ti and ~xi measurement, we require a ver- tex with at least 4 tracks, tracks pT > 15 GeV/c, and |zi| < 60 cm; this also helps to reduce non-collision back- grounds. For events with multiple reconstructed vertices, we pick the vertex with the highest tracks pT . To re- duce cosmic ray background, events are rejected if there are hits in a muon chamber that are not matched to any track and are within 30◦ of the photon. After the above requirements there are 11,932 events in the data sample. There are two major classes of background events: col- lision and non-collision photon candidates. Collision pho- tons are presumed to come from standard model interac- tions, e.g., γ+jet+mismeasured E/T , dijet+mismeasured E/T where the jet is mis-identified as a γ, and W → eν where the electron is mis-identified as a γ. Non-collision Preselection Requirements Cumulative (individual) Efficiency (%) > 30 GeV, E/T > 30 GeV 54 (54) Photon ID and fiducial, |η| < 1.0 39 (74)* Good vertex, tracks pT > 15 GeV/c 31 (79) |ηjet| < 2.0, Ejet > 30 GeV 24 (77) Cosmic ray rejection 23 (98)* Requirements after Optimization E/T > 40 GeV, E > 35 GeV 21 (92) ∆φ(E/T , jet) > 1 rad 18 (86) 2 ns < tγc < 10 ns 6 (33) TABLE I: The data selection criteria and the cumulative and individual requirement efficiencies for an example GMSB model point at mχ̃0 = 100 GeV/c2 and τχ̃0 = 5 ns. The ef- ficiencies listed are, in general, model-dependent and have a fractional uncertainty of 10%. Model-independent efficiencies are indicated with an asterisk. The collision fiducial require- ment of |zi| < 60 cm is part of the good vertex requirement (95%) and is estimated from data. backgrounds come from cosmic rays and beam effects that can produce photon candidates, E/T , and sometimes the reconstructed jet. We separate data events as a func- tion of tγc into several control regions that allow us to estimate the number of background events in the final signal region by fitting to the data using collision and non-collision shape templates as shown in Fig. 1. Collision photons are subdivided in two subclasses: correct and incorrect vertex selection [13]. An incorrect vertex can be selected when two or more collisions occur in one beam bunch crossing, making it possible that the highest reconstructed tracks pT vertex does not produce the photon. While the fraction of events with incorrect vertices depends on the final event selection criteria, the tγc distribution for each subclass is estimated separately using W → eν data where the electron track is dropped from the vertexing. For events with a correctly associ- ated vertex, the tγc distribution is Gaussian and centered at zero with a standard deviation of 0.64 ns [13]. For those with an incorrectly selected vertex the tγc distribu- tion is also Gaussian with a standard deviation of 2.05 ns. The tγc distributions for both non-collision backgrounds are estimated separately from data using events with no reconstructed tracks. Photon candidates from cos- mic rays are not correlated in time with collisions, and therefore their tγc distribution is roughly flat. Beam halo photon candidates are produced by muons that origi- nate upstream of the detector (from the p direction) and travel through the calorimeter, typically depositing small amounts of energy. When the muon deposits significant energy in the EM calorimeter, it can be misidentified as a photon and cause E/T . These photons populate predomi- nantly the negative tγc region, but can contribute to the signal region. Since beam halo muons travel parallel to the beam line, these events can be separated from cosmic ray events by identifying the small energy deposited in the calorimeter towers along the beam halo muon trajec- tory. The background prediction uses control regions out- side the signal time window but well within the 132 ns time window that the calorimeter uses to measure the energy. The non-collision background templates are nor- malized to match the number of events in two time win- dows: a beam halo-dominated window at {−20, −6} ns, selected to be 3σ away from the wrong vertex collision background, and a cosmic rays-dominated window at {25, 90} ns, well away from the standard model and beam halo contributions. The collision background is estimated by fitting events in the {−10, 1.2} ns window with the non-collision contribution subtracted and with the fraction of correct to incorrect vertex events allowed to vary. In this way the background for the signal region is entirely estimated from data samples. The systematic uncertainty on the background estimate is dominated by our ability to calibrate the mean of the tγc distribution for prompt photons. We find a variation of 200 ps on the mean and 20 ps on the standard deviation of the dis- tribution by considering various possible event selection criteria. These contribute to the systematic uncertainty of the collision background estimate in the signal region and are added in quadrature with the statistical uncer- tainties of the final fit procedure. We estimate the sensitivity to heavy, long-lived parti- cles that decay to photons using GMSB models for dif- ferent χ̃01 masses and lifetimes. Events from all SUSY processes are simulated with the pythia Monte Carlo program [16] along with the detector simulation [17]. The acceptance is the ratio of simulated events that pass all the requirements to all events produced. It is used in the optimization procedure and in the final limit setting and depends on a number of effects. The fraction of χ̃01 decays in the detector volume is the dominant effect on the acceptance. For a given lifetime this depends on the boost of the χ̃01. A highly boosted χ̃ 1 that decays in the detector typically does not contribute to the accep- tance because it tends to produce a photon traveling in the same direction as the χ̃01. Thus, the photon’s arrival time is indistinguishable from promptly produced pho- tons. At small boosts the decay is more likely to happen inside the detector, and the decay angle is more likely to be large, which translates into a larger delay for the photon. The fraction of events with a delayed photon ar- rival time initially rises as a function of χ̃01 lifetime, but falls as the fraction of χ̃01’s decaying outside the detector begins to dominates. In the χ̃01 mass region considered (65 ≤ mχ̃0 ≤ 150 GeV/c2), the acceptance peaks at a lifetime of around 5 ns. The acceptance also depends on the mass as the boost effects are mitigated by the ability to produce high energy photons or E/T in the collision, as discussed in Ref. [8]. The total systematic uncertainty of 10% on the ac- Photon Corrected Time of Arrival (ns) -20 0 20 40 60 80 )-1 + Jet data (570 pb E + γ Standard Model Beam Halo Cosmics GMSB Signal MC Photon Corrected Time of Arrival (ns) -20 0 20 40 60 80 FIG. 1: The time distribution for photons passing all but the final timing requirement for the background predic- tions, data, and a GMSB signal for an example point at = 100 GeV/c2, τχ̃0 = 5 ns. A total of 1.3±0.7 back- ground events are predicted and 2 (marked with a star) are observed in the signal region of 2 < tγc < 10 ns. ceptance is dominated by the uncertainty on the mean of the tγc distribution (7%) and on the photon ID effi- ciency (5%). Other significant contributions come from uncertainties on initial and final state radiation (3%), jet energy measurement (3%), and the parton distribution functions (1%). We determine the kinematic and tγc selection require- ments that define the final data sample by optimizing the expected cross section limit without looking at the data in the signal region. To compute the expected 95% confidence level (C.L.) cross section upper limit [18], we combine the predicted GMSB signal and background esti- mates with the systematic uncertainties using a Bayesian method with a flat prior [19]. The expected limits are op- timized by simultaneously varying the selection require- ments for E/T , photon ET , jet ET , azimuth angle be- tween the leading jet and E/T (∆φ(E/T , jet)), and t c . The ∆φ(E/T , jet) requirement rejects events where the E/T is overestimated because of a poorly measured jet. While each point in χ̃01 lifetime vs. mass space gives a slightly different optimization, we choose a single set of require- ments because it simplifies the final analysis, while only causing a small loss of sensitivity. The optimized require- ments are summarized in Table I. As an example, the ac- ceptance for mχ̃0 = 100 GeV/c2 and lifetime τχ̃0 = 5 ns is estimated to be (6.3±0.6)%. After all kinematic requirements, 508 events are ob- served in the data before the final signal region time re- quirement. Their time distribution is shown in Fig. 1. Our fit to the data outside the signal region predicts total backgrounds of 6.2±3.5 from cosmic rays, 6.8±4.9 from beam halo background sources, and the rest from the )2 mass (GeV/c 65 70 75 80 85 90 95 100 105 110 0 1χ∼ 1.0 pb 0.5 pb 0.3 pb 0.2 pb 0.13 pb FIG. 2: The contours of constant 95% C.L. upper cross section limits for a GMSB model [7]. standard model. Inside the signal time region, {2, 10} ns, we predict 1.25±0.66 events: 0.71±0.60 from standard model, 0.46±0.26 from cosmic rays, and 0.07±0.05 from beam halo. Two events are observed in the data. Since the result is consistent with the no-signal hypothesis, we set limits on the χ̃01 lifetime and mass. Figure 2 shows the contours of constant 95% C.L. cross section upper limit. Figure 3 shows the exclusion region at 95% C.L., along with the expected limit for comparison. This takes into account the predicted production cross section at next- to-leading order [20] as well as the uncertainties on the parton distribution functions (6%) and the renormaliza- tion scale (2%). Since the number of observed events is above expectations, the observed limits are slightly worse than the expected limits. These limits extend at large masses beyond those of LEP searches using photon “pointing” methods [11]. In conclusion, we have performed the first search for heavy, long-lived particles that decay to photons at a hadron collider using data collected with the EMTim- ing system at the CDF II detector. There is no excess of events beyond expectations. As our search strategy does not rely on event properties specific solely to GMSB models, we can exclude any γ+jet+E/T signal that would produce more than 5.5 events. We set cross section limits using a supersymmetric model with χ̃01 → γG̃, and find a GMSB exclusion region in the χ̃01 lifetime vs. mass plane with the world-best 95% C.L. lower limit on the χ̃01 mass of 101 GeV/c 2 at τχ̃0 = 5 ns. Future improve- ments with similar techniques should also provide sen- sitivity to new particle decays with a delayed electron signature [2]. By the end of Run II, an integrated lumi- nosity of 10 fb−1 is possible for which we estimate a mass reach of ≃ 140 GeV/c2 at a lifetime of 5 ns. )2 mass (GeV/c 65 70 75 80 85 90 95 100 105 110 0 1χ∼ )-1+1jet analysis with EMTiming (570 pb Predicted exclusion region Observed exclusion region ALEPH exclusion upper limit χ∼GMSB )=15β, tan(Λ=2messM >0µ=1, messN 65 70 75 80 85 90 95 100 105 110 FIG. 3: The exclusion region at 95% C.L. as a function of eχ01 lifetime and mass for a GMSB model [7]. The predicted and the observed regions are shown separately and are compared to the most stringent published limit from LEP searches [11]. We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contribu- tions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Culture, Sports, Science and Technology of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; the Swiss National Science Founda- tion; the A.P. Sloan Foundation; the Bundesministerium für Bildung und Forschung, Germany; the Korean Sci- ence and Engineering Foundation and the Korean Re- search Foundation; the Particle Physics and Astronomy Research Council and the Royal Society, UK; the Russian Foundation for Basic Research; the Comisión Interminis- terial de Ciencia y Tecnoloǵıa, Spain; in part by the Eu- ropean Community’s Human Potential Programme un- der contract HPRN-CT-2002-00292; and the Academy of Finland. [1] We use a cylindrical coordinate system in which the pro- ton beam travels along the z-axis, θ is the polar angle, φ is the azimuthal angle, and η = − ln tan(θ/2). The trans- verse energy and momentum are defined as ET = E sin θ and pT = p sin θ where E is the energy measured by the calorimeter and p the momentum measured in the tracking system. E/T = | − EiT ~ni| where ~ni is a unit vector that points from the interaction vertex to the ith calorimeter tower in the transverse plane. [2] J. L. Feng, A. Rajaraman and F. Takayama, Phys. Rev. D 68, 063504 (2003); M. J. Strassler and K. M. Zurek, arXiv:hep-ph/0605193. [3] S. Ambrosanio et al., Phys. Rev. D 54, 5395 (1996); C. H. Chen and J. F. Gunion, Phys. Rev. D 58, 075005 (1998). [4] F. Abe et al. (CDF Collaboration), Phys. Rev. Lett. 81, 1791 (1998) and Phys. Rev. D 59, 092002 (1999). [5] D. Acosta et al. (CDF Collaboration), Phys. Rev. D 71, 031104 (2005). [6] V. Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 94, 041801 (2005). [7] B. C. Allanach et al., Eur. Phys. J. C25, 113 (2002). We use benchmark model 8 and allow the eG mass factor and the supersymmetry breaking scale to vary independently. [8] D. Toback and P. Wagner, Phys. Rev. D 70, 114032 (2004). [9] P. Wagner, Ph.D. Thesis, Texas A&M University, 2007. [10] D. Acosta et al. (CDF Collaboration), Phys. Rev. D 71, 032001 (2005). [11] A. Heister et al. (ALEPH Collaboration), Eur. Phys. J. C 25, 339 (2002); also see M. Gataullin, S. Rosier, L. Xia and H. Yang, arXiv:hep-ex/0611010; G. Abbiendi et al. (OPAL Collaboration), Proc. Sci. HEP2005 346 (2006); J. Abdallah et al. (DELPHI Collaboration), Eur. Phys. J. C 38 395 (2005). [12] The distribution of the pp̄ collisions has a standard devi- ation of 30 cm and 1.3 ns in zi and ti, respectively. [13] M. Goncharov et al., Nucl. Instrum. Methods A565, 543 (2006). [14] The standard requirement, χ2CES < 20 (see F. Abe et al. (CDF Collaboration), Phys. Rev. D 52, 4784 (1995)), has been removed because there is evidence that it is in- efficient for photons that arrive with large incident angles relative to the face of the detector. [15] See F. Abe et al. (CDF Collaboration), Phys. Rev. D 45, 1448 (1992). We use corrected jets reconstructed with a cone of ∆R = 0.7, see A. Bhatti et al., Nucl. In- strum. Methods A566, 375 (2006). [16] T. Sjöstrand et al., Comput. Phys. Commun. 135, 238 (2001). We use version 6.216. [17] We use the standard geant based detector simulation [R. Brun et al., CERN-DD/EE/84-1 (1987)] and add a parametrized EMTiming simulation. [18] E. Boos, A. Vologdin, D. Toback, and J. Gaspard, Phys. Rev. D 66, 013011 (2002). [19] J. Conway, CERN Yellow Book Report No. CERN 2000- 005, 2000, p. 247. [20] W. Beenakker et al., Phys. Rev. Lett. 83, 3780 (1999). http://arxiv.org/abs/hep-ph/0605193 http://arxiv.org/abs/hep-ex/0611010

0704.0761

Failure of the work-Hamiltonian connection for free energy calculations

Failure of the work-Hamiltonian connection for free energy calculations Failure of the work-Hamiltonian connection for free energy calculations Jose M. G. Vilar1 and J. Miguel Rubi2 1Computational Biology Program, Memorial Sloan-Kettering Cancer Center, 1275 York Avenue, New York, NY 10021 2Departament de Fisica Fonamental, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain Abstract Extensions of statistical mechanics are routinely being used to infer free energies from the work performed over single-molecule nonequilibrium trajectories. A key element of this approach is the ubiquitous expression / ( , )dW dt H x t t/= ∂ ∂ , which connects the microscopic work W performed by a time-dependent force on the coordinate x with the corresponding Hamiltonian (H x t), at time t . Here we show that this connection, as pivotal as it is, cannot be used to estimate free energy changes. We discuss the implications of this result for single-molecule experiments and atomistic molecular simulations and point out possible avenues to overcome these limitations. PACS numbers: 05.40.-a, 05.20.-y, 05.70.Ln Hamiltonians provide two key ingredients to bridge the microscopic structure of nature with macroscopic thermodynamic properties: they completely specify the underlying dynamics and they can be identified with the energy of the system [1]. At equilibrium, the link with the thermodynamic properties is established through the partition function ( )H xZ e dβ−= ∫ x , which here uses the Hamiltonian in the coordinate space ( )H x x as the energy of the system [2]. In particular, the free energy is given by = − Z , where 1 Bk Tβ ≡ / is the inverse of the temperature T times the Boltzmann’s constant . Thermodynamic properties play an important role because they provide information that is not readily available from the microscopic properties, such as whether or not a given process happens spontaneously. The connection between work and Hamiltonian expressed through the relation W H x )t, , or equivalently through its integral representation ( ( ') ') ' W H x t t ∂∫ dt , is typically used to extend statistical mechanics to far-from- equilibrium situations [3-5]. These relations are meant to imply that the work W performed on a system is used to change its energy. The potential advantage of this type of approach is that it would allow one to infer thermodynamic properties even when the relevant details of the Hamiltonian are not known or when they are too complex for a direct analysis. Experiments and computer simulations can thus be performed to probe the microscopic mechanical properties from which to obtain thermodynamic properties. Time-dependent Hamiltonians, however, provide the energy up to an arbitrary factor that typically depends on time and on the microscopic history of the system. Such dependence, as we show below, prevents this approach from being generally applicable to compute thermodynamic properties. To illustrate how work and Hamiltonian fail to be generally connected, we consider a system described by the Hamiltonian under the effects of a time-dependent force 0( )H x ( )f t . The total Hamiltonian is given by 0( ) ( ) ( ) (H x t H x f t x g t), = − + , where is an arbitrary function of time, which leads to a total force . The function does not affect the total force but it changes the Hamiltonian. Therefore, has to be chosen so that the Hamiltonian can be identified with the energy of the system. ( )g t 0 /F H x f= −∂ ∂ + ( )t ( )g t ( )g t In general, the arbitrary time dependence of the Hamiltonian, , cannot be chosen so that the Hamiltonian gives a consistent energy. Consider, for instance, that the system, being initially at ( )g t 0x , is subjected to a sudden perturbation 0( ) ( )f t f t≡ Θ , where 0f is a constant and is the Heaviside step function. The work performed on the system, , where ( )tΘ 0( tW f x x= − 0 ) ( )tx x t≡ represents the value of the coordinate x at time , is in general different from ' 0 0 ( ') ' ( ) (0 tH x t dt f x g t gt , = − + − ∂∫ ) , irrespective of the explicit form of the function . ( )g t To illustrate the consequences of the lack of connection between work and changes in the Hamiltonian, we focus on the domain of validity of nonequilibrium work relations [3] of the type ,EG We eβ β− Δ −= which have been widely used recently to obtain estimates EGΔ of free energy changes from single-molecule pulling experiments [6] and atomistic computer simulations [7]. The promise of this type of relations is that they provide the values of the free energy from irreversible trajectories and therefore do not require equilibration of the system. Yet, in almost all instances in which this approach has been applied, the agreement with the canonical thermodynamic results has not been complete and in some cases the discrepancies have been large. These discrepancies have been attributed to the presence of statistical errors in the estimation of the exponential average We β− [8]. Currently, the mathematical validity of these type of nonequilibrium work relations appears to be well established: they have been derived using approximations [3] and rigorously for systems described by Langevin equations [4, 5]. However, all these derivations rely in different ways on the work-Hamiltonian connection, which as we show below prevents them from giving general estimates of thermodynamic free energies. The free energy difference between two states is defined as revG WΔ = , where is the work required to bring the system from the initial to the final state in a reversible manner [2]. Note that, if the system is not macroscopic, is in general a fluctuating quantity. At quasi-equilibrium, the external force ( )f t balances with the system force . After integration by the displacement, the reversible work done on the system is given by . Therefore, the free energy follows from ( )H x x−∂ /∂ 0 0( ) ( )rev tW H x H x= − 0 0 0( , ) ( ,0)rev eq t eq tG W P x t P x dx dxΔ = ,∫ ∫ where the equilibrium probabilities are obtained, in the usual way, from the Boltzmann distribution ( )( , ) H x t eq Z tP x t e β− ,= . To be explicit, let us consider a harmonic system described by 210 2( )H x kx= and ( ) 0g t = , with a constant. In this case, we can compute exactly the free energy change: G kxΔ = , where ( )eqx f t k≡ / , which leads to a positive value as required for non-spontaneous processes. One might have been tempted to use the partition function to estimate changes in free energy according to the expression 1 ln( ( ) (0))ZG Z t ZβΔ = − / , where ( )( ) H x tZ t e dβ− ,= ∫ x is the time-dependent quasi-equilibrium partition function [3, 4]. However, this relation is not valid when changes in the Hamiltonian cannot be associated with changes in energy. In the case of the harmonic potential, the use of the time-dependent partition function leads to 212Z eqG kΔ = − x , a negative value inconsistent with a process that is not spontaneous. More generally, the Hamiltonian 212( ) ( )(tH x t kx f t x )γ, = − − , where γ is a constant parameter that does not affect the dynamics of the system, leads to 2( )Z eq eqG kx xγΔ = − , which can be positive or negative depending on the value of γ . Therefore, the estimates ZGΔ are not suitable to predict typical thermodynamic properties, such as whether or not a process happens spontaneously. To what extent does the failure of the work-Hamiltonian connection impact nonequilibrium work equalities? In the case of a sudden perturbation and a harmonic potential discussed previously, the following result follows straightforwardly: 0 0( ) 0 0( , ) ( ,0) 1t f x xW eq t eq te e P x t P x dx dx ββ − −− = =∫ ∫ , which is different from . Ge β− Δ An intriguing question then arises: why do experiments and computer simulations sometimes lead to results that agree with nonequilibrium work equalities? Let us consider a situation closer to the experimental and computational setups, with a harmonic time- dependent force that constrains the motion on the coordinate x : 210 2( ) ( ) ( )tH x t H x K x X, = + − . Here is a constant and K tX is the time-dependent equilibrium position for the constraining force. In this case, with 210 2( )H x kx= and 0 0X = , we also have 2rev eq G W kxΔ = = , where now Keq tk Kx X+≡ . For quasi-equilibrium displacements of tX , so that the work performed is equal to the reversible work, , we have 0 0( ) ( )rev tW W H x H x= = − 0 0 0 0( ( ) ( )) 0 0( , ) ( ,0)rev t W H x H x eq t eq te e P x t P x dx β β− − −= ,∫ ∫ dx which leads to ( ) 2 2( 2 ) ( ) k k K eqk K W e ke K k K This result indicates that quasi-equilibrium does not guarantee the accuracy of the exponential estimate of the free energy from nonequilibrium work relations. The free energy change and its exponential estimate GΔ EGΔ agree with each other only for large values of . The reason is that, in this case, work and Hamiltonian are connected to each K other when both quasi-equilibrium and large- conditions are fulfilled simultaneously. Under such conditions, the work-Hamiltonian connection is valid because eq tx x X≈ ≈ implies that the rate of change of the Hamiltonian, ( ) / ( ) /t tH x t t K x X dX dt∂ , ∂ = − − , equals the power associated with the external force, / ( ) /tdW dt K x X dx dt= − − . Interestingly, large values of suppress fluctuations and lead to quasi-deterministic dynamics. Indeed, the experimental data [6] and computer simulations [7] indicate that the agreement between the free energy change GΔ and its exponential estimate EGΔ occurs mainly for relatively slow perturbations that lead to quasi-deterministic trajectories. Bringing thermodynamics to nonequilibrium microscopic processes [9] is becoming increasingly important with the advent of new experimental and computational techniques able to probe the properties of single molecules [6, 7]. Our results show that the classical connection between work and changes in the Hamiltonian cannot be applied straightforwardly to time-dependent systems. As a result, quantities that are based on the work-Hamiltonian connection, such as those obtained from nonequilibrium work relations and time-dependent partition functions, cannot generally be used to estimate thermodynamically consistent free energy changes. A possible avenue to overcome these limitations, as we have shown here, is to identify the particular conditions for which work and changes in the Hamiltonian are connected to each other. References [1] H. Goldstein, Classical mechanics (Addison-Wesley Pub. Co., Reading, Mass., 1980). [2] R. C. Tolman, The principles of statistical mechanics (Oxford University Press, London, 1955). [3] C. Jarzynski, Physical Review Letters 78, 2690 (1997). [4] G. Hummer, and A. Szabo, Proc Natl Acad Sci USA 98, 3658 (2001). [5] A. Imparato, and L. Peliti, Physical Review E 72, 046114 (2005). [6] J. Liphardt et al., Science 296, 1832 (2002). [7] S. Park et al., Journal of Chemical Physics 119, 3559 (2003). [8] J. Gore, F. Ritort, and C. Bustamante, Proc Natl Acad Sci USA 100, 12564 (2003). [9] D. Reguera, J. M. Rubi, and J. M. G. Vilar, Journal of Physical Chemistry B 109, 21502 (2005).

0704.0762

Dependence of the Critical Adsorption Point on Surface and Sequence Disorders for Self-Avoiding Walks Interacting with a Planar Surface

Microsoft Word - Hetsurf_Mac_revision.doc Dependence of the Critical Adsorption Point on Surface and Sequence Disorders for Self-Avoiding Walks Interacting with a Planar Surface Jesse D. Ziebarth1, Yongmei Wang1*, Alexey Polotsky2, Mengbo Luo3 1Department of Chemistry, the University of Memphis, Memphis, Tennessee 38152, USA 2 Sérvice de Physique de l'Etat Condensé CEA Saclay, 91191 Gif-sur-Yvette Cedex, France. 3Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China Abstract. The critical adsorption point (CAP) of self-avoiding walks (SAW) interacting with a planar surface with surface disorder or sequence disorder has been studied. We present theoretical equations, based on ones previously developed by Soteros and Whittington (J. Phys. A.: Math. Gen. 2004, 37, R279-R325), that describe the dependence of CAP on the disorders along with Monte Carlo simulation data that are in agreement with the equations. We also show simulation results that deviate from the equations when the approximations used in the theory break down. Such knowledge is the first step toward understanding the correlation of surface disorder and sequence disorder during polymer adsorption. *Corresponding author: email: [email protected]. Tel: 901-678-2629. Fax: 901-678- 3447. 1. Introduction Adsorption of polymers on surfaces plays a key role in many technological applications and is also relevant to many biological processes. As a result, it has been studied for more than three decades1 and continues to receive intense interest.2 The field is rich and contains a wide variety of topics, from equilibrium properties of adsorbed layers and conformations of adsorbed polymer chains to dynamic properties and non-equilibrium processes in adsorption.2 For polymer adsorption on planar surfaces, it is well-known that there exists a critical adsorption point (CAP) that marks the transition of a polymer chain, in contact with a surface, from a non-adsorbed state to an adsorbed state.3 Scaling laws for a variety of quantities below, above and at the CAP for a homopolymer in contact with a planar surface were developed by Eisenriegler, Kremer, and Binder (EKB).4 For example, when the chain goes from a non-adsorbed state to an adsorbed state, the energy of the chain E changes from an intensive variable independent of chain length N to an extensive variable dependent on N. At the CAP, E is expected to scale with Nφ where φ is the crossover exponent. Numerical studies, including exact enumeration,5 the scanning method6,7 and the multiple Markov chain method8 have been performed to determine the location of the CAP and the crossover exponent φ. The values reported are however not completely in agreement with each other and are still under debate, especially the crossover exponent φ. The disagreement may be traced, as suggested by a recent article,9 to different methods used for determining the CAP and the crossover exponent φ. While many studies focused on adsorption of homopolymers on planar homogeneous surfaces, adsorption of polymers on chemically or physically heterogeneous surfaces has also received a fair amount of studies.10-20 Some were inspired by specific applications such as segregation of polymer chains on patterned surfaces,10 or pattern transfer via surface adsorption,21,22 others were motivated by a desire to understand how the presence of surface or sequence disorders may influence adsorption.13,14,16,17,23-25 For example, Sebastian and Sumithra developed an analytical theory of the adsorption of Gaussian chains on random surfaces using Gaussian variational approach.24,25 They took surface heterogeneity into account by modifying de Genne’s adsorption boundary condition and analyzed influence of randomness on the conformation of the adsorbed chains. Adsorption of heteropolymers on heterogeneous surfaces, in particular, has been studied because of its relevance to molecular recognition in biological process. The concept of “pattern matching” was proposed26 and has been investigated with different approaches.12,20,26,27 Muthukumar for example derived an equation for the critical condition of adsorption of a polyelectrolyte to an oppositely charged patterned surface.26 Golumbfski et al.12 showed that a statistical blocky chain was selectively adsorbed on a patchy surface while a statistically alternating chain was selectively adsorbed on an alternating surface. Jayaraman et al.19 described a simulation method to design surfaces for recognizing specific monomer sequences in heteropolymers. Recently Polotsky et al18 considered adsorption of Gaussian heteropolymer chains onto heterogeneous surface. They found that the presence of correlations between sequence and surface heterogeneity always enhances adsorption. However, the dependence of the critical adsorption point on either surface disorder or sequence disorder is not well-understood. Lack of this knowledge hampers further understanding on the correlation between sequence disorder and surface disorder during adsorption. Here we present theoretical equations that describe the dependence of CAP on the surface disorder or sequence disorder, along with Monte Carlo simulation data in agreement with the derived equations. The current study does not address the correlation between sequence disorder and surface disorder. We only consider cases where the disorder is either present randomly on the surface (i.e. adsorption of homopolymers on random heterogeneous surface) or on the sequence (i.e., adsorption of random copolymer on homogeneous surface). The correlation between sequence disorder and surface disorder will be the subject of future publications. In the following, we first present the theory that predicts the dependence of CAP on surface disorder and sequence disorder. Then we present details of Monte Carlo simulation methods used to determine the CAP, followed by simulation data that agree with the derived equations. Finally, we discuss implications of these results on practical applications such as chromatographic separations of polymers. 2. Theory 2.1 Adsorption of a homopolymer on a homogeneous surface We first consider adsorption of a homopolymer chain on a homogeneous surface. This can be represented by a self-avoiding walk (SAW) in a three-dimensional lattice interacting with a plane and restricted to lie on one side of the plane. The vertices of the walks interact with the surface sites with an attractive energy εw. The partition function for a N-step SAW interacting with a homogeneous surface is given by ( )∑= wNw vvcNZ εε exp)(),(homo (1) where cN(v) is the number of SAWs that lie above the surface with v visits to the surface. Hammersley et al.28 have shown that the model exhibits a phase transition at a critical adsorption energy, εc, with a desorbed state for εw < εc, and an adsorbed state for εw > εc. They have shown that the limiting monomer free energy f(εw) ),(log lim)( homo wNw NZN = (2) exists and is a convex non-decreasing continuous function of εw. Moreover, f(εw)=κ for εw ≤ 0, where κ is the lattice connective constant, and f(εw) is a strictly increasing function of εw when εw > εc. Therefore, f(εw) is non-analytic at εw = εc. εc has also been determined to be greater than zero and, based on the best-known connective constant for the simple cubic lattice29, to have an upper bound of 0.5738. The lattice connective constant κ is also the limiting monomer free energy of the SAWs in bulk solution. Hence the CAP can be understood as the condition where the limiting monomer free energy of a chain attached to the surface becomes equal to the limiting monomer free energy of the chain in the bulk solution. 2.2 Adsorption of a homopolymer on a random heterogeneous surface Now we consider the adsorption of a homopolymer interact with a heterogeneous surface consisting of two types of surface sites, A and B. The interaction energy of the vertices with the two surface sites are εwA and εwB. Following Soteros and Whittington23, and express the partition function of a N-step SAW interacting with a heterogeneous surface that consists of A and B surface sites as: ( ) ( )∑ ∑ ANBAhet BvfAvfAv vcffZ )()( )((exp)(exp )(),( εε (3) where cN(v) is the number of walks that have v surface contacts, v(A) is the number of monomers interacting with the A sites, and v(B) is the number of monomers interacting with the B sites, fA and fB = 1 - fA are the fractions of A and B sites on the surface, respectively. Here the partition function is averaged over random distributions of the surface sites, i.e. the so called annealed approximation. Physically the annealed disorder means that the type of surface sites may change while the system attains equilibrium state. However, it has been previously suggested11,15 that the annealed approximation is valid if the chain can visit a large area of the surface and hence samples all distributions of surface patterns. Furthermore, the surface sites are randomly distributed. If there is a correlation between surface disorders, such as those present in patchy surface or alternating surface, then Eq. (3) will not be valid, as Eq. (3) gives equal weight to all possible surface labelings, while correlations restrict possible labelings. Summing over v(A), equation (3) can be simplified to ( )∑ += wANBAhet ffvcffZ )exp()exp()(),( εε (4) A comparison of equations (1) and (4) reveals that the partition functions for homogeneous and annealed random heterogeneous surface become equivalent if ( ) ( ) )exp(expexp BwBAwAw ff εεε += (5) From Eq. (5), we derive the following equation that gives the dependence of CAP on the surface disorder: ( ) ( ) ))(exp(exp)1()(exp ccffcc BwBAwBhw εεε +−= (6) where εwh(cc) is the CAP of a homopolymer above a homogeneous surface, εwB(cc) is the CAP of a homopolymer above a heterogeneous surface while the surface interaction energy εwA held constant. It can be easily seen from this equation, that the dependence of the CAP on the percentage of attractive sites on the surface is not expected to be linear, in contrast to the conclusion drawn by an earlier study.13 Equation (6) is expected to be valid as long as the two conditions are met: (i) the chain has enough mobility to visit a large area of surface so that the annealed approximation is valid, and (ii) the surface sites are randomly distributed (i.e. uncorrelated). 2.3 Adsorption of a random heteropolymer on a homogeneous surface The same approach can be extended to consider the adsorption of a random heteropolymer interacting with a homogeneous surface. We will use the same notation as in previous section except now fA and fB represent fractions of A and B monomers present on the heteropolymer. We will only consider random copolymers composed by A and B monomers. The sequence of a random copolymer can be represented by χ ={χ1, χ2, … χN} where χi are independently and identically distributed random variables with χi =A with a probability of fA and χi=B with a probability of 1-fA. A sequence order parameter λ can be defined to characterize the sequence randomness.12,27 BAAB pp −−= 1λ (7) where pij is the nearest neighbor transition probabilities which is the probability that a monomer of type i is followed by a monomer of type j. When λ=0, the sequence is random. When λ>0, then the sequence is statistically blocky, and when λ<0, the sequence is statistically alternating. We note that a given random sequence designated by χ may have non-zero values of λ. More discussions will be given in the later section. The partition function of N-step SAWs with the given sequence above a homogenous surface is written as: )exp()|,(),,( BwB BANBAhetpoly vvvvCffZ εεχχ += ∑ (8) There are two different ways to average over different distributions of random sequences, namely the annealed average and the quenched average. With the annealed average, the partition function in Eq. (8) is first averaged over different distributions of χ. This then leads to a partition function, Zhetpoly(fA, fB), which is exactly the same as in Eq. (3). With the annealed approximation, we derive the same equation as given by Eq. (6) for the CAP of a random heteropolymer interacting with a homogeneous surface, provided that fA and fB now represent the fractions of A and B monomers on the chain. In the following, we will present Monte Carlo simulation data that conform to the two equations and also results that do not conform to the equations because of the invalidation of the approximations used in deriving the equations. 3. Monte Carlo Simulation Methods In our simulations, polymer chains are modeled as SAWs with N vertices on a simple cubic lattice of dimensions 250a × 250a × 100a, where a is the lattice spacing. Each vertex represents a monomer on the polymer chain. Chain lengths studied are in the range of N = 25 to 250. There is an impenetrable wall in the z = a plane representing the surface. One monomer, picked randomly from the chain, is first placed on a site adjacent to the wall (in the z = 2a plane). The rest of the chain is then grown using the biased chain insertion method.30 Monomers that are in the z = 2a plane are considered to be adsorbed on the surface. For all adsorbed monomers, an attractive polymer-surface interaction, εw, is applied. The standard chemical potential of the chain (since it does not contain translation entropy), µ0, is calculated from the Rosenbluth- Rosenbluth weighting factor, W(N), which is given by30 0 ln)(lnβµ and )exp( β (9) where z is the lattice coordination number (z = 6 for simple cubic lattice), Ej is the energy of ith inserted monomer in the jth potential direction. We note that µ0 calculated is the free energy per chain, and µ0/N is free energy per monomer discussed in equation (2). Typically, the chemical potential is determined based on about twenty million copies of trial chain conformations. We obtained the standard chemical potentials of a chain with at least one monomer attached to the surface, µads0, and compared that against a chain grown in a bulk solution, µbulk0. The bulk solution is modeled by a 100a × 100a × 100a lattice with periodic boundary conditions applied in all three directions. All chemical potentials calculated are reduced by the Boltzmann factor, β=1/kBT=1. A coefficient K, similar to partition coefficient if the chain was placed in a pore instead of near a surface, is calculated by K =exp(-∆µ0), where ∆µ0 = µads0 − µbulk0. The way we determined the CAP is based on the dependence of K on the chain length N and will be presented in the results section. Heterogeneous surfaces were modeled by making the z = a plane composed of two different types of sites, which have different values for polymer-surface interactions. The designations εwA and εwB will be used to distinguish between interaction energies of different site types. Simulations were performed using surfaces with different fractions of A and B sites. Surfaces were created by randomly assigning each site as A or B based on the probabilities, pA and pB, where pA and pB are, respectively, the desired fractions of A and B sites on the surface. Because of size of the surface, this procedure resulted in the real surface composition percentages matching the desired percentages within 0.1%. For a given surface composition, the surface was randomly created once and was subsequently used in all simulations that determine the chemical potential of a chain above that surface. The surfaces displayed quenched randomness, i.e. the surface pattern remained unchanged throughout the simulations. However, the first bead of chain was placed randomly over the surface during the chain insertion, and hence the chemical potential determined has been averaged over different surface randomness. Therefore, the annealed approximation used in deriving Eq. (6) was met in the simulations. In a few cases, patchy and alternating surfaces were created by simulating a two-dimensional Ising model at appropriate conditions. Heteropolymers were modelled as SAWs consisting of two types of monomers, A and B with specified fractions fA and fB=1-fA.. Chains were created by randomly selecting N*fB different positions along the chain to be B beads, while the remaining beads were assigned as A beads, ensuring that the chain had the exact composition called for by fA and fB. The sequence order parameter, λ, in generated random sequences exhibits a Gaussian distribution with zero mean. Examples of distributions are presented in Figure 1. The longer the chain, the narrower the distribution is. For a given chain length N, we typically generate 5000 copies of random sequences with specified fA. Each sequence is then used in biased insertion for 5000 or more copies to obtain the Rosenbluth-Rosenbluth weighting factor. Letting W(N, χ) stands for the sequence order parameter λ -1.0 -0.5 0.0 0.5 1.0 N=100 N=200 Figure 1: Distribution of sequence order parameters obtained from 5000 copies of random sequences generated with fA = fB = 0.50 for three different chain lengths. Lines are smooth fit to the data. Rosenbluth-Rosenbluth weighting factor obtained for a given sequence χ, the chemical potential of a chain can be obtained using two different averages over sequences: ),(ln)(0 χβµ NWNads −= (10) ),(ln),()( 00 χχβµβµ NWNN adsads −== (11) The first approach is the annealed average, while the second approach is the quenched average. The two chemical potentials calculated differ slightly from each other. More discussion of the quenched versus annealed averages will be given later. For the determination of CAP, we have used annealed chemical potentials. 4. Results and Discussion 4.1. Method Used to Determine the Critical Adsorption Point The method we used to determine the CAP follows our earlier papers31,32 and is briefly sketched out. We obtain the difference in standard chemical potential ∆µ0 at different surface interaction εw for a set of chains with different lengths. An example of data is presented in Figure 2(a) for a homopolymer above a homogeneous surface. The lines for different length N nearly intersect at a common point, which is estimated to be at εc=0.276 ± 0.005. A convenient way to identify this intersection point is to plot the standard deviation of all ∆µ0, σ(∆µ0), for a given range of chain length studied versus εw, which yields a minimum in a plot shown in Figure 2(b). The minimum identified is directly related to the critical condition point employed in liquid chromatography at the critical condition (LCCC) 32-34. In LCCC, the critical condition was defined as the co-elution point of homopolymers with different molecular weights, which, corresponding to computer simulation, is the point where K has least dependence on chain length. If K is truly independent of chain length, then σ(∆µ0) will be zero and will be the minimum in a plot in Figure 2(b). The critical condition point bracketed in this fashion depends slightly on the range of chain length included in the calculation of σ(∆µ0). However, in the current study we fixed the range of chain lengths used. Since this common intersection point does not occur at ∆µ0 =0, one may wonder if it is the critical adsorption point discussed in the literature. We have applied the same method for random walks above a planar surface in simple cubic lattice31. The intersection point found was at εc = 0.183± 0.002, in excellent agreement with expected CAP for random-walks, εc = -ln(5/6)= 0.1823.1 On the other hand, CAP could be understood as the point where the limiting monomer free energy for a chain attached to the surface f(ε) equals to the limiting monomer free energy of an unattached chain in the bulk solution. Therefore, we may define a CAP at a finite chain 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 N=100 N=200 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 Figure 2: (a) Plot of ∆µ0 versus εw for SAW chains with N =25, 50, 100 and 200 above a homogeneous surface. The critical adsorption point is identified as the common intersection point, εw(cc)=0.276±0.005. (b) Plot of deviation in ∆µ0 for the given range of N versus εw. The minimum in the plot is the critical adsorption point. length, εc (N), at which ∆µ0(N)=0. From Figure 2(a), we extract such εc (N). This εc (N) is expected to depend on N in a scaling law, εc (N) = εc(∞) –αN−φ, and εc(∞) is the CAP at infinite chain length limit. Assuming φ = 0.5, Figure 3 shows the linear fitting of εc (N) versus N−0.5 which yields εc (∞) = 0.274 ± 0.005. The εc (∞) identified is within the error bars of the common intersection point. The CAP of SAWs in simple cubic lattice has been studied by others.6-8 The reported literature value for the CAP of SAWs on the simple cubic lattice ranged from ~0.37 by Ma et al.35 down to 0.288 ± 0.02 by Janse van Rensburg and Rechnitzer8. The value reported by Ma et al. was considered to be too high, probably due to chains analyzed being too short. Methods used to determine the CAP varied in the literature. Meirovitch and Livne6 obtained the CAP for SAW in simple cubic lattice with Monte Carlo simulations with the scanning method. They plotted E(T)/N against N and found the exponent α in E(T)/N~Nα over three different ranges of chain length (N = 20-60, 60-170, and 170-350). Then, the critical point was located by finding the N-0.5 0.00 0.05 0.10 0.15 0.20 0.25 Figure 3: Plot of εc(N) versus N-0.5 where εc(N) is extracted from figure 1(a) as the point when ∆µ(N) = 0. The extrapolated εc(∞) =0.274 ± 0.005. value of the reciprocal temperature Θ that resulted in the exponent α being constant for the three different ranges of chain lengths. Their reported Θc, which is equivalent to our εc, was 0.291 + 0.001. Their method for determining Θc was based on the scaling theory developed by EKB.4 As stated earlier, at CAP, E(T)/N is expected to scale with Nφ-1 where φ is the crossover exponent. The value of this crossover exponent was debated. EKB first showed that φ ≈ ν ≈ 0.59, where ν is the Flory’s exponent. Several recent reports suggest that φ = 0.5 even for SAW chains, the same as φ for random-walks.8,36 In Meriovitch and Levin’s study, φ was left as an adjustable parameter. The reported φ value in their study was =0.530+ 0.007, slightly larger than recent reported values φ=0.5. If we were to take φ=0.5, then their data would suggest a lower Θc. Recently Decase et al.9 explored four different ways to determine the CAP, mostly based on the scaling idea. They found that a slight change of εc lead to large deviations in the resulting φ. Therefore, simultaneous determination of εc and φ may not give the true location of CAP. Janse van Rensburg and Rechnitzer8 studied CAP for SAWs in two and three dimensions using a variety of methods, including studying the energy ratios of walks of different lengths and the specific heats of the chains. They found that analysis of the specific heat data in three dimensions were fraught with difficulty. The energy ratios of different lengths and the free energy method yielded εc within the error bars. They reported a value for the CAP, εc=0.288 + 0.020 and a crossover exponent φ = 0.5005 + 0.0036. Our CAP is within the error bars of their reported value. Interestingly, if they assume that the convergence of the energy ratios of different chain lengths is proportional to N1 , the yielded εc = 0.276 + 0.029, exactly the same as in our study. The above discussion suggests that the critical condition determined with our approach is the CAP. Our approach to determine the CAP does not depend on knowledge of φ and therefore does not suffer from the uncertainty in εc when both εc and φ need to be determined simultaneously. In the remainder of the paper, we will use this method to determine the CAP of SAWs above a planar heterogeneous surface and SAWs for heteropolymers above a planar homogeneous surface. 4.2. Homopolymers above Heterogeneous Surfaces with Attractive and Non-Interacting Sites Here we consider adsorption of homopolymers above a heterogeneous surface. The first type of heterogeneous surface studied consists of a surface composed of two types of sites. One type of the surface sites, which will be called A sites, did not interact with the polymer chains; that is, εwA = 0. The other type of surface site, the B sites, had an attractive interaction with the polymer chains, εwB. The value of εwB was varied to locate the CAP. Figure 4 shows a plot of the standard deviations in β∆µ0 over all chain lengths for each value of εwB scanned. The minimum in standard deviations occurs for εwB(cc) = 0.49± 0.01, where the error was based on the energy increment scanned. The same method was used to determine the CAP for surfaces with 10%, 15%, 20%, 25%, and 75% attractive sites. Table I summarizes the CAP of homopolymers over heterogeneous surfaces along with the data over a homogeneous surface. Figure 5 presents the plot of CAP, εwB(cc), as a function of fB along with the theoretical prediction according to Eq. (8) with εwA = 0 and εwh(cc) = 0.276. It is clear that a good agreement between Eq. (6) and simulation data is observed. Also we note that CAP is not linearly dependent on fB over the entire range but is well-described by Eq.(6). Earlier study by Sumithra and Baumgaertner13 focused on surfaces with fB above the percolation threshold. Within that limited range of fB, a linear dependence may be obtained. This study is the first to confirm the dependence of CAP on the surface disorder over a wide range of fB. 0.44 0.46 0.48 0.50 0.52 0.54 0.56 Figure 4: Plot of deviation in ∆µ0 against εwB for a homogeneous chain adsorbing on a surface with 50% attractive sites and 50% non-interacting sites. The CAP occurs at εwB = 0.49 + 0.01. As discussed in the theory section, one of the assumptions used in deriving Eq. (6) is that the interacting surface sites are randomly distributed. We have tested this assumption by studying adsorption of homopolymers over a 50% surface with alternating and patchy patterns. For a surface with 50% of A and B, an order parameter O.P. can be defined (readers are referred to literature for the definition).19 If O.P.=0, the surface is random; if O.P.=+1, then the surface is patchy; and if O.P.=-1, the surface is alternating. The data are also included in Table I and are indicated in Figure 4. The two points deviate from the line described by Eq. (6). The CAP obtained over a 50% alternating surface is larger than that over a 50% random surface. On the other hand, the CAP obtained over a 50% patchy surface is smaller than over a 50% random surface. These results can be easily understood. When a chain is adsorbed on the surface, it forms trains, loops and tails.1 Formation of trains lowers the energy of a chain to overcome the entropy loss during the adsorption. When a chain is in contact with an alternating surface, it is however difficult to form trains as no adsorbing sites are adjacent, while this is possible for Percent B sites 0 20 40 60 80 100 Figure 5: Plot of the CAP, εwB(cc), against the percent of attractive B sites, fB. The symbols are the CAP determined by the simulation, and the solid line is from equation (6) with εWA =0.0 and εwh(cc) =0.276. Circles are CAP over random surfaces, the cross (×) is the CAP over a strictly alternating surface, and the upper triangle (∆) is the CAP over a patchy surface with O.P. =+0.94. random and patchy surfaces. Therefore, chains attraction to the alternating surface is lessened, and adsorption over a 50% alternating surface has to occur at a larger value of εw. On the other hand, a chain over a patchy surface can selectively sample patches of the surface composed of adsorbing sites, so the adsorption over patchy surface can occur at a smaller value of εw. Another assumption used in deriving Eq. (6) is the annealed approximation. This approximation is strictly met if the surface pattern in contact with the chain changes during the chain adsorption,11 hence averaging over different distributions can be performed as done in Eq. (3). The surface in this case is said to contain annealed randomness. If the surface pattern can not change, then the surface is said to contain quenched randomness. In our simulations, the surface contains quenched randomness. In fact, we have used only one realization of a quenched random surface. However, the chain was placed randomly over different surface sites, making the annealed approximation applicable to our simulations. We note that Sumithra and Baumgaertner13, in their studies, averaged over 50 different realizations of quenched randomness and they compared the results with that of a single surface realization. They did not find major difference between these two approaches, especially if the temperature is high. Moghaddam and Whittington16 investigated the difference between the quenched average and the annealed average for homopolymer adsorption on heterogeneous surface and random copolymer adsorption on homogeneous surface. Their data show that there was no difference between the two averages in the case of adsorption on random surfaces but there were differences for adsorption of random copolymers especially at low temperature. It has been argued that quenched and annealed averages are equivalent in cases where the quenched surface is large in comparison with the polymer.11,15 Polotsky et. al18 have also found that the CAP for quenched and annealed surface disorders are the same. In our simulations, the surface is large in comparison with the size of the polymer, and the attachment of the polymer to the surface occurs at many random places on the surface. Therefore, the chain can effectively interact with many different random arrangements of surface sites, and the system approaches the annealed average. 4.3. Homopolymers above Heterogeneous Surfaces with All Sites Interacting In order to assess whether the equation derived for the CAP of random surfaces was valid in more general cases, random surfaces that contained all attractive sites were prepared. For these surfaces, the polymer-surface interaction for the A sites, εwA, was set at a relatively weak attractive strength, 0.10, and the interaction for the B surface sites was varied to find the CAP. Additionally, surfaces with repulsive A sites (εwA = -.10) were also investigated. Percent B Sites 0 20 40 60 80 100 Figure 6: Plot of the critical adsorption point, εwB(cc), against the percent of attractive B sites for surfaces with attractive or repulsive A sites. The dashed line and open symbols are for surfaces with slightly repulsive A sites, εwA=-0.10. The solid line and closed symbols are for surfaces with slightly attractive A sites, εwA=+0.10. The symbols are simulation results, while the lines are from equation (6) with the corresponding εwA values. Figure 6 shows the values of εwB(cc) determined for these two cases, as well as the prediction of the value of εwB(cc) given the values of fB and εwA used in the simulation. As can be seen in the figure, there is a good agreement between the data and the equation, indicating that the equation is valid for surfaces with many different types of surfaces, not just surfaces with attractive and non-interacting sites. 4.4. Random Copolymers above Homogeneous Surfaces Critical adsorption point for random copolymers adsorbing on homogeneous surfaces were also determined. In these systems, polymer chains are considered to be composed of two different types of monomers, A’s and B’s, interacting with a surface composed of only one type of site. B monomers were attracted to the surface, while A monomers do not interact with the surface, i.e. εwA = 0. Table 2 shows the values of the CAP, εwB(cc), for various values of fB along with results obtained for homopolymers, alternating copolymers and block copolymers. Here we have used annealed chemical potentials to determine the CAP. Figure 7 presents the plot of B(cc) as a function of fB along with the theoretical prediction according to Eq. (6) with εwA = 0 and εwh(cc) = 0.276. The data fit the equation well for situations in which sequences are randomly specified. However, similar to homopolymer adsorption on heterogeneous surfaces, the equation does not apply when the chain sequence is not random. For a diblock copolymer, where the first half of the chain is all A monomers while the second half of the chain is all B monomers, a weaker attraction is required to reach the CAP than for a random 50% copolymer chain. An alternating copolymer requires a slightly stronger attraction to reach the CAP. Again, these results can be explained by considering the tendency of forming trains during adsorption. The diblock copolymer is a homogeneous string of adsorbing B monomers attached to a string of A monomers. The B section of the chain is able to interact with the surface like a homogeneous chain, while the A section does not adsorb and slightly repels the chain from the surface, indicating that the value of εwB(cc) for a diblock chain should be similar to a homogeneous chain on a homogeneous surface. In fact, εwB(cc) = 0.30 for diblock copolymers, a value only slightly higher than for homopolymer adsorption, and much lower than εwB(cc) for a 50% random copolymer chain. For an alternating chain, consecutive attractive interactions are not possible, resulting in the necessity of a stronger εwB(cc) than for a random chain. Finally we compare the chemical potential determined with annealed approximation versus quenched average. We found that the chemical potential of a random copolymer above the surface, µ0ads, obtained via the annealed average in Eq. (10) was smaller than the quenched Percent B monomers 0 20 40 60 80 100 Figure 7: Plot of the CAP, εwB(cc), of copolymers over a homogenous surface against the percent of attractive B monomers, fB. The symbols are the CAP determined by the simulation, and the solid line is the plot according to equation (6) with εWA =0.0 and h(cc) =0.276. Circles are CAP of random copolymers, the cross (×) is the CAP of block copolymers, and (∆) is the CAP of alternating copolymers. average in Eq. (11). This has been suggested in the literature.23 Annealed approximation implies that the chain sequence can change when it interacts with the surface. As a result, the chemical potential is lowered when compared with a chain with a fixed sequence. Figure 8 below shows the distribution of µ0ads, obtained based on trial insertions of a given random sequence, against the sequence order parameter λ. As discussed in section 3, a generated random sequence may not correspond to exactly λ=0, therefore resulting a distribution of µ0ads against λ. Figure 8 shows that within the range of λ spanned by random sequences, the chemical potential is seen to depend on λ. The µ0ads is higher for negative λ and is lower for positive λ. This is consistent with the results in Table II. A negative λ implies the random copolymer chain exhibits statistically alternating behaviour. A higher µ0ads implies that the chain is more difficult to be adsorbed on the surface; therefore, it needs a stronger attraction to reach CAP. -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 Figure 8: The distribution of µ0ads versus the sequence order parameter λ of a random copolymer. Each data point represent one µ0ads based on insertion of one given random sequence for 5000 times and the figure contains data for 5000 random sequences. Chain length N =100, fA = fB = 0.5, and εwA =0.0 and εWB =-0.5. 5. Summary Remarks Polymer adsorption at surfaces is relevant to many practical applications and has thus received extensive experimental investigation. However, interest in the CAP, to a large degree, has, until recently, remained a theoretical exercise. There were neither experimental methods that directly measure the CAP, nor were there applications that depended on the exact location of the CAP. This has now changed as interesting applications in liquid chromatography separations have been developed.37,38 In particular, liquid chromatography at the critical condition (LCCC), first reported in the 1980’s, has now widely used for characterization of polymer systems that contain structural and chemical heterogeneities. The critical condition in LCCC experiments was defined as the point where homopolymers of a specific type co-elute regardless of their molecular weights. By erasing the dependence of elution on the molecular weights of one species, other species, differing either chemically or structurally, can then be analyzed. Experimentalists39 have mostly regarded this critical condition as the CAP. Our earlier Monte Carlo simulations largely support this view.31-34 The current study provides knowledge on the dependence of CAP on sequence disorder or surface disorder and such knowledge will be useful to develop chromatographic methods for analyzing random copolymers. We note that several earlier studies 13,14,16,17 have examined the adsorption of polymers on surfaces with either surface disorder or sequence disorder. These studies examined influence of disorder on a variety of properties related to polymer adsorption, such as the change of heat capacity, energy of the chain, and radius of gyration of the chain. Very few, however, have tried to determine the dependence of CAP on the disorder. One of possible reasons that hamper these earlier studies to study the dependence of CAP on the disorder may be due to the lack of a convenient way to determine the CAP. As we have discussed in the theory section, CAP was typically understood as the phase transition of an infinitely long chain near a surface. Earlier studies trying to determine the CAP need to wrestle with the difficulty in extrapolation of results to the limit of infinitely long chain. On the other hand, validity of our studies hinges on the way we determine the CAP. In the case of adsorption of homopolymers over homogeneous surface, we discussed the relationship between the CAP determined by our method with reported literature values. Abundant evidence that supports the validity of our approach was presented in section 4.1. However, for the adsorption over heterogeneous surface, the nature of this CAP is not well-understood. Can a long chain in contact with a surface with few adsorbing sites still exhibit a phase transition similar as that of homopolymers over homogeneous surface? If it does, is the transition first-order or second-order? These questions therefore may cast some doubt on the CAP determined by our approach in the presence of disorder. However, the CAP we determined is directly related to the critical condition point in LCCC. Hence, even though the physical meaning of the CAP determined in this study in the presence of disorder could be subjected to further scrutiny, the importance of our results is not undermined. Table 1: Critical Adsorption Point for Homopolymers above Heterogeneous Surfaces with Attractive B Sites and Non-interacting A Sites. Percentage of Attractive Sites εwB(cc) 100% 0.276 + 0.005 75% 0.35 + 0.01 50% 0.49 + 0.01 25% 0.82 + 0.01 20% 0.96 + 0.01 15% 1.15 + 0.01 10% 1.45 + 0.01 50% alternating surface 0.55+ 0.01 50% patchy surface (O.P=0.94) 0.31+ 0.01 Table 2: Critical Adsorption Point for Heteropolymers with Attractive B Monomers and Non-interacting A Monomers over Homogeneous Surface Percentage of B monomers εwB(cc) 100% 0.276 + 0.005 75% 0.36 + 0.01 50% 0.49 + 0.01 25% 0.84 + 0.01 15% 1.16 + 0.01 50% alternating copolymers 0.55 + 0.01 50% block copolymers 0.30 + 0.01 References: (1) Fleer, G. J.; Cohens Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman & Hall: London, UK, 1993. (2) O'Shaughnessy, B.; Vavylonis, D. J. Phys.: Conden. Matt. 2005, 17, R63-R99. (3) De Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell Univ. Press: Ithaca, 1979. (4) Eisenriegler, E.; Kremer, K.; Binder, K. J. Chem. Phys. 1982, 77, 6296-6320. (5) Ishinabe, T. J. Chem. Phys. 1982, 77, 3171-3176. (6) Meirovitch, H.; Livne, S. J. Chem. Phys. 1988, 88, 4507-4515. (7) Livne, S.; Meirovitch, H. J. Chem. Phys. 1988, 88, 4498-4506. (8) van Rensburg, E. J. J.; Rechnitzer, A. R. J. Phys. A: Math. Gen. 2004, 37, 6875-6898. (9) Decase, R.; Sommer, J.-U.; Blumen, A. J. Chem. Phys. 2004, 120, 8831-8840. (10) Balazs, A. C.; Huang, K.; McElwain, P.; Brady, J. E. Macromolecules 1991, 24, 714- 717. (11) Wu, D.; Hui, K.; Chandler, D. J. Chem. Phys. 1992, 96, 835-841. (12) Golumbfskie, A. J.; Pande, V. S.; Chakraborty, A. K. Proc. Nat. Acad. Sci. 1999, 96, 11707-11712. (13) Sumithra, K.; Baumgaertner. J. Chem. Phys. 1998, 109, 1540-1544. (14) Sumithra, K.; Baumgaertner, A. J. Chem. Phys. 1999, 110, 2727-2731. (15) Charkraborty, A. K. Phys. Rep. 2001, 342, 1-61. (16) Moghaddam, M. S.; Whittington, S. G. J. Phys. A: Math. Gen. 2002, 35, 33-42. (17) Moghaddam, M. S. J. Phys. A: Math. Gen. 2003, 36, 939-949. (18) Polotsky, A.; Schmid, F.; Degenhard, A. J. Chem. Phys. 2004, 121, 4853-4864. (19) Jayaraman, A.; Hall, C. K.; Genzer, J. Phys. Rev. Lett. 2005, 94, 078103. (20) Bogner, T.; Degenhard, A.; Schmid, F. Phys. Rev. Lett. 2004, 93, 268108-268101- 268104. (21) Genzer, J. J. CHem. Phys. 2001, 115, 4873-4881. (22) Genzer, J. Macromol. Theory Simul. 2002, 11, 481-493. (23) Soteros, C. E.; Whittington, S. G. J. Phys. A.: Math. Gen. 2004, 37, R279-R325. (24) Sumithra, K.; Sebastian, K. L. Journal of Physical Chemistry 1994, 98, 9312-9317. (25) Sebastian, K. L.; Sumithra, K. Phys. Rev. E. 1993, 47, R32-R35. (26) Muthukumar, M. J.Chem. Phys. 1995, 103, 4723-4731. (27) Bratko, D.; Chakraborty, A. K.; Shakhnovich, E. I. Chem. Phys. Lett. 1997, 280, 46-52. (28) Hammersly, J. M.; Torrie, G. M.; Whittington, S. G. J. Phys. A: Math. Gen. 1982, 15, 539-571. (29) Arteca, G. A.; Zhang, S. Phys. Rev. E. 1998, 58, 6817-6820. (30) Frenkel, D.; Smit, B. Understanding molecular simulations-from algorithms to applications; Academic Press: San Diego, CA, 2002. (31) Gong, Y.; Wang, Y. Macromolecules 2002, 35, 7492-7498. (32) Orelli, S.; Jiang, W.; Wang, Y. Macromolecules 2004, 37, 10073-10078. (33) Jiang, W.; Khan, S.; Wang, Y. Macromolecules 2005, 38, 7514-. (34) Ziebarth, J.; Orelli, S.; Wang, Y. Polymer 2005, 46, 10450-10456. (35) Ma, L.; Middlemiss, K. M.; Torrie, G. M.; Whittington, S. G. J. Chem. Soc. Frad. Trans. II. 1978, 74, 721-726. (36) Metzger, S.; Muller, M.; Binder, K.; Baschnagel, J. Macromol. Theory Simul. 2002, 11, 985-995. (37) Pasch, H.; Trathnigg, B. HPLC of Polymers; Springer-Verlag Berlin Heidelberg, 1999. (38) Chang, T. J. Polym. Sci. B 2005, 43, 1591-1607. (39) Macko, T.; Hunkeler, D. Adv. Polym. Sci. 2003, 163, 61-136. Graphics to be used for the Table of Contents Percent B sites 0 20 40 60 80 100

0704.0763

Coherent control of atomic tunneling

rhoRR_rhoee.tex Coherent control of atomic tunneling John Martin and Daniel Braun Laboratoire de Physique Théorique, IRSAMC, UMR 5152 du CNRS, Université Paul Sabatier, Toulouse, FRANCE We study the tunneling of a two-level atom in a double well potential while the atom is coupled to a single electromagnetic field mode of a cavity. The coupling between internal and external degrees of freedom, due to the mechanical effect on the atom from photon emission into the cavity mode, can dramatically change the tunneling behavior. We predict that in general the tunneling process becomes quasiperiodic. In a certain regime of parameters a collapse and revival of the tunneling occurs. Accessing the internal degrees of freedom of the atom with a laser allows to coherently manipulate the atom position, and in particular to prepare the atom in one of the two wells. The effects described should be observable with atoms in an optical double well trap. PACS numbers: 73.40.Gk, 37.30.+i I. INTRODUCTION The tunneling effect is considered one of the hallmarks of quantum mechanical behavior. Historically, tunneling was first examined for single particles (e.g. α particles [1], electrons in field emission [2] and later in mesoscopic cir- cuits [3]), for Cooper pairs [4], and for molecular groups [5, 6, 7]. Recently the tunneling of atoms has attracted substantial attention [8, 9, 10, 11]. Dynamical (chaos as- sisted) tunneling of ultracold atoms between different is- lands of stability in phase space was analyzed in [12, 13] and has been observed experimentally [14, 15]. Reso- nantly enhanced tunneling of atoms between wells of a tilted optical lattice has also been observed very recently [16]. In all of these examples, the atoms have been con- sidered internally as inert, and only the center of mass coordinate of the atom was of interest. In [17] it was shown that by taking into account the internal degrees of freedom of atoms, an atom/optical double well poten- tial could be created in which tunneling atoms see their internal and external states correlated (such an effect is also known from other contexts [18]). Mechanical effects of light in optical resonators were also investigated in [19], but no tunneling was considered. Here we show that the tunneling effect can be drasti- cally modified if an internal transition of the atom is cou- pled to a single electromagnetic mode in a cavity, such that photon emission is a reversible and coherent pro- cess. The resulting Rabi oscillations between states with the excitation in the atom and states with a photon in the cavity modulate the periodic tunneling motion. De- pending on the frequencies involved, a rich quasi-periodic behavior can result. If the cavity is fed with a coherent state, collapse and revival of the tunneling effect can oc- cur. Moreover, we show that one may profit from access to the internal degrees of freedom of the atom (e.g. with a laser) to control the atomic motion in the external po- tential. FIG. 1: (Color online) Two-level atom in a double well po- tential interacting with a standing wave inside a cavity. II. MODEL A. Derivation of the Hamiltonian Consider a trapped two-level atom (with levels |g〉, |e〉 of energy ∓~ω0/2 respectively) interacting with a stand- ing wave (with wave number k and frequency ω) inside a cavity as illustrated in Fig. 1. The atom is assumed to be bound in the y − z plane at the equilibrium position y = z = 0 and to experience a symmetric double well po- tential V (x) along the x direction. We denote by ∆ the tunnel splitting, i.e. the energy spacing between the two lowest energy states (the symmetric |−〉 and antisym- metric |+〉 states) of this double well potential. Below we also allow the trapped atom to interact resonantly with an external laser. The Hamiltonian of this system is given by H = HA +HF +HAF , (1) where HA = H A is the Hamiltonian of the trapped atom, HF is the Hamiltonian of the free field and HAF is the interaction Hamiltonian describing the atom-field interaction. We have HexA = + V (x), H inA = σinz , HF = ~ωa HAF = −d.E, http://arxiv.org/abs/0704.0763v2 where d denotes the atomic dipole, E = Eωε a+ a† sin(k(x− x0)) (3) is the electric field operator, with Eω = , where ǫ0 is the permittivity of free space, V the electromag- netic mode volume, x0 the abscissa at the left cavity mirror (x0 < 0), and ε the electric field polarization vec- tor. We have introduced the operators σini (resp. σ for i = x, y, z as the Pauli spin operators in the basis {|e〉, |g〉} (resp. {|+〉, |−〉}). The operator x stands for the center-of-mass position of the atom, px is the conju- gate momentum along the x axis, m denotes the atomic mass, and a (a†) the annihilation (creation) operator of the cavity radiation field. We adopt the two-level approximation which consists of taking into account only the two lowest motional en- ergy states. This requires the Rabi frequency 4g2 + δ2 (with δ = ω − ω0 the detuning between the cavity field and the atomic transition frequencies) to be much smaller than the frequency gap ∆̃ between the upper motional states and the ground state doublet (see Fig. 1). Within this approximation, Hamiltonian HexA becomes HexA = σexz (4) and the position operator takes the form x = b σexx with b/2 = 〈+|x|−〉. We can form states that are mainly con- centrated in the left/right wells, |L〉 = (|+〉 − |−〉)/ |R〉 = (|+〉+ |−〉)/ The average position of a particle localized in the right well is then given by b/2 (see Fig. 1) and σexx = |R〉〈R| − |L〉〈L|. The interaction Hamiltonian HAF can then be written HAF = −~g(a+ a†) sinχ cosκ σinx − cosχ sinκ σexx σinx with the atom-field coupling strength g = −〈e|d|g〉.εEω/~, and χ = kx0, κ = kb/2. (6) For long wavelengths (κ ≪ 1), or κ = nπ with inte- ger n, the left and right sites of the double well are in- distinguishable to the cavity photon and HAF reduces to Jaynes-Cummings Hamiltonian without rotating wave approximation (with a sine varying coupling constant), −~g sinχ (a + a†)σinx . Note that κ ≪ 1 would normally be identified with the Lamb-Dicke regime. Here the sit- uation is more subtle as the level spacings between the tunneling split ground state doublet and the next excited states can be very different such that the recoil energy ~ωrecoil satisfies ∆ ≪ ωrecoil ≪ ∆̃. One may thus be in the Lamb-Dicke regime concerning transitions to higher vibrational states but have a significant mechanical ef- fect on the atomic tunneling. Furthermore, since there is only one photon mode, the recoil energy cannot vary con- tinuously and exciting higher vibrational levels requires ωrecoil close to a level spacing. Our numerical calculations show that even for κ ∼ 1 the two-level approximation can still work very well (see Fig. 4). For δ, ∆ ≪ ω, ω0, a rotating wave approximation is justified, which consists in eliminating the energy non- conserving terms aσex± σ − and a †σex± σ + with σ + = |e〉〈g|, σin− = σ + and σ + = |+〉〈−|, σex− = σ + . Within this approximation, the total Hamiltonian reads σexz + σinz + ~ωa †a (7) +~g(aσin+ + a †σin− ) cosχ sinκ σexx − sinχ cosκ 1ex Thus, depending on the parameters χ and κ, the cavity photon may induce internal transitions in the atom only (cosχ sinκ = 0), or induce transitions between internal and external states at the same time (cosχ sinκ 6= 0) even for a vanishing detuning (δ = ω − ω0 = 0). This is in contrast to conventional sideband transitions of har- monically bound atoms or ions in the Lamb-Dicke regime which require an appropriate value of the detuning. For a fixed potential center (and thus fixed χ), κ can be changed through a modulation of the well-to-well sep- aration b. We will neglect in the following the effects of decoherence, which means that not only g but also ∆ should be much larger than the rate of spontaneous emission Γ, and the cavity decay rate κcav. We denote the global state of the atom-field system by |n, i, j〉 ≡ |n〉⊗|i〉⊗|j〉 where |n〉 stands for the cavity field eigenstates, |i〉 ∈ {|−〉, |+〉} for the external motional states, and |j〉 ∈ {|g〉, |e〉} for the internal states. The total excitation number N is given by a†a+ σin+σ B. Energy levels The states |0,±, g〉 are eigenstates of H with eigen- value (−~ω0 ± ~∆)/2, i.e. these states remain uncou- pled and represent the two lowest energy states in the regime δ, ∆ ≪ ω, ω0. It is straightforward to ver- ify that the Hamiltonian (7) only induces transitions between states with the same number of excitations N , {|N − 1,+, e〉, |N,+, g〉, |N − 1,−, e〉, |N,−, g〉} ≡ {|1〉, |2〉, |3〉, |4〉}. It is therefore sufficient to solve the dynamics in this subspace. In doing so, we obtain the eigenvalues of H , λρµ = (N − 1/2)~ω + ρ , (8) for ρ, µ ∈ {±}, N = 1, 2, . . ., and with 2Ng2(1− cos(2κ) cos(2χ)) + δ2 +∆2 ± 2Ω2 , (9) 4Ng2 cos2 κ sin2 χ(∆2 + 4Ng2 sin2 κ cos2 χ) + δ2∆2 . (10) For a vanishing tunnel splitting (∆ = 0), Ω± reduces to the maximum (minimum) of the two Rabi frequencies of the Jaynes-Cummings models in the right and left wells. For cosκ = 1, the decoupling of external and internal degrees of freedom manifests itself also in the eigenvalues with Ω± = | 4Ng2 sin2 χ+ δ2 ±∆|. C. Evolution operator The whole dynamics of the system can be described by means of the evolution operator U(t) = e−iHt/~ with components Uij = 〈i|U(t)|j〉 = Uji, which can be calcu- lated exactly. In order to simplify the expressions, we restrict ourselves in the following to χ = −π/4−2nπ (in- teger n). We find, up to a an overall phase e−i(N−1/2)ωt, U11 = − µSµΩ−µ ξ + µ(∆− δ)Ω2 − iµΩ+Ω−Cµ(δ∆− µΩ2) U12 = Ng cosκ√ µSµΩ−µ(∆ 2 + 2Ng2 sin2 κ + µΩ2) + iµΩ+Ω−∆Cµ U13 = −iNg2 sin(2κ) µδΩµS−µ + iµΩ+Ω−Cµ U23 = Ng sinκ√ µΩµS−µ(δ∆+ 2Ng 2 cos2 κ− µΩ2) ξ = ∆(δ2 + 2Ng2 cos2 κ− δ∆), Λ = Ω+Ω−Ω and where all time dependence is in the coefficients C± = cos(Ω±t/2), S± = sin(Ω±t/2). (16) The remaining components can be deduced from the relations U22(δ,∆) = U33(−δ,−∆) = U44(δ,−∆) = U11(−δ,∆), U24(δ,∆) = U13(−δ,∆), U14(δ,∆) = U23(δ,−∆) = U23(−δ,∆), and U34(δ,∆) = U12(δ,−∆), valid for any χ, where we have made explicit the depen- dence of the Uij on δ and ∆. III. INTERNAL AND EXTERNAL DYNAMICS The reduced density matrix ρex for the atomic center- of-mass motion alone follows from ρ = |ψ(t)〉〈ψ(t)| by tracing out the field and internal degrees of freedom, where the total wave function at time t reads |ψ(t)〉 = i,j=1 Uij〈j|ψ(0)〉 |i〉. The average position of the atom in the double well potential is then given by 〈x〉 = Trex(ρ exσexx ) = (1− 2ρLL) (17) with ρLL = 〈L|ρex|L〉. Similarly, we obtain the reduced density matrix ρin for the internal atomic state by tracing out the field and external degrees of freedom, and the probability to find the atom in the excited state as ρee = 〈e|ρin|e〉. In the following, we first focus on resonant atom-field interaction (ω = ω0) before moving to the non-resonant case (ω 6= ω0). We distinguish three regimes according to the tunnel splitting compared to the Rabi frequency g : the small tunnel splitting regime (when ∆/g ≪ 1), the intermediate regime (when ∆/g ∼ 1), and the large tunnel splitting regime (when ∆/g ≫ 1). A. Resonant atom-field interaction For resonant atom-field interaction (δ = 0), the ex- pressions for Uij can be greatly simplified. If the system is initially prepared in the state |N − 1, R, e〉 and for κ = π/4, we have ρLL = ∆2 +Ng2 Ωtunt with the tunnel frequency Ωtun = (Ω+ +Ω−) , (19) ρee = Ω2µ −∆2 cos(Ωµt) + 4∆ 2 cos Ω+−Ω− 8(Ng2 +∆2) The atom position oscillates with a single frequency Ωtun given by Eq. (19), whereas ρee evolves with three in general incommensurable frequencies Ω+, Ω−, and (Ω+− Ω−)/2 giving rise to a quasi-periodic signal. For ∆/g ≪ 1, Eq. (18) leads to ρLL ≃ 0 (up to order (∆/g)2), indicating that tunneling is suppressed. This is already obvious from (7), as the term responsible for tun- neling, (~∆/2)σexz = (~∆/2)(|R〉〈L| + |L〉〈R|) becomes very small compared to the last term, diagonal in |R〉, |L〉 which leads to internal Rabi flopping. Note, however, that tunneling is suppressed on all time scales, even for t≫ 1/∆, due to the reduced amplitude in Eq. (18), very much in contrast to tunneling without internal degrees of freedom, where only the period of the tunneling motion, but not the amplitude is affected when ∆ is reduced. For κ approaching π, the situation changes because the term g cosχ sinκ σexx of the interaction Hamiltonian inducing transitions between vibrational states becomes small in comparison with ∆ thereby allowing tunneling again. Because internal and external degrees of freedom are coupled, the tunneling frequency (Eq. (19)) depends on the number of photons inside the cavity. As an exam- ple, let us now consider ∆ ∼ g and a cavity field initially in a coherent state |α〉 = e− 12 |α|2 |n〉 with |α|2 equal to the mean photon number 〈n〉. Figure 2 shows that the average position of the atom in the double well as a function of time for a coherent state exhibits col- lapses and revivals. The oscillation amplitude decreases with increasing mean photon number 〈n〉 and decreasing tunnel splitting ∆ (see Eqs. (18,9)). Since the probabil- ity to find the atom in the excited state oscillates with three frequencies, no collapses and revivals are observed for ρee. The collapse time tc of the tunneling motion can be estimated from the condition [20] (Ωtun(〈n〉 + 〈n〉) − Ωtun(〈n〉− 〈n〉)) tc ∼ 1 with Ωtun(m) given by Eq. (19) for N = m+ 1, which yields, for 〈n〉 ≫ 1, (∆/g)2 + 3/4 +O(〈n〉−2) (21) The time interval between two following revivals, tr, follows from (Ωtun(〈n〉) − Ωtun(〈n〉 − 1)) tr = 2π, and is given for 〈n〉 ≫ 1 by (∆/g)2 + 1/2 +O(〈n〉−2) For the parameters of Fig. 2, Eq. (22) yields gtr ≃ 68.23 for ∆/g = 2 and gtr ≃ 86.70 for ∆/g = 5. Smaller revival times are possible for smaller values of 〈n〉, but in general the observation of revivals will be quite challenging, as they require ∆ ∼ g ≫ κcav. For large tunnel splitting, ∆/g ≫ 1, Ωtun = ∆ + Ng2/(2∆) + O((g/∆)3), and Eq. (18) reduces to ρLL ≃ sin2(∆t/2), which is identical to the tunneling of a parti- cle without internal structure. Equation (20) reduces to a Rabi oscillation ρee ≃ cos2( Ngt/2). B. Non-resonant atom-field interaction For non-resonant atom-field interaction (δ 6= 0), and intermediate tunnel splitting [see Fig. 3 for ∆ = δ = g], ∆/g = 5 ∆/g = 2 100806040200 FIG. 2: (Color online) Average position of the atom in the double well as a function of time for ∆/g = 2 (blue, top curve) and ∆/g = 5 (red), κ = π/4 and a coherent state with α = 5. 100806040200 FIG. 3: (Color online) Average position of the atom in the double well as a function of time for ∆ = δ = g, κ = π/4 and N = 1. The blue solid/red dashed curve corresponds to an excited atom initially located in the left/right well. 〈x(t)〉 involves in general the two non-commensurate fre- quencies Ω+ and Ω− and varies therefore quasiperiod- ically as a function of time. Figure 3 also shows that an atom initially located in one of the two wells remains mostly confined to that well. For small tunnel splitting, ∆/g ≪ 1 and large detuning |δ|/g ≫ 1 (with ∆|δ|/g2 ∼ 1), the matrix elements of U simplify to U13 = i Ng2 sin 2κ δ2∆2 +N2g4 sin2(2κ) (23a) U33 = cos δ2∆2 +N2g4 sin2(2κ) (23b) up to corrections of order O(∆/g) and a phase factor ei[(Ng 2/δ+δ)−(2N−1)ω]t/2 while the components U12 and U23 are of order O(∆/g). In this situation, the system 20151050 FIG. 4: (Color online) Density matrix elements ρRR (top) and ρee (bottom) as a function of the interaction time gt for an initially excited atom located in the right well and for the parameters ∆/g ≃ 0.3336, δ/g = 3, κ = π/4, and N = 1. Numerical results from the propagation of the time dependent Schrödinger equation with Hamiltonian (1) and rotating wave approximation are represented by circles and analytical results by solid curves. The time propagation was done with (~ = m = 1) g = 0.01 and the double well potential V (x) = 0.08x4 − x2 yielding a tunnel splitting ∆ ≃ 0.003336 and a ratio ∆̃/ 4g2 + δ2 ≃ 44.4 ≫ 1. oscillates only between the two states |N − 1,+, e〉 and |N − 1,−, e〉 with a single frequency δ2∆2 +N2g4 sin2(2κ) , (24) just as a three-level atom undergoing a Raman transition in the far detuned regime behaves as a two-level system. If the system is initially in the state |N − 1,−, e〉, we have from Eqs. (23) ρLL = − Nδ∆sin(2κ) 2Ω̄2(δ/g)2 1− cos , (25) and ρee = 1. For a detuning δ = ±Ng2 sin(2κ)/∆, ρLL = 1− cos . (26) This regime may be suitable for coherently manipu- lating the atom position through access to its internal degrees of freedom with a laser. Coherent manipulation of the position of neutral atoms has been proposed and demonstrated before, see e.g. [21, 22, 23]. In these exam- ples, the manipulation is done by modifying the external potential. The mechanism we propose here is very dif- ferent, as the potential remains totally unchanged, and only internal transitions and the tunneling effect are used to move the atom in a controlled way. As an example, we show how the atom can be prepared in the left well starting from the ground state |0,−, g〉 for δ = −g2/∆. We first apply a π-pulse with an external laser resonant with the atomic transition. By using a laser with a wave vector perpendicular to the Ox-direction, only the atomic internal degree of freedom is affected, resulting in the transition |0,−, g〉 → −i|0,−, e〉. We assumed that the laser Rabi frequency ΩR is much larger than the tunnel frequency ∆. Now we use the coupling between the internal and ex- ternal degrees of freedom to create a superposition of the |0,±, e〉 states, and then apply a second resonant π- pulse to get back to the uncoupled states |0,±, g〉. For ∆/g ≪ 1, δ = −g2/∆ and κ = π/4, the initial state transforms according to |0,−, g〉 −−−−→ ΩRt=π |0,−, e〉 −−−−−−→ ∆t=π/ |0, L, e〉 −−−−→ ΩRt=π |0, L, g〉 up to a physically irrelevant phase. Other coherent su- perpositions of |0,+, g〉 and |0,−, g〉 can be obtained by choosing appropriate interaction times. In order to verify that the two-level approximation for the external motion used in the derivation of the Hamilto- nian is a good approximation, we have numerically solved the time dependent Schrödinger equation with Hamil- tonian (1) and rotating wave approximation but with the exact external potential V (x) (i.e. with a large num- ber of vibrational states). Figure 4 shows that provided 4g2 + δ2 as stated before, to take only the two lowest vibrational states into account is indeed a good approximation. We finally comment on possible experimental realiza- tions of our model. Double well potentials with tunable well-to-well separation have been demonstrated with op- tical dipole traps e.g. in [21, 24], and on atom chips e.g. in [25, 26]. For our model, the double well poten- tial has to be realized inside the cavity. Optical trapping and even cooling of atoms close to their ground state inside a cavity has been achieved in several groups by now [27, 28, 29, 30], but up to our knowledge double well potentials have not been realized in a cavity so far. However, some of the cavities developed have a very long lateral opening (up to 222 µm [31]) and should allow more complicated trapping potentials (optical lattices inter- secting a cavity have been realized in Chapman’s group [31]). We remark that it is not essential for our model that the double well potential be aligned with the cavity axes. Any other orientation is possible, and only leads to modified coefficients cosχ sinκ and sinχ cosκ. At certain “magical wavelengths”, Cs, Yb, Sr, Mg, and Ca atoms in optical traps experience the same potential for ground and excited internal states coupled by a dipole transition [27, 32, 33, 34]. In a symmetric potential V (z) the tunneling frequency ∆ is given in WKB approxi- mation by ∆ ∼ ωosc exp(−1/~ 2m(E0 − V (z)) dz) where E0 is the ground state energy, ωosc the single well harmonic oscillation frequency, and z = ±a are the corre- sponding classical turning points delimiting the range of the barrier. The exponential factor can approach unity for a barrier that is only slightly higher than the ground state energy E0, in which case cooling to temperatures kBT < ~∆ should be possible with state of the art techniques [27]. In [27] a trap depth V0/~ = 47 MHz was achieved inside a cavity with 1.2 mW laser power. In any case, the trap frequency and thus the tunneling splitting are determined by the laser power and the fo- cussing (or the wavelength for optical lattices), and can therefore be controlled independently of Γ, κcav, such that there should be no fundamental problem achieving ∆ ≫ Γ, κcav. The detection of the tunneling motion should be possible by optical imaging, i.e. diffusion of laser light from another transition in the optical regime with smaller wavelength than the well separation. Al- ternatively, one might monitor the transmission through the cavity in the case that it differs for the two locations of the wells [35]. Another possibility might be using the atomic spin as a position meter [17]. Acknowledgments We thank Jacques Vigué for an interesting discussion and CALMIP (Toulouse) for the use of their comput- ers. This work was supported by the Agence National de la Recherche (ANR), project INFOSYSQQ, and the EC IST-FET project EUROSQIP. [1] G. Gamow, Z. 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Browaeys, N. R. Hecken- berg, K. Helmerons, C. McKenzie, G. J. Milburn, W. D. Philipps, S. L. Holston, H. Rubinsztein-Dunlop, et al., Nature 412, 52 (2001). [16] C. Sias, A. Zenesini, H. Lignier, S. Wimberger, D. Ciampini, O. Morsch, and E. Arimondo, Phys. Rev. Lett. 98, 120403 (2007). [17] D. L. Haycock, P. M. Alsing, I. H. Deutsch, J. Grondalski, and P. S. Jessen, Phys. Rev. Lett. 85, 3365 (2000). [18] T. Salzburger and H. Ritsch, Phys. Rev. Lett. 93, 063002 (2004). [19] P. Domokos and H. Ritsch, J. Opt. Soc. Am. B 20, 1098 (2003). [20] M. Scully and M. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, UK, 1997). [21] J. Sebby-Strabley, M. Anderlini, P. S. Jessen, and J. V. Porto, Phys. Rev. A 73, 033605 (2006). [22] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hänsch, and I. Bloch, Phys. Rev. Lett. 91, 010407 (2003). [23] J. Mompart, K. Eckert, W. Ertmer, G. Birkl, and M. Lewenstein, Phys. Rev. Lett. 90, 147901 (2003). [24] Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, Phys. Rev. Lett. 92, 050405 (2004). [25] E. A. Hinds, C. J. Vale, and M. G. Boshier, Phys. Rev. Lett. 86, 1462 (2001). [26] W. Hänsel, J. Reichel, P. Hommelhoff, and T. W. Hänsch, Phys. Rev. A 64, 063607 (2001). [27] J. McKeever, J. R. Buck, A. D. Boozer, A. Kuzmich, H.- C. Nägerl, D. M. Stamper-Kurn, and H. J. Kimble, Phys. Rev. Lett. 90, 133602 (2003). [28] J. Ye, D. W. Vernooy, and H. J. Kimble, Phys. Rev. Lett. 83, 4987 (1999). [29] J. A. Sauer, K. M. Fortier, M. S. Chang, C. D. Hamley, and M. S. Chapman, Phys. Rev. A 69, 051804(R) (2004). [30] P. Maunz, T. Puppe, I. Schuster, N. Syassen, P. W. H. Pinske, and G. Rempe, Nature 428, 50 (2004). [31] K. M. Fortier, S. Y. Kim, M. J. Gibbons, P. Ahmadi, and M. S. Chapman, Phys. Rev. Lett. 98, 233601 (2007). [32] H. Katori, M. Takamoto, V. G. Pal’chikov, and V. D. Ovsiannikov, Phys. Rev. Lett. 91, 173005 (2003). [33] A. Brusch, R. LeTargat, X. Baillard, M. Fouche, and P. Lemonde, Phys. Rev. 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0704.0764

Correlation functions and excitation spectrum of the frustrated ferromagnetic spin-1/2 chain in an external magnetic field

Correlation functions and excitation spectrum of the frustrated ferromagnetic spin-1 chain in an external magnetic field T. Vekua,1 A. Honecker,2 H.-J. Mikeska,3 and F. Heidrich-Meisner4 Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris Sud, 91405 Orsay Cedex, France Institut für Theoretische Physik, Universität Göttingen, 37077 Göttingen, Germany Institut für Theoretische Physik, Universität Hannover, Appelstrasse 2, 30167 Hannover, Germany Materials Science and Technology Division, Oak Ridge National Laboratory, Tennessee, 37831, USA and Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA (Dated: April 5, 2007; revised: July 6, 2007) Magnetic field effects on the one-dimensional frustrated ferromagnetic chain are studied by means of effective field theory approaches in combination with numerical calculations utilizing Lanczos diagonalization and the density matrix renormalization group method. The nature of the ground state is shown to change from a spin-density-wave region to a nematic-like one upon approaching the saturation magnetization. The excitation spectrum is analyzed and the behavior of the single spin-flip excitation gap is studied in detail, including the emergent finite-size corrections. I. INTRODUCTION The interest in helical and chiral phases of frustrated low-dimensional quantum magnets has been triggered by recent experimental results. While many copper-oxide based materials predominantly realize antiferromagnetic exchange interactions, several candidate materials with magnetic properties believed to be described by frus- trated ferromagnetic chains have been identified,1,2,3,4,5,6 including Rb2Cu2Mo3O12 (Ref. 1), LiCuVO4 (Refs. 2, 3,4,5), and Li2ZrCuO4 (Ref. 6). The frustrated anti- ferromagnetic chain is well-studied,7 but the magnetic phase diagram of the model with ferromagnetic nearest- neighbor interactions remains a subject of active theoret- ical investigations.8,9,10,11 In this work we consider a parameter regime that is in particular relevant for the low-energy properties of LiCuVO4, corresponding to a ratio of J1 ≈ −0.3 J2 be- tween the nearest neighbor interaction J1 and the frus- trating next-nearest neighbor interaction J2 > 0. As the interchain couplings for this material are an order of magnitude smaller than the intrachain ones,3 we an- alyze a purely one-dimensional (1D) model. Apart from mean-field based predictions,8 the nature of the ground state in a magnetic field h is not yet completely known. Therefore, combining the bosonization technique with a numerical analysis we determine ground-state properties and discuss the model’s elementary excitations. The Hamiltonian for our 1D model reads: J1~Sx · ~Sx+1 + J2~Sx · ~Sx+2 Szx , (1) where ~Sx represents a spin one-half operator at site x. Bosonization has turned out to be the appropriate lan- guage for describing the regime |J1| ≪ J2 of Eq. (1). This result has been established by studying the magne- tization process yielding a good agreement between field theory and numerical data.9 The derivation of the ef- fective field theory is summarized in Sec. II. Here, we extend on such comparison of analytical and numerical results and further confirm the predictions of field the- ory by analyzing several correlation functions in Sec. III. Then, in Sec. IV, we numerically compute the one- and two-spin flip excitation gaps and compare them to field- theory predictions. Finally, Sec. V contains a summary and a discussion of our results. II. EFFECTIVE FIELD THEORY We start from an effective field theory describing the long-wavelength fluctuations of Eq. (1). In the limit of strong next-nearest neighbor interactions J2 ≫ |J1|, the spin operators can be expressed as: Szα(r) ∼ m+ c(m) sin 2kF r + + · · · S−α (r) ∼ (−1)re−iθα π + · · · . (2) kF = ( −m)π is the Fermi-wave vector and α = 1, 2 enu- merates the two chains of the zig-zag ladder. In relation with Eq. (1), note that ~S1(r) = ~S(x+1)/2 (~S2(r) = ~Sx/2) for x odd (even). φα and θα are compactified quantum fields describing the out-of-plane and in-plane angles of fluctuating spins obeying Gaussian Hamiltonians: H = v (∂xφα) 2 +K(∂xθα) , (3) with [φα(x), θα(y)] = iΘ(y − x), where Θ(x) is the Heaviside function. Sub-leading terms are suppressed in Eq. (2). m is the magnetization of decoupled chains, re- lated to the real magnetization M of the zig-zag system M ≃ m 1− 2K(m)J1 πv(m) . (4) K(m) and v(m) are the Luttinger liquid (LL) parame- ter and the spin-wave velocity of the decoupled chains, http://arxiv.org/abs/0704.0764v2 respectively. The nonuniversal amplitude c(m) appear- ing in the bosonization formulas (2) has been determined from density matrix renormalization group (DMRG) calculations.12 Note that in our notation M = 1/2 at saturation. Now we perturbatively add the interchain coupling term to two decoupled chains, each of which is described by an effective Hamiltonian of the form Eq. (3) and fields φi and θi, i = 1, 2. For convenience, we transform to the symmetric and antisymmetric combinations of the bosonic fields φ± = (φ1±φ2)/ 2 and θ± = (θ1±θ2)/ In this basis and apart from terms H±0 of the form (3), the effective Hamiltonian describing low-energy proper- ties of Eq. (1) contains a single relevant interaction term with the bare coupling g1 ∝ J1 ≪ v: Heff = H+0 +H 0 + g1 dx cos , (5) and the renormalized LL parametersK± are, in the weak coupling limit: K± = K 1∓ J1 . (6) K+ is the Luttinger-liquid parameter of the soft mode of the zig-zag ladder. The Hamiltonian (5) represents the minimal effective low-energy field theory describing the region J2 ≫ |J1| of the frustrated FM spin-1/2 chain for M 6= 0.9,13 The relevant interaction term cos opens a gap in the φ− sector. Since S x+1 − Szx ∼ ∂xφ−, relative fluctuations of the two chains are locked. This implies that single-spin flips are gapped with a sine- Gordon gap in the sector describing relative spin fluctua- tions of the two-chain system.9 Gapless excitations come from the ∆Sz = 2 channel, i.e. only those excitations are soft where spins simultaneously flip on both chains. DMRG results show that this picture applies to a large part of the magnetic phase diagram.9 III. CORRELATION FUNCTIONS We now turn to the ground state properties of Eq. (1) as a function of magnetization, concentrating on several correlation functions in order to identify the leading in- stabilities. Note that our analysis is only valid if M 6= 0. Apart from a term representing the magnetization M in- duced by the external field, the longitudinal correlation function shows an algebraic decay with distance r: 〈Szα(0)Szβ(r)〉 ≃ M2+ C1 cos(2kF r + (α− β)kF ) 2π2rK+ 8π2r2 The constants Ci, i = 1, 2, 3, appearing here and in Eq. (9) will be determined through a comparison with numerical results. In contrast to Eq. (7), the transverse xy-correlation functions decay exponentially reflecting the gapped na- ture of the single spin-flip excitations. Here we do not restrict ourselves to the equal-time expression only, be- cause we will need non-equal time correlation functions to extract the finite-size corrections to the gap later on. We obtain: 〈S+α (0, 0)S−β (r, τ)〉 ≃ δα,β(−1)re−∆1(M) τ2+r2/v2 (r2 + v2+τ 8K+ (r2 + v2−τ where τ stands for the Euclidean time, ∆1(M) is the ∆Sz = 1 gap, and v± ∼ v ± J1/π in the weak cou- pling limit. The Kronecker delta strictly applies to the thermodynamic limit, while on the lattice an additional contribution for α 6= β exists. It is noteworthy that, different from Eq. (8), the in- plane correlation functions involving bilinear spin combi- nations decay algebraically. This stems from the gapless nature of ∆Sz = 2 excitations. In fact, these are the slowest decaying correlators close to the saturation mag- netization: 〈S+1 (r)S 2 (r)S 1 (0)S 2 (0)〉 ≃ r1/K+ C3 cos(2kF r) rK++1/K+ This result is reminiscent of a partially ordered state be- cause the ordering tendencies in this correlation function are more pronounced than those of the corresponding single-spin correlation function Eq. (8). Therefore, we call the correlator (9) ‘nematic’. Furthermore, we will refer to a situation where Eq. (9) is the slowest decay- ing one among all correlation functions as a ‘nematic-like phase’. By virtue of the exponential decay in (8), the correlator (9) is proportional to: 〈(S+1 (r) + S 2 (r)) 2 (S−1 (0) + S 2 (0)) 2〉 . (10) The term (Sα1 + S 2 appearing in the case of the S = 1 zig-zag ladder corresponds to the operator (Sα)2 in the case of a S = 1 chain. One can think of an effective S = 1 spin formed from two neighboring S = 1 spins coupled by the ferromagnetic interaction. A similar behavior of correlation functions, namely the exponential decay of in- plane spin components and the algebraic decay of their bilinear combinations, is encountered also in the XY 2 phase of the anisotropic S = 1 chain14 and in the spin-1 chain with biquadratic interactions, see, e.g., Ref. 15. The algebraic decay of the nematic correlator as op- posed to the exponential decay of (8) suggests that there are tendencies towards nematic ordering in this phase. Depending on the value of K+ the dominant instabilities are either spin-density-wave ones for K+ < 1 or nematic ones for K+ > 1. From the result for K+ given in Eq. (6) one can perturbatively evaluate the crossover value of J1: |J1,cr| = πv(m) . (11) For J1 < J1,cr the nematic correlator (9) is the slowest decaying one, i.e. one is in the nematic-like phase. The -0.015 -0.01 -0.005 0.005 0.001 0 4 8 12 16 20 24 28 32 fit, L=32 fit, L=48 fit, L=64 ED, L=32 ED, L=48 ED, L=64 FIG. 1: (Color online) Correlation functions at J1 = −J2 < 0, and magnetization M = 3/8: (a) longitudinal component Sz (b) transverse component S± , (c) spin nematic S± . x is the distance in a single-chain notation. ED results for periodic boundary conditions are shown by symbols, fits by lines. Note the logarithmic scale of the vertical axis in panel (b). behavior of the cross-over line can be read off from the behavior of K(m): K(m) increases monotonically with m, tends to K = 1 for m → 1/2, and satisfies K < 1 for m < 1/2 (see, e.g., Refs. 16,17). Therefore, we have J1,cr = 0 for M = 1/2 with increasing ferromagnetic |J1,cr| for decreasing M . This means that for J1 < 0 a regime opens at high M where nematic correlations given by Eq. (9) dominate over spin-density-wave corre- lations given by Eq. (7), in agreement with Chubukov’s prediction.8 Now we check the correlation functions obtained within bosonization against exact diagonalization (ED) results. Numerical data obtained for J1 = −J2 < 0 and M = 3/8 on finite systems with periodic boundary conditions are shown in Fig. 1. This parameter set allows for a clear test of the above predictions, but represents the generic behavior in the phase of two weakly coupled chains. To take into account finite-size effects we use the observation that for a conformally invariant theory, any power law on a plane becomes a power law in the following variable defined on a cylinder of circumference L: x → L . (12) First we fit the nematic correlator given by Eq. (9), which from bosonization is expected to be the leading instability at high magnetizations. Using the part with x ≥ 5 of the L = 64 data shown in Fig. 1c, we find 1/K+ = 0.904 ± 0.011, C2 = 0.143 ± 0.004, and C3 = −0.326±0.013. Fig. 1c shows that all finite-size results for the nematic correlator are nicely described by this fit with the dependence on L taken into account by substituting Eq. (12) for the power laws. Moreover, from K+ > 1, we see that the system is indeed in the region dominated by nematic correlations for M = 3/8 and J1 = −J2. Now we turn to the longitudinal correlation function which we fit to the bosonization result Eq. (7). Since most numerical parameters have been determined by the previous fit, only one free parameter is left which we de- termine from the numerical results of Fig. 1a for L = 64 and x ≥ 14 as C1 = 0.060± 0.004. Predictions for other system sizes are again obtained by substituting Eq. (12) for the power laws. The agreement in Fig. 1a is not as good as in Fig. 1c. However, it improves at larger dis- tances x and system sizes L, indicating that corrections omitted in Eq. (7) are still relevant on the length scales considered here. Finally, the xy-correlation function is shown in Fig. 1b with a logarithmic scale of the vertical axis of this panel. The exponential decay predicted by Eq. (8) is verified. One further observes that correlations between the in- plane spin-operators belonging to different chains (odd x) are an order of magnitude smaller than on the same chain (even x). This suppression of correlations between different chains corresponds to the δ symbol in (8), which strictly applies only in the thermodynamic limit and for large distances. We summarize the main result of this section: in-plane spin correlators are exponentially suppressed for any fi- nite value of the magnetization in the parameter region |J1| < J2. The ground state crosses over from a spin- density-wave dominated to a nematic-like phase with in- creasing magnetic field, with the crossover line given by Eq. (11). IV. EXCITATIONS We next address the excitation spectrum. Since the gap to ∆Sz = 1 excitations should be directly accessible to microscopic experimental probes such as inelastic neu- tron scattering or nuclear magnetic resonance, we analyze its behavior as a function of magnetization. Sufficiently below the fully polarized state the gap can be calculated analytically using results from sine-Gordon theory. In addition, to leading order of the interchain coupling, one can get qualitative expressions using dimensional argu- ments for the perturbed conformally invariant model: ∆1(m) ∼ c2(m)|J1| sin(πm) v(m)(1 − J1K(m)/πv(m)) , (13) where ν = 2− 2K(m) 1+ J1K(m)/πv(m) . m(h),K(h) and v(h) can be determined numerically from the Bethe ansatz integral equations.16,17,18,19 With this information and Eqs. (4) and (13) we de- termine the qualitative behavior of the single-spin gap ∆1(M) as a function of M : it increases from zero at zero magnetization, reaches a maximum at intermediate magnetization values, then shows a minimum and, upon approaching the saturation magnetization, it increases again. As our formulas do not strictly apply at m = 0, the notion of a vanishing gap at zero magnetization may be a spurious result. Note that when the fully polarized state is approached, the magnetization increases in an un- physical fashion since in this limit bosonization becomes inapplicable. At the point where the magnetization sat- urates the exact value of the gap can be obtained from the following mapping to hard-core bosons:8,11,20 Szi = − a†iai , S i = a i . (14) Comparing Eq. (14) with Eq. (2) one recognizes the lead- ing terms in Haldane’s harmonic fluid transformation for bosons.20 Using a ladder approximation which is exact in the two-magnon subspace we arrive at: 4J22 − 2J1J2 − J21 2(J2 − J1) J21 + 8J1J2 + 16J J21 (3 J2 + J1) J2 (J2 − J1) . (15) In Eq. (15) we have represented the gap as a difference of two terms: the quantum and the classical instability fields emphasizing its quantum origin. In order to verify these field theory predictions, we perform complementary numerical computations using the DMRG method.21 Open boundary conditions are im- posed and we typically keep up to 400 DMRG states. From DMRG we obtain the ground-state energies E(Sz) as a function of total Sz. For those values of Sz that emerge as a ground state in an external magnetic field we compute the single-spin excitation gap from ∆1(M) = E(Sz + 1) + E(Sz − 1)− 2E(Sz) . (16) Fig. 2a shows numerical results for ∆1 at a selected value of J1 = −0.3 J2 < 0 for the largest system sizes investi- gated. We find that the finite-size behavior of the gap ∆1(M,L) for system sizes L ≥ 24 is well described by a 1/L correction. This will be further corroborated by field-theoretical arguments outlined below. Therefore, we 0 0.1 0.2 0.3 0.4 0.5 L=120 L=144 L=156 L=120 L=144 L=156 extrapolated FIG. 2: (Color online) Density matrix renormalization group results for the gaps at J1 = −0.3 J2 < 0 as a function of magnetization M . Panel (a) shows the single-spin excitation gap (16), panel (b) the finite-size gap (19) for two flipped spins multiplied by the chain length L. extrapolate it to the thermodynamic limit using a fit to the form ∆1(M,L) = ∆1(M) + + · · · , (17) allowing for an additional 1/L2 correction for those values of M where at least 4 different system sizes are available. This extrapolation is represented by the full circles in Fig. 2a; errors are estimated not to exceed the size of the symbols. Our extrapolation for ∆1 is consistent with a vanishing gap at M = 0 in agreement with previous nu- merical studies22 although bosonization predicts a non- zero – possibly very small – gap.13,22,23 The behavior of ∆1(M) confirms the picture described above: the gap is non-zero for M > 0, goes first through a maximum and then a minimum and finally approaches ∆1/J2 ≈ 0.023 given by Eq. (15) for M → 1/2. We further wish to point out that for chains with pe- riodic boundary conditions, the coefficient a(M) of the finite-size extrapolation Eq. (17) is determined by the spin-wave velocity and the critical exponent of the soft mode from the ∆Sz = 2 channel. Indeed, using Eq. (8) where we can set r = 0, and use the conformal mapping (12) to the cylinder, we see that the leading finite-size correction to the gap is: ∆1(M,L) = ∆1(M) + πv+(M) 4K+(M) . (18) 0 π/2 π wave vector 1.0 0.0 FIG. 3: (Color online) Numerical dispersion spectrum in the subspaces of odd Sz computed for L = 24 and J1 = −J2 < 0. The wave vector is given relative to the ground state wave vector (0 for Sz = 0, 4, 8 and π for Sz = 2, 6). Note that we have to replace sin with sinh in Eq. (12) in order to extract a gap, since we are dealing with Eu- clidean time. In addition we used the fact that in our ap- proximation the effective Hamiltonian (5) is a direct sum of symmetric and antisymmetric sectors. Moreover, it is only the symmetric sector enjoying conformal invariance and consequently we perform the replacement τ → sinh τ only in the symmetric sector. The antisymmetric sector has a spectral gap and its contribution to the finite-size corrections of the single-spin flip excitation energy are exponentially suppressed with system size.24 With this method one cannot fix the amplitudes of the 1/L2 term and beyond. Note furthermore that there may be ad- ditional surface terms for open boundary conditions as employed in the numerical DMRG computations. Nev- ertheless there is a dominant 1/L correction in any case. Next, we briefly look at the ∆Sz = 2 excitations. Their finite-size gap is, in analogy to Eq. (16), computed with DMRG from ∆2(M) = E(Sz + 2) + E(Sz − 2)− 2E(Sz) . (19) Fig. 2b shows numerical results for L∆2(M,L) again at the value J1 = −0.3 J2 < 0. One observes that the scaled finite-size gaps collapse onto a single curve which shows that ∆2(M,L) scales linearly to zero with 1/L, exactly as expected for gapless excitations in 1D. Furthermore, we observe that the scaled quantity L∆2(M,L) vanishes as one approaches saturation M = 1/2 which indicates a vanishing of the velocity of the corresponding excitations at saturation. We proceed by discussing the wave-vector dependence of the ∆Sz = 1 excitation, while we remind the reader that the low-energy excitations are in the ∆Sz = 2 sec- tor. Fig. 3 shows representative ED results obtained for rings with L = 24 and J1 = −J2 < 0. For ground states with low Sz, the ∆Sz = 1 excitation spectrum looks similar to the continuum of spinons. On the other hand, close to saturation one has single-magnon excita- tions with a minimum given by the classical value of the wave vector kcl = arccos(|J1|/4J2).8,13,25 We read off from Fig. 3 that upon lowering the magnetic field, this minimum shifts from the classical incommensurate value towards π/2, i.e. the value appropriate for two de- coupled chains. This renormalization of the minimum of the magnon excitations towards the value of decoupled chains can be interpreted in terms of quantum fluctua- tions, which are enhanced when the density of magnons increases. A strong quantum renormalization of the pitch angle from its classical value at zero magnetization was previously observed by the coupled-cluster method and DMRG calculations.26 V. SUMMARY We have combined numerical techniques with analyti- cal approaches and mapped out the ground state phase diagram of the frustrated ferromagnetic spin chain in an external magnetic field. We have established that with increasing magnetic field, the ground state crosses over from a spin-density-wave dominated to a nematic-like phase. Single spin flip excitations are gapped, giving rise to an exponential decay of in-plane spin correlation func- tions in both regimes. We have studied the single- and two-spin flip excitation energy numerically. Using tools from conformal field theory we have further shown that the amplitude of the leading 1/L correction term to the single-spin flip gap is determined by the critical exponent and the spin-wave velocity of the soft mode. Finally, in order to apply our findings to the mate- rial LiCuVO4, one should take into account interchain interactions as well as anisotropies, which are expected to be present in this system.3 At low fields, a helical state has been observed experimentally.2,3 On the other hand, for the purely one dimensional case, we have shown that upon increasing the magnetic field there is a competition between spin-density-wave and nematic-like tendencies. Those are the two leading instabilities at high magnetiza- tions and thus they are the natural candidates to become long-range ordered in higher dimensions. The question whether there are true phase transitions at high fields in higher dimensions is beyond the scope of the current work. Acknowledgments We thank A. Feiguin for providing us with his DMRG code used for large scale calculations. Most of T.V.’s work was done during his visits to the Institutes of Theoretical Physics at the Universities of Hannover and Göttingen, supported by the Deutsche Forschungsge- meinschaft. The hospitality of the host institutions is gratefully acknowledged. T.V. also acknowledges support from the Georgian National Science Foundation under grant N 06−81−4−100. LPTMS is a mixed research unit 8626 of CNRS and University Paris-Sud. A.H. is sup- ported by the Deutsche Forschungsgemeinschaft (Project No. HO 2325/4-1), and F.H.-M. is supported by NSF grant No. DMR-0443144. 1 M. Hase, H. Kuroe, K. Ozawa, O. Suzuki, H. Kitazawa, G. Kido, and T Sekine, Phys. Rev. B 70, 104426 (2004). 2 B. J. Gibson, R. K. Kremer, A. V. Prokofiev, W. Assmus and G. J. McIntyre, Physica B 350, e253 (2004). 3 M. Enderle, C. Mukherjee, B. F̊ak, R. K. Kremer, J.-M. Broto, H. Rosner, S.-L. Drechsler, J. Richter, J. Malek, A. Prokofiev, W. Assmus, S. Pujol, J.-L. Raggazzoni, H. Rakoto, M. Rheinstädter, and H. M. Rønnow, Europhys. Lett. 70, 237 (2005). 4 M. G. Banks, F. Heidrich-Meisner, A. Honecker, H. Rakoto, J.-M. Broto, and R. K. Kremer, J. Phys.: Cond. Mat. 19, 145227 (2007). 5 N. Büttgen, H.-A. Krug von Nidda, L. E. Svistov, L. A. Prozorova, A. Prokofiev, and W. Aßmus, Phys. Rev. B 76, 014440 (2007). 6 S.-L. Drechsler, O. Volkova, A. N. Vasiliev, N. Tristan, J. Richter, M. Schmitt, H. Rosner, J. Málek, R. Klingeler, A. A. Zvyagin, and B. Büchner, Phys. Rev. Lett. 98, 077202 (2007). 7 H.-J. Mikeska and A. K. Kolezhuk, Lect. Notes Phys. 645, 1 (2004). 8 A. V. Chubukov, Phys. Rev. B 44, 4693 (1991). 9 F. Heidrich-Meisner, A. Honecker, and T. Vekua, Phys. Rev. B 74, 020403(R) (2006). 10 D. V. Dmitriev, V. Y. Krivnov, and J. Richter, Phys. Rev. B 75, 0114424 (2007). 11 R. O. Kuzian and S.-L. Drechsler, Phys. Rev. B 75, 024401 (2007). 12 See, e.g., T. Hikihara and A. Furusaki, Phys. Rev. B 69, 064427 (2004). 13 D. C. Cabra, A. Honecker, and P. Pujol, Eur. Phys. J. B 13, 55 (2000). 14 H. J. Schulz, Phys. Rev. B 34, 6372 (1986). 15 A. Läuchli, G. Schmid, and S. Trebst, Phys. Rev. B 74, 144426 (2006). 16 K. Totsuka, Phys. Lett. A 228, 103 (1997). 17 D. C. Cabra, A. Honecker, and P. Pujol, Phys. Rev. B 58, 6241 (1998). 18 V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quan- tum Inverse Scattering Method and Correlation Functions (Cambridge University Press, Cambridge, England, 1993). 19 S. Qin, M. Fabrizio, L. Yu, M. Oshikawa, and I. Affleck, Phys. Rev. B 56, 9766 (1997). 20 F. D. M. Haldane, Phys. Rev. Lett 47, 1840 (1981). 21 S. R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev. B 48, 10345 (1993). 22 C. Itoi and S. Qin, Phys. Rev. B 63, 224423 (2001). 23 A. A. Nersesyan, A. O. Gogolin, and F. H. L. Eßler, Phys. Rev. Lett. 81, 910 (1998). 24 S. I. Matveenko and A. M. Tsvelik, private communication. 25 C. Gerhardt, K.-H. Mütter, and H. Kröger, Phys. Rev. B 57, 11504 (1998). 26 R. Bursill, G. A. Gehring, D. J. J. Farnell, J. B. Parkinson, T. Xiang, and C. Zeng, J. Phys. Cond. Mat. 7, 8605 (1995).

0704.0765

Evidence of Spatially Inhomogeous Pairing on the Insulating Side of a Disorder-Tuned Superconductor-Insulator Transition

Evidence of Spatially Inhomogeous Pairing on the Insulating Side of a Disorder-Tuned Superconductor-Insulator Transition K. H. Sarwa B. Tan, Kevin A. Parendo, and A. M. Goldman School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA (Date textdate; Received textdate; Revised textdate; Accepted textdate; Published textdate) Abstract Measurements of transport properties of amorphous insulating InxOy thin films have been interpreted as evidence of the presence of superconducting islands on the insulating side of a disorder-tuned superconductor-insulator transition. Although the films are not granular, the behavior is similar to that observed in granular films. The results support theoretical models in which the destruction of superconductivity by disorder produces spatially inhomogenous pairing with a spectral gap. The interplay between localization and superconduc- tivity can be investigated through studies of disordered superconducting films [1], originally treated by Anderson [2], and Abrikosov and Gor’kov [3], who considered the low-disorder regime. Several approaches have been pro- posed for strong disorder, including fermionic mean field theories [4, 5, 6] and theories that focus on the univer- sal critical properties near the superconductor-insulator transition. The latter consider the transition to belong to the dirty boson universality class [7]. When quantum fluctuations are included in fermionic theories for high levels of disorder a spatially inhomogeneous pairing am- plitude is found which retains a nonvanishing spectral gap [8]. For sufficiently disordered systems inhomoge- neous pairing can also be brought about by thermal fluc- tuations [9]. A similar inhomogeneous regime has also been considered under the rubric of electronic microemul- sions in the context of the metal-insulator transition of two dimensional electron gases [10]. In this letter we pro- vide evidence of a spatially inhomogeneous order param- eter on the insulating side of a superconductor-insulator transition driven by structural and/or chemical disorder. Studies of disorder and magnetic field tuned superconductor-insulator transitions have usually been carried out on films that are either amorphous or gran- ular. In the former the disorder is on an atomic scale, and in the latter, on a mesoscopic scale in which case the films consist of metallic grains or clusters connected by tunneling, that are either embedded in an insulating matrix, or on a bare substrate [1]. Amorphous films can be produced when films of metal atoms such as Pb or Bi are grown at liquid helium temperatures on substrates precoated with a wetting layer of amorphous Ge or Sb [11], or by careful vapor deposition of MoxGey, InxOy, or TiN using a variety of techniques. Granular films, are known to develop superconductiv- ity in stages. If the grains are small and weakly con- nected, the film is an insulator. For grains larger than some characteristic size, and sufficiently close together, “local superconductivity” will develop below some tem- perature. The opening of a spectral gap in the density of states of the grains [12] results in a relatively sharp upturn in the resistance below this temperature, which is usually close to the transition temperature of the bulk material. For well enough coupled grains, there may be a small drop in resistance at that temperature, followed by this upturn. This is in contrast with the “global su- perconductivity” that occurs when a sufficient fraction of the grains or clusters are strongly enough Josephson cou- pled to form a percolating superconducting path across the film. We have measured the temperature and magnetic field dependence of the resistance, and nonlinear conductance- voltage characteristics of amorphous InxOy films pre- pared by electron-beam evaporation. These films are not granular, but nevertheless exhibit local supercon- ductivity at the lowest temperatures. At low temper- atures, the application of a magnetic field results in a dramatic rise in resistance exhibiting a maximum that with decreasing temperatures is found at decreasingly small fields. The conductance-voltage characteristics in this high resistance regime are nonlinear in a manner suggestive of single-particle tunneling between supercon- ductors. We argue that these observations are consistent with the presence of droplets, or islands of superconduc- tivity, characterized by a nonvanishing superconducting pair amplitude and coupled by tunneling. Many of the droplets are Josephson coupled, but their density is not high enough to produce a superconducting path across the film. The 22 nm thick films used in this study were deposited at a rate of 0.4 nm/s by electron beam evaporation onto (001) SrTiO3 epi-polished single crystal substrates. Plat- inum electrodes, 10 nm in thickness, were deposited prior to growth. The starting material was 99.999 % pure In2O3. A shadow mask defined a Hall bar geometry in which the effective area for four-terminal resistance mea- surements was 500 x 500 µm2. As-grown films exhibited sheet resistances of about 4600 Ω at room temperature and about 23 kΩ at 10 K. By annealing at relatively low temperatures (55-70 ◦C) in a high vacuum environment (10−7 Torr), film resistances were lowered, and depending upon the annealing time either insulating or supercon- ducting behavior at low temperatures could be induced Typeset by REVTEX 1 http://arxiv.org/abs/0704.0765v2 FIG. 1: Resistance vs. temperature for Films 1 and 2. [13]. Low-temperature rather than high-temperature an- nealing avoids changes in morphology that would result in granular or microcrystalline films. As reported by Gantmakher, et al. [14], at room temperature the re- sistances of annealed films were found to be unstable. However, at low temperatures (40-1400 mK) and in vac- uum, they were stable. The films of the present study had resistances of 2600 Ω at room temperature. Film structure was studied using atomic force mi- croscopy (AFM), X-ray diffraction (XRD) analysis, and high resolution scanning electron microscopy (SEM). From the XRD there was no indication of the presence of crystalline In or In2O3. The SEM did not detect any In inclusions, and could be correlated with AFM stud- ies which revealed for a 22 nm thick film, roughness in the form of surface features with a height of 8.5 nm, and with bases about 18 nm in diameter. The conclusion from these characterization efforts is that the films were homogeneous and amorphous, and did not contain iso- lated grains or In inclusions. Measurements were carried out in an Oxford Kelvinox- 25 dilution refrigerator housed in a screen room, with measuring leads filtered at room temperature using π- section filters and RC filters. For measurements of resis- tance, the applied current was set in the range of 10-100 pA, to avoid the possibility of heating. Figure 1 shows a plot of R (T ) for two films which were studied in detail. For each, dR/dT is negative at the lowest temperatures. In the case of Film 1 there is a local minimum in R(T ) at about 350 mK. Both films exhibit a sharp upturn in R(T ) between 200 and 300 mK, with the effects to be discussed below, occurring for Film 1 at higher tempera- tures than for Film 2. These behaviors are suggestive of a regime of local superconductivity [12]. The sheet resistances of Films 1 and 2 were both ap- proximately 78 kΩ at 40 mK. In a perpendicular magnetic field of only 0.2 T, their sheet resistances increased by up to a factor of 40 at 40 mK. The maximum in R(B) as shown in Fig. 2(a) for Film 1 is followed, at the lowest temperatures, by a relatively slow decrease in resistance with increasing field. The resistance maximum moves to higher fields, with increasing temperature. The behavior of Film 2 resembled the higher temperature data for Film 1, presumably because Film 2 exhibited weaker traces of superconductivity as evidenced by the absence of a lo- cal minimum in R(T ) in the zero-field. This variation in properties from film to film is expected, as small changes in chemistry and/or morphology can have a large effect on disordered film properties. The temperature depen- dencies of the fields, Bpeak and resistances Rpeak are pre- sented in Fig. 2(b). A qualitatively similar, but weaker enhancement of resistance was previously reported for in- sulating InxOy films by Gantmakher and coworkers [14]. A larger enhancement was reported for ultrathin insulat- ing Be thin films [15]. However neither of these works demonstrated the systematic effects shown in Fig. 2(b). To probe the nature of the high resistance state, differ- ential conductance-voltage characteristics were also stud- ied [15, 16, 17, 18, 19, 20]. These are shown in Fig. 3 for Film 2 which was studied in detail. Film 1 exhibited qualitatively similar features. All of the nonlinear conductance-voltage characteris- tics are reminiscent of the single-particle tunneling char- acteristics of superconductor-insulator-superconductor (SIS) tunneling junctions. The effects of electron heating are found at voltages well above the observed conduc- tance thresholds [21]. The fact that the low voltage non- linearities vanish at temperatures above approximately 200 mK, suggests that they are associated with the pres- ence of a nonvanishing pairing amplitude occurring in the disconnected superconducting regions. We can model these thin disordered films exhibiting spatially inhomogeneous pairing as random networks of tunneling junctions of various (random) levels of conduc- tivity, connecting superconducting clusters imbedded in an insulating matrix. Some of these junctions are su- perconducting because they are Josephson coupled. As a consequence there are “superclusters,” which are ag- gregates of Josephson coupled smaller clusters that may cover a macroscopic fraction of the film area. Charge may flow through “superclusters” with zero electrical re- sistance. However, as long as these do not span the film the resistance will be determined by single-particle tun- neling. The sheet resistance of the resultant network can be inferred using the following simple argument [22]. Dis- connect all of the junctions in the network whose conduc- tance involves single particle tunneling, and then recon- nect them one by one in ascending order of resistance. A stage will be reached at which the next junction com- pletes an infinite cluster connecting the ends of the net- work. Let the normal state resistance of this last junction FIG. 2: (a) Resistance vs. magnetic field for Film 1. The temperatures are 40 (top), 80, 100, 120, 130, 140, 150, 170, 180, 200, 230, 250, 300, 350, 400, and 500 mK (bottom). (b)The fields (left axis) and the resistances (right axis) of the peaks in R(B) are plotted as a function of temperature. The flattening of Rpeak at the lowest temperatures may be the result of a failure to cool the electrons. FIG. 3: Differential conductance vs. voltage of Film 2 at 100 mK for 0 (top), 0.01, 0.02, 0.03, 0.04, 0.06, 0.1, 0.175, 0.25, 0.5, and 1 T (bottom). be Rn. The actual value will depend upon the nature of the distribution of single-particle tunneling resistances in the film. The measured normal-state sheet resistance of the entire network will then be Rn, as this junction is the bottleneck. Junctions with R > Rn are irrelevant since they are always shunted by junctions with resistances of order Rn. Junctions with R < Rn only form finite clusters over macroscopic distances. They don’t affect the conductivity because current must still pass through junctions with resistance of order Rn to get from one “supercluster” to the next. The action of a magnetic field is to quench the Josephson coupling within the “su- perclusters.” When this happens, the resistance at each temperature will be governed by the new, higher, value of the bottleneck resistance as there will no longer be any Josephson-coupled “superclusters,” and the distribution of junction resistances will shift towards higher values of resistance. The fact that the magnetic field that induces higher resistance decreases with decreasing temperature is a counter-intuitive result, implying a divergent magnetic length scale in the zero temperature limit, possibly of the form [φ0/B] where φ0 is the flux quantum. This re- sult suggests enhanced quantum fluctuations in the zero temperature limit. A heuristic argument can be made to demonstrate that this is plausible by treating the in- homogeneous pairing state of the film as a collection of superconducting grains or islands coupled by tunneling junctions. Without the inclusion of quantum fluctuations the argument may not capture all of the features of the data. It is first useful to consider the magnetic field de- pendence of the in-plane Josephson coupling between two planar thin film square islands with an area L2. This is a geometry resembling the grain boundary geometry of high temperature superconductor junctions. A magnetic field applied perpendicular to the plane will completely penetrate both electrodes of such a junction. As a conse- quence the minima of the diffraction pattern will be gov- erned by the field corresponding to a single flux quantum over the full area of the structure or B [L(2L+ d)] = φ0 where d is the width of the barrier or gap [23]. Since L >> d, the field at the first minimum of the diffrac- tion pattern would be found at a value proportional to −2 . For a “supercluster” consisting of a square ar- ray of square islands that are Josephson coupled, with some degree of randomness in the coupling, one would expect coherence to vanish at the first minimum. For a random array and with clusters that are not square, one might expect a similar dependence on L−2. If the char- acteristic size of the islands increased with decreasing temperature, which is a plausible assumption, the field quenching the Josephson coupling, would be expected to decrease as is observed. For the films studied, at the lowest temperatures, the peak in the resistance occurs in a field of 0.2 T, which would correspond to a length of approximately 100 nm. The fall off of the resistance at fields above that pro- ducing the maximum slows with decreasing temperature, consistent with the strengthening of the pairing ampli- tude with decreasing temperature. The fact that this remnant of superconductivity persists to fields up to 12 T, far above the bulk critical field of InxOy, implies that the superconducting islands are much smaller than the penetration depth. It should be possible to develop a detailed percolation model of this effect, similar to that developed for granular superconductors [24], which in- cludes the quenching of the Josephson coupling by mag- netic field and quantum fluctuations. Although the resistance of the films of the present work increases with decreasing temperatures in zero magnetic field there is no guarantee that at some temperature lower than the minimum value accessed in these measurements, the resistance will not fall to zero. This could result from the percolation of Josephson coupling across the film as the size of the clusters increases. In that event the inhomogeneous pairing implied by the data would be more likely governed by a theory including both quantum [8] and thermal [9] fluctuations. The large peaks in the magnetoresistance found at fields above the magnetic field-induced superconductor- insulator transition of superconducting amorphous InxOy thin films may result from a similar inhomogeneity of the pairing amplitude, in that case induced by mag- netic field rather than disorder. Such peaks were first re- ported by Hebard and Paalanen [25] who suggested that the state induced when superconductivity was quenched was a Bose insulator, characterized by localized Cooper pairs. They proposed that the peak was the signature of a crossover to a Fermi insulating state of localized elec- trons. This resistance peak has been the subject of more recent studies of InxOy films [14, 16, 26], of microcrys- talline TiN films where the high field limit appears to be a “quantum metal” [18], and of high-Tc superconductors [27]. The fact that inhomogeneous pairing can be in- duced in disordered superconductors by magnetic fields has been recently established using a Hubbard Model [28]. The notion that disorder implies inhomogeneity of su- perconducting order on some length scale was first dis- cussed by Kowal and Ovadyahu [29], and as was men- tioned earlier emerges naturally in a fermionic model of the superconductor-insulator transition that exhibits a disorder-tuned inhomogeneity of the pairing amplitude [8]. The films of Kowal and Ovadyahu differ from those of the present work in that they are presumably more disordered, and thus further into the insulating regime. Their magnetoresistance is always negative as there is no Josephson coupling between islands and the main effect of magnetic field is to weaken the inhomogeneous pairing amplitude, leading to negative magnetoresistance. This work was supported by the National Science Foundation under grant no. NSF/DMR-0455121. The authors would like to thank Zvi Ovadyahu for providing samples and for critical comments, and Leonid Glazman and Alex Kamenev for useful discussions. [1] A. M. Goldman and N. Marković, Phys. Today 51 (11), 39 (1998). [2] P. W. Anderson, J. Phys. Chem Solids 11, 26 (1959). [3] A. A. Abrikosov and L. P. Gor’kov, Zh. Eksp. Teor. Fiz. 36, 319 (1959) [Sov. Phys. JETP 9, 220 (1959)]. [4] H. Fukuyama, H. Ebisawa, and S. Maekawa, J. Phys. Soc. Jpn. 53, 3560 (1984). [5] M. Ma and P. A. Lee, Phys. Rev. B 32, 5658 (1985). [6] A. M. Finkel’stein, Physica B 197, 636 (1994). [7] M. P. A. Fisher, Phys. Rev. Lett 65, 923 (1990). [8] A. Ghosal, M. Randeria, and N. Trivedi, Phys. Rev. Lett. 81, 3940 (1998); A. Ghosal, M. Randeria, and N. Trivedi, Phys. Rev. B 65, 014501 (2001). [9] L. N. Bulaevskĭı, S. V. Panyukov, and M. V. Sadovskĭı, Zh. Eksp. Teor. Fiz. 92, 672 (1987) [Sov. Phys. JETP 65, 380 (1987)]. [10] Boris Spivak and Steven A. Kivelson, arXiv:cond-mat/0510422 v2. [11] Myron Strongin, R. S. Thompson, O. F. Kammerer, and J. E. Crow, Phys. Rev. B 1, 1078 (1970). [12] B. G. Orr, H. M. Jaeger, and A. M. Goldman, Phys. Rev. B 32, 7586 (1985). [13] Z. Ovadyahu, J. Phys. C 19, 5187 (1986). [14] V. F. Gantmakher, M. V. Golubkov, J. G. S. Lok, and A. K. Geim, JETP 82, 951 (1996). [15] E. Bielejec, J. Ruan, and W. Wu, Phys. Rev. B 63, 100502(R) (2001). [16] G. Sambandamurthy, L. W. Engel, A. Johansson, and D. Shahar, Phys. Rev. Lett. 92, 107005 (2004), G. Samban- damurthy et al., Phys. Rev. Lett. 94, 017003 (2005). [17] C. Christiansen, L. M. Hernandez, and A. M. Goldman, Phys. Rev. Lett. 88, 037004 (2002). [18] T. I. Baturina, C. Strunk, M. R. Baklanov, and A. Satta, Phys. Rev. Lett. 98, 127003 (2007). [19] W. Wu and P. W. Adams, Phys. Rev. B 50, 13065 (1994). [20] R. P. Barber, Jr., Shih-Ying Hsu, J. M. Valles, Jr., R. C. Dynes, and R. E. Glover III, Phys. Rev. B 73, 134516 (2006). [21] M. E. Gershenson, Yu. B. Khavin, D. Reuter, P. Schafmeister, and A. D. Wieck, Phys. Rev. Lett. 85, 1718 (2000). [22] B. G. Orr, H. M. Jaeger, A. M. Goldman, and C. G. Kuper, Phys. Rev. Lett. 56, 378 (1986). [23] K. L. Ngai, Phys. Rev. 182, 555 (1969). [24] Pedro A. Pury and Manuel O. Cáceres, Phys. Rev. B 55, 3841 (1997). [25] A. F. Hebard and M. A. Paalanen, Phys. Rev. Lett. 65, 927 (1990); M. A. Paalanen, A. F. Hebard, and R. R. Ruel, Phys. Rev. Lett. 69, 1604 (1992). [26] M. Steiner and A. Kapitulnik, Physica C 422, 16 (2005). [27] M. A. Steiner, G. Boebinger, and A. Kapitulnik, Phys. Rev. Lett. 94, 107008 (2005). [28] Yonatan Dubi, Yigal Meir, and Yshai Avishai, Nature 449, 876 (2007). [29] D. Kowal and Z. Ovadyahu, Solid State Commun. 90, 783 (1994). http://arxiv.org/abs/cond-mat/0510422

0704.0766

Local de Broglie-Bohm Trajectories from Entangled Wavefunctions

Local de Broglie-Bohm Trajectories from Entangled Wavefunctions Michael Clover∗ Science Applications International Corporation San Diego, CA (Dated: October 6, 2021) We present a local interpretation of what is usually considered to be a nonlocal de Broglie- Bohm trajectory prescription for an entangled singlet state of massive particles. After reviewing various meanings of the term “nonlocal”, we show that by using appropriately retarded wavefunc- tions (i.e., the locality loophole) this local model can violate Bell’s inequality, without making any appeal to detector inefficiencies. We analyze a possible experimental configuration appropriate to massive two-particle singlet wave- functions and find that as long as the particles are not ultra-relativistic, a locality loophole exists and Dirac wave(s) can propagate from Alice or Bob’s changing magnetic field, through space, to the other detector, arriving before the particle and thereby allowing a local interpretation to the 2-particle de Broglie-Bohm trajectories. We also propose a physical effect due to changing magnetic fields in a Stern-Gerlach EPR setup that will throw away events and create a detector loophole in otherwise perfectly efficient detectors, an effect that is only significant for near-luminal particles that might otherwise close the locality loophole. PACS numbers: 03.65.-w, 03.65.Ud I. INTRODUCTION In the EPR literature, it is usually considered that the only local models that can explain EPR experiments are contrived intellectual diversions1,2,3, bearing no relation to physical reality, and that additionally require some “unfair” sampling mechanism to generate the correct experimental results4,5,6,7,8. Somewhat less contrived, Accardi9 has proposed a chameleon model that is claimed to be local and violate Bell’s inequalities, and Christian10 had proposed a local model based on properties of a Clifford algebra. With the exception of the last two, most local hidden variable models assume a “detector” loophole. It also seems to be conceded that – for a single massive particle32 – the de Broglie-Bohm interpretation of quantum mechanics is local, in that the gradients of the wave function at the particle position provide a quantum field that exerts a force on a deterministic, otherwise classical, particle. There seems to be general agreement that when it comes to two entangled particles, the same de Broglie-Bohm interpretation is manifestly nonlocal11,12. It is with this argument that we take issue, and propose to regard the two-particle de Broglie-Bohm relations13 as a local model – which we claim can be done as long as time-retarded wavefunctions are used. We motivate this by an example: an analysis of Hooke’s Law and its apparent nonlocality. We will show that our local de Broglie-Bohm model can violate Bell’s inequality as long as it can use the “locality” loophole. Since QM (Dirac) waves move at a phase velocity of c, and since massive particles move much slower, this loophole appears to be big enough for all conceivable experiments involving massive particles. We will also show that there exists a physical mechanism that could operate on these massive particles to provide a “detection” loophole, which we will see is not significant unless the particles are moving at ultra-relativistic speeds – but for such Stern-Gerlach experiments, means that one could still measure apparent violations of Bell’s inequality with a local mechanism. II. HOW NON-LOCAL IS DE BROGLIE-BOHM? In the single particle case, the de Broglie-Bohm velocity law takes the form ~v(x, t) = − i~ ψ∗(x, t)∇ψ(x, t) − (∇ψ(x, t))∗ψ(x, t) ψ∗(x, t)ψ(x, t) ∇ψ(x, t) ψ(x, t) , (1) where ψ(x, t) is the solution to the Schrödinger equation. This is a non-linear, but not obviously non-local, prescription for the velocity of the particle. If a boundary condition changes far away (e.g., closing a slit at x = −L at time t = 0), http://arxiv.org/abs/0704.0766v2 and the effect is felt at x = +L simultaneously, we will claim that the nonlocality is a result of using the non-relativistic Schrödinger equation; using the relativistic Dirac equation means that the wavefunction at x = +L will not change until after the time ∆t = 2L/c, when the “Dirac wave” has propagated to that position (the velocity prescription for Dirac wavefunctions14 is similar to that of de Broglie-Bohm’s for Schrödinger wavefunctions: dxi/dt = cji/j0 where jµ = cψγµψ). Thus, for a single particle, we claim that de Broglie-Bohm interpretation is not essentially non-local. In the case of two particles, we have ~v1(t) ∼ ∇1ψ(x1, x2, t) ψ(x1, x2, t) = v1(x1, x2, t) 6= v(x1, t) , ~v2(t) ∼ ∇2ψ(x1, x2, t) ψ(x1, x2, t) = v2(x1, x2, t) 6= v(x2, t) , for the velocity of each particle, and see that, formally at least, neither particle’s velocity is strictly a function of it’s own position, as one might expect for classical particles in local potentials. This dependence is taken as a priori evidence of nonlocality of the de Broglie-Bohm interpretation16? . We shall argue below that this dependence is misinterpreted. Bohm’s justification of the physicality of nonlocality (i.e., the transmission of forces at infinite speed) was to argue that it is a “non-signaling” non-locality, since his interpretation had been shown to lead11 (p. 187) to “precisely the same predictions for all physical processes as are obtained from the usual interpretation (which is known to be consistent with relativity).” In other words, by taking the ensemble average of lots of non-local results he gets a local result. We would prefer the grammatically simpler “locality” to the double negative of a non-signaling non-locality. As far as we can see, the only reason most physicists don’t agree, is that Bell has said that certain experimental measurements are evidence of nonlocality. We have argued elsewhere17,18 that Bell’s derivation assumed a certain non-contextuality that was too strict, and that a weaker limit could be derived for hidden variables that are “locally” contextual (by local contextuality we mean hidden variables that prevent Alice from measuring a′ at the same time she measures a). This paper presents an example of a local hidden variable theory that is also “remotely” contextual, in that Alice’s result depends on her causal knowledge of Bob’s detector setting (i.e., at a retarded time, A(a(t), b(t−D/c), λ(t))). A. How non-local is Hooke’s Law? Consideration of a wider relevant context prompts one to ask “How non-local is Hooke’s Law?”, because in that case we also have a force law that looks “manifestly” nonlocal: m1ẍ1 = −k(x1 − x2) , (2a) m2ẍ2 = +k(x1 − x2) . (2b) We are not aware of any textbooks that use this as an example of non-locality (springs being an archetypical example of local, causal wave propagation), and most physicists would expect that even if m2 were suddenly fixed (in Equa- tion (2a)), that m1 wouldn’t react to that event until a sound wave had passed down the spring from m2 to m1. That being the case, perhaps x2 should be retarded in the first equation and x1 in the second; unless it is invalid to use these equations in that context. The de Broglie-Bohm interpretation has been modified in precisely that manner15,19? – to use time-retarded positions in a time-independent wavefunction – in order to force a local interpretation out of the velocity prescriptions. This means, of course, that if the original velocity prescription came from nonlocal non-signaling interpretations of the Dirac equation, that the new interpretation would be the solution to something that isn’t quite the Dirac equation anymore, and sooner or later, for some measurement this new re-interpretation will have to give a theoretical prediction different from the Dirac equation (thereby falsifying this retarded solution). If the positions in Hooke’s law are retarded, we would then have m1ẍ1(t) = −k(x1(t)− x2(t− τ)) , m2ẍ2(t) = +k(x1(t− τ)− x2(t)) , where τ = ∆x/cs, and ∆x is some measure of the separation. One can expand these new expressions about t for small τ , leading to m1ẍ1(t) = −k(x1(t)− x2(t))− τkẋ2(t) , m2ẍ2(t) = +k(x1(t)− x2(t))− τkẋ1(t) , which still has the force on the first particle depending on the instantaneous position of the second particle, and vice versa, but now a damping force (∝ c−1) depends on the instantaneous velocity of the other particle as well. It is possible that this reductio ad infinitum might converge short of absurdum, but we would suggest that retarding something that is not nonlocal in the first place will just make the local equations wrong. Our preferred interpretation of this case is to take seriously the notion that Equations (2) are an idealization only appropriate to an isolated system. If the system is isolated, then no one can grab one particle to initiate a nonlocal signal in the first place, and if they do, then the system is no longer isolated, the equations no longer apply, and there is no issue of whether invalid equations can signal or not. Nonetheless, within the idealization, there is a way for m2 in equation (2b) to show indifference to m1’s fate for at least a while, and that would be if m2 took seriously the notion that it was part of an isolated system. Suppose m2 “knows” not only his current location, but has “calculated” (with Equation (2a)) where the coupled and equally isolated m1 should be and thus what acceleration m2 should be experiencing (similarly, m1 calculates m2’s location in order to update her acceleration, in both cases using initial conditions learnt from earlier causal contact). In that case, m2 will continue to oscillate regardless of whether something grabs m1 at a particular time or not. Of course, m2 will continue to oscillate even after ∆x/cs, but now we know that our (local) mathematical model is in a regime it is not designed to handle (and we can predict when any discrepancies should occur); we would have to go beyond two isolated mass points to the continuum model of Navier-Stokes equations in order to have enough sophistication to propagate sound waves from changing boundary conditions at one location to other locations. Note that we can also transform the equations into another form that suggests more locality and less computational capabilities for mass points: m1ẍ1(t) = −k(x1(t)− x2(t)) , = −2k(x1(t)−Xcm) . As long as the center of mass of the isolated system behaves predictably (Xcm(t) = X0, which can always be done after a Gallilean boost if Ẋ0 6= 0), the only non-parametric variable in the force on m1 is the position of m1 in the “force-field” of the center of mass, i.e., F1 = F (k,X0;x1). In this way we see that an isolated system of particles can be such that one particle can depend on the instantaneous position of another, without requiring anything but a local, causal force field at each point (which effectively calculates m1’s position for m2 and vice versa). Without the concept of a force-field, Newton was flummoxed less by the nonlocality than the action at a distance; with the concept of a quantum potential and its field extending through the aether, there is no problem of m1 acting on m2 from a distance, although the question of whether it does it instantaneously or not requires us to determine what an “isolated” system is. We will suggest that in an EPR experiment, by knowing x1(0), x2(0), and Ψ, Alice can calculate both x1 and x2 given the de Broglie-Bohm velocity prescriptions, and similarly for Bob. For Hooke’s law, the differential equations describing the motion are only correct as long as the system is undisturbed – any attempt to grab one of the mass points destroys the isolation and changes the boundary conditions of an underlying continuum model; for an EPR experiment, the Schrodinger wavefunctions provide valid trajectories as long as the boundary conditions (Alice and Bob’s magnet settings) are unchanged – otherwise, we have to wait for the Dirac waves to re-equilibrate the new settings. We remind the reader that the usual method of calculating de Broglie-Bohm trajectories is to solve i~ψ̇ = Hψ in order to propagate a wave packet from its initial position to the final position, so that the derivatives of that wave-packet wavefunction can provide the pilot instructions to the individual point particles. If the boundary conditions change during that time, it is clear that a more fundamental theory must be invoked – for example the Dirac equation, which will propagate changes to the wavefunction at the speed of light. Even so, this still leaves the x1−x2 type functional dependence, which can be worked around in the same way that Hooke’s m1 and m2 performed redundant calculations, by assuming that individual particles know the (fixed) position of the center of mass. If a more fundamental approach is needed, one can then resort to second quantization and QED, the continuum analogs to Navier-Stokes. As with springs, for two quantum-mechanical particles one can recast a Schrödinger equation using individual coordinates, ψ(x1, x2), into one using center-of-mass and relative coordinates, Ψ(X, x), instead. In the Stern-Gerlach case, there are some practical advantages working with two coordinate systems that are not aligned with respect to each other (where Bob’s x̂2 ‖ B̂bob and Alice’s x̂2 ‖ B̂alice, but B̂bob ∦ B̂alice), and since the singlet state’s relative coordinate wavefunction still entangles spin up and down, there is no advantage to using relative coordinates for this problem. (N.B.: when Alice and Bob’s magnets are at the same angle, so that the two x̂’s are parallel, the relative coordinate formulation does allow one to see that the de Broglie-Bohm velocity law factors into a single function of ~x, independent of ~X, shown in Equation (8) below. In other words, for aligned magnet settings, the de Broglie- Bohm velocity law takes on the same dependence on relative coordinates as a Hooke or Coulomb force law, which we have seen is local as long as the system is isolated.) B. Entanglement and a priori Nonlocality Rice20 presents a geometric argument that the de Broglie-Bohm interpretation must be nonlocal, arguing, essentially, that an entangled wavefunction has nonlocality built into it despite (or in spite of) the local behavior of a single wavefunction. Consider the case of two non-interacting particles, such that H1ψ1 = i~ψ̇1 and H2ψ2 = i~ψ̇2. A basis-state wavefunction for the two-particle system would have the form ψ1(x1, t)ψ2(x2, t), and an entangled, singlet state wavefunction can be written as ψ1(x1, t)ψ2(x2, t)−ψ1(x2, t)ψ2(x1, t). It is this form that Rice claims is nonlocal: start by assuming that Bob makes some change to his boundary conditions at time t = 0. Alice (who we assume is at a distance D from Bob), in order to apply the de Broglie-Bohm method to describe the position of her particle (m1), determines that her v1 is proportional to ψ2(x2, t)∇1ψ1(x1, t) − ψ1(x2, t)∇1ψ2(x1, t); if the time t is less than D/c, then no causal signal from Bob (e.g., changes in his Dirac wavefunction) should have reached Alice. Nonetheless, the velocity that she calculates has a (first) term proportional to ψ2(x2, t) – her velocity depends on Bob’s wavefunction (ψ2) after his change to the boundary condition but before Alice can possibly know about it in any causal manner, even if Alice has been keeping track of where Bob’s m2 is or should be currently located. In other words, it is not the ψ(x2) dependence that is non-local, it is the ψ2(t) that is non-local regardless of where x2 is at. Whether these wavefunctions are solutions to Dirac or Schrödinger equations, it is this form of the entangled wavefunction that Rice claims is a priori nonlocal. In regular quantum mechanics, the only use made of such entangled wavefunctions (or any wavefunction) is to calculate matrix elements, and the issue of nonlocality does not arise in such a case: one integrates over all dx2 at a time t, resulting in something that is only a function of x1, which in turn is integrated over dx1 to produce the number that is compared to some experimental measurement. (Note that during the integration over dx2 and dx1, it is assumed that the wavefunction is static, corresponding exactly to an experiment done with detectors in fixed positions. If Alice and Bob are rapidly changing their settings, then the quantum-mechanical averages need to be done with more care than has yet been shown by anyone in the EPR community.) Only the de Broglie-Bohm interpretation makes use of the “pointwise” values of the wavefunction in its velocity prescriptions, and so only the de Broglie-Bohm interpretation suddenly confronts this peculiar form of nonlocality. There are three possible solutions to this problem. The historical approach (Rice’s) is to assume that Alice is indeed godlike (or at least nonlocally omniscient), and instantaneously knows what Bob’s wavefunction has become, so that she can apply her nonlocal velocity algorithm to determine v1, and we again have a “non-signaling” nonlocality. Another approach (Squires’) is to retard the position of the opposite particle in the (presumably still nonlocal) time dependent wavefunction.33 The alternative which we propose is to insist that Alice can only make use of locally available information, and that while she can use Bob’s wavefunction to calculate her velocity, it must be an appropriately retarded wavefunction – the one that Bob would have had if he had done nothing in the most recent D/c time interval. Unlike the classical Alice, who only had to calculate x1 and x2 from Hooke’s law, the quantum Alice not only has to calculate x1 and x2, she has to calculate both ψ1 and ψ2 using only her causal information about all boundary conditions (ditto for quantum Bob). It is hard to pick a good notation with which to write this; Alice will calculate a contemporaneous wavefunction, ψ2(x2, t), to describe the velocity of Bob’s particle at the current time and non-retarded position, but this PDE will be solved using boundary conditions near Bob that are retarded to the time t−D/c; similarly, Bob will calculate ψ1(x1, t), but using boundary conditions near Alice also retarded by D/c. We will refer to these as “retarded wavefunctions” as a shorthand for “contemporary wavefunctions based on retarded boundary conditions”, and write them as ΨA(x1, x2, t) = ψ1(x1, t)ψ2(x2, t−D/c)− ψ1(x2, t−D/c)ψ2(x1, t) , ΨB(x1, x2, t) = ψ1(x1, t−D/c)ψ2(x2, t)− ψ1(x2, t)ψ2(x1, t−D/c) . If the boundary conditions remain static, the wavefunctions evaluated with time-retarded boundary conditions become identical to ones with instantaneous boundary conditions, which means that Alice and Bob will then generate exactly the entangled results (that violate Bell’s inequality). If the boundary conditions do change, then it will take time for the new “entanglement wave” to propagate from Alice to Bob (or vice versa), and Bob, using “old” information, will calculate a trajectory different from one using non-local instantaneous information (we assume averages of such events will satisfy Bell’s inequality and give a concrete illustration of this in the section below on trajectories for massive singlets). C. Nonlocality involving violation of Bell’s Inequality We17,18 and others21 have shown that Bell’s derivation of Bell’s inequality, |〈AB〉 − 〈A′B′〉|+ |〈AB′〉+ 〈A′B〉| ≤ 2 , (3) depended on a particular form of counterfactual definiteness – that two measurement results can be considered to exist at the same time – e.g. that position (A) and momentum (A′) can be measured simultaneously – something that is forbidden by the Heisenberg Uncertainty Principle.34 The “instruction set” paradigm of hidden variables? can also be viewed this way: “the hidden variables are the results not measured” – not something that might be a proximate cause of a measurement (like a spin vector with some horizontal as well as vertical component, or what instruction sets ought to be), but the measurement result itself, since the instruction is plugged into the formula where the measured results are supposed to go. But what can a measurement result be in the absence of a measurement – the particle going left (unnoticed, in a virtual horizontal magnet?) while we detect that it is going up in our vertically oriented magnet? Clearly, if we cannot physically make the two measurements at the same time, we should not talk about the measurement results existing at the same time, even though such classical habits are hard to break. Avoidance of such counterfactual reasoning results in a weaker inequality, |〈AB〉 − 〈A′B′〉|+ |〈AB′〉+ 〈A′B〉| ≤ 2 + min ( |〈B1B′3 −B′2B4〉| , |〈A1A′4 −A′2A3〉| ) , (4) consistent with the fact that one measurement can only be made strictly before or after the other. This inequality also holds for any classical system for which two measurements cannot be made simultaneously.35 Of course, this should not be a surprise, since one can easily show22,23 that the quantum-mechanical measurement operators (for 2-particle states) obey a similar inequality: Ŝ2Bell = 4Î − [Â, Â′][B̂, B̂′] . (5) A number of philosophical questions arise when we contemplate this relation: since there is no factor of ~multiplying the commutator terms on the right hand side of Equation 5, how is it that a putatively nonlocal quantum mechanics can reduce to a local classical mechanics. Furthermore, why is it only non-commutative experimental measurements that generate nonlocal results – if Alice switches from a to a + 180o (A → −A), then even if Bob continues to switch between b and b′, nonlocality will not show up – what kind of nonlocality only shows up when measured at some angles but not others? The issues vanish if the explanation is that measurements are contextual, as shown, for example, by Christian’s model10. As we are about to see, the issues will also vanish if Bell’s assumption that Alice and Bob’s measurements, made a distance D apart, can’t be written as A(a, λ) or B(b, λ), but must be written as A(a, b(t−D/c), λ) and B(a(t−D/c), b, λ). All derivations of Bell inequalities begin by assuming the A(a, λ) form as the mathematical form of the locality postulate, but this is strictly true only if A(a(t), λ) has no dependence on b(t); a dependence on b(t−D/c), while local, is causally “contextual”. It is not clear if EPR would reject that out of hand given the alternative of true nonlocality. III. DE BROGLIE-BOHM TRAJECTORIES FOR MASSIVE SINGLETS The de Broglie-Bohm analog of waiting for a sound-wave to reach the second mass point in a Hooke-law spring is to be understood, not by retarding the second particle’s position, but by using the appropriately retarded wavefunctions in the velocity law (as we have argued in section II B). Imagine that Alice doesn’t change her magnet, but Bob does. In that case, the wavefunction inside Bob’s magnetic volume will change (adiabatically, we presume) while his magnet is being switched, and a Dirac wave (a wave in the electron aether distinct from any electro-magnetic Maxwell wave) will propagate out from that region at the speed of light toward Alice. As long as Alice uses retarded boundary conditions to calculate Bob’s wavefunction, she can calculate both x1 and x2 in a causal manner for use in her velocity prescription for ẋ1; similarly, if Bob uses his instantaneous wavefunction and retarded boundary conditions for Alice’s wavefunction, his ẋ2 will be local and causal. Durt and Pierseaux13 defined an EPR experiment based, presumably, on typical Stern-Gerlach experiments. For their case of two silver atoms in a singlet spin state, they assumed that the velocity of the atom was 104 cm/s, and if we further assume that the overall size of the system is O(100 cm), then Alice will know about Bob’s new wavefunction in O(10−8 s), while the particles take O(10−2 s) to get from source to detector. Unless ultra-relativistic particles are used, it would seem that there is no way for Alice not to know Bob’s magnetic field setting in time to calculate a local expression for her particle. The locality loophole will be very hard to close for massive particles, and de Broglie-Bohm velocity prescriptions using wavefunctions based on retarded boundary conditions will provide a local hidden variable theory that violates Bell’s inequality in perfect agreement with the quantum results. Holland16 (p. 469) and Durt & Pierseaux13 have calculated the velocities of the two particles (using D&P’s notation, which distinguishes Left from Right, but for which we can read Alice and Bob; with the local z axes parallel to the local magnetic fields, ẑL|R ‖ B̂L|R): 1 + k2t2 1+k2t2 (zL+zR) + c2e 1+k2t2 (zL−zR) − c2e 1+k2t2 (−zL+zR) − s2e 1+k2t2 (−zL−zR) 1+k2t2 (zL+zR) + c2e 1+k2t2 (zL−zR) + c2e 1+k2t2 (−zL+zR) + s2e 1+k2t2 (−zL−zR) 1 + k2t2 , (6a) 1 + k2t2 1+k2t2 (zL+zR) − c2e 1+k2t2 (zL−zR) + c2e 1+k2t2 (−zL+zR) − s2e 1+k2t2 (−zL−zR) 1+k2t2 (zL+zR) + c2e 1+k2t2 (zL−zR) + c2e 1+k2t2 (−zL+zR) + s2e 1+k2t2 (−zL−zR) 1 + k2t2 , (6b) where µ/2m , (7a) β = 2α/δr20 , (7b) k = ~/2mδr20 , (7c) c ≡ cos(θA − θB)/2 , s ≡ sin(θA − θB)/2 , (7d) with µ is the magnetic moment, m the mass of the silver atom, δr0 the initial width of the Gaussian wavepacket and dB/dz the magnetic field gradient. θA and θB are the orientations of the magnetic fields with respect to a third, fixed, frame. When s 6= 0, the two z-axes are not aligned and the arguments of the exponentials are not the z-components of the relative and center-of-mass coordinates but a mixture of the z and x coordinates (assuming the particles separate from the source in the ±ŷ direction). 1. Local vs. Nonlocal Velocity Prescriptions For Alice to retard Bob’s boundary condition in her instantiation of the velocity expressions means that she must use a causal value for θB, and similarly, Bob must use only values for θA that he can know causally. Each then uses both equations to model both trajectories, using only the (jointly known) initial location of the particles. For a nonlocal prescription, Alice is free to use the instantaneous value of Bob’s magnet setting. Whether Alice continues to calculate both trajectories, or just “knows” Bob’s instantaneous result is moot, since the two methods will result in the same answer. Recall that for massive particles, it is almost certain that the Dirac wave from Bob’s changing magnetic field will overtake the Alice-bound particle while it is in the field-free region between the source and Alice’s magnet/detector. In that region, entangled Schrödinger wavefunctions generate the same simple velocity law as unentangled wavefunctions, ~vA = −~vB = −~~k/m. Thus, we hypothesize that the changes in the Dirac wave will not affect the particle as it passes over it, and that when the particle finally reaches Alice, she will know the correct magnetic field setting to use for Bob’s boundary condition and therefore impose the correct acceleration on her particle. This local result of Alice’s will generate a result identical to the “nonlocal”, Bell-violating result. 2. The Locality of de Broglie-Bohm Trajectories for Massive Singlets in Aligned Fields When Alice and Bob’s detectors are both at the same angle, θA = θB, then s = 0, c = 1 and the velocity law, like Hooke’s law, reduces to a function of the relative distance coordinate only: d(zL − zR) 1 + k2t2 t(zL − zR) + αt 1 + k2t2 · tanh 1 + k2t2 (zL − zR) . (8) Locality Normalization Detector Efficiency SBell std. dev. Loc singles Efficient -1.31946 ± .03652 Loc coincidences Inefficient -2.76893 ± .07086 NonL singles Efficient -2.77554 ± .03652 NonL coincidences Inefficient -2.76893 ± .07086 TABLE I: Calculationally measured Bell parameters for various “computational” configurations. Since we have shown that the Hooke (or Coulomb) type force law is not actually nonlocal, being an idealization of a more fundamental local continuum model, we can conclude that the entangled particle velocity law – at least when encountering aligned magnets – is also generating local trajectories, as long as the system is isolated. This is important because a recent paper24 purports to show that “Bell Locality” imposes certain constraints on any hidden variable theory, such that it must then obey Bell’s inequality. Since we have just shown that the de Broglie-Bohm interpretation becomes a local model (and satisfies the constraints Norsen proposed for the case of aligned magnet settings), we have an obvious contradiction since we know that our interpretation is local and the de Broglie-Bohm model will reproduce the quantum mechanical results. The contradiction can be resolved by realizing that Bell’s inequality implicitly requires hidden variable theories to violate Heisenberg’s uncertainty principle17 – whereas de Broglie- Bohm can predict counterfactual results (i.e., A(a, λ, t) and A(a′, λ, t′)), but it can’t predict them simultaneously, but only A(a+a , λ, t) in such a case. The fact that Alice’s velocity depends on c and s (in particular, on θB), makes this de Broglie-Bohm trajectory violate one of Bell’s assumptions,36 that the measurement result, A, be written as A(a, λ), and not as A(a, b, λ), (or A(a(t), b(t − τ), λ)). In the case that a Dirac wave has time to propagate from Bob to Alice, then there is no actual nonlocality, just a nonlinear sensitivity to (remote) initial conditions that is only revealed by entangled wavefunctions – indeed all of the physical phenomena that tout nonlocality only require entanglement in order to be understood, and we might refer to this as “contextuality at a distance”. 3. Computer Experiments We have programmed an EPR experiment, using the Durt and Pierseaux velocity prescription to determine the motion of neutral silver atoms in a 30 cm magnet with a field gradient of 104 G/cm, and initial spread in the Gaussian wavefunction of 10−3 cm, so that α = 2.58 · 105 cm s−2, β = 5.17 · 1011 cm−1s−2, and k = 2.94. The values in table I resulted from the tracking of 4000 pairs of particles, where new magnet settings were randomly chosen as each pair was launched (this means that even though the particles took a millisecond to traverse the magnet, and presumably as long to get to the magnet, we pretend that the speed of light is even slower, so that Alice and Bob are spacelike separated). Starting with the bottom two rows of the table, if Alice and Bob use non-local knowledge of the other’s instantaneous magnet setting, and if there are no physics-induced detector inefficiencies, the coincidence rate will be the same as the singles rate in each detector, so that normalizing by singles (or coincidences) will result in a small standard deviation (0.036, third row); if there are physics-induced inefficiencies (see section IV below), then the number of coincidences will be about 1/4 the number of single events, leading to a standard deviation twice as large (bottom row), in both cases violating Bell’s inequality. If Alice and Bob only use local knowledge of the other’s setting (i.e., retarded boundary conditions), and if there are no detector inefficiencies, then they will calculate the correct position of their partner’s particle only half the time (on average) and the Bell parameter they measure will be less than 2 (1.3, but with a small standard deviation, first row). If there are processes that throw particles out of the beamline before getting to a magnet that has just been changed, then Alice and Bob will find their coincidence rate to be half their singles rate, but by normalizing to the coincidences, they will calculate a Bell parameter of 2.8 with a larger standard deviation (0.07, second row). If Alice and Bob change their magnets without affecting the oncoming particles, and if the particles move at subluminal speeds (v/c ∼ 10−6) so that the Dirac wave outruns the particle, then Alice and Bob will be using the “effectively instantaneous” values of each others magnet settings. In such a case, they will measure a causal (local) violation of Bell’s inequality with a small standard deviation (identical to results in the third row). IV. PHYSICAL BUT UNFAIR SAMPLING EFFECTS In order to experimentally establish nonlocality with massive particles, it is necessary to switch the magnet settings on Alice and Bob’s detectors in a random manner, and at a rate greater than the source emission rate. Clearly if two or more singlets are emitted before either detector changes settings, there will be a much higher percentage of “locally” describable events than the potentially non-local, for which Alice and Bob, using the old (wrong) settings, will predict the wrong event behavior. On the other hand, the settings should not change so frequently as to change while the particle is inside the magnet, nor should they change and change back again before the particle gets to the detector. Using the Durt and Pierseaux expressions with the time-lagged values of the other’s magnet setting, we have shown, via computer experiments, that the Bell parameter drops to a value below 2.0 when Alice and Bob randomly pick a new detector setting after each singlet is launched (compared to 2 2 when using the instantaneous values of both). If such an experiment could actually be performed, we would expect to see experiments measure values significantly less than 2 when (if) the “locality loophole” is closed, but 2 2 if it is not. However, we remind the reader that in such an experiment25, the magnetic field will be of order 1-10 T or ∼ 105 G (although Durt and Pierseaux never mention the field strength), with spatial gradients of the field of order 102 T/m (104 G/cm). The magnet can be switched either mechanically, or by the equivalent of energizing two (opposite) poles of a quadrupole while de-energizing the other two poles. This change of magnetic field will result in the emission of a real electromagnetic (Maxwell) wave travelling down the flight-path toward the source of the particles (as well as in all other directions), and the wave will have a magnetic field in the direction of a− a′ (or b − b′). In other words, if Alice’s settings for her magnetic field (B̂) are a = 0 and a = π/2, then her ∆B̂ will point at −π/4 or +3π/4. For the sake of clarity, let us assume that on all relevant timescales for this problem, it takes no time to switch the magnet from one setting to another. Thus Bob’s full ∆B leaves the front end of his magnet and travels toward Alice, and 30/c seconds later (assuming that the magnet is L = 30 cm long), ∆B drops back to zero as the signal from the back end of the magnet leaves the front on its way to Alice. Within this virtual magnet volume that is sweeping from Bob to Alice, the gradients of the magnetic field are the same size as in the magnet – dBz = 104 G/cm.37 (One might argue that electromagnetic fields should fall off as r−2, making the magnitude of the field too small to have any effect. This would be true in free space, but for a neutral massive particle (e.g., Ag) to cover a distance of order meters from source to detector, it will need to travel in an evacuated beamline, and those are usually made of steel; to the extent that this will act as a wave guide, the ∆B field strength will not attenuate.) Since the Durt and Pierseaux velocity formulae only depend on this derivative of the magnetic field, we can treat the EM wave washing over the particle in flight as if it were in a shorter magnet that would take L/c = 10−9 s to traverse, and use Equations (6) to calculate that the particle will experience a transverse velocity before it even gets to the magnet, proportional to that shorter time. The transverse velocity for our typical silver atom will be about a part per million of what it would have been after leaving the real magnet, since the time in the real magnet is O(10−3 s) compared to the nanosecond “in” the EM wave, resulting in an insignificant effect. Now consider a faster particle, say one moving 103 times faster, so that it only spends O(10−6 s) in the magnet (assuming for the sake of argument that the very small deflections will still be resolvable). The 30 cm long ∆B wave will still pass over the particle in 1 ns, so the transverse velocity will now be a part per thousand of those making the resolvable deflections. This too will probably not result in any lost particles. However, if we now go for the ultimate neutral fermion, and send entangled pairs of neutrino’s at the Stern-Gerlach apparatus, the neutrino will spend only twice as much time in the magnet as under the influence of the ∆B wave. This means the transverse velocity experienced between the source and the detector will be half that it would experience in the detector, and the deflection a quarter. Thus, for light particles moving at near light-speeds, one can expect that the ∆B field will kick any particle in flight entirely out of the experiment, deflecting it so much that it can no longer fit through any collimators at the entrance to the magnet.38 Unfortunately, neutrino’s have their own detector efficiency problems. The simplest experimental signal of such a phenomenon would be to make absolute count-rate measurements, so that if Alice and Bob make no changes to their magnets, one measures a quiescent singles count rate, Q1 somewhat larger than the quiescent councidence rate C2 (the latter being dependent on the square of the efficiency of nearly perfect detectors). If Alice and Bob now randomly switch their magnets once on average during the source-detector flight time, in the absence of this detector inefficiency, their coincidence rate should remain C2; but if this effect is present, then the coincidence rate will drop to C′2 ∼ C2/4, the singles rate to Q′1 ∼ Q1/2, and these relative drops should be the same regardless of the efficiency of the individual detectors. V. PHOTON EPR EXPERIMENTS Tittel et al.26,27 performed an EPR experiment in Geneva that atempted to close the locality loophole by arranging that the photon in one arm of the experiment went through a beam-splitter into one of two analyzers. As the authors point out, a hidden variable could determine into which one of the two it would go, preserving locality, since the setup was static. The authors dismiss this, arguing that a quantum-mechanical random number generator (the beam splitter) is the best that can be imagined. Given that we have argued that entanglement is the underlying causal explanation for the violations of Bell’s inequalities, and entanglement relies on the quantum (Dirac) fields that permeate space, it requires no additional stretch of the imagination to believe that the de Broglie-Bohm-ish behavior of the photons at the beam splitter is also determined by similar (Maxwell) fields that determine the Bell violations in the detectors themselves. We conclude that this experiment has a gaping locality loophole. The Innsbruck experiment of Weihs et al.28 was designed to close the locality loophole, and did so if the assumption of fair sampling is valid. In this experiment, electro-optical modulators were used to rotate the polarization of the photon before it entered the polarizing beam-splitter/detector. These modulators are in some sense the analog of the magnetic field in the Stern-Gerlach apparatus, although no deviation of photon trajectories occurs inside the devices. If a detector loophole were the only flaw in this experiment, then one would have to rely on some property of the modulators, that, for example when switched, induces a loss of total-internal-reflection in the fiber-optic cable, so that photons are lost from the fiber if the switch is made. Although it is highly unlikely that such an effect would have gone unnoticed, it would be interesting to know if the coincidence counting rates vary when the modulators are turned off during the course of a setting measurement (e.g., 〈AB′〉), compared to when the modulators switch between each coincidence event. Given the nature of our local de Broglie-Bohm trajectories for massive particles, there is no conceivable way such a model could account for the spacelike separated EPR results without a detector loophole. On the other hand, there is no de Broglie-Bohm prescription for photons16 (p. 541), and it may be that the Clifford-algebra model of Christian10 may be a more correct idealization in this context than our fermion model, in which case the Innsbruck data may also be explainable by a local model. VI. CONCLUSION A. The Meaning of Nonlocality The controversy over the EPR paradox has depended on ambiguous concepts of locality and contextuality. EPR were wrong (and Bohr was right as our de Broglie-Bohm expressions show) in that Alice’s measurement result can depend on the measurement that Bob makes, if the two particles are entangled. EPR would have been right to argue that if there is such a distance effect, it must be time-retarded in order to be causal (we thus predict that perfect photon detectors will show obedience to Bell’s inequality in Innsbruck-type experiments): time-retarded “action at a distance” is not nonlocal, but it is contextual.39 Because EPR rejected contextuality at a distance, and equated that rejection to locality, Bell and others’ derivations of inequalities, as part of mathmatically translating “no context at a distance” into A(a, λ) and B(b, λ), have implicitly assumed that there can be no “local contextuality” either, so that even if A(a, λ) is measured, A(a′, λ) is assumed to exist, and is used in formulae in a manner to suggest that it could have been measured at the same time and place, thereby limiting their inequality to hidden variable theories that violate Heisenberg’s Uncertainty principle. If hidden variable theories are restricted by assumption to modeling commuting operators, it is not surprising that they fail to model non-commuting ones; Christian’s Clifford algebra10 provides a counterexample of a local model with non-commuting properties. B. Résumé The “manifestly nonlocal” equations of motion, be they Hooke’s or de Broglie-Bohm’s, are valid only as long as the system they are intended to model is isolated, which means only as long as no one tries to signal by grabbing one of the no-longer-isolated parts. Any attempt to change the boundary conditions destroys the assumption of isolation, invalidates the equation(s), and moots the point of whether the equation can signal or not. We have also seen that two-body idealizations are approximations of an underlying local continuum model; Hooke has his Navier-Stokes, but there is still work to bring QED/QCD to an equally satisfactory footing to underlie the de Broglie-Bohm interpretation of Schrödinger/Dirac. By analogy with Hooke’s Law, we have developed a local interpretation of entangled trajectories given time in- dependent wavefunctions. We have also shown that this interpretation, for time dependent wavefunctions, can be made truly nonlocal by using, unjustifiably, non-retarded values of the wavefunctions. This makes the locus of de Broglie-Bohm nonlocality reside in superluminal Dirac waves, which is patently absurd. On the other hand, we can calculate local, causal trajectories by using appropriately time-retarded wavefunction(s) in the “calculation” of the velocities for these particles, making this the first local model (to our knowledge) that is physics-based and not artifi- cially contrived. Noting that the Dirac waves for massive particles are such that any changes in those waves (and their entanglement) can only move at the speed of light while the massive particles guided by them move at subluminal velocities, we have concluded that it will be very difficult to close the locality loophole for such particles29, but that if it is closed, we predict they will satisfy Bell’s inequality. Finally, we have also shown that act of changing the magnetic fields generates an EM wave that can affect the oncoming particles and could, for relativistic particles, kick them out of the detector, thereby creating a detector loophole in exactly the conditions that would otherwise close the locality loophole. ∗ Electronic address: [email protected] 1 M. Clover, The Innsbruck EPR experiment: A time-retarded local description of space-like separated correlations (2003), quant-ph/0304115v2. 2 W. Philipp and K. Hess, A local mathematical model for EPR experiments (2002), quant-ph/0212085. 3 C. H. Thompson and H. Holstein, Found. Phys. Lett. 9, 357 (1996). 4 N. Gisin and H. Zbinden, Bell inequality and the locality loophole: Active versus passive switches (1999). 5 N. Gisin and B. Gisin, A local hidden variable model of quantum correlation exploiting the dectection loophole (1999), quant-ph/9905018. 6 L. E. Szabo and A. Fine, Phys. Lett. A 295, 229 (2002), quant-ph/0007102. 7 J. Larsson, Phys. Lett. A 256, 245 (1999). 8 A. J. Leggett, Found. Phys. 33, 1469 (2003). 9 L. Accardi and A. Khrennikov, Chameleon effect, the range of values hypothesis and reproducing the EPR-Bohm correlations (2006), quant-ph/0611259v1. 10 J. Christian, Disproof of Bell’s Theorem by Clifford Algebra valued Local Variables (2007), quant-ph/0703179. 11 D. Bohm, Phys. Rev. 85, 180 (1952). 12 J. Bell, in Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, 1993), pp. 1–13. 13 T. Durt and Y. Pierseaux, Phys. Rev. A 66, 052109 (2002). 14 P. Holland, Phys. Rev. A 60, 4326 (1999). 15 E. Squires, Lorentz invariant Bohmian mechanics (1995), quant-ph/9508014v2. 16 P. R. Holland, The Quantum Theory of Motion, An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics (Cambridge University Press, 1993). 17 M. Clover, Bell’s Theorem: A new Derivation that preserves Heisenberg and Locality (2004), quant-ph/0409058v2. 18 M. Clover, Bell’s Theorem: A critique (2005), quant-ph/0502016. 19 S. Mackman and E. Squires, Foundations of Physics 25, 391 (1995). 20 D. Rice, Am. J. Phys. 65, 144 (1997). 21 A. Matzkin, Classical statistical distributions can violate Bell’s inequalities (2007), quant-ph/0703251. 22 A. Rizzi, The Meaning of Bell’s Theorem (2003), quant-ph/0310098v1. 23 W. de Baere, A. Mann, and M. Revzen, Found. Phys. 29, 67 (1999). 24 T. Norsen, Bell Locality and the Nonlocal Character of Nature (2006), quant-ph/0601205. 25 M. Gondran and A. Gondran, A complete analysis of the Stern-Gerlach experiment using Pauli spinors (2005), quant- ph/0511276. 26 W. Tittel, J. Brendel, B. Gisin, T. Herzog, H. Zbinden, and N. Gisin, Phys. Rev. A 57, 3229 (1998). 27 W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, Phys. Rev. Lett. 81, 3563 (1998). 28 G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 81, 5039 (1998). 29 M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Windeland, Nature 409, 791 (2001). 30 E. P. Wigner, in Quantum Theory and Measurement, edited by J. A. Wheeler and W. H. Zurek (Princeton University Press, 1983), p. 260. 31 A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). 32 We will confine our discussion to massive particles, for which there is a well-defined de Broglie-Bohm interpretation based on the first-quantized Schrödinger and Dirac wavefunctions. 33 Squires’ work15,19? retarded positions in time independent wavefunctions. 34 This sneaks in when one tries to avoid “contextuality” without making a distinction between distant contexts (Bob’s settings affecting Alice’s results) and nearby ones (Alice not being able to make both settings simultaneously). 35 For example, even if a person could understand two questions that were asked of him simultaneously, there is no way that he could answer them simultaneously. Such psychological “measurements” (i.e.,“trick questions”) can be incompatible, such that mailto:[email protected] different answers result depending on the order the questions are asked, and could require modeling with non-commutative models. 36 Examining Wigner’s derivation of Bell’s inequality30, it is clear that local time-delayed knowledge of one’s partner’s settings is explicitly excluded in his proof as well. 37 Although Durt and Pierseaux assume for simplicity of exposition that no other component of ~B has any gradients, this is not physical – in order for ∇ · ~B = 0, it must be the case that at least dBx = − dBz if there is no gradient in the direction of motion, but more generally, dBx . In this case, as long as any electromagnetic field is traveling, the magnitude of the spatial gradients in any direction will be comparable to any other direction. 38 The fact that a neutrino’s mass is O(1) eV compared to O(1011) eV for Ag108 means that α in Equation (7a) will increase a lot more than the time in the magnet will decrease – (α∆t)ν ∼ 10 5(α∆t)Ag – assuming the magnetic moments are comparable; alternatively, while a silver atom at v/c = 0.99 would require a magnet thousands of kilometers long to generate the same deflections if the mass in the expression for α (Equation (7a)) goes over to γm0 (so that α drops by a factor O(10 −2) and ∆t drops by O(10−6)), it would avoid being kicked out of the beamline before the detector by a changing field. 39 EPR assumed that either QM was incomplete, or that the physical quantities corresponding to non-commuting operators weren’t simultaneously real; they did not distinguish31 (p. 780) between the measurement result’s reality – the positive or negative deflections in a magnet (“...we must consider the quantity P [i.e., σx = ±1] as being an element of reality...”) – and the “hidden” properties of a particle that caused such deflections, even though they referred earlier31 (p. 777) to “...an element of physical reality corresponding to this physical quantity” [my emphases]. A de Broglie-Bohm interpretation of Bohm’s spin example would distinguish between the left or right value of the hidden position coordinate, x, which results in σx = ±1, and the up or down value of the vertical position coordinate, y, that results in σy = ±1. A magnetic field cannot be devised that measures both deflections simultaneously, but both hidden positions exist (simultaneously) even if only one is relevant to a given measurement. Introduction How Non-local is de Broglie-Bohm? How non-local is Hooke's Law? Entanglement and a priori Nonlocality Nonlocality involving violation of Bell's Inequality de Broglie-Bohm Trajectories for Massive Singlets Local vs. Nonlocal Velocity Prescriptions The Locality of de Broglie-Bohm Trajectories for Massive Singlets in Aligned Fields Computer Experiments Physical But Unfair Sampling Effects Photon EPR experiments Conclusion The Meaning of Nonlocality Résumé References

0704.0767

Ground-based Microlensing Surveys

Ground-based Microlensing Surveys Andrew Gould1, B. Scott Gaudi1, and David P. Bennett2 1. Overview Microlensing is a proven extrasolar planet search method that has already yielded the de- tection of four exoplanets. These detections have changed our understanding of planet formation “beyond the snowline” by demonstrating that Neptune-mass planets with sepa- rations of several AU are common. Microlensing is sensitive to planets that are generally inaccessible to other methods, in particular cool planets at or beyond the snowline, very low-mass (i.e. terrestrial) planets, planets orbiting low-mass stars, free-floating planets, and even planets in external galaxies. Such planets can provide critical constraints on models of planet formation, and therefore the next generation of extrasolar planet searches should include an aggressive and well-funded microlensing component. When combined with the results from other complementary surveys, next generation microlensing surveys can yield an accurate and complete census of the frequency and properties of planets, and in particular low-mass terrestrial planets. Such a census provides a critical input for the design of direct imaging experiments. Microlensing planet searches can be carried out from either the ground or space. Here we focus on the former, and leave the discussion of space-based surveys for a separate paper. We review the microlensing method and its properties, and then outline the potential of next generation ground-based microlensing surveys. Detailed models of such surveys have already been carried out, and the first steps in constructing the required network of 1-2m class telescopes with wide FOV instruments are being taken. However, these steps are primarily being taken by other countries, and if the US is to remain competitive, it must commit resources to microlensing surveys in the relatively near future. 2. The Properties of Microlensing Planet Searches If a foreground star (“lens”) becomes closely aligned with a more distant star (“source”), it bends the source light into two images. The resulting magnification is a monotonic function of the projected separation. For Galactic stars, the image sizes and separations are of order µas and mas respectively, so they are generally not resolved. Rather “microlensing events” are recognized from their time-variable magnification (Paczynski 1986), which typically occurs on timescales tE of months, although it ranges from days to years in extreme cases. Presently about 600 microlensing events are discovered each year, almost all toward the Galactic bulge. If one of these images passes close to a planetary companion of the lens star, it further 1Department of Astronomy, The Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA 2Department of Physics, University of Notre Dame, IN 46556, USA http://arxiv.org/abs/0704.0767v1 – 2 – perturbs the image and so changes the magnification. Because the range of gravitational action scales ∝ M , where M is the mass of the lens, the planetary perturbation typically lasts tp ∼ tE mp/M , where mp is the planet mass. That is, tp ∼ 1 day for Jupiters and tp ∼ 1.5 hours for Earths. Hence, planets are discovered by intensive, round-the-clock photometric monitoring of ongoing microlensing events (Mao & Paczynski 1991; Gould & Loeb 1992) 2.1 Sensitivity of Microlensing While, in principle, microlensing can detect planets of any mass and separation, orbiting stars of any mass and distance from the Sun, the characteristics of microlensing favor some regimes of parameter space. • Sensitivity to Low-mass Planets: Compared to other techniques, microlensing is more sensitive to low-mass planets. This is because the amplitude of the perturbation does not de- cline as the planet mass declines, at least until mass goes below that of Mars (Bennett & Rhie 1996). The duration does decline as mp (so higher cadence is required for small planets) and the probability of a perturbation also declines as mp (so more stars must be moni- tored), but if a signal is detected, its magnitude is typically large ( & 10%), and so easily characterized and unambiguous. • Sensitivity to Planets Beyond the Snowline: Because microlensing works by perturb- ing images, it is most sensitive to planets that lie at projected distances where the images are the largest. This so-called “lensing zone” lies within a factor of 1.6 of the Einstein ring, (4GM/c2)Dsx(1− x), where x = Dl/Ds and Dl and Ds are the distances to the lens and source. At the Einstein ring, the equilibrium temperature is TE = T⊕ )1/4( rE )−1/2 → 70K 0.5M⊙ [4x(1− x)]1/4 (1) where we have adopted a simple model for lens luminosity L ∝ M5, and assumed Ds = 8 kpc. Hence, microlensing is primarily sensitive to planets in temperature zones similar to Jupiter/Saturn/Uranus/Neptune. • Sensitivity to Free Floating Planets: Because the microlensing effect arises directly from the planet mass, the existence of a host star is not required for detection. Thus, mi- crolensing maintains significant sensitivity at arbitrarily large separations, and in particular is the only method that is sensitive to old, free-floating planets. See § 4. • Sensitivity to Planets from 1 kpc to M31: Microlensing searches require dense star fields and so are best carried out against the Galactic bulge, which is 8 kpc away. Given that the Einstein radius peaks at x = 1/2, it is most sensitive to planets that are 4 kpc away, but maintains considerable sensitivity provided the lens is at least 1 kpc from both the observer and the source. Hence, microlensing is about equally sensitive to planets in the bulge and disk of the Milky Way. However, specialized searches are also sensitive to closer planets and to planets in other galaxies, particularly M31. See § 5. • Sensitivity to Planets Orbiting a Wide Range of Host Stars: Microlensing is about equally sensitive to planets independent of host luminosity, i.e., planets of stars all along the main sequence, from G to M, as well as white dwarfs and brown dwarfs. By contrast, other – 3 – techniques are generally challenged to detect planets around low-luminosity hosts. • Sensitivity to Multiple Planet Systems: In general, the probability of detecting two planets (even if they are present) is the square of the probability of finding one, which means it is usually very small. However, for high-magnification events, the planet-detection probability is close to unity (Griest & Safizadeh 1998), and so its square is also near unity (Gaudi et al. 1998). In certain rare cases, microlensing can also detect the moon of a planet (Bennett & Rhie 2002). 2.2 Planet and Host Star Characterization Microlensing fits routinely return the planet/star mass ratio q = mp/M and the projected separation in units of the Einstein radius b = r⊥/rE (Gaudi & Gould 1997). Historically, it was believed that, for the majority of microlensing discoveries, it would be difficult to obtain additional information about the planet or the host star beyond measurements of q and b. This is because of the well-known difficulty that the routinely-measured timescale tE is a degenerate combination of M , Dl, and the velocity of the lens. In this regime, individual constraints on these parameters must rely on a Bayesian analysis incorporating priors derived from a Galactic model (e.g., Dong et al. 2006). Experience with the actual detections has demonstrated that the original view was likely shortsighted, and that one can routinely expect improved constraints on the mass of the host and planet. In three of the four microlensing events yielding exoplanet detections, the effect of the angular size of the source was imprinted on the light curve, thus enabling a measurement of the angular size of the Einstein radius θE = rE/Dl. This constrains the statistical estimate of M and Dl (and so mp and r⊥). In hindsight, one can expect this to be a generic outcome. Furthermore, it is now clear that for a substantial fraction of events, the lens light can be detected during and after the event, allowing photometric mass and distance estimates, and so reasonable estimates of mp and r⊥ (Bennett et al. 2007). By waiting sufficiently long (usually 2 to 20 years) one could use space telescopes or adaptive optics to see the lens separating from the source, even if the lens is faint. Such an analysis has already been used the constrain the mass of the host star of the first microlensing planet discovery (Bennett et al. 2006), and similar constraints for several of the remaining discoveries are forthcoming. Finally, in special cases it may also be possible to obtain information about the three-dimensional orbits of the discovered planets. 3. Present-Day Microlensing Searches Microlensing searches today still basically carry out the approach advocated by Gould & Loeb (1992): Two international networks of astronomers intensively follow up ongoing microlens- ing events that are discovered by two other groups that search for events. The one major modification is that, following the suggestion of Griest & Safizadeh (1998), they try to focus on the highest magnification events, which are the most sensitive to planets. Monitoring is done with 1m (and smaller) class telescopes. Indeed, because the most sensitive events are highly magnified, amateurs, with telescopes as small as 0.25m, play a major role. – 4 – Fig. 1.— (Left) Known extrasolar planets detected via transits (blue), RV (red), and microlensing (green), as a function of their mass and equilibrium temperature. (Right) Same as the right panel, but versus semimajor axis. The contours show the number of detections per year from a NextGen microlensing survey. To date, four secure planets have been detected, all with equilibrium temperatures 40K < T < 70K. Two are Jupiter class planets and so are similar to the planets found by RV at these temperatures (Bond et al. 2004; Udalski et al. 2005). However, two are Neptune mass planets, which are an order of magnitude lighter than planets detected by RV at these temperatures (Beaulieu et al. 2006; Gould et al. 2006). See Figure 1. This emphasizes the main advantages that microlensing has over other methods in this parameter range. The main disadvantage is simply that relatively few planets have been detected despite a huge amount of work. 4. NextGen Microlensing Searches Next-generation microlensing experiments will operate on completely different principles from those at present, which survey large sections of the Galactic bulge one–few times per night and then intensively monitor a handful of the events that are identified. Instead, wide- field (∼ 4 deg2) cameras on 2m telescopes on 3–4 continents will monitor large (∼ 10 deg2) areas of the bulge once every 10 minutes around-the-clock. The higher cadence will find 6000 events per year instead of 600. More important: all 6000 events will automatically be monitored for planetary perturbations by the search survey itself, as opposed to roughly 50 events monitored per year as at present. These two changes will yield a roughly 100-fold increase in the number of events probed and so in the number of planetary detections. Two groups (led respectively by Scott Gaudi and Dave Bennett) have carried out detailed – 5 – Fig. 2.— Expectations from a NextGen ground-based microlensing survey. These results represent the average of two independent simulations which include very different input assumptions but differ in their predictions by only ∼ 0.3 dex. (Left) Number of planets detected per year assuming every main-sequence (MS) star has a planet of a given mass and semi-major axis (see §4). (Right) Same as left panel, but assuming every MS has two planets distributed uniformly in log(a) between 0.4-20 AU. The arrows indicate the masses of the four microlensing exoplanet detections. simulations of such a survey, taking account of variable seeing and weather conditions as well as photometry systematics, and including a Galactic model that matches all known constraints. While these two independent simulations differ in detail, they come to similar conclusions. Figure 1 shows the number of planets detected assuming all main-sequence stars have a planet of a given mass and given semi-major axis. While, of course, all stars do not have planets at all these different masses, Gould et al. (2006) have shown that the two “cold Neptunes” detected by microlensing imply that roughly a third of stars have such planets in the “lensing zone”, i.e. the region most sensitive for microlensing searches. Microlensing sensitivity does decline at separations that are larger than the Einstein radius, but then levels to a plateau, which remains constant even into the regime of free-floating planets. In this case, the timescales are similar to those of bound-planet perturbations (1 day for Jupiters, 1.5 hours for Earths) but there is no “primary event”. Again, typical amplitudes are factor of a few, which makes them easily recognizable. If every star ejected f planets of mass mp, the event rate would be Γ = 2 × 10−5f mp/Mj yr −1 per monitored star. Since NextGen experiments will monitor 10s of millions of stars for integrated times of well over a year, this population will easily be detected unless f is very small. Microlensing is the only known way of detecting (old) free-floating planets, which may be a generic outcome of – 6 – planet formation (Goldreich et al. 2004; Juric & Tremaine 2007; Ford & Rasio 2007). 4.1 Transition to Next Generation Although NextGen microlensing experiments will work on completely different principles, the transition is actually taking place step by step. The Japanese/New Zealand group MOA already has a 2 deg2 camera in place on their 1.8m NZ telescope and monitors about 4 deg2 every 10 minutes, while covering a much wider area every hour. The OGLE team has funds from the Polish government to replace their current 0.4 deg2 camera on their 1.3m telescope in Chile with a 1.7 deg2 camera. When finished, they will also densely monitor several square degrees while monitoring a much larger area once per night. Astronomers in Korea and Germany have each made comprehensive proposals to their governments to build a major new telescope/camera in southern Africa, which would enable virtually round-the- clock monitoring of several square degrees. Chinese astronomers are considering a similar initiative. In the meantime, intensive followup of the currently surveyed fields is continuing. 5. Other Microlensing Planet Searches While microlensing searches are most efficiently carried out toward the Galactic bulge, there are two other frontiers that microlensing can broach over the next decade or so. • Extragalactic Planets: Microlensing searches of M31 are not presently sensitive to plan- ets, but could be with relatively minor modifications. M31’s greater distance implies that only more luminous (hence physically larger) sources can give rise to detectable microlensing events. To generate substantial magnification, the planetary Einstein ring must be larger than the source, which generally implies that Jupiters are detectable, but Neptunes (or Earths) are not (Covone et al. 2000; Baltz & Gondolo 2001). Nevertheless, it is astonishing that extragalactic planets are detectable at all. To probe for M31 planets, M31 microlensing events must be detected in real time, and then must trigger intensive followup observa- tions of the type currently carried out toward the Galactic bulge, but with larger telescopes (Chung et al. 2006). This capability is well within reach. • Nearby microlensing events: In his seminal paper on microlensing, Einstein (1936) famously dismissed the possibility that it would ever be observed because the event rate for the bright stars visible in his day was too small. Nevertheless, a Japanese amateur recently discovered such a “domestic microlensing event” (DME) of a bright (V ∼ 11.4), nearby (∼ 1 kpc) star, which was then intensively monitored by other amateurs (organized by Columbia professor Joe Patterson). While intensive observations began too late to detect planets, Gaudi et al. (2007) showed that more timely observations would have been sensitive to an Earth-mass planet orbiting the lens. In contrast to more distant lenses, DME lenses would usually be subject to followup observations, including RV. This would open a new domain in microlensing planet searches. Virtually all such DMEs could be found with two “fly’s eye” telescopes, one in each hemisphere, which would combine 120 10 cm cameras on a single mount to simultaneously monitor the π steradians above airmass 2 to V = 15. A fly’s eye telescope would have many other applications including an all-sky search for transiting – 7 – planets and a 3-day warning system for Tunguska-type impactors. Each would cost ∼$4M. 6. Conclusion and Outlook In our own solar system, the equilibrium-temperature range probed by microlensing (out past the “snow line”) is inhabited by four planets, two gas giants and two ice giants. All have similar-sized ice-rock cores and differ primarily in the amount of gas they have accreted. Systematic study of this region around other stars would test predictive models of planet formation (e.g. Ida & Lin 2004) by determining whether smaller cores (incapable of accreting gas) also form. Such a survey would give clues as to why cores that reach critical gas-grabbing size do or do not actually manage to accrete gas, and if so, how much. In the inner parts of this region, RV probes the gas giants but not the ice giants nor, of course, terrestrial planets. RV cannot make reliable measurements in the outer part of this region at all because the periods are too long. Future astrometry missions (such as SIM) could probe the inner regions down to terrestrial masses, but are also limited by their limited lifetime in the outer regions. Hence, microlensing is uniquely suited to a comprehensive study of this region. Although microlensing searches have so far detected only a handful of planets, these have already changed our understanding of planet formation “beyond the snowline”. Next gen- eration microlensing surveys, which would be sensitive to dozens of “cold Earths” in this region, are well advanced in design conception and are starting initial practical implemen- tation. These surveys play an additional crucial role as proving grounds for a space-based microlensing survey, the results of which are likely to completely revolutionize our under- standing of planets over a very broad range of masses, separations, and host star masses (see the Bennett et al. ExoPTF white paper). Traditionally, US astronomers have played a major role in microlensing planet searches. For example, Bohdan Paczyński at Princeton essentially founded the entire field (Paczynski 1986) and co-started OGLE. Half a dozen US theorists have all contributed key ideas and led the analysis of planetary events. The Ohio State and Notre Dame groups have played key roles in inaugurating and sustaining the follow-up teams that made 3 of the 4 microlensing planet detections possible. 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0704.0768

The luminous infrared composite Seyfert 2 galaxy NGC 7679 through the [O III] 5007 emission line

Astronomy & Astrophysics manuscript no. ngc7679-paper c© ESO 2018 November 1, 2018 The luminous infrared composite Seyfert 2 galaxy NGC 7679 through the [O III] λ 5007 emission line ⋆ I. M. Yankulova1, V. K. Golev1, and K. Jockers2 1 Department of Astronomy, St Kliment Okhridski University of Sofia, 5 James Bourchier Street, BG–1164 Sofia, Bulgaria e-mail: [email protected], [email protected] 2 Max-Planck-Institut für Sonnensystemforschung, Max-Planck-Straße 2, D–37191 Katlenburg-Lindau e-mail: [email protected] Accepted March 29, 2007 ABSTRACT Context. NGC 7679 (Mrk 534) is a nearby (z = 0.0177) nearly face-on SB0 luminous infrared Sy2 galaxy in which starburst and AGN activities co-exist. The ionization structure is maintained by both the AGN power-law continuum and starburst. The galaxy is a bright X-ray source possessing a low X-ray column density NH < 4 × 1020 cm−2. Aims. The Compton-thin nature of such unabsorbed objects infers that the simple formulation of the Unified model for SyGs is not applicable in their case. The absorption is likely to originate at larger scales instead of the pc-scale molecular torus. The main goal of this article is to investigate both gas distribution and ionization structure in the circumnuclear region of NGC 7679 in search for the presence of a hidden Sy1-type nucleus, using the ø3 λ5007 luminosity as a tracer of AGN activity. Methods. NGC 7679 was observed with the 2m RCC reflector of the Ukraine National Astronomical Observatory at peak Terskol, Caucasus, Russia. The observations were carried out on October 1996 with the Focal Reducer of the Max-Planck-Institut für Sonnensystemforschung, Germany. All observations were taken with tunable Fabry-Perot narrow-band imaging with spectral FWHM of the Airy profile δλ between 3 and 4 Å depending of the used wavelength. Results. The ø3 λ5007 emission-line image of the circumnuclear region of NGC 7679 shows elliptical isophotes extended along the PA≈ 80◦ in the direction to the counterpart galaxy NGC 7682. There is a maximum of this emission which is shifted ∼ 4 arcsec from the center defined by the continuum emission. The maximum of ionization by the AGN power-law continuum traced by ø3 λ5007/Hα ratio is displaced by ∼ 13 arcsec eastward from the nucleus. The direction where high ionization is observed at PA≈ 80◦ ± 10◦ coincides with the direction to the companion galaxy NGC 7682 (PA≈ 72◦). On the contrary, at PA∼ 0◦ the ionization in the circumnuclear region is entirely due to hot stars. Conclusions. Both the ratio (Nph/Nion)hν > 55 eV ≈ 0.2 − 20 of the number Nph of photons traced by ø3 to the number Nion of high-energy ionizing photons and the presence of weak and elusive Hα broad wings indicate a hidden AGN. We conclude that the dust and gas in the high ionization direction PA≈ 80◦ has a direct view to the central AGN engine. This possibly results in dust/star-formation decay. A large fraction of the unabsorbed Compton-thin Sy2s with ø3 luminosity >∼ 1041 erg s−1 possesses a hidden AGN source. Key words. galaxies: individual: NGC 7679 (Mrk 534) – galaxies: ISM – galaxies: starburst – galaxies: Seyfert 1. Introduction Luminous infrared galaxies (LIGs) are characterized by ex- treme IR luminosities LIR >∼ 1011L⊙ at mid- to far-infrared (FIR) wavelengths. In their comprehensive spectroscopic sur- vey of LIGs Kim et al. (1995) and Veilleux et al. (1995) have shown a clear tendency for the more luminous objects to be more Seyfert-like. The starburst and AGN are tightly connected phenomena and the interaction between them is a matter of de- bate. Send offprint requests to: Ivanka Yankulova, e-mail: [email protected] ⋆ Based on observations obtained at the Peak Terskol Observatory, Caucasus, Russia. Based on a large spectroscopic optical survey of bright IRAS and X-ray sources from ROSAT All Sky Survey, Moran et al. (1996) extracted low-redshift galaxies with optical spec- tra characterized by the HII regions and X-ray luminosities typical of AGNs and these objects were named Composite Seyfert/Starburst galaxies. Other similar galaxies (i.e. with bright X-ray emission together with the clear predominance of a starburst in the optical and IR regime) have been found also in the deep ROSAT fields (Boyle et al. 1995, Griffiths et al. 1996) and in the Chandra and XMM-Newton deep fields (Rosati et al. 2001). A significant part of the observed FIR-emission of these composites could be associated with circumnuclear starburst events. The nuclear X-ray source there is generally absorbed http://arxiv.org/abs/0704.0768v1 2 Yankulova I., Golev V., and Jockers, K.: The luminous infrared composite Seyfert 2 galaxy NGC 7679 through ... with column density of NH > 10 22 cm−2 and these values range from 1022 cm−2 to higher than 1024 cm−2 for about 96 % of this class of objects (Risaliti et al. 1999, Bassani et al. 1999). The circumnuclear starburst should also play a major role in the ob- scuration processes – see for details Levenson et al. (2001) and references therein. However, there are Sy2 galaxies with col- umn densities lower than 1022 cm−2. Panessa & Bassani (2002, hereafter PB02) present a sample of 17 type 2 SyGs showing such low absorption in X-rays. The Compton thin nature of these sources is strongly suggested by some isotropic indica- tors such as FIR and ø3 emission. The fraction of Composite Seyfert/Starburst objects is esti- mated to be in the range of 10% - 30% of the Sy2 population. The simple formulation of the Unified model for SyGs is not applicable in such sources. The observed absorption is likely to originate at larger scales instead in the pc-scale molecular torus. Probably the Broad Line Regions (BLRs) of these ob- jects are covered by some obscuring dusty material. NGC 7679 is a nearby (z = 0.0177) nearly face-on SB0 Seyfert 2 type galaxy in which starburst and AGN activities co- exist. The IRAS fluxes show that the luminosity of NGC 7679 in the far infrared is about LFIR ≈ 1011L⊙. This object is in- cluded in the large spectroscopic survey of 200 luminous IRAS galaxies (Kim et al. 1995, Veilleux et al. 1995). NGC 7679 is physically associated by a common stream of ionized gas with the Sy2 galaxy NGC 7682 at ∼ 4.5 arcmin eastward (PA ≈ 72◦) forming the pair Arp 216 (VV 329). The tidal interac- tions between both galaxies together with the existence of a bar in NGC 7679 could enhance the gas flow towards the nu- clear regions and possibly trigger the starburst processes (Gu et al. 2001). The X-ray properties of the NGC 7679 based on the BeppoSAX observations and on the ASCA archive were dis- cussed by Della Ceca et al. (2001, hereafter DC01). Their conclusion is that NGC 7679 is a Seyfert-starburst composite galaxy which implies the clear predominance of an AGN in the X-ray regime connected with a starburst in the optical and IR regime. DC01 found that a simple power-law spectral model with Γ ∼ 1.75 and small intrinsic absorption (NH < 4 × 1020 cm−2) provides a good description of the spectral properties of NGC 7679 from 0.1 to 50 keV. The small X-ray absorption and the absence of strong (EW ∼ 1 keV) Fe-lines suggest a Compton thin type 2 AGN in NGC 7679 which clearly distin- guishes this galaxy from the other LIG Seyferts. The main goal of this article is to investigate both gas distri- bution and ionization structure in the circumnuclear regions of the luminous IR unabsorbed Seyfert galaxy NGC 7679 and to look for tracers of the presence of a hidden Sy1-type nucleus. Some information on the observations and data reduction procedures is presented in Section 2. The results are presented in Section 3 and discussed in Section 4. The combination of the data taken from recent literature and our Fabry-Perot ob- servations provides new insight in the circumnuclear region of NGC 7679 and in the phenomena occurring there. Table 1. Observiation details image interference Fabry-Perot frames frame filtera) tuned × wavelength exposure λc/FWHM λFP time (Å)/(Å) (Å) (s) Hα 6662/55 6674.8 1 × 1800 2 × 900 [N II]λ6548 6662/55 6659.9 1 × 900 continuum 6719/33 6720.0 1 × 1800 1 × 900 ø3 λ5007 5094/44 5092.4 2 × 900 continuum 5002/41 4437.7 1 × 1200 Gunn rb) 6800/1110 1 × 60 BG 39/2b) 4720/700 2 × 1500 a) Used to separate Fabry-Perot working orders b) Broad-band image taken without Fabry-Perot to reveal the mor- phology 2. Observations and data reduction NGC 7679 was observed by K. Jockers, T. Bonev, and T. Credner with the 2m RCC reflector of the Peak Terskol Observatory, Caucasus, Russia. The observations were carried out in October 1996 with the Two-channel Focal Reducer of the former Max-Planck-Institut für Aeronomie, Germany (now Max-Planck-Institut für Sonnensystemforschung, MPS). This instrument was primarily intended for cometary studies but it has repeatedly been used for observations of active galactic nu- clei (see for example Golev et al. 1995, 1996, and Yankulova 1999). The technical data and the present capabilities of the MPS Two-channel Focal Reducer are described in Jockers (1997) and Jockers et al. (2000). All observations were taken in Fabry-Perot (FP) mode us- ing tunable FP narrow-band imaging with spectral FWHM of the Airy profile δλ in order of 3 - 4 Å. The details of obser- vations are presented in Table 1 where the central wavelengths λc and the effective width ∆λ of the interference filters used to separate the Fabry-Perot interference orders, the wavelength λFP at which the Fabry-Perot was tuned, and the exposures are listed. The overall “finesse” of the system ∆λ/δλ is ≈ 15, ∆λ is the free spectral range of the FP. As one can see from Table 1 ∆λ is comparable to the filter’s band width and therefore all FP orders except the central one are efficiently suppressed. Two exposures of NGC 7679 were obtained through each filter to eliminate cosmic ray events and to increase the signal-to-noise ratio. Flatfield exposures were obtained using dusk and dawn Yankulova I., Golev V., and Jockers, K.: The luminous infrared composite Seyfert 2 galaxy NGC 7679 through ... 3 Fig. 1. Contours of continuum-subtracted narrow-band ø3 λ5007-image superimposed on the gray-scale ø3 λ5007-emission distribution of the circumnuclear region of NGC 7679. The background noise level is σ = 2.01 × 10−17 ergs cm−2 s−1 arcsec−2. The outermost contour is taken at 5σ above the sky level and the next contours increase by a factor 2. Note East-West elongation and two extrema decentered of about ∼ 4 arcsec from the position of the nucleus marked by cross. North is up, East is to the left. twilight for uniform illumination of the detector. No dark cor- rection was required. The images were reduced following the usual reduction steps for narrow-band imaging. After flatfielding the frames were aligned by rebinning to a common origin. The final align- ment of all the images was estimated to be better than 0.1 px (the scale is 1 px = 0.8 arcsec). A convolution procedure was performed in order to match the Point-Spread Functions (PSFs) of each line-continuum pair which unavoidably degrades the fi- nal FWHM of the images to the mean value ≈ 3 − 3.3 arcsec (shown as ’seeing’ in Fig. 1). At the distance of NGC 7679 one arcsec corresponds to a distance of about 340 pc assuming H0 = 75 km sec −1 Mpc−1. 3. Results 3.1. Narrow-band emission-line images Gray-scale images of the narrow-band flux distribution of the extended circumnuclear region of NGC 7679 in the ø3 λ5007, Hα, and [N II]λ6548 emission lines with superimposed con- tours are presented in Fig. 1, 2, and 3, respectively. The ø3 λ5007 emission shown in Fig. 1 reveals a bright, about 20 arcsec in size, extended emission-line region (EELR) which is elongated approximately in East direction (PA ≈ 80◦± 10◦). This region is similar to the analogous EELRs observed in many Sy2 type galaxies. Most probably it is powered by the Fig. 2. Contours of continuum-subtracted narrow-band Hα- image superimposed on the gray-scale Hα-emission distribu- tion of the circumnuclear region of NGC 7679. The back- ground noise level is σ = 2.77 × 10−18 ergs cm−2 s−1 arcsec−2. The outermost contour is taken at 5σ above the sky level and the next contours increase by a factor of 2. North is up, East is to the left. Fig. 3. Contours of continuum-subtracted narrow-band [N II]λ6548-image superimposed on the gray-scale [N II]λ6548-emission distribution of the circumnuclear region of NGC 7679. The background noise level is σ = 4.75 × 10−18 ergs cm−2 s−1 arcsec−2. The outermost contour is taken at 5σ above the sky level and the next contours increase by a factor 2. North is up, East is to the left. 4 Yankulova I., Golev V., and Jockers, K.: The luminous infrared composite Seyfert 2 galaxy NGC 7679 through ... AGN-type activity of the nucleus. The emission-line peak of ø3 λ5007 is shifted at about ∼ 4 arcsec to the East with respect to the center defined by the continuum emission and marked by cross in Fig. 1. At larger distances (∼ 37 arcsec) the ionized gas forms an envelope which is extended along the direction PA ≈ 72◦ to the NGC 7682, the counterpart of NGC 7679, as it was already noted by Durret & Warin (1990). In Fig. 2 we present our very deep and high-contrast Hα continuum-subtracted image with numerous starburst regions where because of both seeing and pixel size we are able to see only elliptical central isophotes instead of the “double nu- cleus” observed recently by Buson et al (2006). Our analysis of the unpublished Hα images taken from the archive of the Isaak Newton Group of telescopes at La Palma as well as the archive images of Buson et al. (2006) from the ESO La Silla NTT also revealed a “double nucleus” otherwise unseen in the known broad-band images. The separation between the nuclear counterparts (in fact one is the active nucleus itself and the other one is a bright spiral-like extremely powerful starburst region) is ≈ 3 arcsec. The existence of this “double nucleus” in NGC 7679 could enhance the gas flows towards the nuclear regions and possibly trigger the starburst process itself. The “double nucleus” can be also seen at very different wavelength range on 6 cm and 20 cm high-resolution VLA radio continuum map of NGC 7679 published by Stine (1992). The angular dis- tance and PA between two counterparts is quite the same. The radio spectral index is −0.37 and steepens away from the cen- ter which indicates that nonthermal emission leaks out of the starburst region. The low-excitation gas traced by the emission in Hα re- veals different morphology as compared to that of the ø3 λ5007 emission. Inside of the region with radius of 6 – 8 arcsec from the center the contours of the Hα emission are nearly circular. Outside this region to the West of the main body of NGC 7679 a clearly outlined wide arc is observed at 16 arcsec (∼ 5 kpc) from the center. To the East this arc converts into a gaseous envelope which forms a part of a circumnuclear starforming ring mentioned by Pogge (1989). This arc is not detected on the narrow-band continuum image next to the Hα. The same morphology in Hα + [N II] with higher spatial resolution was observed by Buson et al. (2006). The Fabry-Perot technique used by us makes possible to disentangle [N II]λ6548 from Hα. The pure [N II]λ6548 emis- sion (Fig. 3) shows extended structure ∼ 20 arcsec in diameter. The starforming ring revealed by the Hα image is not seen here. As a rule the gas component in the starforming ring is ionized by stellar UV-emission and the [N II]λ6548 is weaker than that one where the gas is ionized by power-low AGN continuum. On the other hand, this could be an effect due to the shorter exposure time of our [N II]λ6548 frame. 3.2. Narrow-band emission-line total fluxes The total emission-line fluxes of Hα, ø3 λ5007and [N II]λ6583 were estimated from our flux calibrated images in an aperture of 2 kpc (r <∼ 3 arcsec) like the one used by the authors cited in Table 2. In this Table we have collected available measure- ments of the emission lines observed by us up to now. Our measured fluxes are in good agreement with those of Kim et al. (1995) and differ from the measurements of Contini et al. (1998). Flux values given by Contini et al. (1998) are twice larger than ours and those given by Kim et al. (1995). Recently Gu et al. (2006) measured the central flux in ø3 λ5007. We found a reasonable coincidence between their value (1.55 × 10−14 ergs cm−2 s−1) and ours (1.94 × 10−14 ergs cm−2 s−1) in the much smaller aperture used by them. We estimated the flux of the continuum near ø3 λ5007 within the central 2 kpc to be F(λcont) = 6.74×10−15 ergs cm−2 s−1Å. Then the equivalent width of the emission line ø3 λ5007 is EW(λ 5007) = 7.6 Å. Baskin and Loar (2005) have used the photoionization code CLOUDY to calculate the dependence of EW(λ 5007) on the electron density ne, the ionization parame- ter U, and the covering factor CF. Following their Fig. 5 and our estimation of EW(λ 5007) we derive for the covering factor CF the range 0.016 ≤ CF ≤ 0.04 with the most probable value CF ≈ 0.024. There is a large quantity of absorbing matter in the central region of NGC 7679 (Telesco et al. 1995) which modifies the Balmer emission lines. The Balmer decrement reported by Kim et al. (1995) in the central 2 kpc is F(Hα)/F(Hβ)≈ 17.4, but fol- lowing Contini et al. (1998) this decrement is 8.5. Kewley et al. (2000) give E(B − V)= 0.47 which results to F(Hα)/F(Hβ)= 5.04. In Table 2 the value of the parameter C is evaluated from the measured Balmer decrement and from the assumption that in AGNs F(Hα)/F(Hβ)= 3.1 and the optical depth τλ = C f (λ) where f (λ) is the reddening curve (Osterbrock 1989). The ex- tinction E(B − V) derived from the Balmer decrement is also given in Table 2. Contini et al. (1998) present measurements of emission- lines fluxes made in the extranuclear region 9 arcsec off the nu- cleus at PA = 207◦ in an aperture of 3 arcsec. We estimated the emission-line fluxes from our images in the same aperture at the same place in order to compare with those given by Contini et al. (1998). The results are given in Table 2. The Contini’s values are about 2 times larger than ours in the extranuclear region as well as at the nucleus. Moustakas & Kennicutt (2006) report total emission-line fluxes of Hα and ø3 λ5007 in a wide rectangular aperture 30 × 80 arcsec oriented at PA = 90◦. Their Hα-flux F(Hα) = (1.535± 0.062)× 10−12 ergs cm−2 s−1 coincides with our value (1.52 × 10−12 ergs cm−2 s−1) in the same wide aperture after a correction for extinction with E(B − V) = 0.065 used by them. In ø3 λ5007 the coincidence is reasonably good (4.72 × 10−13 compared with ours 3.90 × 10−13 ergs cm−2 s−1). 3.3. The ionization map F(ø3 λ5007) / F(Hα) Our flux-calibrated emission-line images are used to form the F(ø3 λ5007)/F(Hα) ionization map in order to analyse the mean level of ionization. This map is shown in the left panel of Fig. 4. All pixels below 4σ of the background noise level were suppressed before the division of the corresponding images. The ionization map infers a presence of a maximum shifted Yankulova I., Golev V., and Jockers, K.: The luminous infrared composite Seyfert 2 galaxy NGC 7679 through ... 5 Table 2. Measured emission lines fluxes in 2 kpc central aperture in NGC 7679 Emission Measured flux F(λ), ergs cm−2 s−1 2 kpc central aperture 9 arcsec off the nucleus 1 2 3 4 5 6 7 Hα 1.92 × 10−13 1.9 × 10−13 4.5 × 10−13 – 3.8 × 10−13 3.73 × 10−14 1.04 × 10−14 [N II]λ6548 9.96 × 10−14 1.08 × 10−13 1.86 × 10−13 – – 9.8 × 10−15 4.5 × 10−15 ø3 λ5007 5.2 × 10−14 5.3 × 10−14 8.8 × 10−14 – – 9 × 10−15 4.6 × 10−15 Hβ – 1.1 × 10−14 5.24 × 10−14 1.0 × 10−14 – 5.9 × 10−15 – F(Hα)/F(Hβ) – 17.4 8.5 5.0 4.58 6.3 – F(Hγ)/F(Hβ) – 0.24 0.32 0.4 0.3 – – C – 4.93 2.88 1.6 1.12 2.02 – E(B − V) – 1.45 0.85 0.47 0.33 0.65 – Columns: 1 - this work; 2 - Kim et al. (1995); 3 - Contini et al. (1998); 4 - Kewley et al. (2000); 5 - Buson et al. (2006); 6 - Contini et al. (1998); 7 - this work, PA = 207◦. Fig. 4. F(ø3 λ5007) / F(Hα) ionization map of NGC 7679. All pixels below 4σ of the background noise level were suppressed before image division (left). The ratio F(ø3 λ5007)/F(Hα) vs the axial distance from the nucleus along PA ≈ 80◦ (right). The positions labeled 1 to 5 are equidistant with step size of 3 arcsec. We refer to them later in the text (see Fig. 6). to the East at PA ≈ 80◦ with respect to the photometric cen- ter defined by the integral light of the continuum images and marked by cross on the figure. A slice of this map along the PA ≈ 80◦ versus the axial dis- tance from the nucleus is presented in the right panel of Fig. 4. Below we will discuss in more detail the behaviour of the ion- ization at positions 1 to 5. 4. Discussion 4.1. The ionizing flux from the central engine In order to estimate the number of ionizing photons emitted from the central engine, we made use of the recent X-ray obser- vations of NGC 7679. This object was observed by ASCA and BeppoSAX in 1998, and by XMM-Newton in 2005. A detailed analysis of ASCA and BeppoSAX data sets is present in DC01. 6 Yankulova I., Golev V., and Jockers, K.: The luminous infrared composite Seyfert 2 galaxy NGC 7679 through ... They show that a single absorbed power-law function (with a photon index 1.75) fits the observed spectrum very well and the X-ray absorption is relatively small (NH ≤ 4 × 1020 cm−2). The data for the X-ray observations in 2005 were taken from the XMM-Newton public archive. The corresponding X- ray spectra for the PN and the two MOS detectors were ex- tracted following the standard procedures using the XMM- Newton Science Analysis System software (SAS version 7.0.0). A single absorbed power-law function gave a good fit (χ2/do f = 201/191) to all the three spectra which were fitted simultaneously. The small X-ray absorption in the nucleus of NGC 7679 was confirmed, NH = 5.6[4.0 ÷ 7.5] × 1020 cm−2, and no change in the shape of the spectrum was found, a pho- ton index of 1.81[1.70÷1.92] (the 90%-confidence intervals are given in brackets). The absorbing X-ray column density along the line of sight is about an order of magnitude smaller than that one estimated from the observed Balmer decrement which is NH ∼ 8 × 1021 cm−2 and ∼ 5 × 1021 cm−2 following Kim et al. (1995) and Contini et al. (1998), respectively. Interestingly, the observed X-ray flux has decreased by a factor ∼ 10 over a time period of ∼ 7 years: FX = 3.8 × 10−13 and 5.8 × 10−13 ergs cm−2 s−1 correspondingly in the 0.1-2.0 keV and 2.0-10.0 keV energy intervals. Since, on the one hand, there is only about 5% scatter of the fluxes for all the three detectors (one PN and two MOS) around the average values given above, and, on the other hand, NGC 7679 shows an appreciable X-ray variability (DC01), it is then likely that the detected decrease of the X-ray flux is real and not an instru- mental effect. The extrapolation of the DC01’s power law to the UV spec- tral domain (that is to hν0 = 13.6 eV) yields F ν = Fν0(ν0/ν) where α = 0.75 and Fν0 = 2.0 × 10 −28 erg cm−2 s−1 Hz−1. The same extrapolation for the XMM-Newton spectrum results in Fntν = Fν0(ν0/ν) α where α = 0.81 and Fν0 = 2.8 × 10 −29 erg cm−2 s−1 Hz−1. The number of ionizing photons with hν > 55 eV provided by the central AGN source is defined as Nion = 55 eV F ntν dν = 4πR2G hν=55 eV where RG is the distance to the NGC 7679. For the BeppoSAX data this estimation is Nion ∼ 1052 ph s−1 and for the XMM- Newton data Nion ∼ 1051 ph s−1. These values are averaged be- tween all BeppoSAX and XMM-Newton bands, respectively. The number of ionizing photons decrease from the BeppoSAX time to the XMM-Newton time in the range of 1051 <∼ Nion <∼ 1052 ph s−1. 4.2. Physical conditions in the circumnuclear region of NGC 7679 The extended emission-line region in NGC 7679 has a rather different morphology when observed in Hα (low ionization emission line) as compared to ø3 λ5007 (high ionization emis- sion line). The Hα image (Fig. 2) contains a compact circum- nuclear region (∼ 20 arcsec in diameter) whose isophotes do not infer any preferred direction. In contrast, the ø3 λ5007 im- age (Fig. 1) of the circumnuclear region of NGC 7679 shows elliptical isophotes extended along the PA ≈ 80◦ ± 10◦. Such difference in morphology of the emission-line images signals the presence of at least two distinct ionization components (see for example Pogge 1989). The extended morphology both of the ø3 λ5007 image (Fig. 1) and of the ø3 λ5007/Hα flux ratio image (Fig. 4) sug- gests an anisotropy of the radiation field. In order to check whether the ionizing field is collimated or not we have to com- pare the number of ionizing photons Nph, absorbed by the ex- tended emission line gas with the number of ionizing photons Nion, emitted by the central AGN engine. Usually, the hydro- gen line flux F(Hα ) or F(Hβ ) is used to find Nph. But the NGC 7679 high resolution Hα image reveals a central circum- nuclear star-forming spiral ring capable of producing about ∼ 75% of the optical line emission within a radius of ∼ 1 kpc (Buson et al. 2006). For this reason it is not quite correct to use the F(Hα) in order to make the Nph estimate. Kauffmann et al. (2003) focus on the luminosity of the ø3 λ5007 as a tracer of AGN activity. We can estimate the number Nph of ionizing photons with energy above hν = 55 eV from the observed ø3 λ5007 luminosity after correction for extinc- tion. A dust correction to ø3 based on the ratio F(Hα) / F(Hβ) should be regarded as best approximation (Kauffmann et al. 2003). According to Draine & Lee (1984) (Fig.7 therein) the optical depth is τ5007 = 0.96 C = 2.76. Here we adopt the value of C= 2.88 following Contini et al. (1998) as a more com- promising reddening value among the different Balmer decre- ment assessments. Then the luminosity, corrected for extinc- tion, Lcorr([O+2]λ5007) = 4.4 × 1041 ergs s−1. We note that PB02 give 5.7×1041ergs s−1 for the ø3 λ5007 luminosity. The total number of ionizing photons that must be available to produce the observed ø3 λ5007 emission is given by the expression Nph = +2, Te)L corr([O+2]λ5007) CF−1 αeff5007(ne, Te) hν5007 ≈ 2 × 1052 ph s−1 (2) where αG(O +2, Te) = 5.1 × 10−12 cm3 s−1 (Aldrovandi & Pequignot 1973) is the recombination coefficient at Te ≈ 104 K and αeff5007(ne, Te) = 1.1× 10 −9 cm3 s−1 is the effective recombi- nation coefficient at ne = 10 5 cm−3 and Te = 10 4 K. This coeffi- cient strongly depends on the electron density and temperature. If we accept Te = 10 4 K then αeff5007(ne) = 5.14 × 10 −3A21/ne cm3 s−1 where A21 = 0.021 s −1. As the critical electron density is ncre (5007) = 5×10 5 cm−3 we assume that the electron density is not lower than ne ≈ 104 cm−3 in order to emit the ø3 λ5007. Then the lower limit for Nph is ≈ 2×1051 ph s−1. For NGC 7679 the covering factor CF = 0.024. The photon ratio Nph/Nion is a probe of the collimation hy- pothesis. In the anisotropic case this ratio is considerably larger than 1. Under the above assumptions about ne and Te we esti- mate for NGC 7679 0.2 <∼ (Nph/Nion)hν>55 eV <∼ 20 but the lower limit could increase if the luminosity L([O+2]λ5007) is inte- grated over the whole image. The increase of the upper limit of Yankulova I., Golev V., and Jockers, K.: The luminous infrared composite Seyfert 2 galaxy NGC 7679 through ... 7 Fig. 5. Spectral energy distribution (SED) from the radio to the X-ray band of the composite Starburst/Sy2 galaxy NGC 7679 (open diamonds). The radio values at 6 cm and 20 cm are from VLA (Stine, 1992). The X-ray band data are from ASCA and BeppoSAX (DC02 and Risaliti, 2002). Filled diamonds repre- sent recent X-ray observations taken from the XMM-Newton archive. All other data are taken from NED. The SED has been compared with a normal spiral galaxy template (dotted line) taken from Elvis et al.(1994), with Starburst and Sy2 galaxy templates (dashed line and thin solid line) taken from Schmitt et al. (1997), and with Sy1 galaxy template (thick solid line) taken from Mas-Hesse et al. (1995). this ratio is due to the XMM-Newton data which are ∼8 times lower than ASCA/BappoSAX ones. Both the ratio (Nph/Nion)hν>55 eV and the presence of weak and elusive broad Hα-wings (Kewley et al. 2000) indicate a hidden AGN in the NGC 7679. Contrary, the NGC 7679 X- ray spectrum is not highly absorbed and NH < 4 × 1020 cm−2 (see discussion in section 4.1). As a matter of fact Bian & Gu (2006) recently found a very high detectability of hidden BLRs (∼ 85%) for Compton-thin Sy2s with higher ø3 luminosity of L([O+2]λ5007) > 1041 erg s−1. We have to note that NGC 7679 resembles in many respects the galaxy IRAS 12393+3520. In this galaxy direct X-ray evi- dence suggests the presence of a hidden AGN (Guainazzi et al., 2000). This homology can be seen in Fig. 5 where the spectral energy distribution (SED) from the radio to the X-ray band of NGC 7679 is shown. The composite nature of NGC 7679 is clearly seen. Whereas the starburst component dominates in the FIR-IR range, the X-ray band emission is well below that of a typi- cal Sy1. The extrapolation of the power-low X-ray spectrum to 13.6 eV shows a much lower value than the typical Sy2 emis- sion at this wavelength. This again favors the idea about a hid- den central engine. Guainazzi et al. (2000) suppose that a dusty ionized absorber is able to obscure selectively the optical emis- sion, leaving the X-rays almost unabsorbed. Fig. 6. The ø3 λ5007/Hβ vs. [N II]λ6583/Hα diagnostic dia- gram of Veilleux & Osterbrock (1987). The dashed and dotted theoretical lines demarcate between Starbursts and AGNs ac- cording to Kauffmann et al. (2003) and Kewley et al. (2001), respectively. The line dividing between LINERs and SyGs is taken according to PA02. The label “Comp” indicates the re- gion of the diagram in which composite objects are expected to be found. The diagnostic value measured by us is denoted by thick triangle. See text for other designations. 4.3. Ionization structure in the circumnuclear region of the NGC 7679 The ionization map (the right panel of Fig. 4) displays the clear signature of highly-excited gas. The ø3 λ5007/Hα-ratio increases in the direction of the counterpart galaxy NGC 7682 reaching a maximum of ≈ 2.5 at about 12 arcsec off the nu- cleus. More than 15 years ago Durret & Warin (1990) also re- ported about the presence of high-ionization gas in this direc- tion (see their Fig.3a) but their result seemingly did not attract attention. On the other hand at PA ≈ 0◦ our map shows values around ø3 λ5007/Hα ≈ 0.3 and the ionization in this direction is en- tirely due to the young hot stars. The ø3 λ5007/Hβ vs. [N II]λ6583/Hα diagnostic diagram (Veilleux & Osterbrock, 1987) helps to delineate the different ionization mechanisms maintaining the ionization of gaseous component in AGNs and in Starbursts. In Fig. 6 such a dia- gram is shown for NGC 7679. Kewley et al. (2001) distinguish between Starbursts and AGN using a theoretical upper limit de- rived from star forming models. This limit is shown as a dotted line in Fig. 6. Objects with emission-line ratios above this limit cannot be explained by any possible combination of parame- ters in a star forming model. Kauffmann et al. (2003) published an updated estimate for the starburst boundary derived from the SDSS observations. In Fig. 6 this boundary is shown as a dashed line. The location of the Composites is expected to lie between these two lines (see e.g. Panessa et al., 2005). 8 Yankulova I., Golev V., and Jockers, K.: The luminous infrared composite Seyfert 2 galaxy NGC 7679 through ... In Fig. 6 we plot the emission-line flux ratios of NGC 7679 measured in an aperture of 3 arcsec in steps of 3 arcsec both along the PA≈ 80◦ (with crosses) and PA= 0◦ (with diamonds). The labels 1 - 5 for PA≈ 80◦ correspond to the labels in the right panel of Fig 4. Using spectra taken from the Smithsonian Astrophysical Observatory data Center Z-Machine Archive obtained with 3 arcsec slit width, we estimate the observed F(Hα)/F(Hβ) ∼ 5 in NLR. On Fig. 6 positions 1 and 2 at PA = 80◦ off the nucleus lie well within the region occupied by the Sy 2 galaxies. The position 5, which is at the same distance from the nucleus but in opposite direction, is located nearly on the dividing line. All points which refer to the PA = 0◦ are situated between Kauffmann’s and Kewley’s demarcation lines in the region of Composites. In Fig. 6 we also plot with asterisks the nuclear diagnostic ratios according to the data of authors presented in Table 2. The thick triangle refers to the nucleus according to our measure- ments under the assumption of F(Hα)/F(Hβ) = 8.5 (Contini et al., 1998). The large scattering of nuclear values is probably due to the variations of the strength of Hβ absorption line of the star-forming stellar population. 4.4. Unabsorbed SyGs with and without hidden BLRs The unabsorbed Sy2 galaxies with low absorption in X-rays (NH < 10 22 cm−2) possess a hidden or nonhidden central en- gine and BLRs. We have used the ø3 λ5007 emission to test the presence of hidden or nonhidden AGN sources in unabsorbed Sy2 galaxies in the sample of PB02 (14 objects) and Panessa et al., 2005 (6 objects selected by Moran et al., 1996) in the same way as it was done for NGC 7679 (Subsections 4.1 and 4.2). We derive the ratio (Nph/Nion)hν>55 eV following equations (1) and (2) under the assumtions of ne ≈ 5 × 104 cm−3 (which is an order of magnitude smaller than the critical electron density for the ø3 λ5007 emission), Te ≈ 104 K, and CF ≈ 10−2. These assumptions refer to the inner circumnuclear clouds of AGNs. The ratios are presented in Table 3. For the objects dis- cussed in Panessa et al. (2005) the most popular (i.e. as in NED) galaxy names are used. The Lcorr([O+2]λ5007) values are taken from PB02 and Panessa et al. (2005). In the case of NGC 7679 we have used both their and our determinations of Lcorr([O+2]λ5007). For three objects with estimated broad Hα component LbroadHα (Panessa et al., 2005, Table 1 therein) we derive also the number of recombinations Nrec resulting in the Hα emission. We assume Te = 10 4 K and CF = 1 which leads to the estima- tion of the lower limit of the value of Nrec. The Nrec/Nion lower limits are also presented in Table 3. One can see that 17 out of 20 objects of the unabsorbed Sy2s discussed here reveal Nph/Nion)hν>55 eV > 0.3. This indi- cates that the central AGN sources in a considerable part of the unabsorbed Sy2s are obscured. The NGC 7679 does not make an exception and also possesses a hidden AGN engine suggested both by the ø3 λ5007 morphology and by the photon deficiency. Table 3. The photon deficiency for unabsorbed Sy2s discussed by Panessa and Bassani (2002) and Panessa et al. (2005) galaxy (Nph/Nion)hν>55 eV Nrec/Nion (lower limit) ESO 540-G001 4.2 13.0 CGCG 551-008 1.0 MCG -03-05-007 2.2 UGC 03134 19.5 IRAS 20051-1117 1.6 1.2 CGCG 303-017 1.3 2.0 IC 1631 0.3 NGC 2992 2.0 NGC 3147 0.4 NGC 4565 6.7 NGC 4579 0.2 NGC 4594 1.7 NGC 4698 0.3 NGC 5033 1.3 MRK 273x 0.4 NGC 5995 0.4 NGC 6221 0.02 NGC 6251 6.0 NGC 7590 0.4 NGC 7679 3.4 (2.0 from our data) It is still not clear what kind of physical process is related to the presence of hidden central engines in Sy2s. PB02 suggest two scenarios for the unabsorbed Sy2s (i) the central engine and their BLR must be hidden by an absorbing medium with high value of the AV/NH ratio, and (ii) the BLR is very weak or absent. 5. Conclusions We present a new ø3 λ5007 emission - line image of the circumnuclear region of NGC 7679 which shows elliptical isophotes extended along the PA≈ 80◦ ± 10◦ in the direction to the counterpart galaxy NGC 7682. The maximum of this emis- sion is displaced by about 4 arcsec from the photometric center defined by the continuum emission. The ratio of the quantity of ionizing photons inferred from the observed extinction corrected ø3 λ5007 luminosity to the number of ionizing photons with hν > 55 eV provided by the central AGN source (Nph/Nion)hν> 55 eV ≈ 0.2 − 20 as well as the presence of weak and elusive Hα broad wings probably indicate a hidden AGN. The high ionization inferred by the flux ratio ø3 λ5007/Hα in the direction of about PA≈ 80◦ ± 10◦ coincides with the di- rection to the counterpart galaxy NGC 7682. It is possible that the dust and gas in this direction has a direct view to the central AGN engine. It suggests that starburst and dust decay in this di- Yankulova I., Golev V., and Jockers, K.: The luminous infrared composite Seyfert 2 galaxy NGC 7679 through ... 9 rection have occurred because of tidal interaction between the two galaxies. In the direction PA≈ 0◦ the ionization is entirely caused by hot stars. A large part of the unabsorbed Compton-thin Sy2s with higher ø3 luminosity (>∼ 1041 erg s−1) possesses a hidden AGN source. Acknowledgements. We are grateful to the referee, Lucio Buson, for his valuable comments which improved both the content and the clar- ity of this manuscript. We would like to thank T. Bonev, Institute of Astronomy of Bulgarian Academy of Sciences, for kindly providing the Fabry-Perot observations and for useful discussions. We are grateful to S. Zhekov, Space Research Institute of Bulgarian Academy of Sciences, for the numerous fruitful discussions and especially for the analysis of the X-ray properties of NGC 7679. Our work was partially based on data from the La Palma ING, ESO NTT, and XMM-Newton Archives. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We acknowledge the support of the National Science Research Fund by the grant No.F-201/2006. References Aldrovandi, S. M. 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0704.0769

The Fermionic Density-functional at Feshbach Resonance

arXiv:0704.0769v1 [cond-mat.other] 5 Apr 2007 The Fermionic Density-functional at Feshbach Resonance Michael Seidl Institute of Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany Rajat K. Bhaduri Department of Physics and Astronomy, McMaster University, Hamilton, Canada L8S 4M1 (Dated: November 17, 2018) We consider a dilute gas of neutral unpolarized fermionic atoms at zero temperature. The atoms interact via a short-range (tunable) attractive interaction. We demonstrate analytically a curious property of the gas at unitarity. Namely, the correlation energy of the gas, evaluated by second order perturbation theory, has the same density dependence as the first order exchange energy, and the two almost exactly cancel each other at Feshbach resonance irrespective of the shape of the potential, provided (µrs) ≫ 1. Here (µ)−1 is the range of the two-body potential, and rs is defined through the number density, n = 3/(4πr3s). The implications of this result for universality is discussed. I. INTRODUCTION Consider a dilute gas ofN ≫ 1 neutral fermionic atoms (massM) at T = 0 interacting with a short-range attrac- tive potential. In general, the properties of the dilute gas are determined by the number density n, and the scatter- ing length a. The Hamiltonian of this N -particle system reads Ĥ = − ~ ∇2i + |ri − rj | . (1) Not written explicitly here, there is also an external po- tential vext(r) that forces the N atoms to stay within a large box with volume Ω [where vext(r) ≡ 0]. The attractive interaction potential is assumed to have the 2-parameter form v(r) = −v0f(µr) (2) where v0 > 0 is the strength of the interaction, R0 = is its range, and f(x) is a dimensionless function. In the true ground state of the Hamiltonian (1) the attractive atoms may form dimers or even clusters. We are, however, looking for a metastable state where there is a dilute gas of separated atoms with uniform density n, satisfying the condition (µrs) ≫ 1 where n = NΩ = Even then, for a weak v0, there will be BCS-type pair- ing, followed by dimer formation as the strength of the interaction increases. This was predicted long back by Leggett [1], and has been observed experimentally [2]. For the density functional analysis of the uniform gas at Feshbach resonance, we shall disregard the BCS con- densed pairs in this paper. To study the effect of the attractive interaction v(r), we consider the corresponding atom-atom scattering prob- lem in the relative s-state. Separating the center-of-mass motion, we are left with the relative Hamiltonian Ĥrel = − − v0f(µr) . (3) Keeping the range of the potential small enough such that (µrs) ≫ 1, the strength v0 is adjusted such that the potential can support a single bound state at zero energy. This happens when the scattering length a→ ∞, leaving no length scale from the interaction. Such a tuning of the interaction is possible experimentally, and gives rise to Feshbach resonance [3]. The scattering cross section in the given partial wave (s-wave in our case) reaches the unitary limit, and the gas is said to be at unitarity. It is then expected to display universal behavior [4]. Note that at Feshbach resonance, there is no length scale left other than the inverse of the Fermi wave number kF , where kF = (3π 2n)1/3. The energy per particle, E/N , as a function of the density n, should therefore scale the same way as the noninteracting kinetic energy, 3 2k2F /2M ∝ n2/3. There has been much interest amongst theorists to calculate the properties of the gas in the unitary regime (kF |a| ≫ 1). In particular, at T = 0, the energy per particle of the gas is calculated to be , (4) where ξ ≃ 0.44 [5]. The experimental value of ξ is about 0.5, but with large error bars [6]. Recently, there have been two Monte Carlo (MC) finite temperature calcula- tions [7, 8] of an untrapped gas at unitarity, where various thermodynamic properties as a function of temperature have been computed. It is clear that at unitarity, the kinetic and potential energies should scale the same way. This has been assumed a priori in a previous density functional treatment of a unitary gas [9]. However, such a scaling behavior is not evident from the density func- tionals for the direct, exchange and correlation energies [10] (see sects. II and III). The aim of the present paper is to examine this point in some detail. In particular, we are able to show analytically that the leading contribu- tion of the correlation energy (calculated in second order perturbation theory), cancels the first order exchange en- rgy almost exactly at Feshbach resonance. This happens irrespective of the shape of the potential as specified by f(µr), provided the condition (µrs) ≫ 1. We show that http://arxiv.org/abs/0704.0769v1 our general Eq.(24) (derived later in the text) that en- sures such a cancellation is satisfied at unitarity for a variety of 2-parameter potentials, including the square well and the delta-shell, as well as the smoothly varying cosh−2(µr) and Gaussian potentials. This is the main re- sult of the present work. The implications of this result for universality is marginal. This is because these po- tential energy terms, in the limit of (µrs) ≫ 1, are very small compared to the kinetic energy [4]. For a moder- ately large value like (µrs) ≃ 3 howevr, these terms are comparable in magnitude to the kinetic energy (sect. IV). Even then, the cancellation of the first order exchange, and the second order perturbative terms leave the di- rect first order term in tact. In the electron gas, this (repulsive) term got cancelled by the interaction of the electrons with the positive ionic background. There is no such mechanism of cancellation here, unless we assume, rather arbitrarily, that the short-range interatomic repul- sion cancels this direct (attractive) contribution. Even without any such assumptions, however, our main result (Table I), applicable at Feshbach reonance, is interesting from the angle of potential theory. II. PERTURBATION EXPANSION Treating the interaction (2) as a weak perturbation in the Hamiltonian (1), the unperturbed energy E(0) is the kinetic energy of a non-interacting Fermi gas, E(0) = Nts(rs) ≡ N . (5) Here, kF = and α3 = 4 . The corresponding ground state |Φ0〉 is a Slater determinant of plane waves. In terms of dimensionless coordinates xi = µri, the Hamiltonian (1) can be written as Ĥ = −1 ∇2i − λ |xi − xj | , λ = This suggests that the perturbation parameter is not re- ally small at unitarity. For example, for the square-well potential, the zero-energy single bound state occurs when λ = π (see sect. III). Nevertheless the low-order terms can point to important information, even when the ex- pansion is divergent [11]. In our problem, there are three parameters, µ, v0, and rs. The unitarity condition re- lates µ and v0, so two independent parameters are left. One of these may be taken to be the small parameter ζ = (µrs) −1. The remaining free parameter v0 may be chosen independently of ζ to fulfill the unitarity condi- tion. A. First order Formally, the first-order correction, E(1) = 〈Φ0|V̂int|Φ0〉, (7) has a direct contribution U(rs, µ) = Nu(rs, µ) with u(rs, µ) = d3r′v |r− r′| (µrs)3 f2. (8) Here, f2 = dxx2f(x). The other first-order contribution is the exchange en- ergy Ex(rs, µ) = Nex(rs, µ), [4] ex(rs, µ) = − drj1(kF r) 2v(r). (9) Here, j1(z) is a spherical Bessel function. Since v(r) is short-range and kF is small in a dilute gas, we can use the small-z expansion j1(z) = +O(z3) to find ex(rs, µ) = (µrs)3 f2 +O(µrs) −5. (10) B. Second order 1. General expressions Also the second-order correction, E(2) = − |〈Φn|V̂int|Φ0〉|2 En − E0 dir + e , (11) has a direct and an exchange contribution [12], dir(rs, µ) = − d3q f̃ d3k1 d q · (q+ k1 − k2) , (12) e(2)ex (rs, µ) = + d3q f̃ d3k1 d |q+ k1 − k2| q · (q+ k1 − k2) . (13) While v20(2M/~ 2µ2) has the dimension energy, the inte- gration variables q, k1, and k2 are dimensionless here. The domain of the integral over d3k1 d 3k2 depends on q, D : |k1|, |k2| < 1; |k1 + q|, |k2 − q| > 1. (14) Furthermore, f̃(y) is a dimensionless transform of f(x), f̃(y) = dxx2 f(x) j0(yx) dxx f(x) sin(yx). (15) To recover Eqs. (8) and (9) of Ref. [12], put M = me, v0 = −e2µ, and f(x) = 1x or f̃(y) = , such that v(r) = becomes the electronic Coulomb repulsion. (Note that Ref. [12] uses Rydberg units, mee 4/2~2 = e2/2aB = 1.) 2. The limit µrs ≫ 1 For a dilute gas (small kF ) with short-range interaction (large µ), Eqs. (12) and (13) can be evaluated in the limit µ/kF ≡ αµrs ≫ 1 where α3 = 49π . Following Ref. [13], we choose a number q1 such that 1 ≪ q1 ≪ µ/kF and split the integrals over d3q into two parts, d3q = d3q + d3q. (16) In the first part with q < q1, we have q ≪ µ/kF and |q + k1 − k2| ≪ µ/kF (note that |k1|, |k2| < 1 ≪ q1). Therefore, we may expand f̃(y) = f2+O(y 2) in Eqs. (12) and (13) and keep the leading term f2 only. The sum of the two resulting q < q1 contributions reads q<q1(rs, µ) = − f22 × d3k1 d q · (q+ k1 − k2) . (17) The number q1 can be chosen independently of µ/kF ≫ 1, despite the condition 1 ≪ q1 ≪ µ/kF . Then, the integral in Eq. (17) is a finite constant and we conclude q<q1(rs, µ) = O(µrs) −4. (18) In the second part q > q1 ≫ 1 of the integral (16), we can put q + k1 − k2 ≈ q, since |k1|, |k2| < 1. The resulting contributions to Eqs. (12) and (13) add up to q>q1(rs, µ) = − where d3k1 d 3k2 = ( )2 has been used. Now, dy f̃(y)2 (20) where y1 = kF q1/µ ≪ 1. If dy f̃(y)2 in Eq. (20) did not depend on y1, expression (19) did rigorously have the order O(µrs) −3. However, using the small-y expansion f̃(y) = f2 +O(y 2), we have dyf̃(y)2 = f22 y1 +O(y Consequently, shifting the lower limit y1 of the integral (20) to zero does not affect the leading-order contribution to expression (19), q>q1(rs, µ) = O(µrs) −3. (21) Therefore, the quantity (18) does not contribute to the leading order of e c = e dir + e ex which is purely due to expression (19), e(2)c (rs, µ) = − (µrs)3 F +O(µrs) −4 (22) where F = dyf̃(y)2. III. DENSITY SCALING AT UNITARITY If the perturbation expansion is convergent [12], the total energy E(rs, µ) = Ne(rs, µ) of the gas can be ex- pressed in the form e(rs, µ) = ts(rs) + ex(rs, µ) + e(n)c (rs, µ). (23) At unitarity, when the relative Hamiltonian (3) has a single bound state at zero energy, the exchange plus cor- relation energy ex + n=2 e c should display the same density scaling as the kinetic energy, ts(rs) ∝ r−2s ∝ ρ2/3. This is obviously not the case with any one of the present (leading-order) results (10) and (22). However, since the exchange energy (10) and the second-order correlation energy (22) have opposite signs, they can cancel each other at some value of µ. This happens when , (24) where f2 = dxx2 f(x) and F = dyf̃(y)2. This is the main result of our paper, and we check it by tak- ing four different potentials. The results of this analysis, summarized in Table I, are discussed in detail below. Generally, we need an eigenfunction ψ(r) = of the relative Hamiltonian (3) with eigenvalue zero. Writing u(r) = φ(µr), the corresponding dimensionless Schrödinger Equation reads φ′′(x) = −λf(x)φ(x), λ ≡ Mv0 . (25) Precisely, we wish to determine that particular value λuty of λ for which this zero-energy solution is the only bound state. Then, φ(x) must obey φ(0) = 0, φ′(x) < 0 for x ≥ 0, and φ(x) → const. for x → ∞. In the following examples (A-D), the solution φ(x) can be found analyti- cally or numerically. (A) Square-well potential of radius R0 = 1/µ: v(r) = −v0Θ(R0 − r) , (26) where Θ(z) denotes the Heavyside step function, Θ(z) = 1 for z > 0 and Θ(z) = 0 for z ≤ 0. By setting the dimensionless variable µr = x, we see that f(x) = Θ(1− x). The square-well potential (26) supports a single zero energy bound state when the LHS of Eq.(24) is λuty = π2/4. It may be easily checked analytically that for the square-well potential (26), f2 = and F = π so that the RHS of Eq.(24) is 5 , very close to its LHS, π2/4 = 2.47. (B) Rosen-Morse hyperbolic potential [5]. This poten- tial is given by v(r) = −v0 sech2(µr) , (27) which suppotrs a single zero energy bound state when the LHS of Eq.(24) is λuty = 2 instead of π 2/4. For this potential, it is easy to check that f2 = π 2/12. The quan- tity F , however, has to be calculated numerically, and is given by F = 0.596. Again, Eq.(24) is approximately satisfied, since its RHS for this potential is 2.17. (C) Delta-shell potential [14]. Consider the potential v(r) = −η ~ δ(r −R0) , = −η ~ = −v0f(µr) . (28) Thus, we have v0 = η , µ = 1 , and f(x) = δ(x− 1). So we get f2 = 1, f̃(y) = sin y , and F = π . Hence the RHS of Eq. (24) is unity. The LHS is (ηR0), which is exactly unity when the s-state scattering length goes to infinity [14]. Thus Eq.(24) is exactly obeyed in this case. (D) Gaussian Potential. v(r) = −v0 exp(−µ2r2) (29) For this example, f(x) = exp(−x2) in Eq. (2). We find π and F = 1 )3/2 so that the RHS of Eq. (24) becomes πf2/2F = 2 3/2. Solving Eq. (25) numerically for this f(x), we obtain a single bound state at zero energy when the LHS of Eq. (24) is λuty = 0.949× 23/2, close to 23/2. Note, however, that contributions O(µrs) −3 may also come from higher order terms of the perturbation expan- sion in section II, since that expansion is carried out with respect to the parameter λ =Mv0/~ 2µ2, but not 1/µrs. IV. DISCUSSION The dimensionless Hamiltonian ĥ from Eq. (6) depends on the dimensionless paramaters TABLE I: The moments f2 and F of four different profiles f(x) for the potential (2). λuty is the value at unitarity of the parameter λ in Eq. (25). At unitarity, the ratio Q of the LHS of Eq. (24) to the RHS is always close to 1. f(x) f2 F λuty Q Θ(1− x) 1 0.987 sech(x)2 π 0.596 2 0.922 δ(1− x) 1 π 1 1.000 exp(−x2) 1 )3/2 2.684 0.949 and, not written explicitly, xs = µrs. The perturbation expansion of the ground-state energy of ĥ reads ε(xs, λ) = εn(xs)λ n. (31) The ground-state energy of the original Hamiltonian Ĥ , with three independent parameters, is then given by E(rs, µ, λ) = ε(µrs, λ) εn(µrs)λ n. (32) For µrs ≫ 1, we may expand εn(µrs) = (µrs)m . (33) From Eq. (5), we have ε02 = N α−2 while ε0m = 0 for m 6= 2. Eqs. (8) and (10) imply that ε1m = 0 for m < 3 and ε13 = N(− 32 + )f2. Eventually, due to Eq. (22), ε2m = 0 for m < 3 and ε23 = N(− 34π )2F . So far as the unitary point is concerned, we are inter- ested in a situation where kF |a| ≫ 1 ≫ kFR0 ∼ (µrs)−1. In view of the fact that the perturbation series above does not converge at unitarity, how significant is our low order perturbation calculation in this situation ? Note that our first order direct and exchange (potential) energy terms given by Eqs. (8,10) are the same as those obtained in the Hartree-Fock calculation (see, for example, Eq.(10) of Heiselberg [4]). How big are these terms at unitarity compared to the kinetic energy per particle ? Taking the example of the square-well potential discussed earlier, it is straight forward to show that our exchange term (10) at Feshbach resonance is ex(rs, µ) = . (34) For the square-well example, (kF a) = (µrs) 1− tan . (35) At unitarity, the RHS diverges for any finite value of (µrs), how ever large. Even in the neighbourhood of unitarity, it is possible to have (kF |a|) ≫ 1 for (µrs) ≫ 1. From Eq.(34), we note that too large a choice for (µrs) would make ex negligible against EF . Instead, taking a modestly large value, µrs = 3, we obtain the ration of ex to kinetic energy per particle to be about 0.56. Noting that ex has a different density-dependence than the kinetic energy per particle, its cancellation with the second order perturbative correlation term helps towards scale invariance, but only if there is a mechanism for the direct first order term to be cancelled. We conclude by emphasizing that the new result in this paper is displayed in Table 1, and should be of interest from the point of view of potential theory. The authors would like to thank Brandon van Zyl for discussions. This research was financed by NSERC of Canada. [1] A.J. Leggett, in Modern Trends in the Theory of Con- densed Matter, Springer-Verlag Lecture Notes, Vol. 115, edited by A. Peklaski and J. Przystawa (Springer-Verlag, Berlin, 1980), p.13 [2] C.A. Regal et al., Nature (London) 424, 47 (2003); M.W. Zwierlein et al., Phys. Rev. Lett. 91, 250401 (2003); C.A. Regal et al., Phys. Rev. Lett. 92, 040403 (2004); M.W. Zwierlein et al., Nature (London) 435, 1046 (2005); G. B. Partridge et al., Science 311, 503 (2006). [3] S. Inouye et al., Nature (London) 392, 151 (1998); Ph. Courteille et al., Phys. Rev. Lett 81, 69 (1998). [4] G.A. Baker, Phys. Rev. C60, 054311 (1999); H. Heisel- berg, Phys. Rev. A63, 043606 (2001); T.-L. Ho. Phys. Rev. Lett. 92, 090402 (2004). [5] J. Carlson, S.-Y. Chang, V. R. Pandharipande, and K. E. Schmidt, Phys. Rev. Lett. 91, 050401 (2003); A. Perali, P. Pieri, and G. C. Strinati, Phys. Rev. Lett. 93, 100404 (2004). [6] M. Bartenstein et al., Phys. Rev. Lett. 92, 120401 (2004); T. Bourdel et al., Phys. Rev. Lett. 93, 050401 (2004). [7] A. Bulgac, J. E. Drut J.E., and P. Magierski, Phys. Rev. Lett. 96, 090404 (2006). [8] E. Burovski, N. Prokof’ev, B. Svistunov, and M. Troyer, Phys. Rev. Lett. 96, 160402 (2006). [9] T. Papenbrock, Phys. Rev. A72, 041603 (R) (2005); A. Bhattacharyya and T. Papenbrock, Phys. Rev. A74, 041602 (R) (2006). [10] R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989); W. Kohn, Rev. Mod. Phys. 71, 1253 (1999). [11] M. Seidl, J. P. Perdew, and S. Kurth, Phys. Rev. Lett. 84, 5070 (2000). [12] M. Gell-Mann, K. A. Brueckner, Phys. Rev. 106, 364 (1957). [13] L. Zecca, P. Gori-Giorgi, S. Moroni, and G. B. Bachelet, Phys. Rev. B 70, 205 127 (2004). [14] K. Gottfried, Quantum Mechanicsvol.I, (W. A. Ben- jamin, Inc., New York, 1966). See sect. (15).

0704.0770

Chemical Evolution

Chemical Evolution Francesca Matteucci Department of Astronomy University of Trieste and Osservatorio Astronomico di Trieste (INAF) Via G.B. Tiepolo, 11, 34124 Trieste Italy ([email protected]) http://arxiv.org/abs/0704.0770v1 Contents 1 Chemical Evolution page 1 1.1 Lecture I: basic assumptions and equations of chem- ical evolution 1 1.1.1 The basic ingredients 1 1.1.2 The Star Formation Rate 2 1.1.3 The Initial Mass Function 3 1.1.4 The Infall Rate 4 1.1.5 The Outflow Rate 4 1.1.6 Stellar evolution and nucleosynthesis: the stellar yields 5 1.1.7 Type Ia SN Progenitors 6 1.1.8 Yields per Stellar Generation 7 1.1.9 Analytical models 8 1.1.10 Numerical Models 9 1.2 Lecture II: the Milky Way and other spirals 11 1.2.1 The Galactic formation timescales 11 1.2.2 The two-infall model 12 1.2.3 Common Conclusions from MW Models 18 1.2.4 Abundance Gradients from Emission Lines 19 1.2.5 Abundance Gradients in External Galaxies 21 1.2.6 How to model the Hubble Sequence 21 1.2.7 Type Ia SN rates in different galaxies 24 1.2.8 Time-delay model for different galaxies 25 1.3 Lecture III: interpretation of abundances in dwarf irregulars 27 1.3.1 Properties of Dwarf Irregular Galaxies 27 1.3.2 Galactic Winds 31 iv Contents 1.3.3 Results on DIG and BCG from purely chemical models 32 1.3.4 Results from Chemo-Dynamical models: IZw18 34 1.4 Lecture IV: Elliptical galaxies-Quasars- ICM Enrich- ment 38 1.4.1 Ellipticals 38 1.4.2 Chemical Properties 38 1.4.3 Scenarios for galaxy formation 39 1.4.4 Ellipticals-Quasars connection 41 1.4.5 The chemical evolution of QSOs 41 1.4.6 The chemical enrichment of the ICM 43 1.4.7 Conclusions on the enrichment of the ICM 46 References 48 Chemical Evolution 1.1 Lecture I: basic assumptions and equations of chemical evolution To build galaxy chemical evolution models one needs to elucidate a num- ber of hypotheses and make assumptions on the basic ingredients. 1.1.1 The basic ingredients • INITIAL CONDITIONS: whether the mass of gas out of which stars will form is all present initially or it will be accreted later on. The chemical composition of the initial gas (primordial or already enriched by a pregalactic stellar generation). • THE BIRTHRATE FUNCTION: B(M, t) = ψ(t)ϕ(M) (1.1) where: ψ(t) = SFR (1.2) is the star formation rate (SFR) and: ϕ(M) = IMF (1.3) is the initial mass function (IMF). • STELLAR EVOLUTION AND NUCLEOSYNTHESIS: stellar yields, yields per stellar generation • SUPPLEMENTARY PARAMETERS : infall, outflow, radial flows. 2 Chemical Evolution 1.1.2 The Star Formation Rate Here we will summarize the most common parametrizations for the SFR in galaxies, as adopted by chemical evolution models: • Constant in space and time and equal to the estimated present time SFR. For example, for the local disk, the present time SFR is SFR=2- 5M⊙pc −2Gyr−1 (Boissier& Prantzos, 1999). • Exponentially decreasing: SFR = νe−t/τ∗ (1.4) with τ∗ = 5− 15 Gyr (Tosi, 1988). The quantity νis a parameter that we call efficiency of SF since it represents the SFR per unit mass of gas and is expressed in Gyr−1. • The most used SFR is the Schmidt (1959) law, which assumes a de- pendence on the gas density, in particular: SFR = νσkgas (1.5) where k = 1.4± 0.15, as suggested by a study of Kennicutt (1998) of local star forming galaxies. • Some variations of the Schmidt law with a dependence also on the total mass have been suggested for example by Dopita & Ryder (1994). This formulation takes into account the feedback mechanism acting between supernovae ( SNe) and stellar winds injecting energy into the interstellar medium (ISM) and the galactic potential well. In other words, the SF process is regulated by the fact that in a region of recent star formation the gas is too hot to form stars and it is easily removed from that region. Before new stars could form the gas needs to cool and collapse back into the star forming region and this process depends on the potential well and therefore on the total mass density: SFR = νσk1totσ gas (1.6) with k1 = 0.5 and k2 = 1.5. • Kennicutt (1998) also suggested, as an alternative to the Schmidt law to fit the data, the following relation: SFR = 0.017Ωgasσgas ∝ R −1σgas (1.7) with Ωgas being the angular rotation speed of gas. 1.1 Lecture I: basic assumptions and equations of chemical evolution3 • Finally a SFR induced by spiral density waves was suggested by Wyse & Silk (1989): SFR = νV (R)R−1σ1.5gas (1.8) with R being the galactocentric distance and V (R) the gas rotation velocity. 1.1.3 The Initial Mass Function The IMF is a probability function describing the distribution of stars as a function of mass. The present day mass function is derived for the stars in the solar vicinity by counting the Main Sequence stars as a function of magnitude and then applying the mass-luminosty relation, holding for Main Sequence stars, to derive the distribution of stars as a function of mass. In order to derive the IMF one has then to make assumptions on the past history of SF. The derived IMF is normally approximated by a power law: ϕ(M)dM = aM−(1+x)dM (1.9) where ϕ(M) is the number of stars with masses in the interval M, M+dM. Salpeter (1955) proposed a one-slope IMF (x = 1.35) valid for stars with M > 10M⊙. Multi-slope (x1, x2, ..) IMFs have been suggested later on always for the solar vicinity (Scalo 1986,1998; Kroupa et al. 1993; Chabrier 2003). The IMF is generally normalized as: ∫ 100 Mϕ(M)dM = 1 (1.10) where a is the normalization constant and the assumed interval of inte- gration is 0.1− 100M⊙. The IMF is generally considered constant in space and time with some exceptions such as the IMF suggested by Larson (1998) with: x = 1.35(1 +m/m1) −1 (1.11) where m1 is variable typical mass and is associated to the Jeans mass. This IMF predicts then that m1 is a decreasing function of time. 4 Chemical Evolution 1.1.4 The Infall Rate For the rate of gas accretion there are in the literature several parametriza- tions: • The infall rate is constant in space and time and equal to the present time infall rate as measured in the Galaxy (∼ 1.0M⊙yr • The infall rate is variable in space and time, and the most common assumption is an exponential law (Chiosi 1980; Lacey & Fall 1985): IR = A(R)e−t/τ(R) (1.12) with τ(R) constant or varying with the galactocentric distance. The parameter A(R) is derived by fitting the present day total surface mass density, σtot(tG), at any specific galactocentric radius R. • For the formation of the Milky Way two episodes of infall have been suggested (Chiappini et al. 1997), where during the first infall episode the stellar halo forms whereas during the second infall episode the disk forms. This particular infall law gives a good representation of the formation of the Milky Way. The proposed two-infall law is: IR = A(R)e−t/τH(R) +B(R)e−(t−tmax)/τD(R) (1.13) where τH(R) is the timescale for the formation of the halo which can be costant or vary with galactocentric distance. The quantity τD(R) is the timescale for the formation of the disk and is a function of the galactocentric distance; in most of the models it is assumed to increase with R (e.g. Matteucci & François, 1989). • More recently, Prantzos (2003) suggested a gaussian law with a peak at 0.1 Gyr and a FWHM of 0.04 Gyr for the formation of the stellar halo. 1.1.5 The Outflow Rate The so-called galactic winds occur when the thermal energy of the gas in galaxies exceeds its potential energy. Generally, gas outflows are called winds when the gas is lost forever from the galaxy. Only detailed dynamical simulations can suggest whether there is a wind or just an outflow of gas which will soon or later fall back again into the galaxy. In chemical evolution models galactic winds can be sudden or continuous. If they are sudden, the mass is assumed to be lost in a very short interval of 1.1 Lecture I: basic assumptions and equations of chemical evolution5 time and the galaxy is devoided from all the gas; if they are continuous, one has to assume the rate of gas loss. Generally, in chemical evolution models (Bradamante et al. 1998) and also in cosmological simulations (Springel & Hernquist, 2003) it is assumed that the rate of gas loss is several times the SFR: W = −λSFR (1.14) where λ is a free parameter with the meaning of wind efficiency. This particular formulation for the galactic wind rate is confirmed by obser- vational findings (see Martin, 1999). 1.1.6 Stellar evolution and nucleosynthesis: the stellar yields Here we summarize the various contribution to the element production by stars of all masses. • Brown Dwarfs (M < ML, ML = 0.08 − 0.09M⊙) are objects which never ignite H and their lifetimes are larger than the age of the Uni- verse. They are contributing to lock up mass. • Low mass stars (0.5 ≤ M/M⊙ ≤ MHeF ) (1.85-2.2M⊙) ignite He ex- plosively but without destroying themselves and then become C-O white dwarfs (WD). If M < 0.5M⊙ they become He WDs. Their lifetimes range from several 109 years up to several Hubble times! • Intermediate mass stars (MHeF ≤ M/M⊙ ≤ Mup) ignite He quies- cently. The mass Mup is the limiting mass for the formation of a C-O degenerate core and is in the range 5-9M⊙, depending on stellar evo- lution calculations. Lifetimes are from several 107 to 109 years. They die as C-O WDs if not in binary systems. If in binary systems they can give rise to cataclysmic variables such as novae and Type Ia SNe. • Massive stars (M > Mup). We distinguish here several cases: -Mup ≤ M/M⊙ ≤ 10 − 12. Stars with Main Sequence masses in this range end up as electron-capture SNe leaving neutron stars as remnants. These SNe will appear as Type II SNe which show H in their spectra. -10 − 12 ≤ M/M⊙ ≤ MWR, (with MWR ∼ 20 − 40M⊙ being the limiting mass for the formation of a Wolf-Rayet (WR) star). Stars in this mass range end their life as core-collapse SNe (Type II) leaving a neutron star or a black hole as remnants. -MWR ≤ M/M⊙ ≤ 100. Stars in this mass range are probably 6 Chemical Evolution exploding as Type Ib/c SNe which do not show H in their spectra. Their lifetimes are of the order of ∼ 106 years. • Very Massive Stars (M > 100M⊙), they should explode by means of instability due to “pair creation” and they are called pair-creation SNe. In fact, at T ∼ 2 · 109 K a large portion of the gravitational energy goes into creation of pairs (e+, e−), the star becomes unstable and explodes. They leave no remnants and their lifetimes are < 106 years. Probably these very massive stars formed only when the metal content was almost zero (Population III stars, Schneider et al. 2004). All the elements with mass number A from 12 to 60 have been formed in stars during the quiescent burnings. Stars transform H into He and then He into heaviers until the Fe-peak elements, where the binding energy per nucleon reaches a maximum and the nuclear fusion reactions stop. H is transformed into He through the proton-proton chain or the CNO- cycle, then 4He is transformed into 12C through the triple- α reaction. Elements heavier than 12C are then produced by synthesis of α- particles. They are called α-elements (O, Ne, Mg, Si and others). The last main burning in stars is the 28Si -burning which produces 56Ni which then decays into 56Co and 56Fe. Si-burning can be quiescent or explosive (depending on the temperature). Explosive nucleosynthesis occurring during SN explosions mainly pro- duces Fe-peak elements. Elements originating from s- and r-processes (with A> 60 up to Th and U) are formed by means of slow or rapid (rel- ative to the β- decay) neutron capture by Fe seed nuclei; s-processing occurs during quiescent He-burning whereas r-processing occurs during SN explosions. 1.1.7 Type Ia SN Progenitors The Type Ia SNe, which do not show H in their spectra, are believed to originate from WDs in binary systems and to be the major producers of Fe in the Universe. The model proposed are basically two: • Single Degenerate Scenario (SDS), with a WD plus a Main Se- quence or Red Giant star, as originally suggested by Whelan and Iben (1973). The explosion (C-deflagration) occurs when the C-O WD reaches the Chandrasekhar mass, MCh =∼ 1.44M⊙, after accreting material from thecompanion. In this model the clock to the explosion is given by the lifetime of the companion of the WD (namely the less 1.1 Lecture I: basic assumptions and equations of chemical evolution7 massive star in the system). It is interesting to define the minimum timescale for the explosion which is given by the lifetime of a 8M⊙ star, namely tSNIamin=0.03 Gyr (Greggio and Renzini 1983). Recent observations in radio-galaxies by Mannucci et al. (2005;2006) seem to confirm the existence of such prompt Type Ia SNe. • Double Degenerate Scenario (DDS), where the merging of two C- OWDs of mass∼ 0.7M⊙, due to loss of angular momentum as a conse- quence of gravitational wave radiation, produces C-deflagration (Iben and Tutukov 1984). In this case the clock to the explosion is given by the lifetime of the secondary star, as above, plus the gravitational time delay, namely the time necessary for the two WDs to merge. The minimum time for the explosion is tSNIamin = 0.03+∆tgrav=0.04 Gyr (see Tornambè 1989). Some variations of the above scenarios have been proposed such as the model by Hachisu et al. (1996; 1999), which is based on the sin- gle degenerate scenario where a wind from the WD is considered. Such a wind stabilizes the accretion from the companion and introduces a metallicity effect. In particular, the wind, necessary to this model, oc- curs only if the systems have metallicity ([Fe/H]< −1.0). This implies that the minimum time for the explosion is larger than in the previous cases. In particular, tSNIamin = 0.33 Gyr, which is the lifetime of the more massive secondary considered (2.3M⊙) plus the metallicity delay which depends on the assumed chemical evolution model. 1.1.8 Yields per Stellar Generation Under the assumption of Instantaneous Recycling Approximation (IRA) which states that all stars more massive than 1M⊙ die immediately, whereas all stars with masses lower than 1M⊙ live forever, one can define the yield per stellar generation (Tinsley, 1980); mpimϕ(m)dm (1.15) where pim is the stellar yield of the element i, namely the newly formed and ejected element i by a star of mass m. The quantity R is the so-called Returned Fraction: (m−Mrem)ϕ(m)dm (1.16) 8 Chemical Evolution and is the total mass of gas restored into the ISM by an entire stellar generation. 1.1.9 Analytical models The Simple Model for the chemical evolution of the solar neighbourhood is the simplest approach to model chemical evolution. The solar neigh- bourhood is assumed to be a cylinder of 1 Kpc radius centered around the Sun. The basic assumptions of the Simple Model are: - the system is one-zone and closed, no inflows or outflows with the total mass present since the beginning, - the initial gas is primordial (no metals), - instantaneous recycling approximation holds, - the IMF, ϕ(m), is assumed to be constant in time, - the gas is well mixed at any time (IMA) The Simple Model fails in describing the evolution of the Milky Way (G-dwarf metallicity distribution, elements produced on long timescales and abundance ratios) and the reason is that at least two of the above assumptions are manifestly wrong, epecially if one intends to model the evolution of the abundance of elements produced on long timescales, such as Fe. In particular the assumptions of the closed boxiness and the However, it is interesting to know the solution of the Simple Model and its implications. Be Xi the abundance by mass of an element i. If Xi << 1, which is generally true for metals, we obtain the solution of the Simple Model. This solution is obtained analytically by ignoring the stellar lifetimes: Xi = yiln( ) (1.17) where µ =Mgas/Mtot and yi is the yield per stellar generation, as defined above, otherwise called effective yield. In particular, the effective yield is defined as: yieff = ln(1/G) (1.18) namely the yield that the system would have if behaving as the simple closed-box model. This means that if yieff > yi, then the actual system has attained a higher abundance for the element i at a given gas fraction G. Generally, in the IRA, we can assume: 1.1 Lecture I: basic assumptions and equations of chemical evolution9 (1.19) which means that the ratio of two element abundances are always equal to the ratio of their yields. This is no more true when IRA is relaxed. In fact, relaxing IRA is necessary to study in detail the evolution of the abundances of single elements. One can obtain analytical solutions also in presence of infall and/or outflow but the necessary condition is to assume IRA. Matteucci & Chiosi (1983) found solutions for models with outflow and infall and Matteucci (2001) found it for a model with infall and outflow acting at the same time. The main assumption in the model with outflow but no infall is that the outflow rate is: W (t) = λ(1−R)ψ(t) (1.20) where λ ≥ 0 is the wind parameter. The solution of this model is: (1 + λ) ln[(1 + λ)G−1 − λ] (1.21) for λ = 0 the equation becomes the one of the Simple Model (1.17). The solution of the equation of metals for a model without wind but with a primordial infalling material (XAi = 0) at a rate: A(t) = Λ(1−R)ψ(t) (1.22) and Λ 6= 1 is : [1− (Λ − (Λ− 1)G−1)−Λ/(1−Λ)] (1.23) For Λ = 1 one obtains the well known case of extreme infall studied by Larson (1972) whose solution is: Xi = yi[1− e −(G−1−1)] (1.24) This extreme infall solution shows that when G→ 0 then Xi → yi. 1.1.10 Numerical Models Numerical models relax IRA and close boxiness but generally retain the constancy of ϕ(m) and the IMA. 10 Chemical Evolution If Gi is the mass fraction of gas in the form of an element i, we can write: Ġi(t) = −ψ(t)Xi(t) ∫ MBm ψ(t− τm)Qmi(t− τm)ϕ(m)dm ∫ MBM ∫ 0.5 f(γ)ψ(t− τm2)Qmi(t− τm2)dγ]dm ∫ MBM ψ(t− τm)Qmi(t− τm)ϕ(m)dm ψ(t− τm)Qmi(t− τm)ϕ(m)dm +XAiA(t)−Xi(t)W (t) (1.25) where B=1-A, A=0.05-0.09. The meaning of the A parameter is the fraction in the IMF of binary systems with those specific features re- quired to give rise to Type Ia SNe, whereas B is the fraction of all the single stars and binary systems in the same mass range of definition of the progenitors of Type Ia SNe. The values of A indicated above are cor- rect for the evolution of the solar vicinity where an IMF of Scalo (1986, 1989) or Kroupa et al.(1993) is adopted. If one adopts a flatter IMF such as the Salpeter (1955) one then A is different. In the above equations the contribution of Type Ia SNe is contained in the third term on the right hand side. The integral is made over a range of masses going from 3 to 16 M⊙ which represents the total masses of binary systems able to produce Type Ia SNe in the framework of the SDS. There is also an inte- gration over the mass distribution of binary systems; in particular, one considers the function f(γ) where γ = M2 M1+M2 , with M1 and M2 being the primary and secondary mass of the binary system, respectively (for more details see Matteucci & Greggio 1986 and Matteucci 2001).The functions A(t) and W(t) are the infall and wind rate, respectively. Fi- nally, the quantity Qmi represents the stellar yields (both processed and unprocessed material). 1.2 Lecture II: the Milky Way and other spirals 11 1.2 Lecture II: the Milky Way and other spirals The Milky Way galaxy has four main stellar populations: 1) the halo stars with low metallicities (the most common metallicity indicator in stars is [Fe/H]= log(Fe/H)∗− log(Fe/H)⊙) and eccentric orbits, 2) the bulge population with a large range of metallicities and is dominated by random motions, 3) the thin disk stars with an average metallicity < [Fe/H ] >=-0.5 dex and circular orbits, and finally 4) the thick stars which possess chemical and kinematical properties intermediate between those of the halo and those of the thin disk. The halo stars have average metallicities of < [Fe/H ] >=-1.5 dex and a maximum metallicity of ∼ −1.0 dex although stars with [Fe/H] as high as -0.6 dex and halo kinematics are observed. The average metallicity of thin disk stars is ∼ −0.6 dex, whereas the one of Bulge stars is ∼ −0.2 dex. 1.2.1 The Galactic formation timescales The kinematical and chemical properties of the different Galactic stel- lar populations can be interpreted in terms of the Galaxy formation mechanism. Eggen et al. (1962) in a cornerstone paper suggested a rapid collapse for the formation of the Galaxy lasting ∼ 3 · 108 years. This suggestion was based on a kinematical and chemical study of so- lar neighbourhood stars. Later on, Searle & Zinn (1979) proposed a central collapse like the one proposed by Eggen et al. but also that the outer halo formed by merging of large fragments taking place over a con- siderable timescale > 1 Gyr. More recently, Berman & Suchov (1991) proposed the so-called hot Galaxy picture, with an initial strong burst of SF which inhibited further SF for few Gyr while a strong Galactic wind was created. From an historical point of view, the modelization of the Galactic chemical evolution has passed through different phases that I summarize in the following. • SERIAL FORMATION The Galaxy is modeled by means of one accretion episode lasting for the entire Galactic lifetime, where halo, thick and thin disk form in sequence as a continuous process. The obvious limit of this approach is that it does not allow us to predict the observed overlapping in metallicity between halo and thick disk stars and between thick and thin disk stars, but it gives a fair representation of our Galaxy (e.g. Matteucci & François 1989). 12 Chemical Evolution • PARALLEL FORMATION In this formulation, the various Galactic components start at the same time and from the same gas but evolve at different rates (e.g. Pardi et al. 1995). It predicts overlapping of stars belonging to the different components but implies that the thick disk formed out of gas shed by the halo and that the thin disk formed out of gas shed by the thick disk, and this is at variance with the distribution of the stellar angular momentum per unit mass (Wyse & Gilmore 1992), which indicates that the disk did not form out of gas shed by the halo. • TWO-INFALL FORMATION In this scenario, halo and disk formed out of two separate infall episodes (overlapping in metallicity is also predicted) (e.g. Chiappini et al. 1997; Chang et al. 1999). The first infall episode lasted no more than 1-2 Gyr whereas the second, where the thin disk formed, lasted much longer with a timescale for the formation of the solar vicinity of 6-8 Gyr (Chiappini et al. 1997; Boissier& Prantzos 1999). • STOCHASTIC APPROACH Here the hypothesis is that in the early halo phases ([Fe/H] < −3.0 dex), mixing was not efficient and, as a consequence, one should ob- serve in low metallicity halo stars the effects of pollution from single SNe (e.g. Tsujimoto et al. 1999; Argast et al. 2000; Oey 2000). These models predict a large spread for [Fe/H] < −3.0dex which is not ob- served, as shown by recent data with metallicities down to -4.0 dex (Cayrel et al. 2004; see later). 1.2.2 The two-infall model The adopted SFR (see Figure 2.1) is eq.(1.6) with different SF efficiencies for the halo and disk, in particular νH = 2.0Gyr −1, νD = 1.0Gyr respectively. A threshold density (σth = 7M⊙pc −2) for the SFR is also assumed in agreement with results from Kennicutt (1989; 1998). In Figure 2.2 we show the predicted SN (II and Ia) rates by the two- infall model. Note that the Type Ia SN rate is calculated according to the SDS (Greggio & Renzini, 1983; Matteucci & Recchi, 2001). There is a delay between the Type II SN rate and the Type Ia SN rate, and while the Type II SN rate strictly follows the SFR, the Type Ia SN rate is smoothly increasing. François et al. (2004) compared the predictions of the two-infall model for the abundance ratios versus metallicity relations ([X/Fe] vs. [Fe/H]), with the very recent and very accurate data of the project “First Stars” 1.2 Lecture II: the Milky Way and other spirals 13 Fig. 1.1. The predicted SFR in the solar vicinity with the two-infall model. Figure from Chiappini et al. (1997). The oscillating behaviour at late times is due to the assumed threshold density for SF. The threshold gas density is also responsible for the gap in the SFR seen at around 1 Gyr. by Cayrel et al. (2004). They adopted yields from the literature both for Type II and Type Ia SNe and noticed that while for some elements (O, Fe, Si, Ca) the yields of Woosley & Weaver (1995) (hereafter WW95) reproduce the data fairly well, for the Fe-peak elements and heaviers none of the available yields give a good agreement. Therefore, they varied empirically the yields of these elements in order to best fit the data. In Figures 2.3 and 2.4 we show the predictions for α-elements (O, Mg, Si, Ca, Ti, K) plus some Fe-peak elements and Zn. In Figure 2.4 we show also the ratios between the yields derived em- pirically by François et al. (2004) in order to obtain the excellent fits shown in the figures, and those of WW95 for massive stars. For some elements it was necessary to change also the yields from Type Ia SNe relative to the reference ones which are those of Iwamoto et al. (1999) (hereafter I99). In Figure 2.5 we show the predictions of chemical evolution models for 12C and 14N compared with abundance data. The behaviour of C shows a roughly constant [C/Fe] as a function of [Fe/H], although C seems to 14 Chemical Evolution Fig. 1.2. The predicted Type II and Ia SN rate in the solar vicinity with the two-infall model. Figure from Chiappini et al. (1997) slightly increase at very low metallicities, indicating that the bulk of these two elements comes from stars with the same lifetimes. The data in these figures, especially those for N are old and do not contain very metal poor stars. Newer data containing stars with [Fe/H] down to ∼ -4.0 dex (Spite et al. 2005; Israelian et al. 2004) indicate that the [N/Fe] ratio continues to be high also at low metallicities, indicating a primary origin for N produced in massive stars. We recall here that we define primary a chemical elements which is produced in the stars starting from the H and He, whereas we define secondary a chemical element which is formed from heavy elements already present in the star at its birth and not produced in situ. The model predictions shown in Figure 2.5 for C and N assume that the bulk of these elements is produced by low and intermediate mass stars (yields from van den Hoeck and Groenewegen, 1997) and that N is produced as a partly secondary and partly primary element. The N production from massive stars has only a secondary origin (yields from WW95). In Figure 2.5 we show also a model prediction where N is considered as a primary element in massive stars with the yields artificially increased. Recently, Chiappini et al. 1.2 Lecture II: the Milky Way and other spirals 15 Fig. 1.3. Predicted and observed [X/Fe] vs. [Fe/H] for several α- and Fe-peak- elements plus Zn compared with a compilation of data. In particular the black dots are the recent high resolution data from Cayrel et al. (2004). For the other data see references in François et al. (2004). The solar value indicated in the upper right part of each figure represents the predicted solar value for the ratio [X/Fe]. The assumed solar abundances are those of Grevesse & Sauval (1998) except that for oxygen for which we take the value of Holweger (2001). 16 Chemical Evolution Fig. 1.4. Upper panel: predicted and observed [X/Fe] vs. [Fe/H] for several elements as in Figure 2.3. In the bottom part of this Figure are shown the ratios between the empirical yields and the yields by WW95 for massive stars. Such empirical yields have been suggested by François et al. (2004) in order to fit at best all the [X/Fe] vs. [Fe/H] relations. In the small panel at the bottom right side are shown also the ratios between the empirical yields for Type Ia SNe and the yields by I99. 1.2 Lecture II: the Milky Way and other spirals 17 Fig. 1.5. Upper panel: predicted and observed [C/Fe] vs. [Fe/H]. Models from Chiappini et al. (2003a). Lower panel, predicted and observed [N/Fe] vs. [Fe/H]. For references to the data see original paper.The thin and thick continuous lines in both panels represent models with standard nucleosynthe- sis, as described in the text, whereas the dashed line represents the predictions of a model where N in massive stars has been considered as a primary element with “ad hoc” stellar yields. (2006) have shown that primary N produced by very metal poor fastly rotating massive stars can well reproduce the observations. In summary, the comparison between model predictions and abun- dance data indicate the following scenario for the formation of heavy elements: • 12C and 14N are mainly produced in low and intermediate mass stars (0.8 ≤ M/M⊙ ≤ 8). The amounts of primary and secondary N is still uncertain and also the fraction of C produced in massive stars. Primary N from massive stars seems to be required to reproduce the N abundance in low metallicity halo stars. • α-elements originate in massive stars: the nucleosynthesis of O is rather well understood (there is agreement between different authors), the yields from WW95 as functions of metallicity produce an excellent agreement with the observations for this particular element. 18 Chemical Evolution • Magnesium is generally underproduced by nucleosynthesis models. Taking the yields of WW95 as a reference, the Mg yields should be increased in stars with masses M ≤ 20M⊙ and decreased in stars with M > 20M⊙ to fit the data. Silicon should be slightly increased in stars with masses M > 40M⊙. • Fe originates mostly in Type Ia SNe. The Fe yields in massive stars are still uncertain, WW95 metallicity dependent yields overestimate Fe in stars < 30M⊙. For this element, it is better to adopt the yields of WW95 for solar metallicity. • Fe-peak elements: the yields of Cr, Mn should be increased in stars of 10-20 M⊙ relative to the yields of WW95, whereas the yield of Co should be increased in Type Ia SNe, relative to the yields of I99, and decreased in stars in the range 10-20M⊙, relative to the yields of WW95. Finally, the yield of Ni should be decreased in Type Ia SNe. • The yields of Cu and Zn from Type Ia SNe should be larger, relative to the standard yields, as already suggested by Matteucci et al. (1993). 1.2.3 Common Conclusions from MW Models Most of the chemical evolution models for the Milky Way existing in the literature conclude that: • The G-dwarf metallicity distribution can be reproduced only by as- suming a slow formation of the local disk by infall. In particular, the time-scale for the formation of the local disk should be in the range τd ∼ 6 − 8 Gyr (Chiappini et al. 1997; Boissier and Prantzos 1999; Chang et al. 1999; Chiappini et al. 2001; Alibès et al. 2001). • The relative abundance ratios [X/Fe] vs. [Fe/H], interpreted as time- delay between Type Ia and II SNe, suggest a timescale for the halo- thick disk formation of τh ∼ 1.5-2.0 Gyr (Matteucci and Greggio 1986; Matteucci and François, 1989; Chiappini et al. 1997). The external halo and thick disk probably formed more slowly or have been accreted (Chiappini et al. 2001). • To fit abundance gradients, SFR and gas distribution along the Galac- tic thin disk we must assume that the disk formed inside-out (Mat- teucci & François, 1989; Chiappini et al. 2001; Boissier & Prantzos 1999; Alibés et al. 2001). Radial flows can help in forming the gra- dients (Portinari & Chiosi 2000) but they are probably not the main cause for them. A variable IMF along the Disk can in principle ex- plain abundance gradients but it creates unrealistic situations: in fact, 1.2 Lecture II: the Milky Way and other spirals 19 in order to reproduce the negative gradients one should assume that in the external and less metal rich parts of the Disk low mass stars form preferentially (see Chiappini et al. 2000 for a discussion on this point). • The SFR is a strongly varying function of the galactocentric distance (Matteucci & François 1989; Chiappini et al, 1997,2001; Goswami & Prantzos 2000; Alibés et al. 2001). 1.2.4 Abundance Gradients from Emission Lines There are two types of abundance determinations in HII regions: one is based on recombination lines which should have a weak temperature de- pendence of the nebula (He, C, N, O), the other is based on collisionally excited lines where a strong dependence is intrinsic to the method (C, N, O, Ne, Si, S, Cl, Ar, Fe and Ni). This second method has predominated until now. A direct determination of the abundance gradients from HII regions in the Galaxy from optical lines is difficult because of extinction, so usually the abundances for distances larger than 3 Kpc from the Sun are obtained from radio and infrared emission lines. Abundance gradients can also be derived from optical emission lines in Planetary Nebulae (PNe). However, the abundances of He, C and N in PNe are giving only information on the internal nucleosynthesis of the star. So, to derive gradients one should look at the abundances of O, S and Ne, unaffected by stellar processes. In Figure 2.6 we show theoretical predictions of abundance gradients along the disk of the Milky Way compared with data from HII regions and B stars. The adopted model is from Chiappini et al. (2001; 2003a) and is based on an inside-out formation of the thin disk with the inner regions forming faster than the outer ones, in particular τ(R) = 0.875R − 0.75 Gyr. Note that to obtain a better fit for 12C, the yields of this element have been increased artificially relative to those of WW95. As already said, most of the models agree on the inside-out scenario for the Disk formation, however not all models agree on the evolution of the gradients with time. In fact, some models predict a flattening with time (Boissier and Prantzos 1998; Alibès et al. 2001), whereas others such as that of Chiappini et al. (2001) predict a steepening. The reason for the steepening is that in the model of Chiappini et al. is included a threshold density for SF,, which induces the SF to stop when the density decreases below the threshold. This effect is particularly strong in the external regions of the Disk, thus contributing to a slower evolution and 20 Chemical Evolution 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Fig. 1.6. Upper panel: abundance gradients along the Disk of the MW. The lines are the models from Chiappini et al. (2003a): these models differ by the nucleosynthesis prescriptions. In particular, the dash-dotted line represents a model with van den Hoeck & Groenewegen (1997, hereafter HG97) yields for low-intermediate mass stars with η (mass loss parameter) constant and Thielemann et al.’s (1996) yields for massive stars, the long- dashed thick line has HG97 yields with variable η and Thielemann et al. yields, the long-dashed thin line has HG97 yields with variable η but WW95 yields for massive stars. It is interesting to note that in all of these models the yields of 12C in stars > 40M⊙ have been artificially increased by a factor of 3 relative to the yields of WW95. Lower panel: the temporal behaviour of abundance gradients along the Disk as predicted by the best model of Chiappini et al. (2001). The upper lines in each panel represent the present time gradient, whereas the lower ones represent the gradient a few Gyr ago. It is clear that the gradients tend to stepeen in time, a still controversial result. 1.2 Lecture II: the Milky Way and other spirals 21 therefore to a steepening of the gradients with time, as shown in Figure 2.6, bottom panel. 1.2.5 Abundance Gradients in External Galaxies Abundance gradients expressed in dex/Kpc are found to be steeper in smaller disks but the correlation disappears if they are expressed in dex/Rd, which means that there is a universal slope per unit scale length (ref). The gradients are generally flatter in galaxies with central bars (ref). The SFR is measured mainly from Hα emission (Kennicutt, 1998) and show a correlation with the total surface gas density (HI+H2), in particular the suggested law is that of eq. (1.5). In the observed gas distributions differences between field and clus- ter spirals are found in the sense that cluster spirals have less gas, probably as a consequence of stronger interactions with the environ- ment.Integrated colors of spiral galaxies (Josey & Arimoto 1992; Jimenez et al. 1998; Prantzos & Boissier 2000) indicate inside-out formation, as also found for the milky Way. As an example of abundance gradients in a spiral galaxy we show in Figure 2.7 the observed and predicted gas distribution and abun- dance gradients for the disk of M101. In this case the gas distribu- tion and the abundance gradients are reproduced with systematically smaller timescales for the disk formation relative to the MW (M101 formed faster), and the difference between the timescales of formation of the internal and external regions is smaller (τM101 = 0.75R−0.5 Gyr, Chiappini et al. 2003a) To conclude this section we like to recall a paper by Boissier et al. (2001) where a detailed study of the properties of disks is presented. They conclude that more massive disks are redder, more metal rich and more gas-poor than smaller ones. On the other hand their estimated SF efficiency (defined as the SFR per unit mass of gas) seem to be similar among different spirals: this leads them to conclude that more massive disks are older than less massive ones. 1.2.6 How to model the Hubble Sequence The Hubble Sequence can be simply thought as a sequence of objects where the SFR proceeds faster in the early than in the late types (see also Sandage, 1986). We take the Milky Way galaxy, whose properties are best known, as a 22 Chemical Evolution Fig. 1.7. Upper panel: predicted and observed gas distribution along the disk of M101. The observed HI, H2 and total gas are indicated in the Figure. The large open circles indicate the models: in particular, the open circles connected by a continuous line refer to a model with central surface mass density of 1000M⊙pc −2, while the dotted line refers to a model with 800M⊙pc −2 and the dashed to a model with 600M⊙pc −2. Lower panel: predicted and observed abundance gradients of C,N,O elements along the disk of M101.The models are the lines and differ for a different threshold density for SF, being larger in the dashed model. All the models are by Chiappini et al. (2003a). 1.2 Lecture II: the Milky Way and other spirals 23 reference galaxy and we change the SFR relatively to the Galactic one, for which we adopt eq. (1.6). The quantity ν in eq. (1.6) is the efficiency of SF which we assume to be characteristic of each Hubble type. In the two-infall model for the Milky Way we adopt νhalo = 2.0Gyr −1 and νdisk = 1.0Gyr −1 (see Figure 2.1). The choice of adopting a dependence on the total surface mass density for the Galactic disk is due to the fact that it helps in producing a SFR strongly varying with the galactocentric distance, as required by the observed SFR and gas density distribution as well as by the abundance gradients. In fact, the inside-out scenario influences the rate at which the gas mass is accumulated by infall at each galactocentric distance and this in turn influences the SFR. For bulges and ellipticals we assume that the SF proceeds like in a burst with very high star formation efficiency, namely: SFR = νσk (1.26) with k = 1.0 for the sake of simplicity; ν = 10 − 20Gyr−1 (see Mat- teucci, 1994; Pipino & Matteucci 2004). For irregular galaxies, on the other hand, we assume that the SFR proceeds more slowly and less efficiently that in the Milky Way disk, in particular we assume the same SF law as for spheroids but with 0.01 ≤ ν(Gyr−1) ≤ 0.1. Among irregular galaxies, a special position is taken by the Blue Compact Galaxies (BCG) namely galaxies which have blue colors as a consequence of the fact that they are forming stars at the present time, have small masses, large amounts of gas and low metallicities. For these galaxies, we assume that they suffered on average from 1 to 7 short bursts, with the SF efficiency mentioned above (see Bradamante et al. 1998 and next Lecture). Finally, dwarf spheroidals are also a special cathegory, characterized by old stars, no gas and low metallicities. For these galaxies we assume that they suffered one long starburst lasting 7-8 Gyr or at maximum a couple of extended SF periods, in agreement with their measued Color- Magnitude diagram. It is worth noting that both ellipticals and dwarf spheroidals should loose most of their gas and therefore one may con- clude that galactic winds should play an important role in their evolu- tion, although ram pressure stripping cannot be excluded as a mecha- nism for gas removal. Also for these galaxies we assume the previous SF law with k = 1 and ν = 0.01− 1.0Gyr−1. Lanfranchi & Matteucci, (2003, 2004) developed more detailed models for dwarf spheroidals by adopting the SF history suggested by the Color-Magnitude diagrams of 24 Chemical Evolution Fig. 1.8. Predicted SFRs in galaxies of different morphological type. Figure from Calura (2004). Note that for the elliptical galaxy the SF stops abruptly as a consequence of the galactic wind. single galaxies and with the same efficiency of SF as above. In Figure 2.8 we show the adopted SFRs in different galaxies and in Figure 2.9 the corresponding predicted Type Ia SN rates. For the irregular galaxy, the predicted Type Ia SN rate refers to a specific galaxy, LMC, with a SFR taken from observations (see Calura et al. 2003) with an early ans a late burts of SF and low SF in between. 1.2.7 Type Ia SN rates in different galaxies Following Matteucci & Recchi (2001) we define the typical timescale for Type Ia SN enrichment as the time when the SN rate reaches the maximum. In the following we will always adopt the SDS for the pro- genitors of Type Ia SNe. A point that is not often understood is that this timescale depends upon the progenitor lifetimes, IMF and SFR and therefore is not universal. Sometimes in the literature the typical Type Ia SN timescale is quoted as being universal and equal to 1 Gyr, whereas this is just the timescale at which the Type Ia SNe start to be important in the process of Fe enrichment in the solar vicinity. Matteucci & Recchi (2001) showed that for an elliptical galaxy or a bulge of spiral with a high SFR the timescale for Type Ia SN enrichment 1.2 Lecture II: the Milky Way and other spirals 25 Fig. 1.9. Predicted Type Ia SN rates for the SFRs of Figure 2.8. Figure from Calura (2004). Note that for the irregular galaxy here the predictions are for the LMC, where a recent SF burst is assumed. is quite short, in particular tSNIa = 0.3 − 0.5 Gyr. For a spiral like the Milky Way, in the two-infall model, a first peak is reached at 1.0- 1.5 Gyr (the time at which SNeIa become important as Fe producers (Matteucci and Greggio 1986) while a second less important peak occurs at tSNIa = 4 − 5 Gyr. For an irregular galaxy with a continuous but very low SFR the timescale is tSNIa > 5 Gyr. 1.2.8 Time-delay model for different galaxies As we have already seen, the time-delay between the production of oxy- gen by Type II SNe and that of Fe by Type Ia SNe allows us to explain the [X/Fe] vs. [Fe/H] relations in an elegant way. However, the [X/Fe] vs. [Fe/H] plots depend not only on nucleosynthesis and IMF but also on other model assumptions, such as the SFR, through the absolute Fe abundance ([Fe/H]). Therefore, we should expect a different behaviour in galaxies with different SF histories. In Figure 2.10 we show the pre- dictions of the time-delay model for a spheroid like the Bulge, for the solar vicinity and for a typical irregular magellanic galaxy. As one can see in this Figure, we predict a long plateau, well above the solar value, for the [α/Fe] ratios in the Bulge (and ellipticals), owing 26 Chemical Evolution LMC (Hill et al. 2000) DLA (Vladilo 2002) Fig. 1.10. Predicted [α/Fe] ratios in galaxies with different SF histories. The top line represents the predictions for the Bulge or for an elliptical galaxy of the same mass (∼ 1010M⊙), the median line represents the prediction for the solar vicinity and the lower line the prediction for an irregular magellanic galaxy. The differences among the various models are in the efficiency of star formation, being quite high for spheroids (ν = 20Gyr−1), moderate for the Milky Way (ν = 1 − 2Gyr−1) and low for irregular galaxies (ν = 0.1Gyr−1). The nucleosynthesis prescriptions are the same in all objects. The time-delay between the production of α-elements and Fe, coupled with the different SF histories produces the differences in the plots. Data for Damped-Lyman-α systems, LMC and Bulge are shown for comparison. to the fast Fe enrichment reached in these systems by means of Type II SNe: when the Type Ia SNe start enriching substantially the ISM, at 0.3-0.5 Gyr, the gas Fe abundance is already solar. The opposite occurs in Irregulars where the Fe enrichment proceeds very slowly so that when Type Ia SNe start restoring the Fe in a substantial way (> 3 Gyr) the Fe in the gas is still well below solar. Therefore, here we observe a steeper 1.3 Lecture III: interpretation of abundances in dwarf irregulars 27 slope for the [α/Fe] ratio. In other words, we have below solar [α/Fe] ratios at below solar [Fe/H] ratios. This diagram is very important since it allows us to recognize a galaxy type only by means of its abundances, and therefore it can be used to understand the nature of high redshift objects. 1.3 Lecture III: interpretation of abundances in dwarf irregulars They are rather simple objects with low metallicity and large gas con- tent, suggesting that they are either young or have undergone discon- tinuous star formation activity (bursts) or a continuous but not efficient star formation. They are very interesting objects for studying galaxy evolution. In fact, in ”bottom-up” cosmological scenarios they should be the first self- gravitating systems to form and they could also be important contributors to the population of systems giving rise to QSO- absorption lines at high redshift (see Matteucci et al. 1997 and Calura et al. 2002). 1.3.1 Properties of Dwarf Irregular Galaxies Among local star forming galaxies, sometimes referred to as HII galax- ies, most are dwarfs. Dwarf irregular galaxies can be divided into two categories: Dwarf Irregular (DIG) and Blue Compact galaxies (BCG). These latter have very blue colors due to active star formation at the present time. Chemical abundances in these galaxies are derived from optical emis- sion lines in HII regions. Both DIG and BCG show a distinctive spread in their chemical properties, altough this spread is decreasing with the new more accurate data, but also a definite mass-metallicity relation. From the point of view of chemical evolution, Matteucci and Chiosi (1983) first studied the evolution of DIG and BCG by means of ana- lytical chemical evolution models including either outflow or infall and concluded that: closed-box models cannot account for the Z-log G(G = Mgas/Mtot) distribution even if the number of bursts varies from galaxy to galaxy and suggested possible solutions to explain the observed spread. In other words, the data show a range of values of the metallicity for a given G ratio, and this means that the effective yield is lower than that of the Simple Model and vary from galaxy to galaxy. The possible solutions suggested to lower the effective yield were: 28 Chemical Evolution • a. different IMF’s • b. different amounts of galactic wind • c. different amounts of infall In Figure 3.1 we show graphically the solutions a), b) and c). Concerning the solution a), one simply varies the IMF, whereas solutions b) and c) have been already descibed (eqs. 1.21 ans 1.23). Later on, Pilyugin (1993) forwarded the idea that the spread observed also in other chemical properties properties of these galaxies such as in the He/H vs. O/H and N/O vs. O/H relations, can be due to self-pollution of the HII regions, which do not mix efficiently with the surrounding medium, coupled with “enriched” or “differential” galac- tic winds, namely different chemical elements are lost at different rates. Other models (Marconi et al. 1994; Bradamante et al. 1998) followed the suggestions of differential winds and introduced the novelty of the contribution to the chemical enrichment and energetics of the ISM by SNe of different type (II, Ia and Ib). Another important feature of these galaxies is the mass-metallicity relation. The existence of a luminosity-metallicity relation in irregulars and BCG was suggested first by Lequeux et al. (1979), then confirmed by Skillman et al. (1989) and extended also to spirals by Garnett & Shields (1987). In particular, Lequeux et al. suggested the relation: MT = (8.5± 0.4) + (190± 60)Z (1.27) with Z being the global metal content. Recently, Tremonti et al. (2004) analyzed 53000 local star-forming galaxies in the SDSS (irregulars and spirals). Metallicity was measured from the optical nebular emission lines. Masses were derived from fitting spectral energy distribution (SED) models. The strong optical nebular lines of elements other than H are produced by collisionally excited transitions. Metallicity was then determined by fitting simultaneously the most prominent emission lines ([OIII], Hβ , [OII], Hα, [NII], [SII]). Tremonti et al. (2004) derived a re- lation indicating that 12+log(O/H) is increasing steeply from M∗ going from 108.5 to 1010.5 but flattening for M∗ > 10 10.5. In particular, the Tremonti et al. relation is: 12 + log(O/H) = −1.492 + 1.847(logM∗)− 0.08026(logM∗) 2. (1.28) This relation extends to higher masses the mass-metallicity relation 1.3 Lecture III: interpretation of abundances in dwarf irregulars 29 Fig. 1.11. The Z-logG diagram.Solutions a), b) and c) from top to bottom, to lower the effective yield in DIG and BCG by Matteucci & Chiosi (1983). Solution a) consists in varying the yield per stellar generation, here indicated by pZ , just by changing the IMF. The solution b) and c) correspond to eqs. (1.21) and (1.23), respectively. 30 Chemical Evolution Fig. 1.12. Figure 3 from Erb et al. (2006) showing the mass-metallicity rela- tion for star forming galaxies at high redshift. The data from Tremonti et al. (2004) are also shown. found for star forming dwarfs and contains very important information on the physics governing galactic evolution. Even more recently, Erb et al. (2006) found the same mass-metallicity relation for star-forming galaxies at redshift z>2, with an offset from the local relation of ∼ 0.3 dex. They used Hα and [NII] spectra. In Figure 3.2 we show the figure from Erb et al. (2006) for the mass-metallicity relation at high redshift which includes the relation of Tremonti et al. (2004) for the local mass- metallicity relation. The most simple interpretation of the mass-metallicity relation is that the effective yield increases with galactic mass. This can be achieved in several ways, as shown in Fig. 3.1.: either by changing the IMF or the stellar yields as a function of galactic mass, or by assuming that the 1.3 Lecture III: interpretation of abundances in dwarf irregulars 31 galactic wind is less efficient in more massive systems, or that the infall rate is less efficient in more massive systems. One of the most common interpretations of the mass-metallicity relation is that the effective yield changes because of the occurrence of galactic winds, which should be more important in small systems. Evidences for galactic winds exist for dwarf irregular galaxies, as we will see next. 1.3.2 Galactic Winds Papaderos et al. (1994) estimated a galactic wind flowing at a velocity of 1320 Km/sec for the irregular dwarf VIIZw403. The escape velocity es- timated for this galaxy is ≃ 50 Km/sec. Lequeux et al. (1995) suggested a galactic wind in Haro2=MKn33 flowing at a velocity of ≃ 200Km/sec, also larger that the escape velocity of this object. More recently, Martin (1996;1998) found also supershells in 12 dwarfs, including IZw18, which imply gas outflow. Martin (1999) concluded that the galactic wind rates are several times the SFR. Finally, the presence of metals in the ICM (revealed by X-ray observations) and in the IGM (Ellison et al. 2000) represents a clear indication of the fact that galaxies lose their metals. However, we cannot exclude that the gas with metals is lost also by ram pressure stripping, especially in galaxy clusters. In models of chemical evolution of dwarf irregulars (e.g. Bradamante et al. 1998) the feedback effects are taken into account and the condition for the development of a wind is: (Eth)ISM ≥ EBgas (1.29) namely, that the thermal energy of the gas is larger or equal to its binding energy. The thermal energy of gas due to SN and stellar wind heating (Eth)ISM = EthSN + Ethw (1.30) with the contribution of SNe being: EthSN = ǫSNRSN (t ‘)dt‘, (1.31) while the contribution of stellar winds is: Ethw = ∫ 100 ϕ(m)ψ(t‘)ǫwdmdt ‘ (1.32) with ǫSN = ηSN ǫo and ǫo = 10 51erg (typical SN energy), and ǫw = 32 Chemical Evolution ηwEw with Ew = 10 49erg (typical energy injected by a 20M⊙ star taken as representative). ηw and ηSN are two free parameters and indicate the efficiency of energy transfer from stellar winds and SNe into the ISM, respectively, quantities still largely unknown. The total mass of the galaxy is expressed as Mtot(t) = M∗(t) +Mgas(t) +Mdark(t) with ML(t) =M∗(t) +Mgas(t) and the binding energy of gas is: EBgas(t) =WL(t) +WLD(t) (1.33) with: WL(t) = −0.5G Mgas(t)ML(t) (1.34) which is the potential well due to the luminous matter and with: WLD(t) = −GwLD Mgas(t)Mdark (1.35) which represents the potential well due to the interaction between dark and luminous matter, where wLD ∼ S(1 + 1.37S), with S = rL/rD, being the ratio between the galaxy effective radius and the radius of the dark matter core. The typical model for a BCG has a luminous mass of 108 − 109M⊙, a dark matter halo ten times larger than the luminous mass and various values for the parameter S. The galactic wind in these galaxies develops easily but it carries out mainly metals so that the total mass lost in the wind is small. 1.3.3 Results on DIG and BCG from purely chemical models Purely chemical models (Bradamante et al. 1998, Marconi et al. 1994) for DIG and BCG have been computed in the last years by varying the number of bursts, the time of occurrence of bursts tburst, the star forma- tion efficiency, the type of galactic wind (differential or normal), the IMF and the nucleosynthesis prescriptions. The best model of Bradamante et al. (1998) suggests that the number of bursts should be Nbursts ≤ 10, the SF efficiency should vary from 0.1 to 0.7 Gyr−1 for either Salpeter or Scalo (1986) IMF (Salpeter IMF is favored). Metal enriched winds are favored. The results of these models also suggest that SNe of Type II dominate the chemical evolution and energetics of these galaxies, whereas stellar winds are negligible. The predicted [O/Fe] ratios tend to be overabundant relative to the solar ratios, owing to the predominance of Type II SNe during the bursts, in agreement with observational data 1.3 Lecture III: interpretation of abundances in dwarf irregulars 33 (see Figure 3.5 upper panel). Models with strong differential winds and Nburst=10 - 15 can however give rise to negative [O/Fe] ratios. The main difference between DIGs and BCGs, in these models, is that the BCGs suffer a present time burst, whereas the DIGs are in a quiescent phase. In Figure 3.3 we show some of the results of Bradamante et al. (1998) compared with data on BCGs: it is evident from the Figure that the spread in the chemical properties can be simply reproduced by different SF efficiencies, which translate into different wind efficiencies. In Fig 3.4 we show the results of the chemical evolution models of Henry et al. (2000). These models take into account exponential infall but not outflow. They suggested that the SF efficiency in extragalactic HII regions must have been low and that this effect coupled with the primary N production from intermediate mass stars can explain the plateau in log(N/O) observed at low 12+log(O/H). Henry et al. (2000) also concluded that 12C is mainly produced in massive stars (yields by Maeder 1992) whereas 14N is mainly produced in intermediate mass stars (yields by HG97). This conclusion, however, should be tested also on the abundances of stars in the Milky Way, where the flat behaviour of [C/Fe] vs. [Fe/H] from [Fe/H] =-2.2 up to [Fe/H]=0 suggest a similar origin for the two elements, namely partly from massive stars and mainly from low and intermediate mass ones (Chiappini et al. 2003b). Concerning the [O/Fe] ratios we show results from Thuan et al. (1995) in Figure 3.5, where it is evident that generally BCGs have overabundant [O/Fe] ratios. Very recently, an extensive study from SDSS of chemical abundances from emission lines in a sample of 310 metal poor emission line galaxies appeared (Izotov et al. 2006). The global metallicity in these galax- ies ranges from ∼ 7.1(Z⊙/30) to ∼ 8.5(0.7Z⊙). The SDSS sample is merged with 109 BCGs containing extremely low metallicity objects. These data, shown in Figure 3.5 lower panel, substantially confirm pre- vious ones, showing how α-elements do not depend on the O abundance suggesting a common origin for these elements in stars withM > 10M⊙, except for a slight increase of Ne/O with metallicity which is inter- preted as due to a moderate dust depletion of O in metal rich galaxies. An important finding is that all the studied galaxies are found to have log(N/O) > −1.6, which indicates that none of these galaxies is a truly young object, unlike the DLA systems at high redshift which show a log(N/O) ∼ −2.3. 34 Chemical Evolution Fig. 1.13. Upper panel : predicted Log(N/O) vs. 12 + log(O/H) for a model with 3 bursts of SF separated by quiescent periods and different SF efficien- cies here indicated with γ = ν. Lower panel: predicted log(C/O) vs. 12 + log(O/H). The data in both panels are from Kobulnicky and Skillman (1996). The models assume a dark matter halo ten times larger than the luminous mass and S=0.3 ( Bradamante et al. 1998, see text). 1.3.4 Results from Chemo-Dynamical models: IZw18 IZw18 is the most metal poor local galaxy, thus resembling to a pri- mordial object. Probably it did not experience more than two bursts of star formation including the present one. The age of the oldest stars 1.3 Lecture III: interpretation of abundances in dwarf irregulars 35 Fig. 1.14. Figure from Henry et al. (2000): a comparison between numerical models and data for extragalactic HII regions and stars (filled circles, filled boxes and filled diamonds); M and S mark the position of the Galactic HII re- gions and the Sun, respectively. Their best model is model B with an efficiency of SF of ν = 0.03. in this galaxy is still uncertain, although recently Tosi et al. (2006) suggested an age possibly > 2 Gyr. The oxygen abundance in IZW18 is 12+log(O/H)= 7.17-7.26, ∼ 15-20 times lower than the solar oxygen (12+ log(O/H)= 8.39, Asplund et al. 2005) and log N/O= -1.54/ -1.60 (Garnett et al. 1997). Recently, FUSE provided abundances also for HI in IZw18: the evi- dence is that the abundances in the HI are lower than in the HII (Aloisi et al. 2003; Lecavelier des Etangs et al. 2003). In particular, Aloisi et al. (2003) found the largest difference relative to the HII data. Chemo-dynamical (2-D) models (Recchi et al. 2001) studied first the case of IZw18 with only one burst at the present time and concluded that the starburst triggers a galactic outflow. In particular, the metals leave the galaxy more easily than the unprocessed gas and among the enriched material the SN Ia ejecta leave the galaxy more easily than other ejecta. In fact, Recchi et al. (2001) had reasonably assumed that Type Ia SNe can transfer almost all of their energy to the gas, since 36 Chemical Evolution Fig. 1.15. Upper panel: [O/Fe] vs. [Fe/H] observed in a sample of BCGs by Thuan et al. (1995) (filled circles), open triangles and asterisks are disk and halo stars shown for comparison.Figure from Thuan et al. (1995). Lower panel: new data from Izotov et al. (2006). The large filled circles represent the BCGs whereas the dots are the SDSS galaxies. Abundances in the left panel are calculated as in Thuan et al. (1995) whereas those in the right panel are calculated as in Izotov et al. (2006) (see original papers for details). Figure from Izotov et al. (2006). 1.3 Lecture III: interpretation of abundances in dwarf irregulars 37 Fig. 1.16. Figure from Recchi et al. (2004): predicted abundances for the HII region in IZw18 (dashed lines represent a model adopting the yields of Meynet & Maeder (2002) for Z = 10−5, whereas the continuous line refers to a higher metallicity (Z=0.004).Observational data are represented by the shaded areas. they explode in an already hot and rarified medium after the SN II explosions. As a consequence of this, they predicted that the [α/Fe] ratios in the gas inside the galaxy should be larger than the [α/Fe] ratios in the gas outside the galaxy. At variance with previous studies, they found that most of the metals are already in the cold gas phase after 8-10 Myr since the superbubble does not break immediately and thermal conduction can act efficiently. In the following, Recchi et al. (2004) extended the model to a two-burst case, always with the aim of reproducing the characteristics of IZw18. The model well reproduces the chemical properties of IZw18 with a relatively long episode of SF lasting 270 Myr plus a recent burst of SF still going on. In Figure 3.6 we show the predictions of Recchi et al. (2004) for the abundances in the HII regions of IZW18 and in Figure 3.7 those for the HI region, showing a little difference between the HII and HI abundances, more in agreement with the data of Lecavelier des Etangs et al. (2004). 38 Chemical Evolution Fig. 1.17. Figure from Recchi et al. (2004): predicted abundances for the HI region. The models are the same as in Figure 3.6. Observational data are represented by the shaded areas. The upper shaded area in the panel for oxygen and the lower shaded area in the panel for N/O represent the data of Lecavelier des Etangs et al. (2003). 1.4 Lecture IV: Elliptical galaxies-Quasars- ICM Enrichment 1.4.1 Ellipticals We recall here some of the most important properties of ellipticals or early type galaxies (ETG) which are systems made of old stars with no gas and no ongoing SF. The metallicity of ellipticals is measured only by means of metallicity indeces obtained from their integrated spectra which are very similar to those of K giants. In order to pass from metal- licity indices to [Fe/H] one needs then to adopt a suitable calibration often based on population synthesis models (Worthey, 1994). We also summarize the most common scenarios for the formation of ellipticals. 1.4.2 Chemical Properties The main properties of the stellar populations in ellipticals are: • There exist the well-known Color-Magnitude and Color - σo (veloc- 1.4 Lecture IV: Elliptical galaxies-Quasars- ICM Enrichment 39 ity dispersion) relations indicating that the integrated colors become redder with increasing luminosity and mass (Faber 1977; Bower et al. 1992). These relations are interpreted as a metallicity effect, although a well known degeneracy exists between metallicity and age of the stellar populations in the integrated colors (Worthey 1994). • The indexMg2 is normally used as a metallicity indicator since it does not depend much upon the age of stellar populations. There exists for ellipticals a well defined Mg2–σo relation, equivalent to the already discussed mass-metallicity relation for star forming galaxies (Bender et al. 1993; Bernardi et al. 1998; Colless et al. 1999). • Abundance gradients in the stellar populations inside ellipticals are found (Carollo et al. 1993; Davies et al. 1993). Kobayashi & Arimoto (1999) derived the average gradient for ETGs from a large compilation of data and this is: ∆[Fe/H ]/∆r ∼ −0.3, with the average metallicity in ETGs of < [Fe/H ] >∗∼ −0.3dex (from -0.8 to +0.3 dex). • A very important characteristic of ellipticals is that their central dom- inant stellar population (dominant in the visual light) shows an over- abundance, relative to the Sun, of the Mg/Fe ratio, < [Mg/Fe] >∗> 0 (from 0.05 to + 0.3 dex) (Peletier 1989; Worthey et al. 1992; Weiss et al. 1995; Kuntschner et al. 2001). • In addition, the overabundance increases with increasing galactic mass and luminosity, < [Mg/Fe] >∗ vs. σo, (Worthey et al. 1992; Mat- teucci 1994; Jorgensen 1999; Kuntschner et al. 2001). 1.4.3 Scenarios for galaxy formation The most common ideas on the formation and evolution of ellipticals can be summarized as: • they formed by an early monolithic collapse of a gas cloud or early merging of lumps of gas where dissipation plays a fundamental role (Larson 1974; Arimoto & Yoshii 1987; Matteucci & Tornambè 1987). In this model SF proceeds very intensively until a galactic wind is developed and SF stops after that. The galactic wind is devoiding the galaxy from all its residual gas. • They formed by means of intense bursts of star formation in merging subsystems made of gas (Tinsley & Larson 1979). In this picture SF stops after the last burst and gas is lost via ram pressure stripping or galactic wind. 40 Chemical Evolution Fig. 1.18. The relation [α/Fe] vs. velocity dispersion (mass) for ETGs. Figure adapted from Thomas et al. (2002).The continuous line represents the predic- tion of the model by Pipino & Matteucci (2004). The shaded area represents the prediction of hierarchical models for the formation of ellipticals.The sym- bols are the observational data. • They formed by early merging of lumps containing gas and stars in which some dissipation is present (Bender et al. 1993). • They formed and continue to form in a wide redshift range and prefer- entially at late epochs by merging of early formed stellar (e.g. Kauff- mann et al. 1993;1996). Pipino & Matteucci (2004), by means of recent revised monolithic models taking into account the development of a galactic wind (see Lec- ture III), computed the relation [Mg/Fe] versus mass (velocity disper- sion) and compared it with the data by Thomas et al. (2002). Thomas (1990) already showed how hierarchical semi-analitycal models cannot 1.4 Lecture IV: Elliptical galaxies-Quasars- ICM Enrichment 41 reproduce the observed [Mg/Fe] vs. mass trend, since in this scenario massive ellipticals have longer periods of star formation than smaller ones. In Figure 4.1, the original figure from Thomas et al. (2002) is shown, where we have plotted also our predictions. In the Pipino & Matteucci (2004) model it is assumed that the most massive galaxies as- semble faster and form stars faster than less massive ones. The adopted IMF is the Salpeter one. In other words, more massive ellipticals seem to be older than less massive ones, in agreement with what found for spirals (Boissier et al. 2001). In particular, in order to explain the observed < [Mg/Fe] >∗> 0 in giant ellipticals the dominant stellar population should have formed on a time scale no longer than 3-5 ·108 yr (Weiss et al. 1995; Pipino & Matteucci 2004). 1.4.4 Ellipticals-Quasars connection We know now that most if not all massive ETGs are hosting an AGN for sometime during their life. Therefore, there is a strict link between the quasar activity and the evolution of ellipticals. 1.4.5 The chemical evolution of QSOs It is very interesting to study the chemical evolution of QSOs by means of the broad emission lines in the QSO region. The first studies by Wills et al. (1985) and Collin-Souffrin et al. (1986) found that the abundance of Fe in QSOs, as measured from broad emission lines, turned out to be ∼ a factor of 10 more than the solar one and this represented a challenge for chemical evolution model makers. Hamman & Ferland (1992) from N V/C IV line ratios in QSOs derived the N/C abundance ratios and inferred the QSO metallicities. They suggested that N is overabundant by factors of 2-9 in the high redshift sources (z > 2). Metallicities 3-14 times the solar one were also suggested in order to produce such a high N abundance, under the assumption of a mainly secondary N. To inter- pret their data they built a chemical evolution model, a Milky Way- like model, and suggested that these high metallicities are reached in only 0.5 Gyr, implying that QSOs are associated with vigourous star forma- tion. At the same time, Padovani & Matteucci (1993) and Matteucci & Padovani (1993) proposed a model for QSOs in which QSOs are hosted by massive ellipticals. They assumed that after the occurrence of a galac- tic wind the galaxy evolves passively and that for massess > 1011M⊙ the gas restored by the dying stars is not lost but it feeds the central 42 Chemical Evolution black hole. They showed that in this context the stellar mass loss rate can explain the observed AGN luminosities. They also found that solar abundances in the gas are reached in no more than 108 years explaining in a natural way the standard emission lines observed in high-z QSOs. The predicted abundances could explain the data available at that time and solve the problem of the quasi-similarity of QSO spectra at differ- ent redshifts. Finally, they suggested also a criterium for establishing the ages of QSOs on the basis of the [α/Fe] ratios observed from broad emission lines (see also Hamman & Ferland 1993). Much more recently, Maiolino et al. (2005, 2006) used more than 5000 QSO spectra from SDSS data to investigate the metallicity of the broad emission line region in the redshift range 2 < z < 4.5 and over the luminosity range −24.5 < MB < −29.5. They found substantial chemical enrichment in QSOs already at z = 6. Models for ellipticals by Pipino & Matteucci (2004) were used as a comparison with the data and they well reproduce the data, as one can see in Figure 4.2. In this Figure the evolution of the abundances of several chemical elements in the gas of a typical elliptical are shown. The elliptical suffers a galactic wind at around 0.4 Gyr since the beginning of star formation. This wind devoids the galaxy of all the gas present at that time. After this time, the SF stops and the galaxy evolves passively. All the gas restored after the galactic wind event by dying stars can in principle feed the central black hole, thus the abundances shown in Figure 4.2, after the time of the wind, can be compared with the abundances measured in the broad emission line region. As one can see, the predicted Fe abundance after the galactic wind is always higher than the O one, owing to the Type Ia SNe which continue to produce Fe even after the stop in the SF. On the other hand, O and α-elements stop to be produced when the SF halts. The comparison between the predicted abundances and those derived from the QSO spectra, are in very good agreement and indicates ages for these objects between 0.5 and 1 Gyr. Finally, in the context of the joint formation of QSOs and ellipticals we recall the work of Granato et al. (2001) who includes the energy feedback from the central AGN in ellipticals. This feedback produces outflows and stops the SF in a down-sizing fashion, in agreement with the chemical properties of ETGs indicating a shorter period of SF for the more massive objects. 1.4 Lecture IV: Elliptical galaxies-Quasars- ICM Enrichment 43 Fig. 1.19. The temporal evolution of the abundances of several chemical ele- ments in the gas of an elliptical galaxy with luminous mass of 1011M⊙. Feed- back effects are taken into account in the model (Pipino & Matteucci 2004), as described in Lecture III. The downarrow indicates the time for the occurrence of the galactic wind. After this time, the SF stops and the elliptical evolves passively. All the abundances after the time for the occurrence of the wind are those that we observe in the broad emission line region region. The shaded area indicates the abundance sets which best fit the line ratios observed in the QSO spectra. Figure from Maiolino et al. 2006. 1.4.6 The chemical enrichment of the ICM The X-ray emission from galay clusters is generally interpreted as ther- mal bremsstrahlung in a hot gas (107-108 K). There are several emission lines (O, Mg, Si, S) including the strong Fe K-line at around 7keV which was discovered by Mitchell et al. (1976). The iron is the best studied element in clusters. For kT ≥ 3 keV the intracluster medium (ICM) Fe abundance is constant and ∼ 0.3Fe⊙ in the central cluster regions; 44 Chemical Evolution the existence of metallicity gradients seems evident only in some clusters (see Renzini 2004). At lower temperatures, the situation is not so simple and the Fe abundance seems to increase. The first works on chemical enrichment of the ICM even preceeded the discovery of the Fe line (Gunn & Gott 1972, Larson & Dinerstein 1975). In the following years other works appeared such as those of Vigroux (1977), Himmes & Biermann (1988) and Matteucci & Vettolani (1988). In particular, Matteucci & Vettolani (1988) started a more detailed approach to the problem fol- lowed by David et al. (1991), Arnaud (1992), Renzini et al. (1993), Elbaz et al. (1995), Matteucci & Gibson (1995), Gibson & Matteucci (1997), Lowenstein & Mushotzky (1996), Martinelli et al. (2000), Chiosi (2000), Moretti et al. (2003). The majority of these papers assumed that galactic winds (mainly from ellipticals and S0 galaxies) are responsible for the ICM chemical enrichment. In fact, ETGs are the dominant type of galaxy in clusters and Arnaud (1992) found a clear correlation be- tween the mass of Fe in clusters and the total luminosity of ellipticals. No such correlation was found for spirals in clusters. Alternatively, the abundances in the ICM are due to ram pressure stripping (Himmes & Biermann 1988) or derive from a chemical enrichment from pre-galactic Pop III stars (White & Rees 1978). In Matteucci & Vettolani (1988) the Fe abundance in the ICM rel- ative to the Sun, XFe/XFe⊙ , was calculated as (MFe)pred/(Mgas)obs to be compared with the observed ratio (XFe/XFe⊙)obs = 0.3 − 0.5 (Rothenflug & Arnaud 1985). They found a good agreement with the observed Fe abundance in clusters if all the Fe produced by ellipticals and S0, after SF has stopped, is eventually restored into the ICM and if the majority of gas in clusters has a primordial origin. Low values for [Mg/Fe] and [Si/Fe] were predicted at the present time, due to the short period of SF in ETGs and to the Fe produced by Type Ia SNe. With Salpeter IMF they found that the Type Ia SNe contribute ≥ 50% of the total Fe in clusters. This leads to a bimodality in the [α/Fe] ratios in the stars and in the gas in the ICM, since the stars have overabundances of [α/Fe]> 0 whereas the ICM should have [α/Fe]≤ 0. The same con- clusion was reached and more highlighted later by Renzini et al. (1993). More recently, Pipino et al. (2002) computed the chemical enrichment of the ICM as a function of redshift by considering the evolution of the cluster luminosity function and an updated treatment of the SN feed- back. They adopted Woosley & Weaver (1995) yields for Type II SNe and Nomoto et al. (1997) W7 model for Type Ia SNe and a Salpeter IMF. They also predicted solar or undersolar [α/Fe] ratios in the ICM. 1.4 Lecture IV: Elliptical galaxies-Quasars- ICM Enrichment 45 Fig. 1.20. Observed Fe abundance and predicted Fe abundance in the ICM as a function of redshift: data from Tozzi et al. (2003), model (continuous line) from Pipino et al. (2002), where the formation of ETGs was assumed to occur at z=8. The observational data on abundance ratios in clusters are still uncertain and vary from cluster centers where they tend to be solar or undersolar to the outer regions where they tend to be oversolar (e.g. Tamura et al. 2004). So, no firm conclusions can be drawn on this point. Concerning the evolution of the Fe abundance in the ICM as a function of redshift, most of the above mentioned models predict very little or no evolution of the Fe abundance from z=1 to z=0 (Pipino et al. 2002). This prediction seemed to be in good agreement with data from Tozzi et al. (2003) as shown Figure. However, more recently, more data of Fe abundance for high redshift clusters appeared showing a different behaviour. In Figure 4.4 we show the data of Balestra et al. (2006) who claim an increase, by at least a factor of two, of the Fe abundance in the ICM from z=1 to z=0. Clearly, if we assume that only ellipticals have contributed to the Fe abundance in the ICM, this effect is difficult to explain unless we assume recent star formation in ellipticals. Another possible explanation could be that spiral galaxies contribute to Fe when 46 Chemical Evolution Fig. 1.21. New data (always relative to Fe) from Balestra et al. (2006) showing an increase of the Fe abundance in the ICM from z=1 to z=0. Error bars refer to 1σ confidence level. The big shaded area represents the rms dispersion. Figure from Balestra et al. (2006). they become S0 as a consequence of ram pressure stripping, and this morphological transformation might have started just at z=1. 1.4.7 Conclusions on the enrichment of the ICM From what said before we can conclude that: • Elliptical galaxies are the dominant contributors to the abundances and energetic content of the ICM. A constant Fe abundance of ∼ 0.3Fe⊙ is found in the central regions of clusters hotter than 3keV (Renzini 2004). • Good models for the chemical enrichment of the ICM should repro- duce the iron mass measured in clusters plus the [α/Fe] ratios in- side galaxies and in the ICM as well as the Fe mass to light ratio (IMLR= MFeICM /LB, with LB being the total blue luminosity of member galaxies, as defined by Renzini et al. (1993). Abundance 1.4 Lecture IV: Elliptical galaxies-Quasars- ICM Enrichment 47 ratios are very powerful tools to impose constraints on the evolution of ellipticals and of the ICM. • Models which do not assume a top-heavy IMF for the galaxies in clusters (a Salpeter IMF can reproduce at best the properties of local ellipticals) predict [α/Fe]> 0 inside ellipticals and [α/Fe] ≤ 0 in the ICM. Observed values are still too uncertain to draw firm conclusions on this point. Acknowledgements This research has been supported by INAF (Italian National Institute for Astrophysics), Project PRIN-INAF-2005-1.06.08.16 References [1] Alibés, A., Labay, J. & Canal, R., 2001, A&A, 370, 1103 [2] Aloisi, A., Savaglio, S., Heckman, T. M., Hoopes, C. G., Leitherer, C. & Sembach, K. R., 2003, ApJ, 595, 760 [3] Argast, D., Samland, M., Gerhard, O.E. & Thielemann, F.-K., 2000, A&A 356, 873 [4] Arimoto, N. & Yoshii, Y. 1987, A&A 173, 23 [5] Arnaud, M., Rothenflug, R., Boulade, O.,Vigroux, L. & Vangioni-Flam, E., 1992, A&A, 254, 49 [6] Asplund, M., Grevesse, N. & Sauval, A.J., 2005, ASP (Astronomical Society of the Pacific) Conf. Series, Vol. 336, p.55 [7] Balestra, I., Tozzi, P., Ettori, S., Rosati, P., Borgani, S., Mainieri, V., Norman, C. & Viola, M., 2006, A&A in press, astro-ph/0609664 [8] Barbuy, B. & Grenon, M., 1990. in :Bulges of Galaxies, eds. B.J. Jarvis & D.M. 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0704.0771

Suppression of 1/f noise in one-qubit systems

Suppression of 1/fα noise in one-qubit systems Pekko Kuopanportti, Mikko Möttönen, Ville Bergholm, Olli-Pentti Saira, Jun Zhang, and K. Birgitta Whaley Laboratory of Physi s, Helsinki University of Te hnology P. O. Box 4100, 02015 TKK, Finland Low Temperature Laboratory, Helsinki University of Te hnology, P.O. Box 3500, 02015 TKK, Finland Department of Chemistry and Pitzer Center for Theoreti al Chemistry, University of California, Berkeley, CA 94720 (Dated: O tober 26, 2018) We investigate the generation of quantum operations for one-qubit systems under lassi al noise with 1/fα power spe trum, where 2 > α > 0. We present an e� ient way to approximate the noise with a dis rete multi-state Markovian �u tuator. With this method, the average temporal evolution of the qubit density matrix under 1/fα noise an be feasibly determined from re ently derived deterministi master equations. We obtain qubit operations su h as quantum memory and the NOT gate to high �delity by a gradient based optimization algorithm. For the NOT gate, the omputed �delities are qualitatively similar to those obtained earlier for random telegraph noise. In the ase of quantum memory however, we observe a nonmonotoni dependen y of the �delity on the operation time, yielding a natural a ess rate of the memory. I. INTRODUCTION In solid-state realization of qubits, material spe i� �u tuations typi ally indu e the major ontribution to the intrinsi noise. Mu h e�ort has been fo used on the preservation of the state in a quantum memory in the presen e of 1/fα noise sin e this is a ubiquitous form of noise en ountered in solid-state qubit appli a- tions [1, 2, 3℄. Both harge and spin qubits are sus- eptible to noise of this form. For Josephson jun tions, both harge noise [4, 5℄ and riti al urrent noise [6, 7℄ have been measured to have 1/fα power spe tral densi- ties. Similar harge �u tuations are responsible for the well-known 1/fα nature of low frequen y noise in sin- gle ele tron transistors [8℄. Ba kground harge �u tua- tions resulting in 1/fα noise spe tra are onsidered to be the most important sour e of dephasing in Joseph- son jun tion qubits [4, 5, 9℄. Spin qubits su h as those formed from donor spins in semi ondu tors are sus epti- ble to nu lear spin noise deriving from dipolar oupling between environmental nu lear spins. The nu lear spin bath ouples to the donor spins by hyper�ne intera - tions, whi h renders the dynami s of the nu lear spins to ause dephasing. Re ent al ulations for a phospho- rus donor in sili on show that the high frequen y om- ponent of the nu lear spin noise is approximately de- s ribed by a 1/fα power spe trum [10℄. Ele tron spin qubits implanted into sili on [11℄ are also a�e ted by relaxation of dangling bonds deriving from oxygen va- an ies at the Si/SiO2 interfa e. This gives rise to a magneti noise with a 1/fα spe trum that is the dom- inant me hanism for phase �u tuations of donor spins near the surfa e [12℄. Another form of noise losely re- lated to 1/fα noise is random telegraph noise (RTN), whi h arises from oupling of individual bistable �u tu- Ele troni address: pekko.kuopanportti�tkk.� ators to a qubit [2, 13, 14, 15, 16, 17, 18, 19℄. Several approa hes to suppress de oheren e based on pulse design have been proposed in the literature. Among them, dynami al de oupling s hemes average out the un- wanted e�e ts of the environmental intera tion through the appli ation of suitable ontrol pulses [20, 21℄. Appli- ation of these s hemes often involves hard pulses with instantaneous swit hings and unbounded ontrol ampli- tudes, resulting in a range of validity restri ted to time s ales for whi h the pulse duration is mu h less than the noise orrelation time [22, 23℄. In Ref. [24℄, a dire t pulse optimization method restri ted to bounded ontrol pulses was developed for implementing one-qubit operations in a noisy environment. This initial work on noise suppression addressed the example of a single qubit system under the in�uen e of lassi ally modeled random telegraph noise, su h as might arise from a single bistable �u tuator. In this paper, we extend the work of Ref. [24℄ to the physi ally relevant situation of 1/fα noise where 2 > α > 0. This kind of noise is known to result, for ex- ample, from a set of bistable �u tuators [25, 26, 27, 28℄, i.e., RTN sour es. We investigate two ways to approxi- mate the 1/fα noise for omputer simulations, namely, the sum of independent RTN �u tuators and a single dis rete multi-state Markovian noise sour e. We show that the single �u tuator provides a mu h more e� ient way to model 1/fα noise than independent RTN �u tu- ators. Furthermore, the average temporal evolution of the density matrix under this Markovian noise an be exa tly des ribed by a set of deterministi master equa- tions derived in Ref. [29℄. Using this approa h, we avoid the heavy omputational task arising from the numeri al evaluation of the density matrix averaged over a large number of di�erent sample paths of the noise as om- puted in Ref. [24℄. This framework will not only signif- i antly a elerate the onvergen e of the ontrol pulse sequen e optimization, but also allows further theoreti- al analysis. Using these master equations, we employ gradient based optimization pro edures to obtain pulse sequen es that suppress 1/fα noise for quantum mem- http://arxiv.org/abs/0704.0771v1 mailto:[email protected] ory and for a NOT gate. Comparisons with omposite pulses designed to eliminate systemati errors and with refo using pulses demonstrate that the numeri ally opti- mized pulse sequen es yield the highest �delities. The remainder of this paper is organized as follows. In Se . II, we show how to e� iently approximate the 1/fα noise by a multi-state Markovian �u tuator. In Se . III, we de�ne the �delity of qubit operations, re- view the master equations des ribing the average evolu- tion of the qubit density matrix in the presen e of the noise and des ribe the numerial optimization pro edure. Se tions IV and V present optimized ontrol pulse se- quen es and the a hieved �delities for quantum memory and for the NOT gate, respe tively. Finally, Se . VI on- ludes and indi ates further appli ations of the method. II. ONE-QUBIT SYSTEM SUBJECT TO 1/fα NOISE We onsider a one-qubit system des ribed by the e�e - tive Hamiltonian a(t)σx + η(t)σz , (1) where a(t) ∈ [−amax, amax] is the external ontrol �eld applied along the x dire tion and η(t) is the lassi al noise signal perturbing the system along the z dire tion. The noise sour e η(t) an be hara terized by its auto- orrelation fun tion C(t) ≡ 〈η(0)η(t)〉 = lim ∫ T/2 η(s)η(s+ t) ds, (2) the Fourier transformation of whi h de�nes the noise power spe tral density as S(f) = C(t)e−i2πftdt. (3) For a single RTN sour e with the amplitude ∆ and or- relation time τc, the auto orrelation fun tion is given by [30℄ (t) = ∆2e−2|t|/τc, (4) and the orresponding power spe tral density by (f) = 1 + (πfτc)2 . (5) A standard way to simulate 1/fα noise is to use an ensemble of K independent un orrelated RTN pro- esses [1, 25, 27℄. Let ηk(t) be a symmetri RTN signal swit hing between values −∆k and ∆k with the orre- lation time τk ≡ 1/γk, where γk is the transition rate between the two states. The total noise pro ess appears in the Hamiltonian (1) as η(t) = k=1 ηk(t). Sin e the RTN sour es are independent, Eqs. (2) and (4) yield the auto orrelation fun tion C(t) = −2|t|/τk = −2γk|t|, (6) and the orresponding power spe tral density is given by S(f) = ∆2kγk γ2k + (πf) . (7) Introdu ing the density of transition rates g(γ) and ex- pressing the noise strength ∆ as a fun tion of the tran- sition rate, we an repla e the summation in Eq. (7) by an integration, whi h yields S(f) = ∫ γmax ∆2(γ)g(γ)γ γ2 + (πf)2 dγ, (8) where γmin and γmax are minimal and maximal transition rates, respe tively. Provided that ∆2(γ)g(γ) = 2A/γ, (9) where A is a onstant, the power spe tral density in Eq. (8) be omes [27℄ S(f) = arctan − arctan , γmin ≪ πf ≪ γmax. (10) Thus Eq. (10) yields an approximation to the 1/f power spe trum. To generate a general 1/fα power spe tral density for 2 > α > 0, we an hoose ∆2(γ)g(γ) = 2Aγ−α (11) as shown in [27℄. Although the above method yields a valid approxima- tion for the 1/fα spe trum, it is omputationally ine�- ient. In parti ular, the number of distin t noise states in reases exponentially with the number of RTN �u tua- torsK, i.e., the number of terms in the sum of Eq. (7) ap- proximating the 1/fα noise. Sin e the size of the di�eren- tial equation system des ribing the average qubit dynam- i s in reases linearly with the number of noise states [29℄, in pra ti e one has to restri t the omputation to a rather small number of independent RTN �u tuators. To over ome this problem, we present a on eptually di�erent way of generating the desired 1/fα noise spe - trum using a single multi-state Markovian �u tuator. Consider a ontinuous-time Markovian noise pro ess with M dis rete noise states. Let Γkj denote the transition rate from the jth state to the kth one. In order to pre- serve total probability, we must have Γjk = 0 for all k = 1, 2, . . . ,M. (12) Let us assume that the transition rates are symmetri , i.e., Γ = ΓT . Under this assumption the noise pro ess has a steady-state solution in whi h the di�erent noise states are equally probable. In order for the noise to be unbiased, i.e., 〈η〉 = 0, the amplitudes bk asso iated with the noise states must satisfy bk = 0. (13) Thus the auto orrelation is given by C(t) = 〈η(t)η(0)〉 = bT eΓ|t|b. (14) Sin e Γ is symmetri , we an diagonalize it with an or- thogonal matrix V as Γ = V ΛV T , where the real diagonal matrix Λ = diag{λk}Mk=1 arries the eigenvalues of Γ in a des ending order. De�ning χ := 1√ V T b, we rewrite Eq. (14) in the form of Eq. (6) as C(t) = χT eΛ|t|χ = λk|t|. (15) In order to use this multi-state Markovian �u tuator to approximate 1/fα noise, we have to hoose the eigen- values λk and the amplitudes χk su h that Eq. (11) is ful�lled. Moreover, we must onstru t the orthogonal matrix V su h that Γ = V ΛV T satis�es Eq. (12), the amplitudes bk satisfy Eq. (13), and the o�-diagonal ele- ments of Γ must be non-negative. One way to satisfy these requirements is to pi k an integer m ≥ 2 and set M = 2m and to hoose the eigen- values as {λk}Mk=1 = −2{0, γmin, γmin + δ, γmin + 2δ, . . . , γmax}, where γmax = (M − 2)δ+ γmin and 0 < δ ≤ γmin. Hen e, the distribution of the transition rates g(γ) is uniform on [γmin, γmax]. Then we set V = H , where H is the Hadamard matrix Expli it al ulation shows that these hoi es ensure that Eq. (12) is satis�ed. To ful�ll Eqs. (11) and (13), we set χ1 = 0 and χk = γ k for k = 2, . . . , M , where γk is equal to γmin + (k − 2)δ. It an be shown that this onstru tion will also produ e transition matri es Γ with non-negative o�-diagonal elements. Hen e we have pro- vided an e� ient way to implement 1/fα noise. Note that the M -state Markovian �u tuator, Eq. (15), or- responds formally to Eq. (6) with M − 1 non-vanishing RTN �u tuators. Thus we have a hieved an exponential improvement in the e� ien y of the noise approximation. Alternatively, we an hoose the eigenvalues of Γ freely and obtain a valid matrix V with numeri al optimization, 0 0.2 0.4 0.6 0.8 1 1.2 1.4 [(2πf)/γ0] FIG. 1: Logarithm of the power spe tral density for �ve independent RTN �u tuators (dash-dotted line), a multi-state Markovian sour e orresponding to 31 RTN �u tuators (solid line), and an ideal 1/f noise (dotted line). The transition rates of the RTN �u tuators are in both ases distributed uniformly on the interval [γ0, 30γ0]. whi h may result in even more faithful approximation of 1/fα noise. Figure 1 ompares the approximation of the spe - tral density of 1/f noise generated by independent RTN sour es and by a multi-state Markovian sour e. For the RTN approa h, we hoose 5 independent noise sour es, for whi h the transition rates γk are uniformly distributed in the range [γmin, γmax] = [γ0, 30γ0], and the strengths are given by ∆k = 1/ γk. This yields a �u tuator with 32 distin t noise states. For the multi-state �u - tuator, we hoose a 32-state noise sour e, for whi h the nonzero eigenvalues λk of its transition rate ma- trix Γ are distributed uniformly on [−60γ0,−2γ0], and χk = 1/ −λk/2. Thus the ondition in Eq. (9) is satis- �ed for both of the approa hes and the multi-state noise sour e has an auto orrelation fun tion and power spe - tral density whi h are equal to those for a ertain ensem- ble of 31 RTN �u tuators. We employ representations of similar omputational omplexity here in order to be able to assess the relative a ura y for a given omputational e�ort. Figure 1 shows that an ensemble of �ve RTN pro esses is not an a urate model for 1/f noise, whereas a single 32-state Markovian noise sour e is quite a urate, espe- ially in the range 3γ0 . ω . 16γ0. The poor quality of the approximation with �ve RTN �u tuators is due to the small number of independent noise sour es employed here, whereas the 32-state Markovian �u tuator ontains more parameters and thereby introdu es more �exibility in the noise approximation. The frequen y range over whi h the approximation is a urate is relatively short if one onsiders that the 1/f noise dete ted in experi- mental appli ations often extends over several frequen y de ades. The width of this frequen y range an of ourse be in reased by in reasing the width of the region from whi h the eigenvalues of the matrix Γ are hosen. In this ase, however, the number of dis rete levels in the Markovian sour e must also be in reased to preserve the desired a ura y. For the main purpose of demonstrating the feasibility of the numeri al optimization algorithm, in the rest of this paper we will ontinue to approximate 1/fα noise by a single Markovian noise sour e with 32 levels. III. QUBIT DYNAMICS AND CONTROL In Ref. [24℄, the temporal evolution of the qubit density matrix was al ulated by averaging over 104�105 unitary quantum traje tories, ea h orresponding to a sample noise path. To ensure a ura y, a large number of uni- tary traje tories are required, whi h results in extensive omputational e�ort. In Ref. [29℄, exa t deterministi master equations des ribing the average temporal evo- lution of quantum systems under Markovian noise were derived. Following Ref. [29℄, we introdu e a onditional density operator ρk(t) whi h orresponds to the density operator of the system averaged over all the noise sample paths o upying the kth state at the time instant t. The on- ditional density operators are normalized su h that the tra e of the operator ρk(t) yields the probability of the kth noise state as Pk(t) = Tr [ρk(t)]. The total average density operator an be expressed as ρ(t) = ρk(t). (16) The dynami s of ρk is obtained from the oupled master equations [29℄ ∂tρk(t) = [Hk(t), ρk(t)] + Γkjρj(t), (17) whereHk(t) is the Hamiltonian of the system orrespond- ing to the kth noise state, and Γkj the transition rate from the jth state to the kth state, as de�ned in Se . II. Spe i� ally, in our one-qubit ase, Hk(t) = a(t)σx + bkσz, (18) where bk is the noise amplitude of the state k. We shall use Ea {ρ} to denote the state ρ evolved under the in�u- en e of noise and the ontrol sequen e a. The �delity fun tion quantifying the overlap between the desired state ρf and the a tual a hieved �nal state is de�ned as φ(ρf , Ea {ρ0}) = Tr Ea {ρ0} , (19) where ρ0 is the initial state of the system. To measure how lose the evolution Ea is to the intended quantum gate operation U , we al ulate the average of the �delity φ(Uρ0U †, Ea {ρ0}) over all pure initial states ρ0, and ob- tain the gate �delity fun tion [24℄ Φ(U) = k=x,y,z †Ea {σk} . (20) We aim to �nd the optimal ontrol pulses whi h max- imize the �delity of the a hieved quantum operation, and hen e apply a typi al gradient based optimization algorithm su h as the gradient as ent pulse engineering (GRAPE) method developed in Ref. [31℄. If the ontin- uous pulse pro�les are approximated by pie ewise on- stant fun tions, the gradient of the �delity fun tion with respe t to these onstant pulse values and durations an be al ulated by the hain rule. This gradient is further used as a proportional adjustment to update the ontrol pulse pro�le. The optimization pro edure is terminated when ertain desired a ura y is a hieved. Note that due to the non- onvex nature of the problem, the gra- dient based algorithm will only yield a lo ally optimal solution. We further employ a multitude of initial on- ditions to �nd a ontrol pulse whi h a hieves the highest �delity. IV. QUANTUM MEMORY In this se tion, we fo us on the implementation of quantummemory, i.e., the identity operator. For the pur- pose of omparison with the optimized pulse sequen es, we introdu e four other kinds of ontrol s hemes whi h generate the identity operator. The �rst referen e sequen e is simply not to apply any external ontrol pulse, i.e., a(t) = 0. This pulse has no ompensation for de oheren e or error. The se ond referen e sequen e is a onstant 2π pulse given by a2π(t) = amax, for t ∈ [0, 2π~/amax]. (21) The third referen e sequen e is the omposite pulse se- quen e known as ompensation for o�-resonan e with a pulse sequen e (CORPSE), whi h was originally designed to orre t systemati errors in the implementation of one- qubit quantum operations and to provide high order on- trol proto ols for systemati qubit bias, i.e., for the noise orrelation time τc → ∞ [32, 33℄. For the identity oper- ation, the CORPSE pulse sequen e an be obtained as aSC2π(t) = amax, for 0 < t ′ < π −amax, for π ≤ t′ ≤ 3π amax, for 3π < t ′ < 4π, where the dimensionless time t′ is de�ned as t′ = amaxt/~. In the absen e of noise, the CORPSE sequen e gen- erates the identity operator exa tly although it requires twi e as long operation time as a 2π pulse, the se ond referen e pulse above. In the presen e of small system- ati errors, the CORPSE sequen e is mu h more a urate than the 2π pulse. For example, onsider a state trans- formation from the north pole ba k to itself on the Blo h sphere. For η(t) ≡ ∆ in Eq. (1), the �delities de�ned in Eq. (19) an be derived to be φ2π = 1− , (23) SC2π = 1− 4π2 . (24) We observe that the error in the �delity of the 2π pulse is fourth order in the relative noise strength ∆/amax, whereas for the CORPSE pulse sequen e it is eighth or- der. Thus the CORPSE sequen e is mu h more a urate than a 2π pulse in orre ting the e�e ts of systemati errors on quantum memory. The fourth standard pulse sequen e whi h we take as a referen e is the Carr-Pur ell-Meiboom-Gill (CPMG) [34℄ sequen e whi h is designed to preserve qubit oheren e. In our ontext, this sequen e onsists of a π/2 pulse fol- lowed by multiple π pulses at intervals tp, followed by a �nal π/2 pulse to bring the system ba k to the original state. This pulse sequen e is designed for T2 measure- ments on spins, starting from the |0〉 state. Thus one does not expe t a CPMG pulse sequen e to perform as well if the initial state is averaged over the Blo h sphere as is done to ompute a gate �delity. We �rst present the �delities obtained for the iden- tity operator using the various ontrol pulse options in the presen e of 1/f noise. The noise is generated here by the single Markovian noise sour e dis ussed in Se . II, with transition rates distributed uniformly over the inter- val [1/τc, 30/τc] . In Fig. 2, the �delities obtained from optimized ontrol pulses, 2π pulse, CORPSE, CPMG, and zero pulse sequen es are plotted as fun tions of the hara teristi orrelation time τc of the approximate 1/f noise. Here, CPMG1 and CPMG2 refer to two CPMG types of pulses with the intervals between π pulses be- ing π and 2π, respe tively. The total duration for these pulses are all 12π~/a . The optimal ontrol pulse is de- signed for 6π, and therefore we repeat it twi e. Similarly, we repeat the 2π pulse 6 times, the CORPSE sequen e 3 times, the CPMG1 sequen e 3 times, and the CPMG2 sequen e twi e. The optimal ontrol pulse yields learly the highest �delity among all these pulses, whereas the zero pulse sequen e has the worst performan e as there are no orre tion me hanisms. Note that due to motional narrowing, all urves approa h unit �delity in the limit τc → 0. The memory a ess rate is an important spe i� ation in modern omputer te hnology [35℄. In our ontext, it orresponds to the total duration of the ontrol pulses. Figure 3 shows the �delity as a fun tion of the dura- tion for the numeri ally optimized ontrol pulses. Equa- tion (1) implies that in the absen e of noise, the quantum system will generate an identity operator for a = amax 0 5 10 15 20 25 30 0.55 0.65 0.75 0.85 0.95 PSfrag repla ements τc/(~/amax) FIG. 2: Fidelity of the quantum memory as a fun tion of the hara teristi orrelation time τc for optimized ontrol pulses (bla k solid), a 2π pulse (bla k dash-dotted), CORPSE pulse sequen e (bla k dotted), CPMG1 pulse sequen e (bla k dashed), CPMG2 pulse sequen e (gray solid), zero pulse se- quen e (gray dash-dotted). The operation time is hosen to be 12π~/a . The noise is produ ed by a single 32-state Marko- vian sour e with the average strength 〈|η|〉 = 0.125 × amax orresponding to 31 RTN �u tuators with the transition rates uniformly distributed over the region [1/τ , 30/τ ] and strengths hosen as des ribed in Se . II. 0 1 2 3 4 5 6 7 8 9 10 PSfrag repla ements T/(π~/amax) FIG. 3: Fidelity of the quantum memory as a fun tion of the operation time for ontrol pulses optimized at ea h point. The noise is produ ed by a similar multi-state Markovian sour e as in Fig. 2, with τc = 3~/amax. and the duration T = 2nπ/amax. In Fig. 3, we observe that, despite an overall de rease, there are peaks in the �delity near these operation times. Thus we an regard 2nπ/amax as the natural periods for quantum memory, and we always hoose the total duration of ontrol pulses orrespondingly. Here, we study the relation between the optimized �- delities a hieved above and the average noise strength 〈|η|〉, for a �xed value of the hara teristi orrelation 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 PSfrag repla ements 〈|η|〉 FIG. 4: Fidelity of the quantum memory as a fun tion of the average absolute noise strength for optimized ontrol pulses (bla k solid), 2π pulse (bla k dash-dotted), CORPSE (bla k dotted), CPMG1 (bla k dashed), CPMG2 (gray solid), and zero (gray dash-dotted). The operation time is hosen to be 12π~/a . Ex ept for its strength, the noise is produ ed by a similar multi-state Markovian sour e as in Fig. 2 with the orrelation time τ = 30~/a time τc = 30~/amax. Figure 4 shows the �delity as a fun tion of the noise strength for the optimized ontrol pulses, 2π pulse, the CORPSE, CPMG1, CPMG2, and zero pulse sequen es. At small values of 〈|η|〉 again, the optimized ontrol pulses onsistently a hieve higher �- delities than all referen e pulses. However, we note that if the noise strength ex eeds ∼0.4, the optimized pulse sequen e redu es to the zero pulse sequen e, i.e., any nonzero pulse sequen e will a tually deteriorate the �- delity performan e. The dis ussion above is based on the spe i� noise den- sity spe trum 1/fα with α = 1. Figure 5 shows the �delities of quantum memory for four optimized ontrol pulses, ea h of whi h is obtained for a di�erent value of α. The noise is produ ed here by a single multi-state Marko- vian sour e with average strength 〈|η|〉 = 0.125 × amax, and the total duration for all ontrol pulses are �xed to 6π. A systemati s aling of the orrelation time axis with respe t to α is learly visible in Fig. 5. This phe- nomenon is explained by the fa t that the on entra- tion of the power spe trum of 1/fα to high frequen ies, i.e., long orrelation times, in reases with α. Hen e, the urves s ale down in τ with in reasing α. V. NOT GATE In this se tion, we fo us on the generation of high- �delityNOT gates, i.e., the σx operator, under 1/f noise. As in the ase of quantum memory, we ompare the nu- meri ally optimized results with referen e pulses. In this 0 5 10 15 20 25 30 0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 PSfrag repla ements τc/(~/amax) FIG. 5: Fidelity of the quantum memory a hieved with op- timized ontrol pulses as a fun tion of the hara teristi or- relation time τc for 1/f noise with α = 1 (solid ), α = 1.25 (dotted), α = 1.5 (dash-dotted), and α = 1.75 (dashed). The operation time is hosen to be 6π~/a . The noise is pro- du ed by a similar multi-state Markovian sour e as in Fig. 2 with variable values of the power α. ase, our �rst referen e pulse is the π pulse given by aπ(t) = amax, for t ∈ [0, π~/amax], (25) whi h in the absen e of noise is the most e� ient way of a hieving a NOT gate. In addition, we will use the two omposite pulse sequen es CORPSE and short CORPSE [32, 33℄ whi h assume here the form aCπ(t) = amax, for 0 < t ′ < π/3 −amax, for π/3 ≤ t′ ≤ 2π amax, for 2π < t ′ < 13π/3, aSCπ(t) = −amax, for 0 < t′ < π/3 amax, for π/3 ≤ t′ ≤ 2π −amax, for 2π < t′ < 7π/3, respe tively. Both of these pulse sequen es orre t for systemati error, CORPSE being more e� ient. How- ever, the operation time of short CORPSE is mu h shorter than that of CORPSE, and hen e it an yield higher �delities in the presen e of noise. Figure 6 shows the NOT gate �delities obtained by the referen e and optimized pulses in the presen e of the same 1/f noise as employed in the analysis of quantum memory in Se . IV. We observe that for long enough orrelation times, the omposite pulse sequen es provide good error orre tion. Furthermore, as observed earlier for RTN [24℄, for intermediate orrelation times, short CORPSE a hieves the highest �delity among the refer- en e pulses. Figure 7 presents the pulse sequen es ob- tained from the numeri al optimizations for three dif- ferent values of the noise orrelation time τc. For the optimized pulse sequen e, we �nd a transition from an approximately onstant pulse to a short CORPSE -like pulse sequen e at hara teristi orrelation time τ 50~/a . This hange in optimal pulse sequen e is responsible for the apparent dis ontinuity in the �rst derivative of the �delity urve in Fig. 6. 0 50 100 150 200 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 τc/(~/amax) FIG. 6: NOT gate �delities as fun tions of the hara teris- ti noise orrelation time τc for a π pulse (dotted), CORPSE (dash-dotted), short CORPSE (dashed), and gradient opti- mized pulse sequen e (solid). The 1/f noise is generated as in Fig. 2. These results for the generation of NOT gates under 1/f noise are qualitatively quite similar to the previous results presented in Refs. [24, 29℄ for a single RTN. This similarity is due to the fa t that 1/f noise an be re- garded as arising from a sum of independent RTN �u - tuators, ea h of whi h having a similar �delity depen- den e on their orrelation times. Note that the s ale for the referen e orrelation time τc of the �delity obtained in presen e of 1/f noise in Fig. 6 is somewhat di�erent from the orresponding s ale for the orrelation time of a single RTN sour e, sin e the 1/f noise involves an en- semble of RTN �u tuators with a range of orrelation times. VI. CONCLUSIONS We have studied a single qubit under the in�uen e of 1/fα noise for 2 > α > 0 and investigated how de o- heren e due to this noise an be suppressed in the im- plementation of single qubit operations. We presented an e� ient way to approximate the noise with a dis rete multi-state Markovian �u tuator. Due to this �nding, the average temporal evolution of the qubit density ma- trix under 1/fα noise an be e� iently determined from a deterministi master equation. Employing these exa t deterministi master equations des ribing the temporal evolution of the qubit density operator under Markovian noise, we applied a gradient FIG. 7: Optimized pulse sequen es yielding the highest gate �delities for orrelation times (a) 45~/a , (b) 100~/a and ( ) 150~/a orresponding to Fig. 6. based optimization pro edure to sear h for optimal on- trol pulses implementing quantum operations. In par- ti ular, we studied the physi al appli ation of quantum memory, i.e., the identity operator, whi h is a funda- mental on ept in the realization of a quantum om- puter. The optimized ontrol pulses signi� antly im- proved the �delity over several referen e sequen es su h as 2π, CORPSE, CPMG, and zero pulses. We observe peaks on �delity urves orresponding to integer multi- ples of 2π~/a in the total durations of ontrol pulses, where a is the maximum magnitude of the external ontrol �eld. We also studied the performan e of opti- mal ontrol pulses under 1/fα noise for several di�erent values of 2 > α ≥ 1, and found a monotoni behavior in the noise frequen y as a fun tion of α, i.e., the �delity urves are s aled down in the orrelation time for in reas- ing α. We also investigated how the �delities degraded as the noise strength in reases. For the generation of high- �delity NOT gates, we obtained results showing qualita- tively similar behavior to the previous results presented in Refs. [24, 29℄ for a single RTN sour e. In parti u- lar, just as for a single noise sour e, in the presen e of 1/fα noise we observed a transition in the optimal ontrol pulse sequen e from a onstant pulse to a CORPSE-like sequen e as the noise hara teristi orrelation time τc is in reased. This approa h of oupled master equations indexed by noise states of the environment, together with an opti- mization te hnique for pulse design an be readily gen- eralized to multiple qubits evolving in the presen e of 1/fα noise and other Markovian noise sour es. Further- more, it an be used to develop realisti pulse sequen es for mitigation of nu lear spin and surfa e magneti noise a ting on donor spins implanted in sili on [11℄, as well as for suppression of ba kground harge noise a ting on super ondu ting qubits [4℄. In future, we will study the implementation of multi-qubit gates, e.g., the ontrolled NOT gate, in noisy systems and the swapping of quan- tum information from a noisy qubit to long term quan- tum memory. We will also onsider more realisti noise with 1/fα spe trum over many frequen y de ades. 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0704.0772

Stability of a colocated finite volume scheme for the incompressible Navier-Stokes equations

arXiv:0704.0772v1 [math.NA] 5 Apr 2007 INTERNATIONAL JOURNAL OF c© 2007 Institute for Scientific NUMERICAL ANALYSIS AND MODELING Computing and Information Volume 1, Number 1, Pages 1–18 STABILITY OF A COLOCATED FINITE VOLUME SCHEME FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS SÉBASTIEN ZIMMERMANN (Communicated by Jean-Luc Guermond) Abstract. We introduce a finite volume scheme for the two-dimensional incom- pressible Navier-Stokes equations. We use a triangular mesh. The unknowns for the velocity and pressure are both piecewise constant (colocated scheme). We use a projection (fractional-step) method to deal with the incompressibility constraint. We prove that the differential operators in the Navier-Stokes equa- tions and their discrete counterparts share similar properties. In particular, we state an inf-sup (Babuška-Brezzi) condition. We infer from it the stability of the scheme. Key Words. Incompressible fluids, Navier-Stokes equations, projection meth- ods and finite volume. 1. Introduction We consider the flow of an incompressible fluid in a open bounded set Ω ⊂ R2 during the time interval [0, T ]. The velocity field u : Ω × [0, T ] → R2 and the pressure field p : Ω× [0, T ] → R satisfy the Navier-Stokes equations ∆u+ (u ·∇)u+∇p = f ,(1.1) div u = 0 ,(1.2) with the boundary and initial condition u|∂Ω = 0 , u|t=0 = u0. The terms ∆u and (u ·∇)u are respectively associated with the physical phenom- ena of diffusion and convection. The Reynolds number Re measures the influence of convection in the flow. For equations (1.1)–(1.2), finite element and finite dif- ference methods are well known and mathematical studies are available (see [10] for example). Numerous computations have also been conducted with finite vol- ume schemes (e.g. [14] and [1]). However, in this case, few mathematical results are available. Let us cite Eymard and Herbin [7] and Eymard, Latché and Herbin [8]. In order to deal with the incompressibility constraint (1.2), these works use a penalization method. Another way is to use the projection methods which have been introduced by Chorin [4] and Temam [15]. This is the case in Faure [9]. In this work, however, the mesh is made of squares, so that the geometry of the problem is limited. Therefore, we introduce in what follows a finite volume scheme on triangular meshes for equations (1.1)–(1.2), using a projection method. An interesting feature of this scheme is that the unknowns for the velocity and Received by the editors April 1, 2007 and, in revised form, April 1, 2007. 2000 Mathematics Subject Classification. 76M12, 76B99. http://arxiv.org/abs/0704.0772v1 2 S. ZIMMERMANN pressure are both piecewise constant (colocated scheme). It leads to an economic computer storage, and allows an easy generalization of the scheme to the 3D case. The layout of the article is the following. We first introduce (section 2) some no- tations and hypotheses on the mesh. We define (section 2.2) the spaces we use to approximate the velocity and pressure. We define also (section 2.3) the operators we use to approximate the differential operators in (1.1)–(1.2). Combining this with a projection method, we build the scheme in section 3. In order to provide a mathematical analysis for the scheme, we prove in section 4 that the differential operators in (1.1)–(1.2) and their discrete counterparts share similar properties. In particular, the discrete operators for the gradient and the divergence are adjoint. Also, the discrete gradient operator is a consistent approximation of its continuous counterpart. The discrete operator for the convection term is positive, stable and consistent. The discrete operator for the divergence satisfies an inf-sup (Babuška- Brezzi) condition. From these properties we deduce in section 5 the stability of the scheme. We conclude with some notations. The spaces (L2, |.|) and (L∞, ‖.‖∞) are the usual Lebesgue spaces and we set L20 = {q ∈ L2 ; q(x) dx = 0}. Their vectorial counterparts are (L2, |.|) and (L∞, ‖.‖∞) with L2 = (L2)2 and L∞ = (L∞)2. For k ∈ N∗, (Hk, ‖·‖k) is the usual Sobolev space. Its vectorial counterpart is (Hk, ‖.‖k) with Hk = (Hk)2. For k = 1, the functions of H1 with a null trace on the boundary form the spaceH10. Also, we set ∇u = (∇u1,∇u2)T if u = (u1, u2) ∈ H1. If X ⊂ L2 is a Banach space, we define C(0, T ;X) (resp. L2(0, T ;X)) as the set of the applications g : [0, T ] → X such that t → |g(t)| is continous (resp. square integrable). The norms ‖.‖C(0,T ;X) and ‖.‖L2(0,T ;X) are defined respectively by ‖g‖C(0,T ;X) = supt∈[0,T ] |g(t)| and ‖g‖L2(0,T ;X) = |g(t)|2 ds . In all calculations, C is a generic positive constant, depending only on Ω, u0 and f . 2. Discrete setting First, we introduce the spaces and the operators needed to build the scheme. 2.1. The mesh. Let Th be a triangular mesh of Ω: Ω = ∪K∈ThK. For each triangle K ∈ Th, we denote by |K| its area and EK the set of his edges. If σ ∈ EK , nK,σ is the unit vector normal to σ pointing outward of K. The set of edges of the mesh is Eh = ∪K∈ThEK . The length of an edge σ ∈ Eh is |σ| and its middle point xσ. The set of edges located inside Ω (resp. on its boundary) is E inth (resp. Eexth ): Eh = E inth ∪ Eexth . If σ ∈ E inth , Kσ and Lσ are the triangles sharing σ as an edge. If σ ∈ Eexth , only the triangle Kσ inside Ω is defined. We denote by xK the circumcenter of a triangle K. We assume that the measure of all interior angles of the triangles of the mesh are below π , so that xK ∈ K. If σ ∈ E inth (resp. σ ∈ Eexth ) we set dσ = d(xKσ ,xLσ ) (resp. dσ = d(xσ ,xKσ)). We define for all edge σ ∈ Eh (2.1) τσ = The maximum circumradius of the triangles of the mesh is h. We assume ([6] p. 776) that there exists C > 0 such that ∀σ ∈ Eh, d(xKσ , σ) ≥ C|σ| and |σ| ≥ Ch. It implies that there exists C > 0 such that (2.2) ∀σ ∈ Eh , τσ ≥ C , A COLOCATED FINITE VOLUME SCHEME FOR THE NAVIER-STOKES EQUATIONS 3 and for all triangles K ∈ Th we have (with σ ∈ EK and hK,σ the matching altitude) (2.3) |K| = 1 |σ|hK,σ ≥ |σ| d(xK ,xσ) ≥ C h2. Lastly, if K ∈ Th and L ∈ Th are two triangles sharing the edge σ ∈ E inth , we define αK,L = d(xL,xσ) d(xK ,xL) Let us notice that αK,L ∈ [0, 1] and αK,L + αL,K = 1. 2.2. The discrete spaces. We first define P0 = {q ∈ L2 ; ∀K ∈ Th, q|K is a constant} , P0 = (P0)2. For the sake of concision, we set for all qh ∈ P0 (resp. vh ∈ P0) and all triangle K ∈ Th: qK = qh|K (resp. vK = vh|K). Although P0 6⊂ H1, we define the discrete equivalent of a H1 norm as follows. For all vh ∈ P0 we set (2.4) ‖vh‖h = σ∈Eint τσ |vLσ − vKσ |2 + σ∈Eext τσ |vKσ |2 where τσ is given by (2.1). We have [6] a Poincaré-like inequality for P0: there exists C > 0 such that for all vh ∈ P0 (2.5) |vh| ≤ C ‖vh‖h. We also have the following inverse inequality. Proposition 2.1. There exists a constant C > 0 such that for all vh ∈ P0 h ‖vh‖h ≤ C |vh|. Proof. According to (2.4) h2 ‖vh‖2h = σ∈Eint h2 τσ |vLσ − vKσ |2 + σ∈Eext h2 τσ |vKσ |2. We deduce from (2.2) and (2.3) that h2 τσ ≤ C |Kσ| and h2 τσ ≤ C |Lσ|. Thus, since |vLσ − vKσ |2 ≤ 2 |vLσ |2 + |vKσ |2 , we get h2 ‖vh‖2h ≤ C σ∈Eint |Kσ| |vKσ |2 + |Lσ| |vLσ |2 σ∈Eext |Kσ| |vKσ |2. Hence h2 ‖vh‖2h ≤ C |K| |vK |2 ≤ C |vh|2. From the norm ‖.‖h we deduce a dual norm. For all vh ∈ P0 we set (2.6) ‖vh‖−1,h = sup (vh,ψh) ‖ψh‖h For all uh ∈ P0 and vh ∈ P0 we have (uh,vh) ≤ ‖uh‖−1,h ‖vh‖h. Now we introduce some operators on P0 and P0. We define the projection operator ΠP0 : L 2 → P0 as follows. For all w ∈ L2, ΠP0w ∈ P0 is given by (2.7) ∀K ∈ Th , (ΠP0w)|K = w(x) dx. We easily check that for all w ∈ L2 and vh ∈ P0 we have (ΠP0w,vh) = (w,vh). It implies that ΠP0 is stable for the L 2 norm. We define also the interpolation operator Π̃P0 : H 2 → P0. For all q ∈ H2, Π̃P0q ∈ P0 is given by ∀K ∈ Th , Π̃P0q|K = q(xK). 4 S. ZIMMERMANN According to the Sobolev embedding theorem, q ∈ H2 is a.e. equal to a continuous function. Therefore the definition above makes sense. We also set Π̃P0 = (Π̃P0 ) The operator Π̃P0 (resp. Π̃P0) is naturally stable for the L ∞ (resp. L∞) norm. One also checks ([2] and [16]) that there exists C > 0 such that (2.8) |v −ΠP0v| ≤ C h ‖v‖1 , |q − Π̃P0q| ≤ C h ‖q‖2 for all v ∈ H1 and q ∈ H2. We introduce the finite element spaces P d1 = {v ∈ L2 ; ∀K ∈ Th, v|K is affine} , 1 = {vh ∈ P d1 ; ∀σ ∈ E inth , vh|Kσ(xσ) = vh|Lσ(xσ) , Pc1 = {vh ∈ (P d1 )2 ; vh is continuous and vh|∂Ω = 0}. We have Pc1 ⊂ H10. We define the projection operator ΠPc1 : H 0 → Pc1. For all v = (v1, v2) ∈ H10, ΠPc1v = (v h) ∈ Pc1 is given by ∀φh = (φ1h, φ2h) ∈ Pc1 , ∇vih,∇φih) = ∇vi,∇φih). The operator ΠPc is stable for the H1 norm and ([2] p. 110) there exists C > 0 such that for all v ∈ H1 (2.9) |v −ΠPc v| ≤ C h ‖v‖1. Let us address now the space Pnc1 . If qh ∈ Pnc1 , we have usually ∇qh 6∈ L2. Thus we define the operator ∇̃h : Pnc1 → P0 by setting for all qh ∈ Pnc1 and all K ∈ Th ∇̃hqh|K = ∇qh dx. The associated norm is given by ‖qh‖1,h = |qh|2 + |∇̃hqh|2 We also have a Poincaré inequality: there exists C > 0 such that for all qh ∈ Pnc1 ∩L20 (2.10) |qh| ≤ C |∇̃hqh|. We define the projection operator ΠPnc . For all qh ∈ Pnc1 , ΠPnc1 qh is given by (2.11) ∀φ ∈ L2 , (ΠPnc qh, φ) = (qh, φ). We have the following result. Proposition 2.2. If qh ∈ P0, ΠPnc qh is given by ∀σ ∈ E inth , (ΠPnc1 qh)(xσ) = |Kσ|+ |Lσ| qKσ + |Kσ|+ |Lσ| qLσ , ∀σ ∈ Eexth , (ΠPnc1 qh)(xσ) = qKσ . Proof. For all edge σ ∈ Eh, we define the function ψσ ∈ Pnc1 by setting ψσ(xσ′ ) = 1 if σ = σ′, 0 otherwise. Let us notice that ψσ vanishes outside Kσ ∪ Lσ if σ ∈ E inth and outside Kσ if σ ∈ Eexth . Let σ ∈ E inth . Using a quadrature formula we get (ΠPnc qh, ψσ) = (ΠPnc qh)(xσ) A COLOCATED FINITE VOLUME SCHEME FOR THE NAVIER-STOKES EQUATIONS 5 (qh, ψσ) = qKσ + qLσ For an edge σ ∈ Eexth we have (ΠPnc1 qh, ψσ) = (ΠPnc qh)(xσ) and (qh, ψσ) = . By plugging these equations into (2.11) with φ = ψσ, we get the result. We finally introduce the Raviart-Thomas spaces ={vh ∈ Pd1 ; ∀σ ∈ EK , vh|K · nK,σ is a constant, and vh · n|∂Ω = 0} , RT0 ={vh ∈ RTd0 ; ∀K ∈ Th, ∀σ ∈ EK , vh|Kσ · nKσ ,σ = vh|Lσ · nKσ ,σ}. For all vh ∈ RT0, K ∈ Th and σ ∈ EK we set (vh ·nK,σ)σ = vh|K ·nK,σ. We define the operator ΠRT0 : H 1 → RT0. For all v ∈ H1, ΠRT0v ∈ RT0 is given by (2.12) ∀K ∈ Th , ∀σ ∈ EK , (ΠRT0v · nK,σ)σ = v dσ. One checks [3] that there exists C > 0 such that for all v ∈ H1 (2.13) |v −ΠRT0v| ≤ C h ‖v‖1. The following result will be useful. Proposition 2.3. For all v ∈ H1 such that divv = 0, we have ΠRT0v ∈ P0. Proof. Let vh = ΠRT0v and K ∈ Th. According to [3] there exists aK ∈ R2 and bK ∈ R such that: ∀x ∈ K , vh(x) = aK + bK x. Thus divvh|K = 2 bK . On the other hand, according to the divergence formula and (2.12) divv dx = v · n dγ = vh · n dγ = divvh dx. Hence bK = 0 and we get: ∀x ∈ K , vh(x) = aK . 2.3. The discrete operators. The equations (1.1)–(1.2) use the differential op- erators gradient, divergence and laplacian. Using the spaces of section 2.2 we define their discrete counterparts. The discrete gradient ∇h : P0 → P0 is built using a linear interpolation on the edges of the mesh (see [16] for details). This kind of construction has also be considered in [5]. We set for all qh ∈ P0 and all K ∈ Th ∇h qh|K = σ∈EK∩E αKσ,Lσ qKσ + αLσ,Kσ qLσ σ∈EK∩E |σ| qKσ nK,σ.(2.14) We have the following result [16]. Proposition 2.4. If qh ∈ L20 is such that ∇hqh = 0, then qh = 0. The discrete divergence operator divh : P0 → P0 is built so that it is adjoint to the operator ∇h (proposition 4.6 below). We set for all qh ∈ P0 and all K ∈ Th (2.15) divh vh|K = σ∈EK∩E αLσ ,Kσ vKσ + αKσ,Lσ vLσ · nK,σ. The first discrete laplacian ∆h : P0 → P0 ensures that the incompressibility con- straint (1.2) is satisfied in a discrete sense (proposition 3.1). We set for all qh ∈ P0 (2.16) ∆hqh = divh(∇hqh). 6 S. ZIMMERMANN The second discrete laplacian ∆̃h : P0 → P0 is the usual operator in finite volume schemes [6]. We set for all vh ∈ P0 and all K ∈ Th ∆̃hvh|K = σ∈EK∩E τσ (vLσ − vKσ )− σ∈EK∩E τσ vKσ . In order to approximate the convection term (u ·∇)u in (1.1) we define a bilinear form b̃h : P0 ×P0 → P0 using the well-known upwind scheme ([6] p. 766). For all uh ∈ P0, vh ∈ P0, and all K ∈ Th we have (2.17) b̃h(uh,vh) σ∈EK∩E (uσ · nK,σ)+ vK + (uσ · nK,σ)− vLσ We have set uσ = αLσ ,Kσ uKσ + αKσ ,Lσ uLσ and a + = max(a, 0), a− = min(a, 0) for all a ∈ R. Lastly, we define the trilinear form bh : P0 × P0 × P0 → R2 as follows. For all uh ∈ P0, vh ∈ P0, wh ∈ P0, we set (2.18) bh(uh,vh,wh) = |K|wK · b̃h(uh,vh) 3. The scheme We have defined in section 2 the discretization in space. We now have to define a discretization in time, and treat the incompressibility constraint (1.2). We use a projection method to this end. This kind of method has been introduced byChorin [4] and Temam [15]. The basic idea is the following. The time interval [0, T ] is split with a time step k: [0, T ] = n=0[tn, tn+1] with N ∈ N∗ and tn = n k for all n ∈ {0, . . . , N}. For all m ∈ {2, . . . , N}, we compute (see equation (3.2) below) a first velocity field ũmh ≃ u(tm) using only equation (1.1). We use a second-order BDF scheme for the discretization in time. We then project ũmh (see equation (3.4) below) over a subspace of P0. We get a a pressure field p h ≃ p(tm) and a second velocity field umh ≃ u(tm), which fulfills the incompressibilty constraint (1.2) in a discrete sense. The algorithm goes as follows. First, for all m ∈ {0, . . . , N}, we set fmh = ΠP0 f(tm). Since the operator ΠP0 is stable for the L2-norm we get (3.1) |fmh | = |ΠP0 f(tm)| ≤ |f(tm)| ≤ ‖f‖C(0,T ;L2). We start with the initial values u0h ∈ P0 ∩RT0 , u1h ∈ P0 ∩RT0 p1h ∈ P0 ∩ L20. For all n ∈ {1, . . . , N}, (ũn+1h , p h ) is deduced from (ũ h) as follows. • ũn+1h ∈ P0 is given by (3.2) 3 ũn+1h − 4unh + u ∆̃hũ h +b̃h(2u h−un−1h , ũ h )+∇hp h = f • pn+1h ∈ Pnc1 ∩ L20 is the solution of (3.3) ∆h(p h − p divh ũ • un+1h ∈ P0 is deduced by (3.4) un+1h = ũ ∇h(pn+1h − p A COLOCATED FINITE VOLUME SCHEME FOR THE NAVIER-STOKES EQUATIONS 7 Existence and unicity of a solution to equation (3.2) is classical ([6] for example). Let us show that equation (3.3) has also a unique solution. Let qh ∈ P0 ∩ L20 such that ∆hqh = 0. According to proposition 4.6 we have for all qh ∈ P0 −(∆hqh, qh) = − divh(∇hqh), qh = (∇hqh,∇hqh) = |∇hqh|2. Therefore we have ∇hqh = 0. Using proposition 2.4 we get qh = 0. We have thus proved the unicity of a solution for equation (3.3). It is also the case for the associated linear system. It implies that this linear system has indeed a solution. Hence it is also the case for equation (3.3). Let us now prove that for all m ∈ {0, . . . , N}, umh fulfills (1.2) in a discrete sense. Lemma 3.1. If vh ∈ RT0 ∩P0 then divh vh = 0. Proof. Let K ∈ Th. Since vh ∈ RT0, definition (2.15) reads divh vh|K = |σ| (αLσ ,K + αK,Lσ )vK · nK,σ. Since αKσ ,Lσ + αLσ,Kσ = 1 we conclude that divh vh|K = |σ|vK · nK,σ = vK · |σ|nK,σ Proposition 3.1. For all m ∈ {0, . . . , N} we have divh umh = 0. Proof. For m ∈ {0, 1} we have u0h ∈ P0 ∩ RT0 and u1h ∈ P0 ∩RT0. Applying the lemma above we get the result. If m ∈ {2, . . . , N}, we apply the operator divh to (3.3) and compare with (3.4). 4. Properties of the discrete operators We prove that the differential operators in (1.1)–(1.2) and the operators defined in section 2.3 share similar properties. 4.1. Properties of the discrete convective term. We define b̃ : H1 ×H1 → L2. For all u ∈ H1 and v = (v1, v2) ∈ H1 we set (4.1) b̃(u,v) = div(v1 u), div(v2 u) We show that the operator b̃h is a consistent approximation of b̃. Proposition 4.1. There exists a constant C > 0 such that for all v ∈ H2 and all u ∈ H2 ∩H10 satisfying divu = 0 ‖ΠP0b̃(u,v) − b̃h(ΠRT0u, Π̃P0v)‖−1,h ≤ C h ‖u‖2 ‖v‖1. Proof. Let uh = ΠRT0u and vh = Π̃P0v. According to proposition 2.3 we have uh ∈ P0. Let K ∈ Th. According to the divergence formula and (2.7) we have ΠP0b̃(u,v)|K = σ∈EK∩E v (u · n) dσ. On the other hand, let us rewrite b̃h(uh,vh). Let σ ∈ EK ∩ E inth . Setting vK,Lσ = vK si (uh · nK,σ)σ ≥ 0 vLσ si (uh · nK,σ)σ < 0 one checks that vK (uσ ·nK,σ)++vLσ (uσ ·nK,σ)− = vK,Lσ (uσ ·nK,σ). By definition uσ · nK,σ = αLσ,K (uK · nK,σ) + αK,Lσ (uLσ · nK,σ) ; since uh ∈ RT0 we get 8 S. ZIMMERMANN uσ · nK,σ = (αLσ ,K + αK,Lσ ) (uK · nK,σ) = (uK · nK,σ) = (uh · nK,σ)σ. Using at last (2.12), we deduce from (2.17) b̃h(uh,vh)|K = σ∈EK∩E vK,Lσ (u · nK,σ) dσ. ΠP0 b̃(u,v) − b̃h(uh,vh) σ∈EK∩E (v − vK,Lσ ) (u · n) dσ. Let ψh ∈ P0. We have ΠP0b̃(u,v) − b̃h(uh,vh),ψh σ∈EK∩E (v − vK,Lσ ) (u · n) dσ σ∈Eint (ψKσ −ψLσ) (v − vKσ ,Lσ) (u · n) dσ.(4.2) Let σ ∈ E inth . We want to estimate the integral over σ. Since we work in a two- dimensional domain, we have the Sobolev injection H2 ⊂ L∞. Thus (v − vKσ,Lσ) (u · n) dσ ∣∣∣∣ ≤ ‖u‖L∞ |v−vKσ ,Lσ | dσ ≤ C ‖u‖2 |v−vKσ ,Lσ | dσ. Let us first assume that v ∈ C1. We set xKσ,Lσ = xKσ si (uh · nK,σ)σ ≥ 0 xLσ si (uh · nK,σ)σ < 0 If x ∈ σ, we have the following Taylor expansion v(x)− vKσ,Lσ = v(x)−v(xKσ ,Lσ) = ∇v (tx+(1− t)xKσ ,Lσ) (x−xKσ ,Lσ) dt. We have |x−xKσ ,Lσ | ≤ h. Thus, integrating over σ and using the Cauchy-Schwarz inequality, we get |v − vKσ ,Lσ | dσ ≤ |∇v (tx+ (1− t)xKσ ,Lσ)|2 h t dt dσ We then use the change of variable (t,x) → y = tx + (1 − t)xKσ ,Lσ . Let Dσ be the quadrilateral domain given by the endpoints of σ, xKσ and xLσ . The domain [0, 1]× σ becomes DKσ,Lσ with DKσ,Lσ = Dσ ∩Kσ si (uh · nK,σ)σ ≥ 0 Dσ ∩ Lσ si (uh · nK,σ)σ < 0 For all t ∈ [0, 1] we have h t ≤ h t ≤ C d(xKσ ,Lσ , σ) t thanks to the hypothesis on the mesh. We check easily that d(xKσ ,Lσ , σ) t dt dσ = dy. Thus we get |v − vKσ ,Lσ | dσ ≤ C h DKσ,Lσ |∇v (y)|2 dy A COLOCATED FINITE VOLUME SCHEME FOR THE NAVIER-STOKES EQUATIONS 9 Since (C1)2 is dense in H2, this estimate still holds for v ∈ H2. Plugging this estimate into (4.2) and using the Cauchy-Schwarz inequality we get ΠP0 b̃(u,v) − b̃h(ΠRT0u, Π̃P0v),ψh ≤ C h ‖u‖H2 σ∈Eint |ψLσ −ψKσ | σ∈Eint DKσ,Lσ |∇v (y)|2 dy so that ΠP0 b̃(u,v)− b̃h(ΠRT0u, Π̃P0v),ψh )∣∣∣ ≤ C h ‖u‖H2 ‖ψh‖1,h ‖v‖1. Using then definition (2.6), we get the result. Let us consider now the operator bh. Let u ∈ H1 and v ∈ L∞∩H1 with divu ≥ 0. Integrating by parts we deduce from (4.1): v · b̃(u,v) dx = divu dx ≥ 0. The discrete operator bh shares a similar property. Proposition 4.2. Let uh ∈ P0 such that divh uh ≥ 0. For all vh ∈ P0 we have bh(uh,vh,vh) ≥ 0. Proof. Remember that for all edges σ ∈ E inth , two triangles Kσ et Lσ share σ as an edge. We denote by Kσ the one such that uσ · nKσ,σ ≥ 0. Using the algebraic identity 2 a (a− b) = a2 − b2 + (a− b)2 we deduce from (2.18) 2 bh(uh,vh,vh) = 2 σ∈Eint |σ|vKσ · (vKσ − vLσ ) (uσ · nKσ,σ) σ∈Eint |vKσ|2 − |vLσ |2 + |vKσ − vLσ |2 (uσ · nKσ,σ) so that 2 bh(uh,vh,vh) ≥ σ∈Eint |vKσ|2−|vLσ |2 (uσ ·nKσ,σ). This sum can be written as a sum over the triangles of the mesh. We get 2 bh(uh,vh,vh) ≥ |vKσ |2 σ∈EK∩E |σ| (uσ · nKσ,σ). Using finally definition (2.15) we get 2 bh(uh,vh,vh) ≥ |K| |vK |2 (divh uh)|K ≥ 0. The following result states that the operator bh is stable for suitable norms. Proposition 4.3. There exists a constant C > 0 such that for all vh ∈ P0, wh ∈ P0, uh ∈ P0 satisfying divh uh = 0 |bh(uh,vh,vh)| ≤ C |uh| ‖vh‖h ‖vh‖h. Proof. For all triangle K ∈ Th and all edge σ ∈ EK ∩ E inth , we have (uσ · nK,σ)+ vK + (uσ · nK,σ)− vLσ = (uσ · nK,σ)vK − |(uσ · nK,σ)| (vLσ − vK). This way, we deduce from (4.7) bh(uh,vh,wh) = S1 + S2 with vK ·wK σ∈EK∩E |σ| (uσ · nK,σ) , S2 = − σ∈EK∩E |σ| |uσ · nK,σ| (vLσ − vK). 10 S. ZIMMERMANN By writing the sum over the edges as a sum over the triangles we get S2 = − σ∈Eint |σ| |uσ · nK,σ| (vLσ − vK) · (wLσ −wK). Using the Cauchy-Schwarz inequality we get |S2| ≤ h ‖uh‖∞ σ∈Eint |vLσ − vKσ |2 1/2  σ∈Eint |wLσ −wKσ |2 Since uh ∈ P0 we have the inverse inequality [6] h ‖uh‖∞ ≤ C |uh|. Using (2.2) and (2.4) we have σ∈Eint |vLσ − vKσ |2 ≤ C σ∈Eint τσ |vLσ − vKσ |2 ≤ C ‖vh‖2h σ∈Eint |wLσ −wKσ |2 ≤ C ‖wh‖2h. Therefore |S2| ≤ C |uh| ‖vh‖h ‖wh‖h. On the other hand we deduce from definition (2.15) |K| (vK ·wK) (divh uh)|K = 0. By combining the estimates for S1 and S2 we get the result. 4.2. Properties of the discrete gradient. Proposition 4.4. There exists a constant C > 0 such that for all qh ∈ P0: h |∇hqh| ≤ C |qh|. Proof. Using (2.14) and the Minkowski inequality, we have for all triangle K ∈ Th |K| |∇hqh |K |2 ≤ σ∈EK∩E 6 |σ|2 (q2K + q σ∈EK∩E 6 |σ|2 q2K . Let us sum over K ∈ Th. Since |σ| ≤ h, using (2.3), we get |∇hqh|2 ≤ σ∈EK∩E |K| q2K + |Lσ| q2Lσ σ∈EK∩E |K| q2K Thus h2 |∇hqh|2 ≤ C |K| q2K ≤ C |qh|2. We now prove that ∇h is a consistent approximation of the gradient. Proposition 4.5. There exists a constant C > 0 such that for all q ∈ H2 |ΠP0(∇q)−∇h(Π̃P0q)| ≤ C h ‖q‖2. Proof. Let K ∈ Th. Using the gradient formula and definition (2.14) we get ΠP0(∇q)−∇h(Π̃P0q) ∇q dx− |K| ∇h(Π̃P0q) where we have set for all edge σ ∈ EK ∩ E inth IσK = αK,Lσ q(xK) + αLσ,K q(xLσ ) nK,σ dσ A COLOCATED FINITE VOLUME SCHEME FOR THE NAVIER-STOKES EQUATIONS 11 and for all edge σ ∈ EK ∩ Eexth : IσK = q − q(xK) nK,σ dσ. Squaring and using (2.3) we get ΠP0(∇q)−∇h(Π̃P0q) |IσK |2 ≤ |IσK |2. Summing over the triangles K ∈ Th we get (4.3) ∣∣∣ΠP0(∇q)−∇h(Π̃P0q) |IσK |2. We must estimate the integral terms IσK . Let K ∈ Th. Let us first assume that q ∈ C2(Ω). Let σ ∈ EK ∩ E inth . For x ∈ σ we have the following Taylor expansions q(xK) = q(x)+∇q(x) · (xK −x)+ H(q) (txK +(1− t)x)(xK −x) · (xK −x) t dt , q(xLσ) = q(x)+∇q(x)·(xLσ −x)+ H(q) (txLσ+(1−t)x)(xLσ−x)·(xLσ−x) t dt , ∇q(x) = ∇q(xK)− txK + (1− t)x (xK − x) dt. Plugging the last expansion into the two others and integrating over σ we get (4.4) q(xK)− q dσ = |σ| ∇q(xK) · (xK − xσ)−AσK +BσK , (4.5) q(xLσ )− q dσ = |σ| ∇q(xK) · (xLσ − xσ)−AσLσ + B We have set for T ∈ {Kσ, Lσ} (4.6) AσT = ∇∇q (txT + (1 − t)x) (xT − x) dt dσ , (4.7) BσT = H(q) (txT + (1− t)x)(xT − x) · (xT − x) t dt dσ. One can bound these terms as in the proof of proposition 4.1. We get (4.8) |AσT |2 ≤ C h2 |∇∇q (y)|2 dy , |BσT |2 ≤ C h4 |H(q)(y)|2 dy. Now, let us multiply (4.4) by −αK,Lσ nK,σ, (4.5) by −αLσ,K nK,σ and sum the equalities. Since αLσ,K + αK,Lσ = 1 we have −αLσ,K q(xK)− q nK,σ dσ − αK,Lσ q(xLσ )− q nK,σ dσ αKσ,Lσ q(xK,σ) + αLσ,Kσ q(xL,σ) nK,σ dσ = I On the other hand −αK,Lσ (xK −xσ) ·nK,σ −αLσ,K (xLσ −xσ) ·nK,σ = −αK,Lσ αLσ,K (dσ − dσ) = 0. Therefore we get IσK = −αLσ,K AσK +B nK,σ −αK,Lσ AσLσ +B nK,σ. Using estimates (4.8) we obtain |IσK |2 ≤ C h4 (|H(q)(y)|2 + |∇∇q(y)|2) dy. 12 S. ZIMMERMANN We now consider the case σ ∈ EK ∩ Eexth . For x ∈ σ we have q(xK) = q(x)+∇q(x) · (xK − x)+ H(q)(txK + (1− t)x)(xK − x) · (xK − x)tdt. Multiplying by nK,σ and integrating over σ, we get −IσK = JσK nK,σ+BσK nK,σ with JσK = ∇q(x) · (xK − x) dx. Since |xK − x| ≤ h if x ∈ σ, using a trace theorem, we have |JσK | ≤ C h2 ‖∇q‖L∞(σ) ≤ C h2 |∇q(y)|2 + |∇∇q(y)|2 By combining this estimate with (4.8), we get |IσK |2 ≤ 2 |Jσ|2 + 2 |BσK |2 ≤ C h4 |H(q)(y)|2 dy + C h4 (|∇q(y)|2 + |∇∇q(y)|2) dy. The space C2(Ω) is dense in H2. Therefore the bounds for IσK still hold for q ∈ H2. Plugging these bounds into (4.3) we get the result. 4.3. Properties of the discrete divergence. The operators divergence and gra- dient are adjoint: if q ∈ H1 and v ∈ H1 with v · n|∂Ω = 0, we get (v,∇q) = −(q, divv) by integrating by parts. For ∇h and divh we state Proposition 4.6. If vh ∈ P0 and qh ∈ P0 we have: (vh,∇hqh) = −(qh, divh vh). Proof. Using (2.14) one checks that (vh,∇hqh) = qK (S1 + S2 + S3) with σ∈EK∩E |σ|αK,Lσ vK · nK,σ , S2 = σ∈EK∩E |σ|αK,Lσ vLσ · nLσ,σ , and S3 = σ∈EK∩E |σ|vK · nK,σ. Since αK,Lσ + αLσ,K = 1 we have σ∈EK∩E |σ| (1− αLσ ,K)vK · nK,σ σ∈EK∩E |σ|vK · nK,σ − σ∈EK∩E |σ|αLσ,K vK · nK,σ. Since nLσ,σ = −nK,σ, we also have σ∈EK∩E |σ|αK,Lσ vLσ · nLσ,σ = − σ∈EK∩E |σ|αK,Lσ vLσ · nK,σ. Therefore (vh,∇hqh) = − σ∈EK∩E |σ| (αL,Kσ vK +αK,Lσ vLσ) ·nK,Lσ + |σ|vK ·nK,Lσ . Using definition (2.15) we get (vh,∇hqh) = − |K| divh vh|K + |σ|vK · nK,Lσ . Since |σ|vK · nK,Lσ = vK · |σ|nK,Lσ = 0 we obtain finally (vh,∇hqh) = − qK |K| divh vh|K = −(qh, divh vh). A COLOCATED FINITE VOLUME SCHEME FOR THE NAVIER-STOKES EQUATIONS 13 The divergence operator and the spaces L20, H 0 satisfy the following property, called inf-sup (or Babuška-Brezzi) condition (see [10] for example). There exists a constant C > 0 such that (4.9) inf − (q, divv) ‖v‖1|q| We will now prove that the operator divh and the spaces P0 ∩ L20, P0 satisfy an analogous property. The proof is based on the following lemma. Lemma 4.1. We assume that the mesh is uniform (i.e. the triangles of the mesh are equilateral). Then we have for all qh ∈ P0 ∇hqh = ∇̃h(ΠPnc Proof. Since the mesh is uniform we have: ∀σ ∈ E inth , αKσ,Lσ = 12 . Let K ∈ Th. Using definition (2.14) and the gradient formula we get ∇hqh − ∇̃h(ΠPnc σ∈EK∩E (qKσ + qLσ)nK,σ σ∈EK∩E |σ| qKσ nK,σ − (ΠPnc qh)nK,σ dσ. Since qh ∈ P0 we deduce from proposition 2.2 qh dσ = |σ| (ΠPnc qh)(xσ) = (qKσ + qLσ) if σ ∈ E inth , |σ| qKσ if σ ∈ Eexth . Plugging this into the equation above, we get ∇hqh|K = ∇̃h(ΠPnc qh)|K . Lemma 4.2. We assume that the mesh is uniform. There exists a constant C > 0 such that ∀ qh ∈ P0 ∩ L20 , sup vh∈P0\{0} − (qh, divh vh) ‖vh‖h ≥ C h ‖ΠPnc qh‖1,h. Proof. If qh = 0 the result is trivial. Let qh ∈ P0 ∩ L20\{0}. Let vh = ∇hqh ∈ P0\{0}. Using proposition 4.6 we have −(qh, divhvh) = (vh,∇hqh) = |∇hqh|2 = |∇hqh| |vh|. Let χΩ be the characteristic function of Ω. Putting ψ = χΩ in (2.11) we get qh ∈ L20. So according to (2.10) and(4.1) we have |∇hqh| = ∣∣∣∇̃h(ΠPnc ∣∣∣ ≥ C ‖ΠPnc qh‖1,h. On the other hand, according to proposition 2.1: |vh| ≥ C h ‖vh‖h. Therefore −(qh, divhvh) ≥ C h ‖ΠPnc qh‖1,h ‖vh‖h. Proposition 4.7. We assume that the mesh is uniform. There exists a constant C > 0 such that for all qh ∈ P0 ∩ L20 vh∈P0\{0} − (qh, divh vh) ‖vh‖h ≥ C |ΠPnc 14 S. ZIMMERMANN Proof. If qh = 0 the result is clear. Let qh ∈ P0 ∩ L20\{0}. According to (4.9) there exists v ∈ H10 such that (4.10) divv = −ΠPnc qh and ‖v‖1 ≤ C |ΠPnc We set vh = ΠPc v. We want to estimate − qh, divh(ΠP0vh) . Since ∇hqh ∈ P0 we deduce from proposition 4.6 qh, divh(ΠP0vh) = (ΠP0vh,∇hqh) = (vh,∇hqh). Splitting the last term we get (4.11) − qh, divh(ΠP0vh) = (v,∇hqh)− (v − vh,∇hqh). One one hand, integrating by parts, we get (v,∇hqh) = −(ΠPnc qh, divv) + (ΠPnc qh) (v · nK,σ) dσ. According to (4.10) we have −(ΠPnc qh, divv) = |ΠPnc qh|2. Moreover (ΠPnc qh) (v · nK,σ) dσ = σ∈Eint (ΠPnc qh) (v · nKσ ,σ) dσ since v|∂Ω = 0. Using [2] p.269 and (4.10) we have ∣∣∣∣∣∣ σ∈Eint (ΠPnc qh) (v · nK,σ) dσ ∣∣∣∣∣∣ ≤ C h ‖v‖1 ‖ΠPnc qh‖1,h ≤ C h |ΠPnc qh| ‖ΠPnc qh‖1,h. So we get (4.12) (v,∇hqh) ≥ (|ΠPnc qh| − C h ‖ΠPnc qh‖1,h) |ΠPnc On the other hand, using lemma 4.1 and the Cauchy-Schwarz inequality |(v − vh,∇hqh)| = |(v − vh, ∇̃h(ΠPnc qh))| ≤ |v − vh| |∇̃h(ΠPnc qh)|. Using (2.9) and (4.10) we get |v − vh| = |v −ΠPc v| ≤ C h ‖v‖1 ≤ C h |ΠPnc |(v − vh,∇hqh)| ≤ C h |ΠPnc qh| |∇̃h(ΠPnc qh)| ≤ C h |ΠPnc qh| ‖ΠPnc qh‖1,h. Let us plug this estimate and (4.12) into (4.11). We get qh, divh(ΠP0vh) ≥ (|ΠPnc qh| − C h ‖ΠPnc qh‖1,h) |ΠPnc We now introduce the norm ‖.‖h. We have vh = ΠPc v ∈ Pc1 ⊂ H1. Thus, using [6] p. 776, we get ‖ΠP0vh‖h ≤ C ‖vh‖1. Since ΠPc1 is stable for the H 1 norm, we deduce from (4.10) ‖vh‖1 = ‖ΠPc v‖1 ≤ ‖v‖1 ≤ C |ΠPnc Therefore ‖ΠP0vh‖h ≤ C |ΠPnc1 qh|. Using this inequality in (4.3) we obtain that there exists constants C1 > 0 and C2 > 0 such that qh, divh(ΠP0vh) C1 |ΠPnc qh| − C2 h ‖ΠPnc qh‖1,h ‖ΠP0vh‖h. We deduce from this vh∈P0\{0} − (qh, divh vh) ‖vh‖h ≥ C1 |ΠPnc qh| − C2 h ‖ΠPnc qh‖1,h. A COLOCATED FINITE VOLUME SCHEME FOR THE NAVIER-STOKES EQUATIONS 15 Let us combine this with lemma 4.2. Since ∀ t ≥ 0 , max C t , C1 |ΠPnc qh| − C2 t ≥ C C1 C + C2 |ΠPnc we get the result. 4.4. Properties of the discrete laplacian. We first prove the coercivity of the discrete laplacian. Proposition 4.8. For all uh ∈ P0 and vh ∈ P0 we have −(∆̃huh,uh) = ‖uh‖2h − (∆̃huh,vh) ≤ ‖uh‖h ‖vh‖h. Proof. Using definition (2.3) and writing the sum over the triangles as a sum over the edges, we have −(∆̃huh,vh) = − σ∈EK∩E τσ (uLσ − uK)− σ∈EK∩E τσ uK σ∈Eint τσ (vLσ − vK) · (uLσ − uK) + σ∈EK∩E τσ uK · vK . We get the first half of the result by taking vh = uh. On the other hand, using the Cauchy-Schwarz inequality and the algebraic identity a b+c d ≤ a2 + c2 b2 + d2, we get the second half. If v ∈ H2, we have |∆v| ≤ ‖v‖2. The operator ∆h shares a similar property. Proposition 4.9. There exists a constant C > 0 such that for all v ∈ H2 ∣∣∣∆̃h(Π̃P0v) ∣∣∣ ≤ C ‖v‖2. Proof. Let vh = Π̃P0v. Let K ∈ Th. According to definition (2.16) (4.13) ∆̃hvh|K = σ∈EK∩E τσ (v(xLσ )− v(xK))− σ∈EK∩E τσ v(xK ). Let us first assume that v = (v1, v2) ∈ (C∞0 )2. Let i ∈ {1, 2}. If σ ∈ EK ∩ E inth and x ∈ σ we have the Taylor expansions vi(xLσ ) = vi(x)+∇vi(x)·(xLσ−x)+ H(vi)(txLσ+(1−t)x)(xLσ−x)·(xLσ−x) t dt , vi(xK) = vi(x)+∇vi(x)·(xK−x)+ H(vi)(txK+(1−t)x)(xK−x)·(xK−x) t dt , ∇vi(x) = ∇vi(xK)− ∇∇vi(txK + (1− t)x)(xK − x) dt. The notation H(vi) refers to the hessian matrix of vi. Plugging the last expansion into the two others and integrating over σ, we get vi(xLσ )− vi(x) dx = ∇vi(xK) · (xLσ − xσ)−A vi(xK)− vi(x) dx = ∇vi(xK) · (xK − xσ)−Aσ,iK +B The terms A T and B T are the same as in (4.6) and (4.7), with vi instead of q. We substract these equations. Since xLσ − xK = dσ nK,σ we infer from (2.1) vi(xLσ )− vi(xK) = ∇vi(xK) · nK,σ + −Aσ,iLσ +B 16 S. ZIMMERMANN Let us consider now the case σ ∈ EK ∩Eexth . If x ∈ σ we have the Taylor expansions vi(xK) = vi(x)+∇vi(x)·(xK−x)+ H(vi)(txK+(1−t)x)(xK−x)·(xK−x) t dt , ∇vi(x) = ∇vi(xK)− ∇∇vi(txK + (1− t)x)(xK − x) dt. Since vi ∈ C∞0 we have vi(x) = 0. We plug the last expansion into the other and integrate over σ. Since xK − xσ = −dσ nK,σ we deduce from (2.1) −τσ vi(xK) = ∇vi(xK) · nK,σ + Thus we get σ∈EK∩E vi(xLσ )− vi(xK) σ∈EK∩E τσ vi(xK) ∇vi(xK) · |σ|nK,σ + where we have set for all edge σ ∈ EK ∩ E inth −Aσ,iLσ +B and for all edge σ ∈ EK ∩ Eexth : Riσ = 1dσ K − B . Since |σ|nK,σ = 0, setting Rσ = (R σ), we get σ∈EK∩E v(xLσ )− v(xK ) σ∈EK∩E τσ v(xK) = Since the space (C∞0 )2 is dense in H2, one checks that this equation still holds for v ∈ H2. Using (4.13) we infer from it ∣∣∣∆̃hvh ∣∣∣∆̃hvh|K |Rσ|2. Using estimates (4.6) and (4.7) we obtain ∣∣∣∆̃hvh |∇∇vi|2 + |H(vi)|2 dx ≤ C ‖v‖22. 5. Stability of the scheme We now use the results of section 4 to prove the stability of the scheme. We first show an estimate for the computed velocity (theorem 5.1). We then state a similar result for the increments in time (lemma 5.2). Using the inf-sup condition (proposition 4.7), we infer from it some estimates on the pressure (theorem 5.2). Lemma 5.1. For all m ∈ {0, . . . , N} and n ∈ {0, . . . , N} we have (umh ,∇hpnh) = 0 , |umh |2 − |ũmh |2 + |umh − ũmh |2 = 0. Proof. First, using propositions 3.1 and 4.6, we get (umh ,∇hpnh) = −(pnh, divhumh ) = 0. Thus we deduce from (3.4) 2 (umh ,u h − ũmh ) = − umh ,∇h(pmh − pm−1h ) A COLOCATED FINITE VOLUME SCHEME FOR THE NAVIER-STOKES EQUATIONS 17 Using the algebraic identity 2 a (a− b) = a2 − b2 + (a− b)2 we get 2 (umh ,u h − ũmh ) = |umh |2 − |ũmh |2 + |umh − ũmh |2 = 0. We introduce the following hypothesis on the initial data. (H1) There exists C > 0 such that |u0h|+ |u1h|+ k|∇hp1h| ≤ C. Hypothesis (H1) is fulfilled if we set u0h = ΠRT0u0 and we use a semi-implicit Euler scheme to compute u1h. We have the following result. Theorem 5.1. We assume that the initial values of the scheme fulfill (H1). For all m ∈ {2, . . . , N} we have |umh |2 + k ‖ũnh‖2h ≤ C. Proof. Let m ∈ {2, . . . , N} and n ∈ {1, . . . ,m− 1}. Taking the scalar product of (3.2) with 4 k ũn+1h we get 3 ũn+1h − 4unh + u , 4 k ũn+1h − 4 k (∆̃hũ h , ũ +4 k bh(2u h − un−1h , ũ h , ũ h ) + 4 k (∇hp h, ũ h ) = 4 k (f h , ũ h ).(5.1) First of all, using lemma 5.1, we get as in [12] ũn+1h , 3 ũn+1h − 4unh + u = |un+1h | 2 − |unh|2 + 6 |ũn+1h − u 2 + |2un+1h − u h |2 − |2unh − un−1h | +|un+1h − 2u h + u According to proposition 4.8 we have − 4 k (∆̃hũ h , ũ h ) = ‖ũn+1h ‖2h. Also, using lemma 5.1 and (3.4), we have 4 k (∇hpnh, ũn+1h ) = 4 k (∇hp h, ũ h − u (|∇pn+1h | 2 − |∇pnh|2 − |∇pn+1h −∇p h|2). Multiplying (3.4) by 4k∇h(pn+1h − pnh) and using the Young inequality we get |∇(pn+1h − p h)|2 ≤ 3 |un+1h − ũ According to proposition 4.2 we have 4 k bh(2u h − u h , ũ h , ũ h ) ≥ 0. At last using the Cauchy-Schwarz inequality, (2.5) and (3.1) we have 4 k (fn+1h , ũ h ) ≤ 4 k |f h | |ũ h | ≤ C k ‖f‖C(0,T ;L2) ‖ũ h ‖h. Using the Young inequality we get 4 k (fn+1h , ũ h ) ≤ 3 k ‖ũ h + C k ‖f‖2C(0,T ;L2). Let us plug these estimates into (5.1). We get |un+1h | 2 − |unh|2 + |2un+1h − u h|2 − |2unh − un−1h | 2 + |un+1h − 2u h + u +3 |ũn+1h − u 2 + k ‖ũn+1h ‖ (|∇hpn+1h | 2 − |∇hpnh|2) ≤ C k. 18 S. ZIMMERMANN Summing from n = 1 to m− 1 we have |umh |2 + |2umh − um−1h | 2 + 3 |ũn+1h − u 2 + k ‖ũn+1h ‖ |∇hpmh |2 ≤ C + 4 |u1h|2 + |2u1h − u0h|2 + k2 |∇hp1h|2. Using hypothesis (H1) we get the result. We now want to estimate the computed pressure. From now on, we make the following hypothesis on the data f ∈ C(0, T ;L2) , ft ∈ L2(0, T ;L2) , u0 ∈ H2 ∩H10 , divu0 = 0. For all sequence (qm)m∈N we define the sequence (δq m)m∈N∗ by setting δq qm − qm−1 for m ≥ 1. We set δ = (δ)2. If the data u0 and f fulfill a compatibility condition [13] there exists a solution (u, p) to the equations (1.1)–(1.2) such that u ∈ C(0, T ;H2) , ut ∈ C(0, T ;L2) , ∇p ∈ C(0, T ;L2). We introduce the following hypothesis on the initial values of the scheme: there exists a constant C > 0 such that (H2) |u0h − u0|+ ‖u1h − u(t1)‖∞ + |p1h − p(t1)| ≤ C h , |u1h − u0h| ≤ C k. One checks easily that this hypothesis implies (H1). We have the following result. Lemma 5.2. We assume that the initial values of the scheme fulfill (H2). Then there exists a constant C > 0 such that for all m ∈ {1, . . . , N} (5.2) |δumh | ≤ C. Proof. We prove the result by induction. The result holds for m = 1 thanks to hypothesis (H2). Let us consider the case m = 2. We set ũ1h = u h. Let u h ∈ P0 given by (5.3) u−1h = 4u h − 3u1h + h − 2 k b̃h(u0h, ũ1h)− 2 k∇hp1h − 2 k f1h . We substract this equation from equation (3.4) written for n = 1. Since b̃h(2u h − u0h, ũ2h)− b̃h(u0h, ũ1h) = b̃(2u1h − 2u0h, ũ2h) + b̃h(u0h, δũ2h) , upon setting δu0h = u h − u h , we get 3 δũ2h − 4 δu1h + δu0h ∆̃h(δũ h) + b̃h(2u h − 2u0h, ũ2h) + b̃h(u0h, δũ2h) = δf2h . Taking the scalar product with 4 k δũ2h we get 3 δũ2h − 4 δu1h + δu0h, δũ2h ∆̃h(δũ h), δũ +4 k bh(u h, δũ h, δũ h) + 4 k bh(2u h − 2u0h, ũ2h, δũ2h) = 4 k (δf2h , δũ2h).(5.4) According to proposition 4.3 we have 4 k |bh(2u1h − 2u0h, ũ2h, δũ2h)| ≤ C k |2u1h − 2u0h| ‖ũ2h‖h ‖δũ2h‖h ; so that, using hypothesis (H2) ∣∣bh(2u1h − 2u0h, ũ2h, δũ2h) ∣∣ ≤ C k2 ‖ũ2h‖h ‖δũ2h‖h. From the Young inequality and theorem 5.1 we deduce ∣∣bh(2u1h − u0h, ũ2h, ũ2h − ũ1h) ∣∣ ≤ k ‖δũ2h‖2h + C k3 ‖ũ2h‖2h ≤ ‖δũ2h‖2h + C k2. A COLOCATED FINITE VOLUME SCHEME FOR THE NAVIER-STOKES EQUATIONS 19 On the other hand δf2h = f h − f1h = ΠP0 f(t2)−ΠP0 f(t1) = ΠP0 (∫ t2 ft(s) ds Since ΠP0 is stable for the L 2 norm, using the Cauchy-Schwarz inequality, we get |δf2h | ≤ |ft(s)| ds ≤ (∫ t2 |ft(s)|2 ds k ‖ft‖L2(0,T ;L2). 4 k |(δf2h , δũ2h)| ≤ 4 k |δf2h | |δũ2h| ≤ C k3/2 |δũ2h|. So that, using (2.5) and the Young inequality 4 k |(δf2h , δũ2h)| ≤ C k3/2 ‖δũ2h‖h ≤ ‖δũ2h‖2h + C k2. The other terms in (5.4) are dealt with as in the prooof of theroem 5.1. We get (5.5) |δu2h|2 ≤ |δu1h|2 + |2 δu1h − δu0h|2. We know ((5.2) for m = 1) that |δu1h|2 ≤ C k2. It remains to estimate the term |2 δu1h − δu0h|2. According to (5.3) 2 δu1h − δu0h = −δu1h + h − 2 k b̃h(u0h,u1h)− 2 k∇hp1h − 2 k f1h ; by taking the scalar product with 2 δu1h − u0h and using the Cauchy-Schwarz in- equality we get |2 δu1h − δu0h|2 ≤ 2 k ( |δu1h| |∆̃hu1h|+ |∇hp1h|+ |f1h | |2 δu1h − δu0h| + 2 k ∣∣b(u0h, ũ h, 2 δu h − δu0h) ∣∣ .(5.6) Let us bound the terms between braces. First, we have h = ∆̃h u1h − Π̃P0u(t1) + ∆̃h ΠP0u(t1) On one hand, according to proposition 4.8 ∣∣∣∆̃h u1h − Π̃P0u(t1) u1h − Π̃P0u(t1) , ∆̃h u1h − Π̃P0u(t1) ≤ ‖∆̃h u1h − Π̃P0u(t1) ‖h ‖u1h − Π̃P0u(t1)‖h. Applying proposition 2.1 we get ∣∣∣∆̃h u1h − Π̃P0u(t1) u1h − Π̃P0u(t1) | |u1h − Π̃P0u(t1)|. Using the embedding L∞ ⊂ L2 we have |u1h − Π̃P0u(t1)| = |Π̃P0(u1h − u(t1))| ≤ ‖Π̃P0(u1h − u(t1))‖∞ ; since Π̃P0 is stable for the L ∞ norm, we get using hypothesis (H2) |u1h − Π̃P0u(t1)| ≤ ‖u1h − u(t1)‖∞ ≤ C h2. Therefore ∣∣∣∆̃h u1h − Π̃P0u(t1) )∣∣∣ ≤ C. And according to proposition 4.9 ∣∣∣∆̃h ΠP0u(t1) )∣∣∣ ≤ C ‖u(t1)‖ ≤ C ‖u‖C(0,T ;H2). Hence |∆̃hu1h| ≤ C. Let us now bound the pressure term in (5.6). We have ∇hp1h = ∇h p1h − Π̃P0p(t1) Π̃P0p(t1) − ΠP0∇p(t1) +ΠP0∇p(t1). 20 S. ZIMMERMANN According to proposition 4.4 we have ∣∣∣∇h p1h − Π̃P0p(t1) )∣∣∣ ≤ Ch |p h − Π̃P0p(t1)|. Using (2.8) we get ∣∣∣∇h p1h − Π̃P0p(t1) )∣∣∣ ≤ C ‖p(t1)‖2 ≤ C ‖p‖C(0,T ;H2). Since P0 is stable for the L 2 norm we have |ΠP0∇p(t1)| ≤ |∇p(t1)| ≤ ‖p‖C(0,T ;H1). Using proposition 4.5 to treat last term we get |∇hp1h| ≤ C. And according to (3.1) and (5.2) for m = 1 we have + |f1h | ≤ C. We are left with the term∣∣bh(u0h, ũ1h, 2 δu1h − δu0h) ∣∣ in (5.6). We use the following splitting b̃h(u h) = b̃h(u h −ΠRT0u0,u1h) + b̃h ΠRT0u0,u h − Π̃P0u(t1) + b̃h ΠRT0u0, Π̃P0u(t1) Let us take the scalar product with 2 δu1h − δu0h. We get h, 2 δu h − δu0h) = B1 +B2 +B3 B1 = bh(u h −ΠRT0u0,u1h, 2 δu1h − δu0h) , B2 = bh ΠRT0u0,u h − Π̃P0u(t1), 2 δu1h − δu0h ΠRT0u0, Π̃P0u(t1) , 2 δu1h − δu0h Applying propositions 2.1 and 4.3 we have |B1| ≤ |u0h −ΠRT0u0| ‖u1h‖h |2 δu1h − δu0h|. According to (2.8) and (2.13) we have have |u0h−ΠRT0u0| = |ΠP0u0 −ΠRT0u0| ≤ |ΠP0u0 −u0|+ |u0−ΠRT0u0| ≤ C h ‖u0‖1. According to proposition 4.8 and (2.5) ‖u1h‖2h = −(∆̃hu1h,u1h) ≤ |∆̃hu1h| |u1h| ≤ C |∆̃hu1h| ‖u1h‖h ; since |∆̃hu1h| is bounded we get ‖u1h‖h ≤ C. Hence |B1| ≤ C |2 δu1h − δu0h|. In a similar way, using propositions 2.1 and 4.3, we get |B2| ≤ |ΠRT0u0| |u1h − Π̃P0u(t1)| |2 δu1h − δu0h|. We have |ΠRT0u0| ≤ |ΠRT0u0 − u0| + |u0| ≤ C h ‖u0‖1 + |u0| ≤ C ‖u0‖1. Using moreover (5) we get |B2| ≤ C |2 δu1h − δu0h|. Lastly using the following splitting ΠRT0u0, Π̃P0u(t1) ΠRT0u0, Π̃P0u(t1) −ΠP0 b̃ u0,u(t1) + ΠP0 b̃ u0,u(t1) we have B3 = B31 +B32 with B31 = ΠRT0u0, Π̃P0u(t1) −ΠP0 b̃ u0,u(t1) , 2δu1h − δu0h B32 = ΠP0 b̃ u0,u(t1) , 2δu1h − δu0h We have |B31| ≤ ‖b̃h ΠRT0u0, Π̃P0u(t1) −ΠP0 b̃ u0,u(t1) ‖−1,h ‖2δu1h − δu0h‖h A COLOCATED FINITE VOLUME SCHEME FOR THE NAVIER-STOKES EQUATIONS 21 So that, using proposition 4.1 |B31| ≤ C h ‖u0‖2 ‖u(t1)‖2 ‖2 δu1h − δu0h‖h. Using proposition 2.1 we obtain |B31| ≤ C ‖u0‖2 ‖u‖C(0,T ;H2) |2 δu1h − δu0h|. Let us now bound B32. Using the Cauchy-Schwarz inequality and the stability of ΠP0 for the L 2 norm, we have |B32| ≤ ∣∣∣ΠP0 b̃ u0,u(t1) )∣∣∣ |2 δu1h − δu0h| ≤ ∣∣∣b̃ u0,u(t1) )∣∣∣ |2 δu1h − δu0h|. Integrating by parts, we deduce from (4.1) ∣∣∣b̃ u0,u(t1) )∣∣∣ ≤ |u0 · ∇ui(t1)| ≤ |u0| ‖u(t1)‖2 ≤ C |u0| ‖u‖C(0,T ;H2). Thus |B32| ≤ C |2 δu1h − δu0h|. By gathering the estimates for B1, B2, B3 we get ∣∣bh(u0h,u h, 2 δu h − δu0h) ∣∣ ≤ C. Thus we have bounded the right-hand side in (5.6). We infer from it |2 δu1h − δu0h| ≤ C k. Plugging this estimate into (5.5) and using (5.2) for m = 1, we get (5.2) for m = 2. Let m ∈ {3, . . . , N − 1}. We assume that the induction hypothesis is satisfied up to rank n = m− 1. Let us substract equation (3.2) with the same for n− 1. Since the operator b̃h is bilinear we get 3 δũn+1h − 4 δunh + δu ∆̃h(δũ h ) + b̃h(2 δu h − δun−1h , ũ + b̃h(2u h − un−1h , δũ h ) +∇h(δp h) = δf Let us take the scalar product with 4 k δũn+1h . We get 3 δũn+1h − 4 δunh + δu , 4 k δũn+1h − 4 k ∆̃h(δũ h ), δũ +4 k bh(2 δu h − δun−1h , ũ h , δũ h ) + 4 k bh(2u h − un−1h , δũ h , δũ ∇h(δpnh), δũn+1h = 4 k (δfn+1h , δũ According to proposition 4.3 we have ∣∣4 k bh(2 δunh − δu h , ũ h , δũ ∣∣ ≤ C k |2 δunh − δu h | ‖ũ h ‖h ‖δũ h ‖h. Using the induction hypothesis we get ∣∣4 k bh(2 δunh − δu h , ũ h , δũ ∣∣ ≤ C k2 ‖ũn+1h ‖h ‖δũ h ‖h. Using the Young inequality and (5.1) we infer that ∣∣4 k bh(2 δunh − δun−1h , ũ h , δũ ∣∣ ≤ k ‖δũn+1h ‖ h + C k The other terms are treated like the case m = 2. We finally obtain (5.2). Theorem 5.2. We assume that the initial values of the scheme fulfull (H2). There exists a constant C > 0 such that for all m ∈ {2, . . . , N} |ΠPnc pnh|2 ≤ C. 22 S. ZIMMERMANN Proof. Let m ∈ {2, . . . , N}. We set n = m− 1. Using the inf-sup condition (4.7) and proposition 4.6, we get that there exists vh ∈ P0\{0} such that (5.7) C ‖vh‖h |ΠPnc h | ≤ −(p h , divh vh) = (∇hp h ,vh). Plugging (3.4) into (3.2) we have ∇hpn+1h = − 3un+1h − 4unh + u ∆̃hũ h − b̃h(2u h − un−1h , ũ h ) + f so that (∇hpn+1h ,vh) = − 3un+1h − 4unh + u ∆̃hũ h ,vh − bh(2unh − un−1h , ũ h ,vh) + (f h ,vh). Using the Cauchy-Schwarz inequality, (2.5) and (3.1) we have 3un+1h − 4unh + u )∣∣∣∣ ≤ C 3un+1h − 4unh + u ∣∣∣∣ ‖vh‖h (fn+1h ,vh) ≤ |f h | |vh| ≤ C |vh| ≤ C ‖vh‖h , Thanks to proposition 4.3 and theorem 5.1 we have ∣∣bh(2unh − un−1h , ũ h ,vh) 2 |unh|+ |un−1h | ‖ũn+1h ‖h ‖vh‖h ≤ C ‖ũ h ‖h ‖vh‖h. And according to proposition 4.8 we have ∆̃hũ h ,vh ≤ ‖ũn+1h ‖h ‖vh‖h. Thus (∇hpn+1h ,vh) ≤ C + C |3un+1h − 4unh + u + ‖ũn+1h ‖h ‖vh‖h. Comparing with (5.7) we get |ΠPnc h | ≤ C + C |3un+1h − 4unh + u + ‖ũn+1h ‖h Squaring and summing from n = 1 to m− 1 we obtain |ΠPnc pnh|2 ≤ C + C k |3un+1h − 4unh + u + C k ‖ũn+1h ‖ The last term on the right-hand side is bounded, thanks to theorem 5.1. And since 3un+1h − 4u h + u h = 3(u h − u h)− (unh − u h ) = 3 δu h − δu we deduce from lemma 5.2 |3un+1h − 4unh + u ≤ C k |δunh|2 References [1] S. Boivin , F. Cayre, J. M. Herard, A finite volume method to solve the Navier-Stokes equations for incompressible flows on unstructured meshes, Int. J. Therm. Sci., 39 (2000) 806-825. [2] S. C. Brenner, L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, 2002. [3] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, 1991. [4] J. Chorin, On the convergence of discrete approximations to the Navier-Stokes equations, Math. Comp. 23 (1969) 341-353. A COLOCATED FINITE VOLUME SCHEME FOR THE NAVIER-STOKES EQUATIONS 23 [5] R. Eymard, T. Gallouët, R. Herbin, A cell-centered finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension, IMA J. Nu- mer. Anal. 26 (2006) 326-353. [6] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods. In Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds, North-Holland, 2000. [7] R. Eymard and R. Herbin, A staggered finite volume scheme on general meshes for the Navier-Stokes equations in two space dimensions, Int.J. Finite Volumes (2005). [8] R. Eymard, J. C. Latché and R. Herbin, Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2D or 3D meshes, preprint LATP (2004). [9] S. Faure, Stability of a colocated finite volume scheme for the Navier-Stokes equations, Num. Methods Partial Differential Equations 21(2) (2005) 242-271. [10] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, 1986. [11] J.L. Guermond, Some implementations of projection methods for Navier-Stokes equations, M2AN 30(5) (1996) 637-667. [12] J. L. Guermond, Un résultat de convergence l’ordre deux en temps pour l’approximation des équations de Navier-Stokes par une technique de projection, M2AN 33(1) (1999) 169-189. [13] J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier- Stokes problem. I. Regularity of solutions and second-order error estimates for spatial dis- cretization, SIAM J. Numer. Anal., 19(26) (1982) 275-311. [14] D. Kim and H. Choi, A second-order time-accurate finit volume method for unsteady incom- pressible flow on hybrid unstructured grids, J. Comput. Phys. 162 (2000) 411-428. [15] R. Temam, Sur l’approximation de la solution des équations de Navier-Stokes par la méthode de pas fractionnaires II, Arch. Ration. Mech. Anal. 33 (1969) 377-385. [16] S. Zimmermann, Étude et implémentation de méthodes de volumes finis pour les fluides incompressibles, PhD, Blaise Pascal University, 2006. Department of Mathematics, Centrale Lyon University, 63177 Ecully, FRANCE E-mail : [email protected]

0704.0773

Collective behavior of stock price movements in an emerging market

Collective behavior of stock price movements in an emerging market Raj Kumar Pan∗ and Sitabhra Sinha† The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai - 600 113 India (Dated: October 23, 2018) To investigate the universality of the structure of interactions in different markets, we analyze the cross-correlation matrix C of stock price fluctuations in the National Stock Exchange (NSE) of India. We find that this emerging market exhibits strong correlations in the movement of stock prices compared to developed markets, such as the New York Stock Exchange (NYSE). This is shown to be due to the dominant influence of a common market mode on the stock prices. By comparison, interactions between related stocks, e.g., those belonging to the same business sector, are much weaker. This lack of distinct sector identity in emerging markets is explicitly shown by reconstructing the network of mutually interacting stocks. Spectral analysis of C for NSE reveals that, the few largest eigenvalues deviate from the bulk of the spectrum predicted by random matrix theory, but they are far fewer in number compared to, e.g., NYSE. We show this to be due to the relative weakness of intra-sector interactions between stocks, compared to the market mode, by modeling stock price dynamics with a two-factor model. Our results suggest that the emergence of an internal structure comprising multiple groups of strongly coupled components is a signature of market development. PACS numbers: 89.65.Gh,05.45.Tp,89.75.-k I. INTRODUCTION Financial markets can be considered as complex sys- tems having many interacting elements and exhibiting large fluctuations in their associated observable proper- ties, such as stock price or market index [1, 2]. The state of the market is governed by interactions among its components, which can be either traders or stocks. In addition, market activity is also influenced significantly by the arrival of external information. Statistical proper- ties of stock price fluctuations and correlations between price movements of different stocks have been analyzed by physicists in order to understand and model finan- cial market dynamics [3, 4]. The fluctuation distribution of stock prices is found to follow a power law with ex- ponent α ∼ 3, the so-called “inverse cubic law” [5, 6]. This property is quite robust, and has been seen in de- veloped as well as emerging markets [7]. On the other hand, it is not yet known whether the cross-correlation behavior between stock price fluctuations has a similar universal nature. Although the existence of collective modes have been inferred from the study of market dy- namics, such studies have almost exclusively focussed on developed markets, in particular, the New York Stock Exchange (NYSE). To uncover the structure of interactions among the elements in a financial market, physicists primarily fo- cus on the spectral properties of the correlation ma- trix of stock price movements. Pioneering studies in- vestigated whether the properties of the empirical cor- relation matrix differed from those of a random matrix ∗Electronic address: [email protected] †Electronic address: [email protected] that would have been obtained had the price movements been uncorrelated [8, 9]. Such deviations from the pre- dictions of random matrix theory (RMT) can provide clues about the underlying interactions between various stocks. It was observed that, while the bulk of the eigen- value distribution for the correlation matrix of NYSE and Tokyo Stock Exchange follow the spectrum predicted by RMT [8, 9, 10, 11], the few largest eigenvalues deviate significantly from this. The largest eigenvalue has been identified as representing the influence of the entire mar- ket, common for all stocks, whereas, the remaining large eigenvalues are associated with the different business sec- tors, as indicated by the composition of their correspond- ing eigenvectors [10, 12]. The interaction structure of stocks in NYSE have been reconstructed using filter- ing techniques implementing matrix decomposition [13] or maximum likelihood clustering [14]. Correlation ma- trix analysis has applications in the area of financial risk management, as mutually correlated price movements may indicate the presence of strong interactions between stocks [15]. Such analyses have been performed using as- set trees and asset graphs to obtain the taxonomy of an optimal portfolio of stocks [16, 17, 18]. While it is generally believed that stock prices in emerging markets tend to be relatively more correlated than the developed ones [19], there have been very few studies of the former in terms of analysing the spectral properties of correlation matrices [20, 21, 22, 23]. Most studies of correlated price movements in emerging mar- kets have looked at the synchronicity which measures the incidence of similar (i.e., up or down) price movements across stocks [19, 24]. Although related to correlation the two measures are not same, as correlation also gives the relative magnitude of similarity. In this paper, we an- alyze the cross-correlations among stocks in the Indian financial market, one of the largest emerging markets in http://arxiv.org/abs/0704.0773v2 mailto:[email protected] mailto:[email protected] the world. Our study spans over 1996-2006, a period which coincides with the decade of rapid transformation of the Indian economy after liberalization in the early 1990s. We find that, in terms of the properties of its collec- tive modes, the Indian market shows significant devia- tions from developed markets. As the fluctuation distri- bution of stocks in the Indian market [7, 21, 25] follows the “inverse cubic law” seen in NYSE [6, 26], the devi- ations observed in the correlation properties should be almost entirely due to differences in the nature of inter- action structure in the two markets. Our observation of a higher degree of correlation in the Indian market com- pared to developed markets is found to be the result of a dominant market mode affecting all the stocks, which is further accentuated by the relative absence of clusters of mutually interacting stocks. This is explicitly shown by reconstructing the network of interactions within the market, using a filtered correlation matrix from which the common market influence and random noise has been removed. This procedure give a more accurate represen- tation of the intra-market structure than the commonly used method of constructing minimal spanning tree from the unfiltered correlation matrix [13, 16, 17]. To sup- port the interpretation of our empirical observations, we present a two-factor model of market dynamics in Sec- tion IV. Multi-factor models of market behavior have been used by other groups for explaining various spectral properties of empirical correlation matrices [27, 28, 29]. In this model, we assume the market to consist of several correlated groups of stocks which are also influenced by a common external signal, i.e., market mode. By vary- ing the relative strength of the factor associated with the market mode to that associated with the groups, we show that decreasing the intra-group interactions result in spectral distribution properties similar to that seen for the Indian market. Our results imply that one of the key features signifying the transition of a market from emerging to developed status is the appearance and con- solidation of distinct group identities. II. DATA ANALYZED The National Stock Exchange (NSE) is the largest stock market in India. Having commenced operations from Nov 1994, it is already the world’s third largest stock exchange (after NASDAQ and NYSE) in terms of transactions [30]. It is thus an excellent source of data for studying the correlation structure of price movements in an emerging market. We have considered the daily closing price data of 201 stocks (see Table I) traded in NSE from Jan 1996 to May 2006, which corresponds to 2607 days. This data is ob- tained from the NSE web-site [31] and has been manually corrected by us for stock splitting. The selected stocks were traded over the entire period 1996-2006 and had the minimum number of missing data points (i.e., days for which no price data is available). If the price value of a stock is missing on a particular day, a problem common to data from emerging markets [20], it is assumed that no trading took place on that day, i.e, the price remained the same as the preceding day. For comparison we also consider the daily closing price of 434 stocks of NYSE belonging to the S&P 500 index over the same period as the Indian data. However, the total number of work- ing days is slightly different, viz., 2622 days. This data was obtained from the Yahoo! Finance website [32]. In all our analysis, while comparing with the NSE data, we have used multiple random samples of 201 stocks each, from the set of 434 NYSE stocks. We verified that the re- sults obtained were independent of the particular sample of 201 stocks chosen. To ensure that the missing closing prices in the Indian market data do not result in artifacts leading to spurious divergence from the US market, we have also performed our analysis on synthetic US market data containing the same number of missing data points. Multiple sets of such data were generated from the actual closing price time series by randomly choosing the required number of data points and replacing them with the same value as the preceding day. The resulting analysis showed no significant difference from the results obtained with the original US data. III. THE RETURN CROSS-CORRELATION MATRIX To observe correlation between the price movements of different stocks, we first measure the price fluctua- tions such that the result is independent of the scale of measurement. If Pi(t) is price of the stock i = 1, . . . , N at time t, then the (logarithmic) price return of the ith stock over a time interval ∆t is defined as Ri(t,∆t) ≡ lnPi(t+∆t)− lnPi(t). (1) As different stocks have varying levels of volatility (mea- sured by the standard deviation of its returns) we define the normalized return, ri(t,∆t) ≡ Ri − 〈Ri〉 , (2) where σi ≡ 〈R2i 〉 − 〈Ri〉2, is the standard deviation of Ri and 〈. . .〉 represents time average over the period of observation. We then compute the equal time cross- correlation matrix C, whose element Cij ≡ 〈rirj〉, (3) represents the correlation between returns for stocks i and j. By construction, C is symmetric with Cii = 1 and Cij has a value in the domain [−1, 1]. Fig. 1 shows that, the correlation among stocks in NSE is larger on the average compared to that among the stocks in NYSE. −0.2 0 0.2 0.4 0.6 0.8 ) NYSE FIG. 1: The probability density function of the elements of the correlation matrix C for 201 stocks in the NSE of India and NYSE for the period Jan 1996-May 2006. The mean value of elements of C for NSE and NYSE, 〈Cij〉, are 0.22 and 0.20 respectively. This supports the general belief that developing markets tend to be more correlated than developed ones. To un- derstand the reason behind this excess correlation, we perform an eigenvalue analysis of the correlation matrix. A. Eigenvalue spectrum of correlation matrix If the N return time series of length T are mutually un- correlated, then the resulting random correlation matrix is called a Wishart matrix, whose statistical properties are well known [33]. In the limit N → ∞, T → ∞, such that Q ≡ T/N ≥ 1, the eigenvalue distribution of this random correlation matrix is given by Prm(λ) = (λmax − λ)(λ − λmin) , (4) for λmin ≤ λ ≤ λmax and, 0 otherwise. The bounds of the distribution are given by λmax,min = [1± (1/ Q)]2. We now compare this with the statistical properties of the empirical correlation matrix for the NSE. In the NSE data, there are N = 201 stocks each containing T = 2606 returns; as a result Q = 12.97. Therefore, it follows that, in the absence of any correlation among the stocks, the distribution should be bounded between λmin = 0.52 and λmax = 1.63. As observed in developed markets [8, 9, 10, 11], the bulk of the eigenvalue spectrum P (λ) for the empirical correlation matrix is in agreement with the properties of a random correlation matrix spec- trum Prm(λ), but a few of the largest eigenvalues deviate significantly from the RMT bound (Fig. 2). However, the number of these deviating eigenvalues are relatively few for NSE compared to NYSE. To verify that these outliers are not an artifact of the finite length of the observation period, we have randomly shuffled the return time se- ries for each stock, and then re-calculated the resulting correlation matrix. The eigenvalue distribution for this surrogate matrix matches exactly with the random ma- trix spectrum Prm(λ), indicating that the outliers are not 0 2 4 6 8 Eigenvalue λ 0 20 40 max 0 2 4 6 8 Eigenvalue λ 0 20 40 FIG. 2: The probability density function of the eigenvalues of the correlation matrix C for NSE (top) and NYSE (bot- tom). For comparison, the theoretical distribution predicted by Eq. (4) is shown using broken curves, which overlaps with the distribution obtained from the surrogate correlation ma- trix generated by randomly shuffling each time series. In both figures, the inset shows the largest eigenvalue. due to “measurement noise” but are genuine indicators of correlated movement among the stocks. Therefore, by analyzing the deviating eigenvalues, we may be able to obtain an understanding of the structure of interactions between the stocks in the market. B. Properties of the “deviating” eigenvalues The largest eigenvalue λ0 for the NSE cross-correlation matrix is more than 28 times greater than the maximum predicted by RMT. This is comparable to NYSE, where λ0 is about 26 times greater than the random matrix up- per bound. Upon testing with synthetic US data contain- ing same number of missing data points as in the Indian market, we observed that λ0 remains almost unchanged compared to the value obtained from the original US data. The corresponding eigenvector shows a relatively uniform composition, with all stocks contributing to it and all elements having the same sign (Fig. 3, top). As this is indicative of a common factor that affects all the stocks with the same bias, the largest eigenvalue is associ- ated with the market mode, i.e., the collective response of the entire market to external information [8, 10]. Of more interest for understanding the market structure are the intermediate eigenvalues, i.e., those occurring between the largest eigenvalue and the bulk of the distribution predicted by RMT. For the NYSE, it was shown that corresponding eigenvectors of these eigenvalues are local- ized, i.e., only a small number of stocks, belonging to similar or related businesses, contribute significantly to each of these modes [10, 12]. However, for NSE, although the Technology and the IT & Telecom stocks are domi- nant contributors to the eigenvector corresponding to the third largest eigenvalue, a direct inspection of eigenvector composition does not yield a straightforward interpreta- tion in terms of a related group of stocks corresponding to any particular eigenvalue (Fig. 3). To obtain a quantitative measure of the number of stocks contributing to a given eigenmode, we calculate the inverse participation ratio (IPR), defined for the kth eigenvector as Ik ≡ [uki] 4, where uki are the com- ponents of eigenvector k. An eigenvector having compo- nents with equal value, i.e., uki = 1/ N for all i, has Ik = 1/N . We find this to be approximately true for the eigenvector corresponding to the largest eigenvalue, which represents the market mode. To see how different stocks contribute to the remaining eigenvectors, we note that if a single stock had a dominant contribution in any eigenvector, e.g., uk1 = 1 and uki = 0 for i 6= 1, then Ik = 1 for that eigenvector. Thus, IPR gives the recipro- cal of the number of eigenvector components (and there- fore, stocks) with significant contribution. On the other hand, the average value of Ik, for eigenvectors of a ran- dom correlation matrix obtained by randomly shuffling the time series of each stock, is 〈I〉 = 3/N ≈ 1.49×10−2. Fig. 4 shows that the eigenvalues belonging to the bulk of the spectrum indeed have this value of IPR. But at the lower and higher end of eigenvalues, both the US and Indian markets show deviations, suggesting the ex- istence of localized modes. However, these deviations are much less significant and fewer in number in the lat- ter compared to the former. This implies that distinct groups, whose members are mutually correlated in their price movement, do exist in NYSE, while their existence is far less clear in NSE. C. Filtering the correlation matrix The above analysis suggests the existence of a market- induced correlation across all stocks, which makes it dif- ficult to observe the correlations that might be due to interactions between stocks belonging to the same sec- tor. Therefore, we now use a filtering method to remove market mode, as well as the random noise [13]. The cor- relation matrix is first decomposed as λiuiu i , (5) where λi are the eigenvalues of C sorted in descending order and ui are corresponding eigenvectors. As only the eigenvectors corresponding to the few largest eigenvalues are believed to contain information on significantly cor- related stock groups, the contribution of the intra-group correlations to the C matrix can be written as a partial sum of λαuαu α , where α is the index of the correspond- ing eigenvalue. Thus, the correlation matrix can be de- composed into three parts, corresponding to the market, group and random components: C = Cmarket +Cgroup +Crandom = λ0u0u λiuiu i=Ng+1 λiuiu i , (6) where, Ng is the number of eigenvalues (other than the largest one) which deviates from the bulk of the eigen- value spectrum. For NSE we have chosen Ng = 5. How- ever, the exact value of this choice is not crucial as small changes in Ng do not alter the results, the error involved being limited to the eigenvalues closest to the bulk that have the smallest contribution to Cgroup. Fig. 5 shows the result of decomposing the correlation matrix into the three components, for both the Indian and US markets. Compared to the latter, the distribution of matrix ele- ments of Cgroup in the former shows a significantly trun- cated tail. This indicates that intra-group correlations are not prominent in NSE, whereas they are compara- ble with the overall market correlations in NYSE. It fol- lows that the collective behavior in the Indian market is dominated by external information that affects all stocks. Correspondingly, correlations generated by interactions between stocks, as would be the case for stocks in a given business sector, are much weaker, and hence, such corre- lated sectors would be difficult to observe. We indeed find this to be true when we use the infor- mation in the group correlation matrix to construct the network of interacting stocks [13]. The adjacency matrix A of this network is generated from the group correlation matrix Cgroup by using a threshold cth such that Aij = 1 group > cth, and Aij = 0 otherwise. Thus, a pair of stocks are connected if the group correlation coeffi- cient C group is larger than a preassigned threshold value, cth. To determine an appropriate choice of cth = c observe the number of isolated clusters (a cluster being defined as a group of connected nodes) in the network for a given cth (Fig. 6). We found this number to be much less in NSE compared to that observed in NYSE for any value of cth [13]. Fig. 7 shows the resultant network for c∗ = 0.09, for which the largest number of isolated clus- ters of stocks are obtained. The network has 52 nodes and 298 links partitioned into 3 isolated clusters. From these clusters, only two business sectors can be properly identified, namely the Technology and the Pharmaceuti- cal sectors. The fact that the majority of the NSE stocks cannot be arranged into well-segregated groups reflect- ing business sectors illustrates our conclusion that intra- group interaction is much weaker than the market-wide correlation in the Indian market. B C D E F G H I J K L 0 20 40 60 80 100 120 140 160 180 200 Stocks ( i ) FIG. 3: The absolute values of the eigenvector components ui(λ) of stock i corresponding to the first four largest eigenvalues of C for NSE. The stocks i are arranged by business sectors separated by broken lines. A: Automobile & transport, B: Financial, C: Technology D: Energy, E: Basic materials, F: Consumer goods, G: Consumer discretionary, H: Industrial, I: IT & Telecom, J: Services, K: Healthcare & Pharmaceutical, L: Miscellaneous. Eigenvalue λ Eigenvalue λ FIG. 4: Inverse participation ratio as a function of eigen- value for the correlation matrix C of NSE (top) and NYSE (bottom). The broken line indicates the average value of 〈I〉 = 1.49×10−2 for the eigenvectors of a matrix constructed by randomly shuffling each of the N time series. D. Relating correlation with market evolution We now compare between two different time intervals in the NSE data. For convenience we divide the data set into two non-overlapping parts corresponding to the pe- riods between Jan 1996-Dec 2000 (Period I) and between Jan 2001-May 2006 (Period II). The corresponding corre- lation matrices C are generated following the same set of steps as for the entire data set. The average value for the −0.2 0 0.2 0.4 0.6 Matrix elements C group C market C random −0.2 0 0.2 0.4 0.6 Matrix elements C random C market C group FIG. 5: The distribution of elements of correlation matrix corresponding to the market, Cmarket, the group, Cgroup, and the random interaction, Crandom. For NSE (top) Ng = 5 whereas for NYSE (bottom) Ng = 10. The short tail for the distribution of the Cgroup elements in NSE indicates that the correlation generated by mutual interaction among stocks is relatively weak. elements of the correlation matrix is slightly lower for the later period, suggesting a greater homogeneity between the stocks at the earlier period (Fig. 8). Next, we look at the eigenvalue distribution of C for the two periods (Fig. 9. The Q value for Period I is 6.21, while for Period II it is 6.77. Thus the bounds for the random distribution is almost same in the two cases. In contrast, the largest deviating eigenvalues, λ0, are dif- 0 0.05 0.1 0.15 0.2 0.25 0.3 FIG. 6: The number of isolated clusters in the interaction net- work for NSE stocks as a function of the threshold value cth. For low cth the network consist of a single cluster which con- tains all the nodes, whereas for high cth the network consists only of isolated nodes. ferent: 48.56 for Period I and 45.88 for Period II. This implies the relative dominance of the market mode in the earlier period, again suggesting that with time the market has become less homogeneous. The number of deviating eigenvalues remain the same for the two peri- When the interaction networks between stocks are gen- erated for the two periods, they show less distinction into clearly defined sectors than was obtained with the data for the entire period. This is possibly because the shorter data sets create larger fluctuations in the correlation val- ues, thereby making it difficult to segregate the existing market sectors. However, we do observe that, using the same threshold value for generating networks in the two periods yield, for the later period, isolated clusters that are distinguishable into distinct sub-clusters connected to each other via a few links only, whereas in the earlier period the clusters are much more homogeneous. This implies that as the Indian market is evolving, the inter- actions between stocks are tending to get arranged into clearly identifiable groups. We propose that such struc- tural re-arrangement in the interactions is a hallmark of emerging markets as they evolve into developed ones. IV. MODEL OF MARKET DYNAMICS To understand the relation between the interaction structure among stocks and the eigenvalues of the cor- relation matrix, we perform a multivariate time series analysis using a simple two-factor model of market dy- namics. We assume that the normalized return at time t of the ith stock from the kth business sector can be decomposed into (i) a market factor rm(t), that contains information or signal common to all stocks, (ii) a sector factor rkg (t), representing effects exclusive to stocks in the kth sector, and (iii) an idiosyncratic term, ηi(t), which corresponds to random variations unique for that stock. Thus, rki (t) = βirm(t) + γ g (t) + σiηi(t), (7) where βi, γ i and σi represent relative strengths of the three terms mentioned above, respectively. For simplic- ity, these strengths are assumed to be time independent. We choose rm(t), r g (t) and ηi(t) from a zero mean and unit variance Gaussian distribution. We further assume that the normalized returns ri, also follow Gaussian dis- tribution with zero mean and unit variance. Although the empirically observed return distributions have power law tails, as these distributions are not Levy stable, they will converge to Gaussian if the returns are calcu- lated over sufficiently long intervals. The assumption of unit variance for the returns ensures that the relative strengths of the three terms will follow the relation: 2 + (γki ) 2 + σi 2 = 1. (8) As a result, for each stock we can assign σi and γi in- dependently, and obtain βi from Eq. (8). We choose σi and γi from a uniform distribution having width δ and centered about the mean values σ and γ, respectively. We now simulate an artificial market with N stocks be- longing to K sectors by generating time series of length T for returns rki from the above model. These K sec- tors are composed of n1, n2, . . . , nK stocks such that n1 + n2 + · · ·+ nK = N . The collective behavior is then analysed by constructing the resultant correlation matrix C and obtaining its eigenvalues. Our aim is to relate the spectral properties of C with the underlying structure of the market given by the relative strength of the factors. We first consider the simple case, where the contribution due to market factor is neglected, i.e., βi = 0 for all i, and the strength of sector factor is equal for all stocks within a sector, i.e., γki = γ k, is independent of i. In this case, the spectrum of the correlation matrix is composed of K large eigenvalues, 1+(nj−1)(γj)2, where j = 1 . . .K, and N−K small eigenvalues, 1−(γj)2, each with degeneracy nj − 1, where j = 1 . . .K [28]. Now, we consider nonzero market factor which is equal for all stocks i.e., βi = β for all i, and the strength of sector factor is also same for all stocks, i.e., γki = γ (independent of i and k). In this case too, there are K large eigenvalues and N −K small eigenvalues. Our numerical simulations suggest that the largest and the second largest eigenvalues are λ0 ∼ Nβ2, λ1 ∼ nl(1− β2), (9) respectively, where nl is the size of the largest sector, while the N − K small degenerate eigenvalues are 1 − β2 − γ2. We now choose the strength γki and σi from a uniform distribution with mean γ and σ respectively and with width δ = 0.05. Fig. 10 shows the variation of the largest and second largest eigenvalues with σ and γ. The strength of the market factor is determined from Eq.8. Note that, decreasing the strength of the sector factor relative to the market factor results in decreasing the second largest eigenvalue λ1. As Q = T/N is fixed, the RMT bounds for the bulk of the eigenvalue distribution, [λmin, λmax], remain unchanged. Therefore, a decrease in Pajek FIG. 7: The structure of interaction network in the Indian financial market at threshold c∗ = 0.09. The left cluster comprises of mostly Technology stocks, while the middle cluster is composed almost entirely of Healthcare & Pharmaceutical stocks. By contrast, the cluster on the right is not dominated by any particular sector. The node labels indicate the business sector to which a stock belongs and are as specified in the caption to Fig 3. −0.2 0 0.2 0.4 0.6 0.8 NSE ’01−’06 NSE ’96−’00 FIG. 8: The probability density functions of the elements in the correlation matrix C for NSE during (a) the period Jan 1996-Dec 2000 and (b) Jan 2001-May 2006. The mean value of the elements of C for the two periods are 0.23 and 0.21, respectively. λ1 implies that the large intermediate eigenvalues occur closer to the bulk of the spectrum predicted by RMT, as is seen in the case of NSE. The analysis of the model supports our hypothesis that the spectral properties of the correlation matrix for the NSE are consistent with a market in which the effect of information common for all stocks (i.e., the market mode) is dominant, resulting in all stocks exhibiting a significant degree of correlation. V. CONCLUSIONS In conclusion, we demonstrate that the stocks in emerging market are much more correlated than in de- veloped markets. Although, the bulk of the eigenvalue spectrum of the correlation matrix of stocks C in emerg- ing market is similar to that observed for developed mar- kets, the number of eigenvalues deviating from the RMT upper bound are smaller in number. Further, most of the observed correlation among stocks is found to be due to effects common to the entire market, whereas correla- tion due to interaction between stocks belonging to the same business sector are weak. This dominance of the market mode relative to modes arising through interac- 0 1 2 3 4 5 Eigenvalue λ 0 20 40 0 1 2 3 4 5 Eigenvalue λ 0 20 40 FIG. 9: The probability density function of the eigenvalues of the NSE correlation matrix C for the periods (top) Jan 1996-Dec 2000 and (bottom) Jan 2001-May 2006. For com- parison, the theoretical distribution predicted by Eq. (4) is shown using broken curves. In both figures, the inset shows the largest eigenvalue. tions between stocks makes an emerging market appear more correlated than developed markets. Using a simple two-factor model we show that a dominant market factor, relative to the sector factor, results in spectral properties similar to that observed empirically for the Indian mar- ket. Our study helps in understanding the evolution of markets as complex systems, suggesting that strong in- teractions may emerge within groups of stocks as a mar- ket evolves over time. How such self-organization occurs and its relation to other changes that a market undergoes during its development, e.g., large increase in transaction volume, is a question worth pursuing in the future with the tools available to econophysicists. Our paper also makes a significant point regarding the 0.4 00.1 0.4 00.1 FIG. 10: The variation of the largest (top) and second largest (bottom) eigenvalues of the correlation matrix of simulated return in the two-factor model (Eq. 7) with the model pa- rameters γ and σ (corresponding to strength of the sector and idiosyncratic effects, respectively). The matrix is con- structed for N = 200 stocks each with return time series of length T = 2000 days. We assume there to be 10 sectors, each having 20 stocks. physical understanding of markets as complex dynami- cal systems. In recent times, the role of the interaction structure within a market in governing its overall dy- namical properties has come under increasing scrutiny. However, such intra-market interactions affect only very weakly certain market properties, which is underlined by the observation of identical fluctuation behaviour in markets having very different interaction structures, viz., NYSE and NSE [7, 25]. The system can be considered as a single homogeneous entity responding only to external signals in explaining these statistical features, e.g., the price fluctuation distribution. This suggests that the ba- sic assumption behind the earlier approach of studying financial markets as essentially executing random walks in response to independent external shocks [34], which ig- nored the internal structure, may still be considered to be accurate for explaining market fluctuation phenomena. In other words, complex interacting systems like finan- cial markets can have simple mean field-like description for some of their properties. Acknowledgments We thank N. Vishwanathan for assistance in preparing the data for analysis. We also thank M. 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TABLE I: The list of 201 stocks of NSE analyzed in this paper. i Company Sector i Company Sector i Company Sector 1 UCALFUEL Automobiles Transport 68 IBP Energy 135 HIMATSEIDE Industrial 2 MICO Automobiles Transport 69 ESSAROIL Energy 136 BOMDYEING Industrial 3 SHANTIGEAR Automobiles Transport 70 VESUVIUS Energy 137 NAHAREXP Industrial 4 LUMAXIND Automobiles Transport 71 NOCIL Basic Materials 138 MAHAVIRSPG Industrial 5 BAJAJAUTO Automobiles Transport 72 GOODLASNER Basic Materials 139 MARALOVER Industrial 6 HEROHONDA Automobiles Transport 73 SPIC Basic Materials 140 GARDENSILK Industrial 7 MAHSCOOTER Automobiles Transport 74 TIRUMALCHM Basic Materials 141 NAHARSPG Industrial 8 ESCORTS Automobiles Transport 75 TATACHEM Basic Materials 142 SRF Industrial 9 ASHOKLEY Automobiles Transport 76 GHCL Basic Materials 143 CENTENKA Industrial 10 M&M Automobiles Transport 77 GUJALKALI Basic Materials 144 GUJAMBCEM Industrial 11 EICHERMOT Automobiles Transport 78 PIDILITIND Basic Materials 145 GRASIM Industrial 12 HINDMOTOR Automobiles Transport 79 FOSECOIND Basic Materials 146 ACC Industrial 13 PUNJABTRAC Automobiles Transport 80 BASF Basic Materials 147 INDIACEM Industrial 14 SWARAJMAZD Automobiles Transport 81 NIPPONDENR Basic Materials 148 MADRASCEM Industrial 15 SWARAJENG Automobiles Transport 82 LLOYDSTEEL Basic Materials 149 UNITECH Industrial 16 LML Automobiles Transport 83 HINDALC0 Basic Materials 150 HINDSANIT Industrial 17 VARUNSHIP Automobiles Transport 84 SAIL Basic Materials 151 MYSORECEM Industrial 18 APOLLOTYRE Automobiles Transport 85 TATAMETALI Basic Materials 152 HINDCONS Industrial 19 CEAT Automobiles Transport 86 MAHSEAMLES Basic Materials 153 CARBORUNIV Industrial 20 GOETZEIND Automobiles Transport 87 SURYAROSNI Basic Materials 154 SUPREMEIND Industrial 21 MRF Automobiles Transport 88 BILT Basic Materials 155 RUCHISOYA Industrial 22 IDBI Financial 89 TNPL Basic Materials 156 BHARATFORG Industrial 23 HDFCBANK Financial 90 ITC Consumer Goods 157 GESHIPPING Industrial 24 SBIN Financial 91 VSTIND Consumer Goods 158 SUNDRMFAST Industrial 25 ORIENTBANK Financial 92 GODFRYPHLP Consumer Goods 159 SHYAMTELE Telecom 26 KARURVYSYA Financial 93 TATATEA Consumer Goods 160 ITI Telecom 27 LAKSHVILAS Financial 94 HARRMALAYA Consumer Goods 161 HIMACHLFUT Telecom 28 IFCI Financial 95 BALRAMCHIN Consumer Goods 162 MTNL Telecom 29 BANKRAJAS Financial 96 RAJSREESUG Consumer Goods 163 BIRLAERIC Telecom 30 RELCAPITAL Financial 97 KAKATCEM Consumer Goods 164 INDHOTEL Services 31 CHOLAINV Financial 98 SAKHTISUG Consumer Goods 165 EIHOTEL Services 32 FIRSTLEASE Financial 99 DHAMPURSUG Consumer Goods 166 ASIANHOTEL Services 33 BAJAUTOFIN Financial 100 BRITANNIA Consumer Goods 167 HOTELEELA Services 34 SUNDARMFIN Financial 101 SATNAMOVER Consumer Goods 168 FLEX Services 35 HDFC Financial 102 INDSHAVING Consumer Goods 169 ESSELPACK Services 36 LICHSGFIN Financial 103 MIRCELECTR Consumer Discretonary 170 MAX Services 37 CANFINHOME Financial 104 SURAJDIAMN Consumer Discretonary 171 COSMOFILMS Services 38 GICHSGFIN Financial 105 SAMTEL Consumer Discretonary 172 DABUR Health Care 39 TFCILTD Financial 106 VDOCONAPPL Consumer Discretonary 173 COLGATE Health Care 40 TATAELXSI Technology 107 VDOCONINTL Consumer Discretonary 174 GLAXO Health Care 41 MOSERBAER Technology 108 INGERRAND Consumer Discretonary 175 DRREDDY Health Care 42 SATYAMCOMP Technology 109 ELGIEQUIP Consumer Discretonary 176 CIPLA Health Care 43 ROLTA Technology 110 KSBPUMPS Consumer Discretonary 177 RANBAXY Health Care 44 INFOSYSTCH Technology 111 NIRMA Consumer Discretonary 178 SUNPHARMA Health Care 45 MASTEK Technology 112 VOLTAS Consumer Discretonary 179 IPCALAB Health Care 46 WIPRO Technology 113 KECINTL Consumer Discretonary 180 PFIZER Health Care 47 BEML Technology 114 TUBEINVEST Consumer Discretonary 181 EMERCK Health Care 48 ALFALAVAL Technology 115 TITAN Consumer Discretonary 182 NICOLASPIR Health Care 49 RIIL Technology 116 ABB Industrial 183 SHASUNCHEM Health Care 50 GIPCL Energy 117 BHEL Industrial 184 AUROPHARMA Health Care 51 CESC Energy 118 THERMAX Industrial 185 NATCOPHARM Health Care 52 TATAPOWER Energy 119 SIEMENS Industrial 186 HINDLEVER Miscellaneous 53 GUJRATGAS Energy 120 CROMPGREAV Industrial 187 CENTURYTEX Miscellaneous 54 GUJFLUORO Energy 121 HEG Industrial 188 EIDPARRY Miscellaneous 55 HINDOILEXP Energy 122 ESABINDIA Industrial 189 KESORAMIND Miscellaneous 56 ONGC Energy 123 BATAINDIA Industrial 190 ADANIEXPO Miscellaneous 57 COCHINREFN Energy 124 ASIANPAINT Industrial 191 ZEETELE Miscellaneous 58 IPCL Energy 125 ICI Industrial 192 FINCABLES Miscellaneous 59 FINPIPE Energy 126 BERGEPAINT Industrial 193 RAMANEWSPR Miscellaneous 60 TNPETRO Energy 127 GNFC Industrial 194 APOLLOHOSP Miscellaneous 61 SUPPETRO Energy 128 NAGARFERT Industrial 195 THOMASCOOK Miscellaneous 62 DCW Energy 129 DEEPAKFERT Industrial 196 POLYPLEX Miscellaneous 63 CHEMPLAST Energy 130 GSFC Industrial 197 BLUEDART Miscellaneous 64 RELIANCE Energy 131 ZUARIAGRO Industrial 198 GTCIND Miscellaneous 65 HINDPETRO Energy 132 GODAVRFERT Industrial 199 TATAVASHIS Miscellaneous 66 BONGAIREFN Energy 133 ARVINDMILL Industrial 200 CRISIL Miscellaneous 67 BPCL Energy 134 RAYMOND Industrial 201 INDRAYON Miscellaneous

0704.0774

Galaxy morphologies and environment in the Abell 901/902 supercluster from COMBO-17

Mon. Not. R. Astron. Soc. 000, 1–7 () Printed 23 July 2021 (MN LATEX style file v2.2) Galaxy morphologies and environment in the Abell 901/902 supercluster from COMBO-17 K. P. Lane1?, M. E. Gray1, A. Aragón-Salamanca1, C. Wolf2, K. Meisenheimer 3 1. School of Physics and Astronomy, The University of Nottingham, University Park, Nottingham, NG7 2RD 2. Department of Physics, Denys Wilkinson Bldg., University of Oxford, Keble Road, Oxford OX1 3RH 3. Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117, Heidelberg, Germany ABSTRACT We present a morphological study of galaxies in the A901/902 supercluster from the COMBO-17 survey. A total of 570 galaxies with photometric redshifts in the range 0.155 < zphot < 0.185 are visually classified by three independent classifiers to MV = −18. These morphological classifications are compared to local galaxy density, distance from the nearest cluster centre, local surface mass density from weak lensing, and photometric classification. At high local galaxy densities log Σ10/Mpc2 > 1.5 a classical morphology-density relation is found. A correlation is also found between morphology and local projected surface mass den- sity, but no trend is observed with distance to the nearest cluster. This supports the finding that local environment is more important to galaxy morphology than global cluster properties. The breakdown of the morphological catalogue by colour shows a dominance of blue galaxies in the galaxies displaying late-type morphologies and a corresponding dominance of red galax- ies in the early-type population. Using the 17-band photometry from COMBO-17, we further split the supercluster red sequence into old passive galaxies and galaxies with young stars and dust according to the prescription of Wolf et al. (2005). We find that the dusty star-forming population describes an intermediate morphological group between late-type and early-type galaxies, supporting the hypothesis that field and group spiral galaxies are transformed into S0s and, perhaps, ellipticals during cluster infall. Key words: galaxies: clusters: general — galaxies: evolution — galaxies: interactions 1 INTRODUCTION The precise role that environment plays in transforming the mor- phological and star-formation properties of galaxies as they are accreted onto groups and clusters remains unclear. Well-known correlations of cluster properties with environment hinting at evo- lutionary effects include the cluster morphology-density relation (Dressler 1980; Dressler et al. 1997) and the increasing fraction of blue galaxies in clusters at higher redshift (Butcher & Oemler 1978, 1984). A galaxy’s encounters with other galaxies, with a hot intracluster medium (ICM) or with a tidal cluster potential could all be pathways to morphological alteration. However, it is also possi- ble that galaxies in the densest regions formed earlier, evolved more quickly, and thus display more mature evolutionary states. Dressler et al. (1997) find that at z ∼ 0.5 the fraction of S0s is significantly lower than at present, with a proportional increase in the spiral fraction. As the fraction of ellipticals is already sim- ilar to that found locally, it is surmised that the low-z S0 popula- tion formed chiefly from spirals. One way to investigate the effect of environment is to search for transitional objects in the process of transformation. For example, ‘passive spirals’ displaying spiral ? email: [email protected] arms but no signatures of star formation could be identified as an intermediary stage between spirals and S0s (Goto et al. 2003; Pog- gianti 2004). It is therefore important to examine the variation of morphological populations with environment (local mass, gas, and galaxy densities) to determine the physical processes at work. Historically both relaxed and irregular clusters have been the focus of morphological analysis. In fact Dressler (1980) showed that the morphology-density relation is present in both. This im- plies that morphological segregation is already in progress before cluster virialisation, if heirarchical models of structure formation are to be believed. An advanced stage of evolution will lead to clus- ter properties being strongly correlated, and corresponding signa- tures of different environmental influences may then be difficult to untangle. In order to decouple the effects of mass density, galaxy density, cluster-centric distance and gas content on galaxy mor- phologies it is necessary to examine systems still in the process of formation. In a system that has not yet reached equilibrium, the different components of the structure may still be segregated. To this end we morphologically classify galaxies in the Abell 901(a,b)/902 supercluster, a structure consisting of three clusters and associated groups all within 5 × 5 Mpc at z = 0.17. We use imaging data from the COMBO-17 survey of this region (Wolf et al. 2001, 2004), which includes observations from the ESO/MPG c© RAS 2 K. P. Lane et al Wide-Field Imager in broad-band UBV RI and a further 12 medium-band optical filters. The 17-band photometry provides pre- cision photometric redshifts (mean error σz/(1 + z) < 0.01 for R < 20 galaxies) and spectral energy distributions (SEDs). Addi- tionally, a deep R-band image (R < 25.5) with 0.7′′seeing pro- vides excellent image quality for visual classifications and weak gravitational lensing. Further multiwavelength coverage from X- ray to MIR of the 0.5◦ × 0.5◦ field includes observations with XMM-Newton, GALEX, Spitzer, a NIR extension to COMBO-17 using Omega2000 on Calar Alto, plus spectra of the 300 brightest cluster members from 2dF. This extensive data set makes the su- percluster a uniquely well-positioned subject for detailed studies of galaxy evolution. The utility of this field will be greatly extended by a forthcoming HST mosaic as part of the STAGES survey. Additional interest comes from the fact that the supercluster is dynamically complex, with no clear scaling relations between mass from weak lensing maps (Gray et al. 2002; Taylor et al. 2004), galaxy number density, velocity dispersion, and X-ray luminosity (Gray et al, in prep). This provides an excellent opportunity to in- vestigate which environmental properties have the greatest influ- ence on galaxy evolution, bearing in mind the important caveat that all such quantities are seen in projection. Here we investi- gate correlations of visually classified galaxy morphologies with environment. Throughout we use a concordance cosmology with Ωm = 0.27, Ωλ = 0.73, and H0 = 71 km s −1 Mpc−1 so that 1 arcmin = 168 kpc at the redshift of the supercluster. 2 VISUAL GALAXY CLASSIFICATION The cluster sample was chosen by a 0.155 < z < 0.185 cut in photometric redshift at an initial absolute magnitude cut of MV = −19 (all magnitudes are Vega). According to Wolf et al. (2005), σz < 0.01 for this sample gives 99% completeness, if a Gaussian distribution is assumed. Using the same redshift range but increasing the magnitude depth to MV = −18 increases the error in the photo-z to σ ≈ 0.015 and therefore the completeness is ∼ 68% at this luminosity limit. The above sample of cluster galax- ies is estimated to have ∼ 60 non-cluster contaminants (Wolf et al. 2005). This represents 8.6% of our sample and so only introduces small uncertainties into our results. Classifications were performed independently by three of the authors (KPL, MEG, AAS) and combined in order to reduce clas- sifier bias. Galaxies were classified according to de Vaucouleurs’ T-type scheme, in which -5 is elliptical, -2 = S0, 1 = Sa, 2 = Sab, 3 = Sb, 4 = Sbc, 5 = Sc, 6 = Scd, 7 = Sd, 8 = Sdm, 9 = Sm/Irr (de Vau- couleurs 1959). However, in terms of real ability to discriminate by the classifiers this version of the T scale is distorted: for example, a gap of three between E and S0s is far too wide. We therefore adopted an alternative T-type scale where the difference between adjacent T-types is set to one and the gap between a given Hub- ble type and its intermediary (e.g. Sa/Sab) is taken as a difference of 0.5. Additionally, the scheme allows combined classifications in the case where a classifier cannot positively separate two adjacent T-types (the most common example being E/S0 or S0/E). The rel- ative weighting assigned to the primary and secondary choices is discussed below. Comments were recorded if the morphology was abnormal in any way and galaxies were flagged if they showed vi- sual signs of asymmetry, merger, or interaction. Combining the three classification sets gives some measure of the reliability of the classifications. Fig. 1 shows the fraction of galaxies with a disagreement of more than two alternative T- Figure 1. Effects of magnitude on classification precision. The fraction of galaxies with a spread in classification of > 2 alternative T-types from 3 independent classifiers is shown as a function of absolute magnitude. By MV = −18 the fraction of galaxies with a spread in classifications > 2 is almost 20%. types over the three independent classifications. The level of dis- agreement increases with greater magnitude as the galaxies be- come fainter and their structure harder to discern. Above an abso- lute magnitude of -18 the disagreement of > 2 alternative T-types reaches ∼ 20% of galaxies. A magnitude limit of MV = −18 was therefore adopted for further morphological analysis resulting in 570 galaxies. These three classification sets were combined into one over- all set according to a set of rules based on the prescription used by the EDisCS group (Desai et al. 2006). For each galaxy, the final classification was computed by an equally weighted combination of the three sets. In the case of a combined classification where a classifier was unable to discriminate between adjacent T-types, the primary and secondary classifications were assigned 3/4 and 1/4 of that classifier’s weighting, respectively. The T-type with the highest combined weighting was taken as the final classification for that galaxy. If two T-types had equal combined weightings the fi- nal classification was randomly selected between them. Finally, in the case where there were more than two equally weighted types the differences between the types were computed. If one difference was 6 3 on the alternative T-type scale the final classification was again chosen randomly between the two types spanning this gap. Otherwise if there was more than one gap of this size or none 6 3 then the median of the equally highly-weighted T-types was taken as the final classification. The final Hubble type classification cata- log was comprised of 275 ellipticals, 126 S0, 59 Sa, 15 Sab, 49 Sb, 8 Sbc, 9 Sc, 2 Scd, 19 Sm/Irr and 6 unknown galaxies. For use in the following morphology-environment studies spirals and Irr were binned together (167 in all), but given the small numbers it would make no difference if we considered only pure spirals. The visual distinction between an S0 galaxy and an elliptical is difficult, especially if the S0 is face on. Following Dressler et al. (1997) we gauge the reliabilty of our S0/E separation by compar- ing the ellipticities of our classified galaxies to the ellipticities of c© RAS, MNRAS 000, 1–7 Galaxy morphologies and environment in the Abell 901/902 supercluster from COMBO-17 3 Figure 2. Comparison of the ellipticities of the samples classified as Ellip- ticals and S0s in the A901/2 field with the ellipticities of the corresponding morphological samples as found in the Coma cluster (Andreon et al. 1996). The good agreement between the distributions of ellipticities in our sam- ples and those in the Coma cluster, combined with the different ellipticity distributions of our Elliptical and S0 samples, provides some confidence in our ability to separate Elliptical and S0 classes. galaxies in the Coma cluster (Andreon et al. 1996, see Fig 2). El- lipticities were measured from the R-band image using SExtractor (Bertin & Arnouts 1996). Kolmogorov-Smirnov tests give confi- dence intervals of 98.9% for ellipticals and 53.1% for S0s being drawn from the same populations as their Andreon et al. (1996) counterparts. This compares with an ∼ 10−6 confidence that our ellipticals and S0s are drawn from the same population. These re- sults provide some confidence that we have reliably separated these two classes. 3 LINKING MORPHOLOGY TO CLUSTER ENVIRONMENT The morphology-density, or Σ − T , relation has been observed in a large range of galaxy environments, from rich cluster cores, groups, to field densities (e.g. Postman & Geller 1984; Treu et al. 2003). Several physical mechanisms may be responsible for trans- forming galaxy morphologies. These will be effective in different regimes: e.g. in groups where galaxy density increases over the field, but where relative velocities are still low, galaxy-galaxy in- teractions including major mergers may play a large role (Barnes 1992). The high velocities reached by galaxies in the cores of rich clusters make mergers less likely, but increase the efficacy of high- speed interactions such as harassment (Moore et al. 1999). Like- wise, cluster-specific mechanisms such as tidal stripping (Merritt 1983) or the removal of a galaxy’s hot (Larson et al. 1980) or cold (Gunn & Gott 1972) gas by ram-pressure processes require a steep potential or high ICM densities and so are more likely to take effect in the inner regions of clusters. In light of the many physical mechanisms that may be at work in different regimes, we examine galaxy morphology as a function of several different proxies for ‘environment’: local galaxy density, projected mass density from lensing, and distance to the nearest cluster centre. 3.1 Linking morphology to local projected galaxy density The projected local galaxy density, Σ10, was found by calculating the area encompassed by the galaxy in question and its 9 nearest neighbours. Only galaxies to MV = −18 were used when calcu- lating Σ10 as fainter galaxies will decrease completeness. Galaxies lying nearer to the edge of the image field than their 9th nearest neighbour were removed from the catalogue, 37 in all. This was to avoid anomalously low densities resulting from missing neighbouring galaxies outside the field-of-view. Due to uncertainties in the photometric redshifts (σ ≈ 0.015 at the magnitude limit used in this study,MV < −18), we estimate that ∼ 95 potential field galaxies cannot be ruled out as cluster members. This implies the minimum density we can measure is ∼ 3.6Mpc−2. However, this is much smaller than the lowest densities considered in our study (∼ 25Mpc−2). The Σ−T relation (Fig. 3) shows a strong increase in the frac- tion of ellipticals at high densities, a corresponding fall in the frac- tion of spirals and the S0 fractions show no correlation. This is the classical morphology-density relation as seen in numerous other studies (e.g. Oemler 1974; Dressler 1980; Dressler et al. 1997; Treu et al. 2003). Error bars are multinomial and were determined using Monte Carlo simulations. By analysing data in the range 0 < z < 1, including Dressler (1980) at z ∼ 0 and Dressler et al. (1997) at z ∼ 0.5, Smith et al. (2005) and Postman et al. (2005) find that the gradient of the early type Σ−T relation increases with lowering z due to reducing num- bers of spirals and increasing numbers of S0s, especially in high density regions. It is difficult to compare the results presented in Fig. 3 with these studies because they use depths comparable to that used by Dressler (1980). To enable a comparison we cut our sample to the same depth as used by Dressler (1980) (MV = −19.6), after correcting for differences in the cosmology used. Doing so reduces the number of sources in our classified sample by more than a fac- tor of 2 which increases the error bars to point where no trends can be seen. However it is noted that in our samples there are smaller fractions of S0s and larger fractions of Es when compared with c© RAS, MNRAS 000, 1–7 4 K. P. Lane et al Dressler (1980). This could be due to cluster to cluster variation or systematic differences in classification. If we compare our full sample with that of Dressler (1980) relative to the cluster core Σ10, the gradient in the Σ − T relation for our elliptical population is shallower than found in the highest density regions by Dressler (1980). There are also smaller fractions of S0s and larger fractions of spirals as compared with Dressler (1980). This is in agreement with the findings of Smith et al. (2005) and Postman et al. (2005). At lower densities we find larger frac- tions of ellipticals than Dressler (1980). That our fractions of S0s are so much smaller than Dressler (1980), and that our elliptical fractions are so much larger, is unexpected. However, S0 fraction does vary by large amounts from cluster to cluster and the crite- ria used to separate E and S0 classes varies from study to study. We also note that the fraction of early types (elliptical + S0) in our highest density bins is in keeping with the positive trend found in the fraction of early type galaxies with decreasing redshift in high density regions (Smith et al. 2005). Again it is noted that this com- parison is made using our full sample to MV = −18 to enable sample errors to be reduced to the point where trends can be seen. With the adopted cosmology, 1 arcsecond corresponds to 2.80kpc at the cluster redshift. Similarly the seeing limited resolu- tion of our image is 1.96kpc at the cluster redshift. This compares with ∼ 700pc for Postman et al. (2005), using HST ACS data, at z ∼ 1, and ∼ 500pc for Dressler et al. (1997), using HST WFPC2 data, at z ∼ 0.5. Our resolution may result in later type spirals be- ing classified as earlier types since fine structure will be harder to discern. This situation will be improved by using HST ACS data obtained as part of the STAGES project. Fasano et al. (2000) use ground based data with resolution 2-4 kpc at the redshifts of their clusters (0.1 . z . 0.25). Again no comparisons can be drawn when our sample is cut to the same abso- lute magnitude (MV = −20). However, when comparing our full sample we find good agreement between our MD relation and the MD relation found by Fasano et al. (2000) in their high elliptical- concentration clusters. Our data fits the trend of rising S0 fraction, falling Sp fraction and no evolution in the fraction of E with red- shift. We also find our data to be consistent with the rising trends in the NS0/NE and NS0/Nsp fractions with lowering redshift, as presented in Fasano et al. (2000). Note that the dynamic range in our Σ10 measurements is somewhat smaller than in the above studies, particularly at low densities. However, this does not have a significant effect on the comparisons discussed above. 3.2 Linking morphology to projected cluster mass As the mass of a cluster is made up predominantly of dark mat- ter, one might expect various environmental properties, such as lo- cal galaxy density and the intracluster medium (ICM), to trace the potential wells described by the dark matter mass of the cluster. This may well be true in a virialised cluster where there is a po- sitional degeneracy between such environmental factors, however maps of projected mass and galaxy distribution (Gray et al. 2002) and extended X-ray emission (Gray et al, in prep) show that in the A901/902 supercluster such scaling relations are not self-consistent from cluster to cluster. For example, Abell 901b displays a promi- nent mass peak and LX = 2.35 × 1044 erg s−1, yet is relatively deficient in galaxy numbers. This then provides an opportunity to ascertain if cluster mass has a direct effect on galaxy morphology, or whether it is merely a tracer of other morphology affecting envi- ronmental properties. Figure 3. Morphological type, as a fraction of the total, with increasing local density. At high density there is a clear upward trend in ellipticals and a downward trend in spirals. There appears to be no trend in S0s. We ig- nored bins with fewer than 5 galaxies due to the large sampling errors. This corresponds to log Σ10 < 1.3. The upper panel shows the total number of galaxies in each density bin. The projected surface mass density for this region was recon- structed by Gray et al. (2002) from an analysis of weak lensing of faint galaxies in the same COMBO-17 image. The surface mass is measured as a dimensionless quantity κ, where κ = Σ/Σcrit is the ratio of the projected surface mass density to the critical surface mass density for lensing for a fixed source and lens redshift. For the supercluster lens at z = 0.17 and a population of faint lensed galaxies with 〈z〉 ∼ 1 we have Σcrit = 5.0 × 1015hM� Mpc The Gray et al. (2002) map includes smoothing with a Gaussian of σ = 60 arcsec, and the rms noise in the map was estimated as σκ = 0.027 through simulations. Fig. 4 shows a clear relation between projected mass and mor- phology (a κ− T relation) similar to the observed Σ− T relation, but only at high mass densities (corresponding to the 3σ κ error regime). An upward trend with κ is found in ellipticals and a down- ward trend in S0s, however spirals do not show any correlation with projected mass. The different trends observed between Σ10 and κ suggest that both are tracers of morphology. Whether they are in- depenent or, as is more likely, they are tracers of different aspects of the same environmental driver for morphology, remains to be determined. Fewer than 30% of galaxies in the highest κ bins are classified as spirals. This is analogous to the star-formation–κ re- lation found for the supercluster by Gray et al. (2004), where the highest density regions were found to be populated almost exclu- sively by galaxies with quiescent SEDs. 3.3 Linking morphology to cluster radius Any correlation between clustercentric radius and the morphology of a galaxy (R − T relation) will most likely be a reflection of global rather than local properties of the cluster since radius is not a localised quantity. c© RAS, MNRAS 000, 1–7 Galaxy morphologies and environment in the Abell 901/902 supercluster from COMBO-17 5 Figure 4. Morphological type, as a fraction of the total, with increasing projected surface mass density, κ, from lensing. At high surface density an upward trend in ellipticals and a downward trend in S0s is observed. No clear trend is found in the spiral population. The upper panel shows the total number of galaxies in each κ bin. The bin width corresponds to the noise of the κ maps, σκ = 0.027. The clustercentric radius is the distance to the nearest cluster, where the cluster centre was defined as the peak of the κ map, al- though BCGs could have been used without much change. Fig. 5 shows that there is no clear trend between galaxy morphology and cluster radius, with only a small rise in elliptical fractions and a small decrease in late-type and S0 fractions at small radii. These small radii correspond to regions which would be the least affected by the large scale cluster merger. This result may then reflect the virialized cluster cores where radius is degenerate with projected mass. The presence of a relation between morphology and lo- cal galaxy density, combined with this apparent lack of a rela- tion between morphology and clustercentric radius adds further weight to the hypothesis that local conditions have more effect on galaxy morphology than global cluster properties. Previous stud- ies (Dressler et al. 1997) have shown that any radial dependence of morphology is most likely a reflection of the Σ − T relation due to the one-to-one correspondence between density and radius in re- laxed clusters. For the A901/902 system in particular the cluster properties and radius are more decoupled than in relaxed systems due to the dynamical complexity. However, one caveat relevant to both the Σ − T and R − T correlations is the possibility of pro- jection effects, particularly in the region between the A901a and A901b clusters. This possibility has been checked by masking out this region and is found to have no appreciable effect on the above results and hence is ruled out as a source of uncertainty. 4 LINKING MORPHOLOGY TO PHOTOMETRIC CLASSIFICATION In Wolf et al. (2005) the Abell 901/902 cluster galaxy sample ex- amined here was divided into three subpopulations: red, passively Figure 5. Morphological type, as a fraction of the total, with increasing distance to nearest cluster centre. No clear trend is observed with increasing radial distance. Data points with zero sample size have undetermined errors. evolving galaxies; blue star-forming galaxies; and a previously un- known third population of red galaxies revealed by the 17-band photometry. This third population consists of cluster galaxies also located along the cluster red sequence, but containing significant amounts of young stars and dust (hereafter referred to as ‘dusty red galaxies’). Examining the photometric class of each morphological type, Fig. 6 shows a clear trend. The majority of morphologically early- type galaxies (E,S0) are photometrically passive and red. Late-type galaxies (Sb,Sc) are predominantly blue. The third population of dusty red galaxies, on the other hand, shows a distinct distribution of intermediate morphologies. In this case intermediate types have been binned with the next highest integer alternative T-type for clar- ity, e.g. Sab is binned with Sb, Sbc with Sc and so on. Two possible origins for this intermediate population of dusty red galaxies are posed by Wolf et al. (2005). Firstly, that they orig- inate in the blue cloud and are in a state of being transformed into red cluster galaxies or, secondly, that they are the result of mi- nor mergers of infalling galaxies with established cluster galaxies. To try and distinguish between these two formation scenarios the merger or interaction state of each galaxy was noted. Of the classified galaxies which are photometrically found to be dusty and red, only 14.2±9.4% were found by at least one clas- sifier to be in a state of interaction with a neighbouring galaxy or undergoing a merger. This compares to 10.6±6.5% of the clas- sified passive red galaxies and 27.5±8.5% of the classified blue galaxies. Within the uncertainties the three groups have consistent merger/interaction fractions, however, the low fraction of interac- tions/mergers for dusty red galaxies is inconsistent with a minor merger scenario for their formation. However, it should be noted that a large fraction of mergers may go undetected by a visual mor- phological analysis since this technique is only sensitive to asym- metries or tidal features in morphology which may not be present in a minor merger at scales larger than our resolution (∼ 2 kpc). c© RAS, MNRAS 000, 1–7 6 K. P. Lane et al Therefore the minor merger scenario for formation of dusty red galaxies cannot be ruled out, but does look unlikely. The dusty red galaxies represent a significant proportion of the overall galaxy population (22.4%) and do not appear to be a sub- set of the blue or passive red galaxy populations. The differences in morphology are paralleled by differences in average spectra and spatial distributions shown in Wolf et al. (2005). In particular, they occupy regions of medium densities, avoiding high densities nearer the cluster core as well as low density regions in the cluster periph- ery (Wolf et al. 2005). These pieces of evidence would then suggest that one ma- jor route in which infalling galaxies can be incorporated into the cluster is via transformative processes that do not necessarily in- volve mergers. Galaxies entering the cluster may have their star- formation ultimately quenched, but after an initial phase of en- hanced star-formation (Milvang-Jensen et al. 2003; Bamford et al. 2005). A triggered starburst, possibly via interaction with the ICM, would introduce dust via supernovae feedback to produce the tran- sitional dusty, red phase. Ultimately the gas supply will be ex- hausted and star formation quenched, leaving the remaining stars to evolve passively on the red sequence. Dressler et al. (1999) and Poggianti et al. (1999) find similar spectroscopic populations of dusty starburst galaxies, or e(a) galax- ies, at z ∼ 0.4 − 0.5. They attribute these to the progenitors of post-starburst k+a/a+k galaxies. For the COMBO-17 A901/2 field spectra have been obtained for 64 cluster galaxies using the 2dF instrument (see Wolf et al. 2005). The average spectra for dusty red galaxies is seen to be inconsistent with that of k+a galaxies. The weak [OII] emission as well as Hδ absorption observed in the average spectra of these dusty red galaxies are consistent with e(a) type galaxies. To find such a large fraction (22.4%) of poten- tial k+a progenitor galaxies at z ∼ 0.17 is surprising given that Dressler et al. (1999) find that 18% of their cluster sample exhibit k+a/a+k spectral types at z ∼ 0.5. However, the continuing cluster- cluster merger seen in the A901/2 system could well produce an increased incidence of e(a) galaxies due to the large number of in- falling galaxies. 5 CONCLUSIONS The complex dynamics present in the Abell 901/902 systems pro- vide an ideal testing ground for the distinction between the local and global processes driving galaxy morphology. In this paper we have examined relations between visual galaxy morphologies and local measures of environment including galaxy density, projected surface mass density from lensing, and clustercentric radius. The presence of a strong Σ − T and κ − T relations and absence of a corresponding Σ − R relation shows that despite the large scale complexities of the clusters, local conditions are still well corre- lated to morphology. Furthermore, the photometric breakdown of morphologies provides a tantalising glimpse of galaxy transformation during in- fall, in action. By determining the interaction and merger state of each classified galaxy it was shown that the population of red se- quence galaxies with young stars and dust of Wolf et al. (2005) is not likely to be explained by minor mergers of infalling galax- ies with cluster members. It is more plausible then that the dusty red galaxies are experiencing additional, cluster-specific phenom- ena during infall, causing their star-formation to become dust ob- scured and reddened. This would then support a picture of cluster formation in which accreted galaxies can be transformed through Figure 6. The photometric colour of each morphological type as a fraction of the total type population. Dusty red galaxies appear to form an interme- diate regime between star-forming late-type galaxies and early-type passive galaxies. Data points with very large sampling error have been omitted. processes other than major or minor mergers, most likely through induced star-bursts and associated dust obscuration before the gas supply is stripped and/or exhausted, and star-formation stops. This study will be extended in the forthcoming STAGES sur- vey of the A901/902 supercluster. The survey consists of an 80 or- bit mosaic using HST/ACS of the 0.5× 0.5 degree region and will be combined with the 17-band photometric redshifts and other de- tailed multiwavelength data sets. This mosaic will be used for mor- phological classifications not only for the bright end of the cluster luminosity function probed here, but also the dwarf galaxy popula- tion, which may be more sensitive to environmental processes due to their lower gravitational potentials. In this way we will build up an even more detailed picture of galaxy evolution within a complex environment. ACKNOWLEDGMENTS KPL was supported by a PPARC studentship. MEG was supported by an Anne McLaren Research Fellowship from the University of Nottingham. C. Wolf was supported by a PPARC Advanced fellow- ship. Thanks go M. Merrifield and O. Almaini for useful and infor- mative discussions. We thank the anonymous referee for comments which greatly improved the reliability of the results presented. REFERENCES Andreon S., Davoust E., Michard R., Nieto J.-L., Poulain P., 1996, A&A Supp., 116, 429 Bamford S. P., Milvang-Jensen B., Aragón-Salamanca A., Simard L., 2005, MNRAS, 361, 109 Barnes J. E., 1992, ApJ, 393, 484 Bertin E., Arnouts S., 1996, A&A Supp., 117, 393 Butcher H., Oemler A., 1978, ApJ, 226, 559 c© RAS, MNRAS 000, 1–7 Galaxy morphologies and environment in the Abell 901/902 supercluster from COMBO-17 7 Butcher H., Oemler A., 1984, ApJ, 285, 426 de Vaucouleurs G., 1959, Handbuch der Physik, 53, 275 Desai V., Dalcanton J. J., Aragón-Salamanca A., Jablonka P., Pog- gianti B., Gogarten S. M., Simard L., Clowe D., Halliday C., Milvang-Jensen B., Pelló R., 2006, ApJ, p. submitted 2006 Dressler A., 1980, ApJ, 236, 351 Dressler A., Oemler A. J., Couch W. J., Smail I., Ellis R. S., Barger A., Butcher H., Poggianti B. M., Sharples R. M., 1997, ApJ, 490, 577 Dressler A., Smail I., Poggianti B. M., Butcher H., Couch W. J., Ellis R. S., Oemler A. J., 1999, ApJS, 122, 51 Fasano G., Poggianti B. M., Couch W. J., Bettoni D., Kjærgaard P., Moles M., 2000, ApJ, 542, 673 Goto T., Okamura S., Sekiguchi M., Bernardi M., Brinkmann J., Gómez P. L., Harvanek M., Kleinman S. 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F., Rix H.-W., Cimatti A., Hasinger G., Szokoly G., 2004, A&A, 421, 913 c© RAS, MNRAS 000, 1–7 Introduction Visual galaxy classification Linking morphology to cluster environment Linking morphology to local projected galaxy density Linking morphology to projected cluster mass Linking morphology to cluster radius Linking morphology to photometric classification Conclusions

0704.0775

K_0-theory of n-potents in rings and algebras

K0-THEORY OF n-POTENTS IN RINGS AND ALGEBRAS EFTON PARK AND JODY TROUT Abstract. Let n ≥ 2 be an integer. An n-potent is an element e of a ring R such that en = e. In this paper, we study n-potents in matrices over R and use them to construct an abelian group (R). If A is a complex algebra, there is a group isomorphism (A) ∼= K0(A) for all n ≥ 2. However, for algebras over cyclotomic fields, this is not true, in general. We consider Kn as a covariant functor, and show that it is also functorial for a generalization of homomorphism called an n-homomorphism. 1. Introduction For more than forty years, K-theory has been an essential tool in studying rings and algebras [1, 7]. Given a ring R, a simple functorial object associated to R is the abelian group K0(R). There are multi- ple ways of defining K0(R), but the most useful characterization when working with operator algebras is to define K0(R) in terms of idempo- tents (or projections, if an involution is present) in matrix algebras over R; i.e., elements e in Mk(R) for some k with the feature that e 2 = e (p = p∗ = p2 in the involutive case). In this paper, we define, for each natural number n ≥ 2, a group which we denote Kn0 (R). This group is constructed from matrices e over R with the property that en = e; we call such matrices n-potents. We define Kn0 (R) for all rings, unital or not, and show that Kn0 determines a covariant functor from rings to abelian groups. Let Q(n− 1) be the cyclotomic field obtained from the rationals by adjoining the (n− 1)-th roots of unity. We show that Kn0 is half-exact on the subcategory of Q(n − 1)-algebras, and given any such algebra A, we show that Kn0 (A) is isomorphic to a direct sum of n − 1 copies of K0(A). Since a C-algebra A is a Q(n − 1)-algebra for all n, what- ever invariants are contained in Kn0 (A) are already contained in K0(A). However, K 0 for p 6= n may generate new groups for cyclotomic alge- bras, e.g., K40(Q(4)) ∼= Z⊕2Z (Theorem 3.15) which is not isomorphic 2010 Mathematics Subject Classification: 18F30, 19A99, 19K99. http://arxiv.org/abs/0704.0775v2 2 EFTON PARK AND JODY TROUT to K40 (Q(3)) ∼= Z3. Thus, K40 distinguishes between the fields Q(3) and Q(4), but idempotent, and also tripotent (n = 3), K-theory does not. The paper is organized as follows. In Section 2, we define various notions of equivalence on the set of n-potents, and explore the rela- tionships between these equivalence relations. Most of our results in this section mirror analogous facts about idempotents, but in many cases the proofs differ or are more delicate for n-potents. In Section 3, we define n-potent K-theory and study its properties and compute some examples. Finally, in Section 4, we consider n-homomorphisms on rings and algebras [2, 3, 4], and show that n-potent K-theory is functorial for such maps; this is a phenomenon that does not appear in ordinary idempotent K-theory. The authors thank Dana Williams and Tom Shemanske for their helpful comments and suggestions. Note: Unless stated otherwise, all rings and algebras have a unit; i.e., a multiplicative identity, and all ring and algebra homomorphisms are unital. 2. Equivalence of n-potents Fix a natural number n ≥ 2. In this section, we develop the ba- sic theory of n-potents, including various equivalence relations among them. We begin by looking at n-potents over general rings, but even- tually we will specialize to get a well-behaved theory. Definition 2.1. Let R be a ring. An element e in R is called an n-potent if en = e. For n = 2, 3, 4, we use the terms idempotent, tripotent, and quadripotent, respectively. The set of all n-potents in R is denoted Pn(R). We begin with a very simple but useful fact about n-potents: Lemma 2.2. Suppose e is an n-potent. Then en−1 is an idempotent. Proof. (en−1)2 = en−1en−1 = enen−2 = een−2 = en−1. � Definition 2.3. Let e and f be n-potents in a ring R. We say that e and f are algebraically equivalent and write e ∼a f if there exist elements a and b in R such that e = ab and f = ba. We say that e and f are similar and write e ∼s f if there exists an invertible element z in R with the property that f = zez−1. Lemma 2.4. Suppose that e and f are algebraically equivalent n- potents in a ring R. Then the elements a and b described in Definition K0-THEORY WITH n-POTENTS 3 2.3 can be chosen so that a = en−1a = afn−1 = en−1afn−1 b = fn−1b = ben−1 = fn−1ben−1. Proof. Choose elements ã and b̃ in R so that ãb̃ = e and b̃ã = f . Set a = en−1ãfn−1 and b = fn−1b̃en−1. Using Lemma 2.2, we have ab = (en−1ãfn−1)(fn−1b̃en−1) = en−1ãfn−1b̃en−1 = en−1(ãb̃)nen−1 = en−1enen−1 = (en−1)2en = en−1e = en = e. Similarly, ba = f . The two strings of equalities in the statement of the lemma then follow easily. � Proposition 2.5. The relations ∼a and ∼s are equivalence relations on Pn(R). Proof. The only nonobvious point to establish is that ∼a is transitive. Let e, f , and g be elements of Pn(R), and suppose that e ∼a f ∼a g. Choose elements a, b, c and d in R so that e = ab, f = ba = cd, and g = dc, and set s = afn−2c and t = db. Then st = afn−2cdb = afn−1b = a(ba)n−1b = (ab)n = en = e ts = dbafn−2c = dfn−1c = d(cd)n−1c = (dc)n = gn = g. Proposition 2.6. If e and f are similar n-potents in a ring R, then they are algebraically equivalent. Proof. Choose an invertible element z in R such that f = zez−1, and set a = ez−1 and b = zen−1. Then ab = en = e and ba = zenz−1 = f . � As is the case with idempotents, algebraic equivalence does not imply similarity in general. However, we do have the following result, just as for idempotents: Proposition 2.7. Suppose that e and f are algebraically equivalent n-potents in a ring R. Then in the ring M2(R) of 2× 2 matrices over R. 4 EFTON PARK AND JODY TROUT Proof. Choose elements a and b in R so that e = ab and f = ba; without loss of generality, we assume that a and b satisfy the conclusions of Lemma 2.4. Define 1− fn−1 b afn−2 1− en−1 1− en−1 en−1 en−1 1− en−1 Straightforward computation yields that both u2 and v2 equal the iden- tity matrix in M2(R), and thus each is its own inverse. Set z = uv. Then we compute that z−1 = 1− fn−1 b afn−2 1− en−1 1− fn−1 b afn−2 1− en−1 beafn−2 0 since beafn−2 = b(ab)a(ba)n−2 = (ba)n = fn = f. � Definition 2.8. We say n-potents e and f in a ring R are orthogonal if ef = fe = 0, in which case we write e ⊥ f . The next result follows immediately by mathematical induction. Proposition 2.9. Let e and f be orthogonal n-potents in a ring R. Then (e+ f)k = ek + fk. In particular, e+ f is an n-potent. Proposition 2.10. For i = 1, 2, let ei and fi be algebraically equivalent n-potents in a ring R. Suppose that e1 and f1 are orthogonal to e2 and f2, respectively. Then e1 + e2 and f1 + f2 are algebraically equivalent. Proof. For i = 1, 2, choose ai and bi so that ei = aibi, fi = biai, and so that ai and bi satisfy the conclusion of Lemma 2.4. Then a1b2 = a1f 2 b2 = 0. Similarly, b2a1, a2b1, and b1a2 are also zero. Thus (a1 + a2)(b1 + b2) = a1b1 + a2b2 = e1 + e2 (b1 + b2)(a1 + a2) = b1a1 + b2a2 = f1 + f2, whence e1 + e2 is algebraically equivalent to f1 + f2. � Proposition 2.11. Let e and f be n-potents in a ring R. K0-THEORY WITH n-POTENTS 5 (b) If e ⊥ f then e+ f 0 Proof. Define and b = 0 fn−1 en−1 0 in M2(R). Then 0 fn−1 en−1 0 0 fn−1 en−1 0 which establishes the first part of (a); to obtain the second part, simply take f to be zero. To prove (b), first observe that if e ⊥ f , then e + f is an n-potent by Proposition 2.9. Define and b = en−1 fn−1 en−1 fn−1 en efn−1 fen−1 fn en−1 fn−1 en + fn 0 e + f 0 whence the result follows. � Later in this paper we will restrict our attention to n-potent K- theory of cyclotomic algebras: Definition 2.12. For each integer n ≥ 2, the cyclotomic field Q(n−1) is the field obtained by adjoining the (n − 1)st primitive root of unity ζn−1 = e 2πi/(n−1) to the field Q of rational numbers. A cyclotomic algebra is a Q(n− 1)-algebra for some n ≥ 2. Observe that Q(n− 1) ⊂ C, and therefore every C-algebra is canon- ically a Q(n− 1)-algebra for all n. Definition 2.13. Let F be a field and let A be an F-algebra with unit. An n-partition of unity is an ordered n-tuple (e0, e1, . . . , en−1) of idem- potents in A such that (1) e0 + e1 + · · ·+ en−1 = 1; 6 EFTON PARK AND JODY TROUT (2) e0, e1, . . . , en−1 are pairwise orthogonal; i.e., ejek = δjkek for all 0 ≤ j, k ≤ n− 1. Note that e0 = 1 − (e1 + · · · + en−1) is completely determined by e1, e2, . . . , en−1 and is thus redundant in the notation for an n-partition of unity. Cyclotomic algebras admit a distinguished n-partition of unity. Set ω0 = 0 and let ωk = e 2πi(k−1)/(n−1) for 1 ≤ k ≤ n − 1. Note that ω1, . . . , ωn−1 are the (n−1)st roots of unity, and Ωn = {ω0, ω1, . . . , ωn−1} is the set of roots of the polynomial equation xn − x = 0. Theorem 2.14. Let A be a Q(n − 1)-algebra with unit, and suppose e is an n-potent in A. Then there exists a unique n-partition of unity (e0, e1, . . . , en−1) in A such that ωkek. Proof. Let p0, p1, . . . , pn−1 ∈ Q(n− 1)[x] be the Lagrange polynomials pk(x) = j 6=k(x− ωj) j 6=k(ωk − ωj) In particular, p0(x) = 1 − xn−1. Each polynomial pk has degree n − 1 and satisfies pk(ωk) = 1 and pk(ωj) = 0 for all j 6= k. We claim that for all numbers α ∈ Q(n− 1) ⊆ C, pk(α) = p0(α) + · · ·+ pn−1(α) = 1 and that (2) α = ωkpk(α). Indeed, these identities follow from the fact that these polynomial equa- tions have degree n− 1 but are satisfied by the n distinct points in Ωn. Now, given any ωni = ωi in Ωn it follows that pk(ωi) 2 = pk(ωi). Hence, for any n-potent e ∈ A, if we define ek = pk(e), then each ek is an idempotent in A, and Equation (1) implies that pk(e) = 1. These idempotents are pairwise orthogonal, because ejek = pj(e)pk(e) = 0 K0-THEORY WITH n-POTENTS 7 for j 6= k. Finally, ωkpk(e) = by Equation (2). � 3. K0-theory with n-potents We can now proceed to construct our n-potent K-theory groups. Definition 3.1. Let R be a ring. For all k ≥ 1, let Pnk (R) denote the set of n-potents in Mk(R), and let ik denote the inclusion ik(a) = ofMk(R) intoMk+1(R), as well as its restriction as a map from Pnk (R) to Pnk+1(R). Define M∞(R) and Pn∞(R) to be the (algebraic) direct limits M∞(R) = Mk(R), Pn∞(R) = Pnk (R) = Pn(M∞(R)). We define a binary operation ⊕ on Pn∞(R) as follows: let e and f be elements of Pn∞(R), choose the smallest natural numbers k and ℓ such that e ∈Mk(R) and f ∈Ml(R), and set e⊕ f = diag(e, f) = ∈ Pnk+l(R) ⊂ Pn∞(R). Definition 3.2. Let R be a ring, and define an equivalence relation ∼ on Pn∞(R) as follows: take e and f in Pn∞(R), and choose a natural number k sufficiently large that e and f are elements of Pnk (R). Then e ∼ f if e ∼a f in Mk(R). We let Vn(R) denote the set of equivalence classes of ∼. Note that if e = ab and f = ba in Mk(R), then and therefore the equivalence relation described in Definition 3.2 is well-defined. 8 EFTON PARK AND JODY TROUT Note that for any n-potent e, f in M∞(R), we get Thus, the binary operation ⊕ induces a binary operation + on Vn∞(R) as follows: take e and f in Pn∞(R), and define [e] + [f ] = [e⊕ f ] = This operation is well-defined and commutative by Propositions 2.9 and 2.11. The next proposition is straightforward and left to the reader. Proposition 3.3. For every ring R and natural number n ≥ 2, Vn(R) is an abelian monoid under the addition defined above, and whose iden- tity element is the class of the zero n-potent. If α : R −→ S is a unital ring homomorphism, then the induced map Vn(α) : Vn(R) −→ Vn(S) given by Vn(α)([(aij)]) = [(α(aij))] is a well-defined homomorphism of abelian semigroups. The correspon- dences R 7→ Vn(R) and α 7→ Vn(α) induce a covariant functor from the category of rings and ring homomorphisms to the category of abelian monoids and monoid homomorphisms. Definition 3.4. Let R be a ring and let n ≥ 2 be a natural number. We define Kn0 (R) to be the Grothendieck completion [6] of the abelian monoid Vn(R). Given an n-potent e in Pn∞(R), we denote its class in Kn0 (R) by [e]. In light of Propositions 2.6 and 2.7, we could have alternatively used similarity to define Vn(R), and hence Kn0 (R). Proposition 3.5. The assignments R 7→ Kn0 (R) determines a covari- ant functor from the category of rings and ring homomorphisms to the category of abelian groups and group homomorphisms. Proof. Proposition 3.3 states that V is a covariant functor from the category of rings to the category of abelian monoids, and Grothendieck completion determines a covariant functor from the category of abelian monoids to the category of abelian groups; we get the desired result by composing these two functors. � The following result shows that for (unital) algebras over a field of characteristic 6= 2, the tripotent K-theory functor K30 offers us no new invariants over ordinary idempotent K-theory. However, we will see later (Theorem 3.15) that the situation is subtly different for K40 . K0-THEORY WITH n-POTENTS 9 Theorem 3.6. Let F be a field with characteristic 6= 2. If A is a unital algebra over F then there is a natural isomorphism K30 (A) K0(A) of abelian groups. Proof. If e = e3 ∈M∞(A) is a tripotent, then one can easily check that (e2 + e) and e2 = (e2 − e) are (unique) idempotents in M∞(A) such that e = e1 − e2. It follows that we have a natural bijection of abelain monoids V3(A) → V2(A)⊕ V2(A) [e] 7→ [e1]⊕ [e2] with inverse map [e1]⊕[e2] 7→ [e1⊕−e2]. Since these maps are additive, the result easily follows. � While Kn0 (R) is well-defined for any ring R, to obtain a well-behaved theory where the usual exact sequences exist, we must restrict our attention to a smaller class of rings. The problem is that unlike the situation for idempotents, it is not generally true that if e is an n- potent, then so is 1− e. However, given an n-potent in an algebra over the cyclotomic field Q(n− 1), there is an adequate substitute: Definition 3.7. Let e be an n-potent in a Q(n − 1)-algebra A, and write as in the conclusion of Theorem 2.14. We define an n-potent ω1(1− e1), ω2(1− e2), . . . , ωn−1(1− en−1) ∈Mn−1(A) and call e⊥ the complementary n-potent of e. Observe that if n = 2, this definition agrees with the usual one for idempotents; i.e., e⊥ = 1− e. Note also that e⊕ e⊥ ∼s ω, where ω = diag(ω11A, . . . , ωn1A) ∈Mn−1(Q(n− 1)) ⊆Mn−1(A). Proposition 3.8 (Standard Picture of Kn0 (A)). Let n ≥ 2 be a natural number and let A be a Q(n−1)-algebra. Then every element of Kn0 (A) can be written in the form [e]−[ω], where e in an n-potent inMk(A) for some natural number k and ω is a diagonal n-potent in Mk(Q(n− 1)). 10 EFTON PARK AND JODY TROUT Proof. Start with an element [ẽ]− [f̃ ] in Kn0 (A), and take f̃⊥ to be the complementary n-potent of f as defined in Definition 3.7. Then [ẽ]− [f̃ ] = [ẽ] + [f̃⊥] [f̃ ] + [f̃⊥] The n-potents f̃ and f̃⊥ are orthogonal, and therefore [f̃ ] + [f̃⊥] = [f̃ + f̃⊥] = [ω], where ω has the desired form. Finally we take e to be ẽ⊕ f̃⊥, and by enlarging the matrix ω, we obtain the desired result. � Proposition 3.9. Let n ≥ 2 and let A be a Q(n− 1)-algebra. Suppose e and f are n-potents in M∞(A). Then [e] = [f ] in K 0 (A) if and only if e⊕ ω is similar to f ⊕ ω for some n-potent ω in M∞(Q(n− 1)). Proof. The “only if” direction is obvious. To show the inference in the opposite direction, suppose that [e] = [f ] in Kn0 (A). By the definition of the Grothendieck completion, e ⊕ ẽ is similar to f ⊕ ẽ for some n- potent ẽ in M∞(A). Then e ⊕ ẽ ⊕ ẽ⊥ is similar to f ⊕ ẽ ⊕ ẽ⊥. But if we write ẽ = k=1 ωkẽk as in Theorem 2.14, then Proposition 2.11(b) implies that ẽ ∼s diag ω1ẽ1, ω2ẽ2, . . . , ωn−1ẽn−1 Therefore ẽ ⊕ ẽ⊥ is similar to an n-potent in M∞(Q(n − 1)), and the proposition follows. � We next turn our attention to n-potent K-theory for nonunital alge- bras. Given a nonunital Q(n − 1)-algebra A, we define its unitization A+ as the unital Q(n−1)-algebra A+ = {(a, λ) : a ∈ A, λ ∈ Q(n−1)}, where addition and scalar multiplication are defined componentwise, and multiplication is given by (a, λ)(b, τ) = (ab+ aτ + bλ, λτ). Definition 3.10. Let A be a nonunital Q(n−1)-algebra, and let A+ be its unitization. Let π : A+ −→ Q(n− 1) be the algebra homomorphism π(a, λ) = λ. Then we define Kn0 (A) = ker π∗. It is easy to see that π∗ is surjective, so by definition of K 0 (A) we have a short exact sequence 0 // Kn0 (A) // Kn0 (A // Kn0 (Q(n− 1)) // with splitting induced by the map ψ : Q(n − 1) −→ A+ defined by ψ(λ) = (0, λ). In addition, it is easy to check that if A already has a unit and we form A+, then ker π∗ is naturally isomorphic to our original definition of Kn0 (A). K0-THEORY WITH n-POTENTS 11 Proposition 3.11. Let A be a nonunital Q(n−1)-algebra. Then every element of Kn0 (A) can be written in the form [e]− [s(e)], where e is an n-potents in Mk(A +) for some integer k ≥ 1, and s = ψ ◦π : A+ → A+ is the scalar mapping [6, Sect. 4.2.1]. Proof. Follows directly from Proposition 3.8 and Definition 3.10. � Proposition 3.12 (Half-exactness). Every short exact sequence 0 // I // A/I // 0 of Q(n− 1)-algebras, with A unital, induces an exact sequence Kn0 (I) // Kn0 (A) // Kn0 (A/I) of abelian n-potent K-theory groups. Proof. Since q ◦ i = 0, we have by functoriality that q∗ ◦ i∗ = 0 and so the image of Kn0 (I) under i∗ in K 0 (A) is contained in the kernel of q∗. To show the reverse inclusion, suppose we have [e]− [λ] in Kn0 (A) such that q∗ [e]− [λ] = 0. Then [q(e)] = [q(λ)] = [λ] in Kn0 (A/I). By Proposition 3.9, there exists an n-potent τ in M∞(Q(n− 1)) so that q(e)⊕ τ ∼s λ⊕ τ. Choose N sufficiently large so that we may view e, λ, and τ as N by N matrices, and choose z in GL2N (A/I) so that q(e)⊕ τ z−1 = λ⊕ τ. By Proposition 3.4.2 and Corollary 3.4.4 in [1], we can lift diag(z, z−1) to an element u in GL4N(A). Set f = u(e⊕ τ)u−1. Then q(f) = diag(z, z−1)(q(e)⊕ τ)diag(z−1, z) = λ⊕ τ, and thus f and λ⊕ τ are in M4N (I+). Therefore [e]− [λ] = [e⊕ τ ]− [λ⊕ τ ] = i∗([f ]− [λ⊕ τ ]) is in the image of Kn0 (I) under i∗ as desired. � Note that our proof of Proposition 3.12 relies critically on Proposi- tion 3.9, which in turn is proved using the standard picture of Kn0 (A). We do not have a standard picture for Kk0 (A) when k 6= n, and it seems likely to the authors that Kk0 is, in fact, not half-exact in this case. However, we do not have a counterexample where half-exactness fails to hold. While it is not at all obvious from its definition, Kn0 (A) can be iden- tified with a more familiar object. 12 EFTON PARK AND JODY TROUT Theorem 3.13. Let n ≥ 2 be a natural number and let A be a not nec- essarily unital Q(n− 1)-algebra. Then there is a natural isomorphism Kn0 (A) K0(A) of abelian groups. Proof. First consider the case where A is unital. We define a homo- morphism ψ̃ : Vn(A) −→ V0(A) in the following way: for each n-potent e = ωkek in M∞(A), set ψ̃[e] = [e1], [e2], . . . , [en−1] It is easy to check that ψ̃ is additive and well-defined. Next, define a homomorphism φ̃ : )n−1 −→ Vn(A) by the formula [f1], [f2], . . . , [fn−1] ω1diag(f1, 0, 0, . . . , 0) + ω2diag(0, f2, 0, . . . , 0) + · · · + ωn−1diag(0, 0, . . . , 0, fn−1) Note that [f1], [f2], . . . , [fn−1] ω1diag(f1, 0, . . . , 0) + · · ·+ ωn−1diag(0, 0, . . . , fn−1) [diag(f1, 0, . . . , 0)], [diag(0, f2, . . . , 0)] . . . [diag(0, 0, . . . , fn−1)] [f1], [f2], . . . , [fn−1] φ̃ψ̃[e] = φ [e1], [e2], . . . , [en−1] ω1diag(e1, 0, . . . , 0) + · · ·+ ωn−1diag(0, 0, . . . , en−1) diag(ω1e1, ω2e2, . . . , ωn−1en−1) = [e], where the last equality is a consequence of Proposition 2.11(b). The universal mapping property of the Grothendieck completion implies that ψ̃ extends uniquely to an abelian group isomorphism ψ : Kn0 (A) −→ K0(A) and thus the theorem is true for unital Q(n− 1)-algebras. K0-THEORY WITH n-POTENTS 13 Now suppose that A does not have a unit. Then we have the following commutative diagram with exact rows: 0 // Kn0 (A) // Kn0 (A +) // Kn0 (Q(n− 1)) // 0 // K0(A) n−1 // K0(A +)n−1 // K0(Q(n− 1))n−1 // 0 An easy diagram chase shows that there is a unique group iso- morphism from Kn0 (A) to K0(A) that makes the diagram com- mute. � Since a complex algebra is a Q(n− 1)-algebra for all values of n, we have the following immediate corollary. Corollary 3.14. If A is a C-algebra, there are natural isomorphisms Kn0 (A) K0(A) of abelian groups for all natural numbers n ≥ 2. We now arrive at the result that suggests why we should consider all Kn0 -functors for algebras over a cyclotomic field. Theorem 3.15. Let Q(4) = Q[i] be the 4th cyclotomic field. Then we have the following isomorphisms of abelian groups: K20(Q(4)) ∼= Z, K30(Q(4)) ∼= Z2, K40(Q(4)) ∼= Z⊕ 2Z, K50(Q(4)) ∼= Z4. Thus, K40 (Q(4)) 6∼= Z3 ∼= K40 (Q(3)). Proof. Since Q(4) is a field [7], we have K20 (Q(4)) = K0(Q(4)) ∼= Z. The field Q(4) has characteristic 0 6= 2, so Theorem 3.6 implies that K30(Q(4)) K0(Q(4) )2 ∼= Z2. Theorem 3.13 implies that we have an isomorphism K50(Q(4)) K0(Q(4) )4 ∼= Z4. However, the spectrum of 4-potents is contained in 0, 1,−1 which is not contained inQ(4) since the two primitive 3rd roots of unity ω = ζ3 = −12 + i and ω̄ = ζ̄3 = −12 − i are not in Q(4) = Q[i]. Given any 4-potent e ∈Mn(Q(4)) ⊂Mn(C) we can uniquely write e = e1 + ωe2 + ω̄e3, 14 EFTON PARK AND JODY TROUT where e1, e2, e3 are orthogonal idempotents in Mn(C) that sum to an idempotent e1+e2+e3 = e 3 inMn(Q(4)) by Lemma 2.2. We thus have e2 = e1 + ω̄e2 + ωe3 e3 = e1 + e2 + e3 because ω2 = ω̄, ω̄2 = ω, and ω3 = ω̄3 = 1. Since ω + ω̄ = −1, this implies that the first idempotent e1 = (e + e 2 + e3)/3 ∈Mn(Q(4)) and the sum of the last two idempotents e2 + e3 = e 3 − e1 ∈Mn(Q(4)) are both inMn(Q(4)). Using a simple trace argument and the fact that ω, ω̄ 6∈ Q(4), we conclude that rank(e2) = trace(e2) = trace(e3) = rank(e3), and so rank(e2 + e3) = trace(e2 + e3) = 2trace(e2) is even. We then have a well-defined map V4(Q(4)) → V2(Q(4))⊕ 2V2(Q(4)) ∼= N⊕ 2N [e] 7→ [e1]⊕ [e2 + e3] ∼= trace(e1)⊕ 2 trace(e2); this is because the classes of e1 and e1 + e2 are preserved by (stable) similarity, and the K0-class of an idempotent in a matrix ring over a number field (or a PID) is the rank (= trace). It is easy to check that this map is injective (using e1 ⊥ e2 + e3 in Mn(Q(4))) and additive. The only question is surjectivity. It suffices to show that there is a 4-potent e over Q(4) whose stable similarity class is mapped to the generator 1⊕ 2 of N⊕ 2N. Consider the block diagonal matrix 1 0 0 0 0 i 0 i −1  ∈M3(Q(4)), which is easily checked to be quadripotent. The lower right quadripo- tent 2× 2 invertible block has the desired eigenvalues ω and ω̄, and so does not diagonalize over Q(4). The result now follows easily. � 4. n-Homomorphisms and Kn0 Functorality We know from Proposition 3.5 that Kn0 is a covariant functor from the category of (unital) rings and ring homomorphisms to the category of abelian groups and group homomorphisms. However, Kn0 is actually functorial for a more general class of ring mappings. K0-THEORY WITH n-POTENTS 15 Definition 4.1. Let R and S be rings. An additive map (not neces- sarily unital) φ : R −→ S is called an n-homomorphism if φ(a1a2 · · · an) = φ(a1)φ(a2) · · ·φ(an) for all a1, a2, . . . , an in R. Obviously every (ring) homomorphism is an n-homomorphism, but the converse is false in general. For example, an AEn-ring is a ring R such that every additive map φ : R → R is an n-homomorphism. Feigelstock [2, 3] classified all unital AEn-rings. The algebraic version of n-homomorphism was introduced for complex algebras in [4] and has been carefully studied in the case of C∗-algebras in [5]. Proposition 4.2. Let φ : R → S be an n-homomorphism between unital rings. Then φ induces a group homomorphism φ∗ : K 0 (R) −→ Kn0 (S). Furthermore, the assignment R 7→ Kn0 (R) is a covariant functor from the category of unital rings and n-homomorphisms to the category of abelian groups and ordinary group homomorphisms. Proof. For each natural number k, we extend φ to a map from Mk(R) to Mk(S) by applying φ to each matrix entry; it is easy to check this also gives us an n-homomorphism. Moreover, φ is compatible with stabilization of matrices; the only nonobvious point to check is that φ respects algebraic equivalence. Let e and f be algebraically equivalent n-potents in Mk(R) for some k, and choose a and b in Mk(R) so that e = ab and f = ba. Define elements a′ = φ(ea)φ(f)n−2 and b′ = φ(b) in Mk(S). We compute: a′b′ = φ(ea)φ(f)n−2φ(b) = φ((ea)fn−2b) = φ(ea(ba)n−2b) = φ(e(ab)n−1) = φ(en) = φ(e). A similar argument shows that b′a′ = φ(f). Therefore φ determines a monoid homomorphism from Vn(R) to Vn(S), and hence a group homomorphism φ∗ : K 0 (R) −→ Kn0 (S). We leave it to the reader to make the straightforward computations to show that we have a covari- ant functor. � Note that while we have an isomorphism Kn0 (A) K0(A) Q(n− 1)-algebras, it is not at all clear from the right hand side of this isomorphism that Kn0 (A) is functorial for n-homomorphisms. 16 EFTON PARK AND JODY TROUT References [1] B. Blackadar,K-theory for Operator Algebras, 2nd ed., MSRI Publication Series 5, Springer-Verlag, New York, 1998. [2] S. Feigelstock, Rings whose additive endomorphisms are N -multiplicative, Bull. Austral. Math. Soc. 39 (1989), no. 1, 11–14. [3] S. Feigelstock, Rings whose additive endomorphisms are n-multiplicative. II, Period. Math. Hungar. 25 (1992), no. 1, 21–26. [4] M. Hejazian, M. Mirzavaziri, M.S. Moslehian, n-homomorphisms, Bull. Iranian Math. Soc. 31 (2005), no. 1, 13-23. [5] E. Park and J. Trout, On the Nonexistence of Nontrivial Involutive n- homomorphisms of C∗-algebras, Trans. Amer. Math. Soc. 361 (2009), no. 4, 1949–1961 [6] M. Rordam, F. Larsen, N. Laustsen, An Introduction to K-theory for C∗- algebras, London Mathematical Society Student Texts, vol. 49. Cambridge Uni- versity Press, Cambridge, 2000. [7] J. Rosenberg, Algebraic K-theory and Its Applications, Graduate Texts in Math- ematics, vol. 147, Springer-Verlag, New York, 1994. Box 298900, Texas Christian University, Fort Worth, TX 76129 E-mail address : [email protected] 6188 Kemeny Hall, Dartmouth College, Hanover, NH 03755 E-mail address : [email protected] 1. Introduction 2. Equivalence of n-potents 3. K0-theory with n-potents 4. n-Homomorphisms and K0n Functorality References

0704.0776

Spin coupling in zigzag Wigner crystals

Spin coupling in zigzag Wigner crystals A. D. Klironomos,1,2 J. S. Meyer,2 T. Hikihara,3 and K. A. Matveev1 Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA Department of Physics, Hokkaido University, Sapporo 060-0810, Japan (Dated: October 28, 2018) We consider interacting electrons in a quantum wire in the case of a shallow confining potential and low electron density. In a certain range of densities, the electrons form a two-row (zigzag) Wigner crystal whose spin properties are determined by nearest and next-nearest neighbor exchange as well as by three- and four-particle ring exchange processes. The phase diagram of the resulting zigzag spin chain has regions of complete spin polarization and partial spin polarization in addition to a number of unpolarized phases, including antiferromagnetism and dimer order as well as a novel phase generated by the four-particle ring exchange. PACS numbers: 73.21.Hb,71.10.Pm I. INTRODUCTION The deviations of the conductance from perfect quanti- zation in integer multiples of G0 = 2e 2/h observed in bal- listic quantum wires at low electron densities have gener- ated great experimental and theoretical interest in recent years.1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27 These conductance anomalies manifest themselves as quasi-plateaus in the conductance as a function of gate voltage at about 0.5 to 0.7 of the conductance quantum G0, depending on the device. Although most experiments are performed with electrons in GaAs wires,1,2,3,4,5,6,7,8,9,10,11 a similar “0.7 structure” was recently observed in devices formed in two-dimensional hole systems.12,13,14 It is widely accepted that the origin of the quasi-plateau lies in correlation effects, but a complete understanding of this phenomenon remains elusive. Although some alternative interpretations have been proposed,11,26,27 most commonly the experimental find- ings are attributed to non-trivial spin properties of quan- tum wires.1,4,5,6,7,8,9,10,14,15,16,17,18,19,20,21,22,23,24,25 In a truly one-dimensional geometry the spin coupling is rel- atively simple: electron spins are coupled antiferromag- netically, and the low energy properties of the system are described by the Luttinger liquid theory. The pic- ture may change dramatically when transverse displace- ments of electrons are important and the system be- comes quasi-one-dimensional. In particular, the spon- taneous spin polarization of the ground state, which was proposed1,6,9,10,14,15,16 as a possible origin of the conduc- tance anomalies, is forbidden in one dimension,28 but allowed in this case. The electron system in a quantum wire undergoes a transition from a one-dimensional to a quasi-one- dimensional state when the energy of quantization in the confining potential is no longer large compared to other important energy scales. In this paper we consider the spin properties of a quantum wire with shallow confin- ing potential. In such a wire the electron system be- comes quasi-one-dimensional while the electron density is still very low, and thus the interactions between elec- trons are effectively strong. At very low densities, elec- trons in the wire form a one-dimensional structure with short-range crystalline order—the so-called Wigner crys- tal. As the density increases, strong Coulomb interac- tions cause deviations from one-dimensionality creating a quasi-one-dimensional zigzag crystal with dramatically different spin properties. In particular, ring exchanges will be shown to play an essential role. We find several interesting spin structures in the zigzag crystal. In a sufficiently shallow confining po- tential, in a certain range of electron densities, the 3- particle ring exchange dominates and leads to a fully spin-polarized ground state. At higher electron densities, and/or in a somewhat stronger confining potential, the 4-particle ring exchange becomes important. We study the phase diagram of the corresponding spin chain us- ing the method of exact diagonalization, and find that the 4-particle ring exchange gives rise to novel phases, including one of partial spin polarization. The paper is organized as follows. The formation of a Wigner crystal in a quantum wire and its evolution into a zigzag chain as a function of electron density are dis- cussed in Sec. II. Spin interactions in a zigzag Wigner crystal which arise through 2-particle as well as ring ex- changes are introduced in Sec. III. The numerical calcu- lation of the relevant exchange constants is presented in Sec. IV. The results of the numerical calculation estab- lish the existence of a ferromagnetic phase at intermedi- ate densities and the dominance of the 4-particle ring ex- change at high densities. Subsequently, a detailed study of the zigzag chain with 4-particle ring exchange is pre- sented in Sec. V. An attempt to construct the phase dia- gram for a realistic quantum wire as a function of electron density and interaction strength is presented in Sec. VI. The paper concludes with a discussion of the relation of our work to recent experiments, given in Sec. VII. A brief summary of some of our results has been reported previously in Ref. 29. http://arxiv.org/abs/0704.0776v1 0 0.05 0.1 0.15 0.2 0.25 (a) ν=0.70 (b) ν=0.90 (c) ν=1.75 FIG. 1: Wigner crystal of electrons in a quantum wire. The structure as determined by the dimensionless distance be- tween rows d/r0 depends on the parameter ν proportional to electron density (see text). As density grows, the one- dimensional crystal (a) gives way to a zigzag chain (b,c). II. WIGNER CRYSTALS IN QUANTUM WIRES We consider a long quantum wire in which the elec- trons are confined by some smooth potential in the direc- tion transverse to the wire axis. Assuming a quadratic dispersion and zero temperature, the kinetic energy of an electron is typically of the order of the Fermi en- ergy EF = (π~n) 2/8m, whereas the Coulomb interaction energy is of the order of e2n/ǫ. Here, n is the (one- dimensional) density of electrons, ǫ is the dielectric con- stant of the host material, and m is the effective electron mass. As the density of electrons is lowered, Coulomb interactions become increasingly more important, and at n ≪ a−1B they dominate over the kinetic energy, where the Bohr radius is given as aB = ~ 2ǫ/me2. (In GaAs its value is approximately aB ≈100Å.) In this low-density limit, the electrons can be treated as classical particles. They will minimize their mutual Coulomb repulsion by occupying equidistant positions along the wire, forming a structure with short-range crys- talline order—the so-called Wigner crystal, Fig. 1(a). Unlike in higher dimensions, the long-range order in a one-dimensional Wigner crystal is smeared by quantum fluctuations, and only weak density correlations remain at large distances.30 However, as will be shown in the following sections, the coupling of electron spins is con- trolled by electron interactions at distances of order 1/n, where the picture of a one-dimensional Wigner crystal is applicable. Henceforth, we speak of a Wigner crystal in a quantum wire with this important distinction in mind. Upon increasing the density, the inter-electron distance diminishes, and the resulting stronger electron repulsion will eventually overcome the confining potential Vconf , transforming the classical one-dimensional Wigner crys- tal into a staggered or zigzag chain31,32, as depicted in Fig. 1(b,c). From the comparison of the Coulomb inter- action energy Vint(r) = e 2/ǫr with the confining potential an important characteristic length scale emerges. Indeed, the transition from the one-dimensional Wigner crystal to the zigzag chain is expected to take place when dis- tances between electrons are of the order of the scale r0 such that Vconf(r0) = Vint(r0). It is therefore necessary to identify the electron equi- librium configuration as a function of density. In order to proceed in a quantitative way we consider a specific model, namely a quantum wire with a parabolic confining potential Vconf(y) = mΩ 2y2/2, where Ω is the frequency of harmonic oscillations in the potential Vconf(y). Within that model the characteristic length scale r0 is given as 2e2/ǫmΩ2 . (1) It is convenient for the following discussion to measure lengths in units of r0. To that respect we introduce a dimensionless density ν = nr0. (2) Then minimization of the energy with respect to the electron configuration31,32 reveals that a one-dimensional crystal is stable for densities ν < 0.78, whereas a zigzag chain forms at intermediate densities 0.78 < ν < 1.75. (If density is further increased, structures with larger numbers of rows appear.31,32) The distance d between rows grows with density as shown in Fig. 1. Note that at ν ≈ 1.46 the equilateral configuration is achieved. There- fore, at higher densities—and in a curious contradiction in terms—the distance between next-nearest neighbors is smaller than the distance between nearest neighbors (see Fig. 1(c)). III. SPIN EXCHANGE In order to introduce spin interactions in the Wigner crystal, it is necessary to go beyond the classical limit. In quantum mechanics spin interactions arise due to ex- change processes in which electrons switch positions by tunneling through the potential barrier that separates them. The tunneling barrier is created by the exchanging particles as well as all other electrons in the wire. The re- sulting exchange energy is exponentially small compared to the Fermi energy EF . Furthermore, as a result of the exponential decay of the tunneling amplitude with distance, only nearest neighbor exchange is relevant in a one-dimensional crystal. Thus, the one-dimensional crys- tal is described by the Heisenberg Hamiltonian H1 =∑ j J1SjSj+1, where the exchange constant J1 is posi- tive and has been studied in detail recently.24,33,34,35 The exchange constant being positive leads to a spin-singlet ground state with quasi-long-range antiferromagnetic or- der, in accordance with the Lieb-Mattis theorem.28 The zigzag chain introduced in the previous section displays much richer spin physics. As the distance be- tween the two rows increases as a function of density, the distance between next-nearest neighbors becomes com- parable to and eventually even smaller than the distance between nearest neighbors, as illustrated in Fig. 1(b,c). Consequently, the next-nearest neighbor exchange con- stant J2 may be comparable to or larger than the nearest neighbor exchange constant J1. Drawing intuition from studies of the two-dimensional Wigner crystal,36,37,38,39 one comes to a further important realization regarding the physics of the zigzag chain: in addition to 2-particle exchange processes, ring exchange processes, in which three or more particles exchange positions in a cyclic fashion, have to be considered in this geometry. It has long been established that, due to symmetry properties of the ground state wavefunctions, ring ex- changes of an even number of fermions favor antiferro- magnetism, while those of an odd number of fermions favor ferromagnetism.40 In a zigzag chain, the Hamilto- nian reads J1Pj j+1 + J2Pj j+2 − J3(Pj j+1 j+2 + Pj+2 j+1 j) +J4(Pj j+1 j+3 j+2 + Pj+2 j+3 j+1 j)− . . . , (3) where Pj1...jl denotes the cyclic permutation operator of l spins. Here the exchange constants are defined such that all Jl > 0. Furthermore, only the dominant l-particle ex- changes are shown. A more familiar form of the Hamilto- nian in terms of spin operators is obtained by noting that Pij = + 2SiSj and Pj1...jl = Pj1j2Pj2j3 . . . Pjl−1jl . Using spin operators and considering the two-spin ex- changes one obtains the Hamiltonian H12 = (J1SjSj+1 + J2SjSj+2) . (4) The competition between the nearest neighbor and next- nearest neighbor exchanges becomes the source of frus- tration of the antiferromagnetic spin order and eventu- ally leads to a gapped dimerized ground state at J2 > 0.24J1. 41,42,43,44 The simplest ring exchange involves three particles and is therefore ferromagnetic. Including the 3-particle ring exchange J3 in addition to the 2-particle exchanges, the Hamiltonian of the corresponding spin chain retains a simple form. The 3-particle ring exchange does not in- troduce a new type of coupling, but rather modifies the 2-particle exchange constants.40 For a zigzag crystal we find the effective 2-particle exchange constants J̃1 = J1 − 2J3, (5) J̃2 = J2 − J3. (6) Thus the total Hamiltonian has the form H123 = J̃1SjSj+1 + J̃2SjSj+2 , (7) where J̃1 and J̃2 can have either sign. ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� ���������������������� J 2~ 0 FM AF Dimers FIG. 2: The phase diagram including nearest neighbor, next- nearest neighbor, and 3-particle ring exchanges. The effective couplings eJ1 and eJ2 are defined in the text. The shaded region between the dimer and ferromagnetic phases corresponds to the exotic phase predicted in Ref. 48. Consequently, regions of negative (i.e. ferromagnetic) nearest and/or next-nearest neighbor coupling become accessible. The phase diagram of the Heisenberg spin chain (7) with both positive and negative couplings has been studied extensively.41,42,43,44,45,46,47,48,49,50 In ad- dition to the antiferromagnetic and dimer phases dis- cussed earlier, a ferromagnetic phase exists for J̃1 < min{0,−4J̃2}.46 An exotic phase called the chiral-biaxial- nematic phase has been predicted48 to appear for J̃1 < 0 and −0.25 < J̃2/J̃1 < −0.38. However, the nature of the system in this parameter region is still controversial. The phase diagram is drawn in Fig. 2. Thus, depending on the relative magnitudes of the var- ious exchange constants, different phases are realized. Extensive studies of the two-dimensional Wigner crys- tal have shown that, at low densities (or strong interac- tions), the 3-particle ring exchange dominates over the 2-particle exchanges. As a result, the two-dimensional Wigner crystal becomes ferromagnetic at sufficiently strong interactions.36,39 Given that the electrons in a two-dimensional Wigner crystal form a triangular lat- tice, by analogy, one should expect a similar effect in the zigzag chain at densities where the electrons form ap- proximately equilateral triangles. More specifically, upon increasing the density and consequently the distance be- tween rows, one would expect the system to undergo a phase transition from an antiferromagnetic to a ferromag- netic phase. To establish this scenario conclusively, the various exchange energies in the zigzag crystal have to be determined. The system differs from the two-dimensional crystal in two important aspects. (i) The electrons are subject to a confining potential as opposed to the flat background in the two-dimensional case. Even more im- portantly, (ii) the electron configuration depends on den- sity, cf. Fig. 1, as opposed to the ideal triangular lattice in two dimensions. In the following section, we proceed with a numerical study of the exchange energies for the specific configurations of the zigzag Wigner crystal in a parabolic confining potential. IV. SEMICLASSICAL EVALUATION OF THE EXCHANGE CONSTANTS The effective strength of interactions is usually de- scribed by the interaction parameter rs which measures the relative magnitude of the interaction energy and the kinetic energy and is of order the distance between elec- trons measured in units of the Bohr radius. For quan- tum wires, it is more appropriate to use the parameter rΩ = r0/aB, which takes into account the confining po- tential. Within our model, the interaction parameter rΩ rΩ = 2 2ǫ2~2 . (8) For rΩ ≫ 1, strong interactions dominate the physics of the system, and a semiclassical description is appli- cable. In order to calculate the various exchange con- stants, we use the standard instanton method, success- fully employed in the study of the two-dimensional36,37,38 and one-dimensional34,35 Wigner crystal. Within this approach, the exchange constants are given by Jl = J∗l exp (−Sl/~). Here Sl is the value of the Euclidean (imaginary time) action, evaluated along the classical ex- change path. By measuring length and time in units of r0 and T = 2/Ω, respectively, the action S[{rj(τ)}] can be rewritten in the form S = ~η rΩ, where the functional η[{rj(τ)}] = + y2j |rj−ri|  (9) is dimensionless. Thus, we find the exchange constants in the form Jl = J l exp (−ηl rΩ), (10) where the dimensionless coefficients ηl depend only on the electron configuration (cf. Fig. 1) or, equivalently, on the density ν. The exponents ηl are calculated nu- merically for each type of exchange by minimizing the action (9) with respect to the instanton trajectories of the exchanging electrons. This procedure is mathemati- cally equivalent to solving a set of coupled, second order in the imaginary time τ , differential equations for the trajectories rj(τ). The boundary conditions at τ = ±∞ are, respectively, the original equilibrium configuration and the configuration where the electrons have exchanged positions according to the exchange process considered. 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 FIG. 3: The exponents η1, η2, η3, and η4 as functions of the dimensionless density ν. ν η1 η2 η3 η4 1.0 1.050 2.427 1.254 1.712 1.1 1.161 2.169 1.261 1.605 1.2 1.255 1.952 1.275 1.532 1.3 1.337 1.754 1.287 1.469 1.4 1.406 1.566 1.293 1.398 1.5 1.456 1.376 1.278 1.299 1.6 1.471 1.169 1.215 1.135 1.7 1.391 0.901 1.022 0.784 TABLE I: The numerically calculated values of the density dependent exponents ηl, see Eq. (10). The computation was carried out including 12 moving spectator particles on either side of the exchanging particles. Corrections to all ηl from the remaining spectators do not exceed 0.1%. In the simplest approximation only the exchanging electrons are included in the calculation while all other electrons, being frozen in place, create the background potential. It turns out, however, that it is important to take into account the motion of “spectators”—the electrons in the crystal to the left and to the right of the exchanging particles—during the exchange process. The results presented here are obtained by successively adding more spectators on both sides until the values ηl converge. We find that including 12 moving spectators on either side of the exchanging particles determines the exponents to an accuracy better than 0.1%. Figure 3 shows the calculated exponents for various ex- changes as a function of dimensionless density ν and the corresponding values are reported in Table I. At strong interactions (rΩ ≫ 1), the exchange with the smallest value of ηl is clearly dominant, and the prefactor J l is of secondary importance to our argument. At low densities, when the zigzag chain is still close to one-dimensional, J1 (c) J3 (b) J(a) J 2 (d) J4 FIG. 4: The calculated particle trajectories for various ex- changes at a representative density ν = 1.5. It is evident that only a few near neighbors of the exchanging particles move appreciably. is the largest exchange constant, and the spin physics is controlled by the nearest-neighbor exchange. In an inter- mediate density regime, when the electron configuration is close to equilateral triangles, the 3-particle ring ex- change dominates. Thus, the numerical calculation con- firms our original expectation, and a transition from an antiferromagnetic to a ferromagnetic state takes place upon increasing the density. Surprisingly, however, at even higher densities the 4-particle ring exchange is the dominant process. The role of the 4-particle ring ex- change and the phase diagram of the associated zigzag spin chain will be the subject of the following section. More complicated exchanges have also been computed, namely multi-particle (l ≥ 5) ring exchanges as well as exchanges involving more distant neighbors. However, the exchanges displayed in Fig. 3 were found to be the dominant ones.29 It is important to note here that spectators contribute to our results in an essential way. Allowing spectators to move results not only in quantitative changes (namely a reduction of the initially overestimated values ηl) but in qualitative changes as well: at high densities, the dom- inance of the 4-particle ring exchange J4 over the next- nearest neighbor exchange J2 is obtained only if specta- tors are taken into account. In particular, it is necessary to include at least 6 moving spectators on each side of the exchanging particles for J4 to take over at high densities. The considerable effect that the spectators have on the values of the exponents raises the question whether a short-ranged interaction potential might cause further quantitative or qualitative changes to the physical pic- ture. In order to investigate that possibility we have repeated the entire calculation for a modified Coulomb interaction of the form V (x) = x2 + (2d)2 . (11) This particular interaction accounts for the presence of a metal gate, modeled by a conducting plane at a distance d from the crystal. The gate screens the bare Coulomb potential, modifying the electron-electron interaction at long distances. Our calculation shows that this modifica- tion affects the values of the exponents only weakly, even when the gate is placed at a distance from the crystal comparable to the inter-particle spacing. Qualitatively, the physical picture remains the same, with the order of dominance of the various exchanges unaffected through- out the range of densities. At the same time, it is particularly noteworthy that (both for the screened and unscreened interaction) the contribution of the spectator electrons saturates rapidly as their number is increased. This is an indication that the destruction of long-range order in the quasi-one- dimensional Wigner crystal by quantum fluctuations will not affect our conclusions. Figure 4 shows the particle trajectories for the dominant exchanges at a represen- tative density of ν = 1.5. The trajectories of both the exchanging particles and a subset of the spectators are shown, and their relative displacements can be readily compared. V. FOUR-PARTICLE RING EXCHANGE We have shown in the preceding section that in a cer- tain range of densities, the 4-particle ring exchange dom- inates. Unlike the 3-particle exchange, the 4-particle ring exchange not only modifies the nearest and next- nearest neighbor exchange constants, but, in addition, introduces more complicated spin interactions.40 For the zigzag chain, we find H4 = J4 SjSj+l + 2 (SjSj+1)(Sj+2Sj+3) +(SjSj+2)(Sj+1Sj+3)− (SjSj+3)(Sj+1Sj+2) . (12) Not much is known about the physics of zigzag spin chains with interactions of this type. We have stud- ied this particular system described by the Hamiltonian H = H123 + H4 using exact diagonalization, consider- ing systems of N = 12, 16, 20, 24 sites. Periodic bound- ary conditions have been imposed, and we have employed the well-known Lanczos algorithm to calculate a few low- energy eigenstates. Figure 5 shows the total spin S of the ground state as a function of the effective couplings J̃1/J4 and J̃2/J4 for the largest system considered, one with N = 24 sites. The darkest region corresponding to maximal total spin is the ferromagnetic phase, which occurs for large negative couplings in direct analogy to the phase diagram for the system without four-spin interactions (see Fig. 2). For all system sizes that we have considered, the obtained phase boundary is almost independent of the system size and agrees very well with the conditions for ferromagnetism J̃1 + 2J4 < 0, (13) J̃1 + 4J̃2 + 10J4 < 0, (14) FIG. 5: Total spin S of the ground state for a chain of N = 24 sites as a function of the effective couplings eJ1/J4 and eJ2/J4. derived by treating the four-spin terms in the Hamilto- nian (12) on a mean field level near the ferromagnetic state. A new phase of partial spin polarization appears adja- cent to the ferromagnetic phase. The partially polarized phase possesses a ground state total spin of S = 2 for N = 12, S = 2 or 4 for N = 16, 20, and S = 4 for N = 24; it appears that total spin of one third of the saturated magnetization N/2 prevails throughout most of that phase. The phase persists, to a significant extent, in range and form as N increases. Therefore, we believe it is not a finite size effect. We note here that it has been shown rigorously that a model described by a Hamilto- nian having a similar form to ours also exhibits a ground state with partial spin polarization.51 On the other hand, the scattered points corresponding to non-zero total spin in the first quadrant (J̃1, J̃2 > 0) appear to shift posi- tion as N increases and the size of the total spin remains small, S ≤ 2, for all system sizes considered. We cannot ascertain at this point whether they persist in a larger system. At large values of |J̃1|/J4 and |J̃2|/J4, one would ex- pect to recover the phases present in the absence of J4. Thus, the large white area in Fig. 5 corresponding to total spin S = 0 should contain the antiferromagnetic phase, analogues of the dimer phases observed in the system without four-spin interactions, and possibly en- tirely new phases as well. In order to distinguish between these phases, we first calculate the overlap between the ground state wavefunctions in our model and the ones representing the dimer and antiferromagnetic phases in the well-studied model with J4 = 0. The representative ground state wavefunctions are obtained for the chain with J4 = 0 and typical parameter sets of (J̃1, J̃2) cho- sen deep in the dimer and antiferromagnetic phases of the phase diagram shown in Fig. 2. The results for the −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 8 1.0 FIG. 6: Overlaps of the ground state wavefunctions in the presence of the 4-particle ring exchange with the wavefunc- tions representing (a) the dimer and (b) the antiferromag- netic phase for J4 = 0. The representative ground states (a) and (b) are obtained for ( eJ1, eJ2, J4) = (1, 10, 0) and ( eJ1, eJ2, J4) = (1,−10, 0), respectively. chain with N = 24 sites are shown in Fig. 6. As can be seen from the figure, the ground states for a broad region of large positive J̃2/J4 have a significant overlap with the representative ground state of the dimer phase while the ground states for large positive J̃1/J4 and/or negative J̃2/J4 resemble very much the one belonging to the antiferromagnetic phase. This behavior indicates the appearance of the expected dimer and antiferromagnetic phases for large effective couplings |J̃1|/J4 and |J̃2|/J4. We have confirmed the existence of these phases in the corresponding parameter regimes by studying the associ- ated structure factors. In order to study and clarify the properties of the sys- tem in more detail, we have calculated the excitation energies ∆En(S,Q) = En(S,Q)− Egs, (15) where En(S,Q) is the energy of n-th lowest level in the subspace characterized by the total spin S and the mo- −2 0 2 4 6 J2 / J4 J1 /J4 = 2, N = 24 : (0, 0) : (0, π) : (1, π) FIG. 7: Excitation energies ∆En(S,Q) in the system of N = 24 sites for eJ1/J4 = 2 as functions of eJ2/J4. The two-lowest levels are plotted for the subspaces of (S,Q) = (0, 0) and (0, π) while only the lowest one is shown for all other subspaces. The energies for (S,Q) = (0, 0), (0, π), and (1, π) are plotted by thick solid, dotted, and dashed curves, respectively. The energies of the levels belonging to other subspaces are shown by thin gray curves. mentum Q, and Egs is the ground state energy. Figure 7 shows the results for the system of size N = 24, obtained along the vertical line in the phase diagram given by J̃1/J4 = 2. At large positive J̃2/J4, the ground and first- excited states belong to the subspace (S,Q) = (0, 0) and (0, π), respectively.52 These states are expected to form the ground state doublet of the dimer phase in the ther- modynamic limit. For J̃2/J4 > (J̃2/J4)c,dim ∼ 3.5, one of the dimer doublet states is the ground state and the sys- tem is in the dimer phase. At smaller J̃2/J4, both states of the dimer doublet shift upward and move away from the low-energy regime, while other states decrease steeply in energy and eventually become the ground state. We therefore take the point (J̃2/J4)c,dim as the boundary of the dimer phase. After the transition, the system enters a region with exotic ground states and a large number of low-lying excitations. We have numerically checked that these exotic ground states have no or, at most, negligibly small overlap with the ground state of either the dimer or antiferromagnetic phases. When J̃2/J4 decreases further, the exotic states leave the low-energy regime and the system predictably enters the antiferromagnetic phase, which occurs for J̃2/J4 < (J̃2/J4)c,AF ∼ 0.1. Performing the same type of analysis for several pa- rameter lines, we can estimate the phase boundaries (J̃2/J4)c,dim and (J̃2/J4)c,AF as functions of J̃1/J4. In the limit of large negative coupling J̃1/J4 → −∞, the boundary of the dimer phase (J̃2/J4)c,dim approaches the line J̃1 = −0.38J̃2, suggesting a smooth connection to the behavior for J̃1 < 0 and J4 = 0 (cf. Ref. 48). In −5 0 5 Dimers FIG. 8: The phase diagram of the Heisenberg chain including nearest neighbor, next-nearest neighbor, and 4-particle ring exchanges. The expected phases consist of a ferromagnetic and an antiferromagnetic phase as well as a dimer phase. In addition, a novel region (4P ) dominated by the 4-particle ring exchange appears. The latter includes a phase of partial spin polarization (M). Triangles, squares and circles correspond to the boundaries obtained for N = 16, 20, and 24 sites, re- spectively. We note that although the phase of partial spin polarization persists as the system size is increased, its bound- ary with the 4P phase has a rather irregular size dependence and is represented approximately in the figure. a similar fashion, at large positive coupling J̃1/J4, we find no indication for the appearance of exotic phases after J̃1/J4 ≥ 6; the data of the energy spectrum and the wavefunction overlaps show essentially the same be- haviors as those at J̃1/J4 → ∞. We therefore conclude that there occurs a direct transition between the dimer and antiferromagnetic phases and estimate the transition line using the method of level spectroscopy, according to which the transition point is determined by the level crossing between the first-excited states in the dimer and antiferromagnetic phases.43 Combining all these phase boundaries and including the boundaries of the ferromagnetic and partially spin polarized phases which were obtained using the total spin of the ground state as a criterion, we determine the phase diagram in the J̃1/J4 versus J̃2/J4 plane. The result is shown in Fig. 8. The phase diagram has similarities to the one obtained without the four-spin interaction term, see Fig. 2. In particular, the expected ferromagnetic, an- tiferromagnetic, and dimer phases appear for large values of the effective couplings, |J̃1|/J4 and |J̃2|/J4. But more importantly, at not too large values of the effective cou- plings, new phases appear as a direct result of the new interaction term. We can identify a phase with partial spin polarization and a region occupied by one or sev- eral novel phases with total spin S = 0. In the region where J4 dominates, the ground state has no similarity at the level of wavefunctions with that of the conventional phases. It is important to note that the region occupied by the new phases becomes broader as the system size N grows, indicating that it survives even in the thermo- dynamic limit. From the analysis of the wavefunction overlaps between the ground states, there are strong in- dications that the novel unpolarized region might consist of several different phases. Unfortunately, it has proven difficult to clarify the nature of the new phases and, in particular, discover the order parameters that character- ize them based solely on the analysis of small systems. Therefore, the issue is relegated to future studies. In the absence of detailed understanding of its properties, we collectively dub the region of the phase diagram the “4P” phase. VI. PHASE DIAGRAM FOR REALISTIC QUANTUM WIRES Having identified possible phases of the zigzag chain, the most interesting question is which of the various phases appearing in the phase diagram Fig. 8 are ac- cessible in quantum wires. At finite rΩ, the calculations of the exchange constants discussed in Sec. IV have to be completed in an important way by computing the prefac- tors Jl in Eq. (10). To that effect it is necessary to take into account Gaussian fluctuations around the classical exchange path. We employ the method introduced by Voelker and Chakravarty38 which, for the sake of com- pleteness, is outlined in the Appendix . The prefactors have the form J∗l = AlFl rΩ , (16) where Fl is density dependent. The factor Al is used to account for multiple classical trajectories corresponding to the same exchange process (see Appendix). Table II contains the values of Fl we calculated for the various exchanges considered in this work. Note that, in order to achieve a comparable level of convergence, a more ac- curate determination of the instanton trajectories was required for the calculation of the prefactors J∗l than for the calculation of the exponents ηl. By including up to 28 moving spectators on either side of the exchanging par- ticles, we have been able to achieve an accuracy better than 2%. We are now in a position to map out the areas of the phase diagram of Fig. 8 that are encountered as one tra- verses the density region of interest for a given rΩ. The resulting phase diagram obtained with the calculated ex- change energies is shown in Fig. 9. Since the semiclassical approximation is applicable only at rΩ ≫ 1, we do not extend the phase diagram to values of rΩ < 10. It turns out that the spin polarized phases are only realized at rΩ & 50. On the other hand, the novel “4P” phase is ν F1 F2 F3 F4 1.0 1.12 ≃ 6 1.22 2.44 1.1 1.04 ≃ 4 1.03 1.73 1.2 1.05 2.38 0.97 1.28 1.3 1.08 1.86 0.97 1.15 1.4 1.19 1.71 1.02 1.13 1.5 1.40 1.63 1.14 1.18 1.6 1.80 1.51 1.26 1.19 1.7 2.07 1.07 0.81 0.50 TABLE II: The numerically calculated values of the density dependent part Fl of the exchange energy prefactor J , see Eq. (16), calculated with mobile spectators. For all the num- bers reported, the accuracy is better than 2%, except for F2 at ν = 1.0, 1.1, for which extrapolated values with an estimated error of ∼ 10% are shown. 1.1 1.2 1.3 1.4 1.5 1.6 AF 4P FIG. 9: The phase diagram as a function of the dimension- less density ν and interaction strength rΩ. The various phases were obtained by first calculating the effective couplings eJ1/J4 and eJ2/J4 for a given point; subsequently, the correspond- ing phase was determined utilizing the calculated boundaries shown in Fig. 8 for a system of N = 24 sites. expected to appear in a certain density range as long as rΩ ≫ 1. VII. DISCUSSION In the preceding sections we have studied the coupling of spins of electrons forming a zigzag Wigner crystal in a parabolic confining potential. We have found that apart from the 2-particle exchange couplings between the near- est and next-nearest neighbor spins, the 3- and 4-particle ring exchange processes have to be taken into account. At relatively low electron densities, when the transverse displacement of electrons is small compared to the dis- tance between particles, Fig. 1(b), the nearest-neighbor 2-particle exchange dominates. In this regime the spins form an antiferromagnetic ground state, with low-energy excitations described by the Tomonaga-Luttinger theory. At relatively high densities, when the transverse displace- ments are large, Fig. 1(c), the 4-particle ring exchange processes dominate. Since the ring exchange processes in- volving even numbers of particles favor spin-unpolarized states, the ground state of the system in this regime has zero total spin. Finally, if the confining potential is suf- ficiently shallow, so that the parameter rΩ & 50, there is an intermediate density range in which the 3-particle exchange processes are important, and the ground state is spontaneously spin-polarized. These results are sum- marized in Fig. 9. We expect that the zigzag Wigner crystal state can be realized in quantum wires. In order for the zigzag crys- tal to form the confining potential of the wire should be rather shallow, so that large values rΩ ≫ 1 of the pa- rameter (8) could be achieved. The exact shape of the confining potential in existing wires is not well known. Using the quoted value of subband spacing ∼ 20 meV we estimate that the parameter rΩ is of order unity in cleaved-edge-overgrowth wires.53 The confining potential in split-gate quantum wires tends to be more shallow. For a typical value 1 meV of subband spacing we es- timate rΩ ≈ 6. Finally, for p-type quantum wires13,54 with subband spacing ∼ 300 µeV we estimate rΩ ≈ 20. These hole systems are the most promising devices for observation of the zigzag Wigner crystal. Given the relatively modest values of rΩ . 20 in the ex- isting quantum wire structures, we do not expect that the spontaneously spin-polarized ground state will be easily observed in experiments. Instead, we expect that as the density of charge carriers is increased, a transition from antiferromagnetism to a state dominated by 4-particle ring exchanges will occur. We have found that the ground state in this phase has a complicated size dependence, which makes it very difficult to identify its nature by exact diagonalization of finite-size chains. To fully un- derstand the spin properties in the high density regime, further studies of zigzag ladders with ring exchange cou- pling are needed. Acknowledgments We acknowledge helpful discussions with A. Läuchli and T. Momoi. This work was supported by the U. S. Department of Energy, Office of Science, under Contract No. DE-AC02-06CH11357. T.H. was supported in part by a Grant-in-Aid from the Ministry of Education, Cul- ture, Sports, Science and Technology (MEXT) of Japan (Grant Nos. 16740213 and 18043003). Part of the calcu- lations were performed at the Ohio Supercomputer Cen- ter thanks to a grant of computing time. APPENDIX: CALCULATION OF THE PREFACTORS In order to find the prefactors J∗l in the expressions for the exchange constants, fluctuations around the in- stanton trajectory have to be taken into account. The Euclidean (imaginary time) path integral for the propa- gator G(R1,R2;T ) = 〈R1|e−TH |R2〉 can be written as G(R1,R2;T ) = ∫ R(T )=R2 R(0)=R1 DR e− S[R], (A.1) where the Euclidean action is given by S[R] = + V (R) . (A.2) Here R is a M -dimensional position vector, where M/2 is the total number of moving particles, including the exchanging particles as well as the spectators. In the semiclassical limit, the integral is dominated by the clas- sical path Rcl(τ) that extremizes the action S for a given exchange process. (The exponents η are given as η = S[Rcl]/(~ rΩ).) The Gaussian quantum fluctua- tions about the classical path can be taken into account by defining fluctuation coordinates u(τ) ≡ R(τ)−Rcl(τ) and subsequently expanding the action to second order. We obtain for the propagator G(R1,R2;T ) = F [Rcl]e S[Rcl], (A.3) F [Rcl] = ∫ u(T )=0 u(0)=0 Du(τ) e− 1~ δS[u(τ)], (A.4) δS[u(τ)] = 2(τ) + uT (τ)H(τ)u(τ) , (A.5) Hkp(τ) = ∂2V (R) ∂Rk∂Rp R=Rcl(τ) . (A.6) In the preceding formulas, R1 and R2 correspond to two configurations of electrons that minimize the electrostatic potential V (R) describing electron-electron interactions as well as the confining potential. The exchange constant is related to the ratio of the propagator for a particular exchange process R1 → R2, divided by the propagator for the trivial path Rcl(τ) = R1: F [Rcl] F [R1] S[Rcl]. (A.7) We start from the expression for the propagator in the semiclassical limit and proceed by partitioning the time interval [0, T ] into N subintervals (τ0, τ1), (τ1, τ2), . . . , (τN−1, τN ), with τ0 = 0, τN = T . The partition is cho- sen sufficiently fine as to enable the approximation that in each subinterval, the Hessian matrix H(τ) of the sec- ond derivative of the potential can be considered time independent, H(τ) ≃ H(τν) ≡ Hν . (In what follows, we use the convention that for the fluctuation coordinates, superscripts denote time subinterval, while subscripts de- note spatial coordinate.) Subsequently the path integral is calculated as a product of path integrals over the par- titioned interval. Moreover, each individual path inte- gral is that of a multidimensional harmonic oscillator, for which analytic results exist. We then have F [Rcl] = du1 G1(u 1,u0; τ1 − τ0) . . . duN−1 GN−1(u N−1,uN−2; τN−1 − τN−2)GN (uN ,uN−1; τN − τN−1), (A.8) and the propagator for each subinterval is ν ,uν−1; τν − τν−1) = ∫ u(τν)=uν u(τν−1)=uν−1 Du(τ) exp 2(τ) + uT (τ)Hνu(τ) . (A.9) Within each imaginary time subinterval, we define or- thonormal eigenvectors qνµ = k=1 U k. The unitary matrix Uν is such that Hν = UνΛν(Uν)T , with Λ a diag- onal matrix of eigenvalues (λνµ) 2, µ = 1 . . .M , where M is the number of spatial coordinates. Then one immedi- ately obtains ν ,qν−1; τν − τν−1) = ∫ q(τν)=qν q(τν−1)=qν−1 Dq(τ) exp 2(τ) + qT (τ)Λνq(τ) = F̄ [qcl]e δS[qcl], (A.10) where qcl is the classical trajectory connecting q ν−1 and qν . Considering the fluctuation part first, we obtain an elementary path integral F̄ [qcl] = ∫ q(τν)=0 q(τν−1)=0 Dq(τ) exp dτ qT (τ) , (A.11) where Bνµ = ~ sinh(λνµ∆τν) , (A.12) and ∆τν = τn − τn−1. The exponent δS[qcl] can now be calculated explicitly δS[qcl] [(qνµ) cl + (q cl] cosh(λ µ∆τν) −2(qνµ)cl(qν−1µ )cl . (A.13) The subscript “cl” used for notational clarity will be sub- sequently dropped from all expressions. With some addi- tional algebra, the remaining integral is easily evaluated. With the following definitions Γνkp = ~ tanh(λνµ∆τν) Uνpµ (A.14) ∆νkp = ~ sinh(λνµ∆τν) Uνpµ, (A.15) we find F [Rcl] = (2π) , (A.16) where the M(N − 1) ×M(N − 1) matrix Ωνλkp has com- ponents Ωνλkp = (Γ kp + Γ kp )δ ν,λ −∆νkpδν,(λ+1) −∆λkpδν,(λ−1). (A.17) The calculation of F [R1] is carried out in an identical manner and the subscript “0” will be used to distinguish the results pertaining to that calculation. Finally, one has to account for the existence of an eigenvalue of the matrix Ω which is identically zero in the continuum limit and corresponds to the zero mode associated with uniform translation of the instanton in imaginary time. The procedure is standard55 and we simply report the result for the prefactor here. One ob- tains G = T Bνµ,0 detΩ0 det′ Ω , (A.18) where the primed determinant implies the exclusion of the eigenvalue corresponding to the zero mode. Revert- ing to the system of units used in this work, the prefactor of the exchange energy is given by J∗l = Al rΩ Bνµ,0 detΩ0 det′ Ω (A.19) The additional factor Al is used to account for multiple classical trajectories corresponding to the same exchange process, as happens for the case of nearest and next- nearest neighbor exchanges (i.e., A1 = A2 = 2, whereas Al = 1 for l ≥ 3). The numerical implementation of the method outlined above is straightforward. In particular, the quantity that needs to be numerically calculated, once for each type of exchange at all densities of interest, is Bνµ,0 detΩ0 det′ Ω . (A.20) We note here that the eigenvalue corresponding to the zero mode is easily calculated with the same procedure used by Voelker and Chakravarty38. In the definition of the prefactor, see Eqs. (A.4) and (A.5), one replacesH(τ) with H(τ)− λ, with λ a free parameter. Subsequently, a numerical search for the smallest eigenvalue that results in 1/F (λ) = 0 is carried out. 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0704.0777

Decoupling Supergravity from the Superstring

arXiv:0704.0777v1 [hep-th] 5 Apr 2007 CALT-68-2636 DAMTP-2007-25 UT-07-11 Decoupling Supergravity from the Superstring Michael B. Green,1 Hirosi Ooguri,2,3 and John H. Schwarz2 1Department of Applied Mathematics and Theoretical Physics Cambridge University, Cambridge CB3 0WA, UK 2California Institute of Technology, Pasadena, CA 91125, USA 3Department of Physics, University of Tokyo, Tokyo 113-0033, Japan Abstract We consider the conditions necessary for obtaining perturbative maximal supergrav- ity in d dimensions as a decoupling limit of type II superstring theory compactified on a (10 − d)-torus. For dimensions d = 2 and d = 3 it is possible to define a limit in which the only finite-mass states are the 256 massless states of maximal supergravity. However, in dimensions d ≥ 4 there are infinite towers of additional massless and finite- mass states. These correspond to Kaluza–Klein charges, wound strings, Kaluza–Klein monopoles or branes wrapping around cycles of the toroidal extra dimensions. We con- clude that perturbative supergravity cannot be decoupled from string theory in dimensions ≥ 4. In particular, we conjecture that pure N = 8 supergravity in four dimensions is in the Swampland. March, 2007 http://arxiv.org/abs/0704.0777v1 There has recently has been some speculation that four-dimensional N = 8 super- gravity might be ultraviolet finite to all orders in perturbation theory [1,2,3]. If true, this would raise the question of whether N = 8 supergravity might be a consistent theory that is decoupled from its string theory extension. A related issue is whether N = 8 supergrav- ity can be obtained as a well-defined limit of superstring theory. Here we argue that such a supergravity limit of string theory does not exist in four or more dimensions, irrespective of whether or not the perturbative approximation is free of ultraviolet divergences. In this paper, we will study limits of Type IIA superstring theory on a (10 − d)- dimensional torus T 10−d for various d. One may regard the following analysis as analogous to the study of the decoupling limit on Dp-branes (the limit where field theories on branes decouple from closed string degrees freedom in the bulk) for various p [4,5]. The decoupling limit on Dp-branes is known to exist for p ≤ 5. On the other hand, subtleties have been found for p ≥ 6, where infinitely many new world-volume degrees of freedom appear in the limit. This has been regarded as a sign that a field theory decoupled from the bulk does not exist on Dp-branes for p ≥ 6. We will find similar subtleties for Type IIA theory on T 10−d ×Rd for d ≥ 4. It will be sufficient for our purposes to consider the torus T 10−d to be the product of (10− d) circles, each of which has radius R. Numerical factors, such as powers of 2π, are irrelevant to the discussion that follows and therefore will be dropped. In ten dimensions, Newton’s constant is given by G10 = g 2ℓ8s , where ℓs is the string scale and g is the string coupling constant. Thus, the effective Newton constant in d dimensions is given by Gd ≡ ℓd−2d = R10−d g2ℓ8s R10−d , (1) where ℓd is the d-dimensional Planck length, so that . (2) We are interested in whether there is a limit of string theory that reduces to maximal supergravity, which is defined purely in terms of the dynamics of the 256 states in the massless supermultiplet. In other words, we are interested in the limit in which all the excited string states, together with the Kaluza–Klein excitations and string winding states associated with the (10− d)-torus, decouple. A necessary condition for this to happen is that these states are all infinitely massive compared to the d-dimensional Planck scale ℓd. This is achieved by taking , and , (3) with ℓd fixed. This is compatible with keeping g fixed for d < 6. If the extra states do decouple then the surviving states are the 256 massless states of maximal supergravity, which is N = 8 supergravity when d = 4. Let us now consider the spectrum of nonperturbative superstring excitations in this limit. First consider a Dp-brane wrapping a p cycle of the torus. The mass of such a state in d dimensions is · ℓ1− . (4) When d ≤ 5, we also need to consider a NS5-brane wrapping a 5 cycle. This has a mass given by MNS5 = g2ℓ6s · ℓ2−d . (5) In order to obtain the pure supergravity theory with 32 supercharges in d dimensions, these nonperturbative states also need to decouple, so their masses must satisfy Mp,MNS5 ≫ 1/ℓd. In the case of d = 4 the nonperturbative BPS particle spectrum also includes Kaluza–Klein monopoles, which are discussed in the next paragraph. Before studying the limit in any dimension, d, we will discuss what to expect on general ground. A Kaluza–Klein momentum state and a wrapped string state have masses 1/R and R/ℓ2s , respectively, and they are half-BPS objects that carry a single unit of a conserved charge. In d-dimensions, their magnetic duals are (d − 4)-branes. The BPS saturation condition together with the Dirac quantization condition implies quite generally that the mass m of a BPS particle and the tension T of its magnetic dual (d − 4)-brane are related by mT ∼ 1 . (6) 1 In this limit, the string length ℓs provides a regularization scale for supergravity. Thus, if string amplitudes depend sensitively on ℓs, it can be taken as evidence for ultraviolet divergences in supergravity. This is seen explicitly, for example, in the one-loop four graviton amplitude, which is ultraviolet divergent in nine dimensions. The corresponding string expression is finite and its low-energy limit is sensitive to the presence of these massive states with momenta ∼ 1/ℓs. Applying this to d = 4, we immediately conclude that there is no limit in four dimensions where we can keep all BPS particles heavier than the Planck scale. In particular, magnetic duals of Kaluza–Klein excitations, which are the well-known Kaluza–Klein monopoles, are BPS states with masses ∼ R/ℓ24 → 0.2 Similarly, magnetic duals of wrapped strings are NS5-branes wrapping 5-cycles of T 6, and their masses go as ℓ2s/Rℓ 4 → 0. Later, we will discuss implications of these light states. When d ≥ 5, at least a subset of the BPS branes become tensionless in the limit (3). By contrast, in three dimensions it is possible to define a limit where all BPS particles become infinitely massive simultaneously. In this case, magnetic duals of BPS particles are (−1)-branes, namely instantons, and their Euclidean actions vanish in the limit. Thus, one would expect nonperturbative effects to be very large in three dimensions even though no singularity is apparent from the spectrum. In two dimensions, there are no magnetic duals of BPS particles, and we expect that there is a smooth limit where all BPS particles are massive and instanton actions remain non-vanishing. Now, let us look at each case in more detail. When d = 2, the conditions we want to impose are and MNS5 = → ∞. (7) On the other hand, the string coupling constant is given by . (8) Thus, the desired limit can be taken by sending R → 0 while keeping the string coupling constant finite. In this limit, all particle masses are much higher than the Planck mass, except for the massless two-dimensionalN = 16 supergravity states [6]. However, Dp-brane and NS5-brane instantons wrapping T 8 have Euclidean actions proportional to (ℓs/R) 3−p ∼ 8 and (ℓs/R) 2 ∼ g− 14 , respectively. Though the actions all remain finite and non-zero in the limit, their effects are not uniformly suppressed for small g. Thus, the resulting theory may not have a weak coupling limit that is dominated by the perturbative contribution. When d = 3, the conditions we need to impose are and MNS5 = → ∞. (9) 2 If the torus has six independent radii Ri, the Kaluza–Klein monopole mass spectrum has the form M2 = (niRi/ℓ Since we now have · ℓ3, (10) we can rewrite (9) as and MNS5 = → ∞. (11) Since p = 0, 2, 4, 6 in Type IIA theory, this can again be arranged by taking R → 0 keeping g finite.3 This is also compatible with the limit (3). Thus, all particle states develop large masses and may decouple, except for those in three-dimensional N = 16 supergravity theory [7]. However, Dp-brane and NS5-brane instanton actions, which are given by g 4 (R/ℓ3) 8 and g− 2 (R/ℓ3) 4 , vanish in the limit R → 0 for any finite value of g. This means that nonperturbative effects are strong and it may be difficult to determine the properties of the resulting three-dimensional supergravity. In view of these observations, it is interesting that gravity theories formulated in terms of a finite number of fields are known to exist in two and three dimensions. In three dimensions, the relation with Chern-Simons gauge theory [8] suggests that pure Einstein gravity is finite to all orders in perturbation theory. However, this theory has no propagating degrees of freedom, and it is not known whether there is a finite quantum gravity theory in three dimensions that includes propagating (scalar or spin-1/2) degrees of freedom. Such degrees of freedom are present, of course, in the examples considered here. The fact that we find limits of string theory compactifications with a finite number of such propagating degrees of freedom in these dimensions may be encouraging, though the implications of the nonperturbative instanton contributions need to be understood. When d = 4, the conditions, (3), necessary for the extra modes to have infinite masses and MNS5 = → ∞. (12) Clearly, this cannot be realized simultaneously for all p = 0, 2, 4, 6. This is in accord with the general argument given earlier, since a wrapped Dp-brane and a wrapped D(6−p)-brane 3 Note that, in the Type IIB theory, a wrapped D7-brane cannot be made heavy unless g ≫ 1. This is not in contradiction with T-duality since g transforms under T-duality in such a way that ℓp given by (1) remains invariant. T-duality along one of the circles on T 10−d transforms the coupling g → gℓs/R so it diverges in the limit R → 0 with the original coupling constant, given by (10), kept finite. are electric–magnetic duals. Similarly, the magnetic duals of Kaluza–Klein excitations and wrapped strings are Kaluza–Klein monopoles and wrapped NS5-branes, whose masses behave as R/ℓ24 and ℓ 4, respectively. There are infinitely many such states since they have arbitrary integer charges. In the limit R, ℓ2s/R → 0, there is no mass gap and the spectrum becomes continuous. To understand the implications of these infinitely many light states, we note that among the elements of the four-dimensional U-duality group E7(Z) is the four-dimensional S-duality transformation that interchanges the 28 types of electric charge with the corre- sponding magnetic charges [9,10]. This duality is described by the following transforma- tions of the moduli, S : R → R̃ = ℓ and ℓs → ℓ̃s = . (13) Note that this transformation inverts the radius R in four-dimensional Planck units (in contrast to T-duality, which inverts R in string units). Since g is related to R and ℓs by (2), this transformation acts as the inversion g → g̃ = 1/g, which maps BPS states into each other. For example, a wrapped Dp-brane is interchanged with a wrapped D(6 − p)- brane. Similarly, a Kaluza–Klein excitation is interchanged with a Kaluza–Klein monopole (whereas T-duality would relate it to a wrapped F-string). Thus, in the dual frame in which the compactification scale R̃ → ∞, the six-torus is decompactified. This explains the continuous spectrum in the limit (3). The fact that an infinite set of states from the nonperturbative sector become massless shows that the limit of interest does not result in pure N = 8 supergravity in four dimensions. Rather, it results in 10-dimensional decompactified string theory with the string coupling constant inverted. This is true in both the type IIA and type IIB cases. The only way of avoiding this would be to relax (3), in which case there would instead be extra finite-mass Kaluza–Klein or winding number states, which would therefore not decouple. One may regard our results on the limit of superstring compactification on T 10−d as examples illustrating the conjectures formulated in [11,12] on the geometry of continuous moduli parameterizing the string landscape. The conjectures concern consistent quantum gravity theories with finite Planck scale in four or more dimensions. Among the conjectures are the statements that, if a theory has continuous moduli, there are points in the moduli space that are infinitely far away from each other, and an infinite tower of modes becomes massless as a point at infinity is approached [12]. Since the limit considered in this paper corresponds to a point in the moduli space of string compactifications at infinite distance from a generic point in the middle of moduli space, the conjectures predict than an infinite number of particles become massless in the limit. For d = 4, we have found that among such particles are Kaluza–Klein monopoles, i.e., Kaluza–Klein modes on T 6 in the dual frame in the limit R̃ → ∞. On the other hand, the moduli space of pureN = 8 supergravity also contains infinite distance points, but it does not take account of new light particles appear near these points. If the BPS particles required by string theory were included one would have string theory and not N = 8 supergravity.4 Thus, the conjectures of [12] imply that the N = 8 supergravity is in the Swampland. Similarly, there are many superstring compactifications with N < 8 supersymmetry, and discarding stringy states in these compactifications results in further supergravity theories in the Swampland. It is interesting to see how scattering amplitudes behave in the limit (3). Consider a four-dimensional graviton scattering amplitude where the graviton momenta are below the four-dimensional Planck scale. According to (1) and (2), the ten-dimensional Planck length, ℓ10, is given by ℓ10 = g 4 ℓs = R . (14) After the S-duality transformation (13), the limit R → 0 turns into R̃ → ∞. Thus, we have ℓ̃10 = R̃ → ∞ in ten dimensions. Since ℓ̃10 ≪ R̃, the extra dimensions decompactify and the theory is effectively ten-dimensional. Furthermore, if we take this limit keeping the graviton momenta fixed (in units of the four-dimensional Planck mass), the scattering process becomes trans-Planckian. Generically, we expect that it will involve formation and evaporation of virtual black holes in ten dimensions. The original motivation of this work was to investigate the relation between super- string theory and N = 8 supergravity to see, in particular, under what conditions super- gravity might be ultraviolet finite. What we have found is that in four or more dimensions (d ≥ 4) there is no limit of compactified superstring theory in which the stringy effects decouple and only the 256 massless supergravity fields survive below the four-dimensional Planck scale. This is true whether or not there are ultraviolet divergences in supergravity perturbation theory. Of course, there is a well-defined procedure for extracting UV finite four-dimensional scattering amplitudes from perturbative string theory. This involves tak- ing g → 0 first, before taking the limit (3). However, this procedure does not keep ℓ4 fixed, and therefore it does not correspond to the limit considered in this paper. 4 One can imagine an alternative history in which type II superstring theory and M-theory were discovered by properly interpreting the BPS solitons of N = 8 supergravity. It might be instructive to compare the situation to that of the conifold limit of Calabi– Yau compactified type II superstring theory studied by Strominger [13]. In that case, certain terms in the low-energy effective theory that are independent of the string coupling constant g, due to the decoupling of vector and hypermultiplet fields, can be computed in string perturbation theory. One can estimate the singularity of these terms using the fact that a brane wrapping a vanishing cycle describes a nonperturbative BPS particle that becomes massless in the conifold limit. If one could identify analogous terms in N = 8 supergravity, one could transform the Feynman diagram computation in four-dimensional supergravity into a corresponding computation in ten dimensions, which might give insight into the question of ultraviolet finiteness. The situation is qualitatively different in two and three dimensions (d = 2, 3), where all non-supergravity states develop masses larger than the Planck scale in the limit (3), and therefore they can decouple. In these cases only the 256 massless supergravity states survive, and a self-contained quantum gravity theory may well exist decoupled from string theory. We have found, however, that in the d = 3 case there are instantons with zero action, which give rise to large nonperturbative contributions. In the d = 2 case the instanton actions do not vanish in the limit (3), but not all of them are small when g is small. Therefore the amplitudes may not be dominated by the perturbative contribution in this case, too. Acknowledgments We thank Z. Bern, N. Dorey, C. Hull, J. Russo, N. Seiberg, A. Sen, M. Shigemori, Y. Tachikawa, D. Tong, P. Vanhove and E. Witten for discussions. H.O. thanks the hospitality of the particle theory group of the University of Tokyo. H.O. and J.H.S. are supported in part by the DOE grant DE-FG03-92-ER40701. The research of H.O. is also supported in part by the NSF grant OISE-0403366 and by the 21st Century COE Program at the University of Tokyo. References [1] M. B. Green, J. G. Russo and P. Vanhove, “Non-renormalisation conditions in type II string theory and maximal supergravity,” JHEP 0702, 099 (2007) [arXiv:hep- th/0610299]. [2] Z. Bern, L. J. Dixon and R. Roiban, “Is N = 8 supergravity ultraviolet finite?,” Phys. Lett. B 644, 265 (2007) [arXiv:hep-th/0611086]. [3] Z. Bern, J. J. Carrasco, L. J. Dixon, H. Johansson, D. A. Kosower and R. Roiban, “Three-loop superfiniteness of N = 8 supergravity,” arXiv:hep-th/0702112. [4] A. Sen, “D0-branes on Tn and matrix theory,” Adv. Theor. Math. Phys. 2, 51 (1998) [arXiv:hep-th/9709220]. [5] N. Seiberg, “Why is the matrix model correct?,” Phys. Rev. Lett. 79, 3577 (1997) [arXiv:hep-th/9710009]. [6] H. Nicolai and N. P. Warner, “The structure of N = 16 supergravity in two dimen- sions,” Commun. Math. Phys. 125, 369 (1989). [7] N. Marcus and J. H. Schwarz, “Three-dimensional supergravity theories,” Nucl. Phys. B 228, 145 (1983). [8] E. Witten, “(2+1)-dimensional gravity as an exactly soluble system,” Nucl. Phys. B 311, 46 (1988). [9] C. M. Hull and P. K. Townsend, “Unity of superstring dualities,” Nucl. Phys. B 438, 109 (1995) [arXiv:hep-th/9410167]. [10] C. M. Hull, “String dynamics at strong coupling,” Nucl. Phys. B 468, 113 (1996) [arXiv:hep-th/9512181]. [11] C. Vafa, “The string landscape and the swampland,” arXiv:hep-th/0509212. [12] H. Ooguri and C. Vafa, “On the geometry of the string landscape and the swampland,” [arXiv:hep-th/0605264]. [13] A. Strominger, “Massless black holes and conifolds in string theory,” Nucl. Phys. B 451, 96 (1995) [arXiv:hep-th/9504090]. http://arxiv.org/abs/hep-th/0610299 http://arxiv.org/abs/hep-th/0610299 http://arxiv.org/abs/hep-th/0611086 http://arxiv.org/abs/hep-th/0702112 http://arxiv.org/abs/hep-th/9709220 http://arxiv.org/abs/hep-th/9710009 http://arxiv.org/abs/hep-th/9410167 http://arxiv.org/abs/hep-th/9512181 http://arxiv.org/abs/hep-th/0509212 http://arxiv.org/abs/hep-th/0605264 http://arxiv.org/abs/hep-th/9504090

0704.0778

Frobenius splitting and geometry of $G$-Schubert varieties

arXiv:0704.0778v2 [math.AG] 10 Sep 2008 FROBENIUS SPLITTING AND GEOMETRY OF G-SCHUBERT VARIETIES XUHUA HE AND JESPER FUNCH THOMSEN Abstract. Let X be an equivariant embedding of a connected reductive group G over an algebraically closed field k of positive characteristic. Let B denote a Borel subgroup of G. A G-Schubert variety in X is a subvariety of the form diag(G) · V , where V is a B × B-orbit closure in X . In the case where X is the wonderful compactification of a group of adjoint type, the G-Schubert vari- eties are the closures of Lusztig’s G-stable pieces. We prove that X admits a Frobenius splitting which is compatible with all G- Schubert varieties. Moreover, when X is smooth, projective and toroidal, then any G-Schubert variety in X admits a stable Frobe- nius splitting along an ample divisors. Although this indicates that G-Schubert varieties have nice singularities we present an example of a non-normal G-Schubert variety in the wonderful compactifi- cation of a group of type G2. Finally we also extend the Frobenius splitting results to the more general class of R-Schubert varieties. 1. Introduction Let G denote a connected and reductive group over an algebraically closed field k, and let B denote a Borel subgroup of G. An equi- variant embedding X of G is a G × G-variety which contains G = (G × G)/diag(G) as an open G × G-invariant subset, where diag(G) is the diagonal image of G in G × G. Any equivariant embedding X of G contains finitely many B × B-orbits. In recent years the geom- etry of closures of B × B-orbits has been studied by several authors. The most general result was obtained in [H-T2] where it was proved that B × B-orbit closures are normal, Cohen-Macaulay and have (F - )rational singularities (actually, even stronger results were obtained). In the present paper we will study (closed) subvarieties inX of the form diag(G) ·V, where V denotes the closure of a B×B-orbit. Subvarieties of equivariant embeddings of G of this form will be called G-Schubert varieties. When G is a semisimple group of adjoint type there exists a canonical equivariant embedding X of G which is called the wonderful compact- ification. The wonderful compactifications are of primary interest in this paper. Actually, this work arose from the question of describing the closures of the so-called G-stable pieces of X. The G-stable pieces makes up a decomposition of X into locally closed subsets. They were introduced by Lusztig in [L] where they were used to construct and http://arxiv.org/abs/0704.0778v2 2 XUHUA HE AND JESPER FUNCH THOMSEN study a class of perverse sheaves which generalizes his theory of charac- ter sheaves on reductive groups. More precisely, these perverse sheaves are the intermediate extensions of the so-called “character sheaves” on a G-stable piece. This motivates the study of closures of G-stable pieces which turns out to coincide with the set of G-Schubert varieties. Before discussing the closures of G-stable pieces in details, let us make a short digression and discuss some other motivations for study- ing G-stable pieces and G-Schubert varieties (in wonderful compactifi- cations): (1) When G is a simple group, the boundary of the closure of the unipotent subvariety of G in the wonderful compactification X, is a union of certain G-Schubert varieties (see [He] and [H-T]). Thus knowing the geometry of these G-Schubert varieties will help us to understand the geometry of the closure of the unipo- tent variety within X. (2) Let Lie(G) denote the Lie algebra of a simple group G over a field of characteristic zero. Let ≪,≫ denote a fixed symmet- ric non-degenerate ad-invariant bilinear form. Let <,> be the bilinear form on Lie(G)⊕ Lie(G) defined by < (x, y), (x′, y′) >=≪ x, x′ ≫ − ≪ y, y′ ≫ . In [E-L], Evens and Lu showed that each splitting Lie(G) ⊕ Lie(G) = l ⊕ l′, where l and l′ are Lagrangian subalgebras of Lie(G)⊕ Lie(G), gives rise to a Poisson structure Πl,l′ on X. If moreover, one starts with the Belavin-Drinfeld splitting, then all the G-stable pieces/G-Schubert varieties and B×B−-orbits of X are Poisson subvarieties, where B− is a Borel subgroup opposite to B. Thus to understand the Poisson structure on X corresponding to the Belavin-Drinfeld splitting, one needs to understand the geometry of the G-stable pieces/G-Schubert varieties. If we start with another splitting, then we obtain a different Poisson structure on X and in order to understand these Poisson structures, one needs to study the R-stable pieces [L-Y] instead (see Section 12), which generalize both the G- stable pieces and the B × B−-orbits. The main technical ingredient in this paper is the positive character- istic notion of Frobenius splitting. Frobenius splitting is a powerful tool which has been proved to be very useful in obtaining strong geometric conclusions for e.g. Schubert varieties and closures of B × B-orbits in equivariant embeddings. In the present paper we obtain two types of results related to G-Schubert varieties over fields of positive character- istic. First of all, if we fix an equivariant embedding X of a reductive group G then we prove that all G-Schubert varieties in X are simul- taneously compatibly Frobenius split by a Frobenius splitting of X . Secondly, concentrating on a single G-Schubert variety X, in a smooth projective and toroidal embedding X , we prove that this admits a stable Frobenius splitting along an ample divisor. Statements of this form put strong conditions on the intertwined behavior of cohomology groups of line bundles on X and its G-Schubert varieties. As this is re- lated to geometric properties it therefore seems natural to expect that G-Schubert varieties should have nice singularities. It therefore comes as a complete surprise that G-Schubert varieties, in general, are not even normal. We only provide a single example of this phenomenon (in the wonderful compactification of a group of type G2), but expect that this absence of normality is the general picture. In obtaining the Frobenius splitting result mentioned above, we have developed some general theory of how to construct Frobenius splitting of varieties of the form G×PX (see Section 4.2 for the definition). This part of the paper is influenced by the theory of B-canonical Frobenius splitting as discussed in [B-K, Chap.4]; in particular the proof of [B-K, Prop.4.1.17]. The presentation we provide is more general and makes it possible to extract even better result from the ideas of B-canonical Frobenius splittings. This theory is presented in Chapter 5 in a general- ity which is more than necessary for obtaining the described Frobenius splitting results for G-Schubert varieties. However, we hope that this theory could be useful elsewhere and we certainly consider it to be of independent interest. This paper is organized in the following way. In Section 2 we intro- duce notation, and in Section 3 we briefly define Frobenius splitting and explain its fundamental ideas. Section 4 is devoted to some results on linearized sheaves which should all be well known. In Section 5 we study the Frobenius splitting of varieties of the form G ×P X for a variety X with an action by a parabolic subgroup P . The main idea is to decompose the Frobenius morphism on G×P X into maps associ- ated to the Frobenius morphism on the base G/P and the fiber X of the natural morphism G×P X → G/P . In Section 6 we relate B-canonical Frobenius splittings to the material in Section 5. Section 7 contains applications of Section 5 to general G × G-varieties. In section 8 we define the G-stable pieces and G-Schubert varieties. In Section 9 we apply the material of the previous sections to the class of equivariant embeddings and obtain Frobenius splitting results for G-Schubert vari- eties. Section 10 contains results related to cohomology of line bundles on G-Schubert varieties. Section 11 contains an example of a non- normal G-Schubert variety. Finally Section 12 contains generalizations and variations of the previous sections. We would like to thank the referee for a careful reading of this paper and for numerous suggestions concerning the presentation. 4 XUHUA HE AND JESPER FUNCH THOMSEN 2. Notation We will work over a fixed algebraically closed field k. The charac- teristic of k will depend on the application. By a variety we mean a reduced and separated scheme of finite type over k. In particular, we allow a variety to have several irreducible components. 2.1. Group setup. We letG denote a connected linear algebraic group over k. We fix a Borel subgroup B and a maximal torus T ⊂ B. The notation P is used for a parabolic subgroup of G containing B. The set of T -characters is denoted by X∗(T ) and we identify this set with the set X∗(B) of B-characters. 2.2. Reductive case. In many cases we will specialize to the case where G is reductive. In this case we will also use the following no- tation : the set of roots determined by T is denoted by R ⊆ X∗(T ) while the set of positive roots determined by (B, T ) is denoted by R+. The simple roots are denoted by α1, . . . , αl, and we let ∆ = {1, . . . , l} denote the associated index set. The simple reflection associated to the simple root αi is then denoted by si. The Weyl group W = NG(T )/T is generated by the simple reflections si, for i ∈ ∆. The length of w ∈ W will be denoted by l(w). For J ⊂ ∆, let WJ denote the subgroup of W generated by the simple reflection associated with the elements in J , and let W J (resp. JW ) denote the set of minimal length coset rep- resentatives for W/WJ (resp. WJ\W ). The element in W of maximal length will be denoted by w0, while w 0 is used for the same kind of element in WJ . For any w ∈ W , we let ẇ denote a representative of w in NG(T ). For J ⊂ ∆, let PJ ⊃ B denote the corresponding standard parabolic subgroup and P−J ⊃ B − denote its opposite parabolic. Let LJ = PJ ∩P J be the common Levi subgroup of PJ and P J containing T . Let UJ (resp. U J ) denote the unipotent radical of PJ (resp. P When J = ∅ we also use the notation U and U− for UJ and U J respec- tively. When G is semisimple and simply connected we may associate a fundamental character ωi to each simple root αi. The sum of the fundamental characters is then denoted by ρ. Then ρ also equals half the sum of the positive roots. 3. The relative Frobenius morphism In this section we collect some results related to the Frobenius mor- phism and to the concept of Frobenius splitting. Compared to other presentations on the same subject, this presentation differs only in its emphasis on the set HomOX′ (FX)∗OX ,OX′ (to be defined below) and not just the set of Frobenius splittings. Thus, the obtained results are only small variations of already known results as can be found in e.g. [B-K]. 3.1. The Frobenius morphism. By definition a variety X comes with an associated morphism pX : X → Spec(k), of schemes. Assume that the field k has positive characteristic p > 0. Then the Frobenius morphism on Spec(k) is the morphism of schemes Fk : Spec(k) → Spec(k), which on the level of coordinate rings is defined by a 7→ ap. As k is assumed to be algebraically closed the morphism Fk is actually an isomorphism and we let F−1k denote the inverse morphism. Composing pX with F k we obtain a new variety p′X : X → Spec(k), with underlying scheme X . In the following we suppress the morphism pX from the notation and simply use X as the notation for the variety defined by pX . The variety defined by p X is then denoted by X The relative Frobenius morphism on X is then the morphism of varieties : FX : X → X which as a morphism of schemes is the identity map on the level of points and where the associated map of sheaves X : OX′ → (FX)∗OX , is the p-th power map. A key property of the Frobenius morphism is the relation (1) (FX) ′ ≃ Lp which is satisfied for every line bundle L on X (here L′ denotes the corresponding line bundle on X ′). 3.2. Frobenius splitting. A variety X is said to be Frobenius split if the OX′-linear map of sheaves : X : OX′ → (FX)∗OX , has a section; i.e. if there exists an element s ∈ HomOX′ (FX)∗OX ,OX′ such that the composition s ◦F X is the identity endomorphism of OX′ . The section s will be called a Frobenius splitting of X . 6 XUHUA HE AND JESPER FUNCH THOMSEN 3.3. Compatibility with line bundles and closed subvarieties. Fix a line bundle L on X and a closed subvariety Y in X with sheaf of ideals IY . Let Y ′ denote the closed subvariety of X ′ associated to Y with sheaf of ideals denoted by IY ′ . The kernel of the natural morphism HomOX′ (FX)∗L,OX′ → HomOX′ (FX)∗(L⊗ IY ),OY ′ induced by the inclusion L ⊗ IY ⊂ L and the projection OX′ → OY ′ , will be denoted by EndLF (X, Y ). The associated space of global sections will be denoted by EndLF (X, Y ). When Y = X we simply denote EndLF (X, Y ) (resp. End F (X, Y )) by End F (X) (resp. End F (X)). The sheaf EndLF (X, Y ) is a subsheaf of End F (X) consisting of the elements compatible with Y . Moreover, there is a natural morphism EndLF (X, Y )|Y → End F (Y ), where the notation |Y means restriction to Y . If Y1, Y2, . . . , Ym is a collection of closed subvarieties of X then the notation EndLF (X, Y1, . . . , Ym) (or sometimes End F (X, {Yi} i=1)) will de- note the intersection of the subsheaves EndLF (X, Yi) for i = 1, . . . , m. The set of global sections of the sheaf EndLF (X, Y1, . . . , Ym) will be de- noted by EndLF (X, Y1, . . . , Ym). When L = OX we remove L from all of the above notation. In particular, the vectorspace EndF (X) denotes the set of morphisms from (FX)∗OX to OX′ and thus contains the set of Frobenius splittings of X . A Frobenius splitting s of X contained in EndF (X, {Yi}i) is said to be compatible with the subvarieties Y1, . . . , Ym. When s is compatible in this sense it induces a Frobenius splitting of each Yi for i = 1 . . . , m. In this case we also say that s compatibly Frobenius splits Y1, . . . , Ym. In concrete terms, this is equivalent to (FX)∗IYi ⊂ IY ′i . for all i. Lemma 3.1. Let Y and Z denote closed subvarieties in X and let s denote a global section of EndLF (X,Z, Y ). (1) s ∈ EndLF (X, Y1) for every irreducible component Y1 of Y . (2) If the scheme theoretic intersection Z ∩ Y is reduced then s is contained in EndLF (X, Y ∩ Z). Proof. Let Y1 denote an irreducible component of Y and let J = s (FX)∗(IY1 ⊗ L) ⊂ OX′ . Let U denote the open complement (in X ′) of the irreducible compo- nents of Y ′ which are different from Y ′1 . Then IY ′1 coincides with IY ′ on U and consequently J|U ⊂ (IY ′)|U as s is compatible with Y . In particular, J|U ⊂ (IY ′1 )|U . We claim that this implies that J ⊂ IY ′1 : let V denote an open subset of X ′ and let f be a section of J over V . As J is a subsheaf of OX′ , we may consider f as a function on V , and it suffices to prove that f vanishes on Y ′1 ∩ V . If Y 1 ∩ V is empty then this is clear. Otherwise, U ∩ V ∩ Y ′1 is a dense subset of Y 1 and it suffices to prove that f vanishes on this set. But this follows from the inclusion J|U ⊂ (IY ′1 )|U . As a consequence s is compatible with Y1. The second claim follows as the sheaf of ideals of the intersection Z ∩ Y is IY + IZ . � The condition that Z ∩ Y is reduced, in Lemma 3.1, only ensures that Z ∩ Y is a variety. When L = OX and s is a Frobenius splitting this is always satisfied [B-K, Prop.1.2.1]. 3.4. The evaluation map. Let k[X ′] denote the space of global reg- ular functions on X ′. Evaluating an element s : (FX)∗OX → OX′ of EndF (X) at the constant global function 1 on X defines an element in k[X ′] which we denote by evX(s). This defines a morphism evX : EndF (X) → k[X with the property that evX(s) = 1 if and only if s is a Frobenius splitting of X . 3.5. Frobenius D-splittings. Consider an effective Cartier divisor D on X , and let σD denote the associated global section of the associated line bundle OX(D). A Frobenius splitting s of X is said to be a Frobe- nius D-splitting if s factorizes as s : (FX)∗OX (FX)∗σD −−−−−→ (FX)∗OX(D) −→ OX′ , for some element sD in End OX(D) . We furthermore say that the Frobenius D-splitting s is compatible with a subvariety Y if sD is com- patible with Y . The following result assures that, in this case, the compatibility with closed subvarieties agrees with the usual definition [R, Defn.1.2]. Lemma 3.2. Assume that s defines a Frobenius D-splitting of X. Then sD is compatible with Y if and only if (i) s compatibly Frobe- nius splits Y and (ii) the support of D does not contain any irreducible components of Y . Proof. The if part of the statement follows from [R, Prop.1.4]. So assume that sD is compatible with Y . Then sD induces a morphism sD : (FY )∗OX(D)|Y → OY ′ , satisfying sD((σD)|Y ) is the constant function 1 on Y ′. As a conse- quence (σD)|Y does not vanish on any of the irreducible components of Y . This proves part (ii) of the statement. Part (i) is clearly satis- fied. � 8 XUHUA HE AND JESPER FUNCH THOMSEN It follows that if s is compatible with Y and, moreover, defines a Frobenius D-splitting of X then D ∩ Y makes sense as an effective Cartier divisor on Y and, in this case, s induces a Frobenius D ∩ Y - splitting of Y . 3.6. Stable Frobenius splittings along divisors. Let X(0) = X and define recursively X(n) = (X(n−1))′ for n ≥ 1. Composing the Frobenius morphisms on X(i) for i = 0, . . . , n, we obtain a morphism X : X → X with an associated map of sheaves ♯ : OX(n) → (F X )∗OX . Let, as in Section 3.5, D denote an effective Cartier divisor on X with associated canonical section σD of OX(D). We say that X admits a stable Frobenius splitting along D if there exists a positive integer n and an element s ∈ HomO X )∗OX(D),OX(n) such that the composed map OX(n) −−−−→ (F X )∗OX −−−−−−→ (F X )∗OX(D) −→ OX(n) , is the identity map on OX(n) . The element s is called a stable Frobenius splitting of X along D. When Y is a closed subvariety of X we say that the stable Frobenius splitting s is compatible with Y if X )∗(IY ⊗ OX(D)) ⊂ IY (n). Notice that this condition necessarily implies that the support of D does not contain any of the irreducible components of Y (cf. proof of Lemma 3.2). Notice also that if X admits a Frobenius D-splitting which is compatible with Y then X admits a stable Frobenius splitting along D which is compatible with Y . The following is well known (see e.g. [T, Lem.4.4]) Lemma 3.3. Let D1 and D2 denote effective Cartier divisors on X and let Y denote a closed subvariety of X. Then X admits stable Frobenius splittings along D1 and D2 which are compatible with Y if and only if X admits a stable Frobenius splitting along D1+D2 which is compatible with Y . The following result explains one of the main applications of (stable) Frobenius splitting. Remember that a line bundle L is nef if L⊗M is ample whenever M is ample. Proposition 3.4. Assume that X admits a stable Frobenius splitting along an effective Cartier divisor D. Then there exists a positive integer n such that for each line bundle L on X we have an inclusion of abelian groups Hi(X,L) ⊂ Hi(X,Lp ⊗ OX(D)). In particular, if D is ample and L is nef, then Hi(X,L) = 0 for i > 0. Moreover, if the stable Frobenius splitting of X is compatible with a subvariety Y , D is ample and L is nef then the restriction morphism H0(X,L) → H0(Y,L), is surjective. Proof. Argue as in the proof [R, Prop.1.13(i)]. � 3.7. Duality for FX. By duality (see [Har2, Ex.III.6.10]) for the finite morphism FX we may to each quasi-coherent OX′-module F associate an OX -module denoted by (FX) !F and satisfying (FX)∗(FX) F = HomOX′ (FX)∗OX ,F Actually, as FX is the identity on the level of points we may define !F as the sheaf of abelian groups HomOX′ (FX)∗OX ,F with OX -module structure defined by (g · φ)(f) = φ(gf), for g, f ∈ OX and φ ∈ HomOX′ (FX)∗OX ,F . When F = OX we will also use the notation End!F (X) for (FX) !OX . This sheaf is par- ticularly nice when X is smooth as (FX) !OX then coincides with the line bundle ω X , where ωX denotes the dualizing sheaf of X (see e.g. [B-K, Sect.1.3]). If Y1, Y2, . . . , Ym is a collection of closed subvarieties of X then End!F (X, Y1, . . . , Ym) (or End F (X, {Yi} i=1)) will denote the subsheaf of End!F (X) consisting of the elements mapping the sheaf of ideals IYi to IY ′i for all i = 1, . . . , m. We say that End F (X, {Yi} i=1) is the subsheaf of elements compatible with Y1, . . . , Ym. More generally, duality for FX implies that we have a natural iden- tification (FX)∗HomOX G, (FX) ≃ HomOX′ (FX)∗G,F whenever G (resp. F) is a quasicoherent sheaf on X (resp. X ′). This leads to the identification HomOX G, (FX) ≃ HomOX′ (FX)∗G,F where a morphism η : G → (FX) !F is identified with the composed morphism η′ : (FX)∗G (FX)∗η −−−−→ (FX)∗(FX) F ≃ HomOX′ (FX)∗OX ,F Here the latter map is the natural evaluation map at the element 1 in OX . From now on we will specialize to the case where F = OX′ 10 XUHUA HE AND JESPER FUNCH THOMSEN and G equals a line bundle L on X . In this case, an element in HomOX L,End!F (X) may also be considered as a global section of the sheaf End!F (X)⊗ L −1. For later use we emphasize Lemma 3.5. Let η be an element in HomOX L,End!F (X) and let η′ denote the corresponding element in HomOX′ (FX)∗L,OX′ by the above identification. Then η′ factors through the morphism (FX)∗L (FX)∗η −−−−→ (FX)∗End F (X). Moreover, the element η′ is compatible with a collection of closed sub- varieties Y1, . . . , Ym of X if and only if the image of η is contained in End!F (X, Y1, . . . , Ym). Proof. The first part of the statement follows directly from the discus- sion above. To prove the second statement we may assume that m = 1. We use the notation Y = Y1. Let σ denote a section of L over an open subset U of X , and consider s = η(σ) as a map s : OX(U) → OX′(U That s is compatible with Y means that s(f) vanishes on Y ′ whenever f vanishes on Y for a function f on U . Alternatively, the evaluation of f · s at 1, which coincides with η′(f · σ), should vanish on Y ′. In particular, the image of η is contained in End!F (X, Y ) if and only if the restriction of η′ to (FX)∗ IY ⊗ L maps into IY ′ . This ends the proof. � We will also need the following remark Lemma 3.6. Let D denote a reduced effective Cartier divisor on X and L denote a line bundle on X. Let M = OX((p − 1)D) ⊗ L and assume that we have an OX-linear morphism η : M → End F (X). Let σD denote the canonical section of OX(D) and consider the map ηD : L → End F (X), induced by σ D . Then the element η′D ∈ HomOX′ (FX)∗L,OX′ induced by ηD, is compatible with the support of D. In particular, the image of ηD is contained in End F (X,D). Proof. Notice that η′D is the composition η′D : (FX)∗L (FX)∗σ −−−−−−→ (FX)∗M −→ OX′ , where η′ is the element corresponding to η. Hence, the restriction of η′D to L⊗ OX(−D) coincides with the map (FX)∗ L⊗ OX(−D) ) (FX)∗σ −−−−−→ (FX)∗M −→ OX′ . But the restriction of η′ to (cf. (1)) (FX)∗ OX(−pD)⊗M ≃ OX′(−D ′)⊗ (FX)∗M, maps by linearity into OX′(−D ′). The in particular part follows by Lemma 3.5. � 3.8. Push-forward operation. Assume that f : X → Z is a mor- phism of varieties satisfying that the associated map f ♯ : OZ → f∗OX is an isomorphism. Let f ′ : X ′ → Z ′ denote the associated morphism. Then f ′∗ induces a morphism f ′∗EndF (X) → EndF (Z). If Y ⊂ X is a closed subset then the subsheaf f ′∗EndF (X, Y ) is mapped to EndF (Z, f(Y )), where f(Y ) denotes the variety associated to the closure of the image of Y . On the level of global sections this means that every Frobenius splitting s of X induces a Frobenius splitting f ′∗s of Z such that when s is compatible with Y then f ′∗s is compatible with f(Y ). Likewise Lemma 3.7. With notation as above, let L denote a line bundle on Z and let s be an element of End f∗(L) . Then f ′∗s is an element of EndLF . Moreover, if s is compatible with a closed subvariety Y of X then f ′∗s is compatible with f(Y ). Proof. This follows easily from the fact that the sheaf of ideals of f(Y ) coincides with f∗IY [B-K, Lem.1.1.8]. � 4. Linearized sheaves In this section we collect a number of well known facts about lin- earized sheaves. The chosen presentation follows rather closely the presentation in [Bri, Sect.2]. Let H denote a linear algebraic group over the field k and let X denote a H-variety with H-action defined by σ : H × X → X . We let p2 : H × X → X denote projection on the second coordinate. A H-linearization of a quasi-coherent sheaf F on X is an OH×X -linear isomorphism φ : σ∗F → p∗2F, satisfying the relation (2) (µ× 1X) ∗φ = p∗23φ ◦ (1H × σ) as morphisms of sheaves on H × H × X . Here µ : H × H → H (resp. p23 : H × H × X → H × X) denotes the multiplication on H (resp. the projection on the second and third coordinate). Based on the fact that σ∗OX = p 2OX we see that the sheaf OX admits a canonical linearization. In the following we will always assume that OX is equipped with this canonical linearization. 12 XUHUA HE AND JESPER FUNCH THOMSEN A morphism ψ : F → F′ of H-linearized sheaves is a morphism of OX -modules commuting with the linearizations φ and φ ′ of F and F′, i.e. φ′ ◦ σ∗(ψ) = p∗2(ψ) ◦ φ. Linearized sheaves on X form an abelian category which we denote by ShH(X). 4.1. Quotients and linearizations. Assume that the quotient q : X → X/H exists and that q is a locally trivial principal H-bundle. Then for G ∈ Sh(X/H), q∗G is naturally a H-linearized sheaf on X . This defines a functor q∗ : Sh(X/H) → ShH(X). On the other hand, for F ∈ ShH(X), q∗F has a natural action of H . Define a functor qH∗ : ShH(X) → Sh( X/H) by qH∗ (F) = (q∗F) H the subsheaf of H- invariants of q∗F. It is known that the functor q ∗ : Sh(X/H) → ShH(X) is an equivalence of categories with inverse functor qH∗ . In general, if H is a closed normal subgroup of G and X is a G- variety such that the quotient X/H exists (as above), then X/H is a G/H- variety and the functor q∗ : ShG/H(X/H) → ShG(X) is an equivalence of categories with inverse functor qH∗ : ShG(X) → ShG/H( X/H). 4.2. Induction equivalence. Consider now a connected linear alge- braic group G and a parabolic subgroup P in G. Let X denote a P -variety. Then G×X is a G× P -variety by the action (g, p)(h, x) = (ghp−1, px), for g, h ∈ G, p ∈ P and x ∈ X . Then the quotient, denoted by G ×P X , of G × X by P exists and the associated quotient map q : G×X → G×P X is a locally trivial principal P -bundle. The quotient of G × X by G also exists and may be identified with the projection p2 : G×X → X . In particular, we may apply the above consideration to obtain equivalences of the categories ShP (X), ShG×P (G × X) and ShG(G×P X). Notice that under this equivalence a P -linearized sheaf F on X corresponds to the G-linearized sheaf IndGP (F) = (q∗p P . In particular, the space of global sections of IndGP (F) equals IndGP (F)(G×P X) = p∗2F(G×X) k[G]⊗k F(X) = IndGP (F(X)), where the second equality follows by the Künneth formula. This also explains the notation IndGP (F). Similarly, starting with a G-linearized sheaf G on G×P X then the associated P -linearized line bundle on X equals G′ = ((p2)∗q ∗G)G. However, by [Bri, Lemma 2(1)] the latter also equals the simpler pull back i∗G by the P -equivariant map i : X → G×P X, sending x to q(1, x). In particular, we conclude that the functor i∗ : ShG(G ×P X) → ShP (X) is an equivalence of categories with inverse functor IndGP . Notice also that the space of global sections of G is G-equivariantly isomorphic to G(G×P X) = Ind (i∗G)(X) which follows by (3) above. 4.3. Duality. Assume that the field k has positive characteristic p > 0. Regard X ′ as a H-variety in the canonical way and let F denote a H- linearized sheaf on X ′. The sheaf (FX) !F, defined in Section 3.7, is then naturally a H-linearized sheaf on X . Moreover, the induced H- linearization of (FX)∗(FX) !F coincides with the natural H-linearization HomOX′ (FX)∗OX ,F When X is smooth the sheaf (FX) !OX′ is canonically isomorphic to X (cf. Section 3.7). We may use this isomorphism to define a H- linearization of ω X . Alternatively we may consider the natural H- linearization of the dualizing sheaf ΩX of X and use this to define a H-linearization of ω X . It may be checked that the two stated ways of defining a H-linearization of ω X coincide. 5. Frobenius splitting of G×P X Let G denote a connected linear algebraic group over an algebraically closed field k of characteristic p > 0. Let P denote a parabolic subgroup of G and let X denote a P -variety. In this section we want to consider Frobenius splittings of the quotient Z = G×PX of G×X by P . We let π : Z → G/P denote the morphism induced by the projection of G×X on the first coordinate. When g ∈ G and x ∈ X we use the notation [g, x] to denote the element in Z represented by (g, x). 5.1. Decomposing the Frobenius morphism. The Frobenius mor- phism FZ admits a decomposition FZ = Fb ◦ Ff where Fb (resp. Ff ) is related to the Frobenius morphism on the base (resp. fiber) of π. More precisely, define Ẑ and the morphisms π̂ and Fb as part of the fiber product diagram (4) Ẑ Fb // // (G/P)′ A local calculation shows that we may identify Ẑ with the quotient G ×P X ′, where the P -action on the Frobenius twist X ′ of X is the natural one. With this identification π̂ : G ×P X ′ → G/P is just the 14 XUHUA HE AND JESPER FUNCH THOMSEN map [g, x′] 7→ gP . It also follows that the natural morphism (induced by the Frobenius morphism on X) Ff : G×P X → G×P X makes the following diagram commutative Fb // // (G/P)′ 5.2. Let M denote a P -linearized line bundle on X and let MZ = IndGP (M) denote the associated G-linearized line bundle on Z. The main aim of this section is to construct global sections of the sheaf F (Z) = HomOZ′ (FZ)∗MZ ,OZ′ To this end we fix a P -character λ and let L denote the associated line bundle on G/P (cf. Section 4). The pull back π̂∗L of L to Ẑ is then denoted by LẐ . We then define the following sheaves F (Z)f := HomOẐ (Ff)∗MZ ,OẐ F (Z)b := HomOZ′ (Fb)∗LẐ ,OZ′ with spaces of global sections denoted by End F (Z)f and End F (Z)b. Notice that whenM is substituted with the P -linearized twistM(−λ) := M⊗ k−λ then M(−λ)Z = MZ ⊗ π ∗(L−1) = MZ ⊗ (Ff) and thus by the projection formula (5) End M(−λ)Z F (Z)f = HomOẐ (Ff)∗MZ ,LẐ Sections of End F (Z) are then constructed as compositions of global sections of the sheaves End M(−λ)Z F (Z)f and End F (Z)b. More precisely, v ∈ HomO (Ff)∗MZ ,LẐ u ∈ HomOZ′ (Fb)∗LẐ ,OZ′ are global sections of the latter sheaves, then the composition u◦(Fb)∗v defines a global section of End F (Z). 5.3. An equivariant setup. We now give equivariant descriptions of the sheaves End F (Z)f and End F (Z)b. 5.3.1. A description of End F (Z)f . Now End F (Z)f is a G-linearized sheaf on Ẑ = G×P X ′. Let Y denote a P -stable subvariety of X and let ZY = G ×P Y denote the associated subvariety of Z with sheaf of ideals IZY ⊂ OZ . Let ẐY denote the subvariety G ×P Y ′ of Ẑ. Then there is a natural morphism of G-linearized sheaves F (Z)f → HomOẐ (Ff )∗(MZ ⊗ IZY ),OẐY induced by the inclusion IZY ⊂ OZ and the projection OẐ → OẐY . We let End F (Z,ZY )f denote the kernel of the above map and arrive at a left exact sequence of G-linearized sheaves 0 → EndMZF (Z,ZY )f → End F (Z)f → HomOẐ (Ff)∗(MZ⊗IZY ),OẐY In particular, the space of global sections of End F (Z,ZY )f is identified with the set of elements in End F (Z)f which map (Ff )∗(MZ ⊗ IZY ) to ⊂ OẐ . Using the observations in Section 4.2 we can give another description of the space of global sections of End F (Z,ZY )f . Let i X ′ → G×P X ′ denote the morphism i′(x′) = [1, x′]. Then the functor i′ is exact on the category of G-linearized sheaves. We want to apply this fact on the left exact sequence (6) above : notice first that (i′)∗End F (Z)f = HomOX′ (i′)∗(Ff )∗MZ ,OX′ where, moreover, (i′)∗(Ff)∗MZ = (FX)∗M. Thus (i ′)∗End F (Z)f = EndMF (X). Similarly, (i′)∗HomO (Ff)∗(MZ ⊗ IZY ),OẐY = HomOX′ ((FX)∗(M⊗ IY ),OY ′). In particular, we see that the P -linearized sheaf on X ′ corresponding to the G-linearized sheaf End F (Z,ZY )f equals the kernel of the natural EndMF (X) → HomOX′ ((FX)∗(M⊗ IY ),OY ′), i.e. it equals EndMF (X, Y ). By Section 4.2 the space of global sections F (Z,ZY )f of End F (Z,ZY )f is then G-equivariantly isomorphic IndGP EndMF (X, Y )). Applying the above conclusions to the sheaf M(−λ) we find: Proposition 5.1. There exists a G-equivariant isomorphism M(−λ)Z F (Z)f ≃ Ind EndMF (X)⊗ kλ) such that when Y is a closed P -stable subvariety of X then the subset of elements of End M(−λ)Z F (Z)f which map (Ff)∗(MZ⊗IZY ) to (IẐY ⊗LẐ) ⊂ LẐ (cf. equation (5)) is identified with M(−λ)Z F (Z,ZY )f ≃ Ind EndMF (X, Y )⊗ kλ). 16 XUHUA HE AND JESPER FUNCH THOMSEN 5.3.2. A description of End F (Z)b. As π ′ in the fibre-diagram (4) is flat the natural morphism (π′)∗(FG/P )∗L → (Fb)∗π̂ ∗L is an isomor- phism ([Har2, Prop.III.9.3]). Thus there is a natural isomorphism of G-linearized sheaves F (Z)b ≃ (π ′)∗HomO(G/P )′ (FG/P )∗L,O(G/P)′ = (π′)∗EndLF ( G/P). Let V denote a closed subvariety of G/P . Then EndLF ( G/P , V ) is the kernel of the natural map EndLF ( G/P) → HomO(G/P )′ (FG/P )∗(IV ⊗ L),OOV ′ In particular, (π′)∗ EndLF ( G/P , V ) maps into the kernel of the induced morphism (7) End F (Z)b → (π ′)∗HomO (G/P )′ (FG/P )∗(IV ⊗ L),OOV ′ Let q : G→ G/P denote the quotient map. Then π̂−1(V ) identifies with the quotient q−1(V )×P X ′. Moreover, as π′ is locally trivial it follows that π̂∗(IV ) = Iq−1(V )×PX′. In particular, the sheaf (π′)∗HomO(G/P )′ (FG/P )∗(IV ⊗ L),OOV ′ is isomorphic to HomOZ′ (Fb)∗(Iq−1(V )×PX′ ⊗ LẐ),O(q−1(V )×PX)′ Thus we see that the kernel of (7) is the subsheaf End F (Z, π −1(V ))b of elements which map (Fb)∗(Iq−1(V )×PX′ ⊗ LẐ) to I(q−1(V )×PX)′ . The global sections of this subsheaf is denote by End F (Z, π −1(V ))b. In conclusion Proposition 5.2. The map π′ induces a G-equivariant morphism (π′)∗ : EndLF ( G/P) → End F (Z)b. Moreover, when V is a closed subvariety of G/P then (π′)∗ maps the subset EndLF ( G/P , V ) into End F (Z, q −1(V )×P X)b. The following is also useful. Lemma 5.3. Let Y denote a closed P -stable subvariety of X and fix notation as above. Then each element of End F (Z)b maps (Fb)∗(IẐY ⊗ LẐ) to I(ZY )′. Proof. It suffices to show that the natural morphism HomOZ′ (Fb)∗LẐ ,OZ′ → HomOZ′ (Fb)∗(IẐY ⊗ LẐ),O(ZY )′ is zero. By linearity, this will follow if the natural morphism I(ZY )′ ⊗ (Fb)∗LẐ → (Fb)∗(IẐY ⊗ LẐ), is an isomorphism, which can be checked by a local calculation. � 5.4. Conclusions. By Proposition 5.1 an element v in the vectorspace IndGP EndMF (X) ⊗ kλ defines an element in End M(−λ)Z F (Z)f . More- over, by Proposition 5.2, an element u ∈ EndLF ( G/P) defines an element (π′)∗(u) in End F (Z)b. Thus by the discussion in Section 5.2 we obtain a G-equivariant map M,λ : End G/P)⊗ IndGP EndMF (X)⊗ kλ → End F (G×P X), defined by M,λ(u⊗ v) = (π ′)∗u ◦ (Fb)∗v. We can now prove Theorem 5.4. Let X denote a P -variety and M denote a P -linearized line bundle on X. Let L denote the equivariant line bundle on G/P associated to the P -character λ. Then the G-equivariant map Φ1 defined above, satisfies (1) When Y is a P -stable closed subvariety of X then the restriction of Φ1 M,λ to the subspace : EndLF ( G/P)⊗ IndGP EndMF (X, Y )⊗ kλ maps to End F (G×P X,G×P Y ). (2) When V denotes a closed subvariety of G/P then the restriction of Φ1 M,λ to the subspace EndLF ( G/P , V )⊗ IndGP EndMF (X)⊗ kλ maps to End F (G ×P X, q −1(V ) ×P X), where q : G → G/P denotes the quotient map. Proof. The first statement follows from Proposition 5.1 and Lemma 5.3. The second statement follows from Proposition 5.2 and Lemma 5.5 below. � Lemma 5.5. Let V denote a closed subset of G/P . Then every element of End M(−λ)Z F (Z)f will map (Ff)∗(MZ ⊗ Iπ−1(V )) to I(π̂)−1(V ) ⊗ LẐ . Proof. It suffices to prove that the natural morphism I(π̂)−1(V ) ⊗ (Ff)∗MZ → (Ff)∗ Iπ−1(V ) ⊗MZ is an isomorphism, which can be checked by a local calculation. � 5.5. Identify IndGP with the space of global sections of MZ (cf. Equation (3)). Then we can define a G-equivariant morphism (8) End F (G×P X)⊗ Ind → EndF (G×P X), by mapping s⊗σ, for σ a global section of MZ and s : (FZ)∗MZ → OZ′ , to the element (FZ)∗OZ (FZ )∗σ −−−−→ (FZ)∗MZ −→ OZ′ , 18 XUHUA HE AND JESPER FUNCH THOMSEN in EndF (G×PX). Combining Φ M,λ with the morphism in (8) we obtain a G-equivariant map ΦM,λ : End G/P)⊗ IndGP EndMF (X)⊗kλ ⊗ IndGP → EndF (Z), where an element u⊗ v⊗ σ in the domain is mapped to the composed (9) (FZ)∗OZ (FZ)∗σ −−−−→ (FZ)∗MZ (Fb)∗v −−−→ (Fb)∗LẐ (π′)∗u −−−→ OZ′. Notice that by Lemma 3.5 the map u ∈ EndLF ( G/P) factors as (10) (FG/P )∗L (FG/P )∗u −−−−−→ (FG/P )∗ω → O(G/P)′ , where u! is some global section of the line bundle Ľ := ω ⊗L−1 as- sociated to u (cf. Section 3.7), and the rightmost map is the evaluation map with domain (FG/P )∗ω = EndF (G/P). It follows that we may extend (9) into a commutative diagram (FZ)∗OZ (FZ)∗σ // (Fb)∗π̂ ∗(u!) (FZ)∗MZ (Fb)∗v // (Fb)∗π̂ ∗(u!) (Fb)∗LẐ (π′)∗u (Fb)∗π̂ ∗(u!) (FZ)∗π ∗Ľ // (FZ)∗(MZ ⊗ π ∗Ľ) //// (Fb)∗(π̂ rrrrrrrrrrr where all the vertical maps are induced by multiplication by π̂∗(u!). Likewise the lower horizontal maps are induced from the upper hori- zontal maps by multiplication with π̂∗(u!). The triangle on the right is induced from (10) by pull-back to Z ′. Theorem 5.6. Let X denote a P -variety and M denote a P -linearized line bundle on X. Let L denote the equivariant line bundle on G/P associated to the P -character λ. Then the G-equivariant map ΦM,λ, defined above, satisfies (1) When Y is a P -stable closed subvariety of X then the restriction of ΦM,λ to the subspace : EndLF ( G/P)⊗ IndGP EndMF (X, Y )⊗ kλ ⊗ IndGP maps to EndF (G×P X,G×P Y ). (2) When V denotes a closed subvariety of G/P then the restriction of ΦM,λ to the subspace : EndLF ( G/P , V )⊗ IndGP EndMF (X)⊗ kλ ⊗ IndGP maps to EndF (G ×P X, q −1(V ) ×P X), where q : G → G/P denotes the quotient map. Moreover, let u ∈ EndLF ( G/P), v ∈ IndGP EndMF (X) ⊗ kλ and σ ∈ IndGP . Then the element ΦM,λ(u⊗ v ⊗ σ) factorizes both as (FZ)∗OZ (FZ)∗σ −−−−→ (FZ)∗MZ −→ OZ′, and as (FZ)∗OZ (FZ)∗(σ⊗π −−−−−−−−→ (FZ)∗(MZ ⊗ π −→ OZ′, where s1 and s2 satisfies i) If v is contained in IndGP EndMF (X, Y ) ⊗ kλ then s1 and s2 are compatible with G×P Y . ii) If u is contained in EndLF ( G/P , V ) then s1 is compatible with q −1(V )×P Proof. Part (1) and (2) follows directly from Theorem 5.4 and the defi- nition of ΦM,λ. The existence of s1 and s2 follows by the diagram (11). Finally the claims about the compatibility of s1 and s2 follows from Theorem 5.4 and Lemma 5.3. � 5.6. We will now describe when an element in the image of ΦM,λ defines a Frobenius splitting of Z. For this we consider the composed map evZ◦ΦM,λ. Recall that an element s ∈ EndF (Z) is a Frobenius splitting of Z if and only if evZ(s) is the constant function 1 on Z Let u ∈ EndLF ( G/P), v ∈ IndGP EndMF (X)⊗kλ and σ ∈ IndGP By Equation (9) the image of u⊗v⊗σ under evZ ◦ΦM,λ coincides with the global section of OZ′ determined by the composed map (12) OZ′ −→ (FZ)∗OZ (FZ )∗σ −−−−→ (FZ)∗MZ (Fb)∗v −−−→ (Fb)∗LẐ (π′)∗u −−−→ OZ′ . We may divide this composition into two parts. The first part −→ (FZ)∗OZ (FZ)∗σ −−−−→ (FZ)∗MZ (Fb)∗v −−−→ (Fb)∗LẐ is defined by σ and v and defines a global section of LẐ . The corre- sponding map M,λ : Ind EndMF (X)⊗ kλ ⊗ IndGP → IndGP k[X ′]⊗ kλ is the map induced by the morphism (13) EndMF (X)⊗M(X) → k[X mapping s : (FX)∗M → OX′ and τ a global section of M, to s(τ). Notice that we here identify IndGP k[X ′]⊗ kλ with the space of global sections of LẐ (cf. Equation (3)). The second part takes a global section τ̃ of LẐ and an element u in End G/P) to the global section of OZ′ defined by −→ (Fb)∗OẐ (Fb)∗τ̃ −−−→ (Fb)∗LẐ (π′)∗u −−−→ OZ′. 20 XUHUA HE AND JESPER FUNCH THOMSEN The corresponding map is Φλ : End G/P)⊗ IndGP k[X ′]⊗ kλ → k[Z ′], which maps u⊗ τ̃ , to ((π′)∗u)(τ̃) (cf. Proposition 5.2). The restriction of Φλ : (14) φλ : End G/P)⊗ IndGP is the map corresponding to Φλ in case X is the one point space Spec(k) (in which case k[X ′] is just k). In combination this defines us a com- mutative diagram EndLF ( G/P)⊗ IndGP EndMF (X)⊗ kλ ⊗ IndGP Id⊗Φ2 ΦM,λ // EndF (Z) EndLF ( G/P)⊗ IndGP k[X ′]⊗ kλ ) Φλ // k[Z ′] EndLF ( G/P)⊗ IndGP φλ // k evG/P 33ggggggggggggggggggggggggggggggg wheremλ is the natural map which makes the lower part of the diagram commutative. Notice that when k[X ′] = k, e.g. if X ′ is a complete and irreducible variety, then φλ and Φλ coincides. Let χ denote the P - character associated to the canonical G-linearization of ω−1G/P (cf. Sec- tion 4.3). Then as noted earlier (Section 5.5) the G-module EndLF ( coincides with the space of global sections of Ľ = ω ⊗L−1 and thus coincides with (16) EndLF ( G/P) = IndGP (p− 1)χ− λ where we abuse notation and write (p− 1)χ− λ for the 1-dimensional P -representation associated with the character (p− 1)χ−λ. It follows that mλ is the natural multiplication map (17) mλ : Ind (p− 1)χ− λ ⊗ IndGP → IndGP (p− 1)χ which is surjective if the domain is nonzero, i.e. if L and ω ⊗ L−1 are effective line bundles on G/P [R-R, Thm.3]. The commutativity of the diagram (15) then implies: Proposition 5.7. Let Ξ denote an element in the domain of ΦM,λ, and assume that the image (Id⊗Φ2 M,λ)(Ξ) is contained in the subspace EndLF ( G/P)⊗IndGP (cf. diagram (15)). Then ΦM,λ(Ξ) is a Frobenius splitting of Z if and only if φλ((Id ⊗ Φ M,λ)(Ξ)) equals the constant 1. In particular, if EndLF ( G/P) ⊗ IndGP is nonzero and IndGP contained in the image of Φ2 M,λ, then Z admits a Frobenius splitting. Proof. The first part of the proof is just a restatement of the fact that the diagram (15) is commutative. The second part follows by the sur- jectivity of mλ and the fact that G/P admits a Frobenius splitting. Corollary 5.8. Assume that X is irreducible and complete. If both IndGP and IndGP (p − 1)χ − λ are nonzero and Φ2 M,λ is surjective, then Z admits a Frobenius splitting. 5.7. In many concrete situation the existence of a P -invariant ele- ment in EndMF (X) ⊗ kλ is given. Notice that this is equivalent to a G-invariant element v in IndGP EndMF (X) ⊗ kλ and thus ΦM,λ defines a G-equivariant map (18) EndLF ( G/P)⊗ IndGP → EndF (Z), u⊗ σ 7→ ΦM,λ(u⊗ v ⊗ σ). Similarly Φ2 M,λ defines a G-equivariant morphism (19) IndGP → IndGP k[X ′]⊗ kλ which makes the diagram (20) EndLF ( G/P)⊗ IndGP // EndF (Z) EndLF ( G/P)⊗ IndGP k[X ′]⊗ kλ ) Φλ // k[Z ′] commutative. We also note Corollary 5.9. Assume that X is irreducible and complete and let v denote a P -invariant element of EndMF (X)⊗ kλ. If the induced map M,λ)|v⊗IndGP (M(X)) : IndGP → IndGP is surjective then Z admits a Frobenius splitting. In particular, if IndGP is an irreducible G-representation then for Z to be Frobenius split it suffices that the latter map is nonzero. Proof. Apply Corollary 5.8. � 6. B-Canonical Frobenius splittings In this section we continue the study of the Frobenius splitting prop- erties of Z = G ×P X . The notation is kept as in Section 5 but we restrict ourselves to the case where G is a connected, semisimple and simply connected linear algebraic group. Moreover, we fix P = B, M = OX and λ = −(p − 1)ρ. Recall that, in this setup, the dualizing sheaf ωG/B is the G-linearized sheaf associated to the B-character 2ρ. 22 XUHUA HE AND JESPER FUNCH THOMSEN Thus, with the notation in Section 5.6, we have χ = −2ρ. Recall also the G-equivariant identity (see (16)) (21) EndLF ( G/B) ≃ IndGB((p− 1)χ− λ) = Ind B(λ) = Ind B((1− p)ρ). The latter G-module is called the Steinberg module of G and will be denoted by St. The Steinberg module is a simple and selfdual G- module. A B-canonical Frobenius splitting ofX is then a B-equivariant (22) θ : St⊗ k(p−1)ρ → EndF (X), containing a Frobenius splitting in its image. Notice that a B-canonical Frobenius splitting of X is not a Frobenius splitting as defined in Sec- tion 3.2. However, there exists a unique nonzero lowest weight vector v− of St such that θ(v−) is a Frobenius splitting in the sense of Sec- tion 3.2. Moreover, as St is a simple G-module the map θ is uniquely determined by θ(v−), and we may thus identify θ with θ(v−). In this way θ(v−) will also be called a B-canonical Frobenius splitting of X . The importance of B-canonical Frobenius splittings was first ob- served by O. Mathieu in connection with good filtrations of G-modules. We refer to [B-K, Chapter 4] for a general reference on B-canonical Frobenius splittings. 6.1. Consider a B-canonical Frobenius splitting as in (22). By Frobe- nius reciprocity this defines a map St → IndGB EndF (X)⊗ kλ and as IndGB k[X ] contains k we may consider the inducedG-equivariant morphism θ̃ : St → IndGB EndF (X)⊗ kλ ⊗ IndGB k[X ] Composing θ̃ with the map Φ2 M,λ of Section 5.6 we end up with a map M,λ ◦ θ̃ : St → Ind k[X ′]⊗ kλ We claim Lemma 6.1. The composed map Φ2 M,λ ◦ θ̃ is an isomorphism on its image IndGB Proof. We first prove that the image of Φ2 M,λ◦θ̃ is contained in Ind For this let EndF (X)c denote the inverse image of k ⊂ k[X ′] under the evaluation map evX . It suffices to prove that the image of θ is contained in EndF (X)c. Notice that EndF (X)c is a B-submodule of EndF (X) containing the set of Frobenius splittings of X . In particular, the image of the lowest weight space of St under θ is contained in EndF (X)c. Moreover, as St is an irreducible G-module it is generated by the lowest weight space as a B-module. Thus, the image of θ will be contained in the B-module EndF (X)c. Now Φ2 M,λ ◦ θ̃ is a map from St to Ind B(λ) = St. Thus, by Frobenius reciprocity, it suffices to prove that Φ2 M,λ ◦ θ̃ is nonzero which is the case as θ contains a Frobenius splitting in its image. � Using Lemma 6.1 we can now combine the diagram (15) with the map Φ2 M,λ ◦ θ̃ and obtain a commutative and G-equivariant diagram (23) St⊗ St ≃ Id⊗(Φ2 Θ // EndF (G×B X) evZ // k[Z ′] EndLF ( G/B)⊗ IndGB φλ // k evG/B 88qqqqqqqqqqqqqq where Θ is the map induced by θ̃ and ΦM,λ. By Proposition 5.7 it follows that Θ(Ξ), for Ξ in St⊗ St, is a Frobenius splitting of Z if and only if the image of Ξ under φλ and Id ⊗ (Φ M,λ ◦ θ̃) equals 1. The latter map from St⊗St to k will be denoted by φ. By construction φ is G-equivariant. Moreover, mλ is surjective and evG/B is nonzero (as G/B admits a Frobenius splitting) and thus φ is nonzero. As St is a simple G-module it follows that (24) φ : St⊗ St → k, defines a nondegenerate G-invariant bilinear form on St. By Frobenius reciprocity such a form is uniquely determined up to a nonzero con- stant. In particular, this provides a very useful way to construct lots of Frobenius splittings of Z. Corollary 6.2. Let θ : St⊗ k(p−1)ρ → EndF (X) denote a B-canonical Frobenius splitting of X. Then the induced morphism (defined above) Θ : St⊗ St → EndF (G×B X), satisfies the following (1) The image Θ(ν) of an element ν in St⊗ St defines a Frobenius splitting of G×BX up to a nonzero constant if and only if φ(ν) is nonzero. (2) If the image of θ is contained in EndF (X, Y ) for a B-stable closed subvariety Y of X, then the image of Θ is contained in EndF (G×B X,G×B Y ). (3) Let v denote an element of St = EndLF ( G/B) which is compatible with a closed subvariety V of G/B. For any element v′ ∈ St we Θ(v ⊗ v′) ∈ EndF (G×B X, q −1(V )×B X), with q : G→ G/B denoting the quotient map. 24 XUHUA HE AND JESPER FUNCH THOMSEN (4) Any element of the form Θ(v ⊗ v′) factorizes as (FZ)∗OZ (FZ )∗π −−−−−→ (FZ)∗π −→ OZ′ , where Z = G×B X and L is the line bundle on G/B associated to the B-character (1 − p)ρ. Moreover, if the image of θ is contained in EndF (X, Y ) then s is compatible with G×B Y . Proof. All statements follows directly from Theorem 5.6 and the con- siderations above. � The first part (1) and (2) of the above result is well known (see e.g. [B-K, Ex. 4.1.E(4)]). However, the second part (3) and (4) seems to be new. 6.2. B-canonical Frobenius splitting when G is not semisim- ple. Although Corollary 6.2 is only stated for connected, semisimple and simply connected groups it also applies in other cases : assume that G is a connected linear algebraic group containing a connected semisimple subgroup H such that the induced map H/H∩B → G/B is an isomorphism. E.g. this is satisfied for any parabolic subgroup of a reductive connected linear algebraic group. Let qsc : Hsc → H denote a simply connected cover of H . Then X admits an action of the par- abolic subgroup Bsc := q sc (B ∩ H) of Hsc. Furthermore, the natural morphism Hsc ×Bsc X → G×B X, is then an isomorphism. We then say that X admits a B-canonical Frobenius splitting if X , as a Bsc-variety, admits a Bsc-canonical Frobe- nius splitting. In this case we may apply Corollary 6.2 to obtain Frobe- nius splitting properties of G×B X . 6.3. Restriction to Levi subgroups. Return to the situation where G is connected, semisimple and simply connected. Let J be a subset of the set of simple roots ∆ and let GJ denote the commutator subgroup of LJ . Then GJ is a connected, semisimple and simply connected linear algebraic group with Borel subgroup BJ = GJ ∩B and maximal torus TJ = T ∩ GJ . We let StJ denote the associated Steinberg module. Notice that StJ = Ind ((1− p)ρJ ) where ρJ denotes the restriction of ρ to BJ . The following should be well known but we do not know a good reference. Lemma 6.3. There exists a GJ -equivariant morphism StJ → St, such that the B−J -invariant line of StJ maps surjectively to the B invariant line of St. In particular, if X is a G-variety admitting a B- canonical Frobenius splitting then X admits a BJ -canonical Frobenius splitting as a GJ -variety. Proof. Let M denote the T -stable complement to the B-stable line in St. Then M is B−-invariant and thus also B−J -invariant. The trans- late ẇJ0M is then invariant under BJ and we obtain a BJ -equivariant morphism St → St/(ẇJ0M) ≃ k(1−p)ρJ . By Frobenius reciprocity this defines a GJ -equivariant map St → StJ such that the B-stable line of St maps onto the BJ -stable line of StJ . Now apply the selfduality of StJ and St to obtain the desired map. This proves the first part of the statement. The second part follows easily by composing the obtained morphism StJ → St with the B-canonical Frobenius splitting St → EndF (X)⊗ k(1−p)ρ, of X and noticing that the restriction of ρ to BJ is ρJ . 7. Applications to G×G-varieties In this section we consider a linear algebraic group G satisfying the conditions of Section 6.2, i.e. we assume that G contains a closed connected semisimple subgroup H such that H/H∩B → G/B is an iso- morphism. We also let Hsc denote the simply connected version of H and let Bsc denote the associated Borel subgroup. 7.1. A well known result. Consider for a moment (i.e. in this sub- section) the case where G is semisimple and simply connected. Re- member that the G-linearized line bundle on G/B associated to the B- character 2ρ coincides with the dualizing sheaf ωG/B. Let L denote the line bundle on G/B associated to the B-character (1−p)ρ and recall from Section 6 the notation St = IndGB((1−p)ρ) for the Steinberg module. As the Steinberg module is a selfdual G-module we may fix a G-invariant nonzero element v∆ in the tensorproduct St⊗ St. We may think of v∆ as a global section of the line bundle L⊠ L on (G/B)2 = G/B × G/B. Identify G/B × G/B with G×B G/B by the isomorphism G×B G/B → G/B × G/B, [g, hB] 7→ (gB, ghB), and let D denote the subvariety of G/B × G/B corresponding to G ×B ∂(G/B), where ∂(G/B) denotes the union of the codimension 1 Schubert varieties in G/B. Then, by [B-K, proof of Thm.2.3.8], the zero scheme of v∆ equals (p− 1)D. Consider then the natural morphism : η : (L⊠ L)⊗ (L⊠ L) → ω (G/B)2 = End!F (( G/B)2) and define ηD : (L⊠ L) → End G/B)2), 26 XUHUA HE AND JESPER FUNCH THOMSEN as in Lemma 3.6, using the identification L ⊠ L = O(G/B)2 (p − 1)D Then by Lemma 3.6 the image of ηD is contained in End G/B)2, D) and thus the associated element η′D ∈ HomO((G/B)2)′ (F(G/B)2)∗(L⊠ L),O((G/B)2)′ is compatible with D. It follows Lemma 7.1. The element in EndL⊠LF (( G/B)2) ≃ St⊠ St defined by v∆ is compatible with the diagonal diag(G/B) in G/B × G/B. Proof. We have to prove that η′D, defined above, is compatible with the diagonal diag(G/B). As η′D is compatible with D it suffices to show that EndL⊠LF (( G/B)2, D) is contained in EndL⊠LF (( G/B)2, diag(G/B) . This fol- lows by an application of Lemma 3.1 and an argument as at the end of the proof of [B-K, Thm.2.3.1]. � 7.2. We return to the setup as in the beginning of this section. We want to apply the results of the preceding sections to the case when the group equals G×G. So let X denote a B ×B-variety and assume that X admits a Bsc ×Bsc-canonical Frobenius splitting defined by θ : (St⊠ St)⊗ (k(p−1)ρ ⊠ k(p−1)ρ) → EndF (X), which is compatible with certain B×B-stable subvarieties X1, . . . , Xm, i.e. the image of θ is contained in EndF (X,Xi) for all i. Then Theorem 7.2. The variety (G × G) ×(B×B) X admits a diag(Bsc)- canonical Frobenius splitting which compatibly Frobenius splits the sub- varieties diag(G)×diag(B) X and (G×G)×(B×B) Xi for all i. Proof. It suffices to consider the case where G = Hsc (cf. discussion in Section 6.2). By Corollary 6.2 there exists a G × G-equivariant morphism Θ : (St⊠ St)⊗ (St⊠ St) → EndF ((G×G)×(B×B) X), satisfying certain compatibility conditions. Let v∆ ∈ St ⊠ St be a nonzero diag(G)-invariant element and let v ∈ St ⊠ St be arbitrary. Then by Corollary 6.2 and Lemma 7.1 the element Θ v∆ ⊗ v is com- patible with diag(G) ×diag(B) X and (G × G) ×(B×B) Xi for all i. In particular, if we define the diag(G)-equivariant morphism Θ∆ : St⊗ St → EndF ((G×G)×(B×B) X), by Θ∆(v) = Θ v∆ ⊗ v , then every element in the image of Θ∆ is compatible with diag(G) ×diag(B) X and (G × G) ×(B×B) Xi for all i. Consider k(p−1)ρ as the highest weight line in St. Then the restriction of Θ∆ to St⊗ k(p−1)ρ defines a diag(B)-canonical Frobenius splitting of (G×G)×(B×B) X with the desired properties. � Notice that by the general machinery of canonical Frobenius split- tings (see e.g. [B-K, Prop.4.1.17]) the existence of a Frobenius splitting of diag(G)×diag(B) X follows if X admits a diag(Bsc)-canonical Frobe- nius splitting. In the above setup X only admits a Bsc ×Bsc-canonical Frobenius splitting which is less restrictive. However, in contrast to the situation when X admits a diag(Bsc)-canonical Frobenius splitting, the present Frobenius splitting is not necessarily compatible with subvari- eties of the form BẇB ×B X , with w denoting an element of the Weyl group and BẇB denoting the closure of BẇB in G. 8. G-Schubert varieties in equivariant Embeddings From now on, unless otherwise stated, we assume that G is a con- nected reductive group. 8.1. Equivariant embeddings. Consider G as a G × G-variety by left and right translation. An equivariant embedding X of G is then a normal irreducible G × G-variety containing an open dense subset which is G × G-equivariantly isomorphic to G. In particular, we may identify G with an open subset of X , and the complement X \ G is then called the boundary. As G is an affine variety the boundary is of pure codimension 1 in X [Har, Prop.3.1]. Any equivariant embedding of G is a spherical variety (with respect to the induced B × B-action) and thus X contains finitely may B ×B-orbits. 8.2. Wonderful compactifications. When G = Gad is of adjoint type there exists a distinguished equivariant embedding X of G which is called the wonderful compactification (see e.g. [B-K, 6.1]). The boundary X \ G is a union of irreducible divisors Xj , j ∈ ∆, which intersect transversely. For a subset J ⊂ ∆, we denote the inter- section ∩j∈JXj by XJ . As a (G×G)-variety, XJ is isomorphic to the variety (G × G) ×P− ×P∆\J Y, where Y denotes the wonderful com- pactification of the group of adjoint type associated to L∆\J . Here the ×P∆\J -action on Y is defined by the quotient maps P∆\J → L∆\J and P− → L∆\J . In particular, X∆ is G×G-equivariantly isomorphic to the variety G/B− × G/B. 8.3. Toroidal embeddings. Let Gad denote the group of adjoint type associated to G, and let X denote the wonderful compactifica- tion of Gad. An embedding X of the reductive group G is then called toroidal if the canonical map G→ Gad admits an extension X → X. 8.4. G-Schubert varieties. By a G-Schubert variety in an equivari- ant embedding X we will mean a subvariety of the form diag(G) · V , for some B × B-orbit closure V . Notice that diag(G) · V is the image of diag(G)×diag(B) V under the proper map diag(G)×diag(B) X → X, 28 XUHUA HE AND JESPER FUNCH THOMSEN [g, x] 7→ g · x, and thus G-Schubert varieties are closed diag(G)-stable subvarieties of If G = Gad and X = X is the wonderful compactification then a G- Schubert variety in X∆ is diag(G)-equivariantly isomorphic to a variety of the form G×BX(w), where X(w) denotes a Schubert variety in G/B. In particular, this explains the name G-Schubert varieties as this is the name used for varieties of the form G×B X(w). In the rest of this section, we will relate G-Schubert varieties to closures of so-called G-stable pieces. Our primary interest are G-stable pieces in wonderful compactifications but below we will also describe the toroidal case in general. 8.5. G-stable pieces in the wonderful compactification. LetG = Gad denote a group of adjoint type and let X denote its wonderful com- pactification. Let J ⊂ ∆ and identify XJ with (G× G)×P− ×P∆\J as in Section 8.2. Using this identification it easily follows that there exists a unique element in XJ which is invariant under U J × UJ and diag(LJ). We denote this element by hJ and note that as an element of (G×G)×P− ×P∆\J Y it equals [(e, e), eJ ], where e (resp. eJ) denotes the identity element of G (resp. the adjoint group associated to L∆\J ). For w ∈ W∆\J , we then let XJ,w = diag(G)(Bw, 1) · hJ , and call XJ,w a G-stable piece of X. A G-stable piece is a locally closed subset of X and by [L, section 12] and [He, section 2], we can use them to decompose X as follows w∈W∆\J XJ,w. Moreover, by the proof of [He2, Theorem 4.5], any G-Schubert variety is a finite union of G-stable pieces. In particular, we may think of G-Schubert varieties as closures of G-stable pieces. 8.6. G-stable pieces in arbitrary toroidal embeddings. We fix a toroidal embedding X of G. The irreducible components of the bound- ary X \G will be denoted by X1, . . . , Xn. For each G×G-orbit closure Y in X we then associate the set KY = {i ∈ {1, . . . , n} | Y ⊂ Xi}, where by definition KY = ∅ when Y = X . Then by [B-K, Prop.6.2.3], Y = ∩i∈KYXi. Moreover, we define I = {KY ⊂ {1, . . . , n} | Y a G×G-orbit closure in X }, and write XK := ∩i∈KXi for K ∈ I. Then (XK)K∈I are the set of closures of G×G-orbits in X . Let now πX : X → X denote the given extension of G → Gad. Then the closure of πX(XK) equals XP (K) for some unique subset P (K) of ∆. This defines a map P : I → P(∆), where P(∆) denotes the set of subsets of ∆. As in [H-T2, 5.4], for K ∈ I we may choose a base point hK in the open G×G-orbit of XK which maps to hP (K). By [H-T2, Proposition 5.3], XK is then naturally isomorphic to (G×G)×P− ×P∆\J L∆\J · hK , where J = P (K) and L∆\J · hK is a toroidal embedding of a quotient (L∆\J )/H by some subgroup H of the center of L∆\J . For K ∈ I and w ∈ W∆\p(K), we then define XK,w = diag(G)(Bw, 1) · hK , and call XK,w a G-stable piece of X . One can then show, in the same way as in [He2, 4.3], that w∈W∆\P (K) XK,w. Also similar to the proof of [He2, Theorem 4.5], for any B × B-orbit closure V in X , the G-Schubert variety diag(G) · V is a finite union of G-stable pieces. In particular, G-Schubert varieties are closures of G-stable pieces. 9. Frobenius splitting of G-Schubert varieties In this section, we assume that X is an equivariant embedding of G. Let Gsc denote a simply connected cover of the semisimple commutator subgroup (G,G) of G. We fix a Borel subgroup Bsc of Gsc which is compatible with the Borel subgroup B in G. Similarly we fix a maximal torus Tsc ⊂ Bsc. Let X1, . . . , Xn denote the boundary divisors of X . The closure within X of the B × B-orbit Bsjw0B ⊂ G will be denoted by Dj . Then Dj is of codimension 1 in X . The translate (w0, w0)Dj of Dj will be denoted by D̃j. By earlier work we know Theorem 9.1. [H-T2, Prop.7.1] The equivariant embedding X admits a Bsc × Bsc-canonical Frobenius splitting which compatibly Frobenius splits the closure of every B ×B-orbit. As a direct consequence of Theorem 7.2 we then obtain Corollary 9.2. The variety (G × G) ×(B×B) X admits a diag(Bsc)- canonical Frobenius splitting which is compatible with all subvarieties of the form (G×G)×(B×B) Y and diag(G)×diag(B)Y , for a B×B-orbit closure Y in X. Proposition 9.3. The equivariant embedding X admits a diag(Bsc)- canonical Frobenius splitting which compatibly splits all G-Schubert va- rieties in X. 30 XUHUA HE AND JESPER FUNCH THOMSEN Proof. By Corollary 9.2 the variety Z = diag(G) ×diag(B) X admits a diag(Bsc)-canonical Frobenius splitting which is compatible with all subvarieties of the form diag(G)×diag(B) Y , with Y denoting a B ×B- orbit closure in X . As X is a diag(G)-stable we may identify Z with G/B ×X using the isomorphism G×B X → G/B ×X, [g, x] 7→ (gB, gx). In particular, we see that the morphism π : Z = diag(G)×diag(B) X → X, [g, x] 7→ g · x, is projective and that π∗(OZ) = OX . As a consequence (see Section 3.8) the diag(Bsc)-canonical Frobenius splitting of Z induces a diag(Bsc)- canonical Frobenius splitting of X which is compatible with all subva- rieties of the form π(diag(G)×diag(B) Y ) = diag(G) · Y, i.e. with all the G-Schubert varieties in X . This ends the proof. � As a direct consequence of Proposition 9.3, we conclude the following vanishing result (see [B-K, Theorem 1.2.8]). Corollary 9.4. Let X denote a projective equivariant embedding of G. Let X denote a G-Schubert variety in X and let L denote an ample line bundle on X. Then Hi(X,L) = 0, i > 0. Moreover, if X̃ ⊂ X is another G-Schubert variety, then the restriction H0(X,L) → H0(X̃,L), is surjective. Later (i.e. Cor. 10.5) we will generalize the vanishing part of this result to nef line bundle. 9.1. F-splittings along ample divisors. In this subsection we as- sume that X is toroidal. The following structural properties of toroidal embeddings can all be found in [B-K, Sect.6.2]. Let X0 denote the com- plement in X of the union of the subsets BsiB− for i ∈ ∆. If we let T̄ denote the closure of T in X , then X0 admits a decomposition defined by the following isomorphism (25) U × U− × (T̄ ∩X0) → X0, (x, y, z) 7→ (x, y) · z. Moreover, every G×G-orbit in X intersects (T̄ ∩X0) in a unique orbit under the left action of T . Notice here that as T is commutative the T × T -orbits and the (left) T -orbit in T will coincide. Lemma 9.5. Let X denote a projective toroidal equivariant embedding of G and let Y denote a G × G-orbit closure in X. Let K denote the subset of {1, . . . , n} consisting of those j such that Y is contained in the boundary component Xj. Then Y ∩ ( j /∈K (1, w0)Di), has pure codimension 1 in Y and contains the support of an ample effective Cartier divisor on Y . Proof. Let XK = ∪j /∈KXj . We claim that Y \X K coincides with the open G × G-orbit Y0 of Y . Clearly Y0 is contained in Y \ X K . On the other hand, let U be a G×G-orbit in Y \XK . Then Xj contains U if and only if j /∈ K. But every G × G-orbit closure in X is the intersection of those Xj which contain it [B-K, Prop.6.2.3]. It follows that the closure of Y0 and U coincide and thus U = Y0. As X is normal we may choose a G × G-linearized very ample line bundle L on X . Then H0(Y,L) is a finite dimensional (nonzero) rep- resentation of G ×G, and it thus contains a nonzero element v which is B × B−-invariant up to constants. The support of v is then the union of B × B−-invariant divisors on Y . As Y0 ∩ (T̄ ∩X0) is a single T × T -orbit it follows that Y0 ∩X0 ≃ U × U − × (Y0 ∩ (T̄ ∩X0)), is an affine variety and a single B×B−-orbit. In particular, the support of v is contained in Y \ (Y0 ∩X0) = Y ∩ (X (1, w0)Di). This shows the second part of the statement. The first part follows as Y0 ∩X0 is affine [Har, Prop.3.1]. � Let now X denote a smooth projective toroidal embedding of G. As the line bundles OX(Di) and OX(D̃i) are isomorphic it follows by [B-K, Prop.6.2.6] that (26) ω−1X ≃ OX (Di + D̃i) + Recall that a X is normal and G is semisimple and simply connected, any line bundle on X will admit a unique G2sc = Gsc×Gsc-linearization. In particular, if we let τi denote the canonical section of the line bundle OX(Di), then we may consider τi as a B sc = Bsc×Bsc-eigenvector of the space of global sections of OX(Di). As in the proof of [B-K, Prop.6.1.11] we find that the associated weight of τi equals ωi ⊠ −w0ωi, where ωi denotes the i-th fundamental weight. Similarly, we may consider the canonical section σj of OX(Xj) as a G sc-invariant element. 32 XUHUA HE AND JESPER FUNCH THOMSEN Let V denote a B ×B-orbit closure in X . As V is B ×B-stable the subset Y = (G×G) ·V is closed in X . Thus we may consider Y as the smallest G×G-invariant subvariety of X containing V . Now define K as in Lemma 9.5 and let M denote the line bundle M = OX (p− 1)( D̃i + j /∈K By Equation (26) and Lemma 3.6 it then follows that multiplication with τ i , for i ∈ ∆, and σ j , for j ∈ K, defines a morphism of B2sc-linearized line bundles M → End!F X, {Di, Xj}i∈∆,j∈K ⊗ kλ⊠λ, where λ = (1 − p)ρ. By [H-T2, Prop.6.5] and Lemma 3.1 any element in End!F (X) which is compatible with the closed subvarieties Di, i ∈ ∆, and Xj, j ∈ K, is also compatible with V and Y . In particular, we have defined a B2sc-equivariant map (27) η : M → End!F (X, Y, V ⊗ kλ⊠λ, which, by Lemma 3.5, is the same as a B2sc-invariant element η EndMF X, Y, V ⊗ kλ⊠λ. In particular, this defines us an element (28) v ∈ Ind EndMF X, Y, V ⊗ kλ⊠λ which is G2sc-invariant. We are then ready to use the ideas explained in Section 5.7. First we use (18) to construct a morphism (29) EndL⊠LF (Gsc/Bsc) ⊗M(X) → EndF G2sc ×B2sc X (u, σ) 7→ ΦM,λ⊠λ(u⊗ v ⊗ σ), where L is the Gsc-linearized line bundle on Gsc/Bsc associated to the character λ = (1− p)ρ. Notice that we here have used that M(X) is a G2sc-module. Lemma 9.6. There exists a G2sc-equivariant map (30) St⊠ St → M(X), which maps the B−sc×B sc-invariant line in St⊠St to a nonzero multiple of the global section j /∈K j ∈ M(X), where τ̃i denotes the canonical section of OX(D̃i). Proof. As OX(D̃i) and OX(Di) are isomorphic as line bundles we may consider the element j /∈K as a global section ofM. Then σ is a B2sc-eigenvector inM(X) of weight (p − 1)ρ ⊠ (p − 1)ρ. In particular, σ induces a Bsc × Bsc-equivariant k(p−1)ρ ⊠ k(p−1)ρ → M(X). Applying Frobenius reciprocity and the selfduality of the Steinberg module St, this defines the desired map St⊠ St → M(X), with the stated properties. � Combining the map (29) with the map (30) in Lemma 9.6 we obtain a G2sc-equivariant map (31) Θ : EndL⊠LF (Gsc/Bsc) St⊠ St → EndF G2sc ×B2sc X We will now study when the map (31) describes a Frobenius splitting of G2sc ×B2sc X . Consider the G sc-equivariant map (32) M(X) → St⊠ St, σ 7→ Φ2 M,λ⊠λ(v ⊗ σ), defined as the map (19) in Section 5.7. We claim Lemma 9.7. The composition of the map (30) in Lemma 9.6 and the map in (32) is an isomorphism on St⊠ St. Proof. By Frobenius reciprocity it suffices to show that the described composed map is nonzero. In particular, it suffices to show that M,λ⊠λ(v ⊗ σ̃) 6= 0, where σ̃ denotes the global section of M defined in Lemma 9.6. For this we use the fact that the global section (τiτ̃i) X defines a Frobenius splitting of X (see e.g. [B-K, proof of Thm.6.2.7]). As a consequence η(σ̃) is a Frobenius splitting ofX , where η is the map defined in (27). Equivalently , the natural G2sc-equivariant morphism EndMF (X)⊗M(X) → k[X ′] = k, defined in (13), will map η′ ⊗ σ̃ to 1. This induces a commutative diagram (33) Ind EndMF ⊗ kλ⊠λ ⊗M(X) M,λ⊠λ ++VVV St⊗ St EndMF (X)⊗ kλ⊠λ ⊗M(X) // kλ⊠λ 34 XUHUA HE AND JESPER FUNCH THOMSEN where the image of v⊗ σ̃ under the diagonal map is nonzero. This ends the proof. � Proposition 9.8. Let Θ denote the map defined in (31). The image Θ(ν) of an element ν defines, up to a nonzero constant, a Frobenius splitting of G2sc ×B2sc X if and only if the image of ν under the map (34) φλ⊠λ : End (Gsc/Bsc) St⊠ St defined in Section 5.6, is nonzero. Proof. Apply Proposition 5.7 and Lemma 9.7. With the identification EndL⊠LF (Gsc/Bsc) ≃ St⊠ St the map φλ⊠λ, defined in (34), must necessarily (up to a nonzero constant) be the G2sc-invariant form on St⊠ St mentioned in Section 6.1. Let v∆ denote the diag(G)-invariant element in EndL⊠LF (Gsc/Bsc)2 defined in Section 7.1. Then the diag(G)-equivariant map St⊗ St → k, ν 7→ φλ⊠λ(v∆ ⊗ ν), is nonzero and thus it must coincide (up to a nonzero constant) with the Gsc-invariant form φ on St defined in (24). Proposition 9.9. Fix notation as above and let D denote the effective Cartier divisor (p− 1) (1, w0)Di + j /∈K on X. Then X admits a Frobenius D-splitting which is compatible with the subvariety Y and the G-Schubert variety diag(G) · V . Proof. Consider the diag(G)-equivariant morphism Θ∆ : St⊠ St → EndF G2sc ×B2sc X ν 7→ Θ(v∆ ⊗ ν), where Θ is the map in (31). By Lemma 9.8 the image Θ∆(ν) of an element ν ∈ St⊗ St is a Frobenius splitting, up to a nonzero constant, if and only if φ(ν) is nonzero. Here φ is the the map defined in (24). Let v+ (resp. v−) denote a nonzero B (resp. B −)-eigenvector of St and let ν = v+ ⊗ v−. After possibly multiplying v+ with a constant we may assume that s = Θ∆(ν) defines a Frobenius splitting of Z = G2sc ×B2sc X . As v is compatible with Y and V (cf. (28)) it follows by Theorem 5.6 and Lemma 7.1 that s factorizes as (35) s : (FZ)∗OZ (FZ )∗σ −−−−→ (FZ)∗MZ −→ OZ′, where s1 is compatible with the subvarieties G sc×B2sc V , G sc×B2sc Y and diag(Gsc)×diag(Bsc) X . Here MZ is the G sc-linearized line bundle on Z associated with the B2sc-linearized line bundle M on X as explained in Section 5.2, and σ is the global section of MZ defined as the image of ν under the map (30) in Lemma 9.6. Notice that as M is a G2sc-linearized line bundle on the G2sc-variety X we may identify the global sections of M and MZ . Actually , as X is a G sc-variety the morphism G2sc ×B2sc X → Gsc/Bsc × Gsc/Bsc ×X, [(g1, g2), x] 7→ (g1B, g2B, (g1, g2) · x), is an isomorphism. Moreover, under this isomorphism, the line bundle MZ is just the pull back of M under projection pX on the third coor- dinate. Thus, by Lemma 9.6 it follows that σ is the pull back from X of the effective Cartier divisor D = (p− 1) (1, w0)Di + j /∈K Applying the functor (pX)∗ to (35) we obtain the Frobenius D-splitting (pX)∗s : (FX)∗OX (FX)∗σD −−−−−→ (FX)∗O(D) (pX)∗s1 −−−−→ OX′ ofX where (pX)∗s1 is compatible with the subvarieties pX(G sc×B2scY ) = Y and pX(diag(Gsc)×diag(Bsc) V ) = diag(G) · V (by Lemma 3.7). This ends the proof. � Corollary 9.10. Let X denote a G-Schubert variety in a smooth pro- jective toroidal embedding of a reductive group G. Then X admits a stable Frobenius splitting along an ample divisor. Proof. Apply Proposition 9.9, Lemma 9.5 and Lemma 3.3. � 10. Cohomology of line bundles The main aim of this section is to obtain a generalizing the vanishing part of Corollary 9.4 to nef line bundles. The concept of a rational morphism is here central and for this we use [B-K, Sect.3.3] as a general reference. First we recall : Definition 10.1. A morphism f : Y → Z of varieties is a called a ra- tional morphism if the induced map f ♯ : OZ → f∗OY is an isomorphism and Rif∗OY = 0, i > 0. The following criterion for a morphism to be rational will be very useful ([R, Lem.2.11]). Lemma 10.2. Let f : Y → Z denote a projective morphism of ir- reducible varieties and let Ŷ denote a closed irreducible subvariety of Y . Consider the image Ẑ = f(Ŷ ) as a closed subvariety of Z. Let L denote an ample line bundle on Z and assume (1) f ♯ : OZ → f∗OY is an isomorphism. (2) Hi(Y, f ∗Ln) = Hi(Ŷ , f ∗Ln) = 0, for i > 0 and n≫ 0. 36 XUHUA HE AND JESPER FUNCH THOMSEN (3) The restriction map H0(Y, f ∗Ln) → H0(Ŷ , f ∗Ln) is surjective for n≫ 0. Then the induced map f̂ : Ŷ → Ẑ is a rational morphism. 10.1. Toric variety. An equivariant embedding Z of the (reductive) group T is called a toric variety (wrt. T ). Notice that, as T is commu- tative, we may consider the T ×T -action on Z as just a T -action. The following result should be well known but, as we do not know a good reference, we include a proof. Lemma 10.3. Let f : Y → Z denote a projective surjective morphism of equivariant embeddings of T . Let T · z denote a T -orbit in Z and let T · y denote a T -orbit in f−1(T · z) of minimal dimension. Then the map T · y → T · z, induced by f , is an isomorphism. Proof. Let T · z and T · y denote the closures of T · z and T · y in Z and Y respectively. Then the induced map f̂ : T · y → T · z, is a projective morphism. Moreover, by the minimality assumption on T · y, the inverse image f̂−1(T · z) equals T · y. In particular, the induced morphism : T · y → T · z is projective. But any T -orbit in a toric variety (wrt. to T ) is isomorphic to a torus T1 satisfying that the cokernel of the induced map of character groups X∗(T1) → X ∗(T ) is a free abelian group ([Ful, Sect.3.1]). In particular, the varieties T ·y and T · z are tori and the cokernel of the induced map of character groups X∗(T · z) → X∗(T · y) is a free abelian group. But T · y → T · z is an affine projective morphism and thus it must be a finite morphism. Thus the cokernel of X∗(T · z) → X∗(T · y) is a finite group and, as it is already a free group, it must be trivial. This ends the proof as tori are determined by their character groups. � Lemma 10.4. Let X denote a projective embedding of a reductive group G and let Y denote a G × G-orbit closure of X. Then there exists a smooth toroidal embedding X̂ of G, a projective G-equivariant morphism f : X̂ → X and a G×G-orbit closure Ŷ in X̂ such that the induced morphism f : Ŷ → Y is a rational morphism. Proof. Assume first that X is toroidal. By [B-K, Prop.6.2.5] there ex- ists a smooth toroidal embedding X̂ of G with a projective morphism f : X̂ → X . Let X0 denote the open subset of X introduced in the beginning of Section 9.1, and let X̂0 denote the corresponding sub- set of X̂ . Then the inverse image f−1(X0) coincides with X̂0 [B-K, Prop.6.2.3(i)]. Let T (resp. T̂ ) denote the closure of T in X (resp. X̂). Then T and T̂ are toric varieties [B-K, Prop.6.2.3], and the induced map f : T̂ → T is a projective morphism of toric varieties. Thus also the induced map X̂0 ∩ T̂ → X0 ∩ T , is a projective morphism of toric varieties. As mentioned in Section 9.1 every G×G-orbit in X will intersect X0 ∩ T in a unique T -orbit. We let T · x denote the open T -orbit in the intersection of Y with X0 ∩ T . By Lemma 10.3 we may find a T -orbit T · x̂ in X̂0 ∩ T̂ which by f is isomorphic to T · x, and we then define Ŷ to be the closure of the G×G-orbit through x̂. By the isomorphism (25) we then conclude that f induces a projective birational morphism Ŷ → Y . By [H-T2, Cor.8.4] the orbit closure Y is normal and thus, by Zariski’s main theorem, we conclude f∗OŶ = OY . By Lemma 10.2 (used on the morphism Ŷ → Y and the closed non-proper subvariety Ŷ of Ŷ ) it now suffices to prove Hi(Ŷ , f ∗L) = 0, i > 0, for a very ample line bundle L on Y . This follows from [H-T2, Prop.7.2] and ends the proof in the case when X is toroidal. Consider now an arbitrary projective equivariant embedding X of G. Let X̂ denote the normalization of the closure of the image of the natural G×G-equivariant embedding G→ X ×X, where X denotes the wonderful compactification of Gad. Then X̂ is a toroidal embedding of G with an induced projective equivariant mor- phism f : X̂ → X . Let Ŷ denote any G×G-orbit closure in X̂. Then f : Ŷ → f(Ŷ ) is a rational morphism [H-T2, Lem.8.3]. In particular, we may find a G × G-orbit closure Ŷ of X̂ with an induced rational morphism f : Ŷ → Y . Finally we may apply the first part of the proof to Ŷ and X̂ and use that a composition of rational morphisms is again a rational morphism. � Corollary 10.5. Let X denote a projective embedding of a reductive group G and let X denote a G-Schubert variety in X. Let Y = (G×G)· X denote the minimal G × G-orbit closure of X containing X. When L is a nef line bundle on X then Hi(X,L) = 0, i > 0. Moreover, when L is a nef line bundle on Y then the restriction mor- phism H0(Y,L) → H0(X,L), is surjective. Proof. Assume first that X is smooth and toroidal. Then by Propo- sition 9.9, Lemma 9.5 and Lemma 3.3 the variety Y admits a stable Frobenius splitting along an ample divisor which is compatibly with X. Thus the statement follows in this case by Proposition 3.4. 38 XUHUA HE AND JESPER FUNCH THOMSEN Let now X denote an arbitrary projective equivariant embedding of G. Choose, using Lemma 10.4, a smooth projective toroidal embedding X̂ with a projective equivariant morphism f : X̂ → X onto X , and a G × G-orbit closure Ŷ in X̂ with an induced rational morphism onto Y . Let V denote a B×B-orbit closure in Y such that X = diag(G) ·V . As Y is the minimal G × G-orbit closure containing X it follows that V will intersect the open G×G-orbit of Y . In particular, there exists a B×B-orbit closure V̂ in X̂ which intersects the open G×G-orbit of Ŷ and which maps onto V . In particular, X̂ := diag(G) · V̂ , is a G-Schubert variety in X̂ which by f maps onto X. Moreover, Ŷ is the minimal G×G-orbit closure containing X̂. We claim that the induced morphism X̂ → X is a rational morphism. To prove this we apply Lemma 10.2 to the rational morphism f : Ŷ → Y . Choose an ample line bundle M on Y . Then it suffices to prove (36) Hi(Ŷ , f ∗Mn) = Hi(X̂, f ∗Mn) = 0, i > 0, n > 0, and that the restriction map (37) H0(Ŷ , f ∗Mn) → H0(X̂, f ∗Mn), is surjective for n > 0. But Mn is an ample, and thus nef, line bundle on Y and therefore the pull back f ∗Mn is a nef line bundle on Ŷ ([Laz, Ex. 1.4.4]). As X̂ is smooth and toroidal, the conclusion of the first part of this proof then shows that conditions (36) and (37) are satisfied. Now both X̂ → X and Ŷ → Y are rational morphisms. In particular, we have identifications Hi(Ŷ , f ∗L) ≃ Hi(Y,L), i ≥ 0, Hi(X̂, f ∗L) ≃ Hi(X,L), i ≥ 0, for any line bundle L on Y or, in the second equation, on X . When L is a nef line bundle the pull back f ∗L is also nef ([Laz, Ex. 1.4.4]). Thus as we have already completed the proof of the statement for smooth toroidal embeddings, in particular for X̂ , this now ends the proof. � By the proof of the above result we also find that any G-Schubert variety X in a projective equivariant embedding of G, will admit a G- equivariant rational morphism f : X̂ → X by a G-Schubert variety X̂ of some smooth projective toroidal embedding of G. Remark 10.6. When X = X is the wonderful compactification of a group G of adjoint type and L is a nef line bundle on X, then the restriction morphism H0(X,L) → H0(Y,L), to any closed G×G-stable irreducible subvariety Y of X is surjective. In particular, also the restriction morphism H0(X,L) → H0(X,L), to any G-Schubert variety X is surjective by the above result. We do not know if the latter is true for arbitrary equivariant embeddings. 11. Normality questions The obtained Frobenius splitting properties of G-Schubert varieties in Section 9 and the cohomology vanishing results in Corollary 10.5 should be expected to have strong implications on the geometry of these varieties. However, in this section we provide an example of a G- Schubert variety in the wonderful compactification of a group of type G2 which is not even normal. In fact, it seems that there are plenty of such examples. 11.1. Some general theory. We keep the notations as in Section 8.5. For J ⊂ ∆ and w ∈ W∆\J , we let XJ,w denote the closure of XJ,w in X. Let K = max{K ′ ⊂ ∆ \ J ;wK ′ ⊂ K ′}. By [He2, Prop. 1.12], we have a diag(G)-equivariant isomorphism diag(G)×diag(PK) (PKẇ, PK)hJ ≃ XJ,w induced by the inclusion of (PKẇ, PK)hJ in X. Let V denote the closure of (PKẇ, PK)hJ within X. Then V is the closure of a B × B- orbit and we find that the induced map (38) f : diag(G)×diag(PK) V → XJ,w, is a birational and projective morphism. Thus, by Zariski’s Main The- orem, a necessary condition for XJ,w to be normal is that the fibers of f are connected. Actually, in positive characteristic, connectedness of the fibers is also sufficient forXJ,w to be normal. This follows asXJ,w is Frobenius split (Prop. 9.3) and thus weakly normal [B-K, Prop.1.2.5]. 11.2. An example of a non-normal closure. Let now, further- more, G be a group of type G2. Let α1 denote the short simple root and α2 denote the long simple root. The associated simple reflections are denoted by s1 and s2. Let J = {α2} and w = s1s2 ∈ W ∆\J . In this case K = ∅ and we obtain a birational map f : diag(G)×diag(B) V ≃ XJ,w where V is the closure of (Bẇ,B)hJ . By [Sp, Prop. 2.4], the part of V which intersect the open G×G-orbit of XJ equals (Bẇ′, B)hJ ∪ ws1≤w′ (Bẇ′, Bṡ1)hJ . 40 XUHUA HE AND JESPER FUNCH THOMSEN In particular, x := (v̇, 1)hJ is an element of V , where v = s2s1s2. We claim that the fiber of f over x is not connected. To see this let y denote a point in the fiber over x. Then we may find g ∈ G and x̃ ∈ V such that y = [g, x̃]. By (39), x̃ = (bẇ′, b′)hJ for some b ∈ B, b ′ ∈ P∆\J and w ′ ≥ w. Then (gbẇ′, gb′)hJ = (v̇, 1)hJ . It follows that (v̇−1gbẇ′, gb′) lies in the stabilizer of hJ . In particular, gb′ ∈ P∆\J and thus also g ∈ P∆\J . If g ∈ B then y = [1, x]. So assume that g = u1(t)ṡ1 where u1 is the root homomorphism associated to α1. Assume that t 6= 0. Then we may find b1 ∈ B and s ∈ k such that g = u−1(s)b1 where u−1 is the root homomorphism associated to −α1. x̃ = (g−1, g−1)(v̇, 1)hJ = (b−11 u−1(−s)v̇, g −1)hJ = (b−11 v̇, g −1)hJ ∈ (Bv̇, Bṡ1)hJ where the third equality follows as v̇−1u−1(−s)v̇ is contained in the unipotent radical of P− . But (Bv̇, Bṡ1)hJ has empty intersection with V (by (39)) which contradicts the assumption that t 6= 0. It follows that the only possibilities for y are [1, x] and [ṡ1, (ṡ 1 v̇, ṡ 1 )hJ ]. As (ṡ−11 v̇, ṡ 1 ) is contained in V (by (39)) we conclude that the fiber of f over x consists of 2 points; in particular the fiber is not connected and thus XJ,w is not normal. Remark 11.1. It seems likely that normalizations of G-Schubert vari- eties should have nice singularities : If we let ZJ,w denote the normal- ization of the closure of XJ,w, then the map (38) induces a birational and projective morphism f̃ : diag(G)×diag(PK) V → ZJ,w. We expect that f̃ can be used to obtain global F -regularity of ZJ,w (see [S] for an introduction to global F -regularity). In fact, by the results in [H-T2] the B×B-orbit closure V is globally F -regular. Thus diag(G)×diag(PK) V is locally strongly F -regular, and as f̃∗Odiag(G)×diag(PK )V = OZJ,w , it seems likely that ZJ,w is also locally strongly F -regular. Moreover, similarly to Corollary 9.10 one may conclude that ZJ,w admits a stable Frobenius splitting along an ample divisor. Thus ZJ,w is globally F - regular if it is locally strongly F -regular. At the moment we do not know if ZJ,w is locally strongly F -regular. 12. Generalizations Fix notation as in Section 2. An admissible triple of G × G is by definition a triple C = (J1, J2, θδ) consisting of J1, J2 ⊂ ∆, a bijection δ : J1 → J2 and an isomorphism θδ : LJ1 → LJ2 that maps T to T and the root subgroup Uαi to the root subgroup Uαδ(i) for i ∈ J1. To each admissible triple C = (J1, J2, θδ), we associate the subgroup RC of G×G defined by RC = {(p, q) : p ∈ PJ1, q ∈ PJ2, θδ(πJ1(p)) = πJ2(q)}, where πJ : PJ → LJ , for a subset J ⊂ ∆, denotes the natural quotient Let X denote an equivariant embedding of the reductive group G. A RC-Schubert variety of X is then a subset of the form RC · V for some B × B-orbit closure V in X . When G = Gad is a group of adjoint type and X = X is the associated wonderful compactification the set of RC-Schubert varieties coincides with closures of the set of RC- stable pieces. By definition [L-Y, section 7], a RC-stable piece in the wonderful compactification X of Gad is a subvariety of the form RC ·Y , where Y = (Bv1, Bv2) · hJ for some J ⊂ ∆, v1 ∈ W J and v2 ∈ (notation as in Section 8.5). Notice that when J1 = J2 = ∆ and θδ is the identity map then a RC-stable piece is the same as a G-stable piece. On the other hand, when J1 = J2 = ∅, then a RC-stable piece is the same as a B × B-orbit. Moreover, any RC-Schubert variety is a finite union of RC-stable pieces [L-Y, Section 7]. The following is a generalization of Proposition 9.3 and Proposition Proposition 12.1. Let C = (J1, J2, θδ) denote an admissible triple of G×G and let X denote an equivariant embedding of G. Then X admits a Frobenius splitting which compatible splits all RC-Schubert varieties in X. If, moreover, X is a smooth, projective and toroidal embedding and Y = XK = (G × G) · V , for some B × B-orbit closure V in X, then X admits a Frobenius splitting along the Cartier divisor D = (p− 1) (wJ10 , 1)D̃i + j /∈K which is compatibly with Y and RC · V . Proof. As the proof is similar to the proof of Proposition 9.3 and Propo- sition 9.9 we only sketch the proof. In the following GJ , for a subset J ⊂ ∆, denotes the commutator of the Levi subgroup in Gsc associated to J . The Borel subgroup GJ ∩Bsc of GJ is denoted by BJ . Define XC to be the G2J1-variety which as a variety is X but where the action is twisted by the morphism GJ1 ×GJ1 −−−→ GJ1 ×GJ2. 42 XUHUA HE AND JESPER FUNCH THOMSEN Then the BJ1 × BJ2-canonical Frobenius splitting of X defined by Theorem 9.1 and Lemma 6.3 induces a B2J1-canonical Frobenius split- ting of XC. In particular, all subvarieties of XC which corresponds to B × B-orbit closures in X will be compatibly Frobenius split by this canonical Frobenius splitting. Now apply an argument as in the proof of Proposition 9.3 and use the identification of RC · V ⊂ X with diag(GJ1) · V ⊂ XC. This ends the proof of the first statement. Assume now that X is a smooth, projective and toroidal embedding and consider the B2sc-equivariant morphism η : M → End!F (X, Y, V )⊗ k(1−p)ρ⊠(1−p)ρ, defined in (27). Let YC and VC be defined similar to XC. Then η induces a B2J1-equivariant morphism ηC : M → End F (XC, YC, VC)⊗ k(1−p)ρJ1⊠(1−p)ρJ1 . Similar to the definition of v in (28) we obtain from ηC an element vC ∈ Ind EndF (XC, YC, VC)⊗ k(1−p)ρJ1⊠(1−p)ρJ1 and from this a G2J1-equivariant morphism (40) End (GJ1/BJ1) ⊗M(XC) → EndF G2J1 ×B2J1 similar to (29). Here LJ1 is the line bundle on GJ1/BJ1 associated to the character (1 − p)ρJ1. Combining Lemma 6.3 and Lemma 9.6 we also obtain a map (41) StJ1 ⊠ StJ1 → M(XC), with properties similar to the ones described in Lemma 9.6. As in (32) we may also use vC to construct a morphism M(XC) → StJ1 ⊠ StJ1 , such that the composition with (41) is an isomorphism on StJ1 ⊠ StJ1 . Finally we may construct ΘC : End (GJ1/BJ1) ⊗ (StJ1 ⊠ StJ1) → EndF G2J1 ×B2J1 similar to (31). In particular, a statement equivalent to Proposition 9.8 is satisfied for ΘC. Let v + (resp. v − ) denote a highest (resp. lowest) weight vector in StJ1 and let v ∆ denote the diag(GJ1)-invariant element of End (GJ1/BJ1) . Imitating the proof of Proposition 9.9 we then find that ΘC(v ∆ ⊗ (v + ⊗ v − )) is a Frobenius splitting of G2J1×B2J1 XC (up to a nonzero constant). Moreover, the push forward of this Frobenius splitting to X has the desired properties. We only have to note that the effective Cartier associated to the image of vJ1+ ⊗ v under the map (41) equals D = (p− 1) (wJ10 , 1)D̃i + j /∈K This ends the proof. � We may also argue as in Corollary 10.5 to obtain Corollary 12.2. Let X denote a projective embedding of a reductive group G and let V denote the closure of a B × B-orbit in X. Let Y = (G×G) · V and XC = RC · V . When L is a nef line bundle on XC Hi(XC,L) = 0, i > 0. Moreover, when L is a nef line bundle on Y then the restriction mor- phism H0(Y,L) → H0(XC,L), is surjective. Remark 12.3. In the case where k = C and X is the wonderful com- pactification, the subvarieties (wJ10 , 1)D̃i, Xj and all the RC-Schubert varieties are Poisson subvarieties with respect to the Poisson structure on X corresponding to the splitting Lie(G)⊕ Lie(G) = l1 ⊕ l2, where l1 = Lie(RC) and l2 is a certain subalgebra of Ad(w 0 )Lie(B Lie(B−). See [L-Y2, 4.5]. References [Bri] M. Brion, Multiplicity-free subvarieties of flag varieties, Contemp. Math. 331 (2003), 13–23. [B-K] M. Brion and S. Kumar, Frobenius Splittings Methods in Geometry and Representation Theory, Progress in Mathematics (2004), Birkhäuser, Boston. [B-T] M. Brion and J. F. Thomsen, F -regularity of large Schubert varieties, Amer. J. Math. 128 (2006), 949–962. [E-L] S. Evens and J.-H. Lu, On the variety of Lagrangian subalgebras, I, II, Ann. Sci. cole Norm. Sup. (4) 34 (2001), no. 5, 631–668; 39 (2006), no. 2, 347–379. [Ful] W. Fulton, Introduction to Toric Varieties, Ann. Math. Studies, 131 (1993), Princeton University Press. [Har] R. Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Math. 156 (1970), Springer-Verlag. [Har2] R. Hartshorne, Algebraic Geometry, GTM 52 (1977), Springer-Verlag. [He] X. He, Unipotent variety in the group compactification, Adv. in Math. 203 (2006), 109-131. [He2] X. He, The G-stable pieces of the wonderful compactification, Trans. Amer. Math. Soc. 359 (2007), 3005-3024. 44 XUHUA HE AND JESPER FUNCH THOMSEN [H-T] X. He and J. F. Thomsen, On the closure of Steinberg fibers in the won- derful compactification, Transformation Groups, 11 (2006), no. 3, 427-438. [H-T2] X. He and J.F.Thomsen, Geometry of B×B-orbit closures in equivariant embeddings, math.RT/0510088. [Laz] R. Lazarsfeld, Positivity in Algebraic Geometry I, classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzge- biete. 3. Folge (2004), Springer-Verlag, Berlin. [L] G. Lusztig, Parabolic character sheaves, I, II, Mosc. Math. J. 4 (2004), no. 1, 153–179; no. 4, 869–896. [L-Y] J.-H Lu and M. Yakimov, Partitions of the wonderful group compactifica- tion, math.RT/0606579. [L-Y2] J.-H Lu and M. Yakimov, Group orbits and regular partitions of Poisson manifolds, math.SG/0609732. [M-R] V.B. Mehta and A. Ramanathan, Frobenius splitting and cohomology van- ishing for Schubert varieties, Ann. of Math. 122 (1985), 27–40. [R] A. Ramanathan, Equations defining Schubert varieties and Frobenius split- ting of diagonals, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 61–90. [R-R] S. Ramanan and A, Ramanathan, Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985), 217–224. [S] K. E. Smith, Globally F -regular varieties: Applications to vanishing the- orems for quotients of Fano varieties, Michigan Math. J. 48 (2000), 553– [Sp] T. A. Springer, Intersection cohomology of B×B-orbits closures in group compactifications, J. Alg. 258 (2002), 71–111. [T] J. F. Thomsen, Frobenius splitting of equivariant closures of regular con- jugacy classes Proc. London Math. Soc. 93 (2006), 570–592. Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA E-mail address : [email protected] Institut for matematiske fag, Aarhus Universitet, 8000 Århus C, Denmark E-mail address : [email protected]

0704.0779

High Energy Variability Of Synchrotron-Self Compton Emitting Sources: Why One Zone Models Do Not Work And How We Can Fix It

High Energy Variability Of Synchrotron-Self Compton Emitting Sources: Why One Zone Models Do Not Work And How We Can Fix It Philip B. Graff∗, Markos Georganopoulos∗,†, Eric S. Perlman∗∗ and Demosthenes Kazanas† ∗Department of Physics, University of Maryland, Baltimore County †NASA, Goddard Space Flight Center ∗∗Department of Physics and Space Sciences, Florida Institute of Technology Abstract. With the anticipated launch of GLAST, the existing X-ray telescopes, and the enhanced capabilities of the new generation of TeV telescopes, developing tools for modeling the variability of high energy sources such as blazars is becoming a high priority. We point out the serious, innate problems one zone synchrotron-self Compton models have in simulating high energy variability. We then present the first steps toward a multi zone model where non-local, time delayed Synchrotron- self Compton electron energy losses are taken into account. By introducing only one additional parameter, the length of the system, our code can simulate variability properly at Compton dominated stages, a situation typical of flaring systems. As a first application, we were able to reproduce variability similar to that observed in the case of the puzzling ‘orphan’ TeV flares that are not accompanied by a corresponding X-ray flare. Keywords: radiation mechanisms: non-thermal, blazars, variability, X-rays, Gamma-rays PACS: 95.30.Gv, 98.54.Cm, 98.54.Gr, 98.62.Nx INTRODUCTION In blazars, radio loud active galaxies with their relativistic jets pointing close to the line of sight, the observed radiation is dominated by relativistically beamed emission from the sub-pc base of the jet. Due to the small size of the emitting region and the large distance of such sources, currently it is not possible to spatially resolve the emitting region. Because of this, we can only obtain information about its structure through multiwavelength variability studies. When modeling blazar radiation, past models have mostly incorporated some form of a one zone model. Such models are limited because high-energy electrons cool faster than the source light crossing time and also because time delay effects must be considered. When neither of these are accounted for, the model produces variability on timescales less than the light crossing time, which is incorrect. One must consider the time that it takes for light to be transmitted from different parts of the source, and as Chiaberge and Ghisellini (1999) showed, the shortest observable variability that can be trusted is the light crossing time of the zone. The basic limitation of one zone models stems from the fact that, by construction, all the high energy variability is produced by electrons with cooling times faster than the light crossing time. One, therefore, cannot use one zone models to infer the source structure from high energy variability. MODEL SETUP Kirk et al. (1998) developed an analytical model in which electrons, after being accelerated, propagate in a pipe geometry and cool through synchrotron radiation. This results in a frequency dependent source size, in agreement with the fact that the variability timescale of synchrotron radiation increases with decreasing frequency. Here we present a numerical code that in addition incorporates the important inverse Compton (IC) process, and in particular the most complicated synchrotron-self Compton (SSC) losses and emissivity. We assume that a power law of relativistic electrons is injected at the base of a flow, and that the electrons flow downstream and cool radiatively. Variations in the injected plasma parameters propagate downstream and manifest themselves as frequency dependent variability. Synchrotron and IC losses from photons external to the source are local processes, in the sense that at a given point in the flow, the energy loss rate only depends on the magnetic field and external photon field energy densities, and not FIGURE 1. TeV (solid line) and X-ray (dashed line) variability from our inhomogeneous model. On the left, an event that produces a TeV flare without an accompanying X-ray flare. On the right, an event that produces an ’echo’ flare in the TeV band. on the conditions throughout the source. This is not the case with SSC losses, because synchrotron photons produced throughout the source in past times - to take into account the light travel time from one point of the source to another - contribute to the photon energy density responsible for the SSC losses and emissivity at a given point and time in the source. To incorporate this, we record the synchrotron emissivity throughout the source as a function of time. ORPHAN TEV FLARES In most cases, TeV and X-ray variability are correlated (e.g. Fossati et al. 2000). A variability pattern that cannot be explained by one zone models is the so-called orphan flares, TeV flares that are not accompanied by X-ray flares (e.g. Krawczynski et al. 2004, Blazejowski et. al. 2005). This cannot be simulated by one zone models because the same electrons that produce X-rays through synchrotron raditation also IC-scatter lower energy photons to produce TeV emission. It follows then, that flares in these spectra would be linked. We present (Fig. 1) two types of orphan flares produced by our model using a ‘pipe’ flow geometry: a TeV flare unaccompanied by an X-ray flare and a combined flare followed by an “echo” flare in the TeV range. We produce both flares with the system being in or near a Compton-dominated state, consistent with observations. The first type of flare occurs when there is an additional injection of low-energy electrons; through synchrotron radiation, these provide additional low energy seed photons for IC radiation, resulting in the TeV flare. The dip in X-ray radiation is due to the additional IC cooling of the high-energy electrons. The second type of flare occurs in the case of an increase in high-energy electron injection. The system responds with an initial flare in both X-ray and TeV energies. As the high- energy electrons travel down the jet and cool, they emit extra radiation in the IR. This travels back toward the base of the jet and allows for an increase in TeV emission by providing additional seed photons. ACKNOWLEDGMENTS This project is part of the senior thesis of Philip Graff, under the supervision of Markos Georganopoulos. The authors acknowledge support from a Chandra theory grant and a NASA long-term space astrophysics grant. REFERENCES 1. Chiaberge, M. and Ghisellini, G. 1999, MNRAS 306, 551 2. Kirk, J. G., Rieger, F. M., & Mastichiadis 1998, A&A, 333, 452 3. Fossati, G. et al. 2000, ApJ, 541, 153 4. Blazejowski, et. al. 2005, ApJ, 630, 130 5. Krawczynski, et. al. 2004, ApJ, 601, 151 Introduction Model setup Orphan TeV flares

0704.0780

Fusion of radioactive $^{132}$Sn with $^{64}$Ni

Fusion of radioactive Sn with J. F. Liang, D. Shapira, J. R. Beene, C. J. Gross, R. L. Varner, A. Galindo-Uribarri, J. Gomez del Campo, P. A. Hausladen, P. E. Mueller, D. W. Stracener Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 H. Amro, J. J. Kolata Department of Physics, University of Notre Dame, Notre Dame, IN 46556 J. D. Bierman Physics Department AD-51, Gonzaga University, Spokane, Washington 99258-0051 A. L. Caraley Department of Physics, State University of New York at Oswego, Oswego, NY 13126 K. L. Jones Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854 Y. Larochelle Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37966 W. Loveland, D. Peterson Department of Chemistry, Oregon State University, Corvallis, Oregon 97331 (Dated: October 28, 2018) Evaporation residue and fission cross sections of radioactive 132Sn on 64Ni were measured near the Coulomb barrier. A large sub-barrier fusion enhancement was observed. Coupled-channel calculations including inelastic excitation of the projectile and target, and neutron transfer are in good agreement with the measured fusion excitation function. When the change in nuclear size and shift in barrier height are accounted for, there is no extra fusion enhancement in 132Sn+64Ni with respect to stable Sn+64Ni. A systematic comparison of evaporation residue cross sections for the fusion of even 112−124Sn and 132Sn with 64Ni is presented. PACS numbers: 25.60.-t, 25.60.Pj I. INTRODUCTION Fusion of heavy ions has been a topic of interests for several decades[1]. One motivation is to understand the reaction mechanisms so that the production yield of heavy elements can be better estimated by model calcula- tions. The formation of a compound nucleus is a complex process. The projectile and target have to be captured inside the Coulomb barrier and subsequently evolve into a compact shape. In heavy systems, the dinuclear system can separate during shape equilibration prior to passing the saddle point. This quasifission process is considered the primary cause of fusion hindrance[2, 3, 4]. At energies near and below the Coulomb barrier, the structure of the participants plays an important role in influencing the fusion cross section[5, 6, 7]. Sub-barrier fusion enhancement due to nuclear deformation and in- elastic excitation has been observed[8, 9, 10, 11, 12]. Coupled-channel calculations have successfully repro- duced experimental data by including nuclear deforma- tion and inelastic excitation. Nucleon transfer is another important channel to be considered[13, 14]. Recently available radioactive ion beams offer the op- portunity to study fusion under the influence of strong nucleon transfer reactions. Several theoretical works have predicted large enhancement of sub-barrier fusion involv- ing neutron-rich radioactive nuclei[15, 16, 17, 18, 19]. In addition, the compound nucleus produced in such re- actions is predicted to have a higher survival probabil- ity and longer lifetimes. This is encouraging for super- heavy element research. If high-intensity, neutron-rich radioactive beams become available in the future, new neutron-rich heavy nuclei may be synthesized with en- hanced yields. The longer lifetime of new isotopes of heavy elements would enable the study of their atomic and chemical properties[20]. However, the current inten- sity of the radioactive beams is several orders of mag- nitude lower than that of stable beams. It is thus not practical to use such beams for heavy element synthesis experiments, but they do provide excellent opportuni- ties for studying reaction mechanisms of fusion involving neutron-rich radioactive nuclei. Fusion enhancement, with respect to a one- dimensional barrier penetration model prediction, has been observed in experiments performed with neutron- rich radioactive ion beams at sub-barrier energies[21, 22, 23, 24, 25]. For instance, the effect of large neutron excess on fusion enhancement can be seen in 29,31Al+197Au[23]. http://arxiv.org/abs/0704.0780v2 However, when comparing reactions involving stable iso- topes of the projectile or target, the fusion excitation functions are very similar if the change in nuclear sizes is accounted for. This paper reports results of fusion excitation func- tions measured with radioactive 132Sn on 64Ni. The dou- bly magic (Z=50, N=82) 132Sn has eight neutrons more than the heaviest stable 124Sn. Its N/Z ratio (1.64) is larger than that of stable doubly magic nuclei 48Ca (1.4) and 208Pb (1.54) which are commonly used for heavy element production[26]. Evaporation residue (ER) and fission cross sections were measured. The sum of ER and fission cross sections are taken as the fusion cross section. The experimental apparatus is described in Sect. II and data reduction procedures in Sect. III. The results and comparison with model calculations are presented in Sect. IV. In Sect. V a comparison of ER and fusion cross sections with those resulting from stable Sn isotopes on 64Ni is discussed. A summary is given in Sect. VI. II. EXPERIMENTAL METHODS The experiment was carried out at the Holifield Ra- dioactive Ion Beam Facility. A 42 MeV proton beam produced by the Oak Ridge Isochronous Cyclotron was used to bombard a uranium carbide target. The fission fragments were ionized by an electron beam plasma ion source. The largest yield of mass A=132 fragments was 132Te. Therefore, it was necessary to suppress 132Te. This was accomplished by introducing sulfur into the ion source then selecting the mass 164 XS+ molecular ions from the extracted beam. The 132Te to 132Sn ratio in the ion beam was found to be suppressed by a large factor (∼ 7 × 104) compared to that observed with the mass 132 atomic beam. The mass 164 SnS+ beam was con- verted into a Sn− beam by passing it through a Cs va- por cell where the molecular ion underwent breakup and charge exchange[27]. The negatively charged Sn was sub- sequently injected into the 25 MV electrostatic tandem accelerator to accelerate the beam to high energies. The measurement was performed at energies between 453 and 620 MeV. The average beam intensity was 50,000 parti- cles per second (pps) with a maximum of 72,000 pps. The ER cross sections measured between 453 and 560 MeV have been reported previously[24]. The purity of the Sn beam was measured by an ioniza- tion chamber mounted at zero degrees. Figure 1 displays the energy loss spectra of a 560 MeV A=132 beam with and without the sulfur purification. The dashed curves are the results of fitting the spectrum with Gaussian dis- tributions to estimate the composition of the beam. In the upper panel, the beam is primarily 132Te without sulfur in the ion source. When sulfur was introduced in the ion source, the beam was 96% 132Sn, as shown in the lower panel. The small amount of Sb and Te had a neg- ligible impact on the measurement because their atomic number is higher. Fusion of the target with these iso- baric contaminants at sub-barrier energies should have been suppressed due to the higher Coulomb barriers. FIG. 1: (Color online) Composition of a 560 MeV mass A=132 beam measured by the ionization chamber. Top panel: The mass A=132 beam without purification where Te and Sb are the major components of the beam. Bottom panel: Sulfur was introduced into the target ion source and SnS was selected by the mass separator. The dashed curves are results of fitting the spectrum with three Gaussian distributions. The isobar contaminants 132Sb and 132Te were suppressed considerably. The apparatus for the fusion measurement is shown in Fig. 2. A thick 64Ni target (1.0 mg/cm2) was used to compensate for the low beam intensity. Since the com- pound nucleus decays by particle evaporation and fission, the evaporation residue (ER) and fission cross sections were measured. The ERs were detected by the ioniza- tion chamber at zero degrees and the fission fragments were detected by an annular double-sided silicon strip detector. time−of−flight Si det beam defining Timing Timing Timing target ionization chamber FIG. 2: (Color online) Apparatus for measuring fission and evaporation residues cross sections induced by low intensity beams in inverse kinematics. The ERs were identified by the time-of-flight measured with the microchannel plate timing detector located in front of the ionization chamber and by energy loss in the ionization chamber. The two microchannel plate timing detectors located before the target were used to monitor the beam intensity and to provide the timing reference for the time-of-flight measurement. The microchannel plate timing detector in front of the ionization chamber was position sensitive and was used to monitor the beam position. It was located 200 mm from the target and had a 25 mm diameter Mylar foil. The ionization chamber was filled with CF4 gas so that it could function at rates up to 50,000 pps. Higher beam intensities occurred in some of the fission measurements, requiring the ioniza- tion chamber to be turned off. The data acquisition was triggered by either the beam signal rate down scaled by a factor of 1000, the coincidence of the delayed beam signal and ER signal, or the silicon detector signal. A 350 MeV Au beam that resembled ERs was measured by the ion- ization chamber to calibrate the energy loss spectrum. The ER cross section was obtained by taking the ratio of the ER yield to the target thickness and the integrated beam particles in the ionization chamber. A detailed de- scription of the ER measurement technique used in this experiment can be found in Ref. [28]. The annular double-sided silicon strip detector (Mi- cron Semiconductor Design S2) was located 42 mm from the target. It had 48 concentric strips on one side and 16 pie-shaped sectors on the other side. The inner diame- ter was 35 mm and the outer diameter was 70 mm. The thickness of the detector was 300 µm. The detection an- gles spanned 15.6◦ to 39.6◦. The fission fragments were identified by requiring a coincidence of events in the Si detector and by the folding angle distributions of the de- tected particles. III. DATA REDUCTION PROCEDURES A. Evaporation residues Since this was an inverse kinematics reaction, the ERs recoiled in the forward direction in a narrow cone. The apparatus was designed to have high efficiency for detect- ing ERs. The efficiency of the apparatus was estimated by Monte Carlo simulations. The angular distribution of the ERs was generated by statistical model calculations using the code PACE2[29]. The input parameters for the statistical model calculations will be discussed later in this paper. The calculated efficiency for the lowest bom- barding energy is 93±1%. It increases as the reaction energy increases and reaches 98±1% at the highest en- ergy. A relatively thick target was used in this experiment. The beam lost approximately 40 MeV after passing through the target (13 MeV in the center of mass). For this reason, the measured cross section is an average of the contributions from the beam interacting throughout the thickness of the target. The variation of ER cross sections is not very large at energies above the Coulomb barrier because the shape of the excitation function is al- most flat. Therefore, the measured cross section is close to that would be measured at an energy corresponding to the middle of the target. However, at energies below the barrier the ER cross section falls off exponentially. The cross section near the entrance of the target has more weight than that near the exit. Smooth curves fitting the excitation function in this rapidly varying region were used to determine the reaction energy associated with the measured cross section. An iterative method was used to determine the effec- tive reaction energy for the thick target measurement . First, the measured cross sections and the beam energies calculated at the middle of the target were fitted by a tensioned spline[30] where the smoothness of the curve could be adjusted. The resulting curve was then used to calculate the thick target cross section for each measure- ment, according to dE/dx where σ(E) is the curve generated by the spline fit, dE/dx is the stopping power of 132Sn in 64Ni, and ρ is the target thickness. The integration limits were the energies of the beam at the exit of the target and at the entrance of the target. The energy, Ei, corresponding to the cross section, σi, was obtained by interpolation using the fit- ted curve. This set of energies was used as the input for the next iteration of the fit. The result converged very quickly. After five iterations, the energies differed from the previous iteration energies by less than 0.2 MeV. The validity of this method was checked by generating data from a known function such as the Wong formula [31] and folding in the effects of target thickness. Comparing to the cross-section-weighted-average method described in Ref. [28], the differences in energies determined by these two methods are not noticeable at high energies because the excitation function is fairly flat. However, at energies below the barrier, the energy determined by the cross-section-weighted-average method is larger than that determined by the method described above and disagrees with the measurement in Ref. [32], as can be seen in Fig. 3. Furthermore, it is found that using data generated from a known function the effective energy obtained by the cross- section-weighted-average method is shifted to too high an energy in the exponential falloff region. The uncertainty of the energy determination was esti- mated by comparison with the method using the cross section weighted average. The average uncertainty of the effective reaction energy is 2.3 MeV in the region where the excitation function is almost flat and increases to 3.9 MeV in the exponential fall off region. The uncer- tainty is larger, 5.8 MeV, for the lowest energy data point because an extrapolation is required for calculating the thick target cross section and the extrapolation region is influenced by the location of the next higher energy point. To verify our measurement technique, the ER cross sections for 124Sn+64Ni in inverse kinematics were mea- sured and compared to those published by Freeman et al. measured with a thin target[32]. It is noted that some of our measurements were performed at energies differ- ent from those of Ref. [32]. The comparison is shown in Fig. 3. Our data (open triangles) are in good agreement with those measured by Freeman et al. [32] (filled stars). FIG. 3: (Color online) Comparison of ER cross sections for 64Ni+124Sn measured in this work and by Freeman et al.[32] (filled stars). The filled circles are for energies determined by the method described in Ref. [28] and the open triangles are for energies determined by the method described in this paper. The solid circles are for energy determined by the cross- section-weighted-average method described in Ref. [28]. B. Fission Fission fragments were identified by requiring a coin- cidence of two particles detected by the pie-shaped sec- tors of the Si strip detector on either side of the beam. Figure 4(a) and (b) present two-dimensional histograms of particle energy and strip number of the Si detector for coincident events taken from 560 MeV and 620 MeV 132Sn+64Ni, respectively. They were compared to the kinematics calculation displayed in Fig. 4(c) and (d) where the fission fragments, elastically scattered Sn and Ni are shown by the solid, dash-dotted and dotted curves, respectively. The angular range of the Si strip detector is between the two vertical dashed lines. The elastically scattered Ni and Sn appear in the upper right hand cor- ner and center of the histogram, respectively. The fission events are located in the gated area. The folding angle distributions of the fragments were used to distinguish fission from other reactions, such as deep inelastic reactions. Since there are two solutions for the kinematics of the inverse reaction, as shown in Fig. 4(c) and (d), the fragment angular correlation is not as simple as that in normal kinematics. Monte Carlo simulations were performed to provide guidance. It was assumed that only fusion-fission results from a full mo- mentum transfer. The width of the mass distribution was taken from the 58Ni+124Sn measurement[33]. The width of the mass distribution was varied to estimate the uncer- (d)(c) Strip No. 0 10 20 30 40 50 60 Strip No. 0 10 20 30 40 50 60 FIG. 4: (Color online) (a) and (b) Two dimensional his- tograms of energy and strip number for coincident events from 560 and 620 MeV 132Sn+64Ni, respectively, measured by the annular double-sided silicon strip detector. The gated area shows events from fission and other reactions. (c) and (d) Kinematics of energy as a function of scattering angle for 560 and 620 MeV 132Sn+64Ni, respectively, elastic scattering and fission fragments.The dash-dotted and dotted curves are for the elastically scattered Sn and Ni, respectively whereas the solid curve is for the fission fragments. The angular range of the Si strip detector is between the two vertical dashed lines. tainty of the simulation. The transition state model[34] was used to predict the fission fragment angular distribu- tion. In Fig. 5 the simulated fission fragment folding an- gle distributions for 550 MeV 124Te+64Ni are compared with a stable beam test measurement. The folding angle distributions for one of the fragments detected in strip 2 (16.2◦), strip 22 (27.7◦), and strip 41 (36.8◦) are shown. The gap in the spectra at strip 14, 30, 44, 46, and 47 are malfunctioning strips in the detector. The Monte Carlo simulated folding angle distributions for fission are shown in the middle panels of Fig. 5 and compared to those of measurements shown in the left panels. For one of the fragments detected at forward angles, strip 2 for example, the predicted angular dis- tribution of the other fragment is similar to that of the measurement. Most of these events are considered as re- sulting from fission. For one of the fragments detected near the middle part of the detector, strip 22 for instance, there are differences between measurement and simula- tion in the shapes of the angular distributions of the other fragment. It is predicted that the other fission fragment is distributed around strip 40. The measured distribution spreads to more forward angles. For one of the fragments detected at the backward angles, the yield of the other fragment is predicted to be small and they are equally FIG. 5: (Color online) Left panels: Folding angle distributions for 550 MeV 124Te+64Ni for one of the fragments detected at 16.2◦ (strip 2), 27.7◦ (strip 22), and 36.8◦ (strip 41) by the annular double-sided silicon strip detector. The elastic scat- tering events are excluded. The dotted and dashed histograms are the results of fitting the data with simulated fission and deep inelastic collisions with Q=–20 MeV, respectively (see text). Middle panels: Results of Monte Carlo simulations for fission events. Right panels: Results of Monte Carlo simula- tions for deep inelastic scattering events. The solid curves are for reaction Q value of –10 MeV, the dashed curves are for Q=–20 MeV, and the dotted curves are for Q=–40 MeV. distributed between the middle part of the detector and the outer edge of the detector. But the measured events appear in the middle part of the detector. There are no events in the region where fission events are expected. These differences are attributed to the contribution from other reaction mechanisms, most likely deep inelastic col- lisions. An attempt was made to simulate these deep inelastic collision events. It was assumed that the mass of these products were projectile- and target-like and the angular distribution at forward angles followed a 1/sin(θ) depen- dence. The right panels of Fig. 5 show the results of sim- ulations performed for reaction Q values of –10 (solid), –20 (dashed), and –40 MeV (dotted). It can be seen that the overlap of fission and deep inelastic collisions becomes larger at more backward angles. At strip 41 (36.8◦), deep inelastic collisions account for all the events. The relative contribution of fission and deep inelas- tic collisions were obtained by fitting the simulated fold- ing angle distributions to the measured distributions for all the detector strips using the CERN library program MINUIT[35]. In the fits, the normalization coefficients for the simulated distributions were the only two variable parameters. The results of the fits are shown in the left panels of Fig. 5 by the dotted and dashed histograms for fission and deep inelastic collisions with Q=–20 MeV, re- spectively. The number of fission events in the measured distributions were taken as the summed events in each strip multiplied by the relative contribution of fission. The folding angle distributions for 132Sn+64Ni are shown in Fig. 6. Due to the low statistics, it was not prac- tical to extract the fission events by fitting the folding angle distributions. As an alternative, the fission events were extracted by setting gates on the folding angle dis- tributions using the simulated distributions as references. This gating method was also tested with the 124Te+64Ni measurement. The fission cross sections obtained by the fitting method and the gating method agreed within 10%. FIG. 6: (Color online) Left panels: Folding angle distributions for 560 MeV 132Sn+64Ni for one of the fragments detected at 16.2◦ (strip 2), 27.7◦ (strip 22), and 36.8◦ (strip 41) by the annular double-sided silicon strip detector. The elastic scat- tering events are excluded. Middle panels: Results of Monte Carlo simulations for fission events. Right panels: Results of Monte Carlo simulations for deep inelastic scattering events. The solid curves are for reaction Q value of –10 MeV, and the dotted curves are for Q=–40 MeV. The Monte Carlo simulation was also employed to cal- culate the coincidence efficiency of the detector. The effi- ciency increased from 5.7±0.9% at 530 MeV to 7.6±0.8% at 620 MeV bombarding energy. In the present work, the dynamic range of the ampli- fiers was not sufficiently large resulting in the distortion of the high energy signals. In the future, new amplifiers that are more suitable for measuring the energy of fission fragments will be used so that the mass ratio of reac- tion products can be obtained to help distinguish fission events from other reaction channels. The formation of a compound nucleus depends on whether the interacting nuclei are captured inside the fusion barrier and whether the dinuclear system can sub- sequently evolve into a compact shape. Quasifission oc- curs when the dinuclear system fails to cross the saddle level density parameter (a) A/8 MeV−1 af/an 1.04 diffuseness of spin distribution (∆l) 4 h̄ fission barrier Sierk[45] TABLE I: Input parameters for statistical model calculations. point to reach shape equilibrium. Since the beam inten- sity was several orders of magnitude lower than that of stable beams and the reaction was in inverse kinemat- ics, making separation of fusion-fission and quasifission very difficult, there was no attempt to distinguish quasi- fission from fusion-fission in this work. Furthermore, the experimental results are compared to barrier penetration models which describe the capture process, making it un- necessary to separate these two processes. IV. COMPARISON WITH MODEL CALCULATIONS A. Statistical model The compound nucleus formed in 132Sn+64Ni decays by particle evaporation and fission. Statistical models have successfully described compound nucleus decay for a wide range of fusion reactions. The measured ER and fission cross sections are compared with the predictions of the statistical model code PACE2[29]. The input param- eters were obtained by simultaneously fitting the data from stable Sn on 64Ni[32, 36] and the measured fusion cross sections[36] were used for the calculations. Fig- ure 7(a), (b), and (c) displays the comparison of calcula- tions and data for 112,118,124Sn+64Ni, respectively. The calculations reproduce the measurements well except for the ER cross sections of 112Sn+64Ni. Table I lists the in- put parameters for the calculations. Without adjusting the parameters, calculations for 132Sn+64Ni were per- formed. The results are shown in Fig. 7(d). Very good agreement between the calculation and the data can be seen. It is noted that some of the parameters used in our calculations are different from those used by Lesko et al. [36]. In their calculations, the code CASCADE[37] was used. The mass of the nuclei in the decay chain was calculated using the Myers droplet model[38]. The dif- fuseness of the spin distribution was ∆l = 15h̄ and the ratio of level density at the saddle point to the ground state, af/an, was set to 1.0. In this work, a compilation of measured masses[39], ∆l = 4h̄, and af/an = 1.04 were used. FIG. 7: (Color online) Comparison of measured ER (filled circles) and fission (open circles) cross sections with statistical model calculations. The solid and the dotted curves are the calculated fission and ER cross sections, respectively, using the measured fusion cross sections as input. (a) 64Ni+112Sn, (b) 64Ni+118Sn, (c) 64Ni+124Sn (Freeman et al.[32]), and (d) 132Sn+64Ni (this work) B. Coupled-channel calculation In general, sub-barrier fusion enhancement can be de- scribed by coupled-channel calculations. The fusion cross section of 132Sn+64Ni, the sum of ER and fission cross sections, is compared with coupled-channel calculations using the code CCFULL[40]. The interaction potential (V◦=82.46 MeV, r◦=1.18 fm, and a=0.691 fm) was taken from the systematics of Broglia and Winther[41]. The result of the calculations are compared with the data in Fig. 8. The dotted curve is the prediction of a one- dimensional barrier penetration model and it can be seen that it substantially underpredicts the sub-barrier cross sections. The coupled-channel calculation including in- elastic excitation of 64Ni to the first 2+ and 3− states and 132Sn to the first 2+ state is shown by the dashed curve. The transition matrix elements, B(Eλ), of 64Ni were obtained from Ref. [42, 43] and the B(E2) of 132Sn was obtained from a recent measurement by Varner et al.[44]. This calculation overpredicts the data at energies near the barrier and underpredicts the data well below the barrier. The neutron transfer reactions have positive Q values for transferring two to six neutrons from 132Sn to 64Ni. Since there is no neutron transfer data available for this reaction, the transfer coupling form factor is unknown. Thus, the coupled-channel calculation including transfer and inelastic excitation was performed with one effec- FIG. 8: (Color online) Comparison of 132Sn+64Ni fusion data (filled circles) with a one-dimensional barrier penetra- tion model calculation (dotted curve). The coupled-channel calculation including inelastic excitation of the projectile and target is shown by the dashed curve and the calculation in- cluding inelastic excitation and neutron transfer is shown by the solid curve. tive transfer channel using the Q value for two-neutron transfer. The coupling constant was adjusted to fit the data. The calculation with the coupling constant set to 0.48 is shown by the solid curve. It reproduces the data very well except for the lowest energy data point which has large uncertainties in energy and in cross section. A better treatment of the transfer channels based on exper- imental transfer data would help improve understanding of the influence of transfer on fusion. Experimental neu- tron transfer data on 132Sn+64Ni in the future would be very useful. V. DISCUSSION The ER cross section can be described by σER = πλ (2l+ 1)σl, where λ is the de Broglie wave length, lc the maximum angular momentum for ER formation and σl the partial cross section. The reduced ER cross sections for 64Ni on stable-even Sn isotopes[32] are compared with that for 132Sn+64Ni in Fig. 9. The reduced ER cross sec- tion is defined as the ER cross section divided by the kinematic factor πλ2. It can be seen that the ER cross sections saturate at high energies as fission becomes a significant fraction of the fusion cross section. In addi- tion, the saturation value increases as the neutron excess in Sn increases. This is consistent with the fact that the fission barrier height increases for the more neutron-rich compound nuclei. FIG. 9: (Color online) The reduced ER cross section as a function of the center of mass energy for 64Ni on stable even Sn isotopes[32] and radioactive 132Sn. In Fig. 10, the measured reduced ER cross sections for Ni+Sn as a function of the calculated average mass of the ERs, predicted by PACE2, are presented. In the same re- action, the higher mass ERs are produced at lower beam energies because of the lower excitation energies of the compound nucleus. As the neutron excess in the com- pound nucleus increases, neutron evaporation becomes the dominant decay channel. The PACE2 calculation pre- dicts that a compound nucleus made with Sn isotopes of mass number greater than 120 decays essentially 100% by neutron evaporation and Pt isotopes are the primary ERs. The mass of the compound nucleus is different when it is produced with different Sn isotopes. How- ever, it can be seen that Pt of a particular mass can be produced with different Sn isotopes if different num- bers of neutrons are evaporated. The reaction with a more neutron-rich Sn produces the same Pt isotope at a higher rate. With 132Sn as the projectile, the ERs are so neutron-rich that they cannot be produced by stable Sn induced reactions. This suggests that it may be benefi- cial to use neutron-rich radioactive ion beams to produce new isotopes of heavy elements. The fusion excitation functions of 64Ni on stable even Sn isotopes[36] are compared with that of 132Sn+64Ni in Fig. 11. In order to remove the effects of the difference in nuclear sizes, the cross section is divided by πR2 with R=1.2(A t ) fm, where Ap (At) is the mass num- ber of the projectile (target). The reaction energy in the center of mass is divided by the barrier height predicted by the Bass model[46]. It can be seen that the fusion of 132Sn and 64Ni is not enhanced with respect to the stable- even Sn isotopes when the difference in nuclear sizes is FIG. 10: (Color online) The reduced ER cross section as a function of the calculated average mass of ERs predicted by PACE2[29] for 64Ni on stable even Sn isotopes[36] and radioac- tive 132Sn. considered. FIG. 11: (Color online) Comparison of fusion excitation func- tions for 64Ni on stable even Sn isotopes[36] and radioactive 132Sn. The change in nuclear sizes are corrected by factor- ing out the area and the Bass barrier height[46] in the cross section and energy, respectively. The lowest energy data point has large uncertainties. The cross section seems enhanced comparing to the sta- ble beam measurements in Fig. 9 and Fig. 11. A more pronounced enhancement appears when the data point is compared to our coupled-channel calculations (Fig. 8) and to a time-dependent Hartree-Fock calculation[47]. To further explore if fusion is enhanced at this low energy region, we plan to repeat the measurement with an im- proved apparatus where the thickness of the Mylar foil in the microchannel plate timing detector located in front of the ionization chamber will be reduced. This will al- low a better separation of the energy loss signals from ERs and scattered beams in the ionization chamber at low bombarding energies. The Q values for transferring two to six neutrons from 132Sn to 64Ni are positive. It is necessary to include neutron transfer in coupled-channel calculations to re- produce experimental results. As the neutron excess in the Ni isotopes decreases, the number of neutron transfer channels with positive Q values increases for 132Sn+Ni. In 132Sn+58Ni, the Q values for transferring one to six- teen neutrons from 132Sn to 58Ni are positive and range from 1.7 to 17.4 MeV. A large sub-barrier fusion enhance- ment due to the coupling to neutron transfer is expected to occur in 132Sn+58Ni. An experiment to measure the fusion excitation function of 132Sn on 58Ni is in prepara- tion. Although 132Sn is unstable, its neutron separation en- ergy is 7.3 MeV. This is not very low compared to stable nuclei. The sub-barrier fusion enhancement observed in 132Sn+64Ni with respect to stable Sn nuclei can be ac- counted for by the change in nuclear sizes. No extra en- hancement was found. However, an increased ER yield at energies above the barrier was observed as compared to stable Sn. As the shell closure is crossed, the binding energy for 133Sn decreases by a factor of two. The nu- clear surface of 133Sn and even more neutron-rich Sn may be more diffused. The number of neutron transfer chan- nels with positive Q values increases by a factor of two or more. Larger sub-barrier fusion enhancement beyond the nuclear size effect may be expected. VI. SUMMARY Neutron-rich radioactive 132Sn beams were incident on a 64Ni target to measure fusion cross sections near the Coulomb barrier. With an average intensity of 5×104 pps beams and a high efficiency apparatus for ER detec- tion, the uncertainty of the measured ER cross section is small and comparable to that achieved in stable beam ex- periments. The efficiency for fission fragment detection was low but the detector had a very fine granularity. By requiring a coincident detection of the fission fragments and performing folding angle distribution analysis, fission events were identified. The excitation functions of ER and fission can be described by statistical model calcula- tions using parameters that simultaneously fit the stable even Sn isotopes on 64Ni fusion data. A large sub-barrier fusion enhancement with respect to a one-dimensional barrier penetration model prediction was observed. The enhancement is attributed to the coupling of the projec- tile and target inelastic excitation and neutron transfer. The reduced ER cross sections at energies above the bar- rier are larger for the 132Sn induced reaction than those induced by stable Sn nuclei, as expected from the higher fission barrier of the more neutron-rich compound nu- cleus. For a specific mass of ER, reactions with a more neutron-rich Sn have higher cross sections. When the fusion excitation functions are compared on a reduced scale, where the effects of nuclear size and barrier height are factored out, no extra fusion enhancement is observed in 132Sn+64Ni with respect to stable Sn induced fusion. The fusion cross section measured at the lowest energy seems to be enhanced. Experiments to investigate this with an improved apparatus is planned. VII. ACKNOWLEDGMENT We would like to thank D. J. Hinde for helpful and stimulating discussions. 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0704.0781

Tri-layer superlattices: A route to magnetoelectric multiferroics?

Tri-layer superlattices: A route to magnetoelectric multiferroics? Alison J. Hatt and Nicola A. Spaldin Materials Department, University of California (Dated: October 22, 2018) We explore computationally the formation of tri-layer superlattices as an alternative approach for combining ferroelectricity with magnetism to form magnetoelectric multiferroics. We find that the contribution to the superlattice polarization from tri-layering is small compared to typical polar- izations in conventional ferroelectrics, and the switchable ferroelectric component is negligible. In contrast, we show that epitaxial strain and “negative pressure” can yield large, switchable polar- izations that are compatible with the coexistence of magnetism, even in materials with no active ferroelectric ions. PACS numbers: The simultaneous presence of ferromagnetism and fer- roelectricity in magnetoelectric multiferroics suggests tremendous potential for innovative device applications and exploration of the fundamental physics of coupled phenomena. However, the two properties are chemically contra-indicated, since the transition metal d electrons which are favorable for ferromagnetism disfavor the off- centering of cations required for ferroelectricity [1]. Con- tinued progress in this burgeoning field rests on the iden- tification of alternative mechanisms for ferroelectricity which are compatible with the existence of magnetism [2, 3]. Mechanisms discovered to date include the incor- poration of stereochemically active lone pair cations, for example in BiMnO3 [4, 5] and BiFeO3 [6, 7], geometric ferroelectricity in YMnO3 [8], BaNiF4 [9, 10] and related compounds, charge ordering as in LuFe2O4 [11, 12], and polar magnetic spin-spiral states, of which TbMnO3 is the prototype [13]. However, there are currently no single phase multiferroics with large and robust magnetization and polarization at or near room temperature [14]. The study of ferroelectrics has been invigorated over the last few years by tremendous improvements in the ability to grow high quality ferroelectric thin films with precisely controlled composition, atomic arrangements and interfaces. In particular, the use of compositional ordering that breaks inversion symmetry, such as the layer-by-layer growth of three materials in an A-B-C- A-B-C... arrangement, has produced systems with en- hanced polarizations and large non-linear optical re- sponses [15, 16, 17, 18]. Here we explore computation- ally this tri-layering approach as an alternative route to magnetoelectric multiferroics. Our hypothesis is that the magnetic ions in such a tri-layer superlattice will be con- strained in a polar, ferroelectric state by the symmetry of the system, in spite of their natural tendency to remain centrosymmetric. We note, however, that in previous tri-layering studies, at least one of the constituents has been a strong ferroelectric in its own right, and the other constituents have often contained so-called second-order Jahn-Teller ions such as Ti4+, which have a tendency to off-center. Therefore factors such as electrostatic effects from internal electric fields originating in the strong fer- roelectric layers [19], or epitaxial strain, which is well established to enhance or even induce ferroelectric prop- erties in thin films with second-order Jahn-Teller ions [6, 20, 21], could have been responsible for the enhanced polarization in those studies. We choose a [001] tri-layer superlattice of perovskite- structure LaAlO3, LaFeO3 and LaCrO3 as our model system (see Fig. 1, inset.) Our choice is motivated by three factors. First, all of the ions are filled shell or filled sub-shell, and therefore insulating behavior, a prerequi- site for ferroelectricity, is likely. Second, the Fe3+ and Cr3+ will introduce magnetism. And third, none of the parent compounds are ferroelectric or even contain ions that have a tendency towards ferroelectric distortions, al- lowing us to test the influence of trilayering alone as the driving force for ferroelectricity. For all calculations we use the LDA+U method [22] of density functional the- ory as implemented in the Vienna Ab-initio Simulation Package (VASP) [23]. We use the projector augmented wave (PAW) method [24, 25] with the default VASP po- tentials (La, Al, Fe pv, Cr pv, O), a 6x6x2 Monkhorst- Pack mesh and a plane-wave energy cutoff of 450 eV. Po- larizations are obtained using the standard Berry phase technique [26, 27] as implemented in VASP. We find that U/J values of 6/0.6 eV and 5/0.5 eV on the Fe and Cr ions respectively, are required to obtain insulating band structures; smaller values of U lead to metallic ground states. These values have been shown to give reasonable agreement with experimental band gaps and magnetic moments in related systems [28] but are somewhat lower than values obtained for trivalent Fe and Cr using a con- strained LDA approach [29]. We therefore regard them as a likely lower limit of physically meaningful U/J values. (Correspondingly, since increasing U often decreases the covalency of a system, our calculated polarizations likely provide upper bounds to the experimentally attainable polarizations). We begin by constraining the in-plane a lattice con- stant to the LDA lattice constant of cubic SrTiO3 (3.85 Å) to simulate growth on a substrate, and adjust the out- of-plane c lattice constant until the stress is minimized, with the ions constrained in each layer to the ideal, high- symmetry perovskite positions. We refer to this as our reference structure. (The LDA (LDA+U) lattice con- http://arxiv.org/abs/0704.0781v3 stants for cubic LaAlO3 (LaFeO3, LaCrO3) are 3.75, 3.85 and 3.84 Å, respectively. Thus, LaAlO3 is under tensile strain and LaFeO3/LaCrO3 are unstrained.) The cal- culated total density of states, and the local densities of states on the magnetic ions, are shown in Figure 2; a band gap of 0.32 eV is clearly visible. The polarization of this reference structure differs from that of the corresponding non-polar single-component material (for example pure LaAlO3) at the same lattice parameters by 0.21 µC/cm . Note, however, that this polarization is not switch- able by an electric field since it is a consequence of the tri-layered arrangement of the B-site cations. Next, we remove the constraint on the high symmetry ionic posi- tions, and relax the ions to their lowest energy positions along the c axis by minimizing the Hellmann-Feynman forces, while retaining tetragonal symmetry. We obtain a ground state that is significantly (0.14 eV) lower in en- ergy than the reference structure, but which has a simi- lar value of polarization. Two stable ground states with different and opposite polarizations from the reference structure, the signature of a ferroelectric, are not ob- tained. Thus it appears that tri-layering alone does not lead to a significant switchable polarization in the ab- sence of some additional driving force for ferroelectricity. In all cases, the magnetic ions are high spin with negligi- ble energy differences between ferro- and ferri-magnetic orderings of the Fe and Cr ions; both arrangements lead to substantial magnetizations of 440 and 110 emu/cm3 respectively. Such magnetic tri-layer systems could prove useful in non-linear-optical applications, where a break- ing of the inversion center is required, but a switchable polarization is not. Since epitaxial strain has been shown to have a strong influence on the polarization of some ferroelectrics (such as increasing the remanent polarization and Curie tem- perature of BaTiO3 [20] and inducing room temperature ferroelectricity in otherwise paraelectric SrTiO3 [21]) we next explore the effect of epitaxial strain on the polar- ization of La(Al,Fe,Cr)O3. To simulate the effects of epi- taxial strain we constrain the value of the in-plane lat- tice parameter, adjust the out of plane parameter so as to maintain a constant cell volume, and relax the atomic positions. The volume maintained is that of the calcu- lated fully optimized structure, 167 Å3, which has an in-plane lattice constant of 3.82 Å. As shown in Figure 3, we find that La(Al,Fe,Cr)O3 undergoes a phase transi- tion to a polar state at an in-plane lattice constant of 3.76 Å, which corresponds to a (compressive) strain of -0.016 (calculated from (a‖−a0)/a0 where a‖ is the in-plane lat- tice constant and a0 is the calculated equilibrium lattice constant). A compressive strain of -0.016 is within the range attainable by growing a thin film on a substrate with a suitably reduced lattice constant. We find that significant ferroelectric polarizations can be induced in La(Al,Fe,Cr)O3 at even smaller strain val- ues by using negative pressure conditions. We simulate negative pressure by increasing all three lattice constants and imposing the constraint a=b=c/3; such a growth condition might be realized experimentally by growing the film in small cavities on the surface of a large-lattice- constant substrate, such that epitaxy occurs both hori- zontally and vertically. As in the planar epitaxial strain state, the system becomes strongly polar; this time the phase transition to the polar state occurs at a lattice con- stant of 3.85 Å, at which the strain is a negligible 0.001 relative to the lattice constant of the fully optimized sys- In Fig. 1 we show the calculated energy versus dis- tortion profile and polarization for negative pressure La(Al,Fe,Cr)O3 with in-plane lattice constant = 3.95 Å, well within the ferroelectric region of the phase diagram shown in Fig. 3. The system has a characteristic ferro- electric double well potential which is almost symmetric in spite of the tri-layering; the two ground states have polarizations of 38.9 and -39.9 µC cm−2 respectively, rel- ative to the reference structure at the same lattice con- stant. Since the energies of the two minima are almost identical, the effective electric field Eeff=∆E/∆P, intro- duced in Ref [15], is close to zero and there is no tendency to self pole. The origin of the symmetry is seen in the calculated Born effective charges (3.6, 3.5 and 3.3 for Al, Fe and Cr respectively) which show that the system is largely ionic, with the ions showing very similar trivalent cationic behavior. A similar profile is observed under planar epitaxial strain, although the planar strained sys- tem is around 0.15 eV lower in energy than the negative pressure system for the same in-plane lattice constant. To decouple the effects of interfacial strain and tri- layering we calculate the polarization as a function of strain and negative pressure for the individual compo- nents, LaAlO3, LaFeO3 and LaCrO3. We find that all three single-phase materials become polar at planar epi- taxial strains of -0.03 (LaAlO3), -0.02 (LaFeO3), and -0.01 (LaCrO3). Likewise, all three components be- come polar at negative pressure, under strains of +0.03 (LaAlO3), +0.001 (LaFeO3), and +0.001 (LaCrO3). (The higher strains required in LaAlO3 reflect its smaller equilibrium lattice constant.) These results confirm our earlier conclusion that the large polarizations obtained in strained and negative pressure La(Al,Fe,Cr)O3 are not a result of the tri- layering. We therefore suggest that many perovskite ox- ides should be expected to show ferroelectricity provided that two conditions imposed in our calculations are met: First, the ionic radii of the cation sites in the high sym- metry structure are larger than the ideal radii, so that structural distortions are desirable in order to achieve an optimal bonding configuration. This can be achieved by straining the system epitaxially or in a “negative pres- sure” configuration. And second, non-polar structural distortions, such as Glazer tiltings [30], are de-activated relative to polar, off-centering distortions. These have been prohibited in our calculations by the imposition of tetragonal symmetry; we propose that the symmetry con- straints provided experimentally by hetero-epitaxy in two or three dimensions should also disfavor non-polar tilting and rotational distortions. A recent intriguing theoretical prediction that disorder can be used to disfavor cooper- ative tilting modes is awaiting experimental verification [31]. Finally, we compare the tri-layered La(Al,Fe,Cr)O3 with the polarization of its individual components. Cal- culated separately, the remnant polarizations of LaAlO3, LaFeO3 and LaCrO3, all at negative pressure with a=c=3.95 Å, average to 40.4 µC cm−2. This is only slightly larger than the calculated polarizations of the heterostructure, 38.9 and 39.9 µC cm−2, indicating that tri-laying has a negligible effect on the polarity. This sur- prizing result warrants further investigation into how the layering geometry modifies the overall polarization. In conclusion, we have shown that asymmetric layering alone is not sufficient to produce a significant switchable polarization in a La(Al,Fe,Cr)O3 superlattice, and we suggest that earlier reports of large polarizations in other tri-layer structures may have resulted from the intrinsic polarization of one of the components combined with epi- taxial strain. We find instead that La(Al,Fe,Cr)O3 and its parent compounds can become strongly polar under reasonable values of epitaxial strain and symmetry con- straints, and that tri-layering serves to modify the re- sulting polarization. Finally, we suggest “negative pres- sure” as an alternative route to ferroelectricity and hope that our prediction motivates experimental exploration of such growth techniques. This work was funded by the NSF IGERT program, grant number DGE-9987618, and the NSF Division of Materials Research, grant number DMR-0605852. The authors thank Massimiliano Stengel and Claude Ederer for helpful discussions. [1] N. A. 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Ohsumi, K. Ohwada, K. Ishii, T. Inami, K. Kakurai, Y. Murakami, K. Yoshii, S. Mori, Y. Horibe, et al., Nature 436, 1136 (2005). [12] M. A. Subramanian, H. Tao, C. Jiazhong, N. S. Rogado, T. G. Calvarese, and A. W. Sleight, Adv. mater. 18, 1737 (2006). [13] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y. Tokura, Nature 426, 55 (2003). [14] R. Ramesh and N. A. Spaldin, Nat. Mater. 6, 21 (2007). [15] N. Sai, B. Meyer, and D. Vanderbilt, Phys. Rev. Lett. 84, 5636 (2000). [16] H. N. Lee, H. M. Christen, M. F. Chisholm, C. M. Rouleau, and D. H. Lowndes, Nature 433, 395 (2005). [17] M. P. Warusawithana, E. V. Colla, J. N. Eckstein, and M. B. Weissman, Phys. Rev. Lett. 90, 036802 (2003). [18] Y. Ogawa, H. Yamada, T. Ogasawara, T. Arima, H. Okamoto, M. Kawasaki, and Y. Tokura, Phys. Rev. Lett. 90, 217403 (2003). [19] J. B. Neaton and K. M. Rabe, Appl. Phys. Lett. 82, 1586 (2003). [20] K. J. Choi, M. Biegalski, Y. L. Li, A. Sharan, J. Schubert, R. Uecker, P. Reiche, Y. B. Chen, X. Q. Pan, V. Gopalan, et al., Science 306, 1005 (2004). [21] J. H. Haeni, P. Irvin, W. Chang, R.Uecker, P. Re- iche, Y. L. Li, S. Choudhury, W. Tian, M. E. Hawley, B. Craigo, et al., Nature 430, 758 (2004). [22] V. I. Anisimov, F. Aryasetiawan, and A. I. Liechtenstein, J. Phys.: Condens. Mat. 9, 767 (1997). [23] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). [24] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994). [25] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). [26] R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993). [27] D. Vanderbilt and R. D. King-Smith, Phys. Rev. B 48, 4442 (1993). [28] Z. Yang, Z. Huang, L. Ye, and X. Xie, Phys. Rev. B 60, 15674 (1999). [29] I. Solovyev, N. Hamada, and K. Terakura, Phys. Rev. B 53, 7158 (1996). [30] A. M. Glazer, Acta Crystallogr. B 28, 3384 (1972). [31] D. I. Bilc and D. J. Singh, Phys. Rev. Lett. 96, 147602 (pages 4) (2006). http://link.aps.org/abstract/PRB/v74/e024102 FIG. 1: Energy and polarization as a function of displacement from the centrosymmetric structure for La(Al,Fe,Cr)O3 under negative pressure with a = c/3 = 3.95 Å. Inset: Schematic representation of the centrosymmetric unit cell (center) and displacements of the metal cations corresponding to the en- ergy minima. Displacements are exaggerated for clarity. FIG. 2: Density of states for Fe and Cr ions in La(Al,Fe,Cr)O3 with U/J values of 6/0.6 eV and 5/0.5 eV respectively. The dashed line at 0 eV indicates the position of the Fermi energy. FIG. 3: Calculated polarizations of negative pressure (cir- cles) and epitaxially strained (triangles) La(Al,Fe,Cr)O3 as a function of change in (a) in-plane and (b) out-of-plane lattice constants relative to the lattice constants of the fully relaxed structures. The polarizations are reported relative to the ap- propriate corresponding reference structures in each case.

0704.0782

Testing outer boundary treatments for the Einstein equations

Testing outer boundary treatments for the Einstein equations Oliver Rinne, Lee Lindblom and Mark A. Scheel Theoretical Astrophysics 130-33, California Institute of Technology, Pasadena, CA 91125, USA Abstract. Various methods of treating outer boundaries in numerical relativity are compared using a simple test problem: a Schwarzschild black hole with an outgoing gravitational wave perturbation. Numerical solutions computed using different boundary treatments are compared to a ‘reference’ numerical solution obtained by placing the outer boundary at a very large radius. For each boundary treatment, the full solutions including constraint violations and extracted gravitational waves are compared to those of the reference solution, thereby assessing the reflections caused by the artificial boundary. These tests are based on a first-order generalized harmonic formulation of the Einstein equations and are implemented using a pseudo-spectral collocation method. Constraint- preserving boundary conditions for this system are reviewed, and an improved boundary condition on the gauge degrees of freedom is presented. Alternate boundary conditions evaluated here include freezing the incoming characteristic fields, Sommerfeld boundary conditions, and the constraint-preserving boundary conditions of Kreiss and Winicour. Rather different approaches to boundary treatments, such as sponge layers and spatial compactification, are also tested. Overall the best treatment found here combines boundary conditions that preserve the constraints, freeze the Newman-Penrose scalar Ψ0, and control gauge reflections. PACS numbers: 04.25.Dm, 02.60.Lj, 02.60.Cb 1. Introduction A fundamental problem in numerical relativity is the need to solve Einstein’s equations on spatially unbounded domains with finite computer resources. There are various ways of addressing this issue. Most often, the spatial domain is truncated at a finite distance and suitable boundary conditions are imposed at the artificial boundary. A different approach is to compactify the domain by using spatial coordinates that bring spatial infinity to a finite location on the computational grid. Another method often used for wave-like problems (although it is not commonly used in numerical relativity) includes so-called sponge layers which damp the waves near the outer boundary of the computational domain. The purpose of this paper is to compare these various methods by testing their ability to accurately reproduce dynamical solutions of Einstein’s equations. An ideal boundary treatment would produce a solution to Einstein’s equations that is identical (within the computational domain) to the corresponding solution obtained on an unbounded domain. In particular, no spurious gravitational radiation or constraint violations should enter the computational domain through the artificial Testing outer boundary treatments for the Einstein equations 2 boundary. We can use this principle to test the various boundary treatments in the following way. First we compute a reference solution using a very large computational domain, large enough that its boundary remains out of causal contact with the interior spacetime region where comparisons are being made. Next we compute the same solution using a domain truncated at a smaller distance where one of the boundary treatments is used: we either impose boundary conditions there, compactify spatial infinity, or add a sponge layer. Finally we compare the solution on the smaller domain with the reference solution, measuring the reflections and constraint violations caused by the boundary treatment. Assessing boundary conditions by comparing with a reference solution on a much larger domain or a known analytic solution is a common practice in computational science. For applications to numerical relativity see e.g. [1], chapter 8 of [2], and [3, 4, 5]. The particular test problem used in this paper is a Schwarzschild black hole with an outgoing gravitational wave perturbation. The interior of the black hole is excised; all the characteristic fields propagate into the black hole (and out of the computational domain) at the inner boundary and hence no boundary conditions are needed there. Our numerical implementation uses a pseudo-spectral collocation method. See Appendix A for details on the initial data, the numerical methods, and the quantities that we compare between the solutions. We perform all of these tests using a first-order generalized harmonic formulation of the Einstein equations (see [6] and references therein). In section 2 we discuss the construction of boundary conditions for this system that prevent the influx of constraint violations, and that limit the spurious incoming gravitational radiation by controlling the Newman-Penrose scalar Ψ0 at the boundary. We also improve the boundary conditions on the gauge degrees of freedom by studying small gauge perturbations of flat spacetime. We then evaluate the performance of these boundary conditions on our test problem: measuring the reflections and constraint violations caused by the computational boundary, and determining how these reflections vary with the radius of the boundary. Section 3 evaluates the performance of a variety of other widely used boundary conditions on our test problem. First we test the simple boundary conditions that freeze all the incoming characteristic fields at the boundary. We also test the commonly used variant of this, the Sommerfeld boundary conditions, used in many binary black hole simulations [7, 8, 9, 10, 11] based on the BSSN [12, 13] formulation of Einstein’s equations. Finally in section 3 we evaluate the constraint-preserving boundary conditions proposed by Kreiss and Winicour [14], which differ from those discussed in section 2 mainly by our use of a physical boundary condition that controls In section 4 we evaluate two boundary treatments that are alternatives to imposing local boundary conditions at a finite outer boundary. The first is the spatial compactification method used e.g. by Pretorius [15, 16, 17] in his groundbreaking binary black hole evolutions. In this treatment a coordinate transformation maps spatial infinity to a finite location on the computational grid. As waves travel out, they become increasingly blue-shifted with respect to the compactified coordinates and ultimately they fail to be resolved. Hence numerical dissipation is applied, which damps away these short-wavelength features. We measure the reflections and the constraint violations generated by the waves in our test problem as they interact with this boundary treatment. Finally in section 4 we implement and test a sponge layer method for Einstein’s equations. Testing outer boundary treatments for the Einstein equations 3 One of the main objectives of current binary black hole simulations is the computation of reliable waveforms for gravitational wave data analysis. Therefore it is important to evaluate how the various boundary treatments affect the accuracy of the extracted waveforms. In section 5, we compute the Newman-Penrose scalar Ψ4 (which describes the outgoing waves) on an extraction sphere close to the outer boundary (or compactified region, or sponge layer, respectively) and compare it with the analogous Ψ4 from the reference solution. We also compare the measured reflections caused by our Ψ0 controlling boundary condition with the analytical predictions of these reflections made by Buchman and Sarbach [18, 19]. Finally we discuss the implications of our results in section 6, and we also describe briefly a number of other boundary treatments which we do not test here. 2. Constraint-preserving boundary conditions In this section, we briefly review the generalized harmonic form of the Einstein evolution system used in our tests. The method of constructing constraint-preserving boundary conditions (CPBCs) for this system is also discussed, and an improved boundary condition for the gauge degrees of freedom is derived. The numerical performance of these boundary conditions is evaluated using our test problem, and the dependence of the spurious reflections as a function of the boundary radius is measured. 2.1. The generalized harmonic evolution system The formulation of Einstein’s equations employed here uses generalized harmonic gauge conditions, in which the coordinates xa obey the wave equation �xa = Ha(x, ψ), (1) where � = ψab(∂a∂b − Γcab∂c) is the covariant scalar wave operator, with ψab the spacetime metric and Γcab the associated metric connection. In this formulation of the Einstein system the gauge source function Ha may be chosen freely as a function of the coordinates and of the spacetime metric ψab (but not derivatives of ψab). As is well known, the Einstein equations reduce to a set of coupled wave equations when the gauge is specified by equation (1). We write this system in first-order form, both in time and space, by introducing the additional variables Φiab ≡ ∂iψab and Πab ≡ −tc∂cψab, where tc is the future directed unit normal to the t = const. hypersurfaces. Here lower-case Latin indices from the beginning of the alphabet denote four-dimensional spacetime quantities, whereas lower-case Latin indices from the middle of the alphabet are spatial. The principal parts of these evolution equations are given by ‡ ∂tψab ' 0, ∂tΠab ' Nk∂kΠab −Ngki∂kΦiab − γ2Nk∂kψab, (2) ∂tΦiab ' Nk∂kΦiab −N∂iΠab +Nγ2∂iψab, where ' indicates that purely algebraic terms have been omitted, gij is the spatial metric of the t = const. slices, and N and N i are the lapse function and shift vector, ‡ The parameter γ1 of [6] is chosen to be −1, which ensures that the equations are linearly degenerate. Testing outer boundary treatments for the Einstein equations 4 respectively. The parameter γ2 was introduced in [6] in order to damp violations of the three-index constraint Ciab ≡ ∂iψab − Φiab = 0. (3) We also include terms of lower derivative order that are designed to damp violations of the harmonic gauge constraint [20] Ca ≡ −�xa +Ha = ψbcΓabc +Ha = 0. (4) The system (2) is symmetric hyperbolic. The characteristic fields in the direction ni (where nata = 0) are given by u0ab = ψab, speed 0, (5) u1±ab = Πab ± Φnab − γ2ψab, speed −N n ±N, (6) u2Aab = ΦAab, speed −Nn. (7) For future reference, we also define ũ1±ab ≡ Πab ± Φnab. (8) Here and in the following, an index n denotes contraction with ni, while upper- case Latin indices A,B, . . . are orthogonal to n, e.g. vA = PAivi where Pab ≡ ψab − nanb + tatb. For further details, we refer the interested reader to [6]. 2.2. Construction of boundary conditions Our construction of boundary conditions for the generalized harmonic evolution system can be divided into three parts: constraint-preserving, physical, and gauge boundary conditions. In order to impose constraint-preserving boundary conditions, we derive the subsidiary evolution system that the constraints (3) and (4) obey as a consequence of the main evolution equations (2). The incoming modes of the subsidiary system are then required to vanish at the boundary (cf. [21, 22, 23, 24, 25, 26, 27, 28, 29]). For instance, the harmonic gauge constraint (4) obeys a wave equation �Ca = (lower-order terms homogeneous in the constraints) (9) and the corresponding incoming fields will involve first derivatives of Ca. In terms of the incoming modes u1−ab (6) of the main evolution equations, the resulting constraint- preserving boundary conditions can be written in the form PC cdab ∂nu cd ≡ ( 12PabP cd − 2l(aPb)(ckd) + lalbkckd)∂nu1−cd = (tangential derivatives), (10) where PC is a projection operator of rank 4 (cf. [6]). Here ni now refers to the outward- pointing unit spatial normal to the boundary, la = (ta + na)/ 2, ka = (ta − na)/ = denotes equality at the boundary. If the shift vector points towards the exterior at the boundary (Nn >̇ 0), the fields u2Aab (7) are incoming as well and we obtain a boundary condition on them by requiring the components CnAab of the four-index constraint Cijab ≡ −2∂[iΦj]ab (11) to vanish at the boundary. An acceptable physical boundary condition should require that no gravitational radiation enter the computational domain from the outside (except for backscatter Testing outer boundary treatments for the Einstein equations 5 off the spacetime curvature, an effect that is a first-order correction in M/R). Gravitational radiation may be described by the evolution system that the Weyl tensor obeys by virtue of the Bianchi identities (see e.g. [27]). Our boundary condition requires the incoming characteristic fields of this system to vanish at the outer boundary. These incoming fields are proportional to the Newman-Penrose scalar Ψ0 (evaluated for a Newman-Penrose null tetrad containing the vectors la and ka). Hence the physical boundary condition we use is [27, 22, 30, 29, 31] = 0, (12) which can be written in a form similar to (10), PP cdab ∂nu cd ≡ (Pa d − 1 cd)∂nu = (tangential derivatives). (13) Here PP is a projection operator of rank 2 that is orthogonal to PC [6]. We remark that (12) still causes some, albeit very small, spurious reflections of gravitational radiation. It can be viewed as the lowest level in a hierarchy of perfectly absorbing boundary conditions for linearized gravity [18, 19]. The constraint-preserving (10) and physical (13) boundary conditions together constrain six components of the main incoming fields u1−ab . The remaining four components correspond to gauge degrees of freedom. In the past we chose simply to freeze those components in time [6], PG cdab ∂tu = 0, (14) where PG ≡ I− PC − PP. The initial-boundary value problem (IBVP) for the boundary conditions discussed so far was shown in [32] to be boundary-stable, which is a (rather strong) necessary condition for well posedness. These boundary conditions have been successfully used in long-term stable evolutions of single and binary black hole spacetimes [6, 33, 34]. In the following subsection, we present an improvement to the gauge boundary condition (14) motivated by the evolution of gauge perturbations about flat spacetime. 2.3. Improved gauge boundary condition Let us assume that near the outer boundary, the spacetime is close to Minkowski space in standard coordinates (Ha = 0) so that the Einstein equations may be linearized about that background. This assumption is reasonable because for the dominant wavenumber of the outgoing pulse (k = 1.6/M) and the boundary radius we typically consider (R = 41.9M), we have kR � 1 and R � M . Furthermore, we assume that the outer boundary is a coordinate sphere of radius r = R. We begin by noting that harmonic gauge does not fix the coordinates completely: infinitesimal coordinate transformations xa → xa + ξa (15) are still allowed provided the displacement vector satisfies the wave equation, �ξa = 0. (16) Under such a coordinate transformation, the metric changes by δψab = −2∂(aξb). (17) A closer inspection [32] of the projection operator PG in (14) shows that the gauge boundary conditions control the components laδψab of the perturbations, where Testing outer boundary treatments for the Einstein equations 6 la ≡ (ta + na)/ 2 is the outgoing null vector normal to the boundary. It is interesting to observe that these components vanish in the ingoing radiation gauge [35]. However, imposing radiation gauge on the entire spacetime is not possible in spacetimes containing strong-field regions, which will always generate perturbations laδψab that propagate into the far field. A reasonable condition to require then is that these perturbations pass through the boundary without causing strong reflections. Each Cartesian component of the vector laδψab obeys the scalar wave equation �ψ = 0. (18) Solutions to this equation can be written in the form Ylm(θ, φ)ψl(t, r), (19) where the Ylm are the standard spherical harmonics and the ψl are linear combinations of outgoing (+) and incoming (−) solutions ψ±l (t, r) = r F±l (r ∓ t), (20) F±l (x) being arbitrary functions. A boundary condition is needed on ψ that eliminates the incoming part of these solutions. In [36], a hierarchy of boundary conditions is constructed that accomplish this task for all l 6 L. This idea was applied to the evolution of the Weyl curvature in [18] in order to construct improved physical boundary conditions. For the gauge boundary conditions considered here, we restrict ourselves to the L = 0 member of the hierarchy, which corresponds to the Sommerfeld condition § (∂t + ∂r + r = 0. (21) In contrast, our old gauge boundary condition that froze the incoming characteristic field, as in (14), is given by (∂t + ∂r + γ2)ψ = 0, (22) where γ2 is the constraint damping parameter. This Sommerfeld boundary condition (21) is much less reflective than the freezing condition (22). To see this, we consider a solution of the form ψl = ψ l + ρlψ l (23) with generating functions F±l (x) = e ±ikx, (24) where k ∈ R is the wave number. Substituting this solution into the boundary conditions (21) resp. (22), we solve for the reflection coefficient ρl. Figure 1 shows |ρl| for a typical range of wave numbers k and outer boundary radii R used for the numerical tests in this paper. (The dominant wave number of the outgoing pulse is k ≈ 1.6/M and in most cases, we place the outer boundary at R = 41.9M .) We see that |ρl| is much smaller (by about 3 orders of magnitude) for the Sommerfeld condition than for the freezing condition. § To avoid confusion, we remark that in [5, 14], the term ‘Sommerfeld condition’ is used in reference to a condition of the form (∂t +∂r)u = 0, i.e. without the extra r−1 term due to our polar coordinates. Testing outer boundary treatments for the Einstein equations 7 0 0.5 1 1.5 2 l = 1 R = 41.9 M 50 100 150 200 R / M l = 1 k = 1 / M Figure 1. Predicted reflection coefficients ρl for freezing (dotted) and Sommerfeld (solid) boundary conditions as functions of wave number k and outer boundary radius R. The curves for different l are visually indistinguishable in the freezing case. Note also that ρ0 = 0 for the Sommerfeld condition. In the notation of the previous subsection, the improved gauge boundary condition (21) reads (after taking a time derivative), PG cdab ∂t[u cd + (γ2 − r −1)ψcd] = 0. (25) We remark that the extra terms in (25) as compared with the old condition (14) are of lower derivative order, so that the high-frequency stability result of [32] extends immediately to these modified gauge boundary conditions. 2.4. Numerical results The numerical tests of the various boundary conditions performed in this paper are described in some detail in Appendix A. Figure 2 compares the numerical performance of our new CPBCs (10), (11), (13), (25) with our old ones (10), (11), (13), (14). The outer boundary is placed at radius R = 41.9M for these particular tests. Shown are the discrete L∞ and L2 norms of the difference ∆U between the numerical solution and the reference solution, and also the violations of the constraints C (see Appendix A.4 for precise definitions of these quantities). The reference solution has an outer boundary at radius 961.9M and is computed using our old CPBCs; thus for t < 920M the outer boundary of the reference solution is out of causal contact with the region where ∆U and C are computed. In the difference ∆U we see a reflection that originates when the wave reaches the boundary at t ≈ R and then amplifies as it moves inward in the spherical geometry, assuming its maximum at t ≈ 2R. This feature is much more prominent in the L∞ norm than in the L2 norm, which is why we display only the L∞ norm in subsequent plots. The reflection is much smaller (by a factor of ≈ R/M) for the new boundary conditions as compared with the old ones. Even at later times, the new boundary conditions result in a smaller ∆U , which in contrast to the old conditions appears to decrease as resolution is increased. We would like to stress that ∆U is a coordinate dependent quantity. Hence a smaller ∆U does not necessarily mean that the boundary treatment is ‘better’ in a physically meaningful sense. If however the aim is to produce a solution that is as Testing outer boundary treatments for the Einstein equations 8 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) Figure 2. Old (solid) vs. new (dotted) CPBCs. Four different resolutions are shown: (Nr, L) = (21, 8), (31, 10), (41, 12), and (51, 14). The outer boundary is at R = 41.9M . close to the reference solution in the same coordinates, the choice of gauge boundary conditions does become important. Gauge reflections can in principle also impair the numerical accuracy of gauge-invariant quantities because much numerical resolution is wasted on resolving the gauge reflections. This is particularly the case when the gauge excitations in question are high-frequency modes such as those produced along with the so-called ‘junk radiation’ in binary black hole initial data. There is no discernible difference between the two sets of boundary conditions as far as constraint violations are concerned, which is what we expect because both of them are constraint-preserving. We close this section by investigating the dependence of the reflections on the radius of the outer boundary (figure 3). The amplitude of the first peak in ||∆U||∞ decreases as the boundary is moved outward, roughly like 1/R. At late times, there appears to be a power-law growth of that quantity at a rate that increases slightly with resolution. Inspection of the constraints (also in figure 3) and Ψ4 (figure 10) suggests that this is a pure gauge effect. This blow-up is completely dominated by the innermost domain, which contains a long-wavelength feature that is growing in time. We speculate that this problem might be cured by a more clever choice of gauge source function close to the black hole horizon. Testing outer boundary treatments for the Einstein equations 9 0 200 400 600 800 1000 t / M R/M = 21.9 161.9 201.9 121.9 = 51, L = 14) 0 200 400 600 800 1000 t / M = 51, L = 14) 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) R = 121.9 M 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) R = 121.9 M Figure 3. New CPBCs at different radii. Top half: all radii at the highest resolution, bottom half: R = 121.9M at all resolutions. In the top right panel, curves for all outer boundary radii coincide. 3. Alternate boundary conditions In this section, we consider several alternate boundary conditions that are often used in numerical relativity. All of these are local conditions imposed at a finite boundary radius, then in section 4 we consider some additional non-local boundary treatments. We run the alternate boundary conditions on our test problem and compare the results with the CPBCs (using the new gauge boundary condition (25)). 3.1. Freezing the incoming fields A very simple boundary condition is obtained by freezing in time all the incoming fields at the boundary, i.e., = 0 (and ∂tu = 0 if Nn >̇ 0). (26) This boundary condition is attractive from a mathematical point of view because it is of maximally dissipative type and hence, together with the symmetric hyperbolic evolution equations (2), yields a strongly well-posed IBVP [37, 38, 39]. However, in general this boundary condition is not compatible with the constraints. Testing outer boundary treatments for the Einstein equations 10 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) Figure 4. Freezing (solid) vs. new CPBCs (dotted). Four different resolutions are shown: (Nr, L) = (21, 8), (31, 10), (41, 12), and (51, 14). For freezing boundary conditions, both ||∆U|| and C converge to a nonzero function with increasing resolution. The outer boundary is at R = 41.9M . The left side of figure 4 demonstrates that freezing boundary conditions cause a significantly larger (by ≈ 3 orders of magnitude) initial reflection than our CPBCs. The difference with respect to the reference solution remains large in the subsequent evolution and unlike for the CPBCs does not decrease with increasing resolution. Furthermore, the violations of the constraints (right side of figure 4) do not converge away. This means that a solution to the Einstein equations is not obtained in the continuum limit. 3.2. Sommerfeld boundary conditions A boundary condition that is often imposed in conjunction with the BSSN [12, 13] formulation of the Einstein equations is a Sommerfeld condition on all the components of the spatial metric gij and extrinsic curvature Kij , (∂t + ∂r + r gij − δij = 0. (27) This condition has been used for example in many recent binary black hole simulations [7, 8, 9, 10, 11]. We cannot impose precisely the conditions (27) in our simulations because there is no one-to-one relationship between gij and Kij , and the incoming characteristic fields of our generalized harmonic formulation of Einstein’s equations. Instead we consider the similar condition (∂t + ∂r + r −1)(ψab − ηab) = 0 (28) on all the components of the spacetime metric (ηab being the Minkowski metric). A very similar boundary condition (without the r−1 term) has recently been used in the generalized harmonic evolutions of [40]. In our formulation, boundary conditions are required not on the spacetime metric itself but only on certain combinations of its derivatives. By taking a time derivative of (28) and rewriting in terms of incoming characteristic fields, we obtain ab + (γ2 − r −1)ψab] = 0. (29) Testing outer boundary treatments for the Einstein equations 11 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) Figure 5. Sommerfeld (solid) vs. new CPBCs (dotted). Four different resolutions are shown: (Nr, L) = (21, 8), (31, 10), (41, 12), and (51, 14). The outer boundary is at R = 41.9M . This then is our version of the Sommerfeld boundary condition (cf. (25)), to be imposed on a spherical boundary in the far field (where linearized theory is assumed to be valid). Because the BSSN formulations using (27) are usually second-order in space, there is no analogue of our three-index constraint (3) in that system. To mimic this situation in our tests of equation (29), we also impose a CPBC on u2Aab as discussed in section 2.2, which together with our constraint damping terms ensures that violations of the three-index constraint (3) are exponentially damped. Our version of Sommerfeld boundary conditions performs similarly on our test problem (figure 5) to the freezing boundary conditions (26) (figure 4). The initial pulse of reflections is smaller by ≈ 2 orders of magnitude, but later ||∆U|| grows to a similar level as for freezing boundary conditions. Again the constraints do not converge away, although this non-convergence appears only at somewhat higher resolutions than in the freezing case. 3.3. Kreiss-Winicour boundary conditions Recently, Kreiss and Winicour [14] proposed a set of ‘Sommerfeld-like’ CPBCs for the harmonic Einstein equations and showed that they result in an IBVP that is well-posed in the generalized sense. Their boundary conditions were implemented and tested in [5]; here we compare their performance with the various other boundary treatments. The Kreiss-Winicour boundary conditions are obtained by requiring the harmonic constraint to vanish at the boundary, = 0. (30) In our notation, this can be written as an algebraic condition on part of the incoming fields u1−, = Fa, (31) where [2k(cδad) − kaψcd], lbu1+ab − bcu1+bc + P iju2ija − 12Pa iψbcu2ibc (32) Testing outer boundary treatments for the Einstein equations 12 − γ2ta +Ha. The range of the projection operator PC is identical with that of PC defined in (10). For the unconstrained incoming fields ũ1− (i.e. u1− without the γ2 term, equation (8)), Kreiss and Winicour [14] specify certain free boundary data qPab and q ab. In our notation, PP cdab ũ cd = q ab, P ab ũ cd = q ab. (33) In the linearized wave and gauge wave tests of [5], these boundary data are obtained from the known exact solutions. In the absence of an exact solution, it is suggested that the data could be obtained from an exterior Cauchy-characteristic or Cauchy- perturbative code. However, since we do not have such an exterior code, we compute the boundary data from the background solution, i.e. Schwarzschild spacetime. As in the Sommerfeld case (section 3.2), we use a constraint-preserving boundary condition on u2Aab to emulate the second-order formulation of [5, 14], and this value of u Aab is then used to compute Fa in (32). Figure 6 shows the numerical results for our test problem. The magnitude of the initial reflections lies between that of freezing and Sommerfeld boundary conditions and is somewhat smaller at later times, though still larger than for our CPBCs at the higher resolutions. The constraints converge away with increasing resolution, as they should for a boundary condition that is consistent with the constraints. In a numerical simulation, violations of the constraints are in general present in the interior of the computational domain. These propagate as described by the constraint evolution system (9) and some may hit the outer boundary. The Dirichlet boundary conditions (30) might be expected to cause more reflections of constraint violations than our no-incoming-field conditions (10), however, no indications of this are seen in figure 6. Probably the constraint damping we use is sufficiently effective in eliminating the source of these reflections. We shall see in section 5.1 that the Kreiss-Winicour boundary conditions also cause larger errors in the physical degrees of freedom than our CPBCs. Since the main difference between the two sets of boundary conditions is our use of a physical boundary condition ∂tΨ0 = 0, we conclude that such a condition is crucial in reducing the reflections from the outer boundary. 4. Alternate approaches So far we have only considered boundary conditions that are local algebraic or differential conditions imposed at the boundary of some finite computational domain. There are of course many ways of treating the outer boundary that do not fall into that category. In this section, we evaluate two such approaches: spatial compactification and sponge layers. 4.1. Spatial compactification Spatial compactification is a method that has been widely used in numerical relativity, for instance in [41, 42] or more recently in the generalized harmonic binary black hole simulations of Pretorius [15, 16, 17]. The basic idea is to introduce spatial coordinates that map spacelike infinity to a finite location. Here we consider mappings that are functions of coordinate radius only (whereas Pretorius applies the mapping to each Cartesian coordinate separately). We Testing outer boundary treatments for the Einstein equations 13 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) Figure 6. Kreiss-Winicour (solid) vs. new CPBCs (dotted). Four different resolutions are shown: (Nr, L) = (21, 8), (31, 10), (41, 12), and (51, 14). The outer boundary is at R = 41.9M . have used two such mappings, named Tan and Inverse, as detailed in Appendix B.1. Each map has a scaleR across which the mapping is (essentially) linear. The outermost grid point is placed at a very large but finite uncompactified radius (r = 1017M). With respect to the compactified radial coordinate, the characteristic speeds are below numerical roundoff there and hence no boundary condition should be needed. The following results were produced using constraint-preserving boundary conditions; we have checked for one simulation that using no boundary condition at all yields results that are visually indistinguishable from the ones presented here on the scales of figures 7, 8, and 10. As the waves travel outward, they become more and more blue-shifted with respect to the computational grid and are eventually no longer properly resolved. However, some form of artificial numerical dissipation is applied that acts as a low- pass filter and causes the waves to be damped as they become increasingly distorted. We have experimented with various such filters; see Appendix B.1 for details. One of them (referred to as number 2 in the following) is designed to emulate as closely as possible the fourth-order Kreiss-Oliger dissipation used by Pretorius. In the following numerical comparisons, we evaluate the differences with respect to the reference solution only in the part of the domain where the compatification map is essentially linear, i.e. for r 6 R. First we compare the various filtering methods at a fixed resolution, using the Tan compactification mapping (figure 7). The filters that are applied to the right side of the evolution equations (numbers 1 and 3, cf. table B1) do somewhat better than those applied to the solution itself (numbers 2 and 4), and the Exponential filters (numbers 3 and 4) are slightly better than the Kreiss-Oliger filters (numbers 1 and 2). All of them are outperformed by the CPBCs (imposed at r = R). For our closest approximation to the dissipation used by Pretorius (number 2), ||∆U|| is comparable to constraint-preserving boundary conditions at the peak of reflections (at t ≈ 2R) but becomes larger by about 2 orders of magnitude at later times. The compactification methods also generate considerable constraint violations. Next we focus on the best filter (number 4) of the previous test but vary the resolution (figure 8). We do see convergence of ||∆U|| initially but the convergence degrades at later times. This is surprising at first because with increasing resolution, Testing outer boundary treatments for the Einstein equations 14 0 200 400 600 800 1000 t / M New CPBC TAN, Filter 1 TAN, Filter 2 TAN, Filter 3 TAN, Filter 4 0 200 400 600 800 1000 t / M New CPBC TAN, Filter 1 TAN, Filter 2 TAN, Filter 3 TAN, Filter 4 Figure 7. Tan compactification with various filters vs. new CPBCs. Only the highest resolution (Nr, L) = (51, 14) is shown. The compactification scale (and the radius of the outer boundary in the CPBC case) is R = 41.9M . 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) Figure 8. Tan compactification with filter 4 (solid) vs. new CPBCs (dotted). Four different resolutions are shown: (Nr, L) = (21, 8), (31, 10), (41, 12), and (51, 14). The compactification scale (and the radius of the outer boundary in the CPBC case) is R = 41.9M . the waves travel a longer distance before they fail to be resolved. Note however that the high-frequency filter is applied at each time step, as is done in the simulations of Pretorius. For higher resolutions, the time steps are smaller because of the CFL condition and the filter is applied more often, thus leading to a stronger damping of the waves. This may well lead to the observed loss of convergence with increasing resolution. The constraints appear to converge away in this test, although from figure 8 it appears that this will not persist for even higher resolutions. We have also evaluated the Inverse mapping described in Appendix B.1. The results are similar, but somewhat worse than the Tan mapping results shown here. Testing outer boundary treatments for the Einstein equations 15 4.2. Sponge layers A method that has been used for a long time in computational science, in particular for spectral methods (see e.g. section 17.2.3 of [43] and references therein), involves so-called sponge layers. A sponge layer is introduced by modifying the evolution equations according to ∂tu = . . .− γ(r)(u− u0), (34) where u0 refers to the unperturbed background solution (Schwarzschild spacetime in our case) and the smooth sponge function γ(r) > 0 is large only close to the outer boundary of the computational domain. (Here we use uncompactified coordinates as in sections 2 and 3.) In this way, the waves are damped exponentially as they approach the outer boundary. Details on our particular choice of γ(r) can be found in Appendix We compare the sponge layer method with our CPBCs in figure 9. For the CPBCs, the boundary is either placed at R = 41.9M (the outer edge of the sponge- free region) or at R = 121.9M (the outer edge of the sponge). At early times (t . 2R), the ||∆U||∞ of the sponge layer method lies between that of the CPBCs for the two choices of outer boundary radius, whereas at later times, it is much larger than both versions of CPBCs. The constraint violations in the sponge runs do not converge away. 5. Physical gravitational waves Perhaps the most important predictions of numerical relativity simulations at the present time are the gravitational waveforms produced by astrophysical systems like binary black holes. It is important therefore to understand how the accuracy of these waveforms is affected by the choice of boundary treatment. Physical gravitational radiation can be described by the Newman-Penrose scalars Ψ4 and Ψ0. The scalar Ψ4 is dominated by the outgoing radiation (its ingoing part is suppressed by a factor of (kr)4, where k is the wavenumber), whereas Ψ0 is dominated by the ingoing radiation (its outgoing part is suppressed by a factor of (kr)4). In this section we compare the gravitational waves extracted from the various boundary treatment solutions on a sphere of radius r = Rex, using the methods described in Appendix A.5. We note that Ψ4 (Ψ0) has a coordinate-invariant meaning only in the limit as future (past) null infinity is approached. The quantities computed at finite radius r will differ from those observed at infinity by terms of the order O(1/r). In the particular case of perturbed Schwarzschild spacetime considered here, a gauge-invariant wave extraction method does exist even at finite radius (see e.g. [44] and references therein) but we do not adopt it here. Our purpose in this paper is merely to measure the effects on Ψ4 caused by the various boundary treatments. 5.1. Difference of Ψ4 with respect to the reference solution We begin by evaluating ∆Ψ4 ≡ Ψ4 − Ψref4 , where Ψ4 is the Newman-Penrose scalar computed using one of the various boundary methods and Ψref4 is the same quantity computed from the reference solution at the same extraction radius. The curves shown in figure 10 plot the maximum value of |∆Ψ4| over time intervals of length 20M (this time filtering averages over the high frequency quasi-normal oscillations of the black hole), normalized by the maximum value of |∆Ψ4| over the entire evolution. The Testing outer boundary treatments for the Einstein equations 16 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) CPBC at R = 41.9 M 0 200 400 600 800 1000 t / M ∞ (Nr,L)=(21,8) (31,10) (41,12) (51,14) CPBC at R = 41.9 M 0 200 400 600 800 1000 t / M ,L)=(21,8) (31,10) (41,12) (51,14) CPBC at R = 121.9 M 0 200 400 600 800 1000 t / M ∞ (Nr,L)=(21,8) (31,10) (41,12) (51,14) CPBC at R = 121.9 M Figure 9. Sponge layer method (solid) vs. new CPBCs at two different radii (dotted). Four different resolutions are shown: (Nr, L) = (21, 8), (31, 10), (41, 12), and (51, 14). The size of the sponge-free region is R = 41.9M and ||∆U||∞ is only computed for r 6 R. radius of the outer boundary (or the compactification scale, or the size of the sponge- free region, respectively) used for these comparisons is R = 41.9M , and the radiation is extracted nearby at Rex = 40M . The first peak in |∆Ψ4| seen in figure 10 arises as the wave in our test problem passes outward through the extraction sphere at t ≈ Rex. This peak is caused by a presently unknown (probably gauge) interaction between the outer boundary (or compactified region etc.) and the spacetime near the extraction sphere. We have verified that this interaction and its influence on the peak in ∆Ψ4 goes away if we move the outer boundary (or the extraction surface) so that they are not in causal contact as the outgoing wave pulse passes the extraction surface. Some of the outgoing radiation is reflected off the boundary. Most of this reflected radiation is subsequently absorbed by the black hole, but some of it excites the hole, which then emits quasi-normal mode radiation of exponentially decaying amplitude. This exponential decay can be clearly seen for most of the boundary treatments. In the case of freezing boundary conditions, nearly all of the outgoing quasi- normal mode radiation is reflected from the boundary because the reflection coefficient is nearly 1 for the wave number of the dominant mode, k = 0.376/M (cf. figure 1). Testing outer boundary treatments for the Einstein equations 17 It then re-excites the black hole, which again radiates and so forth. On average the amplitude of the reflections remains roughly constant in time for this case. This behaviour is consistent with the result shown in figure 3 of [6] for a similar perturbed black hole simulation. For the Sommerfeld and Kreiss-Winicour boundary conditions, the reflections are much smaller but still considerably larger (by 2 to 3 orders of magnitude) than for our CPBCs. We attribute this difference largely to our use of the physical boundary condition (12). The spatial compactification method has the largest difference |∆Ψ4|, particularly at early times t ∼ R (about 4 orders of magnitude larger than for the CPBCs). We suspect that this may be a consequence of the use of artificial dissipation, as discussed in section 4.1. The sponge layer method has the smallest errors at early times. This is not surprising because the outer boundary of the sponge layer is much further out at R = 121.9M . However at later times when the waves begin to interact with the sponge layer, this method causes reflections comparable in amplitude to those using Sommerfeld boundary conditions. We also note that at late times the level of |∆Ψ4| decreases significantly with resolution for the CPBCs, but not generally for the other boundary treatments. We think it is remarkable that the maximum relative error in the extracted physical radiation is quite small (10−5 to 10−3) in these tests, even for the less sophisticated boundary treatments such as the freezing or Sommerfeld boundary conditions. This success is due in part to the fact that the extraction radius, Rex = 40M , for this test problem is about ten wavelengths (of the initial radiation pulse) away from the central black hole. Our results are likely to be more accurate than those from typical binary black hole simulations, which place the outer boundary at two or three wavelengths. This suggests that current binary black hole codes using, for instance, Sommerfeld boundary conditions, can still produce waveforms that are useful for some aspects of gravitational wave data analysis provided the outer boundary is placed sufficiently far out. Data analysis applications needing high precision waveforms, however, such as source parameter measurement or high- amplitude supermassive binary black hole signal subtraction for LISA, will need to use a more sophisticated boundary treatment that produces smaller errors in Ψ4. 5.2. Comparison with the predicted reflection coefficient Buchman and Sarbach [18, 19] have recently developed a hierarchy of increasingly absorbing physical boundary conditions for the Einstein equations by analyzing the equations describing the evolution of the Weyl curvature on both a flat and a Schwarzschild background spacetime. Their analysis predicts, in particular, the reflection coefficient ρ (defined as the ratio of the ingoing to the outgoing parts of the solution) that arises from the ∂tΨ0 = 0 physical boundary condition that we use. For quadrupolar radiation (as in our numerical tests), this reflection coefficient is given by equation (89) of [18], ρ(kR) = 3 (kR)−4 +O(kR)−5, (35) where k is the wave number of the gravitational radiation and R is the boundary radius. (As explained at the beginning of section 2.3, we assume the background spacetime to be flat; effects due to the backscattering would only enter at O(M/R).) Testing outer boundary treatments for the Einstein equations 18 0 200 400 600 800 1000 t / M ,L)=(31,10) (51,14) Freezing 0 200 400 600 800 1000 t / M ,L)=(31,10) (51,14) Sommerfeld 0 200 400 600 800 1000 t / M ,L)=(31,10) (51,14) Kreiss-Winicour 0 200 400 600 800 1000 t / M ,L)=(31,10) (51,14) TAN compactification (filter 4) 0 200 400 600 800 1000 t / M ,L)=(31,10) (51,14) Sponge layer Figure 10. Difference of Ψ4 for the various alternate methods (solid) vs. the new CPBCs (dotted). Two resolutions are shown: (Nr, L) = (31, 10) and (51, 14). The radius of the outer boundary (or the compactification scale, or the size of the sponge-free region, respectively) is R = 41.9M and the waves are extracted at Rex = 40M . Testing outer boundary treatments for the Einstein equations 19 0 1 2 3 4 5 6 7 (k R) R = 21.9 M 0 1 2 3 4 5 6 7 (k R) R = 121.9 M Figure 11. Comparison of the time Fourier transform of the measured Ψ0(t) with 3 (kR)−4Ψ4, which is the predicted value using the reflection coefficient of [18]. By evaluating Ψ0 and Ψ4 at the extraction radius of our test, we find that the ratio Ψ0/Ψ4 agrees with their predicted ρ to leading order in 1/(kR). We note that the tetrad we use for wave extraction (Appendix A.5) does not agree exactly with that of [18]. However, the tetrads do agree for the unperturbed Schwarzschild solution, so that the errors introduced into Ψ0 and Ψ4 due to our different choice of tetrad are second-order small in perturbation theory and hence the comparison with [18] is consistent. For a numerical solution using our new CPBCs, we evaluate the Newman-Penrose scalars Ψ0(t) and Ψ4(t) on extraction spheres located 1.9M inside the outer boundary. In figure 11 we plot the time Fourier transforms of these quantities. We also plot (kR)−4Ψ4, which by the above argument should agree with Ψ0 to leading order in 1/(kR). Figure 11 shows that the numerical agreement is reasonably good: roughly at the expected level of accuracy. The overall dependence of the predicted reflection coefficient ρ on k and R is captured very well. We surmise that the levelling off of our numerical Ψ0 for k & 3 is due to numerical roundoff effects. (Note the magnitude of Ψ0 at those frequencies.) For radii R & 200M , Ψ0 is at the roundoff level for all frequencies. 6. Discussion The purpose of this paper is to compare various methods of treating the outer boundary of the computational domain. We evaluate the performance of several often- used boundary treatments in numerical relativity by measuring the amount of spurious reflections and constraint violations they generate. To this end, we consider as a test problem an outgoing gravitational wave superimposed on a Schwarzschild black hole spacetime. First we compute this numerical solution on a reference domain, large enough that the influence of the outer boundary can be neglected. Then we repeat the evolution on smaller domains using one of the boundary treatments, either imposing local boundary conditions, compactifying the domain using a radial coordinate map, or installing a sponge layer. We use a first-order generalized harmonic formulation of the Einstein equations, although these boundary methods can be applied to other Testing outer boundary treatments for the Einstein equations 20 formulations as well. We believe our results are fairly independent of the particular formulation used. Our main conclusion is that our version of constraint-preserving boundary conditions performs better than any of the alternate treatments that we tested. Our boundary conditions include a limitation on the influx of spurious gravitational waves by freezing the Newman-Penrose scalar Ψ0 at the boundary. We also introduce and test an improved boundary condition for the gauge degrees of freedom. For some of the simple boundary conditions, such as freezing or Sommerfeld conditions, we find constraint violations that do not converge away with increasing resolution. The continuum limit does not satisfy Einstein’s equations in these cases. Most of the alternate boundary conditions also generate considerable reflections as measured by ∆U , the norm of the difference with respect to the reference solution. In many cases, these reflections do not decrease significantly with increasing resolution. The difference norm ∆U that we use to measure boundary reflections includes the entire spacetime metric, not just the physical degrees of freedom. It is important then to evaluate separately the effects of the various boundary treatments on the physical degrees of freedom. We use the extracted outgoing radiation as approximated by the Newman-Penrose scalar Ψ4 for this purpose. Here our conclusions are somewhat different. Rather surprisingly, most of the boundary methods we consider generate relatively small errors in Ψ4. This suggests that if gravitational waveforms are only needed to an accuracy of, say, 1% (which is comparable to the discrepancies between recent binary black hole simulations [45]) then even the simple Sommerfeld conditions might be good enough. (For those, we find relative errors ∼ 10−5.) The largest relative errors in Ψ4 we find (∼ 10−2) occur with our implementation of the spatial compactification method used by Pretorius [15, 16, 17]. We attribute these largely to the use of artificial dissipation. Undesirable effects of dissipation might be somewhat less severe in binary black hole evolutions, which have much larger wavelengths (λ ∼ 20 − 100M) than ours (λ ∼ 4M). Our tests suggest that the errors in Ψ4 can be made to decrease significantly with resolution only by using more sophisticated constraint preserving and physical boundary conditions. The importance of using a physical boundary condition on Ψ0 is illustrated in particular by the difference between the performance of our boundary conditions and those of Kreiss and Winicour [14]. Some caveats regarding the interpretation of our results must be stated. First, the ratio of the dominant wavelength to the radius of the outer boundary is typically much larger for binary black hole evolutions (where λ/R & 0.5) than for the simple test problem considered here (where λ/R ∼ 0.1). Boundary treatments generally work better for smaller λ/R, i.e. when the boundary is well out in the wave zone. Hence the results presented here are likely to be more accurate than those from typical binary black hole simulations. Second, we use spectral methods rather than finite-difference methods, which are more commonly used in numerical relativity at this time. This complicates the implementation of the kind of numerical dissipation that is crucial for the spatial compactification method to work. While we have attempted to construct a filter that mimics the finite-difference dissipation as closely as possible, a direct comparison is clearly impossible. In finite-difference methods, the error introduced by the type of numerical dissipation considered here is below the truncation error. Hence tests similar to ours but performed with a finite-difference method would not be able to detect the effect of dissipation. There are several directions in which the present work could be extended. For large values of the outer boundary radius, we observe a non-convergent power- Testing outer boundary treatments for the Einstein equations 21 law growth of the error in our test problem when constraint-preserving boundary conditions are used; the origin of this growth should be investigated further. It would be interesting to implement and test the hierarchy of physical boundary conditions that are perfectly absorbing for linearized gravity (including leading-order corrections due to the curvature and backscatter) found recently by Buchman and Sarbach [18, 19]. Our boundary conditions could also be tested using known exact solutions such as gauge waves, and comparisons could be made with the results found in [5]. For completeness we also mention a number of additional outer boundary approaches that were not addressed in this paper, but would also be interesting future extensions of this research. In [46, 47], boundary conditions for the full nonlinear Einstein equations on a finite domain are obtained by matching to exact solutions of the linearized field equations at the boundary. Alternatively, the interior code could be matched to an ‘outer module’ that solves the linearized field equations numerically [48, 49, 50, 51]. Other approaches involve matching the interior nonlinear Cauchy code to an outer characteristic code (see [52] for a review) or using hyperboloidal spacetime slices that can be compactified towards null infinity (see [53] for a review). Appendix A. Details on the numerical test problem Appendix A.1. Initial data The initial data used for our numerical tests are the same as in [27]. The background solution is a Schwarzschild black hole in Kerr-Schild coordinates, ds2 = −dt2 + 2M (dt+ dr)2 + dr2 + r2dΩ2. (A.1) Throughout the paper, M refers to the bare black hole mass of the unperturbed background. We superpose an odd-parity outgoing quadrupolar wave perturbation constructed using Teukolsky’s method [54]. Its generating function is taken to be a Gaussian G(r) = A exp[−(r − r0)2/w2] with A = 4× 10−3, r0 = 5M , and w = 1.5M . The full non-linear initial value equations in the conformal thin sandwich formulation are then solved to obtain initial data that satisfy the constraints [55]. This yields initial values for the spatial metric, extrinsic curvature, lapse function, and shift vector. We note that after the superposition, the resulting solution is still nearly but not completely outgoing. Our generalized harmonic formulation of Einstein’s equations requires initial data for the full spacetime metric and its first time derivative. These can be computed from the 3+1 quantities obtained above, provided we also choose initial values for the time derivatives of the lapse function and shift vector. These initial time derivatives are freely specifiable and are equivalent to the initial choice of the gauge source function Ha; we choose ∂tN = 0 and ∂tN i = 0 at t = 0. Appendix A.2. Numerical method We use a pseudospectral collocation method as described for example in [27]. The computational domain for the test problem considered here is taken to be a spherical shell extending from r = 1.9M (just inside the horizon) out to some r = R. This domain is subdivided into spherical-shell subdomains of extent ∆r = 10M . On each subdomain, the numerical solution is expanded in Chebyshev polynomials in the radial direction and in spherical harmonics in the angular directions Testing outer boundary treatments for the Einstein equations 22 (where each Cartesian tensor component is expanded in the standard scalar spherical harmonics). Typical resolutions are Nr ∈ {21, 31, 41, 51} coefficients per subdomain for the Chebyshev series and l 6 L with L ∈ {8, 10, 12, 14} for the spherical harmonics. We change the outer boundary radius R by changing the number of subdomains while keeping the width ∆r of each subdomain fixed; this facilitates direct comparisons between runs with different values of R. For example, the innermost four subdomains of the reference solution (which has a total of 96 subdomains and R = 961.9M) are identical to the four subdomains used to compute the solution with R = 41.9M . The evolution equations are integrated in time using a fourth-order Runge-Kutta scheme, with a Courant factor ∆t/∆xmin of at most 2.25, where ∆xmin is the smallest distance between two neighbouring collocation points. As described in [27], the top four coefficients in the tensor spherical harmonic expansion of each of our evolved quantities is set to zero after each time step; this eliminates an instability associated with the inconsistent mixing of tensor spherical harmonics in our approach. We use two methods of numerically implementing boundary conditions; the choice of method depends on the type of boundary conditions. Boundary conditions that can be expressed as algebraic relations involving the characteristic fields are implemented using a penalty method (see [56] and references therein; in the context of finite- difference methods see also [57] and references therein). In particular, we use a penalty method to implement the Kreiss-Winicour boundary conditions (cf. section 3.3) and to impose boundary conditions at the internal boundaries between neighbouring subdomains. Boundary conditions that are expressed in terms of the time derivatives of the characteristic fields are implemented using the method of Bjørhus [58], where the time derivatives of the incoming characteristic fields are replaced at the boundary with the relevant boundary condition. All boundary conditions in this paper besides those mentioned above are implemented using the Bjørhus method. Appendix A.3. Gauge source functions Our generalized harmonic formulation [6] of Einstein’s equations allows for gauge source functions that depend arbitrarily on the coordinates and the spacetime metric: Ha = Ha(t, x, ψ). The generalized harmonic evolution equations are equivalent to Einstein’s equations only if the constraint (4) remains satisfied. We choose the time derivatives of lapse and shift to be zero at the beginning of the simulation; this determines the initial value of Ha via the constraint (4). For the subsequent evolution, we hold this Ha fixed in time. Appendix A.4. Error quantities We use two different measurements of the errors in our solutions, which we monitor during our numerical evolutions. First, given a numerical solution (ψab,Πab,Φiab), the difference between that solution and the reference solution (ψ(ref)ab ,Π (ref) ab ,Φ (ref) iab ) is computed with the following norm at each point in space, δabδcd(M−2∆ψac∆ψbd + ∆Πac∆Πbd +gij∆Φiac∆Φjbd) , (A.2) where ∆ψab means ψab−ψ(ref)ab , and similarly for ∆Πab and ∆Φiab. Second, we define a quantity C that measures the violations in all of the constraints of our system, δab(FaFb + gij(CiaCjb + gklδcdCikacCjlbd) Testing outer boundary treatments for the Einstein equations 23 +M−2(CaCb + gijδcdCiacCjbd)) , (A.3) where Fa and Cia are first derivatives of Ca defined in [6]. To compute global error measures, a spatial norm ||·||, either the L∞ norm or the L2 norm, is applied separately to ∆U and C. The question often arises as to the significance of particular values of ||∆U|| and ||C||. For example, is a simulation with ||C|| = 10−2 good to one percent accuracy? To make it easier to answer such questions, we normalize both ||∆U|| and ||C|| as follows, and we always plot normalized quantities. We divide ||∆U|| by a normalization factor ||∆U0||, defined as the difference between a given solution at t = 0 and the unperturbed Schwarzschild background; i.e., the quantity ||∆U0|| is computed from (A.2) using the unperturbed Schwarzschild solution instead of the reference solution. Since ||∆U0|| is evaluated at t = 0, it depends only on the initial data used in the simulation, and is a measure of the amplitude of the superposed gravitational wave perturbation. For the initial data used here, ||∆U0||∞ = 6× 10−3 and ||∆U0||2 = 1.4× 10−4. The quantity ||∆U||/||∆U0|| is more easily interpreted than ||∆U||; for example, ||∆U||/||∆U0|| is unity when the difference from the reference solution is of the same size as the initial perturbation. Similarly, the constraint energy norm ||C|| is divided by the norm of the first derivatives ||∂U|| (at the respective time), gijδabδcd(M−2∂iψac∂lψbd + ∂iΠac∂jΠbd +gkl∂iΦkac∂jΦlbd) . (A.4) The constraints for our system are linear combinations of the first derivatives of the fields, hence ||C||/||∂U|| ∼ 1 corresponds to a complete violation of the constraints. Appendix A.5. Wave extraction For evaluating gravitational waveforms, we compute the Newman-Penrose scalars Ψ0 = −Cabcdlamblcmd, Ψ4 = −Cabcdkam̄bkcm̄d, (A.5) where Cabcd is the Weyl tensor, la and ka are outgoing and ingoing null vectors normalized according to laka = −1, ma is a complex unit null spatial vector orthogonal to la and ka, and m̄a is the complex conjugate of ma. For perturbations of flat spacetime, there is a standard choice for the vectors la, ka, and ma. In general curved spacetimes, however, no such prescription for the tetrad exists that would produce coordinate-independent quantities Ψ0 and Ψ4 at finite radius. We choose the null vectors according to la = 1√ (ta + na) , ka = 1√ (ta − na) , (A.6) where ta is the future-pointing unit timelike normal to the t = const. slices and na is the unit spacelike normal to the extraction sphere. Finally, we choose sin θ , (A.7) where (r, θ, φ) are spherical coordinates on the r = Rex = const. extraction sphere. Note that our choice of ma is not exactly null nor of unit magnitude at finite extraction radius. However, the tetrad is orthonormal for the unperturbed Schwarzschild solution, so that the errors introduced into Ψ0 and Ψ4 because of the lack of tetrad orthonormality will be second-order small in perturbation theory. Testing outer boundary treatments for the Einstein equations 24 The quantity Ψ4 corresponds to outgoing radiation in the limit of r → ∞, t − r = const., i.e. as future null infinity is approached. Similarly Ψ0 corresponds to ingoing radiation as past null infinity is approached. At finite extraction radius, Ψ4 and Ψ0 will disagree with the waveforms observed at infinity by terms of the order O(Rex)−1. We decompose the quantities Ψ4 and Ψ0 in terms of spin-weighted spherical harmonics of spin-weight −2 on the extraction surface. Since our perturbation is an odd-parity quadrupole wave, the imaginary part of the (l = 2, m = 0) spherical harmonic is by far the dominant contribution to Ψ4, and we only display that mode in our plots. We normalize the curves in our graphs by the maximum (in time) value of |Ψ4| at the extraction radius Rex, which for Rex = 40M is max |Ψ4| = 6× 10−4. Appendix B. Details of the alternate approaches In this appendix, we provide some more details on the alternate boundary treatments discussed in section 4: spatial compactification and sponge layers. Appendix B.1. Spatial compactification We implement spatial compactification by introducing a radial coordinate transformation x → r(x) that maps a compact ball on the computational grid with x ∈ [0, xmax] to the full unbounded physical slice with r ∈ [0,∞]. We consider two such mappings. The Tan mapping is similar to the one used by Pretorius [15, 16, 17] and is given by rTan(x) = R tan , 0 6 x < 2R. (B.1) The scale R determines the range in physical radius r across which the map is essentially linear (see figure B1). When comparing compactification with other boundary treatments, we compare quantities only in the region r < R. (The scale R is equal to unity in the work of Pretorius. He uses mesh refinement to obtain the appropriate resolution close to the origin, while we fix the resolution and choose the scale R appropriately.) We also tested an Inverse map defined by rInverse(x) = x, 0 6 x 6 R , 2R− x, R < x < 2R, (B.2) see figure B1. This map is only C1 at x = R, but we maintain spectral accuracy in our tests by placing this surface at the boundary between spectral subdomains. Dissipation is needed to remove the short wavelength components of the waves as they travel outward on the compactified computational grid and become unresolved. We apply this dissipation only in the radial direction, but everywhere in the computational domain. In spectral methods, dissipation can be conveniently implemented in the form of a spectral filter. This filter is applied by multiplying each spectral expansion coefficient of index k by a function f(k). (See Appendix A.2 for details on the pseudospectral method we use.) Higher values of k correspond to shorter wavelengths in the numerical approximation; let kmax be the highest index used in the spectral expansion. The first filter function we consider is the closest analogue in the context of our spectral methods to Kreiss-Oliger [59] dissipation, fKreiss-Oliger(k) = 1− � sin4 2kmax , 0 6 � 6 1. (B.3) Testing outer boundary treatments for the Einstein equations 25 0 1 2 x / R INVERSE 0 0.5 1 k / k Filters 1 and 2 Filters 3 and 4 Figure B1. Compactification mappings (left) and filter functions (right). The dashed line indicates the boundary of the region in where the compactification mapping is (essentially) linear. Typical values of the parameter � used by Pretorius are � ∈ [0.2, 0.5]; we use � = 0.25. This filter was derived via a comparison with finite-difference methods as follows. In the finite-difference approach, a numerical solution u is represented on a set of equidistant grid points xj . (It suffices to consider the one-dimensional case here.) Some form of numerical dissipation is usually required for the finite-difference method to be stable. The one that is most often used for second-order accurate methods is fourth-order Kreiss-Oliger dissipation [59]. One possible implementation of this, used e.g. by Pretorius, amounts to replacing u→ F [u] ≡ u (B.4) at each time step, where h is the grid spacing and D4 is the second-order accurate centred finite difference operator approximating the fourth derivative, D4ui = h −4(uj−2 − 4uj−1 + 6uj − 4uj+1 + uj+2). (B.5) Taking u to be a Fourier mode u(k)j = exp(ikxj), it follows that the mode is damped by a frequency-dependent factor, u(k) → F [u(k)] ≡ 1− � sin4 2kmax u(k), (B.6) where kmax = π/(2h) is the Nyquist frequency. Thus we obtain the filter function (B.3). Strictly speaking, the above analysis only applies to Fourier expansions and not to the Chebyshev expansions we use. Nevertheless, we apply the filter in the form (B.3) to our Chebyshev expansion coefficients. Note that in (B.6), each spectral coefficient u(k) is filtered separately; this is not true for the analogous calculation for a Chebyshev expansion. We also use a different filter function, which we call the Exponential filter, that is often used in spectral methods (see [60] and references therein), fExponential(k) = exp σkmax . (B.7) Testing outer boundary treatments for the Einstein equations 26 No. Type Parameters Applied to 1 Kreiss-Oliger � = 0.25 right side 2 Kreiss-Oliger � = 0.25 solution 3 Exponential σ = 0.76, p = 13 right side 4 Exponential σ = 0.76, p = 13 solution Table B1. Details of the filtering methods Typical values of the parameters are σ = 0.76 and p = 13. This choice of parameters gives less dissipation at small values of k than the Kreiss-Oliger filter, and also ensures that f(kmax) ≈ 10−16 is at the level of the numerical roundoff error. There are various ways the filters can be applied in a numerical evolution. We have experimented with two different methods. In the first method, the filter is applied to the right side of the equations, i.e. the evolution equations ∂tu = S are modified according to ∂tu = F [S], where F [S] is the filtered right side. In the second method, the filter is instead applied to the solution itself, i.e. after each substep of the time integrator (cf. Appendix A.2), the numerical solution u is replaced with its filtered version F [u]. This second method is closest to how the Kreiss-Oliger filter is applied by Pretorius. For our numerical tests, we have used four different combinations of the various options described above. They are summarized in table B1. Appendix B.2. Sponge layers For sponge layers we must specify a sponge profile function γ(r), as defined in (34). We choose γ(r) to be nonzero only outside some sponge-free region of radius R, and when comparing sponge layers with other boundary treatments, we compare quantities only in the sponge-free region r < R. The sponge profile function γ(r) we use is a Gaussian centred at the outer boundary, which we choose to place at r = 3R, γ(r) = γ0 exp r − 3R . (B.8) The amplitude of the Gaussian is taken to be γ0 = 1. The width σ is chosen so that γ(r) 6 10−16 (the numerical roundoff error) for r 6 R, which requires σ . R/3. In our numerical example, we take R = 41.9M and σ = 13.3M . Hence σ is considerably larger than the wavelength λ ≈ 4M of the gravitational wave, which is required in order to avoid reflections from the sponge layer (cf. section 17.2.3 of [43]). Figure B2 shows a plot of this sponge profile. Acknowledgments We thank Luisa Buchman, Jan Hesthaven, Larry Kidder, Harald Pfeiffer, Olivier Sarbach, and Jeff Winicour for helpful discussions concerning this work. The numerical simulations presented here were performed using the Spectral Einstein Code (SpEC) developed at Caltech and Cornell primarily by Larry Kidder, Mark Scheel and Harald Pfeiffer. This work was supported in part by grants from the Sherman Fairchild Foundation, and from the Brinson Foundation; by NSF grants Testing outer boundary treatments for the Einstein equations 27 0 1 2 3 r / R Figure B2. The sponge profile function γ(r). The dashed line indicates the boundary of the region where γ is below the numerical roundoff error. PHY-0099568, PHY-0244906, PHY-0601459, DMS-0553302 and NASA grants NAG5- 12834, NNG05GG52G. References [1] Novak J and Bonazzola S 2004 Absorbing boundary conditions for simulation of gravitational waves with spectral methods in spherical coordinates J. Comput. Phys. 197 86–196 [2] Rinne O 2005 Axisymmetric Numerical Relativity Ph.D. thesis Univ. of Cambridge Preprint http://www.arxiv.org/abs/gr-qc/0601064 [3] Lau S L 2004 Rapid evaluation of radiation boundarz kernels for time-domain wave propagation on black holes: implementation and numerical tests Class. 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Math. 128 83–131 Introduction Constraint-preserving boundary conditions The generalized harmonic evolution system Construction of boundary conditions Improved gauge boundary condition Numerical results Alternate boundary conditions Freezing the incoming fields Sommerfeld boundary conditions Kreiss-Winicour boundary conditions Alternate approaches Spatial compactification Sponge layers Physical gravitational waves Difference of 4 with respect to the reference solution Comparison with the predicted reflection coefficient Discussion Details on the numerical test problem Initial data Numerical method Gauge source functions Error quantities Wave extraction Details of the alternate approaches Spatial compactification Sponge layers

0704.0783

Stability of a finite volume scheme for the incompressible fluids

arXiv:0704.0783v2 [math.NA] 20 Nov 2007 Mathematical Modelling and Numerical Analysis Will be set by the publisher Modélisation Mathématique et Analyse Numérique STABILITY OF A FINITE VOLUME SCHEME FOR THE INCOMPRESSIBLE FLUIDS Sébastien Zimmermann1 Abstract. We introduce a finite volume scheme for the two-dimensional incompressible Navier-Stokes equations. We use a triangular mesh. The unknowns for the velocity and pressure are respectively piecewise constant and affine. We use a projection method to deal with the incompressibility constraint. We show that the differential operators in the Navier-Stokes equations and their discrete counterparts share similar properties. In particular we state an inf-sup (Babuška-Brezzi) condition. Using these properties we infer the stability of the scheme. Résumé. Nous introduisons ici un schéma volumes finis pour les équations de Navier-Stokes in- compressibles en deux dimensions. Les maillages considérés sont formés de triangles. Les inconnues associées à la vitesse et la pression sont respectivement constantes et affines par morceaux. Nous utilisons une méthode de projection pour traiter la contrainte d’incompressibilité. Nous vérifions que les opérateurs différentiels apparaissant dans les équations de Navier-Stokes et leurs analogues dis- crets vérifient des propriétés similaires. Nous prouvons en particulier une condition inf-sup. Nous en déduisons la stabilité du schéma. 1991 Mathematics Subject Classification. 76D05, 74S10, 65M12. Received: 13 august 2007. 1. Introduction We consider the flow of an incompressible fluid in a polyhedral set Ω ⊂ R2 during the time interval [0, T ]. The velocity field u : Ω× [0, T ] → R2 and the pressure field p : Ω× [0, T ] → R satisfy the Navier-Stokes equations ∆u+ (u · ∇)u+∇p = f , (1.1) div u = 0 , (1.2) with the boundary and initial conditions u|∂Ω = 0 , u|t=0 = u0. The terms ∆u and (u·∇)u are associated with the physical phenomena of diffusion and convection, respectively. The Reynolds number Re measures the influence of convection in the flow. For equations (1.1)–(1.2), finite element and finite difference methods are well known and mathematical studies are available (see [9] for example). Keywords and phrases: Incompressible fluids, Navier-Stokes equations, projection methods, finite volume. 1 17 rue Barrème - 69006 LYON. e-mail: [email protected] c© EDP Sciences, SMAI 1999 http://arxiv.org/abs/0704.0783v2 2 TITLE WILL BE SET BY THE PUBLISHER For finite volume schemes, numerous computations have been conducted ( [12] and [1] for example). However, few mathematical results are available in this case. Let us cite Eymard and Herbin [6] and Eymard, Latché and Herbin [7]. In order to deal with the incompressibility constraint (1.2), these works use a penalization method. Another way is to use the projection methods which have been introduced by Chorin [4] and Temam [13]. This is the case in Faure [8] where the mesh is made of squares. In Zimmermann [14] the mesh is made of triangles, so that more complex geometries can be considered. In the present paper the mesh is also made of triangles, but we consider a different discretization for the pressure. It leads to a linear system with a better-conditioned matrix. The layout of the article is the following. We first introduce in section 2 the discrete setting. We state (section 2.1) some notations and hypotheses on the mesh. We define (section 2.2) the spaces we use to approximate the velocity and pressure. We define also (section 2.3) the operators we use to approximate the differential operators in (1.1)–(1.2). Combining this with a projection method, we build the scheme in section 3. In order to provide a mathematical analysis, we show in section 4 that the differential operators in (1.1)–(1.2) and their discrete counterparts share similar properties. In particular, the discrete operators for the gradient and the divergence are adjoint. The discrete operator for the convection term is positive, stable and consistent. The discrete operator for the divergence satisfy an inf-sup (Babuška-Brezzi) condition. From these properties we deduce in section 5 the stability of the scheme. We conclude with some notations. The spaces (L2, |.|) and (L∞, ‖.‖∞) are the usual Lebesgue spaces and we set L20 = {q ∈ L q(x) dx = 0}. Their vectorial counterparts are (L2, |.|) and (L∞, ‖.‖∞) with L 2 = (L2)2 and L∞ = (L∞). For k ∈ N∗, (Hk, ‖ ·‖k) is the usual Sobolev space. Its vectorial counterpart is (H k, ‖.‖k) with Hk = (Hk)2. For k = 1, the functions of H1 with a null trace on the boundary form the space H10. Also, we set ∇u = (∇u1,∇u2) T if u = (u1, u2) ∈ H 1. If X ⊂ L2 is a Banach space, we define C(0, T ;X) (resp. L2(0, T ;X)) as the set of the applications g : [0, T ] → X such that t → |g(t)| is continuous (resp. square integrable). The norm ‖.‖C(0,T ;X) is defined by ‖g‖C(0,T ;X) = sups∈[0,T ] |g(s)|. In all calculations, C is a generic positive constant, depending only on Ω, u0 and f . 2. Discrete setting First, we introduce the spaces and the operators needed to build the scheme. 2.1. The mesh Let Th be a triangular mesh of Ω. The circumscribed circle of a triangle K ∈ Th is centered at xK and has the diameter hK . We set h = maxK∈Th hK . We assume that all the interior angles of the triangles of the mesh are less than π , so that xK ∈ K. The set of the edges of the triangle K ∈ Th is EK . The symbol nK,σ denotes the unit vector normal to an edge σ ∈ EK and pointing outward K. We denote by Eh the set of the edges of the mesh. We distinguish the subset E inth ⊂ Eh (resp. E h ) of the edges located inside Ω (resp. on ∂Ω). The middle of an edge σ ∈ Eh is xσ and its length |σ|. For each edge σ ∈ E h , let Kσ and Lσ be the two triangles having σ in common. We set dσ = d(xKσ ,xLσ ). For all σ ∈ E h , only the triangle Kσ located inside Ω is defined and we set dσ = d(xKσ ,xσ). Then for all σ ∈ Eh we set τσ = . As in [5] we assume the following on the mesh: there exists C > 0 such that ∀σ ∈ Eh , dσ ≥ C |σ| and |σ| ≥ C h. It implies that there exists C > 0 such that ∀σ ∈ E inth , τσ = |σ|/dσ ≥ C. (2.1) 2.2. The discrete spaces We first define P0 = {q ∈ L 2 ; ∀K ∈ Th, q|K is a constant} , P0 = (P0) TITLE WILL BE SET BY THE PUBLISHER 3 For the sake of concision, we set for all qh ∈ P0 (resp. vh ∈ P0) and all triangle K ∈ Th: qK = qh|K (resp. vK = vh|K). Although P0 6⊂ H 1, we define the discrete equivalent of a H1 norm as follows. For all vh ∈ P0 we set ‖vh‖h = σ∈Eint τσ |vLσ − vKσ | σ∈Eext τσ |vKσ | . (2.2) We have [5] a Poincaré-like inequality: there exists C > 0 such that for all vh ∈ P0 |vh| ≤ C ‖vh‖h. (2.3) We also have [14] an inverse inequality: there exists C > 0 such that for all vh ∈ P0 h ‖vh‖h ≤ C |vh|. (2.4) From the norm ‖.‖h we deduce a dual norm. For all vh ∈ P0 we set ‖vh‖−1,h = sup (vh,ψh) ‖ψh‖h . (2.5) For all uh ∈ P0 and vh ∈ P0 we have (uh,vh) ≤ ‖uh‖−1,h ‖vh‖h. We define the projection operator ΠP0 : L2 → P0 as follows. For all w ∈ L 2, ΠP0w ∈ P0 is given by ∀K ∈ Th , (ΠP0w)|K = w(x) dx. (2.6) We easily check that for all w ∈ L2 and vh ∈ P0 we have (ΠP0w,vh) = (w,vh). We deduce from this that ΠP0 is stable for the L2 norm. We define also the operator Π̃P0 : H 2 → P0. For all w ∈ H 2, Π̃P0w ∈ P0 is given by ∀K ∈ Th , Π̃P0w|K = w(xK). According to the Sobolev embedding theorem, w ∈ H2 is a.e. equal to a continuous function. Therefore the definition above makes sense. We introduce also the finite element spaces P d1 = {v ∈ L 2 ; ∀K ∈ Th, v|K is affine} , Pnc1 = {vh ∈ P 1 ; ∀σ ∈ E h , vh|Kσ (xσ) = vh|Lσ(xσ) , Pc1 = {vh ∈ (P 2 ; vh is continuous and vh|∂Ω = 0}. We have Pc1 ⊂ H 0. We define ΠPc1 : H 0 → P 1. For all v = (v1, v2) ∈ H 0, ΠPc1v = (v h) ∈ P 1 is given by ∀φh = (φ h) ∈ P ∇vih,∇φ ∇vi,∇φ The operator ΠPc is stable for the H1 norm. One checks ( [2] p. 110) that there exists C > 0 such that for all v ∈ H1 |v −ΠPc v| ≤ C h ‖v‖1. (2.7) Let us address now the space Pnc1 . If qh ∈ P 1 , we have usually ∇qh 6∈ L 2. Thus we define the operator ∇h : P 1 → P0 by setting for all qh ∈ P0 and all triangle K ∈ Th ∇hqh|K = ∇qh dx. (2.8) 4 TITLE WILL BE SET BY THE PUBLISHER The associated norm is defined by ‖qh‖1,h = 2 + |∇hqh| We have a Poincaré-like inequality : there exists C > 0 such that for all qh ∈ P 1 ∩ L |qh| ≤ C |∇hqh|. (2.9) We define the projection operator ΠPnc . For all q ∈ H1, ΠPnc q is given by ∀σ ∈ Eh , (ΠPnc q) dσ = q dσ. One checks ( [2] p.110) that there exists C > 0 such that |p−ΠPnc p| ≤ C h ‖p‖1 , ∣∣∣∇̃h(p−ΠPnc ∣∣∣ ≤ C ‖p‖1. (2.10) Finally, we use the Raviart-Thomas spaces (see [3]) = {vh ∈ P 1 ; ∀σ ∈ EK , vh|K · nK,σ is a constant, and vh · n|∂Ω = 0} , RT0 = {vh ∈ RT ; ∀K ∈ Th, ∀σ ∈ EK , vh|Kσ · nKσ ,σ = vh|Lσ · nKσ ,σ}. For all vh ∈ RT0, K ∈ Th and σ ∈ EK we set (vh ·nK,σ)σ = vh|K ·nK,σ. We define the operator ΠRT0 : H RT0. For all v ∈ H 1, ΠRT0v ∈ RT0 is given by ∀K ∈ Th , ∀σ ∈ EK , (ΠRT0v · nK,σ)σ = v dσ. (2.11) 2.3. The discrete operators The equations (1.1)–(1.2) use the differential operators gradient, divergence and laplacian. Using the spaces of section 2.2, we define their discrete counterparts. The discrete gradient ∇h : P 1 → P0 is defined by (2.8). The discrete divergence operator divh : P0 → P 1 is built so that it is adjoint to the operator ∇h. We set for all vh ∈ P0 and all triangle K ∈ Th ∀σ ∈ E inth , (divh vh)(xσ) = 3 |σ| |Kσ|+ |Lσ| (vLσ − vKσ ) · nK,σ ; ∀σ ∈ Eexth , (divh vh)(xσ) = − 3 |σ| |Kσ|+ |Lσ| vKσ · nK,σ. (2.12) The first discrete laplacian ∆h : P 1 → P 1 ensures that the incompressibility constraint (1.2) is satisfied in a discrete sense (see the proof of proposition 3.1 below). We set for all qh ∈ P ∆hqh = divh(∇hqh). The second discrete laplacian ∆̃h : P0 → P0 is the usual operator in finite volume schemes [5]. We set for all vh ∈ P0 and all triangle K ∈ Th ∆̃hvh|K = σ∈EK∩E τσ (vLσ − vKσ )− σ∈EK∩E τσ vKσ . TITLE WILL BE SET BY THE PUBLISHER 5 In order to approximate the term (u · ∇)u in (1.1) we define a bilinear form b̃h : RT0 × P0 → P0 using the well-known upwind scheme [5]. For all uh ∈ P0, vh ∈ P0, and all triangle K ∈ Th we set b̃h(uh,vh) σ∈EK∩E (u · nK,σ) σ vK + (u · nK,σ) σ vLσ . (2.13) We have set a+ = max(a, 0), a− = min(a, 0) for all a ∈ R. Lastly, we define the trilinear form bh : RT0 ×P0 × P0 → R 2 as follows. For all uh ∈ RT0, vh ∈ P0, wh ∈ P0, we set bh(uh,vh,wh) = |K|wK · b̃h(uh,vh) . (2.14) 3. The scheme In order to deal with the incompressibility constraint (1.2) we use a projection method. This kind of method has been introduced by Chorin [4] and Temam [13]. The basic idea is the following. The time interval [0, T ] is split with a time step k: [0, T ] = n=0[tn, tn+1] with N ∈ N ∗ and tn = n k for all n ∈ {0, . . . , N}. For all m ∈ {2, . . . , N}, we compute (see equation (3.2) below) a first velocity field ũmh ≃ u(tm) using only equation (1.1). We use a second-order BDF scheme for the discretization in time. We then project ũmh (see equation (3.4) below) over a subspace of P0. We get a a pressure field p h ≃ p(tm) and a second velocity field u h ≃ u(tm), which fulfills the incompressibility constraint (1.2) in a discrete sense. The algorithm goes as follows. For all m ∈ {0, . . . , N}, we set fmh = ΠP0 f(tm). Since the operator ΠP0 is stable for the L 2-norm we get |fmh | = |ΠP0 f(tm)| ≤ |f(tm)| ≤ ‖f‖C(0,T ;L2). (3.1) We start with the initial values u0h ∈ P0 ∩RT0 , u h ∈ P0 ∩RT0 p h ∈ P0 ∩ L For all n ∈ {1, . . . , N}, (ũn+1h , p h ) is deduced from (ũ h , p h) as follows. • ũn+1h ∈ P0 is given by 3 ũn+1h − 4u h + u ∆̃hũ h + b̃h(2u h − u h , ũ h ) +∇hp h = f h , (3.2) • pn+1h ∈ P 1 ∩ L 0 is the solution of h − p divh ũ h , (3.3) • un+1h ∈ P0 is deduced by un+1h = ũ h − p h). (3.4) Existence and unicity of a solution to equation (3.2) is classical ( [5] for example). The convection term in (3.2) is well defined thanks to the following result. Proposition 3.1. For all m ∈ {0, . . . , N} we have umh ∈ RT0 . Proof. If m ∈ {0, 1} the result holds by definition. If m ∈ {2, . . . , N} we apply the operator divh to (3.3) and compare with (3.4). We get divh u h = 0. Using definition (2.12) we get u h ∈ RT0. 6 TITLE WILL BE SET BY THE PUBLISHER Let us show that equation (3.3) also has a unique solution. Let qh ∈ P 1 ∩ L 0 such that ∆hqh = 0. According to proposition 4.4 we have for all qh ∈ P0 −(∆hqh, qh) = − divh(∇hqh), qh = (∇hqh,∇hqh) = |∇hqh| Therefore we have ∇hqh = 0, so that qh = 0 since qh ∈ L 0. We have thus proved the unicity of a solution for (3.3). It is also the case for the associated linear system. It implies that this linear system has indeed a solution. Hence it is also the case for equation (3.3). Note finally that since umh ∈ P0 ∩RT0, we have divu h = 0 for all m ∈ {0, . . . , N}. Hence the incompressibility condition (1.2) is fulfilled. 4. Properties of the discrete operators We show that the differential operators in (1.1)–(1.2) and the operators defined in section 2.3 share similar properties. 4.1. Properties of the discrete convective term We define b̃ : H1 ×H1 → L2. For all u ∈ H1 and v = (v1, v2) ∈ H 1 we set b̃(u,v) = div(v1 u), div(v2 u) We show that the operator b̃h is a consistent approximation of b̃. Proposition 4.1. There exists a constant C > 0 such that for all v ∈ H2 and all u ∈ H2 ∩ H10 satisfying divu = 0 ‖ΠP0 b̃(u,v)− b̃h(ΠRT0u, Π̃P0v)‖−1,h ≤ C h ‖u‖2 ‖v‖1. Proof. We set uh = ΠRT0u and vh = Π̃P0v. Let K ∈ Th. According to the divergence formula and (2.6) we ΠP0 b̃(u,v)|K = σ∈EK∩E v (u · n) dσ. On the other hand, let us rewrite b̃h(uh,vh). Let σ ∈ EK ∩ E h . Setting vK,Lσ = vK si (uh · nK,σ)σ ≥ 0 vLσ si (uh · nK,σ)σ < 0 one checks that vK (uh · nK,σ) σ + vLσ (uh · nK,σ) σ = vK,Lσ (uh · nK,σ)σ. Using (2.11), we deduce from (2.13) b̃h(uh,vh)|K = σ∈EK∩E vK,Lσ (uh · nK,σ) dσ. ΠP0 b̃(u,v)− b̃h(uh,vh) σ∈EK∩E (v − vK,Lσ) (uh · n) dσ. Let ψh ∈ P0. We have ΠP0 b̃(u,v) − b̃h(uh,vh),ψh σ∈EK∩E (v − vK,Lσ) (uh · n) dσ σ∈Eint (ψKσ −ψLσ) (v − vKσ ,Lσ) (uh · n) dσ. TITLE WILL BE SET BY THE PUBLISHER 7 Let σ ∈ E inth . We consider the quadrilateral Dσ defined by xKσ , xLσ and the vertex of σ. We set DK,Lσ = Dσ ∩K si (uh · nK,σ)σ ≥ 0 Dσ ∩ Lσ si (uh · nK,σ)σ < 0 Using a Taylor expansion and a density argument (see [14]) one checks that |v − vKσ ,Lσ | dσ ≤ C h DKσ,Lσ |∇v (y)|2 dy ΠP0 b̃(u,v) − b̃h(ΠRT0u, Π̃P0v),ψh ≤ C h ‖u‖H2 σ∈Eint |ψLσ −ψKσ | σ∈Eint DKσ,Lσ |∇v (y)|2 dy so that ΠP0b̃(u,v)− b̃h(ΠRT0u, Π̃P0v),ψh )∣∣∣ ≤ C h ‖u‖H2 ‖ψh‖1,h ‖v‖1. Using then definition (2.5), we get the result. Let v ∈ L∞ ∩H1 and u ∈ H1 with divu ≥ 0 a.e. in Ω. Integrating by parts one checks that v · b̃(u,v) dx = divu dx ≥ 0. The operator bh shares a similar property. Proposition 4.2. Let uh ∈ RT0 such that divuh ≥ 0. For all vh ∈ P0 we have bh(uh,vh,vh) ≥ 0. Proof. Remember that for all edges σ ∈ E inth , two triangles Kσ et Lσ share σ as an edge. We denote by Kσ the one such that uσ · nKσ ,σ ≥ 0. Using the algebraic identity 2 a (a− b) = a 2 − b2 + (a − b)2 we deduce from (2.14) 2 bh(uh,vh,vh) = 2 σ∈Eint |σ|vKσ · (vKσ − vLσ) (uh · nKσ,σ) σ∈Eint |vKσ| 2 − |vLσ | 2 + |vKσ − vLσ | (uh · nKσ,σ) so that 2 bh(uh,vh,vh) ≥ σ∈Eint |vKσ| 2 − |vLσ | (uh · nKσ,σ). This sum can be written as a sum over the triangles of the mesh. We get 2 bh(uh,vh,vh) ≥ |vKσ | σ∈EK∩E |σ| (uh · nKσ,σ). Using finally the divergence formula we get 2 bh(uh,vh,vh) ≥ |K| |vK | divuh dx ≥ 0. The following result states that the operator bh is stable for suitable norms. 8 TITLE WILL BE SET BY THE PUBLISHER Proposition 4.3. There exists a constant C > 0 such that for all vh ∈ P0, wh ∈ P0, uh ∈ P0 satisfying divuh = 0 |bh(uh,vh,vh)| ≤ C |uh| ‖vh‖h ‖vh‖h. Proof. For all triangle K ∈ Th and all edge σ ∈ EK ∩ E h , we have (uh · nK,σ) σ vK + (uh · nK,σ) σ vLσ = (uh · nK,σ)σ vK − |(uh · nK,σ)σ| (vLσ − vK). Using this splitting, we deduce from (2.14) bh(uh,vh,wh) = S1 + S2 with vK ·wK σ∈EK∩E |σ| (uh · nK,σ)σ , S2 = − σ∈EK∩E |σ| |(uh · nK,σ)σ| (vLσ − vK). By writing the sum over the edges as a sum over the triangles we have S2 = − σ∈Eint |σ| |(uh · nK,σ)σ| (vLσ − vK) · (wLσ −wK). Using the Cauchy-Schwarz inequality we get |S2| ≤ h ‖uh‖∞ σ∈Eint |vLσ − vKσ | 1/2  σ∈Eint |wLσ −wKσ | Since uh ∈ RT0 we have [5] the inverse inequality h ‖uh‖∞ ≤ C |uh|. Using (2.1) and (2.2) we get σ∈Eint |vLσ − vKσ | 2 ≤ C σ∈Eint τσ |vLσ − vKσ | 2 ≤ C ‖vh‖ and in a similar way σ∈Eint |wLσ −wKσ | 2 ≤ C ‖wh‖ h. Thus |S2| ≤ C |uh| ‖vh‖h ‖wh‖h. On the other hand, according to the divergence formula |K| (vK ·wK) divuh dx = 0. By gathering the estimates for S1 and S2 we get the result. 4.2. Properties of the discrete divergence The operators gradient and divergence are adjoint: if q ∈ H1 , v ∈ H1 with v · n|∂Ω = 0, we get (v,∇q) = −(q, divv) by integrating by parts. For ∇h and divh we state the following. Proposition 4.4. For all vh ∈ P0 and qh ∈ P 1 we have: (vh,∇hqh) = −(qh, divh vh). Proof. According to (2.8) (vh,∇hqh) = |K|vK · ∇hqh|K = |σ|qh(xσ)nK,σ TITLE WILL BE SET BY THE PUBLISHER 9 By writing this sum as a sum over the edges we get (vh,∇hqh) = − σ∈Eint |σ| qh(xσ) (vLσ − vKσ) · nKσ,σ + σ∈Eext |σ| qh(xσ)vKσ · nKσ,σ. (4.1) On the other hand, using a quadrature formula −(qh, divh vh) = − qh(xσ) (divh vh)(xσ). By writing this sum as a sum over the edges of the mesh we get −(qh, divh vh) = − σ∈Eint ( |Kσ| qh(xσ) (divhvh)(xσ)− σ∈Eext qh(xσ) (divh vh)(xσ). Using definition (2.12) and comparing with (4.1) we get the result. The divergence operator and the spaces L20, H 0 satisfy the following property, called inf-sup (or Babuška-Brezzi) condition (see [9] for example). There exists a constant C > 0 such that (q, divv) ‖v‖1|q| ≥ C. (4.2) We will now show that the operator divh and the spaces P0 ∩ L 0, P0 satisfy an analogous property. The proof uses the following lemma. Lemma 4.1. There exists a constant C > 0 such that ∀ qh ∈ P 1 ∩ L 0 , sup vh∈P0\{0} (qh, divh vh) ‖vh‖h ≥ C h ‖qh‖1,h. Proof. If qh = 0 the result is trivial. Let qh ∈ P 0\{0}. Let vh = ∇hqh ∈ P0\{0}. Using proposition 4.4 we have −(qh, divhvh) = (vh,∇hqh) = |∇hqh| 2 = |∇hqh| |vh|. Using (2.3) and (2.4) we get −(qh, divhvh) ≥ C h ‖qh‖1,h ‖vh‖h. We now state the result. Proposition 4.5. There exists a constant C > 0 such that for all qh ∈ P 1 ∩ L vh∈P0\{0} (qh, divh vh) ‖vh‖h ≥ C |qh|. Proof. If qh = 0 the result is trivial. Let qh ∈ P 0\{0}. According to (4.2) there exists v ∈ H 0 such that divv = −qh and ‖v‖1 ≤ C |qh|. (4.3) We set vh = ΠPc v. We want to estimate − qh, divh(ΠP0vh) . Since ∇hqh ∈ P0 we deduce from proposition qh, divh(ΠP0vh) = (ΠP0vh,∇hqh) = (vh,∇hqh). By splitting the last term we get qh, divh(ΠP0vh) = (v,∇hqh)− (v − vh,∇hqh). (4.4) 10 TITLE WILL BE SET BY THE PUBLISHER We bound the right-hand side of (4.4). Using (2.7) and (4.3) we have |v − vh| = |v −ΠPc v| ≤ C h ‖v‖1 ≤ C h |qh|. Thus, using the Cauchy-Schwarz inequality, we get |(v − vh,∇hqh)| ≤ C h |qh| |∇hqh| ≤ C h |qh| ‖qh‖1,h. We estimate the other term as follows. Integrating by parts we get (v,∇hqh) = −(qh, divv) + qh (v · nK,σ) dσ. We have −(qh, divv) = |qh| 2 thanks to (4.3). On the other hand qh (v · nK,σ) dσ = σ∈Eint qh (v · nKσ,σ) dσ since v|∂Ω = 0. Using [2] p.269 and (4.3) we have ∣∣∣∣∣ qh (v · nK,σ) dσ ∣∣∣∣∣ ≤ C h ‖v‖1 ‖qh‖1,h ≤ C h |qh| ‖qh‖1,h. Hence we get (v,∇hqh) ≥ (|qh| − C h ‖qh‖1,h) |qh|. Thus we deduce from (4.4) qh, divh(ΠP0vh) ≥ (|qh| − C h ‖qh‖1,h) |qh|. (4.5) We now introduce the norm ‖.‖h. We have vh = ΠPc v ∈ Pc1 ⊂ H 1. From [5] p. 776 we deduce ‖ΠP0vh‖h ≤ C ‖vh‖1. Since ΠPc is stable for the H1 norm, using (4.3), we get ‖vh‖1 = ‖ΠPc v‖1 ≤ ‖v‖1 ≤ C |qh|. Therefore ‖ΠP0vh‖h ≤ C |qh|. Using this inequality in (4.5) we obtain that there exists C1 > 0 and C2 > 0 such that qh, divh(ΠP0vh) ≥ (C1 |qh| − C2 h ‖qh‖1,h) ‖ΠP0vh‖h. We deduce from this vh∈P0\{0} (qh, divh vh) ‖vh‖h ≥ C1 |qh| − C2 h ‖qh‖1,h. Let us combine this result with lemma 4.1. Since ∀ t ≥ 0 , max C t , C1 |qh| − C2 t C + C2 |qh| , we finally get the result. 4.3. Properties of the discrete laplacian We recall from [14] the coercivity of the laplacian operator. Proposition 4.6. For all uh ∈ P0 and vh ∈ P0 we have −(∆̃huh,uh) = ‖uh‖ h , −(∆̃huh,vh) ≤ ‖uh‖h ‖vh‖h. TITLE WILL BE SET BY THE PUBLISHER 11 5. Stability of the scheme We first prove an estimate for the computed velocity (theorem 5.1). We show a similar result for the increments in time (lemma 5.2). Using the inf-sup condition (proposition 4.5), we infer from it some estimates on the pressure (theorem 5.2). Lemma 5.1. For all m ∈ {0, . . . , N} et n ∈ {0, . . . , N} we have (umh ,∇hp h) = 0 , |u 2 − |ũmh | 2 + |umh − ũ 2 = 0. Proof. First, using propositions 3.1 and 4.4, we get (umh ,∇hp h) = −(p h, divhu h ) = 0. Also, we deduce from (3.4) 2 (umh ,u h − ũ h ) = − umh ,∇h(p h − p Using the algebraic identity 2 a (a− b) = a2 − b2 + (a− b)2 we get 2 (umh ,u h − ũ h ) = |u 2 − |ũmh | 2 + |umh − ũ 2 = 0. We introduce the following hypothesis on the initial data. (H1) There exists C > 0 such that |u0h|+ |u h|+ k|∇hp h| ≤ C. Hypothesis (H1) is fulfilled if we set u0h = ΠRT0u0 and we use a semi-implicit Euler scheme to compute u We have the following stability result. Theorem 5.1. We assume that the initial values of the scheme fulfill (H1). For all m ∈ {2, . . . , N} we have |umh | 2 + k ‖ũnh‖ h ≤ C. (5.1) Proof. Let m ∈ {2, . . . , N} and n ∈ {1, . . . ,m− 1}. Taking the scalar product of (3.2) with 4 k ũn+1h we get 3 ũn+1h − 4u h + u , 4 k ũn+1h (∆̃hũ h , ũ +4 k bh(2u h − u h , ũ h , ũ h ) + 4 k (∇hp h, ũ h ) = 4 k (f h , ũ h ). (5.2) First of all, using lemma 5.1 and proceeding as in [10], we get ũn+1h , 3 ũn+1h − 4u h + u = |un+1h | 2 − |unh| 2 + |2un+1h − u 2 − |2unh − u + |un+1h − 2u h + u 2 + 6 |ũn+1h − u According to proposition 4.6 we have − 4 k (∆̃hũ h , ũ h ) = ‖ũn+1h ‖ h. Also, according to lemma 5.1 and (3.4) 4 k (∇hp h, ũ h ) = 4 k (∇hp h, ũ h − u (|∇pn+1h | 2 − |∇pnh| 2 − |∇pn+1h −∇p 12 TITLE WILL BE SET BY THE PUBLISHER Multiplying equation (3.4) by 4 k∇h(p h − p h) and using the Young inequality we get |∇(pn+1h − p 2 ≤ 3 |un+1h − ũ According to proposition 4.2, we have 4 k bh(2u h − u h , ũ h , ũ h ) ≥ 0. At last using the Cauchy-Schwarz inequality, (2.3) and (3.1) we have 4 k (fn+1h , ũ h ) ≤ 4 k |f h | |ũ h | ≤ C k ‖f‖C(0,T ;L2) ‖ũ h ‖h. Using the Young inequality we get 4 k (fn+1h , ũ h ) ≤ 3 k ‖ũ h + C k ‖f‖ C(0,T ;L2). Thus we deduce from (5.2) |un+1h | 2 − |unh | 2 + |2un+1h − u 2 − |2unh − u 2 + |un+1h − 2u h + u +3 |ũn+1h − u 2 + k ‖ũn+1h ‖ (|∇hp 2 − |∇hp 2) ≤ C k. Summing from n = 1 to m− 1 we have |umh | 2 + |2umh − u 2 + 3 |ũn+1h − u 2 + k ‖ũn+1h ‖ ≤ C + 4 |u1h| 2 + |2u1h − u 2 + k2 |∇hp Using hypothesis (H1) we get (5.1). We now want to estimate the computed pressure. From now on, we make the following hypothesis on the data f ∈ C(0, T ;L2) , ft ∈ L 2(0, T ;L2) , u0 ∈ H 2 ∩H10 , divu0 = 0. One shows that if the data u0 and f fulfill a compatibility condition [11] there exists a solution (u, p) to the equations (1.1)–(1.2) such that u ∈ C(0, T ;H2) , ut ∈ C(0, T ;L 2) , ∇p ∈ C(0, T ;L2). We introduce the following hypothesis on the initial values of the scheme: there exists a constant C > 0 such (H2) |u0h − u0|+ ‖u1h − u(t1)‖∞ + |p h − p(t1)| ≤ C h , |u h − u h| ≤ C k. One checks easily that this hypothesis implies (H1). We have the following result. Lemma 5.2. We assume that the initial values of the scheme fulfill (H2). Then there exists a constant C > 0 such that for all m ∈ {1, . . . , N} |umh − u h | ≤ C. Proof. Using proposition 4.1 one proceeds as in [14]. The difference lies in the way we bound the term ∇hp We use the splitting p1h = (p h −ΠPnc1 p(t1)) + (ΠP p(t1)− p(t1)) + p(t1). TITLE WILL BE SET BY THE PUBLISHER 13 Using an inverse inequality [2] we have p1h −ΠPnc1 p(t1) )∣∣ ≤ ∣∣p1h −ΠPnc1 p(t1) (∣∣p1h − p(t1) ∣∣p(t1)−ΠPnc p(t1) ∣∣) . Using (2.10) and hypothesis (H2) we get p1h −ΠPnc1 p(t1) )∣∣ ≤ C ‖p(t1)‖1 ≤ C ‖p‖C(0,T ;H1). According to (2.10) we also have ∣∣∇h(p(t1)−ΠPnc p(t1)) ∣∣ ≤ C ‖p(t1)‖1 ≤ C ‖p‖C(0,T ;H1). Lastly |∇p(t1)| ≤ ‖p‖C(0,T ;H1). Thus we get |∇hp h| ≤ C. Theorem 5.2. We assume that the initial values of the scheme fulfull (H2). There exists a constant C > 0 such that for all m ∈ {2, . . . , N} |pnh| 2 ≤ C. Proof. Let m ∈ {2, . . . , N}. We set n = m− 1. Using the inf-sup condition (4.5) and proposition 4.4, we get that there exists vh ∈ P0\{0} such that C ‖vh‖h |p h | ≤ −(p h , divh vh) = (∇hp h ,vh). (5.3) Plugging (3.4) into (3.2) we have h = − 3un+1h − 4u h + u ∆̃hũ h − b̃h(2u h − u h , ũ h ) + f so that h ,vh) = − 3un+1h − 4u h + u ∆̃hũ h ,vh − bh(2u h − u h , ũ h ,vh) + (f h ,vh). Thanks to proposition 4.3 and theorem 5.1 we have ∣∣bh(2unh − u h , ũ h ,vh) 2 |unh|+ |u ‖ũn+1h ‖h ‖vh‖h ≤ C ‖ũ h ‖h ‖vh‖h. According to proposition 4.6 we have ∆̃hũ h ,vh ≤ ‖ũn+1h ‖h ‖vh‖h. Using the Cauchy-Schwarz inequality, (2.3) and (3.1) we have (fn+1h ,vh) ≤ |f h | |vh| ≤ C |vh| ≤ C ‖vh‖h and in a similar way 3un+1h − 4u h + u )∣∣∣∣ ≤ C 3un+1h − 4u h + u ∣∣∣∣ ‖vh‖h. Thus we get h ,vh) ≤ C + C |3un+1h − 4u h + u + ‖ũn+1h ‖h ‖vh‖h. By comparing with (5.3) we get |pn+1h | ≤ C + C |3un+1h − 4u h + u + ‖ũn+1h ‖h 14 TITLE WILL BE SET BY THE PUBLISHER Squaring and summing from n = 1 to m− 1 we obtain |pnh| 2 ≤ C + C k |3un+1h − 4u h + u + C k ‖ũn+1h ‖ The last term on the right-hand side is bounded, thanks to theorem 5.1. And since 3un+1h − 4u h + u h = 3(u h − u h)− (u h − u h ) = 3 δu h − δu we deduce from lemma 5.2 |3un+1h − 4u h + u ≤ C k |δunh | References [1] S. Boivin, F. Cayre, J. M Herard, A finite volume method to solve the Navier-Stokes equations for incompressible flows on unstructured meshes, Int. J. Therm. Sci. 39 (2000) 806–825. [2] S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Springer, 2002. [3] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer-Verlag, 1991. [4] J. Chorin, On the convergence of discrete approximations to the Navier-Stokes equations, Math. Comp. 23 (1969) 341–353. [5] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, P.G. Ciarlet and J.L. Lions eds, North-Holland, 2000. [6] R. Eymard and R. Herbin, A staggered finite volume scheme on general meshes for the Navier-Stokes equations in two space dimensions, Int.J. Finite Volumes (2005). [7] R. Eymard, J. C. Latché and R. Herbin, Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2 or 3D meshes, SIAM J. Numer. Anal. 45(1) (2007) 1–36. [8] S. Faure, Stability of a colocated finite volume scheme for the Navier-Stokes equations, Num. Meth. PDE 21(2) (2005) 242–271. [9] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes equations: Theory and Algorithms, Springer, 1986. [10] J. L. Guermond, Some implementations of projection methods for Navier-Stokes equations, M2AN 30(5) (1996) 637–667. [11] J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19(26) (1982) 275–311. [12] D. Kim and H. Choi, A second-order time-accurate finite volume method for unsteady incompressible flow on hybrid unstruc- tured grids, J. Comp. Phys. 162 (2000) 411–428. [13] R. Temam, Sur l’approximation de la solution des équations de Navier-Stokes par la méthode de pas fractionnaires II, Arch. Rat. Mech. Anal. 33 (1969) 377–385. [14] S. Zimmermann, Stability of a colocated finite volume for the incompressible Navier-Stokes equations, arXiv :0704.0772 (2006). [15] S. Zimmermann, Étude et implémentation de méthodes de volumes finis pour les fluides incompressibles, PhD, Blaise Pascal University, France (2006).

0704.0784

D-brane Instantons on the T^6/Z_3 orientifold

arXiv:0704.0784v2 [hep-th] 31 May 2007 ROM2F/2007/09 D-brane Instantons on the T 6/Z3 orientifold Massimo Bianchi, Francesco Fucito and Jose F. Morales Dipartimento di Fisica, Universitá di Roma “Tor Vergata” I.N.F.N. Sezione di Roma II Via della Ricerca Scientifica, 00133 Roma, Italy Abstract We give a detailed microscopic derivation of gauge and stringy instan- ton generated superpotentials for gauge theories living on D3-branes at Z3- orientifold singularities. Gauge instantons are generated by D(-1)-branes and lead to Affleck, Dine and Seiberg (ADS) like superpotentials in the ef- fective N = 1 gauge theories with three generations of bifundamental and anti/symmetric matter. Stringy instanton effects are generated by Euclidean ED3-branes wrapping four-cycles on T 6/Z3. They give rise to Majorana masses, Yukawa couplings or non-renormalizable superpotentials depending on the gauge theory. Finally we determine the conditions under which ADS like superpotentials are generated in N = 1 gauge theories with adjoints, fundamentals, symmetric and antisymmetric chiral matter. http://arxiv.org/abs/0704.0784v2 Contents 1 Introduction 1 2 The Gauge Theory 4 3 D(-1) Instantons 6 3.1 D3-D(-1) in flat space . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 D(-1)-D3 at the C3/Z3-orientifold . . . . . . . . . . . . . . . 7 4 ADS-like superpotential 9 4.1 D3-D(-1) one-loop vacuum amplitudes . . . . . . . . . . . . . 9 4.2 Sp(6)× U(2) superpotential . . . . . . . . . . . . . . . . . . . 14 4.3 U(4) superpotential . . . . . . . . . . . . . . . . . . . . . . . 15 5 ED3-instantons 17 5.1 D3-ED3 one-loop vacuum amplitudes . . . . . . . . . . . . . 19 5.2 The superpotential . . . . . . . . . . . . . . . . . . . . . . . . 21 6 ADS superpotentials: a general analysis 22 7 Conclusions 24 1 Introduction Our understanding of non perturbative effects in four dimensional supersym- metric gauge theories (SYM) has dramatically improved in recent years. This is due mainly to the observation that integrals over the moduli space of gauge connections localize around a finite number of points[1]. These techniques have been applied to the study of multi-instanton corrections to N = 1, 2, 4 supersymmetric gauge theories in R4 [2, 3, 4, 5, 6, 7, 8, 9, 10] (see [11, 12] for reviews of multi-instanton techniques before localization and complete lists of references). In the D-brane language language, the dynamics of the gauge theory around the instanton background is described by an effective theory governing the interactions of the lowest energy excitations of open strings ending on a bound state of Dp-D(p+4) branes. For the case of N = 2, 4 SYM the multi-instanton action has been derived via string techniques in [13, 14]. In [15], D-brane techniques have been applied to the computation of the Affleck, Dine and Seiberg (ADS) superpotential [16, 17] for N = 1 SQCD with gauge group SU(Nc) and Nf = Nc − 1 massless flavours and Sp(2Nc) with 2Nf = 2Nc flavours. The N = 1 gauge theory is realized on the four- dimensional intersection of Nc coloured and Nf flavour D6 branes. Chiral matter comes from strings connecting the flavor and color D6 branes. Instan- tons in the U(Nc) gauge theory are realized in terms of ED2 branes parallel to the stack of Nc D6-branes. By careful integrating the supermoduli (massless strings with at least one end on the ED2) the precise form of the ADS su- perpotential was reproduced in the low energy, field theory limit α′ → 0. In the recent literature ED2-brane instantons in intersecting D6-brane models have received particular attention in connection with the possibility of gen- erating a Majorana mass for right handed neutrinos and their superpartners [18, 19, 20, 21, 22]. The field theory interpretation of this new instanton effect is far from clear and it is the subject of active investigation. In this paper we present a detailed derivation of these new non perturbative superpoten- tials in N = 1 Z3-orientifold models. Investigations of stringy instantons on N = 1 Z2 × Z2 orientifold singularities appeared recently in [23]. We study SYM gauge theories living on D3 branes located at a Z3- orientifold singularity. There are two choices for the orientifold projection [24, 25, 26, 27, 28] realized by two types of O3-planes1. They lead to anomaly free2 chiral N = 1 gauge theories with gauge groups SO(N − 4)× U(N) or Sp(N +4)×U(N) and three generations of chiral matter in the bifundamen- tal and anti/symmetric representation of U(N). The archetype of this class can be realized as a stack of 3N + 4 D3-branes and one O3−-plane sitting on top of an R6/Z3 singularity. This system can be thought of as a T-dual local description (near the origin) of the T 6/Z3 type I string vacuum found in [40]. The lowest choices of N lead to U(4) or U(5) gauge theories with three generations of chiral matter in the 6 and 10 + 5∗ that are clearly of phenomenological interest in unification scenarios [45, 46, 47, 48] 3. In [49] the U(4) case was studied and the form of the ADS-like super- 1We will only consider O3±-planes, not the more exotic Õ3 -planes [29, 30, 31]. 2Factorizable U(1) anomalies are cancelled by a generalization of the Green-Schwarz mechanism [32, 33, 34, 35, 36, 37, 38, 37] that may require the introduction of generalized Chern-Simons couplings [39]. 3Only the U(4) case can be realized in the compact Z3 orientifold. In general the Chan- Paton group is SO(8 − 2n) × U(12 − 2n) ×Hn where Hn = U(n) 3, SO(2n), U(n), U(1)n depending on the choice of Wilson lines [28, 41, 42, 43, 44]. potential was determined combining holomorphicity, U(1) anomaly, dimen- sional analysis and flavour symmetry. Stringy instanton effects were also considered. Very much as for worldsheet instantons in heterotic strings [50, 51, 52, 53, 54], these genuinely stringy instantons give rise to super- potentials that do not vanish at large VEV’s of the open string (charged) ‘moduli’. Here we derive the non-perturbative superpotentials from a direct in- tegration over the D-instanton super-moduli space. Gauge instantons are described in terms of open strings ending on D(-1) branes while stringy in- stantons are given by open strings ending on euclidean ED3 branes wrapping a four cycle inside the Calabi Yau . The open strings connecting the stack of D3 branes to D(-1) and ED3 branes have four and eight mixed Neumann- Dirichelet directions respectively. This ensures that the bound state is su- persymmetric. The superpotential receives contribution from disk, one-loop annulus and Möbius amplitudes ending on the D(-1) or ED3 branes. We find that ADS superpotentials are generated only for two gauge theory choices U(4) and Sp(6)×U(2) inside the Z3-orientifold class. Stringy instantons leads to Majorana masses in the U(4) case, Yukawa couplings in the U(6)×SO(2) gauge theory and non-renormalizable couplings for U(2N + 4) × SO(2N) gauge theories with N > 3. The plan of the paper is as follows. In section 2 we review the gauge theories coming from a stack of D3 branes at a C3/Z3 orientifold singularity. In Section 3 we consider non- perturbative effects generated by D(-1) gauge instantons, corresponding to ADS-like superpotential in the low energy limit. A detailed analysis of one- loop vacuum amplitudes and the integrals over the supermoduli is presented for SYM theories with gauge groups Sp(6) × U(2) and U(4). In section 5, we consider stringy instanton effects generated by ED3-branes. Once again a detailed analysis of the the one-loop string amplitudes and the integrals over the supermoduli is presented. In section 6 we present a “complete” list of N = 1 SYM theories with matter in the adjoint, fundamental, symmetric and antisymmetric representation of the gauge groups (U, SO, Sp) which exhibit a non perturbatively generated ADS superpotential. We conclude with some comments and directions for future investigation in Section 7. 2 The Gauge Theory The low energy dynamics of the open strings living on a stack of N D3-branes in flat space is described by a N = 4 U(N) SYM gauge theory. In the N = 1 language the fields are grouped into a vector multiplet V = (Aµ, λα, λ̄α̇) and 3 chiral multiplets ΦI = (φI , ψIα), I = 1, 2, 3 , all in the adjoint of the gauge group. We consider the D3-brane system at a R6/Z3 singularity. At the singular- ity the N D3-branes group into stacks of Nn fractional branes with n = 0, 1, 2 labelling the conjugacy classes of Z3. The gauge group U(N) decomposes n U(Nn). More precisely, denoting by γθ,N the projective embedding of the orbifold group element θ ∈ Z3 in the Chan-Paton group and imposing = 1 and γ† = γ−1 one can write N0×N̄0 , ωh 1 N1×N̄1 , ω̄h 1 N2×N̄2 ) (2.1) with N = nNn. The resulting gauge theory can be found by projecting the N = 4 U(N) gauge theory under the Z3 orbifold group action: V → γ V γ−1 ΦI → ω γ ΦI γ−1 ω = e2πi/3 (2.2) Keeping only invariant components under (2.2) one finds the N = 1 quiver gauge theory V : N0N̄0 +N1N̄1 +N2N̄2 ΦI : 3× N0N̄1 +N1N̄2 +N2N̄0 (2.3) with gauge group n U(Nn) and three generations of bifundamentals. More precisely V and ΦI are N ×N block matrices (N = Nn) with non trivial Nn × N̄m blocks given by (2.3). Under Z3 a block Nn × N̄m transform as ωn−m. These non-trivial transformation properties are compensated by the space-time eigenvalues of the corresponding field (ω0 for V and ω for ΦI ) making the corresponding component invariant under Z3. Next we consider the effect of introducing an O3±-plane. Woldsheet parity Ω flips open string orientations and act on Chan-Paton indices as Nn ↔ N̄−n where subscripts are always understood mod 3. This prescription leads to Ω : N0 ↔ N̄0 N1 ↔ N̄2 (2.4) The choices of O3±-planes correspond to keep states with eigenvalues Ω = ±1 and lead to symplectic or orthogonal gauge groups4. We start by considering the O3− case. Keeping Ω = − components from (2.3) one finds V : 1 N0(N0 − 1) +N1N̄1 ΦI : 3× N0N̄1 + N1(N1 − 1) (2.5) This follows from (2.3) after identifying the mirror images N̄0 = N0, N̄2 = N1, and antisymmetrizing the resulting block matrix. (2.5) describes the field content of a N = 1 SYM with gauge group SO(N0)×U(N1) and three chiral multiplets in the ( , ¯) + (•, ) For general N0, N1 the U(N1) gauge theory is anomalous. The anomaly is a signal of the presence of a twisted RR tadpole [34, 35]. Focusing on a local description near the orientifold singularity one can relax the global tadpole cancellation condition [55, 56]. These models can be thought as local descriptions of a more complicated Calabi Yau near a Z3 sigularity. Cancellation of the twisted RR tadpole can be written as [40] = −4 ⇒ N0 = N1 − 4 (2.6) and ensures the cancellation of the irreducible four-dimensional anomaly I(F ) ∼ [−N0 + (N1 − 4)] trF 3 = 0 (2.7) Finally the running of the gauge coupling constants is governed by the β functions with one-loop coefficients5 β0 = 3 ℓ( N0(N0 − 1))− 3N1 ℓ(N0) (N0 −N1 − 2) = −9 (IR free) β1 = 3 ℓ(N1N̄1)− 3N0 ℓ(N̄1)− 3 ℓ( N1(N1 − 1)) (−N0 +N1 + 2) = +9 (UV free) (2.8) with βn refering to the n th-gauge group. The last equalities arise after im- posing the anomaly cancellation (2.6). As expected, β0 + β1 = 0 since the ten-dimensional dilaton does not run. 4In the compact case, realized in terms of D9-branes and O9-plane on T 6/Z3, the orthogonal choice is dictated by global tadpole cancellation. Turning on a quantized NS- NS antisymmetric tensor [28, 41, 29] leads to symplectic groups. 5Here trRT aT b = ℓ(R), i.e. ℓ(N) = 1 , ℓ(NN̄) = N and ℓ(1 N(N± 1)) = 1 (N ± 2). The case Ω = + works in a similar way. The resulting N = 1 quiver has gauge group Sp(N0) × U(N1) and three chiral multiplets in the [( , ¯) + (•, )]. The U(N1) is anomaly free for N0 = N1 + 4 and the one- loop β function coefficients are given by β0 = +9 (UV free) and β1 = −9 (IR free). 3 D(-1) Instantons There are two sources of supersymmetric instanton corrections in the D3 brane gauge theory: D(-1)-instantons and Euclidean ED3-branes wrapping four cycles on T 6/Z3. Both are point-like configurations in the space-time and can be thought of as D(-1)-D3 and ED3-D3 bound states with four and eight directions with mixed Neumann-Dirichlet boundary conditions. 3.1 D3-D(-1) in flat space Gauge instantons in SYM can be efficiently described in terms of D(-1)- branes living on the world-volume of D3-branes [57]. As before, we start from the N = 4 case: a bound state of N D3 and k D(-1) branes in flat space. In this formalism instanton moduli are described by the lowest energy modes of open strings with at least one end on the D(-1)-brane stack. The gauge theory dynamics around the instanton background can be described in terms of the U(k) × U(N) 0-dimensional matrix theory living on the D- instanton world-volume. In particular, the ADHM constraints [58] defining the moduli space of self-dual YM connections follow from the F- and D- flatness condition in the matrix theory [57]. The instanton moduli space is given by the D(-1)D3 field content (aµ, θ α , χa, D c, θ̄Aα̇) kk̄ (wα̇, ν A) kN̄ (w̄α̇, ν̄ A) Nk̄ (3.1) with µ = 1, . . . , 4, α, α̇ = 1, 2 (vector/spinor indices of SO(4)), a = 1, . . . , 6, A = 1, . . . , 4 (vector/spinor indices of SO(6)R), c = 1, . . . , 3. The matrices aµ, χa describe the positions of the instanton in the directions parallel and perpendicular to the D3-brane respectively, wα is given by the NS open D3- D(-1) string (instanton sizes and orientations), Dc are auxiliary fields and θAα , θ̄Aα̇, ν A are the fermionic superpartners. The D3-D(-1) action can be written as [59] Sk,N = trk SG + SK + SD (3.2) SG = −[χa, χb] 2 + iθ̄α̇A[χ AB, θ̄ cDc (3.3) SK = −[χa, aµ] 2 + χaw̄ α̇wα̇χa − iθ αA[χABθ α ] + 2iχAB ν̄ SD = i −[aαα̇, θ αA] + ν̄Awα̇ + w̄α̇ν θ̄α̇A +D w̄σcw − iη̄cµν [a µ, aν ] with χAB ≡ T aABχa, T AB = (η AB, iη̄ AB) given in terms of the t’Hooft symbols and g20 = 4π(4π 2α′)−2 gs. The action (3.3) follows from the dimensional reduction of the D5-D9 action in six dimensions down to zero dimension. As a consequence our subsequent results hold up to some computable non vanishing numerical constant. In the presence of a v.e.v. for the six U(N) scalars ϕa in the D3-D3 open string sector we must add to Sk,N Sϕ = trk w̄α̇(ϕaϕa + 2χaϕa)wα̇ + 2iν̄ AϕABν (3.4) The multi-instanton partition function is Zk,N = e−Sk,N−Sϕ = VolU(k) dχ dD da dθ dθ̄dw dν e−Sk,N−Sϕ In the limit g0 ∼ (α ′)−1 → ∞, gravity decouples from the gauge theory and the contributions coming from SG are suppressed. The fields θ̄α̇A, D c become Lagrange multipliers implementing the super ADHM constraints θ̄α̇A : ν̄ Awα̇ + w̄α̇ν A − [aαα̇, θ αA] = 0 Dc : w̄σcw − iη̄cµν [a µ, aν ] = 0 (3.5) 3.2 D(-1)-D3 at the C3/Z3-orientifold Let us now consider in turn the Ω and then the Z3 projection. The effect of introducing an O3±-plane in the D(-1)-D3 system corre- sponds to keep open string states with eigenvalue ΩI = ±, Ω being the worldsheet parity and I a reflection along the Neumann-Dirichlet directions of the Dp-O3 system [8]. On D(-1) string modes, I acts as a reflection in the spacetime plane I : aµ → −aµ θ α → −θ α (3.6) leaving all other moduli invariant. In addition consistency with the D3-O3 projection requires that the D(-1) strings are projected in the opposite way with respect to the D3-branes[60] . From the gauge theory point of view this corresponds to the well known fact that SO(N) and Sp(N) gauge instantons have ADHM constraints invariant under Sp(k) and SO(k) respectively. We start by considering the O3− case. After the ΩI projection the sur- viving fields are (aµ, θ k(k− 1) (Dc, χI , χ̄I , θ̄Aα̇) k(k+ 1) (wα̇, ν) kN . (3.7) Since we are dealing with a SO(N) gauge theory the Dc moduli are projected in the adjoint of Sp(k). This is also the case for all the other moduli even under I while the odd ones, (aµ, θ α ), turn out to be antisymmetric. Let us now consider the Z3 projection. Out of the six χa one can form three complex fields χI with eigenvalues ω under Z3 and their conjugate χ̄I . To embed the Z3 projection into SU(4) we decompose the spinor index A = (0, I), with I = 1, . . . , 3 and the zeroth direction along the surviving N = 1 supersymmetry. The D3 and D(-1) gauge groups SO(N) and Sp(k) break into SO(N0)×U(N1) and Sp(k0)×U(k1) respectively with N0 (k0) the number of fractional D3 (D(-1)) branes invariant under Z3 and N1 (k1) those transforming with eigenvalue ω. More precisely, the projective embedding of the Z3 basic orbifold group element θ in the Chan-Paton group can be written N0×N0 , ωh 1 N1×N̄1 , ω̄h 1 N̄1×N1 k0×k0 , ωh 1 k1×k̄1 , ω̄h 1 k̄1×k1 ) (3.8) After projecting under Z3 the symmetric/antisymmetric matrices in (3.7) break into km × k̄n, km × N̄n or Nm × k̄n each transforming with eigenvalue ωm−n. In addition fields with up(down) index I transform like ω(ω̄). Keeping only the invariant components one finds (aµ; θ k0(k0 − 1) + k1k̄1 k1(k1 − 1) + k0k̄1 (Dc; θ̄0α̇) k0(k0 + 1) + k1k̄1 (χ̄I ; θ̄Iα̇) k̄1(k̄1 + 1) + k0k1 k1(k1 + 1) + k0k̄1 (wα̇; ν0) k0N0 + k1N̄1 + k̄1N1 νI k0N̄1 + k̄1N0 + k1N1 (3.9) Notice that the Z3 eigenvalues of the Chan-Paton indices in the r.h.s. of (3.9) compensate for those of the moduli in the l.h.s. making the field invariant under Z3. In addition (odd)even components under I are (anti)symmetrized ensuring the invariance under ΩI. The multi-instanton action follows from that of N = N0+2N1 D3 branes and k = k0 + 2k1 D(-1) instanton in flat space (3.2) with U(N) and U(k) matrices restricted to the invariant blocks (3.9). The results for O3+ can be read off from (3.9) by exchanging symmetric and antisymmetric representations. 4 ADS-like superpotential 4.1 D3-D(-1) one-loop vacuum amplitudes Non-perturbative superpotentials can be computed from the instanton mod- uli space integral [11, 13, 18] SW = e 〈1〉D+〈1〉A+〈1〉M µβnkn e−Sk,N−Sϕ (4.1) The integration is over the instanton moduli space, M, 〈1〉D is the disk amplitude and 〈1〉A,M are the one-loop vacuum amplitudes with at least one end on the D(-1)-instanton. The factor µβnkn , µ being the energy scale, comes from the quadratic fluctuations around the instanton background and as we will see it combines with a similar contribution coming from the moduli measure to give a dimensionless SW . The terms in front of the integral in (4.1) combine into SW = Λ e−Sk,N−Sϕ (4.2) Λknβn = e2πiknτn(µ) µβnkn τn(µ) = τn − (4.3) the one-loop renormalization group invariant and the running coupling con- stant respectively and τn refers to the complexified coupling constant of the nth gauge group. µ0 is a reference scale. More precisely, the disk amplitude and one-loop amplitudes yields e〈1〉D = e2πiknτn τn = e〈1〉A+〈1〉M = )−βnκn + . . . (4.4) with dots refering to threshold corrections that will not be considered here. To see (4.4) we should compute the following one-loop amplitudes 〈1〉A = − Tr[(1 + (−)F )(1 + θ + θ2) qL0−a] AD(−1)D3 = −A0,D(−1)D3 ln + . . . 〈1〉M = − Tr[ ΩI (1 + (−)F )(1 + θ + θ2) qL0−a] MD(−1) = −M0,D(−1) ln + . . . (4.5) In the above formula µ enters as a UV regulator in the open string chan- nel (see [61] for details) and A0,M0 are the massless contributions to the amplitudes. We start by considering the O3− projection. It is important to notice that only the annulus with one end on the D(-1) and one on the D3 contributes to these amplitudes since D(-1)-D(-1) amplitudes cancel due to the Riemann identity. One finds AD(−1),D3 = trγθ,ktrγθ,N ϑ[αβ ] ϑ[αβ+hi ] (k0 − k1)(N0 −N1) + . . . MD(−1) = trγθ2,k ϑ[αβ ] ϑ[αβ+hi ] = −3(k0 − k1) + . . . (4.6) The sum runs over the even spin structures and cαβ = (−) 2(α+β). The term comes from the (b, c) and (β, γ) ghosts while the extra five thetas in the numerator and denominator describe the contributions of the ten fermionic and bosonic worldsheet degrees of freedom. We adopt the shorthand notation h ] ≡ ϑ[ h ]/(2 cosπh) to describe the massive contribution of a periodic boson to the partition function. hi = ( ) denote the Z3-twists while the extra 1 -shifts in the annulus account for the D(-1)-D3 open string twist along Neuman-Dirichlet directions while 1 twists in the Möbius come from the I-projection. In addition we used the fact that the contribution of the unprojected sector is zero after using the Riemann identity while that of the θ- and θ2-projected sectors are identical explaining the overall factor of 2. The extra factor of 2 in the annulus comes from the two orientations of the string. The second line displays the massless contributions. We use the Chan Paton traces 1,k = k0 + 2k1 trγθ,k = k0 − k1 1,N = N0 + 2N1 trγθ,N = N0 −N1 (4.7) that follows from (2.1) and the first few terms in the theta expansions ϑ[0h] = 1 + q 2 2 cos 2πh+ . . . ϑ[ h ] = q 8 2 cosπh+ . . . η = q 24 + . . . (4.8) From (4.6) one finds A0 +M0 = (k0 − k1)(N0 −N1 − 2) = knβn (4.9) with βn the one-loop β coefficients given in (2.8). Plugging (4.9) into (4.5) results into (4.4). The fact that the β function coefficients are reproduced by the instanton vacuum amplitudes is a nice test of the instanton field content (3.9). Now let us determine the dependence of the instanton measure on the string scaleMs ∼ α ′ −1/2. The scaling of the various instanton moduli follows from (3.3): D, g0 ∼M s χa, ϕa ∼ Ms wα̇, aµ ∼M νA, θAα ∼M s θ̄Aα̇ ∼M s (4.10) Collecting from (3.9) the number of components of the various moduli enter- ing in the instanton measure one finds 6 e−Sk,N−Sϕ ∼ M−βnkns knβn = −2nD − nχ + na + nw + nθ̄ − (k0 − k1)(N0 −N1 − 2) (4.11) Notice that this factor precisely combines with that in (4.2) leading to a di- mensionless SW as expected. This simple dimensional analysis can be used to determine the form of the allowed ADS superpotentials in the gauge theory. A superpotential is generated if and only if the integral over the instanton moduli space reduce to an integral over x 0 describing the center of the in- stanton and θα its superpartner. More precisely SW = Λ e−Sk,N−Sϕ = c d4x0d Λknβn ϕknβn−3 (4.12) where c is a numerical constant. Whether c is zero or not depends on the presence or not of extra fermionic zero modes besides θ. Notice that the power of ϕ is completely fixed requiring that SW is dimensionless. The precise form of the superpotential requires the evaluation of the moduli space integral and will be the subject of the next section. The superpotential follows from (4.12) after promoting ϕI to the chiral superfield ΦI and x0, θα to the measure of the superspace SW = c d4xd2θ Λknβn Φknβn−3 (4.13) 6We recall that fermionic differentials scale as the inverse of the dimension of the fermion itself. This explains the extra minus sign in (4.11). A superpotential of type (4.13) is generated whenever [63, 64, 16, 17] 〈λ2 ϕknβn−3〉 6= 0 (4.14) Each scalar ϕ soaks two fermionic zero-modes and each gaugino λ one zero mode7. The condition (4.14) translates into dimMF = 2knβn − 4 (4.15) with dimMF the fermionic dimension of the instanton super-moduli space. The number of fermionic zero modes can be read off from (3.9) dimMF = nθ + nν − nθ̄ = k0(3N1 +N0 − 2) + k1[2N1 + 3(N0 +N1 − 2)] = k0(4N0 + 10) + k1(8N0 + 14) (4.16) where we used the fact that θ̄α̇A enter as a Lagrangian multiplier imposing the fermionic ADHM constraint and therefore subtracts degrees of freedom. The last line in (4.16) follows from using the anomaly cancellation condition N1 = N0 + 4. The result (4.16) is consistent with the Atiyah-Singer index theorem that states dimMF = 2k0 N0(N0 − 1)) + 3N1ℓ(N0) ℓ(N1N̄1) + 3N0ℓ(N1) + 3ℓ( N1(N1 − 1)) = k0(3N1 +N0 − 2) + k1[2N1 + 3(N0 +N1 − 2)] (4.17) Combining (4.15) and (4.16) one finds k1 − 7k0 − 1 k0 + 2k1 (4.18) One can easily see that the only non-negative solution for N0 is N0 = 0 k0 = 0 k1 = 1 We conclude that in the class of U(N0+4)×SO(N0) SYM theories describing the low-energy dynamics of D3-branes on the Z3 orientifold only the U(4) 7This can be seen by explicitly solving the equations of motion of the gaugino and the ϕ-field in the instanton background [59]. In particular the source for the scalar field comes from the Yukawa coupling LY uk = gYMϕ †ψλ in the gauge theory action. theory with three chiral multiplets in the antisymmetric leads to an ADS-like superpotential generated by gauge instantons. The counting can be easily repeated for the Sp(N1+4)×U(N1) cases by exchanging symmetric and antisymmetric representations in (3.9) as required by the presence of the O3+-plane. The results are knβn = 9(k0 − k1) dimMF = k0(4N1 + 6) + k1(8N1 + 18) 3k0 − 9k1 − 1 k0 + 2k1 (4.19) One can easily see that the only non-negative solution is N1 = 2 k0 = 1 k1 = 0 We conclude that in this class, only the gauge theory Sp(6)×U(2) with three chiral multiplets in the ( , ¯) + (•, ) admits an ADS-like superpotential generated by instantons. The aim of the rest of this section is to compute SW . The integral (4.12) will be evaluated in turn for the Sp(6)× U(2) and U(4) case. 4.2 Sp(6)× U(2) superpotential We first consider the O3+ case, i.e. the Sp(6) × U(2) gauge theory with three chiral multiplets in the [(6, 2̄) + (1, 3)]. The instanton moduli is given by (3.9) after flipping symmetric/antisymmetric representations in order to deal with the symplectic projection. Plugging k0 = 1, k1 = 0, N0 = 6, N1 = 2 into (3.9) one finds the the surviving fields aµ, w , θ0α, ν 0u0 , νIu1 (4.20) with u0 = 1, ..6, and u1 = 1, 2, whose position from lower to upper has been switched in this section for notational convenience as we will momentarily see. In particular both θ̄0α̇ andD c are projected out (the D(-1) “gauge” group is O(1) ≈ Z2 in this case) and therefore no ADHM constraint survives. The instanton action reduces then to S = SK + Sϕ = w α̇ ϕ̄Iu0u1ϕ I u1v0wα̇v0 + ν Iu1ν0u0ϕ̄Iu0u1 (4.21) Here and below we omit numerical coefficients that can be always reabsorbed at the end in the definition of the scale. The integrations over wα̇u0 , ν , νIu1 are gaussian and the final result, up to a non vanishing numerical constant, can be written as SW = Λ d4ad2θ det6×6 (ϕ̄Iu1,u0) det6×6 (ϕ̄Iu1,u0ϕ Iu1,v0) d4ad2θ det6×6 (ϕIu1,u0) (4.22) where we have exploited the possibility of combining I and u1 in one ‘bi- index’ Iu1 so as to get a range of six values. For the sake of simplicity we have dropped the subscript 0 denoting bare scalar fields. In the following scalar fields entering in formulae involving Λ will be always understood to be bare. The last step makes use of det(AB) = det(A)det(B). 4.3 U(4) superpotential We now consider the O3− case, i.e. the U(4) gauge theory with three chiral multiplets in the 6. Setting k0 = 0, k1 = 1, N0 = 0, N1 = 4 in (3.9) the surviving fields can be written as I[uv] , ϕ̄I[uv](0) , aµ(0) , χ̄I(−2) , χ (+2) , D (0) , w u(+1) , w̄ α̇(−1) θ0α(0) , θ̄0α̇(0) , θ̄α̇I(−1) ; ν u(+1) , ν̄ (−1) , ν (+1) (4.23) with u = 1, ..4 and the charge q under U(1)k1 is denoted in parentheses. Plugging into (3.3) (after taking α′ → 0) one finds S = SB + SF (4.24) where ν̄0uwuα̇ + w̄ θ̄α̇0 + ν Iu wuα̇ θ̄ I + χ̄Iν Iu + νIuϕ̄Iuvν̄ SB = w̄ α̇ϕ̄Iuwϕ Iwvwα̇v + ϕ Iuvwα̇uwvα̇χ̄I + ϕ̄Iuvw̄ uα̇w̄vα̇χ I + w̄uα̇wuα̇χ̄Iχ +Dc w̄σcw (4.25) As before we omit numerical coefficients. The integral over Dc leads to a δ function on the ADHM constraints d8wd8w̄δ3(w̄σcw) = dρ ρ9d12U (4.26) In the r.h.s of (4.26) we have solved the ADHM constraints in favor of w and U defined by wuα̇ = ρUuα̇ w̄ uα̇ = ρ Ūuα̇ Ūuα̇Uuβ̇ = δ (4.27) The coset representatives Uα̇u parameterizes the SU(4)/SU(2) orientations of the instanton inside the gauge group. The fermionic integrations lead to the determinant ∆F = ρ 8 ǫu1u2u3u4ǫv1v2u5u6ǫv3v4v5v6Xu1v1u2v2 Xu3v3u4v4 Yu5v5 Yu6v6 (4.28) Xu1v1u2v2 = ǫ I1I2I3χ̄I1ϕ̄I2u1v1ϕ̄I3u2v2 Yuv = U u Uα̇u (4.29) The bosonic integrals are more involved. For arbitrary choices of the scalar VEV’s ϕ̄I and ϕ I , even along the flat directions of the potential, the inte- gration over U represents a challenging if not a prohibitive task. Fortunately choosing ϕIuv = ϕηIuv, ϕ̄Iuv = ϕ̄η Iuv, the full ϕ-dependence can be factor- ized. SU(4) gauge and SU(3) ‘flavor’ invariance can then be used to recover the full answer. After the rescaling ρ2 → ρ2/(ϕϕ̄) χI → ϕχI χ̄I → ϕ̄χ̄I (4.30) The integral becomes SW = Λ d4x0d (4.31) with I the ϕ-independent integral dρρ9 d12U d3χd3χ̄∆F e S̃B = −ρ 2(1 + ηIuvYuvχI + η̄IuvȲ uvχI + χ̄Iχ I) (4.32) and ∆F given again by (4.28) but now in terms of Xu1v1u2v2 = ǫ I1I2I3χ̄I1 η̄I2u1v1 η̄I3u2v2 Finally one can restore the gauge covariance of (4.31) by noticing that there is a unique SU(4)c × SU(3)f singlet in the symmetric tensor of six ϕ det3×3[ǫu1..u4ϕ Iu1u2ϕJu3u4] Therefore one can replace ϕ6 in (4.31) by this singlet. The superpotential follows after replacing ϕI → ΦI SW = c d4xd2θ det3×3[ǫu1..u4Φ Iu1u2ΦJu3u4] (4.33) where c is a computable non-zero numerical coefficient. 5 ED3-instantons Let us now consider the ED3-D3 system. We restrict ourselves to the compact case T 6/Z3 and consider the ED3 fractional instanton wrapping a four-cycle Cn inside T 6/Z3. We start by considering the O3 −-orientifold projection. The zero modes of the Yang-Mills fields in the instanton background can be described as before in terms of open strings with at least one end on the ED3. Open strings connecting ED3 and D3 branes have 8 Neumann- Dirichlet directions therefore the zero-mode dynamics of the ED3-D3 system is equivalent to that of the D7-D(-1) bound state. The instanton action can be found starting from that of the N = (8, 0) sigma model describing the low energy dynamics of a D1-D9 bound state in type I [65] reduced down to zero dimensions. In flat space the D(-1)-D7 action reads S = trk Sg + SK + SD (5.1) Sg = −[χ, χ̄] 2 + Θ̃ȧχΘ̃ȧ +DcDc SK = −[χ,Xm][χ̄, Xm] + Θ aχ̄Θa + ν(χ + ϕ)ν SD = Θ̃ ȧXmΓ a +DcΓ̂cmn[Xm, Xn] (5.2) with m = 1, . . . , 8v, a = 1, . . . , 8s, ȧ = 1, . . . , 8c, c = 1, , . . . , 7. We denote by ϕ = mI(Cn)ϕ I , the gauge scalar parametrizing the position of the D3- brane along the direction perpendicular to the 4-cycle Cn. Here Γ ȧa, Γ̂ are gamma matrices of SO(8) and SO(7) respectively. The introduction of the auxiliary fields Dc has broken the manifest SO(8) invariance of the action that will be further broken by the Z3-projection. In (5.2), Xm and χ, χ̄ describe the position of the D(-1)-instanton in the directions longitudinal and perpendicular to the D7-brane respectively while Θa, Θ̃ȧ are the fermionic superpartners grouped according to the their chirality along the Dirichlet- Dirichlet χ-plane. Unlike the D(-1)-D3 case, in the case of 8 Neumann- Dirichlet directions Ω acts in the same way on the D(-1) and D7 Chan-Paton indices. This implies that Dc transform in the adjoint of SO(k) if we take the D7 gauge symmetry to be SO(N). In addition I acts as I : Xm → −Xm Θ a → −Θa (5.3) Fields with eigenvalues ΩI = − are then in the following representations of SO(k)× SO(N) (χ, χ̄, Dc, Θ̃ȧ) 1 k(k− 1) (Xm,Θ k(k+ 1) ν kN (5.4) Fields even under I transform in the adjoint of SO(k) while odd fields tran- form in the symmetric representation. For k = 1, N = 32 the D(-1)-D7 system or equivalently the D1-D9 bound state describes the S-dual version of the fundamental heterotic string on T 2. k > 1 bound states correspond to multiple windings of the heterotic string [65]. The field Dc implements the one-real D and three complex F flatness conditions V = − DcDc = −g20 m,n=1 [Xm, Xn] 2 = 0 (5.5) Dc = −1 mn[Xm, Xn] (5.6) An explicit choice of Γ matrices in D = 7 is given by (a = 1, 2, 3) Γa8×8 = iσ1 ⊗ η 4×4 Γ 8×8 = iσ3 ⊗ η̄ 4×4 Γ 8×8 = iσ2 ⊗ 14×4 (5.7) As in section 3 Z3 acts both on spacetime and Chan-Paton indices. Chan- Paton indices decompose as N → N0 + N1 + N̄1 and k → k0 + k1 + k̄1. Spacetime indices on the other hand decompose as 8v = 4 + 2ω + 2ω̄ 8s = 2 + 2ω + 4ω̄ 8c = 2 + 2ω̄ + 4ω 7 = 3 + 2ω + 2ω̄ (5.8) In addition χ, ν transform with eigenvalue ω under Z3. Combining with (5.4) one finds the Z3-invariant components χ, χ̄ 1 k1(k1 − 1) + k0k̄1 + h.c. Dc 3(1 k0(k0 − 1) + k1k̄1) + 2 k1(k1 − 1) + k0k̄1 + h.c. Θ̃ȧ 2 k0(k0 − 1) + k1k̄1 k̄1(k̄1 − 1) + k0k1) k1(k1 − 1) + k0k̄1) k0(k0 + 1) + k1k̄1 k1(k1 + 1) + k0k̄1 + h.c. k0(k0 + 1) + k1k̄1 k1(k1 + 1) + k0k̄1) k̄1(k̄1 + 1) + k0k1) ν k0N̄1 + k1N1 + k̄1N0 (5.9) 5.1 D3-ED3 one-loop vacuum amplitudes ED3 generated superpotentials can be computed following the same steps as in section 4.1. The disk amplitude can be written as e〈1〉D = e2πiknτ̃n τ̃n = i 4πV4(Cn) g2n α (C4 + C0 ∧R ∧ R) (5.10) τ̃n describes the coupling of closed string moduli to the ED3 instanton wrap- ping the 4-cycle Cn with volume V4(Cn). We remark that closed string states in the Z3-twisted sectors flow in the ED3-ED3 cylinder amplitude and there- fore τ̃n is function of both untwisted and twisted closed twisted moduli. This is not surprising since the volume of the cycle depends also on the volume of the exceptional cycles that the ED3 wraps. The annulus and Möbius amplitudes are given by AED3,D3 = ϑ[αβ ] 2trγθ,ktrγθ,N ϑ[αβ−2h1 ] + trγ 1,ktrγ1,N ϑ[αβ ] k0N1 − k1(N0 +N1) + . . . MED3 = − ϑ[αβ ] 2trγθ2,k ϑ[αβ−2h1 ] + trγ ϑ[αβ ] = 3k0 + k1 + . . . (5.11) The origin of the various contributions is the same that in the D(-1)-D3 system. Now the D3-ED3 open strings have 8 Neumann-Dirichlet directions explaining the extra 1 twists in the annulus amplitude. On the other side, the I projection accounts for the 1 -shift in the Möbius amplitude. Notice that unlike the D(-1)-D3 case, the unprojected amplitude tr1, now gives a non-trivial contribution. Collecting the contributions from (5.11) one finds Λ̃knbn = µknbn e〈1〉D+〈1〉A+〈1〉M = µknbn e2πikn τ̃n(µ) (5.12) τ̃n(µ) = τ̃n − (5.13) A0 +M0 = knbn = k0(6−N1) + k1(2−N0 −N1) (5.14) The interpretation of the bn as the one-loop β function coefficients of the τ̃n coupling, though tantalizing, is not clear to us. We will now check that knbn reproduces the right scale dependence of the instanton measure. The scaling of the various instanton moduli follows from (5.2): D, g0 ∼M s χ, χ̄, ϕ ∼Ms Xm ∼ M ν, Θa ∼M−1/2s Θ̃ ȧ ∼ M3/2s (5.15) Collecting from (5.9) the number of degrees of freedom entering in the instanton supermoduli measure one finds e−Sk,N ∼ M−knbns knbn = −2nD − nχ + nX + nΘ̃ − k0(6−N1) + k1(2−N0 −N1) (5.16) As in the previous case we write the instanton generated superpotential as the moduli space integral SW = Λ̃ e−Sk,N−Sϕ = d4x0d 2θ Λ̃knbn ϕ−knbn+3 (5.17) After promoting ϕ→ Φ and x0, θα to the measure of the superspace one finds the ED3 generated superpotential d4xd2θ Λ̃knbn Φ−knbn+3 (5.18) The main difference with respect to the D(-1) instantons is that now ϕ enters into Sϕ (5.2) only through the coupling to the ν-fermions. This implies that in order to get a non zero result from the fermionic integral in (5.17) only the ν’s and the two fermionic zero modes θα ∈ Θ a should survive the orientifold projections. From (5.9) one can easily see that this implies k0 = 1, k1 = 0. The same counting shows that no solutions are allowed in the Sp(N) case. 5.2 The superpotential Here we evaluate the instanton moduli space integral for the SO(N0)×U(N1) case. From our analysis above the relevant cases are k0 = 1, k1 = 0. The surviving fields in (5.9) are θα ∈ Θ 0 ∈ Xm νu (5.19) with u = 1, ...N1. The instanton action reduces to S = νuϕ uvνv (5.20) The superpotential is then given by the integral SW = Λ̃ d4x d2θ dN1ν e−νϕν (5.21) After integration over ν and lifting ϕ→ Φ to the superfield one finds SW = c Λ̃ d4x d2θ ǫu1....uN1Φ u1u2Φu3u4...ΦuN1−1uN1 (5.22) where c is a non vanishing numerical constant. Notice that the result is non-trivial only when N1 is even. The superpotentials (5.22) are non- renormalizable for N1 > 6 and grow for large vacuum expectation values where the low energy approximation breaks down. The only exceptions are Majorana masses U(4) + 3 Yukawa couplings SO(2)× U(6) + 3 ( , ¯) + 3 (•, ) (5.23) Notice that both instanton generated Yukawa couplings involve only the mat- ter in the antisymmetric representation. 6 ADS superpotentials: a general analysis Here we consider a general N = 1 gauge theory with gauge group U(N) and nAdj, nf/n̄f , nS/n̄S, nA/n̄A number of chiral multiplets in the adjoint, fun- damental, symmetric and anti-symmetric representations (and their complex conjugates) respectively. The cubic chiral anomaly, one-loop β function and number of fermionic zero modes in the instanton background of the gauge theory can be written Ianom = nf− + nS−(N + 4) + nA−(N − 4) = 0 (6.1) β1−loop = 3N −NnAdj − nf+ − nS+(N + 2)− nA+(N − 2) dimMF = k [2N + 2NnAdj + nf+ + nS+(N + 2) + nA+(N − 2)] nf± = nf ± n̄f nS± = nS ± n̄S nA± = nA ± n̄A (6.2) The condition for an Affleck, Dine and Seiberg like superpotential [16, 17] to be generated was determined in section 4.1 to be dimMF = 2kβ − 4 (6.3) Combining (6.1) and (6.3) one finds β1−loop = 2N + (6.4) nf− = −nS−(N + 4)− nA−(N − 4) nf+ = 2N − − 2NnAdj − nS+(N + 2)− nA+(N − 2) Remarkably the β function in a theory admitting an instanton generated su- perpotential depends only on the rank of the gauge group. A simple inspec- tion shows that a superpotential is generated only for k = 1 and nAdj = 0. The complete list follows from a scan of any choice of nS±,nA± such that n+ ≥ |n−| and n+ ≥ 0. One finds U(N) +Nf ( + ¯ ) Nf ≤ N − 1 U(N) + + (N − 4)¯ +Nf ( + ¯ ) Nf ≤ 2 U(4) + 2 +Nf ( + ¯ ) Nf ≤ 1 U(4) + 3 U(5) + 2 + 2¯ (6.5) The inequalities are saturated for gauge theories satisfying (6.3) and (6.4), while the lower cases are found by decoupling quark-antiquark pairs via mass deformations. The generalization to SO(N)/Sp(N) gauge groups is straightforward. In these cases there is no restriction coming from anomalies since representations are real. The β function and the number of fermionic zero modes in the instanton background are given by β1−loop = (N ± 2)− 1 nS(N + 2)− nA(N − 2) dimMF = k [N ± 2 + nf + nS(N + 2) + nA(N − 2)] with upper sign for Sp(N) and lower sign for SO(N) gauge groups. Imposing (6.3) one finds β1−loop = N ± 2 + (6.6) nf = N ± 2− − nS(N + 2)− nA(N − 2) The list of solutions is even shorter SO(N) +Nf Nf ≤ N − 3 k = 2 Sp(N) +Nf Nf ≤ N k = 1 Sp(N) + + 2 k = 1 (6.7) Notice that k = 1, respectively k = 2, are the basic instantons in Sp(N), respectively SO(N), since the instanton symmetry groups are in these cases SO(k), respectively Sp(k). 7 Conclusions In the present paper, we have given a detailed microscopic derivation of non- perturbative superpotentials for chiral N = 1 D3-brane gauge theories living at Z3-orientifold singularities. We considered both unoriented projections leading to SO(N1− 4)×U(N1) and Sp(N1+4)×U(N1) gauge theories with three generations of chiral matter in the representations ( , ¯) + (•, ) and ( , ¯) + (•, ) respectively. The U(4) case was studied in details in [49] and describes the local physics of type I theory near the origin of T 6/Z3 with SO(8)× U(12) gauge group broken by Wilson lines. In the present T-dual setting, there are two sources of non-perturbative effects: D(-1) and ED3 instantons. The former realize the standard gauge instantons and lead to Affleck, Dine and Seiberg like superpotentials. The latter lead to Majorana masses or non-renormalizable superpotentials and were ignored till very recently [18, 19, 20, 21, 15, 49, 22, Our explicit instanton computations confirm the form of ADS and stringy superpotentials proposed in [49] on the basis of holomorphicity, dimensional analysis U(1) anomaly and flavour symmetry. We show that ADS super- potentials are generated only for the U(4) and Sp(6)× U(2) gauge theories in the Z3-orientifold list. The precise form of the superpotential is derived from an integration over the instanton super-moduli space. Like in [15], the β function running of gauge couplings are reproduced from vacuum ampli- tudes given in terms of annulus and Möbius amplitudes ending on the in- stantons. The same analysis is performed for “stringy instantons” generated by Euclidean ED3-branes (dual to ED1-strings in type I theory) wrapping holomorphic four-cycles on T 6/Z3. A detailed microscopic analysis of the multi-instanton super-moduli space encompasses massless open string states with a least one end on the ED3-instanton. We show the generation of Ma- jorana mass terms for the open string chiral multiplets in the U(4) case, Yukwa couplings for the SO(2)×U(6) gauge theory and non-renormalizable superpotentials for SO(N0)×U(N0 +4) gauge theories. The field theory in- terpretation of the β function coefficients generated by the one-loop vacuum amplitudes for open strings ending on the ED3-instantons is one of the most interesting open question left by our instanton super-moduli space analysis. As previously observed, the invariance under anomalous U(1)’s results from a detailed balance between the charges of the open strings involved and the axionic shift of a closed string R-R modulus from the twisted sector. Our present analysis has some analogies with the recent ones [18, 19, 20, 21, 15, 22, 23] that have focussed on ED2-branes at D6-brane intersection. As stressed in [49], one immediate advantage of the viewpoint advocated here is the consistency of the local description. Indeed, imposing twisted tadpole cancellation [34, 35] the models presented here and all closely related settings of D-branes at singularities (not necessarily of the Zn kind) give rise to anomaly free theories, while this is not necessarily the case for the ‘local’ models with intersecting D-branes. We can envisage the possibility of extending our analysis to other Zn singularities [55, 56] or even to Gepner models [66, 67, 68] where many if not all ingredients, such as the brane actions from gauge kinetic functions including one-loop threshold effects [69, 70, 71, 72], are available. In the present paper we have not addressed phenomenological implica- tions of the stringy instanton effects we have analyzed in detail. We hope to be able to investigate these issues in this or similar contexts with D-branes at singularities, where the rigidity of the cycles is well understood and allows for the correct number of fermionic zero-modes. Clearly additional (closed string) fluxes neeeded for moduli stabilization [73, 74] may change some of our present conclusions. Acknowledgments It is a pleasure to thank P. Anastasopoulos, R. Argurio, C. Bachas, M. Bertolini, M. Billo, G. Ferretti, M.L. Frau, A. Kumar, E. Kiritsis, I. Kle- banov, S. Kovacs, A. Lerda, L. Martucci, I. Pesandro, R. Russo and M. Wijnholt for valuable discussions. Special thanks go to G. Pradisi for collab- oration on the computation of the string amplitudes and useful exchanges. During completion of this work, M.B. was visiting the Galileo Galilei In- stitute in Arcetri (FI) and thanks INFN for hospitality and support. M.B. is very grateful to the organizers and participants to the workshop “String and M theory approaches to particle physics and cosmology” for creating a very stimulating atmosphere. This work was supported in part by the CNRS PICS no. 2530 and 3059, INTAS grant 03-516346, MIUR-COFIN 2003-023852, NATO PST.CLG.978785, the RTN grants MRTNCT- 2004- 503369, EU MRTN-CT-2004-512194, MRTN-CT-2004-005104 and by a Eu- ropean Union Excellence Grant, MEXT-CT-2003-509661. References [1] N. A. Nekrasov, Seiberg-witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831–864, [hep-th/0206161]. [2] R. Flume and R. 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0704.0785

Viscoplastic Properties and Tribological Behavior of Diamond-Like Carbon Films Using Nanoindentation and Nanoscratch Tests

Microsoft Word - essai2.doc VISCOPLASTIC PROPERTIES AND TRIBOLOGICAL BEHAVIOR OF DIAMOND-LIKE CARBON FILMS USING NANOINDENTATION AND NANOSCRATCH TESTS V.Turq, J. Fontaine, J. L. Loubet, D. Mazuyer Ecole Centrale de Lyon, Laboratoire de Tribologie et Dynamique des Systèmes, UMR CNRS 5513 69134 Ecully Cedex, France TEL : 33-472-186-287, Email: [email protected] Introduction Diamond-Like Carbon (DLC) films have been shown to demonstrate various tribological behaviors: in ultra-high vacuum (UHV), with either friction coefficients as low as 0.01 or less and very mild wear, or very high friction coefficients (>0.4) and drastic wear. These behaviors depend notably on gaseous environment, hydrogen content of the film [1], and on its viscoplastic properties [2,3]. A relation between superlow friction in UHV and viscoplasticity has indeed been established for a-C:H films and confirmed for a fluorinated sample (a-C:F:H). In this study, nanoindentation and nanoscratch tests were conducted in ambient air, using a nanoindentation apparatus, in order to evaluate tribological behaviors, as well as mechanical and viscoplastic properties of different amorphous carbon films. Experimental The samples were deposited on a Si (100) substrate by Plasma Enhanced Chemical Vapor Deposition (PECVD) process at different bias voltages, either from acetylene, cyclohexane precursors by d.c.- PECVD, or hexafluorobenzene mixed with hydrogen precursor by r.f.-PECVD for the fluorinated sample (a- C:F:H, noted FDLC). Details of the deposition process can be found in [4,5]. Thickness of the coating is 1µm, except for the FDLC, which is 0.4µm. Nanoindentation and nanoscratch tests were carried out in ambient air, at room temperature, with a MTS NanoIndenter® XP apparatus. A spherical (radius 10µm) and a Berkovich diamond indenter were used. Mechanical and viscoplastic properties were evaluated from nanoindentation tests in continuous stiffness mode, using the Berkovich diamond indenter, with a maximum load of 100mN. As the load P is applied exponentially as a function of time, the ratio between loading rate P’ and load P is kept constant during indentation, and thus the strain rate ε is also constant. Five different ratios P’/P, from 3.10-3 up to 3.10-1Hz, were used. Nanotribological evaluation of the samples was conducted from nanoscratch tests at ramping load (0.1 to 10mN, 3 passes) and at constant load (5mN, 10 passes) with spherical diamond indenter. Results The strain rate sensitivity of the materials is estimated and fitted by a Norton-Hoff law: xH H ε= ⋅ where H is the hardness, H0 a constant, ε the strain and x a constant called viscoplastic exponent (Table 1). Contrary to UHV, no evidence of correlation between friction coefficients in ambient air and viscoplasticity can be made. But even in this environment, some very low friction coefficient values, as low as 0.04 (FDLC), with very mild wear have been evidenced (Figure 1). Hardness H0 seems to be the key parameter: wear resistance in the air is improved with higher H0 and friction coefficient decreases with H0. Note that H0 is also roughly linked with the hydrogen content of the coating for the non fluorinated samples, as it has been shown in [2]. The number of passes seems also to lead to a decrease of friction coefficient. Sample H content (at. %) (GPa) x µ ramping load µ constant load Wear FDLC 5/18(F) 16 0.060 0.04-0.14 0.080 ~ none AC8 34 13 0.014 0.06-0.11 0.083 ~ none AC5 40 11 0.068 0.05-0.16 0.078 mild CY6.5 42 6.8 0.028 0.07-0.13 0.083 mild CY5 42 1.3 0.076 0.10-0.22 0.183 severe Table 1: Summary of nanofriction tests results and viscoplastic properties Conclusion This study shows that in ambient air, wear resistance and frictional behavior of a-C:H and a-C:F:H samples is improved with hardness H0. In UHV, the achievement of super-low friction is linked with the viscoplastic character. Thus, intermediary coating, with high hardness and viscoplastic exponent, as a-C:F:H will demonstrate satisfactory tribological behavior both in ambient air and in UHV. References [1] C. Donnet et al., Tribo. Lett., 9 (2000) 137. [2] J. Fontaine et al., Tribo. Lett., 17 (2004) 709. [3] J. Fontaine et al., Thin Solid Films, (2005) in press. [4] C. Donnet et al., Surf. Coat. Tech., 94-95 (1997) 456. [5] C. Donnet et al., Surf. Coat. Tech., 94-95 (1997) 531. Figure 1: Constant load scratch micrograph FDLC CY5

0704.0786

Implication of the D^0 Width Difference On CP-Violation in D^0-\bar D^0 Mixing

IPPP/07/10 DCPT/07/20 Implication of the D0 Width Difference On CP-Violation in D0-D̄0 Mixing Patricia Ball IPPP, Department of Physics, University of Durham, Durham DH1 3LE, UK Abstract Both BaBar and Belle have found evidence for a non-zero width difference in the D0-D̄0 system. Although there is no direct experimental evidence for CP-violation in D mixing (yet), we show that the measured values of the width difference y ∼ ∆Γ already imply constraints on the CP-odd phase in D mixing, which, if significantly different from zero, would be an unambiguous signal of new physics. ∗[email protected] http://arxiv.org/abs/0704.0786v2 The highlight of this year’s Moriond conference on electroweak interactions and unified theories arguably was the announcement by BaBar and Belle of experimental evidence for D0-D̄0 mixing [1, 2, 3], which was quickly followed by a number of theoretical anal- yses [4, 5, 6, 7, 8, 9]. While Refs. [4, 7, 8, 9] focused on the constraints posed, by the experimental results, on various new-physics models, Ref. [5] presented a first analysis of the implications of these results for the fundamental parameters describing D mixing. The purpose of this letter is to show that the present experimental results already imply constraints on a sizeable CP-odd phase in D mixing, which could only be due to new physics (NP). To start with, let us shortly review the theoretical formalism of D mixing and the experimental results, see Refs. [10, 11] for more detailed reviews. In complete analogy to B mixing, D mixing in the SM is due to box diagrams with internal quarks andW bosons. In contrast to B, though, the internal quarks are down-type. Also in contrast to B mixing, the GIM mechanism is much more effective, as the contribution of the heaviest down-type quark, the b, comes with a relative enhancement factor (m2b −m2s,d)/(m2s −m2d), but also a large CKM-suppression factor |VubV ∗cb|2/|VusV ∗cs|2 ∼ λ8, which renders its contribution to D mixing ∼ 1% and hence negligible. As a consequence, D mixing is very sensitive to the potential intervention of NP. On the other hand, it is also rather difficult to calculate the SM “background” to D mixing, as the loop-diagrams are dominated by s and d quarks and hence sensitive to the intervention of resonances and non-perturbative QCD. The quasi-decoupling of the 3rd quark generation also implies that CP violation in D mixing is extremely small in the SM, and hence any observation of CP violation will be an unambiguous signal of new physics, independently of hadronic uncertainties. The theoretical parameters describing D mixing can be defined in complete analogy to those for B mixing: the time evolution of the D0 system is described by the Schrödinger equation D0(t) D̄0(t) M − i Γ D0(t) D̄0(t) with Hermitian matrices M and Γ. The off-diagonal elements of these matrices, M12 and Γ12, describe, respectively, the dispersive and absorptive parts of D mixing. The flavour-eigenstates D0 = (cū), D̄0 = (uc̄) are related to the mass-eigenstates D1,2 by |D1,2〉 = p|D0〉 ± q|D̄0〉 (2) M12 − i2 Γ12 ; (3) |p|2 + |q|2 = 1 by definition. The basic observables in D mixing are the mass and lifetime difference of D1,2, which are usually normalised to the average lifetime Γ = (Γ1 + Γ2)/2: x ≡ ∆M M2 −M1 , y ≡ ∆Γ Γ2 − Γ1 . (4) In this letter we follow the sign convention of Ref. [5], according to which x is positive by definition. The sign of y then has to be determined from experiment. In addition, if there is CP-violation in the D system, one also has 6= 1, φ ≡ arg(M12/Γ12) 6= 0. (5) While previously only bounds on x and y were known, both BaBar and Belle have now found evidence for non-vanishing mixing in the D system. BaBar has obtained this evidence from the measurement of the doubly Cabibbo-suppressed decay D0 → K+π− (and its CP conjugate), yielding y′ = (0.97± 0.44(stat)± 0.31(syst))× 10−2, x′2 = (−0.022± 0.030(stat)± 0.021(syst))× 10−2, (6) while Belle obtains yCP = (1.31± 0.32(stat)± 0.25(syst))× 10−2 (7) from D0 → K+K−, π+π− and x = (0.80±0.29(stat)±0.17(syst))×10−2, y = (0.33±0.24(stat)±0.15(syst))×10−2 (8) from a Dalitz-plot analysis of D0 → K0Sπ+π−. Here yCP → y in the limit of no CP violation in D mixing, while the primed quantities x′, y′ are related to x, y by a rotation by a strong phase δKπ: y′ = cos δKπ − x sin δKπ, x′ = x cos δKπ + y sin δKπ. (9) Limited experimental information on this phase has been obtainted at CLEO-c [12]: cos δKπ = 1.09± 0.66 , (10) which can be translated into δKπ = (0 ± 65)◦. An analysis with a larger data-set is underway at CLEO-c, with an expected uncertainty of ∆ cos δKπ ≈ 0.1 in the next couple of years [13]; BES-III is expected to reach ∆ cos δKπ ≈ 0.04 after 4 years of running [14]. The experimental result (10) agrees with theoretical expectations, δKπ = 0 in the SU(3)-limit and |δKπ| <∼ 15 ◦ from a calculation of the amplitudes in QCD factorisation [15]. Based on these experimental results, a preliminary HFAG-average was presented at the 2007 CERN workshop “Flavour in the Era of the LHC” [13]: x = (8.5+3.2 −3.1)× 10−3, y = (7.1+2.0−2.3)× 10−3. (11) Adding errors in quadrature, this implies = 1.2± 0.6. (12) The exact relations between ∆M , ∆Γ, M12 and Γ12 are given by (∆M)2 − 1 (∆Γ)2 = 4|M12|2 − |Γ12|2, (∆M)(∆Γ) = 4Re(M∗ Γ12) = 4|M12||Γ12| cosφ . (13) Eq. (13) implies x/y > 0 for |φ| < π/2 and x/y < 0 for π/2 < |φ| < 3π/2. In view of the above experimental results, we assume |φ| < π/2 from now on. As for the CP-violating observables, |q/p| 6= 1 characterises CP-violation in mixing and can be measured for instance in flavour-specific decays D0 → f , where D̄0 → f is possible only via mixing. The prime example is semileptonic decays with ASL = Γ(D0 → ℓ−X)− Γ(D̄0 → ℓ+X) Γ(D0 → ℓ−X) + Γ(D̄0 → ℓ+X) |q/p|2 − |p/q|2 |q/p|2 + |p/q|2 . (14) Although the B factories may have some sensitivity to this asymmetry, its measurement is severely impaired by the fact that D mixing proceeds only very slowly, resulting in a large suppression factor of the mixed vs. the unmixed rate: Γ(D0 → ℓ−X) Γ(D0 → ℓ+X) x2 + y2 2 + x2 + y2 ≈ 6× 10−5. (15) Both in the K and the B system the quantity − 1 (16) is very small, which however need not necessarily be the case forD’s. From (3) one derives the general expression 4 + r2 + 4r sin φ 4 + r2 − 4r sinφ with r = |Γ12/M12| and the weak phase φ defined in (5). In the B system, one has r ≪ 1 (the current up-to-date numbers are r ≈ 7 × 10−3 for Bd and r ≈ 5 × 10−3 for Bs [16]), so that upon expansion in r = 1 + sin φ+O(r2). (18) Note that this formula refers to the definition φ = arg(M12/Γ12), which differs by +π from the one used in Ref. [16], φ = arg(−M12/Γ12). For the K system, one finds r ≈ |∆Γ/∆M | ≈ 2 from experiment, but now the phase φ turns out to be small, so that = 1 + 4 + r2 φ+O(φ2) ≈ 1 + φ. (19) In both cases, |q/p| ≈ 1 to a very good approximation. In the D system, however, there is no natural hierarchy r ≪ 1, and of course one hopes that NP-effects induce |φ| ≫ 0. In -1.5 -1. -0.5 0. 0.5 1. 1.5 Figure 1: |q/p|2, Eq. (20), as a function of the CP-odd phase φ for the central experimental value r̃ = 7.1/8.5. Solid line: full expression, dashed line: first order expansion around φ = 0. this case, and because x and y have been measured, while |M12| and |Γ12| are difficult to calculate, it is convenient to express |q/p|D in terms of x, y, φ, using the exact relations (13). From (3), and defining r̃ = y/x, we then obtain 2(1 + r̃2) 2(1 + r̃2)2 + 16r̃2 tan2 φ +8r̃ tanφ secφ (1 + r̃2)2 − (1− r̃2)2 sin2 φ . (20) Note that for finite xy and φ = ±π/2, |q/p| diverges because xy → 0 for φ → ±π/2 from (13). In Fig. 1 we plot |q/p|2 as function of φ, for the central experimental value from HFAG, r̃ = 7.1/8.5, Eq. (11). It is obvious that even for moderate values of φ the small-φ expansion is not really reliable. What is the currently available experimental information on CP-violating in D mixing, i.e. |q/p| and φ? As already mentioned, the semileptonic CP-asymmetry (14) has not been measured yet. What has been measured, though, is the effect of CP-violation on the time-dependent rates of D0 → K+π− and D̄0 → K−π+. The BaBar collaboration has parametrised these rates as Γ(D0(t) → K+π−) ∝ e−Γt x′2+ + y (Γt)2 Γ(D̄0(t) → K−π+) ∝ e−Γt + y′2 (Γt)2 and fit the D0 and D̄0 samples separately. They find [2] = (9.8± 6.4(stat)± 4.5(syst))× 10−3, = (9.6± 6.1(stat)± 4.3(syst))× 10−3. (22) Adding errors in quadrature, this means y′ = 1.0±1.1. BaBar also obtains values for which we do not quote here, because the sensitivity to the quadratic term in (21) is -1.5 -1. -0.5 0. 0.5 1. -1.5 -1. -0.5 0. 0.5 1. 1.5 Figure 2: Left: y′+/y as function of φ for x/y = 1.2 (solid line) and x/y = {0.6, 1.8} (dashed lines), from Eq. (11). δKπ = 0. Right: y as function of φ for x/y = 1.2 for δKπ = 0 (solid line) and δKπ = ±65◦ (dashed lines). less than that to the linear term in y′ D is the ratio of the doubly Cabbibo-suppressed to the Cabibbo-favoured amplitude, R D = |A(D0 → K+π−)/A(D0 → K−π+)|. δKπ is the relative strong phase in the Cabibbo-favoured and suppressed amplitudes: A(D0 → K+π−) A(D̄0 → K+π−) −iδKπ ; (23) the minus-sign comes from the relative sign between the CKM matrix elements Vcd and Vus. In the limit of no CP-violation in the decay amplitude, one has |A(D0 → K−π+)| = |A(D̄0 → K+π−)|, which is expected to be a very good approximation, in view of the fact that the decay is solely due to a tree-level amplitude. Then the relation of y′ to x, y and φ is given by {(y cos δKπ − x sin δKπ) cosφ+ (x cos δKπ + y sin δKπ) sinφ} , {(y cos δKπ − x sin δKπ) cosφ− (x cos δKπ + y sin δKπ) sinφ} . (24) Presently, the experimental result for y′+/y is compatible with 1, although with con- siderable uncertainties. Any significant deviation from 1 would be a sign for new physics. In Fig. 2 we plot y′+/y as function of φ, for different values of x/y and δKπ. The figures clearly show that the value of y′ is very sensitive to the phase φ, at least if δKπ is not too close to −65◦, which corresponds to the nearly constant dashed line in Fig. 2b. The reason for this dependence on δKπ becomes clearer if y is expanded to first order in = 1− 2φ x(x 2 + 2y2) cos δKπ + y 3 sin δKπ (x2 + y2)(x sin δKπ − y cos δKπ) +O(φ2) . (25) For the central values of x and y, Eq. (11), this amounts to 1+3.4φ for δKπ = 0, 1− 3.3φ for δKπ = +65 ◦ and 1 + 0.45φ for δKπ = −65◦, which explains the shape of the curves in Fig. 2b. Evidently it is important to reduce the uncertainty of δKπ, which, as mentioned x/ySM Figure 3: Plot of |∆Γ/∆ΓSM|, Eq. (26), as a function of x/ySM and φ. earlier, will be achieved within the next few years. On the other hand, as shown in Fig. 2a, y′ , which depends only on the ratio x/y, but not x and y separately, is not very sensitive to the precise value of that ratio, but very much so to φ. The conclusion is that, even if x/y itself cannot be determined very precisely, y′ will nonetheless be a powerful tool to constrain φ, at least once δKπ will be known more precisely. Already now very large values φ ∼ π/2 are excluded. Another, more theory-dependent constraint on φ can be derived from the value of y. This argument centers around the fact that (a) the experimental result (11) is at the top end of theoretical predictions ySM ∼ 1% [17] and (b) new physics indicated by a non-zero value of φ always reduces the lifetime difference, independently of the value of x. This observation is similar to what was found, some time ago, for the Bs system [18]. In order to derive it, we assume that new physics does not affect Γ12, 1 so that Γ12 = Γ 12 . We then have 2|Γ12| = ∆ΓSM and hence |ySM| = |Γ12|/Γ. Using the relations (13), we can then express the ratio |∆Γ/∆ΓSM| in terms of ySM, x and φ: + x2/ cos2 φ . (26) This implies that new physics always reduces the lifetime difference, independently of the value of x (and any new physics in the mass difference). In particular one has y = 0 for φ = ±π/2 and x 6= 0, which follows from the 2nd relation (13). Eq. (26) is the manifestation of the fact that one does not need to observe CP-violation in order to constrain it. A famous example for this is the unitarity triangle in B physics, whose sides are determined from CP-conserving quantities only, but nonetheless allow a precise measurement of the size of CP-violation in the SM, via the angles and the area of the triangle. In Fig. 3, we plot |∆Γ/∆ΓSM| as a function of r = x/ySM. The zero at φ = ±π/2 is clearly visible. The experimental value |y/ySM| = O(1) then excludes phases φ close to ±π/2. In order to make more quantitative statements, apparently a more precise calculation of ySM is needed. 1See, however, Ref. [19] for a discussion of the effect of tiny NP admixtures to Γ12. -1.5 -1. -0.5 0. 0.5 1. 1.5 5.yCP -1.5 -1. -0.5 0. 0.5 1. 1.5 Figure 4: Left: yCP/y as function of φ, for x/y = 1.2 (solid line) and x/y = {0.6, 1.8} (dashed lines), see Eq. (12). Right: AΓ/yCP as function of φ. Two more CP-sensitive observables related to D0 → K+K− have been measured by the Belle collaboration [3]: yCP = [Γ(D0 → K+K−) + Γ(D̄0 → K+K−)]− 1 y cosφ+ x sin φ, (27) [Γ(D0 → K+K−)− Γ(D̄0 → K+K−)]− 1 y cos φ+ x sin φ. (28) The present experimental value of yCP is given in (7), that for AΓ is (0.01± 0.30(stat)± 0.15(syst)) × 10−2. Again, we can study the dependence of these observables on φ. In Fig. 4a we plot the ratio yCP/y, which is a function of x/y and φ, in dependence on φ. As it turns out, this quantity is far less sensitive to φ than y′ , the reason being that its deviation from 1 is only a second-order effect in φ: yCP = y 1 + φ2 x4 + x2y2 − y4 2(x2 + y2)2 +O(φ4) . (29) Hence, unless the experimental accuracy is dramatically increased, and because the results on y′ and y/ySM already exclude a large CP-odd phase φ ≈ ±π/2, it is safe to interpret yCP as measurement of y. In Fig. 4b we plot the quantity AΓ/yCP. Also here there is a distinctive dependence on φ, with AΓ/y ∝ φ for small φ, but the effect is less dramatic than that in y′ In conclusion, we find that the experimental results on D mixing reported by BaBar and Belle already exclude extreme values of the CP-odd phase φ close to ±π/2. This follows from the result for y, which is close to the top end of theoretical predictions and can only be reduced by new physics, and from y′ ∼ 1. While y′ − 1 vanishes in the limit of no CP-violation, y ∼ ∆Γ is a CP-conserving observable, which demonstrates the usefulness of such quantities in constraining CP-odd phases. Also yCP, AΓ and the ratio AΓ/yCP can be useful in constraining φ. As long as there is no major breakthrough in theoretical predictions for D mixing, which are held back by the fact that the D meson is at the same time too heavy and too light for current theoretical tools to get a proper grip on the problem, the long-distance SM contributions to x will completely obscure any NP contributions and their detection. The observation of CP violation, however, presents a theoretically clean way for NP to manifest itself and it is to be hoped that in the near future, i.e. at the B factories or the LHC, at least one of the plentiful opportunities for NP to show up in CP violation [20] will be realised. Acknowledgments This work was supported in part by the EU networks contract Nos. 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0704.0787

Convergence of a finite volume scheme for the incompressible fluids

Untitled Numerische Mathematik manuscript No. (will be inserted by the editor) Sébastien Zimmermann Convergence of a finite volume scheme for the incompressible fluids Received: date / Revised: date Abstract We consider a finite volume scheme for the two-dimensional in- compressible Navier-Stokes equations. We use a triangular mesh. The un- knowns for the velocity and pressure are respectively piecewise constant and affine. We use a projection method to deal with the incompressibility con- straint. The stability of the scheme has been proven in [15]. We infer from it its convergence. Mathematics Subject Classification (2000) Incompressible fluids · Navier-Stokes equations · projection methods · finite volume 1 Introduction We consider the flow of an incompressible fluid in a open bounded polyhedral set Ω ⊂ R2 during the time interval [0, T ]. The velocity field u : Ω× [0, T ] → 2 and the pressure field p : Ω × [0, T ] → R satisfy the Navier-Stokes equa- tions ∆u+ (u ·∇)u+∇p = f , (1) div u = 0 , (2) with the boundary and initial condition u|∂Ω = 0 , u|t=0 = u0. The terms ∆u and (u ·∇)u are associated with the physical phenomena of diffusion and convection, respectively. The Reynolds number Re measures the S. Zimmermann 17 rue Barrème, 69006 Lyon - FRANCE Tel.: (+33)0472820337 E-mail: [email protected] http://arxiv.org/abs/0704.0787v1 2 Sébastien Zimmermann influence of convection in the flow. For equations (1)–(2), finite element and finite difference methods are well known and mathematical studies are avail- able (see [9] for example). For finite volume schemes, numerous computations have been conducted ([12] and [1] for example). However, few mathematical results are available in this case. Let us cite Eymard and Herbin [5] and Eymard, Latché and Herbin [6]. In order to deal with the incompress- ibility constraint, these works use a penalization method. Another way is to use the projection methods which have been introduced by Chorin [4] and Temam [13]. This is the case in Faure [8] where the mesh is made of squares. In Zimmermann [14] the mesh is made of triangles, which allows more complex geometries. In the present paper the mesh is also made of tri- angles, but we consider a different discretisation for the pressure. It leads to a linear system with a better-conditioned matrix. The layout of the article is the following. We first introduce (section 2.1) some notations and hypotheses on the mesh. We define (section 2.2) the spaces we use to approximate the velocity and pressure. We define also (section 2.3) the operators we use to approximate the differential operators in (1)–(2). By combining this with a projection method, we build the scheme in section 3. In order to provide a mathematical analysis, we state in section 4 that the differential operators in (1)–(2) and their discrete counterparts share similar properties. In particular, the discrete operators for the gradient and the divergence are adjoint. We then prove in section 5 the convergence of the scheme. We conclude with some notations. We denote by χI the characteristic func- tion of an interval I ⊂ R. We denote by C∞0 = C∞0 (Ω) the set of the functions with a compact support in Ω. The spaces (L2, |.|) and (L∞, ‖.‖∞) are the usual Lebesgue spaces and we set L20 = {q ∈ L2 ; q dx = 0}. Their vectorial counterparts are (L2, |.|) and (L∞, ‖.‖∞) with L2 = (L2)2 and L∞ = (L∞). For k ∈ N∗, (Hk, ‖ · ‖k) is the usual Sobolev space. Its vectorial counterpart is (Hk, ‖.‖k) with Hk = (Hk)2. For k = 1, the functions of H1 with a null trace on the boundary form the space H10. Also, we set ∇u = (∇u1,∇u2)T if u = (u1, u2) ∈ H1. If X ⊂ L2 is a Banach space, we define C(0, T ;X) (resp. L2(0, T ;X)) as the set of the applications g : [0, T ] → X such that t → |g(t)| is continous (resp. square integrable). The norm ‖.‖C(0,T ;X) is defined by ‖g‖C(0,T ;X) = sups∈[0,T ] |g(s)|. Finally in all calculations, C is a generic positive constant, depending only on Ω, u0 and f . 2 Discrete setting First, we introduce the spaces and operators needed to build the mesh. 2.1 The mesh Let Th be a triangular mesh of Ω: Ω = ∪K∈ThK. For each triangle K ∈ Th, we denote by |K| its area and EK the set of his edges. If σ ∈ EK , nK,σ is the unit vector normal to σ pointing outwards of K. The set of edges of the mesh is Eh = ∪K∈ThEK . The length of an edge σ ∈ Eh is |σ|. The set of edges inside Ω (resp. on the boundary) is E inth (resp. Eexth ): Convergence of a finite volume scheme for the incompressible fluids 3 Eh = E inth ∪ Eexth . If σ ∈ E inth , Kσ and Lσ are the triangles sharing σ as an edge. If σ ∈ Eexth , only the triangle Kσ inside Ω is defined. We denote by xK the circumcenter of a triangle K. We assume that the measure of all interior angles of the triangles of the mesh are below π that xK ∈ K. If σ ∈ E inth (resp. σ ∈ Eexth ) we set dσ = d(xKσ ,xLσ ) (resp. dσ = d(xσ,xKσ )). We define for all edge σ ∈ Eh: τσ = . The maximum circumradius of the triangles of the mesh is h. We assume that there exists C > 0 such that ∀σ ∈ Eh, d(xKσ , σ) ≥ C|σ| and |σ| ≥ Ch. It implies that there exists a constant C > 0 such that for all edge σ ∈ Eh τσ ≥ C (3) and for all triangles K ∈ Th we have (with σ ∈ EK and hK,σ the matching altitude) |K| = 1 |σ|hK,σ ≥ |σ| d(xK ,xσ) ≥ C h2. (4) 2.2 The discrete spaces We first define P0 = {q ∈ L2 ; ∀K ∈ Th, q|K is a constant} , P0 = (P0)2. For the sake of concision, we set for all qh ∈ P0 (resp. vh ∈ P0) and all triangle K ∈ Th: qK = qh|K (resp. vK = vh|K). Although P0 6⊂ H1, we define the discrete equivalent of a H1 norm as follow. For all vh ∈ P0 we set ‖vh‖h = σ∈Eint τσ |vLσ − vKσ |2 + σ∈Eext τσ |vKσ |2 . (5) We have [7] a discrete Poincaré inequality for P0: there exists C > 0 such that for all vh ∈ P0 |vh| ≤ C ‖vh‖h. (6) From the norm ‖.‖h we deduce a dual norm. For all vh ∈ P0 we set ‖vh‖−1,h = sup (vh,ψh) ‖ψh‖h . (7) For all uh ∈ P0 and vh ∈ P0 we have (uh,vh) ≤ ‖uh‖−1,h ‖vh‖h. We define the projection operator ΠP0 : L 2 → P0 as follows. For all w ∈ L2, ΠP0w ∈ P0 is given by ∀K ∈ Th , (ΠP0w)|K = w(x) dx. (8) 4 Sébastien Zimmermann We easily check that for all w ∈ L2 and vh ∈ P0 we have (ΠP0w,vh) = (w,vh). We deduce from this that ΠP0 is stable for the L 2 norm. We define also the operator Π̃P0 : H 2 → P0. For all v ∈ H2, Π̃P0v ∈ P0 is given by ∀K ∈ Th , Π̃P0v|K = v(xK). According to the Sobolev embedding theorem, v ∈ H2 is a.e. equal to a continuous function. Therefore the definition above makes sense. One checks [14] that there exists C > 0 such that |v − Π̃P0v| ≤ C h ‖v‖2 (9) for all v ∈ H1. We introduce also the finite element spaces 1 = {v ∈ L2 ; ∀K ∈ Th, v|K is affine} , Pnc1 = {vh ∈ P d1 ; ∀σ ∈ E inth , vh|Kσ (xσ) = vh|Lσ(xσ) , Pnc1 = (Pnc1 )2. If qh ∈ Pnc1 , we have usually ∇qh 6∈ L2. Therefore we define the operator ∇h : Pnc1 → P0 by setting for all qh ∈ P0 and all triangle K ∈ Th ∇hqh|K = ∇qh dx. (10) We define the projection operator ΠPnc . For all q ∈ H1, ΠPnc q is given by ∀σ ∈ Eh , (ΠPnc q) dσ = q dσ. (11) We also set ΠPnc = (ΠPnc )2. One checks that there exists C > 0 such that ∣∣∇q −∇h(ΠPnc ∣∣ ≤ C h ‖q‖2 , |v −ΠPnc v| ≤ C h ‖v‖1 , (12) for all q ∈ H1 and v ∈ H1. We also use the Raviart-Thomas spaces = {vh ∈ Pd1 ; ∀σ ∈ EK , vh|K · nK,σ is constant, and vh · n|∂Ω = 0} , RT0 = {vh ∈ RTd0 ; ∀K ∈ Th, ∀σ ∈ EK , vh|Kσ · nKσ,σ = vh|Lσ · nKσ,σ}. For all vh ∈ RT0, K ∈ Th and σ ∈ EK we set (vh · nK,σ)σ = vh|K · nK,σ. We define the operator ΠRT0 : H 1 → RT0. For all v ∈ H1, ΠRT0v ∈ RT0 is given by ∀K ∈ Th , ∀σ ∈ EK , (ΠRT0v · nK,σ)σ = v dσ. (13) One checks [3] that there exists a constant C > 0 such that for all v ∈ H1 |v −ΠRT0v| ≤ C h ‖v‖1. (14) Convergence of a finite volume scheme for the incompressible fluids 5 2.3 The discrete operators The equations (1)–(2) use the differential operators gradient, divergence and laplacian. Using the spaces of section 2.2, we now define their discrete coun- terparts. The discrete gradient ∇h : Pnc1 → P0 is defined by (10). The discrete divergence operator divh : P0 → Pnc1 is built so that it is adjoint to the operator ∇h (proposition 3 below). We set for all vh ∈ P0 and all triangle K ∈ Th ∀σ ∈ E inth , (divh vh)(xσ) = 3 |σ| |Kσ|+ |Lσ| (vLσ − vKσ ) · nK,σ ; ∀σ ∈ Eexth , (divh vh)(xσ) = − 3 |σ| |Kσ|+ |Lσ| vKσ · nK,σ. (15) The first discrete laplacian ∆h : P 1 → Pnc1 is given by ∀ qh ∈ Pnc1 , ∆hqh = divh(∇hqh). The second discrete laplacian ∆̃h : P0 → P0 is the usual operator in finite volume schemes [7]. We set for all vh ∈ P0 and all triangle K ∈ Th ∆̃hvh|K = σ∈EK∩E τσ (vLσ − vKσ)− σ∈EK∩E τσ vKσ . (16) In order to approximate the convection term (u · ∇)u in (1), we define a bilinear form b̃h : RT0 ×P0 → P0 using the well-known [7] upwind scheme. For all uh ∈ P0, vh ∈ P0, and all triangle K ∈ Th we set b̃h(uh,vh) σ∈EK∩E (u · nK,σ)+σ vK + (u · nK,σ)−σ vLσ We have set a+ = max(a, 0), a− = min(a, 0) for all a ∈ R. Lastly, we define the trilinear form bh : RT0 × P0 × P0 → R as follows. For all uh ∈ RT0, vh ∈ P0, wh ∈ P0, we set bh(uh,vh,wh) = |K|wK · b̃h(uh,vh) 3 The scheme We have defined in section 2 the discretization in space. We now have to define the discretization in time, and treat the incompressibility constraint (2). We use a projection method to this end. This kind of method has been introduced by Chorin [4] and Temam [13]. The time interval [0, T ] is split with a time step k: [0, T ] = n=0[tn, tn+1] with N ∈ N∗ et tn = n k for all n ∈ {0, . . . , N}. We start with the initial values u0h ∈ P0 ∩RT0 , u1h ∈ P0 ∩RT0 , p1h ∈ Pnc1 ∩ L20. For all n ∈ {1, . . . , N}, (ũn+1h , p h ) is deduced from (ũ h) as follows. 6 Sébastien Zimmermann – ũn+1h ∈ P0 is given by 3 ũn+1h − 4unh + u ∆̃hũ + b̃h(2u h − un−1h , ũ h ) +∇hp h = f h , (17) – pn+1h ∈ Pnc1 ∩ L20 is the solution of h − p divh ũ – un+1h ∈ P0 is given by un+1h = ũ ∇h(pn+1h − p h). (18) We have proven in [14] that the scheme is well defined. In particular the term b̃h(2u h − u h ) in (17) is defined thanks to the following result. Proposition 1 For m ∈ {0, . . . , N} we have umh ∈ RT0. Note also that for m ∈ {0, . . . , N} we have divumh = 0, since umh ∈ P0. Thus the incompressibility condition (2) is fullfilled. 4 Properties of the discrete operators The operators defined in section 2.3 have the following properties [14]. Proposition 2 There exists a constant C > 0 such that for all uh ∈ RT0 satisfying divuh = 0, vh ∈ P0, wh ∈ P0: |bh(uh,vh,vh)| ≤ C |uh| ‖vh‖h ‖wh‖h. Proposition 3 For all vh ∈ P0 and qh ∈ Pnc1 : (vh,∇hqh) = −(qh, divh vh). Proposition 4 For all uh ∈ P0 and vh ∈ P0: −(∆̃huh,uh) = ‖uh‖2h and −(∆̃huh,vh) ≤ ‖uh‖h ‖vh‖h. If v ∈ H1 we have |divv| ≤ ‖v‖1. The operator divh has a similar property. Proposition 5 There exists a constant C > 0 such that for all vh ∈ P0 |divh vh| ≤ C ‖vh‖h. Proof. Using a quadrature formula we have |divh vh|2 = |(divh vh)(xσ)|2 . Convergence of a finite volume scheme for the incompressible fluids 7 Let K ∈ Th. Using definition and (4) we have ∀σ ∈ EK ∩ E inth , |(divh vh)(xσ)| |K| |vLσ − vK | ∀σ ∈ EK ∩ Eexth , |(divh vh)(xσ)| 2 ≤ C |vK |2. Thus: |divh vh|2 ≤ C σ∈EK∩E |vLσ − vK |2 + σ∈EK∩E |vK |2 Writing the sum over the triangles as a sum over the edges, we get |divh vh|2 ≤ C σ∈Eint τσ |vLσ − vK |2 + σ∈Eext τσ |vK |2  ≤ C ‖vh‖2h. Proposition 6 If uh ∈ P0 and vh ∈ P0 we have (∆̃huh,vh) = (uh, ∆̃hvh). Proof. Using definition (2.3) one checks that (∆̃huh,vh) = σ∈Eint τσ (vLσ − vKσ ) · (uLσ − uKσ)− σ∈Eext τσ vKσ · uKσ = (∆̃hvh,uh). Proposition 7 There exists C > 0 such that for all v ∈ H2 satisfying ∇v · n|∂Ω = 0 ‖ΠP0(∆v)−∆h(Π̃P0v)‖−1,h ≤ C h ‖v‖2. Proof. Let ψh ∈ P0. We have ΠP0(∆q)−∆h(Π̃P0v),ψh ΠP0(∆q)−∆h(Π̃P0v) ·ψK . For all K ∈ Th, using (2.3) and the divergence formula, we get ΠP0(∆v) −∆h(Π̃P0v) ) ∣∣∣ σ∈EK∩E ∇v · nK,σ dσ − τσ v(xLσ )− v(xK) Thus, by writing the sum over the triangles as a sum over the edges, we get ΠP0(∆v)−∆h(Π̃P0v),ψh σ∈Eint (ψLσ −ψKσ)Rσ with Rσ = ∇v · nKσ,σ − 1dσ v(xLσ ) − v(xKσ ) dσ. We denote by Dσ the quadrilatere defined by xKσ , xLσ and the endpoints of σ. Using a Taylor expansion and a density argument, we get as in [7] that |Rσ| ≤ C h |H(vi)(y)|2 dy 8 Sébastien Zimmermann Thus, using the Cauchy-Schwarz inequality, we get ΠP0(∆q)−∆h(Π̃P0v),ψh ≤ C h σ∈Eint |ψLσ −ψKσ | σ∈Eint |H(vi)(y)|2 dy According to (3) σ∈Eint |ψLσ −ψKσ | 2 ≤ C σ∈Eint τσ |ψLσ −ψKσ | 2 ≤ C ‖ψ‖2h. ΠP0(∆v)−∆h(Π̃P0v),ψh )∣∣∣ ≤ C h ‖ψh‖h ‖v‖2. Using (7) we get the result. 5 Convergence of the scheme We first recall the stability result that has been proven in [15]. We deduce from it an estimate on the Fourier transform of the computed velocity (lemma 1). Using a result on space P0, we infer from it the convergence of the scheme (theorem 2). One shows that if the data u0 et f fulfill a compati- bility condition [11], there exists a solution (u, p) to equations (1)–(2) such that u ∈ C(0, T ;H2) , ∇p ∈ C(0, T ;L2). We assume from now on that there exists C > 0 such that (HI) |u0h−u0|+ ‖u1h−u(t1)‖∞+|p1h−p(t1)| ≤ C h , |u1h−u0h| ≤ C k. Let us recall the following result [15]. Theorem 1 We assume that the initial values of the scheme fulfill (HI). There exists a constant C > 0 such that for all m ∈ {2, . . . , N} |umh |+ k ‖ũnh‖2h + |umh − um−1h |+ ‖ũnh − ũn−1h ‖ h ≤ C |pnh|2 + k |∇hpmh |+ |∇h(pmh − pm−1h )| ≤ C. From now on we set ũ1h = u h for the sake of conveniance. One deduces from hypothesis (HI) [14] that |p1h| ≤ C and ‖ũ1h‖h ≤ C. Now, let ε = max(h, k). We study the behaviour of the scheme as ε → 0. We define the applications uε : R → P0 , ũε : R → P0 , ũcε : R → P0, pε : R → Pnc1 and fε : R → P0 as follows. For all n ∈ {0, . . . , N − 1} and all t ∈ [tn, tn+1] we set uε(t) = u h , ũε(t) = ũ h , ũ ε(t) = ũ (t− tn) (ũn+1h − ũ pε(t) = p h, fε(t) = ΠP0 f(tn+1) , Convergence of a finite volume scheme for the incompressible fluids 9 and for all t 6∈ [0, T ] we set uε(t) = ũε(t) = ũcε(t) = fε(t) = 0, pε(t) = 0. We recall that the Fourier transform v̂ of a function v ∈ L1(R) is defined by ∀ τ ∈ R, v̂(τ) = e−2iπτt v(t) dt. (19) We have the following result. Lemma 1 Let 0 < γ < 1 . There exists C > 0 such that for all ε > 0 |τ |2γ |̂̃uε(τ)|2 dτ ≤ C. Proof. Let χI be the characteristic function of an interval I ⊂ R. We define the application gε : R → P0 as follows. For all t 6∈ [t1, T ] we set gε(t) = 0. For all t ∈ [t1, T ], gε(t) ∈ P0 is the solution of ∆̃hgε = ∆̃hũε + fε − b̃h 2uε − uε(t− k), ũε with Pε = −∇hpε− 2 ũε(t−k)−uεk + ũε(t−2 k)−uε(t−k) χ[t2,T ]. We have omitted most of the time dependancies for the sake of concision. Let us estimate gε. We have −(∆̃hgε,gε) = −(∆̃hũε,gε)− (fε,gε) (20) 2uε − uε(t− k), ũε,gε + (Pε,gε). According to proposition 4 we have −(∆̃hgε,gε) = ‖gε‖2h , −(∆̃hũε,gε) ≤ ‖ũε‖h ‖gε‖h. Using the Cauchy-Schwarz inequality and (6) we have −(fε,gε) ≤ |fε| |gε| ≤ C |fε| ‖gε‖h. According to proposition 2 and theorem 1 2uε − uε(t− k), ũε,gε ≤ C |2uε − uε(t− k)| ‖ũε‖h ‖gε‖h ≤ C ‖ũε‖h ‖gε‖h. Using (18) we have Pε = − χ[t3,T ] ∇hpε+ χ[t3,T ]∇hpε(t−k)− χ[t3,T ] ∇hpε(t−2 k). Using proposition 3 and the Cauchy-Schwarz inequality we get |(Pε,gε)| ≤ C |pε|+ χ[t3,T ] |pε(t− k)|+ χ[t3,T ] |pε(t− 2 k)| |divh gε|. Using proposition 5 we have |(Pε,gε)| ≤ C |pε|+ χ[t3,T ] |pε(t− k)|+ χ[t3,T ] |pε(t− 2 k)| ‖gε‖h. 10 Sébastien Zimmermann Let us plug these estimates into (20). By simplifying by ‖gε‖h and integrating from t = t1 to T we get ‖gε‖h dt ≤ C + C |pε| dt+ |fε| dt+ ‖ũε‖h dt According to the Cauchy-Schwarz inequality and theorem 1 |pε(t)| dt ≤ |pε(t)|2 dt |pnh|2 Thanks to the stability of ΠP0 for the L 2 norm we have |fε(t)| dt = k |ΠP0 f(tn)| ≤ k |f(tn)| ≤ k ‖f‖C(0,T ;L2) ≤ C. And thanks to the Cauchy-Schwarz inequality and theorem 1 ‖ũε(t)‖h dt ≤ ‖ũε(t)‖2h dt ≤ C k ‖ũnh‖2h ≤ C. Thus, since gε(t) = 0 for t ∈ [0, t1], we get ‖gε(t)‖h = ‖gε(t)‖h dt ≤ C. Using definition (19) we obtain finally ∀ τ ∈ R , ‖ĝε(τ)‖h ≤ C. (21) With this estimate we can now prove the result. Since the function ũcε is piecewise C1 on R, and discontinous for t = 0 and t = T , equation (17) reads dũcε dũcε (t− k) = ∆hgε + (ũ0h δ0 − ũNh δT )− (ũ1h δt1 − ũNh δT+k) ũ1h − ũ0h χ[0,t1] − ũNh − ũ χ[T,T+k] where δ0, δt1 , δT and δT+k are Dirac distributions located respectively in 0, t1, T and T + k. Let τ ∈ R. Applying the Fourier transform we get −2iπτ −2iπτk ̂̃uε(τ) = ∆hĝε(τ) +α (ũ0h − ũNh e−2iπT )− (ũ1h − ũNh e−2iπT ) e−2iπk − 1 ũ1h − ũ0h ũNh − ũ −2iπT e−2iπk − 1 Convergence of a finite volume scheme for the incompressible fluids 11 Taking the scalar product with i ̂̃uε(τ) we get e−2iπτk |̂̃uε(τ)|2 = i ∆hĝε(τ), ̂̃uε(τ) α, ̂̃uε(τ) Let us bound the right-hand side. According to proposition 4 and (21) ∆hĝε(τ), ̂̃uε(τ) )∣∣∣ ≤ ‖ĝε(τ)‖h ‖̂̃uε(τ)‖h ≤ C ‖̂̃uε(τ)‖h. On the other hand, using theorem 1, one checks that α is bounded. Thus, according to the Cauchy-Schwarz inequality and (6) α, ̂̃uε(τ) )∣∣∣ ≤ |α| |̂̃uε(τ)| ≤ C |α| ‖̂̃uε(τ)‖h ≤ C ‖̂̃uε(τ)‖h. Hence we have ∀ τ ∈ R, |τ | |̂̃uε(τ)|2 ≤ C ‖̂̃uε(τ)‖h. If τ 6= 0, multiplying this estimate by |τ |2 γ−1, we get |τ |2 γ |̂̃uε(τ)|2 ≤ C |τ |2 γ−1 ‖̂̃uε(τ)‖h. Using the Young inequality and integrating over {τ ∈ R ; |τ | > 1} we obtain |τ |>1 |τ |2 γ |̂̃uε(τ)|2 dτ ≤ |τ |>1 |τ |4 γ−2 dτ + C |τ |>1 ‖̂̃uε(τ)‖2h dτ. For |τ | ≤ 1 we have |τ |2 γ |̂̃uε(τ)|2 ≤ |̂̃uε(τ)|2 ≤ C ‖̂̃uε(τ)‖2h thanks to (6). |τ |2 γ |̂̃uε(τ)|2 dτ ≤ |τ |>1 |τ |4 γ−2 dτ + C ‖̂̃uε(τ)‖2h dτ. Since 4 γ−2 < −1 we have |τ |>1 |τ |4 γ−2 dτ ≤ C. On the other hand, thanks to the Parseval theorem and thorem 1 ‖̂̃uε(τ)‖2h dτ ≤ ‖̂̃uε(τ)‖2h dt ≤ k ‖ũnh‖2h ≤ C. Hence the result. We introduce the following spaces H = {v ∈ L2 ; divv ∈ L2 et v · n|∂Ω = 0} , V = {v ∈ H10 ; divv = 0}. We also set ((u,v)) = (∇ui,∇vi) , b(u,v,w) = − (vi,u · ∇wi) for all u = (u1, u2) ∈ H1, v = (v1, v2) ∈ H1, w = (w1, w2) ∈ H1. We have the following result. 12 Sébastien Zimmermann Theorem 2 We assume that the initial values of the scheme fulfill hypothesis (HI). We also assume that the space step h and the time step k are such that h ≤ C kα with α > 1. Then we have uε → u in L2(0, T ;L2) with u ∈ C(0, T ;H) ∩ L2(0, T ;V) , ∈ L2(0, T ;L2). (22) We also have u(0) = u0 and for all ψ ∈ C∞0 ([0, T ]) ∀v ∈ V, (u,v) + ((u,v)) + b(u,u,v) − (f ,v) dt = 0. (23) Proof. In what follows, sub-sequences of a sequence (vε)ε>0 will still be noted (vε)ε>0 for the sake of convenience. All the limits are for ε → 0. According to theorem 1 and hypothesis (HI) we have ‖uε‖2L2(0,T ;L2) = k (|u h|2 + |u1h|2) + k |unh|2 ≤ C. We also deduce from (6), hypothesis (HI) and theorem 1 ‖ũε‖2L2(0,T ;L2) = k |u h|2 + k |ũnh|2 ≤ C + C k ‖ũnh‖2h ≤ C. A simple computation shows that there exists C > 0 such that ‖ũcε‖L2(0,T ;L2) ≤ C ‖ũε‖L2(0,T ;L2) ≤ C. Thus the sequences (uε)ε>0, (ũε)ε>0 and (ũ ε)ε>0 are bounded in L 2(0, T ;L2). Therefore there exists u ∈ L2(0, T ;L2), ũ ∈ L2(0, T ;L2) and ũc ∈ L2(0, T ;L2) such that, up to a sub-sequence, we have uε ⇀ u , ũε ⇀ ũ , ũ ε ⇀ ũ c weakly in L2(0, T ;L2). We claim that the limits u, ũ, ũc are the same. Indeed, let us consider uε−ũε. Since unh − ũ h = (u h − u h ) + (u h − ũ h ) we have ‖uε − ũε‖2L2(0,T ;L2) ≤ 2 k |unh − un+1h | 2 + 2 k |un+1h − ũ According to theorem 1 we have k n=0 |unh−u h |2 ≤ C n=0 k 3 ≤ C k2. Thanks to (18) we also have |un+1h − ũ |∇h(pn+1h − p h)|2 ≤ C 3 ≤ C k2. Thus ‖uε − ũε‖L2(0,T ;L2) → 0 and u = ũ. One checks in a simililar way that ũ = ũc. Now, using the Fourier transform, we prove the strong convergence Convergence of a finite volume scheme for the incompressible fluids 13 of the sequence (uε)ε>0 in L 2(0, T ;L2). We set vε = uε −u. Let M > 0. We use the splitting |v̂ε(τ)|2 dτ = |τ |≤M |v̂ε(τ)|2 dτ + |τ |>M |v̂ε(τ)|2 dτ = IMε + JMε . Let us estimate JMε . Since |v̂ε(τ)|2 ≤ 2 |ûε(τ)|2 + 2 |û(τ)|2 we have JMε ≤ 2 |τ |>M |ûε(τ)|2 dτ + 2 |τ |>M |û(τ)|2 dτ. According to lemma 1 we have |τ |>M |ûε(τ)|2 dτ ≤ |τ |>M |τ |2γ |ûε(τ)|2 dτ ≤ |τ |>M |û(τ)|2 dτ. Therefore, for all ε > 0, we have JMε → 0 when M → ∞. We now consider IMε . Let τ ∈ R. Since uε ⇀ u in L2(0, T ;L2), we deduce from definition (19) ̂̃uε(τ)⇀ û(τ) weakly in L2. For all t ∈ R we have ũε(t) ∈ P0. From definition (19) we infer that ̂̃uε(τ) ∈ P0. Now, prolonging ̂̃uε(τ) by 0 outside Ω, we deduce from lemma 4 in [7] that there exists a constant C > 0 such that ∀η∈ R2 , |̂̃uε(τ)(· + η)− ̂̃uε(τ)|2 ≤ ‖̂̃uε(τ)‖2h |η| (|η|+ C h). Using definition (19), the Cauchy-Schwarz inequality and theorem 1, we have ‖̂̃uε(τ)‖2h ≤ C ‖ũε(t)‖2h dt ≤ C k ‖ũnh‖2h ≤ C. Thus, using the compactness criterium given by theorem 1 in [7], we get ̂̃uε(τ) → û(τ) in L2. Thus ̂̃vε(τ) = ̂̃uε(τ)− û(τ) → 0 in L2. Therefore for all M > 0 we have IMε → 0. Using the Parseval inequality, and gathering the limits for IMε and J ε , we get |v̂ε(τ)|2 dτ = |vε|2 dt = |uε − u|2 dτ → 0. We have proven that uε → u in L2(0, T ;L2). We now check the properties of u. First, proceeding as in [7], one checks easily that u ∈ L2(0, T ;H10). Now let q ∈ L2(0, T ; C∞0 ). According to (12) we have ∇h(ΠPnc q) → ∇q in L2(0, T ;L2). Since uε → u in L2(0, T ;L2) we get ∇h(ΠPnc q),uε → (∇q,u) = −(q, divu). 14 Sébastien Zimmermann On the other hand, according to propositions 1 and 3, we have for all ε > 0 ∇h(ΠPnc q),uε = −(ΠPnc q, divh uε) = 0. Thus we have q divu dt = 0 for all q ∈ L2(0, T ; C∞0 ). Since the space C∞0 is dense in L2, we get divu = 0. Hence u ∈ L2(0, T ;V). Let us now check the regularity of . Using hypothesis (HI), (6) and theorem 1, we have dũcε L2(0,T ;L2) |u1h − u0h|2 + k |δũnh|2 ≤ C + C k ‖δũnh‖2h Thus the sequence dũcε is bounded in L2(0, T ;L2). Since uε → u in L2(0, T ;L2) with u ∈ L2(0, T ;H1), proceeding as in, we get dũcε weakly ∈ L2(0, T ;L2) and u ∈ C(0, T ;H). Let us now prove that u satisfies (23). For the sake of simplicity, we omit to note some time dependencies. According to (17) we have for all t ∈ [t1, T ] dũcε dũcε (t− k)− 1 ∆hũε + b̃h 2uε − uε(t− k), ũε χ[t3,T ] ∇hpε + χ[t3,T ] ∇hpε(t− k)− χ[t3,T ] ∇hpε(t− 2 k). Let v ∈ V ∩ (C∞0 )2 and ψ ∈ C∞([0, T ]) with ψ(T ) = 0. We set vh = Π̃P0v. Multiplying the former equation by ψ vh and integrating over [t1, T ] we get dũcε dũcε (t− k),vh dt− 1 ψ (∆̃hũε,vh) dt 2uε − uε(t− k), ũε,vh ψ (fε,vh) dt χψ (∇hpε,vh) dt (24) with χ = −χ[t3,T−2 k] + 13 χ[t2,t3] − χ[t1,t2] − 73 χ[T−k,T ] − χ[T−2 k,T−k]. We now check the limits of the terms in this equation. First, according to (9), we have vh → v in L2. We will use this limit in the computations below without mentioning it. Since ψ(T ) = 0 we obtain by integrating by parts dũcε dt− ψ(t1) (ũ1h,vh)− ψ′ (ũcε,vh) dt dũcε (t− k),vh dt = −ψ(0) (ũ0h,vh)− ∫ T−k ′(t+ k) (ũcε,vh) dt. Convergence of a finite volume scheme for the incompressible fluids 15 According to hypothesis (HI) we have ũ0h = u h → u0 in L2 and ũ1h = u1h → u0 in L2. It implies that (u0h,vh) → (u0,v) and ψ(t1) (ũ1h,vh) = ψ(k) (ũ1h,vh) → ψ(0) (u0,v). On the other hand ψ′ (ũcε,vh) dt = χ[t1,T ] ψ ′ (ũcε,vh) dt→ ψ′ (u,v) dt and since χ[0,T−k] ψ ′(·+ k) → ψ′ in L∞(0, T ) ∫ T−k ′(t+k) (ũcε,vh) dt = χ[0,T−k] ψ ′(t+k) (ũcε,vh) dt→ ′ (u,v) dt. Thus we have dũcε dũcε (t− k),vh dt→ −ψ(0) (u,v)− ψ′ (u,v) dt. (25) Let us now consider the discrete laplacian. Using proposition 6 and the split- ting ∆̃hvh = ∆̃hvh −ΠP0(∆̃v) +ΠP0(∆̃v) we have ∆̃hũε,vh) dt = Aε +Bε with Aε = ũε, ∆̃h(Π̃P0v)−ΠP0(∆̃v) dt,Bε = ũε, ΠP0(∆v) Since |Aε| ≤ ‖∆̃h(Π̃P0v) −ΠP0(∆̃v)‖−1,h ψ ‖ũε‖h dt , using proposition 7 and the Cauchy-Schwarz inequality, we get |Aε| ≤ C h ‖v‖2 ψ ‖ũε‖h dt ≤ C h ψ2 dt )1/2(∫ T ‖ũε‖2h dt Therefore, using theorem 1: |Aε| ≤ C h n=1 ‖ũnh‖2h ≤ C h. Hence Aε → 0. On the other hand, using an integration by parts, we have ψ (ũε, ∆v) dt → ψ (u, ∆v) dt = − ψ ((u,v)) dt. By gathering the limits for Aε and Bε we get ψ (∆̃hũε,vh) dt → − ψ ((u,v)) dt. Let us now consider the pressure. We use the splitting (∇hpε,vh) = (∇hpε,vh − v) + (∇hpε,v −ΠRT0v) + (∇hpε, ΠRT0v). (26) 16 Sébastien Zimmermann First, integrating by parts, we have (∇hpε, ΠRT0v) = − pε, div (ΠRT0v) pε (ΠRT0v · nK,σ). Since divv = 0, using the divergence formula and definition (13), one checks that div (ΠRT0v) = 0. Thus − pε, div (ΠRT0v) = 0. On the other hand pε (ΠRT0v ·nK,σ) = σ∈Eint (ΠRT0v)σ ·nKσ ,σ) (pε|Lσ −pε|Kσ ) dσ and since pε ∈ Pnc1 we get pε (ΠRT0v · nK,σ) = 0. Thus the last term in (26) vanishes. To bound the other terms, we use the Cauchy-Schwartz inequality together with estimates (9), (14) and theorem 1. We get |(∇hpε,v − vh)|+ |(∇hpε,v −ΠRT0v)| ≤ C h |∇hpε| ‖v‖2 ≤ C ‖v‖2. Plugging these estimates into (26) we get |χψ (∇hpε,vh)| dt ≤ C hk . By hypothesis we have h ≤ kα−1 with α− 1 > 0. Thus for ε = max(h, k) → 0 χψ (∇hpε,vh) dt→ 0. Let us now consider the convection term. We set uε = 2uε − uε(t − k) and want to find the limit of ψ bh(uε, ũε,vh) dt. We use the splitting −bh(uε, ũε,vh) + b(u,u,v) = Aε1 +Aε2 +Aε3 with Aε1 = b(u− uε,u,v) , Aε2 = b(uε,u,v) − div(ui uε), v Aε3 = div(ui uε), v − bh(uε,uε,vh). By definition Aε1 = − ui, (u − uε) · ∇vi . Using the Cauchy-Schwarz inequality we get ψ |Aε1| dt ≤ ‖ψ‖∞ ‖v‖W1,∞ ‖u‖L2(0,T ;L2) ‖u− uε‖L2(0,T ;L2). Since uε → u in L2(0, T ;L2) we also have ‖u − uε‖L2(0,T ;L2) → 0. Thus∫ T ψAε1 dt → 0. Let us now consider Aε2. Since uε · n|∂Ω = 0 we obtain by integrating by parts b(uε,u,v) = vi, div(ui uε) . Thus Aε2 = vi − vih, div(ui uε) vi − vih,uε · ∇ui Convergence of a finite volume scheme for the incompressible fluids 17 Using the Cauchy-Schwarz inequality we get ψ |Aε2| dt ≤ C ‖ψ‖L∞ ‖v − vh‖L∞ ‖uε‖L2(0,T ;L2) ‖u‖L2(0,T ;H1). Using a Taylor expansion one checks that ‖v − vh‖L∞ ≤ ‖v‖W1,∞ h. We recall also that ‖uε‖L2(0,T ;L2) ≤ C. Therefore ψAε2 dt → 0. Let us now bound Aε3. For all triangle K ∈ Th and all edge σ ∈ EK ∩ E inth , we set ũεK,Lσ = ũε|K if (uε · nK,σ) ≥ 0 ũε|Lσ if (uε · nK,σ) < 0 Using the divergence formula one checks that Aε3 = σ∈EK∩E (u− ũεK,Lσ ) (uε · nK,σ) dσ. By writing this sum as a sum on the edges we get Aε3 = σ∈Eint (vKσ − vLσ) · (u− ũεKσ ,Lσ) (uε · nKσ,σ) dσ. Thus, using definition (11) and a quadrature formula σ∈Eint (vKσ − vLσ ) (ΠPnc u− ũεKσ,Lσ) (uε · nKσ,σ) dσ σ∈Eint (vKσ − vLσ ) |σ| (ΠPnc u)(xσ)− ũεKσ,Lσ (uε · nKσ,σ)σ. We have |σ| ≤ h and, using a Taylor expansion, one checks that |vKσ−vLσ | ≤ h ‖v‖W1,∞ . Thus, thanks to the Cauchy-Schwarz inequality, we get |Aε3| ≤ C h2 σ∈Eint |uε(xσ)|2 σ∈Eint |(ΠPnc u)(xσ)− ũεKσ ,Lσ | Using (4) we get |Aε3| ≤ C σ∈EK∩E |uε(xσ)|2 σ∈EK∩E |(ΠPnc u)(xσ)− ũε|K |2 Therefore, using a quadrature formula |Aε3| ≤ C |uε| |ΠPnc1 u− uε| ≤ C |uε| |ΠPnc1 u− uε|. 18 Sébastien Zimmermann By writing ΠPnc u − uε = (ΠPnc u − u) + (u − uε) and using (12), we get |Aε3| ≤ C |uε| (h ‖u‖1+ |u−uε|). Thus, using the Cauchy-Schwarz inequality, we have |ψ| |Aε3| dt ≤ C ‖uε‖L2(0,T ;L2) (h ‖u‖L2(0,T ;H1) + ‖u− uε‖L2(0,T ;L2)). ψAε3 dt → 0. By gathering the limits for Aε1, Aε2, Aε3 we obtain∫ T bh(uε,uε,vh)− b(u,u,v) dt→ 0. Since b(u,u,v) dt → 0, we get bh(uε,uε,vh), dt → ψ b(u,u,v) dt. Finally, since vh ∈ P0, we have: (fε,vh) = (ΠP0 f ,vh) = (f ,vh). Therefore ψ (fε,vh) dt = χ[t1,T ] ψ (f ,vh) dt→ ψ (f ,v) dt. We now gather the limits we have obtained into (24). The space V∩(C∞0 )2 is dense in V. Hence we obtain for all v ∈ V and ψ ∈ C∞([0, T ]) with ψ(T ) = 0 −ψ(0) (u0,v)− ′ (u,v) dt+ ((u,v)) +b(u,u,v)− (f ,v) dt = 0. Taking ψ = φ ∈ C∞0 ([0, T ]), we have φ(0) = 0 and from the definition of the derivative in the distributional sense φ′ (u,v) dt = − (u,v) dt. Thus we have proven (23). At last, let us show that the initial condition holds. We have proven before that dũcε weakly in L2(0, T ;L2). Let v ∈ V ∩ (C∞0 )2 and ψ ∈ C∞([0, T ]) such that ψ(T ) = 0. We have dũcε dũcε (t− k),vh Integrating by parts the limit we get dũcε dũcε (t− k),vh dt → −ψ(0) u(0),v ψ′ (u,v) dt. By comparing this limit with (25), we get ψ(0) (u(0) − u0,v) = 0 for all ψ ∈ C∞([0, T ]) with ψ(T ) = 0. Therefore u(0) = u0. At last, note that we have proven so far the convergence of a sub-sequence of (uε)ε>0 towards u. But the application u such that (22), (23) and u(0) = u0 hold is unique ([13], p. 254). Thus the whole sequence (uε)ε>0 converges towards u. Convergence of a finite volume scheme for the incompressible fluids 19 References 1. Boivin, S., Cayre, F., Herard, J. M.: A finite volume method to solve the Navier- Stokes equations for incompressible flows on unstructured meshes. Int. J. Therm. Sci. 39 806–825 (2000). 2. Brenner, S. C., Scott, L.R.: The mathematical theory of finite element methods. Springer, 2002. 3. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer- Verlag, 1991. 4. Chorin, J.: On the convergence of discrete approximations to the Navier-Stokes equations. Math. Comp. 23 341–353 (1969). 5. Eymard, R., Herbin, R.: A staggered finite volume scheme on general meshes for the Navier-Stokes equations in two space dimensions. Int.J. Finite Volumes (2005). 6. Eymard, R., Latché, J. C., Herbin, R.: Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2 or 3D meshes. preprint LATP (2004). 7. Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. P.G. Ciarlet and J.L. Lions eds, North-Holland, 2000. 8. Faure, S.: Stability of a colocated finite volume scheme for the Navier-Stokes equations. Num. Meth. PDE 21(2) 242–271 (2005). 9. Girault, V., Raviart, P. A.: Finite Element Methods for Navier-Stokes equations: Theory and Algorithms. Springer (1986). 10. Guermond, J. L.:Some implementations of projection methods for Navier- Stokes equations. M2AN 30(5) 637–667 (1996). 11. Heywood, J. G., Rannacher, R.: Finite element approximation of the nonsta- tionary Navier-Stokes problem. I. Regularity of solutions and second-order er- ror estimates for spatial discretization. SIAM J. Numer. Anal. 19(26) 275–311 (1982). 12. Kim, D., Choi, H.: A second-order time-accurate finite volume method for unsteady incompressible flow on hybrid unstructured grids. J. Comp. Phys. 162, 411–428 (2000). 13. Temam, R.: Sur l’approximation de la solution des équations de Navier-Stokes par la méthode de pas fractionnaires II. Arch. Rat. Mech. Anal. 33 377–385 (1969). 14. Zimmermann, S.: Stability of a colocated finite volume for the incompressible Navier-Stokes equations. preprint (2006). 15. Zimmermann, S.: Stability of a finite volume scheme for the incompressible fluids. preprint (2006).

0704.0788

Optimal Synthesis of Multiple Algorithms

COMBINING SEVERAL ALGORITHMS INTO A SUPERIOR ONE OPTIMAL SYNTHESIS OF MULTIPLE ALGORITHMS KERRY M. SOILEAU [email protected] JULY 27, 2004 ABSTRACT In this paper we give a definition of “algorithm,” “finite algorithm,” “equivalent algorithms,” and what it means for a single algorithm to dominate a set of algorithms. We define a derived algorithm which may have a smaller mean execution time than any of its component algorithms. We give an explicit expression for the mean execution time (when it exists) of the derived algorithm. We give several illustrative examples of derived algorithms with two component algorithms. We include mean execution time solutions for two-algorithm processors whose joint density of execution times are of several general forms. For the case in which the joint density for a two-algorithm processor is a step function, we give a maximum-likelihood estimation scheme with which to analyze empirical processing time data. mailto:[email protected] 1 INTRODUCTION It can categorically be said that no algorithm is unique. By this we mean that for a given task, invariably more than one algorithm exists which will accomplish that task. One strategy is to select one algorithm deemed generally superior to the rest, and to use that algorithm exclusively. This paper examines an alternative strategy. We ask, given two or more equivalent algorithms, is it ever possible to create a new derived algorithm whose mean execution time is less than that of all of the original algorithms? If so, how can such an algorithm be derived? First we define clearly what we mean by the term “algorithm:” Algorithm: An algorithm α is a pair ( ),α αρ π , where :αρ Ω→Γ is a Turing- computable mapping of a countable set Ω (tasks) into a countable set Γ (outputs), and :απ Ω→ is a mapping of into the positive real numbers. The function Ω αρ specifies the algorithm’s output ( )αρ ω when presented with the task ω∈Ω . The function απ specifies the execution time ( )απ ω required to compute the output ( )αρ ω . Note that under this definition, given a task ω∈Ω , an algorithm will always produce a definite output, namely ( )αρ ω , and will always produce this output after a definite amount of time has passed, namely ( )απ ω . We do not address procedures which are nondeterministic or whose execution time is unpredictable. Definition: We say that an algorithm ( ),α αα ρ π= is finite if and only if for every ( )0 απ ω< < ∞ ω∈Ω . Note that “ ( ),α αα ρ π= is finite” does not imply “ απ is bounded.” For example, Quicksort and Bubblesort are finite. Definition: We say that two algorithms ( ),α αα ρ π= and ( ),β ββ ρ π= are equivalent if and only if Dom Dom α βρ ρ= and ( ) ( )α βρ ω ρ ω= for every ω∈Ω . Notice that equivalent algorithms may require different times to process a given task. For example, Quicksort and Bubblesort are equivalent. Definition: Let { }1 2, , , Nα α α be a set of equivalent algorithms. We say that nα dominates { }1 2, , , Nα α α if and only if for every ω∈Ω , ( ) (n iα α )π ω π ω≤ for every { }1,2, ,i N∈ . Now suppose we are given a set of finite equivalent algorithms { }1 2, , , Nα α α such that no nα dominates { }1 2, , , Nα α α . Suppose further that there exists a probability space over such that ( ), , PΩ ℑ Ω , , , Nα α α π π π are random variables. Let be the joint density of the random variables , , , : α α απ π π , , , Nα α α π π π . Definition of Derived Algorithm: From a set of finite equivalent algorithms { }1 2, , , Nα α α , and a given point ( ) [ ) 1 2 1, , , 0, Nτ τ τ − ∈ ∞ , the function is defined as follows. For each , we define the random variable 1 2 11 2 1 | | | : NN Nτ τ τ α α α α ⎡ Ω→ Γ⎣ ⎤⎦ ( ) [ ) 11 2 1, , , 0, Nτ τ τ − ∈ ∞ , , , 1 2 1, , , : NT α α απ π π τ τ τ − Ω→ as follows: ( )( ) , , , 1 2 1 1 2 2 3 1 2 2 2 1 2 1 1 , , , α α απ π π τ τ τ ω π ω ω τ π ω ω τ τ π ω ω τ τ τ π ω ω τ τ τ π ω ω ⎪ + + ∈⎪ ⎪ + + + + ∈ ⎪ + + + + ∈⎩ (1) 1 2 1 1 2 1 | | | τ τ τ ρ ω ω ρ ω ω ρ ω ω α α α α ω ρ ω ω ρ ω ω ⎪ ∈⎪⎡ ⎤ = ⎨⎣ ⎦ (2) where ( ) ( ) ( ){ } 1 21 2 ; , , , S α α αω τ π ω τ π ω τ π ω= ∈Ω < < < for 1 1n N≤ ≤ − . Each is the event consisting of the points ω∈Ω on which none of the algorithms 1 2, , , nα α α completes processing within each algorithm’s permitted run time limit. The derived algorithm is then defined to be the pair . ( )( )1 2 1 1 21 2 1 , , , 1 2 1| | | , , , ,N NN N NT α α ατ τ τ π π πα α α α τ τ τ−− −⎡ ⎤⎣ ⎦ ( )( ) , , , 1 2 1, , , NT α α απ π π τ τ τ ω− represents the time taken for the derived algorithm to execute when presented with the task ω , and 1 2 11 2 1 | | | NNτ τ τ N α α α α ⎡ ⎤⎣ ⎦ represents the derived algorithm’s output when presented with the task ω . We may envision an implementation of this algorithm as follows. When presented with a task ω∈Ω , a timer is started, and 1α is applied. If 1α has not completed by time 1τ , 1α is abandoned and 2α is applied. If 2α has not completed by time 1 2τ τ+ , 2α is abandoned and 3α is applied, and so on. If 1Nα − has not completed by time 1 2 1Nτ τ τ −+ + + 1N, α − is abandoned and Nα is applied and (unlike the other algorithms) is allowed to run without time limit. ( ) ρ ω is returned as output , where iα is the algorithm which completed execution on the task ω∈Ω . The expected value (if it exists) of the random variable ( ) , , , 1 2 1, , , NT α α απ π π τ τ τ − is given by the following Theorem 1: ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , 1 2 1 1 1 1 1 1 , , , n n n n N N ET E S P S E S S P S S E S P S α α απ π π α τ τ τ π − − − = Ω Ω ∼ ∼ 1− Proof: Recall that ( )( ) , , , 1 2 1 1 2 2 3 1 2 2 2 1 2 1 1 , , , α α απ π π τ τ τ ω π ω ω τ π ω ω τ τ π ω ω τ τ τ π ω ω τ τ τ π ω ω ⎪ + + ∈⎪ ⎪ + + + + ∈ ⎪ + + + + ∈⎩ It follows immediately that ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , 1 2 1 1 1 2 1 2 1 2 2 3 2 3 1 2 2 2 1 2 1 1 2 1 1 1 , , , N N N N N N N E S P S E S S P S S E S S P S S E S S P E S P S α α απ π π τ τ τ τ τ π τ τ τ π τ τ τ π − − − − − − − = Ω Ω + + + + + + + + + + + + + ∼ NS S −∼ (4) This may be written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , 1 2 1 1 1 1 1 2 1 2 1 2 1 2 2 3 2 3 2 3 1 2 2 2 1 2 1 2 1 1 2 1 1 1 1 , , , N N N N N N N N N N N ET E S P S P S S E S S P S S P S S E S S P S S P S S E S S P S S P S E S P S α α απ π π α τ τ τ π τ π τ τ τ τ τ τ τ τ π − − − − − − − − − − − = Ω Ω + + + + + + + + + + + + + + ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ Telescoping sums yield ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )( ) , , , 1 2 1 1 1 1 2 1 2 2 3 2 3 2 1 2 1 1 1 2 2 3 2 1 1 2 2 3 2 1 1 1 1 , , , N N N N N N N N N N N N ET E S P S E S S P S S E S S P S S E S S P S S E S P S P S S P S S P S S P S P S S P S S P S P S α α απ π π α τ τ τ π − − − − − − − − − − − − = Ω Ω + + + + + + + + + + + ∼ ∼ ∼ Next, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , 1 2 1 1 1 1 2 1 2 2 3 2 3 2 1 2 1 1 1 1 1 2 2 1 1 , , , N N N N N N N N ET E S P S E S S P S S E S S P S S E S S P S S E S P S P S P S P S α α απ π π α τ τ τ π π τ τ τ − − − − − − − = Ω Ω + + + + + + + + ∼ ∼ ∼ ∼ − whence ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( , , , 1 2 1 1 1 1 1 1 1 , , , n n n n N N n ET E S P S )nE S S P S S E S P S P S α α απ π π α τ τ τ π − − − − = Ω Ω + +∑ ∑ ∼ ∼ τ+ as desired. 2 CASE (TWO ALGORITHMS) 2N = In this case ( )( ) ( ) ( ) ( ) ( ) π ω π ω τ π ω τ π ω + <⎪⎩ ( ) ( ) ( ) ( ) ( ) ( ) 1 21 2 , 1 1 1 1 1 1 1ET E S P S E S P S P Sα απ π α ατ π π τ= Ω Ω + +∼ ∼ and ( ){ } ;S αω τ π ω= ∈Ω < , (11) ( ) ( ) ( ) ( )( ) ( )1 1 1 2 1 11 2, 1 1 1 1 1 1ET E P E Pα απ π α α α α α ατ π π τ π τ π τ π τ τ π= ≤ ≤ + < + < (12) 2.1 ( ) , 1ET α απ π τ WHEN JOINT DENSITY IS ( )1 2, ,f x yα απ π In this case, ( ) ( ) ( ) ( ) 1 2 1 2 1 2 , 1 , 1 , 0 0 0 ,ET xf x y dydx y f x y dydx α α α α α α π π π π π π ∞ ∞ ∞ = + +∫ ∫ ∫ ∫ , (13) 2.11 EXAMPLE Suppose the joint density of completion times for the two algorithms is given by ( ) ( )( , , 12 exp )f x y xy x yα απ π = − + (14) Figure 1. ,f α απ π Then ( ) ( )( )( ) ( ) ( ) 4 2 3 23 3 , 1 1 1 1 1 1 12 8erf 1 expET α απ π τ π τ τ τ τ τ τ= − + + + − + π (15) Figure 2. ,ET α απ π 2.12 EXAMPLE Suppose the joint density of completion times for the two algorithms is given by ( ) ( ) , , 48 exp 4 3f x y xy x yα απ π = − − Figure 3. ,f α απ π Then ( ) ( ) ( ) , 1 1 14 6erf 2 3 exp 4ET α απ π τ π τ π τ= + − (17) Figure 4. ,ET α απ π 2.13 EXAMPLE Suppose the joint density of completion times for the two algorithms is given by ( ) ( )( ) ( ) ( )( )1 2 , 2 2 exp 1 7 , .022179119694367830844 exp 7 1 f x y xy α απ π ⎛ ⎞− − − − ⎜ ⎟= ⎜ ⎟ + − − − −⎜ ⎟ (18) Figure 5. ,f α απ π Figure 6. ,ET α απ π The minimum occurs at 1 2.492τ . Note that if 1 2.492τ , then ( ) , 1 2.854ET α απ π τ , while 4.260E απ and 2 4.260E απ . In this case the derived algorithm has better mean execution time than either of the original algorithms. Its mean execution time is approximately 33% less than that of either of the original algorithms. Notation: In the following, wherever 1 2, , , MB B B is found in a context requiring a Boolean expression, it means the conjunction 1 2 MB B B∧ ∧ ∧ of the Boolean expressions 1 2, , , MB B B . Notation: In the following, if B is a Boolean expression, then ( ) . 1 is tru 0 is fals In particular we define ( ) a x b a x x b ≤ < ≡ ≤ <⎨ 2.14 EXAMPLE Suppose the joint density of completion times for the two algorithms is given by ( ) ( )( ) ( )( ) , 1 3 4 5 8 2 4 f x y x y α απ π = ≤ < ≤ < + ≤ < ≤ < (19) Figure 7. ,f α απ π Figure 8. ,ET α απ π The minimum occurs at 1 3τ = . Note that if 1 3τ = , then ( ) , 1 4ET α απ π τ = , while 4.25E απ = and 2 4.25E απ = . In this case the derived algorithm has better mean execution time than either of the original algorithms. Its mean execution time is approximately 6% less than that of either of the original algorithms. 2.15 EXAMPLE Suppose the joint density of completion times for the two algorithms is given by ( ) ( ) , , expf x y x yα απ π = − − Figure 9. ,f α απ π Figure 10. ,ET α απ π Note that for any choice of 1τ , then ( ) , 1 1ET α απ π τ = , while 1 1E απ = and 2 1E απ = . In this case the derived algorithm has exactly the same mean execution time as do the original algorithms, so a derived algorithm would be of no benefit. 2.16 ( ) , 1ET α απ π τ DOES NOT ALWAYS EXIST If we take ( ) ( ) ( )1 2 f x y x yα α π π = , then ( ) ( ) ( ) 1 2 1 2 , 1 , 0 0 0 ,xf x y dydx y f x y dydx α α α α π π π π ∞ ∞ ∞ + + =∫ ∫ ∫ ∫ (21) so in this case ( ) , 1ET α απ π τ does not exist. 2.2 ( ) , 1ET α απ π τ WHEN JOINT DENSITY IS OF THE FORM ( ) ( ) ( 1 2 1 2 , , )f x y f x f yα α α απ π π π= ( ) ( ) ( )( ) ( ) ( ) ( ) ( 1 21 2 1 1 1 1 2 1 , 1 1 1 1 1 1 1 ET xf x dx P E E P E P α α α π π π α α α α α α α τ τ π τ π π τ π τ τ π τ π = + < + = ≤ ≤ + + < 2.3 ( ) , 1ET α απ π τ WHEN JOINT DENSITY IS OF THE FORM ( ) ( ) 1 2 2 ax by cx dy xpf x y x y a b c dα απ π + + + + + + + + If then 0 , , ,a b c d≤ ( ) ( 1 2 2 ax by cx dy )xpf x y x y a b c dα απ π + + + + = − − + + + + is a density function over ( ) . Accordingly, [ ) [ ), 0, 0,x y ∈ ∞ × ∞ ( ) ( ) 1 2 6 2 2 4 4 exp 1 2 2 a b c d c a b c d a b c dα απ π + + + + − + − + − − + + + + (23) 2.4 ( ) , 1ET α απ π τ WHEN JOINT DENSITY IS OF THE FORM ( ) ( ) ( )( ) ( ) ( )1 2 , 2 3 3 1 1 1 1 f x y d m n x y x yα α + + + + If , 0 , , then ( )0 ,d m n≤ c≤ ( ) c d m n + =∑∑ 1 ( ) ( ) ( )( ) ( ) ( )1 2 , 2 3 3 1 1 1 1 f x y d m n x y x yα α + + + + is a density function over ( ) [ ) [ ), 0, 0,x y ∈ ∞ × ∞ . A straightforward calculation yields ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 2 0 01 1 1, , 0, 2 1 1 1 1 1 1 2, 1, ln 1 n m n n ET d n d m n d d m n d m n c ∞ ∞ ∞ ∞ = = = = = + + + + + + + − − − +⎜ ⎟+⎝ ⎠+ ∑ ∑ ∑ ∑ (25) 2.5 ( ) , 1ET α απ π τ WHEN JOINT DENSITY IS OF THE FORM ( ) ( )( n n n n n )f x y k a x b c y d α απ π = ≤ < ≤ <∑ Theorem 2: If and ( )( ) n n n n n k b a d c − −∑ = 0 nk< for 1, 2, ,n N= , and ( ) ( )( n n n n n )f x y k a x b c y d α απ π = ≤ < ≤ <∑ , then ( ) ( ) ( )( ) ( )( )( ) ( )( ) ( )( )( ) ( )( ) , 1 1 2 n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n ET k d c k b a d c k b d c d c k b a d c k b d c a d c k b a d c α απ π ⎜ ⎟= − −⎜ ⎟⎜ ⎟ − −⎜ ⎟ + ⎜ ⎟ ⎜ ⎟+ − + −⎜ ⎟ + − − + + − + − − (26) Notation: In the following, if is a Boolean expression, then . ( )B n ( )( ) s B n ≡∑ ∑ s Proof: If and 0( )( ) n n n n n k b a d c − −∑ = nk< for 1, 2, ,n N= , then ( ) ( )( n n n n n )f x y k a x b c y d α απ π = ≤ < ≤ <∑ is a density function over . Now note that ( ) [ ) [ ), 0, 0,x y ∈ ∞ × ∞ ( ) ( ) ( ) ( ) 1 2 1 2 1 2 , 1 , 1 , 0 0 0 ,ET xf x y dydx y f x y dydx α α α α α α π π π π π π ∞ ∞ ∞ = + +∫ ∫ ∫ ∫ , ( )( ) ( ) ( )( ) n n n n n n n n n n x k a x b c y d dydx y k a x b c y d dydx = ≤ < ≤ < + + ≤ < ≤ < (27) ( )( ) ( )( )( ) ( ) ( ) ( ) ( ) 1 0 0 n n n n n n n n n n n n n n n n n n k x a x b c y d dydx k y a x b c y d dy k d c x a x b dx k a x b dx y dy = ≤ < ≤ < + + ≤ < ≤ < = − ≤ < + ≤ < + ∑ ∫ ∫ (28) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n k d c x a x b dx k a x b dx k d c x a x b dx k d c x a x b dx k d c x a x b dx k d c c d a x b dx = − ≤ < ⎛ ⎞+ + ⎜ ⎟+ ≤ < − = − ≤ < + − ≤ < + − ≤ < + − + + ≤ < ( )( ) ( ) ( )( ) ( ) n n n n n n n n n n n n n n k d c c d a x b dx k d c c d a x b dx + − + + ≤ < + − + + ≤ < (29) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) n n n n n n n n n n n n n n n n n n n n n n n n n n n n n ET k d c x a x b dx k d c x a x b dx k d c c d a x b dx k d c c d a x b dx k d c xdx k d = − ≤ < + − ≤ < + − + + ≤ < + − + + ≤ < = − + − ∑ ∫ ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( )( ) 2 2 21 1 1 12 2 2 n n n n n n n n n n n n n n n n n n n n n n a b b n n n n n n n n n n c xdx k d c c d dx k d c c d dx k d c a k d c b a k d c c d b a k d c < < ≤ + − + + + − + + = − − + − + − + + − ( )( ) n n n c d b + + −∑ ( )( ) ( )( )( ) ( )( ) ( )( )( ) ( )( ) n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n k d c k b a d c k b d c d c k b a d c k b d c a d c k b a d c ⎜ ⎟= − −⎜ ⎟⎜ ⎟ − −⎜ ⎟ + ⎜ ⎟ ⎜ ⎟+ − + −⎜ ⎟ + − − + + − + − − (31) ( ) ( ) ( )( ) ( )( )( ) ( )( ) ( )( )( ) ( )( ) , 1 1 2 n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n ET k d c k b a d c k b d c d c k b a d c k b d c a d c k b a d c α απ π ⎜ ⎟= − −⎜ ⎟⎜ ⎟ − −⎜ ⎟ + ⎜ ⎟ ⎜ ⎟+ − + −⎜ ⎟ + − − + + − + − − (32) In particular, ( ) ( ) ( )( ) ( )( )( ) ( )( ) ( )( )( ) ( )( ) n i n n i n n i n i i n n n a a b n n n n n n n n n n n a a b n n n n n n n n n n n n a a b n n n n n ET a a k d c k b a d c k b d c d c k b a d c k b d c a d c k b a d c α απ π ⎜ ⎟= − −⎜ ⎟⎜ ⎟ − −⎜ ⎟ + ⎜ ⎟ ⎜ ⎟+ − + −⎜ ⎟ + − − + + − + − − (33) ( ) ( ) ( )( ) ( )( )( ) ( )( ) ( )( )( ) ( )( ) n i n n i n n i n i i n n n a b b n n n n n n n n n n n a b b n n n n n n n n n n n n a b b n n n n n ET b b k d c k b a d c k b d c d c k b a d c k b d c a d c k b a d c α απ π ⎜ ⎟= − −⎜ ⎟⎜ ⎟ − −⎜ ⎟ + ⎜ ⎟ ⎜ ⎟+ − + −⎜ ⎟ + − − + + − + − − (34) It is straightforward to show that ( ) , 1ET α απ π τ attains a global minimum at one of the points { }1 2 1 2, , , , ,Na a a b b bN . Indeed, ( ) , 1ET α απ π τ is a continuous, piecewise quadratic function. Notice that the set of points of connection of the pieces is a subset of { }1 2 1 2, , , , ,Na a a b b bN . Each piece is either linear or is quadratic with a negative second derivative. We can thus replace each quadratic piece with a linear piece connecting the endpoints of the quadratic piece, without altering the global minimum of , 1ET α απ π τ . After replacing each quadratic piece with the appropriate linear piece, we then have a continuous piecewise linear function whose global minimum is the same as that of ( ) , 1ET α απ π τ . But of course the global minimum of a continuous piecewise linear function is attained at one of its vertices. These vertices are a subset of the set of points of connection { }1 2 1 2, , , , ,Na a a b b bN , as desired. This global minimum is given by ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 2 1 2 1 2 1 2 , 1 , 2 , , 1 , 2 , , , , , , , , ET a ET a ET a ET b ET b ET b α α α α α α α α α α α α π π π π π π π π π π π π ⎪ ⎪⎩ ⎭ ⎬ (35) This minimum can be computed in ( )2O N time. MAXIMUM LIKELIHOOD ESTIMATION Suppose ( ) ( )( n n n n n )f x y k a x b c y d α απ π = ≤ < ≤ <∑ with the following conditions: 1. for 1 , 0nk > n N≤ ≤ 2. for 1 , na b< n n N≤ ≤ 3. for 1 , nc d< n N≤ ≤ 4. The boxes for 1[ ) [ ), ,n n n n nB a b c d≡ × n N≤ ≤ are disjoint, 5. . ( )( ) n n n n n k b a d c − −∑ = Then ,f α απ π is a joint density function. Suppose next that we have observed the performance of two equivalent algorithms α and β over a (finite) sample set sΩ ⊂ Ω . That is, for each task sω∈Ω we have observed the values ( )απ ω and ( )βπ ω representing the time that algorithms α and β actually took to process the task ω . We now present a maximum-likelihood procedure to find the “best fitting” joint density function of the form ( ) ( )( n n n n n )f x y k a x b c y d α απ π = ≤ < ≤ <∑ , subject to the five conditions above. Let ( ) ( ) ( ){ }1 1 2 2, , , , , ,P Px y x y x y be the data observed, where jx and are the durations required by algorithms jy α and β respectively, to process j sω ∈Ω , for 1 j P≤ ≤ , with sP = Ω . Our performance function is defined as ( ) ( ) ( )( ) 1 2 1 2 , 1 2 , , , , N P P N N m m n n m n n m n g k k k f x y k a x b c y d k k k α απ π ≡ = ≤ < ≤ < =∑∏ ∏ N where ( ) ( ) ( ) ( ){ }{ }1 1 2 2, , , , , , , ; ,j P P j j j jS x y x y x y x y a x b c y d≡ ∈ ≤ < ≤ < We form as usual the Lagrange multiplier equations ( )( )1 0j j j j j S b a d c λ + − − = for 1 j P≤ ≤ . We have immediately that ( )( ) j j j j b a d c and recalling the constraint n n n n n k b a d c = − −∑ ) we infer S Pλ λ = − = −∑ whence λ = − thus ( )( ) j j j j P b a d c Substituting into (32), we get ( )( ) , 1 1 2 n n n n n ET d c P b a d cα απ π ⎜ ⎟= − −⎜ ⎟− −⎜ ⎟ ( )( ) ( )( ) ( )( ) ( )( )( ) n n n n n n n n n n n n n n n n n n n b a d c P b a d c b d c d c P b a d c − −⎜ ⎟− −⎜ ⎟ + ⎜ ⎟ ⎜ ⎟+ − +⎜ ⎟− −⎜ ⎟ ( )( ) ( )( ) n n n n n n n n n b a d c P b a d c + − − − −∑ ( )( ) ( )( )( ) n n n n n n n n n n n b d c a d c P b a d c − −∑ − − ( )( ) ( )( ) n n n n n n n n n b a d c P b a d c − −∑ − ( )( ) n n n n a b n n n n n n P b a S b d c P b a ⎜ ⎟⎛ ⎞ ⎜ ⎟⎜ ⎟= − + ⎜ ⎟⎜ ⎟− ⎜ ⎟⎜ ⎟ + −⎝ ⎠ ⎜ ⎟−⎜ ⎟ ( )( ) ( ) 1 1 1 21 1 1 2 2 2 1 1 1 n n n n N N N n n n n n n n n n n nn n a a b b d c b d c a b a P P b a P τ τ τ = = = ≤ < < ≤ + + + + − + ∑ ∑ ∑ n 1 1 1 , 1 1 12 1 1 1 n n n n n N N N n n nn n n nn n n n a b a a b b d cS ET S S b a P b aα απ π τ τ τ τ τ τ = = = < < ≤ < < − +⎜ ⎟= − + +⎜ ⎟− −⎜ ⎟ ∑ ∑ ∑ ( )( ) ( ) ( ) 2 2 2 n n n n n n n n n n n n n nP P n nn n a b a b b d c a S d c S b a < < ≤ ≤ + + − + + + 3 CONCLUSIONS In this paper, we asked the following questions: Given two or more equivalent algorithms, is it ever possible to create a new derived algorithm whose mean execution time is less than that of all of the original algorithms? If so, how can such an algorithm be derived? By giving examples in Section 2, we have shown that the answer to the first question is “yes.” In Section 1, we gave an explicit construction of the derived algorithm.

0704.0789

New Close Binary Systems from the SDSS-I (Data Release Five) and the Search for Magnetic White Dwarfs in Cataclysmic Variable Progenitor Systems

Draft version October 22, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 NEW CLOSE BINARY SYSTEMS FROM THE SDSS–I (DATA RELEASE FIVE) AND THE SEARCH FOR MAGNETIC WHITE DWARFS IN CATACLYSMIC VARIABLE PROGENITOR SYSTEMS Nicole M. Silvestri , Mara P. Lemagie , Suzanne L. Hawley , Andrew A. West , Gary D. Schmidt , James Liebert , Paula Szkody , Lee Mannikko , Michael A. Wolfe , J. C. Barentine , Howard J. Brewington Michael Harvanek , Jurik Krzesinski , Dan Long , Donald P. Schneider , and Stephanie A. Snedden (Received 2007 February 9) Draft version October 22, 2018 ABSTRACT We present the latest catalog of more than 1200 spectroscopically–selected close binary systems observed with the Sloan Digital Sky Survey through Data Release Five. We use the catalog to search for magnetic white dwarfs in cataclysmic variable progenitor systems. Given that approximately 25% of cataclysmic variables contain a magnetic white dwarf, and that our large sample of close binary systems should contain many progenitors of cataclysmic variables, it is quite surprising that we find only two potential magnetic white dwarfs in this sample. The candidate magnetic white dwarfs, if confirmed, would possess relatively low magnetic field strengths (BWD < 10 MG) that are similar to those of intermediate–Polars but are much less than the average field strength of the current Polar population. Additional observations of these systems are required to definitively cast the white dwarfs as magnetic. Even if these two systems prove to be the first evidence of detached magnetic white dwarf + M dwarf binaries, there is still a large disparity between the properties of the presently known cataclysmic variable population and the presumed close binary progenitors. Subject headings: binaries: close — cataclysmic variables — stars: low-mass — stars: magnetic fields — stars: white dwarfs 1. INTRODUCTION The evolution of stars in close binary systems leads to interesting stellar end-products such as cataclysmic variables (CVs), Type 1a supernovae, and helium–core white dwarfs (WDs). The period in which an evolved star ascends the asymptotic giant branch and engulfs a close companion in its evolving atmosphere, referred to as the common envelope phase, probably plays a dominant role in the evolution of these systems and as yet is poorly understood. The angular momentum of the system is believed to aid in the eventual ejec- tion of the common envelope to reveal the remnant WD and close companion. After the common envelope has been ejected, gravitational and magnetic braking work to decrease the orbital separation of the detached system (de Kool & Ritter 1993). This orbital evolution contin- ues through to the CV phase. The effect of the common envelope on the secondary star in these systems is an- other aspect of close binary evolution which is not well characterized. Plausible scenarios for the secondary com- 1 Department of Astronomy, University of Washington, Box 351580, Seattle, WA 98195, USA; [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]. 2 Astronomy Department, 601 Campbell Hall, University of California, Berkeley, CA 94720, USA; [email protected] 3 Department of Astronomy and Steward Observatory, Univer- sity of Arizona, Tucson, AZ 85721, USA; [email protected], [email protected]. 4 Apache Point Observatory, P.O. Box 59, Sunspot, NM 88349, USA; [email protected], [email protected], [email protected], [email protected], sned- [email protected]. 5 Mt. Suhora Observatory, Cracow Pedagogical University, ul. Podchorazych 2, 30-084 Cracow, Poland; [email protected]. 6 Department of Astronomy, Penn State University, PA 16802 USA; [email protected] panion range from accreting as much as 90% of its mass during this phase to escaping relatively unscathed from the common envelope, emerging in the same state as it entered (see Livio 1996, and references therein). Recently, studies of close binary systems with WD companions (see for example Farihi et al. 2005b,a; Pourbaix et al. 2005; Silvestri et al. 2006) have revealed yet another puzzling property of these systems. None of the WDs in close binary systems with low–mass, main se- quence companions appear to be magnetic (Liebert et al. 2005). Close, non–interacting binary systems with WD primaries are quite common and are believed to be the direct progenitors to CVs (Langer et al. 2000, and ref- erences therein). Magnetic WDs, stellar remnants with magnetic fields in excess of ∼ 1 MG, comprise only a small percentage of the isolated WD population (∼ 2%, Liebert et al. 2005). Note that the 2% magnetic WD fraction applies to magnitude–limited samples like the Palomar–Green (Liebert et al. 1988). However, the same paper notes that magnetic WDs may generally have smaller radii than non–magnetic white dwarfs, due to higher mass. In a given volume, the density of mag- netic WDs may be ∼ 10% of all WDs (Liebert et al. 2003). The SDSS is also a magnitude limited sample so we assume a similar expected value for the close bi- naries. Our sample (as discussed in detail in §2) con- tains 1253 potential close binary systems. Therefore we assume approximately 24 of these binaries to harbor a magnetic WD. Possible implications of the small radii for magnetic WD + main sequence pairs will be dis- cussed in §5. However, more than 25% of the WDs in the currently identified CV population are classified as magnetic, and many have magnetic fields in excess of 10 MG (see Wickramasinghe & Ferrario 2000). Holberg et al. (2002) have compiled a list of 109 known http://arxiv.org/abs/0704.0789v1 2 Silvestri et al. WDs within 20pc (and complete to within 13pc) that have nearly complete information about the presence of a companion. Of the 109 WDs in their sample, 19 ± 4 have nondegenerate companions. Table 7 in Kawka et al. (2007) lists all known magnetic WDs as of June 2006. Of the magnetic WDs listed in their table, 149 have field strengths identifiable in SDSS-resolution spectra (BWD ≥ 3 MG). If the magnetic WDs in the Kawka et al. (2007) sample are assumed to be drawn from a similar sample then 28 ± 5.3 would be expected to have nondegenerate companions, and yet none have been detected in the Kawka et al. (2007) sample. This is nearly a 5σ deficit in magnetic WDs with nondegenerate companions. Holberg & Magargal (2005) looked at the 2MASS JHKs photometry of 347 WDs in the Palomar–Green sample. Of the 347 WDs, 254 had reliable infrared mea- surements of at least J magnitude. Of these, 59 had excesses indicative of a nondegenerate companion and another 15 showed “probable” excesses (Liebert et al. 2005). This gives a WD+dM fraction of 23% (definite excess) and 29% (including all probable excesses). If the Kawka et al. (2007) sample had the same frequency of of nondegenerate companions as the Palomar–Green sam- ple, they should have 34 and 43, respectively. This is nearly as 6σ deficit! This apparent lack of magnetic WDs with main se- quence companions is not restricted to studies of close binaries. Low resolution spectroscopic surveys of more than 500 common proper motion binary systems dis- covered by Luyten et al. (1964); Luyten (1968, 1972) and Giclas et al. (1971, 1978) revealed no magnetic WDs paired with main sequence companions in these wide pairs (Smith 1997; Silvestri et al. 2005). In addition, Schmidt et al. (2003) and Vanlandingham et al. (2005) have identified over 100 magnetic WDs in the Sloan Digital Sky Survey (SDSS, Gunn et al. 1998; York et al. 2000; Stoughton et al. 2002; Pier et al. 2003; Gunn et al. 2006). As discussed by Liebert et al. (2005), this implies essentially no overlap between the close binary and mag- netic WD samples. A new class of short–period, low accretion–rate polars (LARPS) identified by Schmidt et al. (2005b) may ex- plain, in part, these “missing” magnetic WD systems. In these systems, the donor star has not filled its Roche Lobe. The WD accretes material by capturing the stel- lar wind of the secondary. These CVs have accretion rates that are less than 1% of accretion rates normally associated with CVs. The discovery of these systems sheds some light on the whereabouts of magnetic WD binaries, though as Schmidt et al. (2005b) point out, this still does not explain the apparent lack of long– period, detached magnetic WD systems. Thought to be the first detached binary with a magnetic WD, SDSS J121209.31+013627.7, a magnetic WD with a proba- ble brown dwarf (L dwarf) companion (Schmidt et al. 2005a) has been shown to be one of these LARP systems (Debes et al. 2006; Koen & Maxted 2006; Burleigh). To date, magnetic WDs have only been found as isolated ob- jects, in binaries with another degenerate object (WD or neutron star companion), or in CVs; none have a clearly main sequence companion. In this study, we investigate a new large sample of close binary systems in an effort to uncover these “missing” magnetic WD binary systems. The sample comprises more than 1200 close binary systems containing a WD and main sequence star drawn from the SDSS, many of which were originally presented in Silvestri et al. (2006, hereafter, S06). We find that only two of the WDs in these pairs appear to be magnetic. Even if confirmed, neither of these WDs has magnetic field strength compa- rable to those observed in the majority of magnetic (Po- lar) CV systems. We confirm that the current CV and close binary populations are indeed disparate and show that more work is necessary to unravel this mystery. In §2 we introduce the catalog of close binary sys- tems through the public SDSS Data Release Five (DR5; Adelman-McCarthy et al. 2007). We discuss our anal- ysis techniques in §3 and we present our results in §2. Our discussion and concluding remarks are given in §5 and §6, respectively. 2. THE SDSS CLOSE BINARY CATALOG THROUGH DR5 The combined properties of the majority of close binaries in this paper are discussed in detail in Raymond et al. (2003) and S06. The S06 cata- log was based on a preliminary list of spectroscopic plates released internally to the collaboration and as such does not include objects from ∼ 200 plates re- leased with the final public Data Release Four (DR4; Adelman-McCarthy et al. 2006). The additional systems from both DR4 and DR5 do not change the overall re- sults from analysis performed in S06, hence no new anal- ysis is presented here. We include this list in its en- tirety to complete the DR4 catalog introduced by S06 and add over 300 new systems from the now public DR5 (Adelman-McCarthy et al. 2007). This completes the catalog of close binary systems with a WD identified through SDSS–I. More close binaries are being targeted in the SDSS–II (SEGUE) survey which will continue to increase the sample through 2008. The list of 1253 potential close binary systems given in Table 1 includes objects from all plates released with the public DR5, thereby superseding the S06 DR4 cat- alog. The technique used to search for these objects is the same as described in S06. As with that study, we do not include systems with low signal-to-noise ratios (S/N < 5) and do not search for systems with non–DA WDs. We emphasize that our sample is not complete (or bias free) due to the selection effects imposed by our detection methods and due to the sporadic targeting of these objects in the SDSS spectroscopic survey as dis- cussed in S06. Thus, our sample represents primarily bright, DAWD +M dwarf binary systems. As evidenced by Smolčić et al. (2004), there are potentially thousands more WD + M dwarf binaries observed photometrically in the SDSS but not targeted for spectroscopy. Our cat- alog represents an interesting and statistically significant sampling of these systems, the properties of which can be used to test models of close binary evolution (see Politano & Weiler 2006, for example). The list of plate numbers from which this sample has been drawn can be found at http://das.sdss.org/DR5/data/spectro/1d 23/. This plate list includes both “extra” and “special” plates. The extra plates are repeat observations of survey plates taken during normal operation. The special plates are observations for special programs (e.g. SEGUE, F stars, http://das.sdss.org/DR5/data/spectro/1d_23/ Magnetic WDs in CV Progenitors 3 Fig. 1.— Example of an M dwarf with excess blue flux (:+dM) from Table 1. The companion is seen as little more than excess blue flux in the M dwarf spectrum. Follow-up spectroscopy to resolve the companion is necessary to rule out the presence of a magnetic WD. Note: spectrum has been boxcar smoothed with a filter size of seven. main sequence turnoff stars, quasar selection efficiency, etc.) that are not part of the original SDSS–I survey. The first four columns of Table 1 list the SDSS identi- fier, the plate number, fiber identification, and modified Julian date (MJD) of the observation, followed by the spectral type of the components (determined visually) where Sp1 represents the blue object and Sp2 is the red object. Columns 6 and 7 give the J2000 coordinates (in decimal degrees) for the object. The next 15 columns give the ugriz PSF photometry (Fukugita et al. 1996; Hogg et al. 2001; Ivezić et al. 2004; Smith et al. 2002; Tucker et al. 2006), photometric uncertainties (σugriz), and reddening (Augriz). The magnitudes are not cor- rected for Galactic extinction. Column 23 lists the SDSS data release in which the object was discovered as well as additional references in the literature. Additional notes for the objects are listed in column 24. The objects identified in Table 1 as :+dM are likely M dwarfs with faint, cool WD companions. The dis- covery spectra for these objects reveal little more than excess blue flux at wavelengths shorter than 5000 Å, as shown in Figure 1. It is possible that some of these pairs may contain a magnetic WD; however, much higher S/N spectra are required to adequately characterize the blue component of these systems. Similarly, the thirty nine objects identified as WD+: or WD+:e (see Figure 8 of S06) have either some ex- cess flux in the red or have emission at Balmer wave- lengths indicative of a faint, active, low–mass or sub– stellar companion. The companion to the magnetic WD in Schmidt et al. (2005a) was first identified by emission at Hα in the SDSS discovery spectrum. Other than the emission at Hα this object had no other optical signa- ture of a companion. We are performing followup obser- vations using the ARC 3.5–m telescope at Apache Point Observatory to obtain radial velocities and near–infrared imaging of these objects to measure the orbital periods and categorize the probable low–mass companion’s spec- tral type. We have already confirmed that none of these systems contain a magnetic WD. 3. THE SEARCH FOR MAGNETIC WDS Schmidt et al. (2003) and Vanlandingham et al. (2005) demonstrated that magnetic WDs with field strengths as low as ∼ 3 MG can be effectively measured using SDSS spectra. Visual inspection of the systems in our sample reveals no obvious magnetic WDs in spectra with good S/N (> 10) (Lemagie et al. 2004). Most are classical Fig. 2.— A Typical WD+dM System: SDSS J140723.03+003841.7, the superposition of a DA (hydrogen atmosphere) WD and a M4 red dwarf star. Hα emission is visible in many of these systems and is a result of chromspheric activity on the surface of the M star, perhaps enhanced due to the influence of the WD. The lack of broad Zeeman absorption features in the hydrogen lines indicates that the magnetic field strength of the WD is very low (compare with Figure 4). WD + M dwarf close binaries as shown in Figure 2. Of interest are the lower quality spectra, where the features of the WD are less obvious because of low S/N and/or contamination by the spectral features of the close M dwarf companion. These effects make it difficult to iden- tify small magnetic field effects on the WD absorption features. Thus, relatively low magnetic fields (BWD < 10 MG) are not easily recognized in the combined spectrum. 3.1. The Simulated Magnetic Binary Systems Given the difficulties associated with visually identi- fying features in these systems, we developed a method to search for the characteristic Zeeman splitting of the DA WD absorption features that is also sensitive to low magnetic field WDs. We use a program that attempts to match absorption features in magnetic DA WD mod- els (see Kemic 1974b,a; Schmidt et al. 2003, and refer- ences therein for details on the models) through an it- erative method of smoothing and searching the stellar spectrum. To develop a robust program to search for magnetic WDs in close binaries we first tested our pro- gram on WDs of known magnetic field strength. We used the magnetic DA WDs with field strengths between 1.5 MG ≤ BWD ≤ 30 MG from Schmidt et al. (2003) and Vanlandingham et al. (2005) as our test sample. We then constructed model spectra at every half–MG be- tween 1.5 MG ≤ BWD ≤ 30 MG, each with magnetic field inclinations of 30◦, 60◦, and 90◦. The program was able to match (using a χ2 minimization) the magnetic field strength of each of the magnetic WDs to within ±5 MG of the value quoted in Schmidt et al. (2003). We then constructed a sample of simulated SDSS spec- tra of magnetic binary systems. The simulated binaries were created by adding the spectra of magnetic WDs used in our initial test from Schmidt et al. (2003) and Vanlandingham et al. (2005) to the M star templates of Hawley et al. (2002). We first normalized all spectra at a wavelength of 6500 Å, and then combined them with flux ratios of 1:4 (WD:M dwarf) to 4:1 to replicate the range of flux ratios observed in the close binary sample (see Figure 3)7. This created a sample of binaries which represent the average brightness and spectral type dis- tribution of the majority of the systems in Table 1 (i.e. 7 Note that 6500 Å is the midpoint of the SDSS combined blue and red spectra, as plotted in Figure 3. In reality the SDSS spectra extend to below 3900 Å and to nearly 10000 Å. 4 Silvestri et al. Fig. 3.— Comparison of simulated and observed pre–cataclysmic variable (PCV) systems. Left Hand Column: Simulated magnetic PCVs produced by adding WD spectra from Schmidt et al. (2003) to M dwarf spectra from Hawley et al. (2002) with brightness ratios as specified at 6500 Å. Right Hand Column: Observed PCVs from Silvestri et al. (2006). Fig. 4.— A Simulated System. Top Left Panel: A 13 MG mag- netic WD from Schmidt et al. (2003). Top Right Panel: Template M4 dwarf star from Hawley et al. (2002). Bottom Panel: addition of the magnetic WD and template M dwarf, assuming equal flux density at 6500 Å. DA WDs and M0–M5 dwarfs). Figure 4 is an example of one of the simulated magnetic binary systems. The upper left hand panel is the SDSS spectrum of a 13 MG magnetic WD, the upper right hand panel is the spectrum of a template M4 dwarf star. The bottom panel is the addition (superposition) of the two spectra with a flux ratio of 1:1 at 6500 Å. As shown, this WD with a relatively moderate magnetic field, when combined with the spectrum of an average M dwarf, is clearly detected at the resolution of the SDSS spectra (R ∼ 1800). 3.2. Results from the Simulated Systems We found that detecting the presence of a WD mag- netic field depends most strongly on the spectral type and relative flux of the M dwarf companion. Due to the selection effects of the close binary sample (see S06 for details), the majority of the M dwarfs in these binaries have spectral sub–types between M0–M4. In SDSS spec- tra, early M dwarf spectral types contribute nearly as much flux in the blue portion of the spectrum (4000–7000 Å) as they do in the red (7000–10000 Å). The spectrum of the blue magnetic WD is then superimposed onto the numerous blue molecular features of the M dwarf. This makes the small absorption features stemming from the subtle influence of a weak magnetic field difficult to de- tect. We plot a subset of our simulated pairs to demon- strate some of these issues in Figure 5 and Figure 6. In Figure 5 we selected four early–type template M dwarfs (WD+M0 = open squares, WD+M1 = open circles, WD+M2 = open triangles, and WD+M3 = Fig. 5.— Left Hand Panel: Subset of the simulated binary sys- tems comprised of early–M dwarfs from Hawley et al. (2002) paired with magnetic WDs and literature values from Schmidt et al. (2003) and Vanlandingham et al. (2005) with BWD ≤ 10 MG. Right Hand Panel: Same M dwarfs from Left Panel paired with magnetic WDs with BWD ≥ 10 MG. The measured values are from our program. In both panels, the filled triangles represent single WDs, open squares are WD+M0, open circles are WD+M1, open triangles are WD+M2, and crosses are WD+M3. The solid line has a slope of one and the dashed lines are ±5 MG. Refer to § 3.2 of the text for details. Fig. 6.— Left Hand Panel: Subset of the simulated binary sys- tems comprised of late-M dwarfs from Hawley et al. (2002) paired with magnetic WDs and literature values from Schmidt et al. (2003) and Vanlandingham et al. (2005) with BWD ≤ 10 MG. Right Hand Panel: Same M dwarfs from Left Panel paired with magnetic WDs with BWD ≥ 10 MG. The measured values are from our program. In both panels, the filled triangles represent single WDs, open squares are WD+M4, open circles are WD+M5, open triangles are WD+M6, and crosses are WD+M7. The solid line has a slope of one and the dashed lines are ±5 MG. Refer to § 3.2 of the text for details. crosses) from Hawley et al. (2002) and added them to a range of magnetic WDs from Schmidt et al. (2003) and Vanlandingham et al. (2005). The quoted value from Schmidt et al. (2003) for the magnetic field strength of each of these WDs represents the “Literature BWD” value on the x–axis. The “Measured BWD” is the value returned by the program. Values returned by the pro- gram that matched the literature values fall along the solid line. The dashed lines represent ±5 MG of the literature value. Figure 6 is the same except we add the same magnetic WDs to later–type M dwarf tem- plates (WD+M4 = open squares, WD+M5 = open cir- cles, WD+M6 = open triangles, andWD+M7 = crosses). The solid triangles represent the tests using the isolated WD spectra. In both Figures 5 and 6 the program returns the value of the single WD to within ∼ ±2 MG for the large ma- jority of the systems. The uncertainty of the fitted value and the spread in values increases for magnetic fields of 3 MG or less when the magnetic WD is paired with an M dwarf of comparable brightness. The flux minima associ- ated with the Zeeman features for such low field strengths are just barely resolvable in high S/N spectra of isolated SDSS WDs (see Schmidt et al. 2003). The added com- Magnetic WDs in CV Progenitors 5 Fig. 7.— Here, we plot the flux ratio (WD flux/ M dwarf [dM]) versus the difference between the literature value (from Schmidt et al. 2003; Vanlandingham et al. 2005) of the magnetic field strength (BLit) and the measured magnetic field strength (BMea) as determined from the WD Hα (top panel), Hβ (center panel), and Hγ (bottom panel) absorption features. Error bars are from the χ2 fit. Refer to § 3.2 of the text for details. plexity of the M dwarf molecular features and the gen- erally lower S/N spectra make it difficult to measure the magnetic features for low magnetic field strengths. How- ever, WDs with magnetic fields ≥ 4 MG were easily mea- sured at all M dwarf spectral types. In both Figures, the largest discrepancies between the literature and measured values occur when the WD’s magnetic field is between 12 MG ≤ BWD ≤ 18 MG; this is true when the WD is paired with both early– and late–type M dwarfs. Inspection of the model results in- dicates that at these field strengths, the Zeeman features overlap on wavelengths with strong M dwarf molecular features, causing confusion in the identification of the fea- ture. However, WD spectra with these and larger field strengths are quite easily recognized visually so we are confident that no systems with ≥ 10 MG have escaped notice, though the exact value of the field strength would be more uncertain. In Figure 7, we demonstrate the effect of the relative flux ratio (WD: M dwarf [dM]) on the identification of the magnetic field strength of WDs in the simulated binary sample. The Figure gives the relative flux ratio versus the difference between the magnetic fields quoted in the literature and those returned by the program. We use the same BWD distribution in Figure 7 as used in Fig- ure 5 and Figure 6. The literature values (BLit) are from Schmidt et al. (2003) and Vanlandingham et al. (2005). The three panels show ratios determined using Hα (top), Hβ (center), and Hγ (bottom). The program consis- tently returns the quoted BWD as determined from Hβ until the flux contribution from the M dwarf is nearly double the flux contribution from the WD. The program returns the magnetic field from the Hα feature to within ±5 MG until the flux contribution from the M dwarf is nearly 1.5× the flux from the WD. The BWD as mea- sured by Hγ is consistently 15–25 MG larger than the BWD value in the literature at any flux ratio. The con- tribution of a relatively clean spectral region near Hβ, to- gether with the fairly strong Zeeman signal at this wave- length makes Hβ a reliable indicator of WD magnetic field strength for binaries with flux ratios up to 1:2. 4. TWO POSSIBLE MAGNETIC WDS IN THE DR5 CLOSE BINARY SAMPLE The method employed by S06 to split the binary sys- tem into its two component spectra through an itera- tive method of fitting and subtracting WD model at- mospheres and template M dwarf spectra was not used Fig. 8.— Two potential magnetic DA WD + M dwarf pairs as identified by our program. The tentative magnetic field strengths are 8 MG ±5 MG (top) and 3 MG +5/−3 MG (bottom) as deter- mined from the Hα and Hβ WD absorption features. on these objects. There are no obviously strong mag- netic WDs in the sample, suggesting that any possibly magnetic WDs must possess relatively weak fields. The subsequent fitting and subtraction of model WDs and template M dwarfs adds noise to the spectrum which would make detection of an already weak magnetic field even more difficult. Also, we would be subtracting a non–magnetic WD model from the spectrum of a poten- tially magnetic WD in our attempt to improve the M dwarf template fit. This adds absorption features where none actually exist, further corrupting the WD spectrum. Given these complications, we chose to work with the original composite SDSS discovery spectra. Table 2 lists the properties of the only two close bi- nary systems flagged by our program as containing po- tential magnetic WDs: SDSS J082828.18+471737.9 and SDSS J125250.03−020608.1. The first four columns are the same as for Table 1, followed by the R.A. and Decl. (J2000 coordinates). The tentative magnetic field strengths (in MG), inclination of the WD magnetic field to the line of sight (in degrees) and the spectral types of the components are listed in Columns 7–9. For each of these systems the magnetic field strength estimate is based upon a match to at least two of the three Balmer features (Hα, Hβ, and/or Hγ) to within ±5 MG of the model minima. The last six columns give the ugriz pho- tometry and the SDSS data release for the objects. Refer to Table 1 for a full listing of photometric errors, redden- ing and alternate literature sources. Figure 8 displays the spectra of these two objects, which have relatively low S/N (∼ 5 at Hα) The iden- tification of the magnetic field strength was determined from the Hα and Hβ features in each spectrum, which upon closer inspection may show some Zeeman splitting. The best fit model for SDSS J082828.18+471737.9 has a magnetic field strength of 8 MG and an inclination of 90◦, while the best fit model for SDSS J125250.03−020608.1 has a magnetic field strength of 3 MG and an inclina- tion of 90◦. Hβ appears to be distorted in both systems, indicating a potential broadening of a few MG field, how- ever Hγ and Hδ would show more splitting than Hβ but both appear to be relatively sharp in comparison. Hβ may be affected by TiO features from the M dwarf and there does appear to be a minor glitch in the blue por- tion of the spectrum, indication difficulty with SDSS flux calibration. 5. DISCUSSION Of the 1253 potential close binary systems in the DR5 catalog, there were 168 systems that we could not mea- 6 Silvestri et al. sure with our program. These include the :+dM systems and binaries with non-DA WDs. We were not able to unambiguously determine if the :+dM systems have a magnetic or a non–magnetic WD as the blue component is barely visible in most of the :+dM SDSS spectra. Un- til we can identify the companion, we can not make any statement about magnetism in these objects. The :+dM cases where a blue component is seen in the spectrum which must be a WD, but too faint even to classify the type may include (a) cases where the WD is simply very cool, but also (b) magnetic WDs of suitably warmer ef- fective temperature but with smaller radii. These need to be reobserved in the blue with a spectrograph and tele- scope of large aperture. We made no attempt to measure the DB WDs because we lack viable magnetic DB WD models; however, all of the DB spectra matched well to non–magnetic DB WD models, so we believe it is un- likely that any of the WDs in these pairs are magnetic. We could not measure the pairs with DC WDs because there are no features with which we can detect a magnetic field and therefore cannot rule out magnetism without employing polarimetry or other methods of identifying a magnetic field in these objects. Of the remaining binary systems, we find only two that may contain WDs with weak magnetic fields. Our automatic detection methods are sensitive to magnetic fields between 3 MG ≤ BWD ≤ 30 MG; field strengths larger than this are easily identified by visual inspec- tion. Therefore, there is a significant shortage of close binary systems that could be the progenitors of the large Intermediate–Polar and Polar CV populations. As mentioned in §1, Schmidt et al. (2005b) discuss six newly identified low accretion rate magnetic binary sys- tems as being the probable progenitors to magnetic CVs. The magnetic field strengths of the WDs in these sys- tems are fairly high, with most around 60 MG. These objects are clearly pre-Polars and provide an obvious link between post–common envelope, detached binaries and Polars. The existence of these objects, however only adds to the mystery. If observations of these objects are possi- ble then why have no detached binary systems with large magnetic field WDs been detected? Perhaps selection effects are to blame. Schmidt et al. (2005b) discuss the various selection effects associated with targeting these pre–Polars with the SDSS. As is the case with the majority of the close binary systems, the pre–Polars were targeted by the SDSS QSO target- ing pipeline (Richards et al. 2002) which accounts for the narrow range of magnetic field strengths found in these objects. In the case of significantly lower or higher mag- netic field strengths, the pre–Polars resemble an ordinary WD + M dwarf binary in color–color space and are re- jected by the QSO targeting algorithm. It is possible that this selection effect accounts for the lack of close binary magnetic systems targeted by the SDSS as well. Arguing against this explanation is the large number of detached close binary systems in our sample, and the fact that the pre–Polars were observed by the SDSS. It is quite sur- prising that a detached system with a WD magnetic field in the range required to detect these pre–Polars has not been observed, if such objects exist. Another selection effect discussed by Liebert et al. (2005) argues that magnetic WDs, on average, are more massive than non–magnetic WDs; this implies smaller WD radii and therefore less luminous WDs. Faint, mas- sive WDs in competition with the flux from an M star companion might go undetected in an optical survey because they are hidden by the more luminous, non– degenerate companion. This would imply an unusually small mass ratio (q = M2/M1) for the initial binary if the progenitor of the magnetic white dwarf were mas- sive (3-8 M⊙). Thus, the magnetics may usually have been paired with an A-G star. However, the vast major- ity of polars and intermediate polars with strongly mag- netic primaries have M dwarf companions. Perhaps they were whittled down from more massive stars by mass transfer. The LARPS are selected for spectroscopy be- cause of their peculiar colors, which arise because of the isolated cyclotron harmonics. As Schmidt et al. (2005b, 2007) point out, the WDs in LARPS are generally rather faint (cool) and, in one case, undetected. So the large mass/small radius selection effect would also apply to the pre–Polars which have been observed by SDSS. 6. CONCLUSIONS We present a new sample of close binary systems through the Data Release Five of the SDSS. This cat- alog includes more than 1200 WD + M dwarf binary systems and represents the largest catalog of its kind to date. We have fit magnetic DA WD models (see Schmidt et al. 2003, and references therein) to the 1100 DA WD + M dwarf close binaries in the DR5 sample. Only two have been found to potentially harbor a mag- netic DA WD of low (BWD < 10 MG) magnetic field strength. Neither of these potential magnetic WDs are convincing cases, though follow–up spectroscopy to im- prove the S/N or polarimetry on these objects should be performed to completely rule out the presence of a magnetic field. The remaining ∼ 100 close binaries comprised of M dwarfs with excess blue flux (:+dM) and binaries with non–DA WDs require other means of detecting mag- netic fields. Methods that are sensitive to magnetic fields weaker than 3 MG should also be employed on this sample to detect possible Intermediate–Polar progenitors that may have escaped detection with our methods. Even if future spectroscopic or polarimetric observa- tions reveal the two DA WD candidates to be magnetic, their field strengths will likely prove to be quite low. A sample of two, detached, low magnetic field WD binaries is not representative of the majority of known magnetic WDs in CVs nor would it comprise an adequate progen- itor population for the newly discovered magnetic pre– Polars described in Schmidt et al. (2005b). The question of where the progenitors to magnetic CVs are remains unanswered by the current spectroscopically identified close binary population. This work was supported by NSF Grant AST 02–05875 (NMS, SLH), a University of Washington undergraduate research grant (MPL), NSF grant AST 03–06080 (GDS), and NSF grant AST 03–07321 (JL). Funding for the SDSS and SDSS–II has been pro- vided by the Alfred P. Sloan Foundation, the Partic- ipating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronau- Magnetic WDs in CV Progenitors 7 tics and Space Administration, the Japanese Monbuka- gakusho, the Max Planck Society, and the Higher Educa- tion Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Par- ticipating Institutions are the American Museum of Nat- ural History, Astrophysical Institute Potsdam, Univer- sity of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel Uni- versity, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sci- ences (LAMOST), Los Alamos National Laboratory, the Max–Planck–Institute for Astronomy (MPIA), the Max– Planck–Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton Uni- versity, the United States Naval Observatory, and the University of Washington. http://www.sdss.org/ 8 Silvestri et al. 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T., et al. 2002, AJ, 123, 2945 Schmidt, G. D., Szkody, P., Henden, A., Anderson, S. F., Lamb, D. Q., Margon, B., & Schneider, D. P. 2007, ApJ, 654, 521 Schmidt, G. D., Szkody, P., Silvestri, N. M., Cushing, M. C., Liebert, J., & Smith, P. S. 2005a, ApJ, 630, L173 Schmidt, G. D., et al. 2003, ApJ, 595, 1101 —. 2005b, ApJ, 630, 1037 Schuh, S., & Nagel, T. 2006, in ASP Conf. Ser., The 15th European Workshop on White Dwarfs, ed. R. Napiwotzki & M. Burleigh, accepted [astro–ph/0610324] Silvestri, N. M., Hawley, S. L., & Oswalt, T. D. 2005, AJ, 129, Silvestri, N. M., et al. 2006, AJ, 131, 1674 Smith, J. A. 1997, Ph.D. Thesis, Florida Institute of Technology Smith, J. A., et al. 2002, AJ, 123, 2121 Smolčić, V., et al. 2004, ApJ, 615, L141 Stoughton, C., et al. 2002, AJ, 123, 485 Tucker, D. L., et al. 2006, Astronomische Nachrichten, 327, 821 van den Besselaar, E. J. M., Roelofs, G. H. A., Nelemans, G. A., Augusteijn, T., & Groot, P. J. 2005, A&A, 434, L13 Vanlandingham, K. M., et al. 2005, AJ, 130, 734 Wickramasinghe, D. T., & Ferrario, L. 2000, PASP, 112, 873 York, D. G., et al. 2000, AJ, 120, 1579 http://arxiv.org/abs/astro-ph/0609366 http://arxiv.org/abs/astro-ph/0606700 http://arxiv.org/abs/astro--ph/0610324 Magnetic WDs in CV Progenitors 9 TABLE 1 The SDSS–I DR5 Catalog of Close Binary Systems. Identifier Plate FiberID MJD Sp1+Sp2a R.A.b Decl. upsf σu Au gpsf σg Ag rpsf σr Ar ipsf σi Ai zpsf σz Az Refs c Notesd (SDSS J) (deg) (deg) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) 001029.87+003126.2 0388 545 51793 DZ:+dM 2.62448 00.52396 21.93 0.19 0.14 20.85 0.04 0.10 19.98 0.03 0.08 19.00 0.02 0.06 18.42 0.04 0.04 EDR 001726.63−002451.2 0687 153 52518 DA+dMe 4.36099 −00.41422 19.68 0.04 0.14 19.29 0.03 0.10 19.03 0.02 0.07 18.19 0.02 0.06 17.54 0.03 0.04 R03 001733.59+004030.4 0389 614 51795 DA+dM 4.38996 00.67511 22.10 0.40 0.13 20.79 0.14 0.10 19.59 0.03 0.07 18.17 0.02 0.05 17.39 0.02 0.04 EDR/R03 001749.24−000955.3 0389 112 51795 DA+dMe 4.45519 −00.16539 16.57 0.02 0.13 16.87 0.02 0.10 17.03 0.01 0.07 16.78 0.01 0.05 16.47 0.02 0.04 EDR/R03 002620.41+144409.5 0753 079 52233 DA+dMe 6.58505 14.73597 17.57 0.01 0.27 17.35 0.01 0.20 17.34 0.02 0.15 16.65 0.01 0.11 16.04 0.02 0.08 DR2 Note. — Table 1 is published in its entirety in the electronic edition of the AJ. A portion is shown here for guidance regarding its form and content. ugriz photometry has not been corrected for Galactic extinction. Sp1: Spectral type of the WD, Sp2: Spectral type of the low–mass dwarf (see Silvestri et al. 2006, for details on Sp determination); e: emission detected visually. R.A. and Decl. are J2000.0 equinox. EDR: Stoughton et al. (2002); DR[1,2,3]: Abazajian et al. (2003, 2004, 2005); DR[4,5]: Adelman-McCarthy et al. (2006, 2007); R03: published in Raymond et al. (2003); K04: published in Kleinman et al. (2004); B05: published in van den Besselaar et al. (2005); Sc05: published in Schmidt et al. (2005a); S06: published in Silvestri et al. (2006); E06: published in Eisenstein et al. (2006); P05: published in Pourbaix et al. (2005); KM: published in Koen & Maxted (2006); SN: published in Schuh & Nagel (2006); da06: R. da Silva (priv. comm., 2006). low: potential low gravity (log g < 7) white dwarf. 10 Silvestri et al. TABLE 2 Two Potential Magnetic White Dwarf Binary Systems. Identifier Plate Fiber MJD R.A. Decl. B i Sp1+Sp2 u g r i z Release SDSS J (deg) (deg) (MG) (deg) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) 082828.18+471737.9 0549 338 51981 127.11742 +47.29387 8 90 DA+dM 20.41 20.35 20.33 19.58 19.02 DR1 125250.03−020608.1 0338 343 51694 193.20846 −02.10227 3 90 DA+dM 19.25 19.12 18.89 18.31 17.82 DR1

0704.0790

Dynamical Casimir effect for gravitons in bouncing braneworlds

arXiv:0704.0790v3 [hep-th] 15 Nov 2007 Dynamical Casimir effect for gravitons in bouncing braneworlds Marcus Ruser∗ and Ruth Durrer† Département de Physique Théorique, Université de Genève, 24 quai Ernest Ansermet, 1211 Genève 4, Switzerland. We consider a two-brane system in five-dimensional anti-de Sitter space-time. We study particle creation due to the motion of the physical brane which first approaches the second static brane (contraction) and then recedes from it (expansion). The spectrum and the energy density of the generated gravitons are calculated. We show that the massless gravitons have a blue spectrum and that their energy density satisfies the nucleosynthesis bound with very mild constraints on the parameters. We also show that the Kaluza-Klein modes cannot provide the dark matter in an anti- de-Sitter braneworld. However, for natural choices of parameters, backreaction from the Kaluza- Klein gravitons may well become important. The main findings of this work have been published in form of a Letter [R.Durrer and M.Ruser, Phys. Rev. Lett. 99, 071601 (2007), arXiv:0704.0756]. PACS numbers: 04.50.+h, 11.10.Kk, 98.80.Cq I. INTRODUCTION In recent times, the possibility that our observed Universe might represent a hypersurface in a higher- dimensional space-time has received considerable attention. The main motivation for this idea is the fact, that string theory [1, 2], which is consistent only in ten spac-etime dimensions (or 11 for M–theory) allows for solutions where the standard model particles (like fermions and gauge bosons) are confined to some hypersurface, called the brane, and only the graviton can propagate in the whole space-time, the bulk [2, 3]. Since gravity is not well constrained at small distances, the dimensions normal to the brane, the extra dimensions, can be as large as 0.1mm. Based on this feature, Arkani-Hamed, Dimopoulos and Dvali (ADD) proposed a braneworld model where the presence of two or more flat extra-dimensions can provide a solution to the hierarchy problem, the problem of the huge difference between the Planck scale and the electroweak scale [4, 5]. In 1999 Randall and Sundrum (RS) introduced a model with one extra dimension, where the bulk is a slice of five-dimensional anti de-Sitter (AdS) space. Such curved extra dimensions are also referred to as warped extra dimensions. While in the RS I model [6] with two flat branes of opposite tension at the edges of the bulk the warping leads to an interesting solution of the hierarchy problem, it localizes four-dimensional gravity on a single positive tension brane in the RS II model [7]. Within the context of warped braneworlds, cosmological evolution, i.e., the expansion of the Universe, can be understood as the motion of the brane representing our Universe through the AdS bulk. Thereby the Lanczos- Sen-Darmois-Israel-junction conditions [8, 9, 10, 11], relate the energy-momentum tensor on the brane to the ∗Electronic address: [email protected] †Electronic address: [email protected] extrinsic curvature and hence to the brane motion which is described by a modified Friedmann equation. At low energy, however, the usual Friedmann equations for the expansion of the Universe are recovered [12, 13]. Since gravity probes the extra dimension, gravita- tional perturbations on the brane, i.e. in our Universe, carry five-dimensional effects in form of massive four- dimensional gravitons, the so-called Kaluza-Klein (KK) tower. Depending on the particular brane trajectory, these perturbations may be significantly amplified lead- ing to observable consequences, for example, a stochastic gravitational wave background. (For a review of stochas- tic gravitational waves see [14].) This amplification mechanism is identical to the dynamical Casimir effect for the electromagnetic field in cavities with dynamical walls (moving mirrors); see [15, 16, 17] and references therein. In the quantum field theoretical language, such an amplification corresponds to the creation of particles out of vacuum fluctuations. Hence, in the same way a moving mirror leads to production of photons, the brane moving through the bulk causes creation of gravitons. Thereby, not only the usual four-dimensional graviton might be produced, but also gravitons of the KK tower can be excited. Those massive gravitons are of particular interest, since their energy density could dom- inate the energy density of the Universe and spoil the phenomenology if their production is sufficiently copious. The evolution of cosmological perturbations under the influence of a moving brane has been the subject of many studies during recent years. Since one has to deal with partial differential equations and time-dependent boundary conditions, the investigation of the evolution of perturbations in the background of a moving brane is quite complicated. Analytical progress has been made based on approximations like the “near brane limit” and a slowly moving brane [18, 19, 20, 21]. The case of de Sitter or quasi-de Sitter inflation on the brane has been investigated analytically in [22, 23, 24, 25, 26]. In [25] it is demonstrated that dur- http://arxiv.org/abs/0704.0790v3 mailto:[email protected] mailto:[email protected] yb(t) FIG. 1: Two branes in an AdS5 spacetime, with y denot- ing the fifth dimension and L the AdS curvature scale. The physical brane is on the left at time dependent position yb(t). While it is approaching the static brane its scale factor is decreasing and when it moves away from the static brane it is expanding [cf. Eq. (2.3)]. The value of the scale factor of the brane metric as function of the extra dimension y is also indicated. ing slow-roll inflation (modeled as a period of quasi-de Sitter expansion) the standard four-dimensional result for the amplitude of perturbations is recovered at low energies while it is enhanced at high energies. However, most of the effort has gone into numerical simulations [27, 28, 29, 30, 31, 32, 33, 34, 35, 36], in particular in order to investigate the high-energy regime. Thereby different coordinate systems have been used for which the brane is at rest, and different numerical evolution schemes have been employed in order to solve the partial differential equation. In this work we chose a different way of looking at the problem. We shall apply a formalism used to describe the dynamical Casimir effect to study the production of gravitons in braneworld cosmology. This approach and its numerical implementation offers many advantages. The most important one is the fact that this approach deals directly with the appearing mode couplings by means of coupling matrices. (In [19] a similar approach involving coupling matrices has been used. However, perturbatively only, and not in the complexity presented here.) Hence, the interaction between the four-dimensional graviton and the KK modes is not hidden within a numerical simulation but can directly be investigated making it possible to reveal the underlying physics in a very transparent way. We consider a five-dimensional anti-de Sitter spacetime with two branes in it; a moving positive tension brane representing our Universe and a second brane which, for definiteness, is kept at rest. This setup is depicted in Fig. 1. For this model we have previously shown that in a radiation dominated Universe, where the second, fixed brane is arbitrarily far away, no gravitons are produced [37]. The particular model which we shall consider is strongly motivated by the ekpyrotic or cyclic Universe and similar ideas [38, 39, 40, 41, 42, 43, 44, 45, 46]. In this model, roughly speaking, the hot big bang corresponds to the collision of two branes; a moving bulk brane which hits “our” brane, i.e. the observable Universe. Within such a model, it seems to be possible to address all major cosmological problems (homogeneity, origin of density perturbations, monopole problem) without invoking the paradigm of inflation. For more details see [38] but also [39] for critical comments. One important difference between the ekpyrotic model and standard inflation is that in the latter one tensor perturbations have a nearly scale invariant spectrum. The ekpyrotic model, on the other hand, predicts a strongly blue gravitational wave spectrum with spectral tilt nT ≃ 2 [38]. This blue spectrum is a key test for the ekpyrotic scenario since inflation always pre- dicts a slightly red spectrum for gravitational waves. One method to detect a background of primordial gravitational waves of wavelengths comparable to the Hubble horizon today is the polarization of the cosmic microwave background. Since a strongly blue spectrum of gravitational waves is unobservably small for large length scales, the detection of gravitational waves in the cosmic microwave background polarization would falsify the ekpyrotic model [38]. Here we consider a simple specific model which is generic enough to cover important main features of the generation and evolution of gravitational waves in the background of a moving brane whose trajectory involves a bounce. First, the physical brane moves towards the static brane, initially the motion is very slow. During this phase our Universe is contracting, i.e. the scale factor on the brane decreases, the energy density on the brane increases and the motion becomes faster. We suppose that the evolution of the brane is driven by a radiation component on the brane, and that at some more or less close encounter of the two branes which we call the bounce, some high-energy mechanism which we do not want to specify in any detail, turns around the motion of the brane leading to an expanding Universe. Modeling the transition from contraction to subsequent expansion in any detail would require assumptions about unknown physics. We shall therefore ignore results which depend on the details of the transition. Finally the physical brane moves away from the static brane back towards the horizon with expansion first fast and then becoming slower as the energy density drops. This model is more similar to the pyrotechnic Universe of Kallosh, Kofman and Linde [39] where the observable Universe is also represented by a positive tension brane rather than to the ekpyrotic model where our brane has negative tension. We address the following questions: What is the spec- trum and energy density of the produced gravitons, the massless zero mode and the KK modes? Can the gravi- ton production in such a brane Universe lead to limits, e.g. on the AdS curvature scale via the nucleosynthesis bound? Can the KK modes provide the dark matter or lead to stringent limits on these models? Similar results could be obtained for the free gravi-photon and gravi-scalar, i.e. when we neglect the perturbations of the brane energy momentum tensor which also couple to these gravity wave modes which have spin-1 respectively spin-0 on the brane. The reminder of the paper is organized as follows. After reviewing the basic equations of braneworld cos- mology and tensor perturbations in Sec. II, we discuss the dynamical Casimir effect approach in Sec. III. In Sec. IV we derive expressions for the energy density and the power spectrum of gravitons. Thereby we show that, very generically, KK gravitons cannot play the role of dark matter in warped braneworlds. This is explained by the localization of gravity on the moving brane which we discuss in detail. Section V is devoted to the presentation and discussion of our numerical results. In Sec. VI we reproduce some of the numerical results with analytical approximations and we derive fits for the number of produced gravitons. We discuss our main results and their implications for bouncing braneworlds in Sec. VII and conclude in Sec. VIII. Some technical aspects are collected in appendices. The main and most important results of this rather long and technical paper are published in the Letter [47]. II. GRAVITONS IN MOVING BRANEWORLDS A. A moving brane in AdS5 We consider a AdS-5 spacetime. In Poincaré coordi- nates, the bulk metric is given by ds2 = gABdx AdxB = −dt2 + δijdxidxj + dy2 (2.1) The physical brane (our Universe) is located at some time dependent position y = yb(t), while the 2nd brane is at fixed position y = ys (see Fig. 1). The induced metric on the physical brane is given by ds2 = y2b (t) dt2 + δijdx = a2(η) −dη2 + δijdxidxj , (2.2) where a(η) = yb(t) (2.3) is the scale factor and η denotes the conformal time of an observer on the brane, dt ≡ γ−1dt . (2.4) We have introduced the brane velocity v ≡ dyb = − LH√ 1 + L2H2 and (2.5) 1− v2 1 + L2H2 . (2.6) Here H is the usual Hubble parameter, H ≡ ȧ/a2 ≡ a−1H = −L−1γv , (2.7) and an overdot denotes the derivative with respect to conformal time η. The bulk cosmological constant Λ is related to the curvature scale L by Λ = −6/L2. The junction conditions on the brane lead to [37, 48] (ρ+ T ) = 6 1 + L2H2 , (2.8) (ρ+ P ) = − 2LḢ 1 + L2H2 . (2.9) Here T is the brane tension and ρ and P denote the energy density and pressure of the matter confined on the brane. Combining (2.8) and (2.9) results in ρ̇ = −3Ha(ρ+ P ) , (2.10) while taking the square of (2.8) leads to . (2.11) These equations form the basis of brane cosmology and have been discussed at length in the literature (for re- views see [49, 50]). The last equation is called the mod- ified Friedmann equation for brane cosmology [13]. For usual matter with ρ+ P > 0, ρ decreases during expan- sion and at sufficiently late time ρ ≪ T . The ordinary four-dimensional Friedmann equation is then recovered if and we set κ = 8πG4 = . (2.12) Here we have neglected a possible four-dimensional cos- mological constant. The first of these equations is the RS fine tuning implying κ5 = κ4 L . (2.13) Defining the string and Planck scales by = L3s , κ4 = = L2Pl , (2.14) respectively, the RS fine tuning condition leads to . (2.15) As outlined in the introduction, we shall be interested mainly in a radiation dominated low-energy phase, hence in the period where ρ and |v| ≪ 1 so that γ ≃ 1 , dη ≃ dt . (2.16) In such a period, the solutions to the above equations are of the form a(t) = |t|+ tb , (2.17) yb(t) = |t|+ tb , (2.18) v(t) = − sgn(t)L (|t|+ tb)2 ≃ −HL . (2.19) Negative times (t < 0) describe a contracting phase, while positive times (t > 0) describe radiation dominated expansion. At t = 0, the scale factor exhibits a kink and the evolution equations are singular. This is the bounce which we shall not model in detail, but we will have to in- troduce a cutoff in order to avoid ultraviolet divergencies in the total particle number and energy density which are due to this unphysical kink. We shall show, that when the kink is smoothed out at some length scale, the pro- duction of particles (KK gravitons) of masses larger than this scale is exponentially suppressed, as it is expected. The (free) parameter tb > 0 determines the value of the scale factor at the bounce ab, i.e. the minimal interbrane distance, as well as the velocity at the bounce vb ab = a(0) = , |v(0)| ≡ vb = . (2.20) Apparently we have to demand tb > L which implies yb(t) < L. B. Tensor perturbations in AdS5 We now consider tensor perturbations on this back- ground. Allowing for tensor perturbations hij(t,x, y) of the spatial three-dimensional geometry at fixed y, the bulk metric reads ds2 = −dt2 + (δij + 2hij)dxidxj + dy2 . (2.21) Tensor modes satisfy the traceless and transverse condi- tions, hii = ∂ih j = 0. These conditions imply that hij has only two independent degrees of freedom, the two polar- ization states • = ×,+. We decompose hij into spatial Fourier modes, hij(t,x, y) = (2π)3/2 •=+,× eik·xe•ij(k)h•(t, y;k) , (2.22) where e•ij(k) are unitary constant transverse-traceless po- larization tensors which form a basis of the two polariza- tion states • = ×,+. For hij to be real we require h∗•(t, y;k) = h•(t, y;−k). (2.23) The perturbed Einstein equations yield the equation of motion for the mode functions h , which obey the Klein- Gordon equation for minimally coupled massless scalar fields in AdS5 [25, 51, 52] ∂2t + k 2 − ∂2y + (t, y;k) = 0 . (2.24) In addition to the bulk equation of motion the modes also satisfy a boundary condition at the brane coming from the second junction condition, LH∂th• − 1 + L2H2∂yh• − γ (v∂t + ∂y)h•|yb = aPΠ(T ) . (2.25) Here Π(T ) denotes possible anisotropic stress perturba- tions in the brane energy momentum tensor. We are in- terested in the quantum production of free gravitons, not in the coupling of gravitational waves to matter. There- fore we shall set Π(T ) = 0 in the sequel, i.e. we make the assumption that the Universe is filled with a perfect fluid. Then, (2.25) reduces to 1 (v∂t + ∂y)h•|yb(t) = 0 . (2.26) This is not entirely correct for the evolution of gravity modes since at late times, when matter on the brane is no longer a perfect fluid (e.g., free-streaming neutrinos) and anisotropic stresses develop which slightly modify the evolution of gravitational waves. We neglect this sub- dominant effect in our treatment. (Some of the difficul- ties which appear when Π(T ) 6= 0 are discussed in [48].) The wave equation (2.24) together with the boundary condition (2.26) can also be obtained by variation of the action Sh = 2 yb(t) |∂th•|2 − |∂yh•|2 − k2|h•|2 , (2.27) which follows from the second order perturbation of the gravitational Lagrangian. The factor 2 in the action is 1 In Equations (4) and (8) of our Letter [47] two sign mistakes have creeped in. due to Z2 symmetry. Indeed, Equation (2.26) is the only boundary condition for the perturbation amplitude h• which is compatible with the variational principle δSh = 0, except if h• is constant on the brane. Since this issue is important in the following, it is discussed more detailed in Appendix A. C. Equations of motion in the late time/low energy limit In this work we restrict ourselves to relatively late times, when ρT ≫ ρ2 and therefore |v| ≪ 1. (2.28) In this limit the conformal time on the brane agrees roughly with the 5D time coordinate, dη ≃ dt and we shall therefore not distinguish these times; we set t = η. We want to study the quantum mechanical evolution of tensor perturbations within a canonical formulation similar to the dynamical Casimir effect for the electro- magnetic field in dynamical cavities [15, 16, 17]. In or- der to pave the way for canonical quantization, we have to introduce a suitable set of functions allowing the ex- pansion of the perturbation amplitude h• in canonical variables. More precisely, we need a complete and or- thonormal set of eigenfunctions φα of the spatial part −∂2y + 3y∂y = −y y−3∂y of the differential opera- tor (2.24). The existence of such a set depends on the boundary conditions and is ensured if the problem is of Sturm-Liouville type (see, e.g.,[53]). For the junc- tion condition (2.26), such a set does unfortunately not exist due to the time derivative. One way to proceed would be to introduce other coordinates along the lines of [54] for which the junction condition reduces to a sim- ple Neumann boundary condition leading to a problem of Sturm-Liouville type. This transformation is, however, relatively complicated to implement without approxima- tions and is the subject of future work. Here we shall proceed otherwise, harnessing the fact that we are interested in low energy effects only, i.e. in small brane velocities. Assuming that one can neglect the time derivative in the junction condition since |v| ≪ 1, Eq. (2.25) reduces to a simple Neumann boundary con- dition. We shall therefore work with the boundary con- ditions ∂yh•|yb = ∂yh•|ys = 0 . (2.29) Then, at any time t the eigenvalue problem for the spatial part of the differential operator (2.24) −∂2y + φα(t, y) = −y3∂y y−3∂yφα(t, y) = m2α(t)φα(t, y) (2.30) is of Sturm-Liouville type if we demand that the φα’s are subject to the boundary conditions (2.29). Consequently, the set of eigenfunctions {φα(t, y)}∞α=0 is complete, φα(t, y)φα(t, ỹ) = δ(y − ỹ)y3 , (2.31) and orthonormal with respect to the inner-product (φα, φβ) = 2 yb(t) φα(t, y)φβ(t, y) = δαβ . (2.32) Note the factor 2 in front of both expressions which is necessary in order to take the Z2 symmetry properly into account. The eigenvalues mα(t) are time-dependent and discrete due to the time-dependent but finite distance between the branes and the eigenfunctions φα(t, y) are time- dependent in particular because of the time dependence of the boundary conditions (2.29). The case α = 0 with m0 = 0 is the zero mode, i.e. the massless four- dimensional graviton. Its general solution in accordance with the boundary conditions is just a constant with re- spect to the extra dimension, φ0(t, y) = φ0(t), and is fully determined by the normalization condition (φ0, φ0) = 1: φ0(t) = ysyb(t)√ y2s − y2b (t) . (2.33) For α = i ∈ {1, 2, 3, · · · , } with eigenvalues mi > 0, the general solution of (2.30) is a combination of the Bessel functions J2 (mi(t) y) and Y2 (mi(t) y). Their particular combination is determined by the boundary condition at the moving brane. The remaining boundary condition at the static brane selects the possible values for the eigen- values mi(t), the KK masses. For any three-momentum k these masses build up an entire tower of momenta in the y-direction; the fifth dimension. Explicitely, the so- lutions φi(t, y) for the KK modes read φi(t, y) = Ni(t)y 2C2 (mi(t) y) (2.34) Cν(miy) = Y1(miyb)Jν(miy)−J1(miyb)Yν(miy). (2.35) The normalization reads Ni(t, yb, ys) = y2sC22(mi ys)− (2/(miπ)) (2.36) where we have used that C2(mi yb) = πmi yb . (2.37) 2 Note that we have changed the parameterization of the solutions with respect to [37] for technical reasons. There, we also did not take into account the factor 2 related to Z2 symmetry. It can be simplified further by using C2(mi ys) = Y1(mi yb) Y1(mi ys) πmi ys (2.38) leading to Y 21 (miys) Y 21 (miyb)− Y 21 (miys) . (2.39) Note that it is possible to have Y 21 (mi ys)− Y 21 (mi yb) = 0. But then both Y 21 (miys) = Y 1 (miyb) = 0 and Eq. (2.39) has to be understood as a limit. For that rea- son, the expression (2.36) for the normalization is used in the numerical simulations later on. Its denominator remains always finite. The time-dependent KK masses {mi(t)}∞i=1 are deter- mined by the condition C1 (mi(t)ys) = 0 . (2.40) Because the zeros of the cross product of the Bessel func- tions J1 and Y1 are not known analytically in closed form, the KK-spectrum has to be determined by solving Eq. (2.40) numerically 3. An important quantity which we need below is the rate of change ṁi/mi of a KK mass given by m̂i ≡ = ŷb m2i π N2i (2.41) where the rate of change of the brane motion ŷb is just the Hubble parameter on the brane ŷb(t) ≡ ẏb(t) yb(t) ≃ −Ha = − ȧ = −H . (2.42) On account of the completeness of the eigenfunctions φα(t, y) the gravitational wave amplitude h•(t, y;k) subject to the boundary conditions (2.29) can now be expanded as h•(t, y;k) = qα,k,•(t)φα(t, y) . (2.43) The coefficients qα,k,•(t) are canonical variables describ- ing the time evolution of the perturbations and the fac- κ5/L3 has been introduced in order to render the qα,k,•’s canonically normalized. In order to satisfy (2.23) we have to impose the same condition for the canonical variables, i.e. q∗α,k,• = qα,−k,•. (2.44) 3 Approximate expressions for the zeros can be found in [55]. One could now insert the expansion (2.43) into the wave equation (2.24), multiplying it by φβ(t, y) and integrating out the y−dependence by using the orthonormality to de- rive the equations of motion for the variables qα,k,•. How- ever, as we explain in Appendix A, a Neumann boundary condition at a moving brane is not compatible with a free wave equation. The only consistent way to implement the boundary conditions (2.29) is therefore to consider the action (2.27) of the perturbations as the starting point to derive the equations of motion for qα,k,•. Inserting (2.43) into (2.27) leads to the canonical action S = 1 |q̇α,k,•|2 − ω2α,k|qα,k,•|2 Mαβ (qα,k,•q̇β,−k,• + qα,−k,•q̇β,k,•) +Nαβqα,k,•qβ,−k,• . (2.45) We have introduced the time-dependent frequency of a graviton mode ω2α,k = k2 +m2α , k = |k| , (2.46) and the time-dependent coupling matrices Mαβ = (∂tφα, φβ) , (2.47) Nαβ = (∂tφα, ∂tφβ) = MαγMβγ (2.48) which are given explicitely in Appendix B (see also [37]). Consequently, the equations of motion for the canonical variables are q̈α,k,• + ω α,kqα,k,• + [Mβα −Mαβ ] q̇β,k,• Ṁαβ −Nαβ qβ,k,• = 0 . (2.49) The motion of the brane through the bulk, i.e. the expansion of the Universe, is encoded in the time-dependent coupling matrices Mαβ , Nαβ. The mode couplings are caused by the time-dependent boundary condition ∂yh•(t, y)|yb = 0 which forces the eigenfunctions φα(t, y) to be explicitly time-dependent. In addition, the frequency of a KK mode ωα,k is also time-dependent since the distance between the two branes changes when the brane is in motion. Both time- dependencies can lead to the amplification of tensor perturbations and, within a quantum theory which is developed in the next section, to graviton production from vacuum. Because of translation invariance with respect to the directions parallel to the brane, modes with different k do not couple in (2.49). The three-momentum k enters the equation of motion for the perturbation only via the frequency ωα,k, i.e. as a global quantity. Equation (2.49) is similar to the equation describing the time-evolution of electromagnetic field modes in a three-dimensional dynamical cavity [16] and may effectively be described by a massive scalar field on a time-dependent interval [17]. For the electromagnetic field, the dynamics of the cavity, or more precisely the motion of one of its walls, leads to photon creation from vacuum fluctuations. This phenomenon is usually referred to as dynamical Casimir effect. Inspired by this, we shall call the production of gravitons by the moving brane as dynamical Casimir effect for gravitons. D. Remarks and comments In [37] we have already shown that in the limit where the fixed brane is sent off to infinity, ys → ∞, only the M00 matrix element survives with M00 = −H[1 + O(ǫ)] and ǫ = yb/ys. M00 expresses the coupling of the zero mode to the brane motion. Since all other couplings dis- appear for ǫ → 0 all modes decouple from each other and, in addition, the canonical variables for the KK modes de- couple from the brane motion itself. This has led to the result that at late times and in the limit ys ≫ yb, the KK modes with non-vanishing mass evolve trivially, and only the massless zero mode is coupled to the brane motion q̈0,k,• + k2 − Ḣ −H2 q0,k,• = 0 . (2.50) Since φ0 ∝ 1/a [cf. Eqs. (4.2),(4.5)] we have found in [37] that the gravitational zero mode on the brane h0,•(t;k) ≡√ κ5/L3q0,k,•φ0(t, yb) evolves according to ḧ0,•(t;k) + 2Hḣ0,•(t;k) + k2h0,•(t;k) = 0 , (2.51) which explicitely demonstrates that at low energies (late times) the homogeneous tensor perturbation equation in brane cosmology reduces to the four-dimensional tensor perturbation equation. An important comment is in order here concerning the RS II model. In the limit ys → ∞ the fixed brane is sent off to infinity and one ends up with a single positive tension brane in AdS, i.e. the RS II model. Even though we have shown that all couplings except M00 vanish in this limit, that does not imply that this is necessarily the case for the RS II setup. Strictly speaking, the above arguments are only valid in a two brane model with ys ≫ 1. Starting with the RS II model from the beginning, the coupling matrices do in general not vanish when calculated with the corresponding eigenfunctions which can be found in, e.g., [22]. One just has to be careful when taking those limits. But what the above consideration demonstrates is that, if the couplings of the zero mode to the KK modes vanish, like in the ys ≫ 1 limit or in the low energy RS II model as observed in numerical simulations (see below) the standard evolution equation for the zero mode emerges automatically from five-dimensional perturbation theory. Starting from five-dimensional perturbation theory, our formalism does imply the usual evolution equation for the four-dimensional graviton in a FLRW-Universe in the limit of vanishing couplings. This serves as a very strong indication (but certainly not proof!) for the fact that the approach based on the approximation (2.29) and the expansion of the action in canonical variables rather than the wave equation is consistent and leads to results which should reflect the physics at low energies. As already outlined, if one would expand the wave equation (2.24) in the set of functions φα, the resulting equation of motion for the corresponding canonical variables is different from Eq. (2.49) and cannot be derived from a Lagrangian or Hamiltonian (see Appendix A). Moreover, in [30] the low energy RS II scenario has been studied numerically including the full junction condition (2.26) without approximations (see also [27]). Those numerical results show that the evolution of tensor perturbations on the brane is four-dimensional, i.e. described by Eq. (2.51) derived here analytically. Combining these observations gives us confidence that the used approach based on the Neumann boundary condition approximation and the action as starting point for the canonical formulation is adequate for the study of tensor perturbations in the low energy limit. The many benefits this approach offers will become visible in the following. III. QUANTUM GENERATION OF TENSOR PERTURBATIONS A. Preliminary remarks We now introduce a treatment of quantum generation of tensor perturbations. This formalism is an advance- ment of the method which is presented in [15, 16, 17] for the dynamical Casimir effect for a scalar field and the electromagnetic field to gravitational perturbations in the braneworld scenario. The following method is very general and not restricted to a particular brane motion as long as it complies with the low energy approach [cf. Eq. (2.28)]. We assume that asymptotically, i.e. for t → ±∞, the physical brane approaches the Cauchy horizon (yb → 0), moving very slowly. Then, the coupling matrices vanish and the KK masses are constant (for yb close to zero, Eq. (2.40) re- duces to J1(miys) = 0): Mαβ(t) = 0 , lim mα(t) = const. ∀α, β . (3.1) In this limit, the system (2.49) reduces to an infinite set of uncoupled harmonic oscillators. This allows to intro- duce an unambiguous and meaningful particle concept, i.e. notion of (massive) gravitons. As a matter of fact, in the numerical simulations, the brane motion has to be switched on and off at finite times. These times are denoted by tin and tout, respectively. We introduce vacuum states with respect to times t < tin < 0 and t > tout > 0. In order to avoid spurious effects influ- encing the particle creation, we have to chose tin small, respectively tout large enough such that the couplings are effectively zero at these times. Checking the indepen- dence of the numerical results on the choice of tin and tout guarantees that these times correspond virtually to the real asymptotic states of the brane configuration. B. Quantization, initial and final state Canonical quantization of the gravity wave amplitude is performed by replacing the canonical variables qα,k,• by the corresponding operators q̂α,k,• ĥ•(t, y;k) = q̂α,k,•(t)φα(t, y) . (3.2) Adopting the Heisenberg picture to describe the quantum time-evolution, it follows that q̂α,k,• satisfies the same equation (2.49) as the canonical variable qα,k,•. Under the assumptions outlined above, the operator q̂α,k,• can be written for times t < tin as q̂α,k,•(t < tin) = (3.3) 2ωinα,k âinα,k,•e −i ωinα,k t + âin†α,−k,•e i ωinα,k t where we have introduced the initial-state frequency ωinα,k ≡ ωα,k(t < tin) . (3.4) This expansion ensures that Eq. (2.44) is satisfied. The set of annihilation and creation operators {âinα,k,•, â α,k,•} corresponding to the notion of gravitons for t < tin is subject to the usual commutation relations âinα,k,•, â α′,k′,•′ = δαα′δ••′δ (3)(k− k′) , (3.5) âinα,k,•, â α′,k′,•′ α,k,•, â α′,k′,•′ = 0. (3.6) For times t > tout, i.e. after the motion of the brane has ceased, the operator q̂α,k,• can be expanded in a similar manner, q̂α,k,•(t > tout) = (3.7) 2ωoutα,k âoutα,k,•e −i ωout t + â out † α,−k,•e i ωout with final state frequency ωoutα,k ≡ ωα,k(t > tout) . (3.8) The annihilation and creation operators {âoutα,k,•, â out † α,k,•} correspond to a meaningful definition of final state gravi- tons (they are associated with positive and negative fre- quency solutions for t ≥ tout) and satisfy the same com- mutation relations as the initial state operators. Initial |0, in〉 ≡ |0, t < tin〉 and final |0, out〉 ≡ |0, t > tout〉 vacuum states are uniquely defined via 4 âinα,k,•|0, in〉 = 0 , âoutα,k,•|0, out〉 = 0 , ∀ α, k, • . (3.9) The operators counting the number of particles defined with respect to the initial and final vacuum state, respec- tively, are N̂ inα,k,• = â α,k,•â α,k,• , N̂ α,k,• = â out † α,k,•â α,k,• . (3.10) The number of gravitons created during the motion of the brane for each momentum k, quantum number α and polarization state • is given by the expectation value of the number operator N̂outα,k,• of final-state gravitons with respect to the initial vacuum state |0, in〉: N outα,k,• = 〈0, in|N̂outα,k,•|0, in〉. (3.11) If the brane undergoes a non-trivial dynamics between tin < t < tout it is â α,k,•|0, in〉 6= 0 in general, i.e. graviton production from vacuum fluctuations takes place. From (2.22), the expansion (3.2) and Eqs.(3.3), (3.7) it follows that the quantized tensor perturbation with respect to the initial and final state can be written ĥij(t < tin,x,y) = (2π)3/2 âinα,k,• e −i ωinα,k t 2ωinα,k × u•ij,α(t < tin,x, y,k) + h.c. (3.12) ĥij(t > tout,x,y) = (2π)3/2 âoutα,k,• e −i ωoutα,k t 2ωoutα,k × u•ij,α(t > tout,x, y,k) + h.c. . (3.13) We have introduced the basis functions u•ij,α(t,x, y,k) = e ik ·x e•ij(k)φα(t, y). (3.14) which, on account of (e•ij(k)) ∗ = e•ij(−k), satisfy (u•ij,α(t,x, y,k)) ∗ = u•ij,α(t,x, y,−k). 4 Note that the notations |0, t < tin〉 and |0, t > tout〉 do not mean that the states are time-dependent; states do not evolve in the Heisenberg picture. C. Time evolution During the motion of the brane the time evolution of the field modes is described by the system of coupled differential equations (2.49). To account for the inter- mode couplings mediated by the coupling matrix Mαβ the operator q̂α,k,• is decomposed as q̂α,k,•(t) = 2ωinβ,k âinβ,k,•ǫ α,k(t) + â β,−k,•ǫ α,k (t) (3.15) The complex functions ǫ α,k(t) also satisfy the system of coupled differential equations (2.49). With the ansatz (3.15) the quantized tensor perturbation at any time dur- ing the brane motion reads ĥij(t,x, y) = (3.16) âinβ,k,•√ 2ωinβ,k α,k(t)u ij,α(t,x, y,k) + h.c. . Due to the time-dependence of the eigenfunctions φα, the time-derivative of the gravity wave amplitude con- tains additional mode coupling contributions. Using the completeness and orthnormality of the φα’s it is readily shown that h•(t, y;k) = p̂α,−k,•(t)φα(t, y) (3.17) where p̂α,−k,•(t) = ˙̂qα,k,•(t) + Mβαq̂β,k,•(t). (3.18) The coupling term arises from the time dependence of the mode functions φα. Accordingly, the time derivative hij reads hij(t,x, y) = âinβ,k,•√ 2ωinβ,k × (3.19) × f (β)α,k(t)u ij,α(t,x, y,k) + h.c. where we have introduced the function α,k(t) = ǫ̇ α,k(t) + Mγα(t)ǫ γ,k(t) . (3.20) By comparing Eq. (3.12) and its time-derivative with Eqs. (3.16) and (3.19) at t = tin one can read off the initial conditions for the functions ǫ α,k(tin) = δαβ Θ α,k , (3.21) α,k(tin) = −iωinα,kδαβ −Mβα(tin) Θinβ,k (3.22) with phase Θinα,k = e −iωinα,k tin . (3.23) The choice of this phase for the initial condition is in principle arbitrary, we could as well set Θinα,k = 1. But with this choice, ǫ α,k(t) is independent of tin for t < tin and therefore it is also at later times independent of tin if only we choose tin sufficiently early. This is especially useful for the numerical work. D. Bogoliubov transformations The two sets of annihilation and creation operators {âinα,k,•, â α,k,•} and {âoutα,k,•, â out † α,k,•} corresponding to the notion of initial-state and final-state gravitons are re- lated via a Bogoliubov transformation. Matching the expression for the tensor perturbation Eq. (3.16) and its time-derivative Eq. (3.19) with the final state expres- sion Eq. (3.13) and its corresponding time-derivative at t = tout one finds âoutβ,k,• = Aαβ,k(tout)âinα,k,• + B∗αβ,k(tout)â α,−k,• (3.24) Aβα,k(tout) = ωoutα,k ωinβ,k α,k(tout) + ωoutα,k α,k(tout) (3.25) Bβα,k(tout) = Θoutα,k ωoutα,k ωinβ,k α,k(tout)− ωoutα,k α,k(tout) (3.26) where we shall stick to the phase Θoutα,k defined like Θ in (3.23) for completeness. Performing the matching at tout = tin the Bogoliubov transformation should become trivial, i.e. the Bogoliubov coefficients are subject to vac- uum initial conditions Aαβ,k(tin) = δαβ , Bαβ,k(tin) = 0. (3.27) Evaluating the Bogoliubov coefficients (3.25) and (3.26) for tout = tin by making use of the initial conditions (3.21) and (3.22) shows the consistency. Note that the Bogoliubov transformation (3.24) is not diagonal due to the inter-mode coupling. If during the motion of the brane the graviton field departs form its vacuum state one has Bαβ,k(tout) 6= 0, i.e. gravitons have been generated. By means of Eq. (3.24) the number of generated final state gravitons (3.11), which is the same for every polarization state, is given by N outα,k (t ≥ tout) = •=+,× 〈0, in|N̂outα,k,•|0, in〉 |Bβα,k(tout)|2. (3.28) Later we will sometimes interpret tout as a continuous variable tout → t such that N outα,k → Nα,k(t), i.e. it becomes a continuous function of time. We shall call Nα,k(t) the instantaneous particle number [see Appendix C 2], however, a physical interpretation should be made with caution. E. The first order system From the solutions of the system of differential equa- tions (2.49) for the complex functions ǫ α,k, the Bogoli- ubov coefficient Bαβ,k, and hence the number of cre- ated final state gravitons (3.28), can now be calculated. It is however useful to introduce auxiliary functions α,k(t), η α,k(t) through α,k(t) = ǫ α,k(t) + ωinα,k α,k(t) (3.29) α,k(t) = ǫ α,k(t)− ωinα,k α,k(t) . (3.30) These are related to the Bogoliubov coefficients via Aβα,k(tout) = (3.31) ωoutα,k ωinβ,k ∆+α,k(tout)ξ α,k(tout) + ∆ α,k(tout)η α,k(tout) Bβα,k(tout) = (3.32) Θoutα,k ωoutα,k ωinβ,k ∆−α,k(tout)ξ α,k(tout) + ∆ α,k(tout)η α,k(tout) where we have defined ∆±α,k(t) = ωinα,k ωα,k(t) , (3.33) Using the second order differential equation for ǫ α,k, it is readily shown that the functions ξ α,k(t), η α,k(t) satisfy the following system of first order differential equations: α,k(t) = −i a+αα,k(t)ξ α,k(t)− a αα,k(t)η α,k(t) c−αγ,k(t)ξ γ,k(t) + c αγ,k(t)η γ,k(t) (3.34) α,k(t) = −i a−αα,k(t)ξ α,k(t)− a αα,k(t)η α,k(t) c+αγ,k(t)ξ γ,k(t) + c αγ,k(t)η γ,k(t) (3.35) a±αα,k(t) = ωinα,k ωα,k(t) ωinα,k  , (3.36) c±γα,k(t) = Mαγ(t)± ωinα,k ωinγ,k Mγα(t) . (3.37) The vacuum initial conditions (3.27) entail the initial conditions α,k(tin) = 2 δαβ Θ α,k , η α,k(tin) = 0. (3.38) With the aid of Eq. (3.32), the coefficient Bαβ,k(tout), and therefore the number of produced gravitons, can be directly deduced from the solutions to this system of coupled first order differential equations which can be solved using standard numerics. In the next section we will show how interesting observables like the power spectrum and the energy den- sity of the amplified gravitational waves are expressed in terms of the number of created gravitons. The system (3.34, 3.35) of coupled differential equations forms the basis of our numerical simulations. Details of the applied numerics are collected in Appendix D. IV. POWER SPECTRUM, ENERGY DENSITY AND LOCALIZATION OF GRAVITY A. Perturbations on the brane By solving the system of coupled differential equations formed by Eqs. (3.34) and (3.35) the time evolution of the quantized tensor perturbation ĥij(t,x, y) can be com- pletely reconstructed at any position y in the bulk. Ac- cessible to observations is the imprint which the pertur- bations leave on the brane, i.e. in our Universe. Of par- ticular interest is therefore the part of the tensor pertur- bation which resides on the brane. It is given by eval- uating Eq. (2.22) at the brane position y = yb (see also [36]) ĥij(t,x, yb) = (2π)3/2 •=+,× eik·xe•ij(k)ĥ•(t, yb,k) . (4.1) The motion of the brane (expansion of the Universe) en- ters this expression via the eigenfunctions φα(t, yb(t)). We shall take (4.1) as the starting point to define ob- servables on the brane. The zero-mode function φ0(t) [cf. Eq. (2.33)] does not depend on the extra dimension y. Using Eq. (2.37), one reads off from Eq. (2.34) that the eigenfunctions on the brane φα(t, yb) are φα(t, yb) = yb Yα(yb) = Yα(a) (4.2) where we have defined Y0(a) = y2s − y2b and (4.3) Yn(a) = Y 21 (mnys) Y 21 (mnyb)− Y 21 (mnys) , (4.4) for the zero- and KK modes, respectively. One immedi- ately is confronted with an interesting observation: the function Yα(a) behaves differently with the expansion of the Universe for the zero mode α = 0 and the KK modes α = n. This is evident in particular in the asymptotic regime ys ≫ yb, i.e. yb → 0 (|t|, a → ∞) where, exploit- ing the asymptotics of Y1 (see [55]), one finds Y0(a) ≃ 1 , Yn(a) ≃ |Y1(mnys)| ≃ (4.5) Ergo, Y0 is constant while Yn decays with the expansion of the Universe as 1/a. For large n one can approximate mn ≃ nπ/ys and Y1(mnys) ≃ Y1(nπ) ≃ (1/π) 2/n [55], so that Yn(a) ≃ , Y2n(a) ≃ πL2mn 2 ysa2 . (4.6) In summary, the amplitude of the KKmodes on the brane decreases faster with the expansion of the Universe than the amplitude of the zero mode. This leads to interest- ing consequences for the observable power spectrum and energy density and has a clear physical interpretation: It manifest the localization of usual gravity on the brane. As we shall show below, KK gravitons which are traces of the five-dimensional nature of gravity escape rapidly from the brane. B. Power spectrum We define the power spectrum P(k) of gravitational waves on the brane as in four-dimensional cosmology by using the restriction of the tensor amplitude to the brane position (4.1): (2π)3 P(k)δ(3)(k− k′) (4.7) •=×,+ 0, in ∣∣∣ĥ•(t, yb;k)ĥ†•(t, yb;k′) ∣∣∣0, in i.e. we consider the expectation value of the field operator ĥ• with respect to the initial vacuum state at the position of the brane y = yb(t). In order to get a physically mean- ingful power spectrum, averaging over several oscillations of the gravitational wave amplitude has to be performed. Equation (4.7) describes the observable power spectrum imprinted in our Universe by the four-dimensional spin-2 graviton component of the five-dimensional tensor per- turbation. The explicit calculation of the expectation value involv- ing a “renormalization” of a divergent contribution is car- ried out in detail in Appendix C 2. The final result reads P(k) = 1 (2π)3 Rα,k(t)Y2α(a). (4.8) The function Rα,k(t) can be expressed in terms of the Bogoliubov coefficients (3.25) and (3.26) if one considers tout as a continuous variable t: Rα,k(t) = Nα,k(t) +ONα,k(t) ωα,k(t) . (4.9) Nα,k(t) is the instantaneous particle number [cf. Ap- pendix C 1] and the function ONα,k(t) is defined in Eq. (C9). It is important to recall thatNα,k(t) can in general not be interpreted as a physical particle number. For example zero modes with wave numbers such that kt < 1 can- not be considered as particles. They have not performed several oscillations and their energy density cannot be defined in a meaningful way. Equivalently, expressed in terms of the complex functions α,k, one finds Rα,k(t) = |ǫ(β)α,k(t)|2 ωinβ,k ωα,k(t) +Oǫα,k(t), (4.10) with Oǫα,k given in Eq. (C10). Equation (4.8) together with (4.9) or (4.10) holds at all times. If one is interested in the power spectrum at early times kt ≪ 1, it is not sufficient to take only the instantaneous particle number Nα,k(t) in Eq. (4.9) into account. This is due to the fact that even if the mode functions ǫ are already oscillating, the coupling matrix entering the Bogoliubov coefficients might still undergo a non-trivial time dependence [cf. Eq. (6.16)]. In the next section we shall show explicitly, that in a radiation dominated bounce particle creation, especially of the zero mode, only stops on sub-Hubble times, kt > 1, even if the mode functions are plane waves right after the bounce [cf, e.g., Figs. 6, 7, 9]. Therefore, in order to determine the per- turbation spectrum of the zero mode, one has to make use of the full expression expression (4.10) and may not use (4.11), given below. At late times, kt ≫ 1 (t ≥ tout) when the brane moves slowly, the couplings Mαβ go to zero and particle cre- ation has come to an end, both functions ONα,k and Oǫα,k do not contribute to the observable power spectrum after averaging over several oscillations. Furthermore, the in- stantaneous particle number then equals the (physically meaningful) number of created final state gravitons N outα,k and the KK masses are constant. Consequently, the ob- servable power spectrum at late times takes the form P(k, t ≥ tout) = (2π)3 N outα,k ωoutα,k Y2α(a) , (4.11) where we have used that κ5/L = κ4. Its dependence on the wave number k is completely determined by the spec- tral behavior of the number of created gravitons N outα,k . It is useful to decompose the power spectrum in its zero- mode and KK-contributions: P = P0 + PKK . (4.12) In the late time regime, using Eqs. (4.11) and (4.5), the zero-mode power spectrum reads P0(k, t ≥ tout) = (2π)3 N out0,k . (4.13) As expected for a usual four-dimensional tensor perturba- tion (massless graviton), on sub-Hubble scales the power spectrum decreases with the expansion of the Universe as 1/a2. In contrast, the KK mode power spectrum for late times, given by PKK(k, t ≥ tout) = N outn,k ωoutn,k Y 21 (mnys), (4.14) decreases as 1/a4, i.e. with a factor 1/a2 faster than P0. The gravity wave power spectrum at late times is therefore dominated by the zero-mode power spectrum and looks four dimensional. Contributions to it arising from five-dimensional effects are scaled away rapidly as the Universe expands due to the 1/a4 behavior of PKK. In the limit of large masses mnys ≫ 1, n ≫ 1 and for wave lengths k ≪ mn such that ωn,k ≃ mn, the late-time KK-mode power spectrum can be approximated by PKK(k, t ≥ tout) = 16π2ys N outn,k (4.15) where we have inserted Eq. (4.6) for Y2n(a). Note that the formal summations over the particle num- ber might be ill defined if the brane trajectory contains unphysical features like discontinuities in the velocity. An appropriate regularization is then necessary, for example, by introducing a physically motivated cutoff. C. Energy density For a usual four-dimensional tensor perturbation hµν on a backgroundmetric gµν an associated effective energy momentum tensor can be defined unambiguously by (see, e.g., [14, 56]) Tµν = 〈hαβ‖µhαβ‖ν〉 , (4.16) where the bracket stands for averaging over several pe- riods of the wave and “‖” denotes the covariant deriva- tive with respect to the unperturbed background metric. The energy density of gravity waves is the 00-component of the effective energy momentum tensor. We shall use the same effective energy momentum tensor to calculate the energy density corresponding to the four-dimensional spin-2 graviton component of the five-dimensional ten- sor perturbation on the brane, i.e. for the perturbation hij(t,x, yb) given by Eq. (4.1). For this it is important to remember that in our low energy approach, and in particular at very late times for which we want to cal- culate the energy density, the conformal time η on the brane is identical to the conformal bulk time t. The en- ergy density of four-dimensional spin-2 gravitons on the brane produced during the brane motion is then given by [see also [36]] κ4 a2 0, in| ˙̂hij(t,x, yb) ˙̂hij(t,x, yb)|0, in . (4.17) Here the outer bracket denotes averaging over several os- cillations, which (in contrast to the power spectrum) we embrace from the very beginning. The factor 1/a2 comes from the fact that an over-dot indicates the derivative with respect t. A detailed calculation is carried out in Appendix C 3 leading to (2π)3 ωα,kNα,k(t)Y2α(a) (4.18) where againNα,k(t) is the instantaneous particle number. At late times t > tout after particle creation has ceased, the energy density is therefore given by (2π)3 ωoutα,k N outα,k Y2α(a). (4.19) This expression looks at first sight very similar to a “naive” definition of energy density as integration over momentum space and summation over all quantum num- bers α of the energy ωoutα,k N outα,k of created gravitons. (Note that the graviton number N outα,k already contains the contributions of both polarizations [see Eq. (3.28)].) However, the important difference is the appearance of the function Y2α(a) which exhibits a different dependence on the scale factor for the zero mode compared to the KK modes. Let us decompose the energy density into zero-mode and KK contributions ρ = ρ0 + ρKK . (4.20) For the energy density of the massless zero mode one then obtains (2π)3 kN out0,k . (4.21) This is the expected behavior; the energy density of stan- dard four-dimensional gravitons scales like radiation. On contrast, the energy density of the KK modes at late times is found to be ρKK = (2π)3 ωoutn,k N outn,k m2nY 21 (mnys), (4.22) which decays like 1/a6. As the Universe expands, the en- ergy density of massive gravitons on the brane is there- fore rapidly diluted. The total energy density of gravita- tional waves in our Universe at late times is dominated by the standard four-dimensional graviton (massless zero mode). In the large mass limit mnys ≫ 1,n ≫ 1 the KK- energy density can be approximated by ρKK ≃ 2a6ys (2π)3 N outn,k ωoutn,kmn . (4.23) Due to the factor mn coming from the function Y2n, i.e. from the normalization of the functions φn(t, y), for the summation over the KK-tower to converge, the number of produced gravitons N outn,k has to decrease faster than 1/m3n for large masses and not just faster than 1/m one might naively expect. D. Escaping of massive gravitons and localization of gravity As we have shown, the power spectrum and energy density of the KK modes scale, at late times when par- ticle production has ceased, with the expansion of the Universe like PKK ∝ 1/a4 , ρKK ∝ 1/a6. (4.24) Both quantities decay by a factor 1/a2 faster than the corresponding expressions for the zero-mode graviton. In particular, the energy density of the KK particles on the brane behaves effectively like stiff matter. Mathemat- ically, this difference arises from the distinct behavior of the functions Y0(a) and Yn(a) [cf. Eq. (4.5)] and is a direct consequence of the warping of the fifth dimension. But what is the underlying physics? As we shall discuss now, this scaling behavior for the KK particles has indeed a very appealing physical interpretation which is in the spirit of the RS model. First, the mass mn is a comoving mass. The (in- stantaneous) ’comoving’ frequency or energy of a KK graviton is ωn,k = k2 +m2n, with comoving wave number k. The physical mass of a KK mode measured by an observer on the brane with cosmic time dτ = adt is therefore mn/a, i.e. the KK masses are redshifted with the expansion of the Universe. This comes from the fact that mn is the wave number corresponding to the y-direction with respect to the bulk time t which corresponds to conformal time η on the brane and not to physical time. It implies that the energy of KK particles on a moving AdS brane is redshifted like that of massless particles. From this alone one would expect that the energy density of KK modes on the brane decays like 1/a4 (see also Appendix D of [22]). Now, let us define the “wave function” for a gravi- Ψα(t, y) = φα(t, y) (4.25) which, by virtue of (φα, φα) = 1, satisfies dyΨ2α(t, y) = 1 (4.26) From the expansion of the gravity wave amplitude Eq. (2.43) and the normalization condition it is clear that Ψ2α(t, y) gives the probability to find a graviton of mass mα for a given (fixed) time t at position y in the Z2- symmetric AdS-bulk. Since φα satisfies Equation (2.30), the wave function Ψα satisfies the Schrödinger like equa- − ∂2yΨα + Ψα = m αΨα (4.27) and the junction conditions (2.29) translate into Ψα|y={yb,ys} = 0. (4.28) In Fig. 2 we plot the evolution of Ψ21(t, y) under the influence of the brane motion Eq. (2.18) with vb = 0.1. For this motion, the physical brane starting at yb → 0 for t → −∞ moves towards the static brane, corresponding to a contracting Universe. After a bounce, it moves back to the Cauchy horizon, i.e. the Universe expands. The second brane is placed at ys = 10L and y ranges from yb(t) to ys. We set Ψ 1 ≡ 0 for y < yb(t) . The time-dependent KK mass m1 is determined numerically from Eq. (2.40). As it is evident from this Figure, Ψ21 is effectively localized close to the static brane, i.e. the weight of the KK-mode wave function lies in the region of less warping, far from the physical brane. Thus the probability to find a KK mode is larger in the region with less warping. Since the effect of the brane motion on Ψ21 is hardly visible in Fig. 2, we show the behavior of Ψ21 close to the physical brane in Fig. 3. This shows that Ψ21 peaks also at the physical brane but with an amplitude roughly ten times smaller than the amplitude at the static brane. While the brane, coming from t → −∞, approaches the point of closest encounter Ψ21 slightly increases and peaks at the bounce t = 0 where, as we shall show in the next Section, the production of KK particles takes place. Afterwards, for t → ∞, when the brane is moving back towards the Cauchy horizon, the amplitude Ψ21 decreases again and so does the probability to find a KK particle at the position of the physical brane, i.e. in our Universe. The parameter settings used in Figures 2 and 3 are typical parameters which we use in the numerical simulations described later on. However, the effect is illustrated much better if the second brane is closer to the moving brane. In Figure 4 we show Ψ21 for the same parameters as in Figures 2 and 3 but now with ys = L. In this case, the probability to find a KK particle on the physical brane is of the same order as in the region close to the second brane during times close to the bounce. However, as the Universe expands, Ψ21 rapidly decreases at the position of the physical brane. From Eqs. (4.2) and (4.5) it follows that Ψ2n(t, yb) ∝ 1/a. The behavior of the KK-mode wave function suggests the following interpretation: If KK gravitons are created on the brane, or equivalently in our Universe, they escape from the brane into the bulk as the brane moves back to the Cauchy horizon, i.e. when the Universe undergoes expansion. This is the reason why the power spectrum and the energy density imprinted by the KK modes on the brane decrease faster with the expansion of the Universe than for the massless zero mode. The zero mode, on the other hand, is localized at the position of the moving brane. The profile of φ0 does not depend on the extra dimension, but the zero-mode wave function Ψ0 does. Its square is Ψ20(t, y) = y2s − y2b if ys ≫ yb , (4.29) such that on the brane (y = yb) it behaves as Ψ20(t, yb) ≃ . (4.30) Equation (4.29) shows that, at any time, the zero mode is localized at the position of the moving brane. For a better illustration we show Eq. (4.29) in Fig. 5 for the same parameters as in Fig. 4. This is the “dynamical analog” of the localization mechanism for four-dimensional gravity discussed in [7]. To establish contact with [7] and to obtain a intu- itive physical description, we rewrite the boundary value problem (4.27), (4.28) as a Schrödinger-like equation − ∂2yΨα(t, y) + V (y, t)Ψα(y, t) = mα(t)Ψα(y, t) (4.31) V (y, t) = yb(t) δ(|y| − yb(t)) − 3a(t) δ(|y| − yb(t)) , (4.32) where we have absorbed the boundary condition at the moving brane into the (instantaneous) volcano potential V (y, t) and made use of Z2 symmetry. Similar to the static case [7], at any time the potential (4.32) supports a single bound state, the four-dimensional graviton (4.29), and acts as a barrier for the massive KK modes. The potential, ensuring localization of four-dimensional gravity on the brane and the repulsion of KK modes, moves together with the brane through the fifth dimen- sion. Note that with the expansion of the Universe, the “depth of the delta-function” becomes larger, expressing the fact that the localization of four-dimensional gravity becomes stronger at late times [cf. Eq. (4.30), Fig. 5]. In summary, the different scaling behavior for the zero- and KK modes on the brane is entirely a conse- quence of the geometry of the bulk space-time, i.e. of the warping L2/y2 of the metric (2.1) 5. It is simply a manifestation of the localization of gravity on the brane: as time evolves, the KK gravitons, which are traces of the five-dimensional nature of gravity, escape into the bulk and only the zero mode which corresponds to the usual four-dimensional graviton remains on the brane. This, and in particular the scaling behavior (4.24), remains also true if the second brane is removed, i.e. in the limit ys → ∞, leading to the original RS II model. By looking at (4.15) and (4.23) one could at first think that then the KK-power spectrum and energy density vanish and no traces of the KK gravitons could be observed on the brane since both expressions behave as 1/ys. But this is not the case since the spectrum of KK masses becomes continuous. In the continuum limit ys → ∞ the summation over the discrete spectrum mn has to be replaced by an integration over continuous masses m in the following way: f(mn) −→ dmf(m) . (4.33) f is some function depending on the spectrum, for example f(mn) = N outn,k . The pre-factor 1/ys in (4.15) and (4.23) therefore ensures the existence of the proper continuum limit of both expressions. Another way of seeing this is to repeat the same calculations but using the eigenfunctions for the case with only one brane from the beginning. Those are δ-function normalized and can be found in, e.g., [22]. They are basically the same as (2.34) except that the normalization is different since it depends on whether the fifth dimension is compact or not. In particular, on the brane, they have the same scale factor dependence as (4.2). At the end, the behavior found for the KK modes should not come as a surprise, since the RS II model has attracted lots of attention because of exactly this; it localizes usual four-dimensional gravity on the brane. As we have shown here, localization of standard four- dimensional gravity on a moving brane via a warped geometry automatically ensures that the KK modes escape into the bulk as the Universe expands because their wave function has its weight in the region of less warping, resulting in an KK-mode energy density on the brane which scales like stiff matter. An immediate consequence of this particular scaling behavior is that KK gravitons in an AdS braneworld 5 Note that it does not depend on a particular type of brane motion and is expected to be true also in the high energy case which we do not consider here. FIG. 2: Evolution of Ψ21(t, y) = φ 1(t, y)/y 3 corresponding to the probability to find the first KK graviton at time t at the position y in the AdS-bulk. The static brane is at ys = 10L and the maximal brane velocity is given by vb = 0.1. FIG. 3: Evolution of Ψ21(t, y) as in Fig. 2 but zoomed into the bulk-region close to the moving brane. cannot play the role of dark matter. Their energy density in our Universe decays much faster with the expansion than that of ordinary matter which is restricted to reside on the brane. V. NUMERICAL SIMULATIONS A. Preliminary remarks In this section we present results of numerical simula- tions for the bouncing model described by the equations (2.17)-(2.19). In the numerical simulations we set L = 1, i.e. all FIG. 4: Evolution of Ψ21(t, y) for ys = L and vb = 0.1. FIG. 5: Localization of four-dimensional gravity on a moving brane: Evolution of Ψ20(t, y) for ys = L = 1 and vb = 0.1 which should be compared with Ψ21(t, y) shown in Fig. 4. dimensionful quantities are measured in units of the AdS5 curvature scale. Starting at initial time tin ≪ 0 where the initial vacuum state |0, in〉 is defined, the system (3.34,3.35) is evolved numerically up to final time tout. Thereby we set tin = −2πNin/k with 1 ≤ Nin ∈ N, such that Θin0,k = 1 [cf. Eq. (3.23)]. This implies 0 (tin) = 2, i.e. independent of the three-dimensional momentum k a (plane wave) zero-mode solution always performs a fixed number of oscillations between tin and the bounce at t = 0 [cf. Eq. (3.38)]. The final graviton spectrum at N outα,k is calculated at late times tout ≫ 1 when the brane approaches the Cauchy horizon and graviton creation has ceased. This quantity is physically well defined and leads to the late-time power spectrum (4.11) and energy density (4.19) on the brane. For illustrative purposes, we also plot the instantaneous particle number Nα,k,•(t) which also determines the power spectrum at all times [cf Eq.(4.9)]. In this section we shall use the term particle number respectively graviton number for both, the instantaneous particle number Nα,k,•(t) as well as the final state graviton number N outα,k,•, keeping in mind that only the latter one is physically meaningful. There are two physical input parameters for the numerical simulation; the maximal brane velocity vb (i.e. tb) and the position of the static brane ys. The latter determines the number of KK modes which fall within a particular mass range. On the numerical side one has to specify Nin and tout, as well as the maximum number of KK modes nmax which one takes into account, i.e. after which KK mode the system of differential equations is truncated. The independence of the numerical results on the choice of the time parameters is checked and the convergence of the particle spectrum with increasing nmax is investigated. More detailed information on numerical issues including accuracy considerations are collected in Appendix D. One strong feature of the brane motion (2.18) is its kink at the bounce t = 0. In order to study how particle production depends on the kink, we shall compare the motion (2.18) with the following motion which has a smooth transition from contraction to expansion (L = 1): yb(t) = (|t|+ tb − ts)−1 if |t| > ts a+ (b/2)t2 + (c/4)t4 if |t| ≤ ts (5.1) with the new parameter ts in the range 0 < ts < tb. This motion is constructed such that its velocity at |t| = ts is the same as the velocity of the kink motion at the bounce. This will be the important quantity determin- ing the number of produced gravitons. For ts → 0 the motion with smooth transition approaches (2.18). The parameters a, b and c are obtained by matching the mo- tions and the first and second derivatives. Matching also the second derivative guarantees that possible spurious effects contributing to particle production are avoided. The parameter ts has to be chosen small enough, ts ≪ 1, such that the maximal velocity of the smooth motion is not much larger than vb in order to have comparable sit- uations. For reasons which will become obvious in the next two sections we shall discuss the cases of long k ≪ 1 and short wavelengths k ≫ 1, separately. B. Generic results and observations for long wavelengths k ≪ 1 Figure 6 displays the results of a numerical simula- tion for three-momentum k = 0.01, static brane position ys = 10 and maximal brane velocity vb = 0.1. Depicted is FIG. 6: Evolution of the graviton number Nα,k,•(t) for the zero mode and the first ten KK modes for three-momentum k = 0.01 and vb = 0.1, ys = 10. FIG. 7: Nn,k,•(t) for the zero mode and the first ten KK modes for the parameters of Fig. 6, but without coupling of the zero mode to the KK modes, i.e. Mi0 ≡ 0. the graviton number for one polarizationNα,k,•(t) for the zero mode and the first ten KK modes as well as the evo- lution of the scale factor a(t) and the position of the phys- ical brane yb(t). Initial and final times are Nin = 5 and tout = 2000, respectively. The KK-particle spectrum will be discussed in detail below. One observes that the zero- mode particle number increases slightly with the expan- sion of the Universe towards the bounce at t = 0. Close to the bounce N0,k,•(t) increases drastically, shows a lo- cal peak at the bounce and, after a short decrease, grows again until the mode is sub-horizon (kt ≫ 1). Inside the horizon N0,k,•(t) is oscillating around a mean value with diminishing amplitude. This mean value which is reached asymptotically for t → ∞ corresponds to the number of generated final state zero-mode gravitonsN out0,k,•. Produc- tion of KK-mode gravitons takes effectively place only at the bounce in a step-like manner and the graviton num- ber remains constant right after the bounce. In Fig. 7 we show the numerical results obtained for the same parameters as in Fig. 6 but without coupling of the zero mode to the KK modes, i.e. Mi0 = 0 (and thus also Ni0 = N0i = 0). One observes that the production of zero-mode gravitons is virtually not affected by the artificial decoupling 6. Note that even if M0j ≡ 0 (see Eqs. B2), which is in general true for Neumann bound- ary conditions, the zero mode q0,k,• couples in Eq. (2.49) to the KK modes via N0j = M00Mj0 and through the anti-symmetric combination Mαβ −Mβα. In contrast, the production of the first ten KK modes is heavily suppressed if Mi0 = 0. The corresponding final- state graviton numbers N outn,k,• are reduced by four orders of magnitude. This shows that the coupling to the zero mode is essential for the production of massive gravitons. Later we will see that this is true for light KK gravitons only. If the KK masses exceed mi ∼ 1, they evolve in- dependently of the four-dimensional graviton and their evolution is entirely driven by the intermode couplings Mij . It will also turn out that the time-dependence of the KK mass mi plays only an inferior role for the gen- eration of massive KK modes. On the other hand, the effective decoupling of the evolution of the zero mode from the KK modes occurs in general as long as k ≪ 1 is satisfied, i.e. for long-wavelengths. We will see that it is no longer true for short wavelengths k ≫ 1. The effective decoupling of the zero-mode evolution from the KK modes makes it possible to derive analytical ex- pressions for the number of zero-mode gravitons, their power spectrum and energy density. The calculations are carried out in section VIA In summary we emphasize the important observation that for long wavelengths the amplification of the four dimensional gravity wave amplitude during the bounce is not affected by the evolution of the KK gravitons. We can therefore study the zero mode separately from the KK modes in this case. C. Zero mode: long wavelengths k ≪ 1 In Figure 8 we show the numerical results for the num- ber of generated zero-mode gravitons N0,k,•(t) and the evolution of the corresponding power spectrum P0(k) on the brane for momentum k = 0.01, position of the static brane ys = 10 and maximal brane velocity vb = 0.1. The results have been obtained by solving the equations for the zero mode alone, i.e. without the couplings to the KK modes, since, as we have just shown, the evolution of the 6 Quantitatively it is N0,k,•(t = 2000) = 965.01 with and N0,k,•(t = 2000) = 965.06 without Mi0. Note that this differ- ence lies indeed within the accuracy of our numerical simulations (see Appendix D.) four-dimensional graviton for long wavelengths is not in- fluenced by the KK modes. Thereby the power spectrum is shown before and after averaging over several oscilla- tions, i.e. employing Eq. (4.9) with and without the term ON0,k, respectively. Right after the bounce where the gen- eration of gravitons is initiated and which is responsible for the peak in N0,k,• at t = 0, the number of gravitons first decreases again. AfterwardsN0,k,• grows further un- til the mode enters the horizon at kt = 1. Once on sub- horizon scales kt ≫ 1, the number of produced gravitons oscillates with a diminishing amplitude and asymptoti- cally approaches the final state graviton number N out0,k,•. During the growth of N0,k,• after the bounce, the power spectrum remains practically constant. Within the range of validity it is in good agreement with the analytical pre- diction (6.22) yielding (L2(2π)3/κ4)P0(k, t) = 4vb(kL)2. When particle creation has ceased, the full power spec- trum Eq.(4.8) starts to oscillate with an decreasing am- plitude. The time-averaged power spectrum obtained by using Eq. (4.9) without the ON0,k-term is perfectly in agreement with the analytical expression Eq. (6.20) which gives (L2(2π)3/κ4)P0(k, t) = 2vb/t2. Note that at early times, the time-averaged power spectrum behaves not in the same way as the full one, demonstrating the importance of the term ON0,k. Figure 9 shows a summary of numerical results for the number of created zero-mode gravitons N0,k,•(t) for dif- ferent values of the three-momentum k. The maximum velocity at the bounce is vb = 0.1 and the second brane is at ys = 10. These values are representative. Other values in accordance with the considered low-energy regime do not lead to a qualitatively different behavior. Note that the evolution of the zero mode does virtually not depend on the value of ys as long as ys ≫ yb(0) (see below). Ini- tial and final integration times are given by Nin = 5 and tout = 20000, respectively. For sub-horizon modes we compare the final graviton spectra with the analytical prediction (6.17). Both are in perfect agreement. On super-horizon scales where parti- cle creation has not ceased yet N0,k,• is independent of k. The corresponding time-evolution of the power spec- tra P0(k, t) is depicted in Fig. 10. For the sake of clarity, only the results for t > 0, i.e. after the bounce, are shown in both figures. The numerical simulations and the calculations of sec- tion VIA reveal that the power spectrum for the four- dimensional graviton for long wavelengths is blue on super-horizon scales, as expected for an ekpyrotic sce- nario. The analytical calculations performed in section VIA rely on the assumption that yb ≪ ys and tin → −∞. Figure 11 shows the behavior of the number of generated zero-mode gravitons of momentum k = 0.01 in depen- dence on the inter-brane distance and the initial integra- tion time. The brane velocity at the bounce is vb = 0.1 which implies that at the bounce the moving brane is at yb(0) = vb ≃ 0.316 (L = 1). In case of a close en- counter of the two branes as for ys = 0.35, the production FIG. 8: Time evolution of the number of created zero-mode gravitons N0,k,•(t) and of the zero-mode power spectrum (4.8): (a) for the entire integration time; (b) for t > 0 only. Parameters are k = 0.01, ys = 10 and vb = 0.1. Initial and fi- nal time of integration are given by Nin = 10 and tout = 4000, respectively. The power spectrum is shown with and without the term ON0,k,•, i.e. before and after averaging, respectively, and compared with the analytical results. FIG. 9: Numerical results for the time evolution of the num- ber of created zero-mode gravitons N0,k,•(t) after the bounce t > 0 for different three-momenta k. The maximal brane velocity at the bounce is vb = 0.1 and the second brane is positioned at ys = 10. In the final particle spectrum the nu- merical values are compared with the analytical prediction Eq. (6.17). Initial and final time of integration are given by Nin = 5 and tout = 20000, respectively. of massless gravitons is strongly enhanced compared to the analytical result. But as soon as ys ≥ 1, (i.e. ys ≥ L) the numerical result is very well described by the analyt- ical expression Eq. (6.16) derived under the assumption ys ≫ yb. For ys ≥ 10 the agreement between both is very good. From panels (b) and (c) one infers that the numerical result becomes indeed independent of the ini- FIG. 10: Evolution of the zero-mode power spectrum after the bounce t > 0 corresponding to the values and parameters of Fig. 9. The numerical results are compared to the analytical predictions Eqs. (6.20) and (6.22). 0 1000 2000 time t analytical =0.35 10 100 1000 2nd brane pos. y 0.1 1 10 100 1000 2nd brane pos. y analytical (a) (b) FIG. 11: Dependence of the zero-mode particle number on inter-brane distance and initial integration time for momen- tum k = 0.01, maximal brane velocity vb = 0.1 in comparison with the analytical expression Eq. (6.16). (a) Evolution of the instantaneous particle number N0,k,•(t) with initial integra- tion time given by Nin = 5 for ys = 0.35, 0.5 and 1. (b) Final zero-mode graviton spectrum N0,k,•(tout = 2000) for various values of ys and Nin. (c) Close-up view of (b) for large ys. tial integration time when increasing Nin. Note that in the limit Nin ≫ 1 the numerical result is slightly larger than the analytical prediction but the difference between both is negligibly small. This confirms the correctness and accuracy of the analytical expressions derived in Sec- tion VIA for the evolution of the zero-mode graviton. 0.01 0.1 1 10 Kaluza-Klein mass m t nmax=60 = 0.3 = 0.1 = 0.5● FIG. 12: Final state KK-graviton spectra for k = 0.001, ys = 100, different maximal brane velocities vb and Nin = 1, tout = 400. The numerical results are compared with the analytical prediction Eq. (6.34) (dashed line). D. Kaluza-Klein-modes: long wavelengths k ≪ 1 Because the creation of KK gravitons ceases right after the bounce [cf Fig. 6] one can stop the numerical simulation and read out the number of produced KK gravitons N outn,k,• at times for which the zero mode is still super-horizon. Even though Eq. (2.40) cannot be solved analytically, the KK masses can be approximated by mn ≃ nπ/ys. This expression is the better the larger the mass. Consequently, for the massive modes the position of the second brane ys determines how many KK modes belong to a particular mass range ∆m. In Figure 12 we show the KK-graviton spectra N outn,k,• for three-momentum k = 0.001 and second brane position ys = 100 for maximal brane velocities vb = 0.1, 0.3 and 0.5. For any velocity vb two spectra obtained with nmax = 60 and 80 KK modes taken into account in the simulation are compared to each other. This reveals that the numerical results are stable up to a KK mass mn ≃ 1. One infers that first, N outn,k,• grows with increasing mass until a maximum is reached. The position of the maximum shifts slightly towards larger masses with increasing brane velocity vb. Afterwards, N outn,k,• declines with growing mass. Until the maximum is reached, the numerical results for the KK-particle spectrum are very stable. This already indicates that the KK-intermode couplings mediated by Mij are not very strong in this mass range. In Figure 13 we show the final KK-particle spectrum for the same parameters as in Fig. 12 but for three-momentum k = 0.01 and 0.01 0.1 1 10 Kaluza-Klein mass m =60, M =0 for all i,j = 0.1 = 0.5 = 0.3 = 0.9 FIG. 13: Final state KK-graviton spectra for k = 0.01, ys = 100, different vb and Nin = 1, tout = 400. The numerical results are compared with the analytical prediction Eq. (6.34) (dashed line). For vb = 0.3, 0.5 the spectra obtained without KK-intermode and self-couplings (Mij ≡ 0 ∀ i, j) are shown as well. the additional velocity vb = 0.9 7. We observe the same qualitative behavior as in Fig. 12. In addition we show numerical results obtained for vb = 0.3 and 0.5 without the KK-intermode and self couplings, i.e. we have set Mij ≡ 0 ∀ i, j by hand. One infers that for KK masses, depending slightly on the velocity vb but at least up to mn ≃ 1, the numerical results for the spectra do not change when the KK-intermode coupling is switched off. Consequently, the evolution of light, i.e. mn <∼ 1, KK gravitons is virtually not affected by the KK-intermode coupling. In addition we find that also the time-dependence of the KK masses is not important for the production of light KK gravitons which is explicitly demonstrated below. Thus, production of light KK gravitons is driven by the zero-mode evolution only. This allows us to find an analytical expression, Eq. (6.34), for the number of produced light KK gravitons in terms of exponential integrals. The calculations which are based on several approximations are performed in Section VIC. In Figs. 12 and 13 the analytical prediction (6.34) for the spectrum of final state gravitons has already been included (dashed lines). Within its range of validity it is in excellent agreement with the numerical results obtained by including the full KK-intermode coupling. It perfectly describes the dependence of N outn,k,• on the three-momentum k and the maximal velocity vb. For small velocities vb <∼ 0.1 it is also able to reproduce the position of the maximum. This reveals that the KK- 7 Such a high brane velocity is of course not consistent with a Neumann boundary condition Eq. (2.29) at the position of the moving brane. intermode coupling is negligible for light KK gravitons and that their production is entirely driven by their coupling to the four-dimensional graviton. The analytical prediction is very precious for testing the goodness of the parameters used in the simulations, in particular the initial time tin (respectively Nin). Since it has been derived for real asymptotic initial conditions, tin → −∞, its perfect agreement with the numerical results demonstrates that the values for Nin used in the numerical simulations are large enough. No spurious initial effects contaminate the numerical results. Note, that the numerical values for N outn,k,• in the ex- amples shown are all smaller than one. However, for smaller values of k than the ones which we consider here for purely numerical reasons, the number of generated KK-mode particles is enhanced since N outn,k,• ∝ 1/k as can be inferred from Eq. (6.34) in the limit k ≪ mn. If we go to smaller values of ys, fewer KK modes belong to a particular mass range. Hence, with the same or similar number of KK modes as taken into account in the simulations so far, we can study the behavior of the final particle spectrum for larger masses. These simulations shall reveal the asymptotical behavior of N outn,k,• for mn → ∞ and therefore the behavior of the total graviton number and energy density. Due to the kink in the brane motion we cannot expect that the energy density of produced KK-mode gravitons is finite when summing over arbitrarily high frequency modes. Eventually, we will have to introduce a cutoff setting the scale at which the kink-approximation [cf. Eqs. (2.17) - (2.19)] is no longer valid. This is the scale where the effects of the underlying unspecified high-energy physics which drive the transition from contraction to expansion become important. The dependence of the final particle spectrum on the kink will be studied later on in this section in detail. In Figures 14 and 15 we show final KK-graviton spectra for ys = 10 and three-momentum k = 0.01 and k = 0.1. The analytical expression Eq. (6.34) is depicted as well and the spectra are always shown for at least two values of nmax to indicate up to which KK mass stability of the the numerical results is guaranteed. Now, only two KK modes are lighter than m = 1. For these modes the analytical expression Eq. (6.34) is valid and in excellent agreement with the numerical results, in particular for small brane velocities vb ∼ 0.1. As before, the larger the velocity vb the more visible is the effect of the truncation of the system of differential equations at nmax. For k = 0.01 the spectrum seems to follow a power law decrease right after the maximum in the spectra. In case of vb = 0.1 the spectrum is numerically stable up to masses mn ≃ 20. In the region 5 <∼ mn <∼ 20 the spec- trum is very well fitted by a power law N outn,k,• ∝ m−2.7n . Also for larger velocities the decline of the spectrum is given by the same power within the mass ranges 0.1 1 10 Kaluza-Klein mass m = 0.5 = 0.3 = 0.1 FIG. 14: Final state KK-graviton spectra for k = 0.01, ys = 10, different maximal brane velocities vb and Nin = 2, tout = 400. The numerical results are compared with the analytical prediction Eq. (6.34) (dashed line). 0.1 1 10 Kaluza - Klein mass m = 0.5 = 0.1 = 0.3 FIG. 15: Final state KK-graviton spectra for k = 0.1, ys = 10, different maximal brane velocities vb and Nin = 2, tout = 400. The numerical results are compared with the analytical prediction Eq. (6.34) (dashed line). where the spectrum is numerically stable. For k = 0.1, however, the decreasing spectrum bends over at a mass around mn ≃ 10 towards a less steep decline. This is in particular visible in the two cases with vb = 0.1 and 0.3 where the first 100 KK modes have been taken into account in the simulation. The behavior of the KK-mode particle spectrum can therefore not be described by a single power law decline for masses mn > 1. It shows more complicated features instead, which depend on the parameters. We shall demonstrate that this bending over of the decline is related to the coupling properties of the KK modes and to the kink in the brane motion. But before we come to a detailed discussion of these issues, let us briefly confront numerical results of different ys to FIG. 16: Upper panel: Final state KK-particle spectra for k = 0.01, vb = 0.1 and different ys = 3, 10, 30 and 100. The analytical prediction Eq. (6.34) is shown as well (dashed line). Lower panel: Energy ωoutn,kN n,k,• of the produced fi- nal state gravitons binned in mass intervals ∆m = 1 for ys = 10, 30, 100. demonstrate a scaling behavior. In the upper panel of Figures 16 and 17 we com- pare the final KK-spectra for several positions of the second brane ys = 3, 10, 30 and 100 obtained for a maximal brane velocity vb = 0.1 for k = 0.01 and 0.1, respectively. One observes that the shapes of the spectra are identical. The bending over in the decline of the spectrum at masses mn ∼ 1 is very well visible for k = 0.1 and ys = 3, 10. For a given KK mode n the number of particles produced in this mode is the larger the smaller ys. But the smaller ys, the less KK modes belong to a given mass interval ∆m. The energy transferred into the system by the moving brane, which is determined by the maximum brane velocity vb, is the same in all cases. Therefore, the total energy of the pro- duced final state KK gravitons of a given mass interval ∆m should also be the same, independent of how many KK modes are contributing to it. This is demonstrated in the lower panels of Figs. 16 and 17 where the energy ωoutn,kN outn,k,•(in units of L) of the generated KK gravitons binned in mass intervals ∆m = 1 is shown 8. One observes that, as expected, the energy transferred into the production of KK gravitons of a particular mass range is the same (within the region where the numerical results are stable), independent of the number of KK modes lying in the interval. This is in particular evident for ys = 30, 100. The discrepancy for ys = 10 is due to the binning. As we shall discuss below in detail, the 8 The energy for the case ys = 3 is not shown because no KK mode belongs to the first mass interval. FIG. 17: Upper panel: Final state KK-particle spectra for k = 0.1, vb = 0.1 and different ys = 3, 10, 30 and 100. The analytical prediction Eq. (6.34) is shown as well (dashed line). Lower panel: Energy ωoutn,kN n,k,• of the produced fi- nal state gravitons binned in mass intervals ∆m = 1 for ys = 10, 30, 100. particle spectrum can be split into two different parts. The first part is dominated by the coupling of the zero mode to the KK modes (as shown above), whereas the second part is dominated by the KK-intermode couplings and is virtually independent of the wave number k. As long as the coupling of the zero mode to the KK modes is the dominant contribution to KK-particle production it is N outn,k,• ∝ 1/k [cf. Eq. (6.34)]. Hence, Eoutn,k,• = ωoutn,kN outn,k,• ∝ 1/k if mn ≫ k. This explains why the energy per mass interval ∆m is one order larger for k = 0.01 (cf Fig. 16) than for k = 0.1 (cf Fig. 17) . Let us now discuss the KK-spectrum for large masses. The qualitative behavior of the spectrum N outn,k,• and the mass at which the decline of the spectrum changes are independent of ys. This is demonstrated in Figure 18 where KK-spectra for vb = 0.1, k = 0.1, ys = 10 [cf Fig. 15] and ys = 3 [cf Fig. 17] are shown. The results obtained by taking the full intermode coupling into account are compared to results of simulations where we have switched off the coupling of the KK modes to each other as well as their self-coupling (Mij ≡ 0 ∀ i, j). Furthermore we display the results for the KK-spectrum obtained by taking only the KK-intermode couplings into account, i.e. Mi0 = Mii = 0 ∀ i. One infers that for the lowest masses the spectra obtained with all couplings are identical to the ones obtained without the KK-intermode (Mij = 0, i 6= j) and self-couplings (Mii = 0). Hence, as already seen before, the primary source for the production of light KK gravitons is their coupling to the evolution of the four-dimensional gravi- ton. In this mass range, the contribution to the particle creation coming from the KK-intermode couplings is very much suppressed and negligibly small. 1 10 100 Kaluza-Klein mass m t full coupling =0 for all i,j =0, M FIG. 18: KK-particle spectra for three-momentum k = 0.1, maximum brane velocity vb = 0.1 and ys = 3 and 10 with different couplings taken into account. The dashed lines indi- cates again the analytical expression Eq. (6.34). For masses mn ≃ 4 a change in the decline of the spectrum sets in and the spectrum obtained without the coupling of the KK modes to the zero mode starts to diverge from the spectrum computed by taking all the couplings into account. While the spectrum without the KK-intermode couplings decreases roughly like a power law N outn,k,• ∝ m−3n the spectrum corresponding to the full coupling case changes its slope towards a power law decline with less power. At this point the KK-intermode couplings gain importance and the coupling of the KK modes to the zero mode looses influence. For a particular mass mc ≃ 9 the spectrum obtained including the KK-intermode couplings only, crosses the spectrum calculated by taking into account exclusively the coupling of the KK modes to the zero mode. After the crossing, the spectrum obtained by using only the KK-intermode couplings approaches the spectrum of the full coupling case. Both agree for large masses. Thus for large masses mn > mc the production of KK gravitons is dominated by the couplings of the KK modes to each other and is not influenced anymore by the evolution of the four-dimensional graviton. This crossing defines the transition between the two regimes mentioned before: for masses mn < mc the production of KK gravitons takes place due to their coupling to the zero mode Mi0, while it is entirely caused by the intermode couplings Mij for masses mn > mc. Decoupling of the evolution of the KK modes from the dynamics of the four-dimensional graviton for large masses implies that KK-spectra obtained for the same maximal velocity are independent of the three- momentum k. This is demonstrated in Fig. 19 where we compare spectra obtained for vb = 0.1 and ys = 3 but different k. As expected, all spectra converge towards the same behavior for masses mn > mc. 1 2 3 4 5 10 20 30 1000.3 Kaluza-Klein masses m t k=0.01 k=0.03 k=0.1 FIG. 19: Comparison of KK-particle spectra for ys = 3, vb = 0.1 and three-momentum k = 0.01, 0.03, 0.1 and 1 demonstrating the independence of the spectrum on k for large masses. nmax = 60 KK modes have been taken into account in the simulations. 1 10 100 Kaluza-Klein mass m t full coupling =0 for all i, j =0, i = j =0, M Mαβ = 0 for all α, β FIG. 20: KK-particle spectra for three-momentum k = 0.1, maximum brane velocities vb = 0.1 and ys = 3 for nmax = 40 obtained for different coupling combinations. Figure. 20 shows KK-particle spectra for k = 0.1,vb = 0.1 and ys = 3 obtained for different couplings. This plot visualizes how each particular coupling combination contributes to the production of KK gravitons. It shows, as already mentioned before but not shown explicitly, that the Mii coupling which is the rate of change of the corresponding KK mass [cf. Eqs. (2.41) and (B4)] is not important for the production of KK gravitons. Switching it off does not affect the final graviton spectrum. We also show the result obtained with all couplings but with α+ii(t) = ω i,k and α ii (t) = 0, i.e. the time-dependence of the frequency [cf. Eq. (3.36)] has been neglected. One observes that in this case the spectrum for larger masses is quantitatively slightly 0.1 1 10 100 Kaluza-Klein mass m t full coupling =0 for all i, j =0, M k=0.01 k=0.1 k=0.1 FIG. 21: KK-particle spectra for ys = 10, vb = 0.1, nmax = 100 and three-momentum k = 0.01 and 0.1 with different couplings taken into account. The thin dashed lines indicates Eq. (6.34) and the thick dashed line Eq. (5.4). different but has a identical qualitative behavior. If, on the other hand, all the couplings are switched off Mαβ ≡ 0 ∀α, β and only the time-dependence of the frequency ωi,k is taken into account, the spectrum changes drastically. Not only the number of produced gravitons is now orders of magnitude smaller but also the spectral tilt changes. For large masses it behaves as Nn,k,• ∝ m−2n . Consequently, the time-dependence of the graviton frequency itself plays only an inferior role for production of KK gravitons. The bottom line is that the main sources of the produc- tion of KK gravitons is their coupling to the evolution of the four-dimensional graviton (Mi0) and their couplings to each other (Mij , i 6= j) for small and large masses, respectively. Both are caused by the time-dependent boundary condition. The time-dependence of the oscilla- tor frequency ωj,k = m2j(t) + k 2 is virtually irrelevant. Note that this situation is very different from ordinary inflation where there are no boundaries and particle production is due entirely to the time dependence of the frequency 9. The behavior of the KK-spectrum, in particular the mass mc at which the KK-intermode couplings start to dominate over the coupling of the KK modes to the zero mode depends only on the three-momentum k = |k| and the maximal brane velocity vb. This is now discussed. In Figure 21 we show KK-particle spectra for ys = 10, vb = 0.1, nmax = 100 and three-momenta k = 0.01 and 0.1. Again, the spectra obtained by taking 9 Note, however, that the time-dependent KK mass mj(t) enters the intermode couplings. 0.1 1 10 Kaluza-Klein mass m t full coupling =0 for all i, j =0, M = 0.1 = 0.1 = 0.3 = 0.03 FIG. 22: KK-particle spectra for three-momentum k = 0.1,ys = 10 and maximum brane velocities vb = 0.03, 0.1 and 0.3 with nmax = 100. As in Fig. 21 different couplings have been taken into account and thin dashed lines indicates Eq. (6.34) and the thick dashed line Eq. (5.4). all the couplings into account are compared to the case where only the coupling to the zero mode is switched on. One observes that for k = 0.01 the spectrum is dominated by the coupling of the KK modes to the zero mode up to larger masses than it is the case for k = 0.1. For k = 0.01 the spectrum obtained taking into account Mi0 only is identical to the spectrum obtained with the full coupling up to mn ≃ 10. In case of k = 0.1 instead, the spectrum is purely zero mode dominated only up to mn ≃ 5. Hence, the smaller the three-momentum k the larger is the mass range for which the KK-intermode coupling is suppressed, and the coupling of the zero mode to the KK modes is the dominant source for the production of KK gravitons. As long as the coupling to the zero mode is the primary source of particle production, the spectrum declines with a power law ∝ m−3n . Therefore, in the limiting case k → 0 when the coupling of the zero mode to the KK modes dominates particle production also for very large masses it is N outn≫1,k→0,• ∝ 1/m3n. Figure 22 shows KK-graviton spectra obtained for the same parameters as in Fig. 21 but for fixed k = 0.1 and different maximal brane velocities vb. Again, the spectra obtained by taking all the couplings into account are compared with the spectra to which only the coupling of the KK modes to the zero mode contributes. The mass up to which the spectra obtained with different couplings are identical changes only slightly with the maximal brane velocity vb. Therefore, the dependence of mc on the velocity is rather weak even if vb is changed by an order of magnitude, but nevertheless evident. This behavior of the spectrum can indeed be understood qualitatively. In Section VIC we demonstrate that the coupling strength of the KK modes to the zero mode 1 10 100 1 10 100 1 10 100 1 10 100 k=0.01 k=0.03 k=0.1 k=1 =25.54 mc=14.11 =8.24 mc=7.31 FIG. 23: KK-particle spectra for three-momentum k = 0.01, 0.03, 0.1 and 1 for ys = 3 and maximum brane velocity vb = 0.1 with different couplings taken into account where the notation is like in Fig. 22. From the crossing of the Mii = Mij = 0- and Mii = Mi0 = 0 results we determine the k-dependence of mc(k, vb). The thick dashed line indi- cates Eq. (5.4). at the bounce t = 0, where production of KK gravitons takes place, is proportional to . (5.2) The larger this term the stronger is the coupling of the KK modes to the zero mode, and thus the larger is the mass up to which this coupling dominates over the KK-intermode couplings. Consequently, the mass at which the tilt of the KK-particle spectrum changes depends strongly on the three-momentum k but only weakly on the maximal brane velocity due to the square root behavior of the coupling strength. This explains qualitatively the behavior obtained from the numerical simulations. An approximate expression for mc(k, vb) can be obtained from the numerical simulations. In Figure 23 we depict the KK-particle spectra for three-momentum k = 0.01, 0.03, 0.1 and 1 for ys = 3 and maximum brane velocity vb = 0.1 with different couplings taken into account. The legend is as in Fig. 22. From the crossings of the Mij = 0, i 6= j and Mii = Mi0 = 0 results one can determine the k-dependence of mc. Note that the spectra are not numerically stable for large masses, but they are stable in the range where mc lies [cf., e.g., Fig. 25, for k = 0.1]. Using the data for k = 0.01, 0.03 and 0.1 one finds mc(k, vb) ∝ 1/ In Fig. 24 KK-graviton spectra are displayed for k = 0.1, ys = 3 and maximal brane velocities vb = 0.3, 0.2, 0.1, 0.08, 0.05 and 0.03 with different couplings taken into account. It is in principle possible to determine the vb-dependence of mc from the crossings 1 10 100 1 10 100 1 10 100 = 0.3 =9.50 = 0.2 v = 0.1 = 0.08 =8.24 = 0.05 v = 0.03 =9.04 =8.06 m =7.72 m =7.52 FIG. 24: KK-graviton spectra for three-momentum k = 0.1, ys = 3 and maximum brane velocities vb = 0.3, 0.2, 0.1, 0.08, 0.05 and 0.03 with different couplings taken into account where the notation is like in Fig. 22. From the crossing of the Mii = Mij = 0- and Mii = Mi0 = 0 results we determine the vb-dependence of mc. of the Mij = 0, i 6= j- and Mii = Mi0 = 0 results as done for the k-dependence. However, the values for mc displayed in the Figures indicate that the dependence of mc on vb is very weak. From the given data it is not possible to obtain a good fitting formula (as a simple power law) for the vb-dependence of mc. (In the range 0.1 ≤ vb ≤ 0.3 a very good fit is mc = 1.12πv0.13b / The reason is twofold. First of all, given the complicated coupling structure, it is a priori not clear that a simple power law dependence exists. Recall that also the ana- lytical expression for the particle number Eq. (6.34) has not a simple power law velocity dependence. Moreover, for the number of modes taken into account (nmax = 40) the numerical results are not stable enough to resolve the weak dependence of mc on vb with a high enough accuracy. (But it is good enough to perfectly resolve the k-dependence.) The reason for the slow convergence of the numerics will become clear below. As we shall see, the corresponding energy density is dominated by masses much larger than mc. Consequently the weak dependence of mc on vb is not very important in that respect and therefore does not need to be determined more precisely. However, combining all the data we can give as a fair approximation mc(k, vb) ≃ π vαb , with α ≃ 0.1. (5.3) Taking α = 0.13 for 0.1 ≤ vb ≤ 0.3 and α = 0.08 for 0.03 ≤ vb ≤ 0.1 fits the given data reasonably well. As we have seen, as long as the zero mode is the dominant source of KK-particle production, the final KK-graviton spectrum can be approximated by a power law decrease m−3n . We can combine the presented numerical results to obtain a fitting formula valid in this regime: N outn≫1,k≪1,• = (Lmn)3 , for < mn < mc. (5.4) This fitting formula is shown in Figs. 21 22 and 23 and is in reasonable good agreement with the numerical results. Since Eq. (5.4) together with (5.3) is an impor- tant result, we have reintroduced dimensions, i.e. the AdS scale L which is set to one in the simulations, in both expressions. Let us now investigate the slope of the KK-graviton spectrum for masses mn → ∞ since it determines the contribution of the heavy KK modes to the energy den- sity. In Figure 25 we show KK-graviton spectra obtained for three-momentum k = 0.1, second brane position ys = 3 and maximal brane velocities vb = 0.01, 0.03 and 0.1. Up to nmax = 100 KK modes have been taken into account in the simulations. One immediately is confronted with the observation that the convergence of the KK-graviton spectra for large mn is very slow. This is since those modes, which are decoupled from the evolution of the four-dimensional graviton, are strongly affected by the kink in the brane motion. Recall that the production of light KK gravitons with masses mn ≪ mc is virtually driven entirely by the evolution of the massless mode. Those light modes are not so sensitive to the discontinuity in the velocity of the brane motion. To be more precise, their primary source of excitation is the evolution of the four-dimensional graviton but not the kink which, as we shall discuss now, is responsible for the production of heavy KK gravitons mn ≫ mc. A discontinuity in the velocity will always lead to a di- vergent total particle number. Arbitrary high frequency modes are excited by the kink since the acceleration diverges there. Due to the excitation of KK gravitons of arbitrarily high masses, one cannot expect that the numerical simulations show a satisfactory convergence behavior which allows to determine the slope by fit- ting the data. However, it is nevertheless possible to give a quantitative expression for the behavior of the KK-graviton spectrum for large masses. The studies of the usual dynamical Casimir effect on a time-dependent interval are very useful for this purpose. For the usual dynamical Casimir effect it has been shown analytically that a discontinuity in the velocity will lead to a divergent particle number [57, 58]. In Appendix E we discuss in detail the model of a massless real scalar field on a time-dependent interval [0, y(t)] for the boundary motion y(t) = y0 + v t with v = const, and present numerical results for final particle spectra (Fig. 34). For this motion it was shown in [58] that the particle spectrum behaves as ∝ v2/ωn where ωn = nπ/y0 is the frequency of a massless scalar particle. This di- vergent behavior is due to the discontinuities in the velocity when the motion is switched on and off, and are responsible for the slow convergence of the numerical 1 2 3 4 5 10 20 30 100 Kaluza-Klein mass m t nmax = 40 = 60 = 80 =0.03 =0.01 FIG. 25: KK-particle spectra for k = 0.1, ys = 3 and max- imal brane velocities vb = 0.01, 0.03, 0.1 up to KK masses mn ≃ 100 compared with an 1/mn decline. The dashed lines indicate the approximate expression (5.6) which describes the asymptotic behavior of the final KK-particle spectra reason- ably well, in particular for vb < 0.1. results shown in Fig. 34 for this scenario. At the kink in the brane-motion the total change of the velocity is 2vb, similar to the case for the linear motion where the discontinuous change of the velocity is 2v. Consequently we may conclude that for large KK masses mn ≫ mc for which the evolution of the KK modes is no longer affected by their coupling to the four-dimensional graviton the KK-graviton spectrum behaves as 10 N outn,k,• ∝ for mn ≫ mc . (5.5) If we assume that the spectrum declines like 1/mn and use that the numerical results for masses mn ≃ 20 are virtually stable one finds N outn,k,• ∝ v2.08b /mn which de- scribes the asymptotics of the numerical results well. As for the dynamical Casimir effect for a uniform motion discussed in Appendix E [cf. Fig. 34], the slow conver- gence of the numerical results towards the 1/mn behav- ior is well visible for large masses mn ≫ mc which do no longer couple to the four-dimensional graviton. This is a strong indication for the statement that the final gravi- ton spectrum for large masses behaves indeed like (5.5). It is therefore possible to give a single simple expression for the final KK-particle spectrum for large masses which 10 Note that the discussion in Appendix E refers to Dirichlet bound- ary conditions. For Neumann boundary conditions considered here, the zero mode and its asymmetric coupling play certainly a particular role. However, as we have shown, for large masses only the KK-intermode couplings are important. Consequently, there is no reason to expect that the qualitative behavior of the spec- trum for large masses depends on the particular kind of boundary condition. -100 -50 0 50 100 time t -100 -50 0 50 100 time t 1 10 50 KK mass m 1 10 50 KK mass m =40 n FIG. 26: Evolution of the zero-mode particle numberN0,k,•(t) and final KK-graviton spectra N outn,k,• for ys = 3, maximal brane velocity vb = 0.1 and three-momenta k = 10 and 30. The dashed line in the upper plots indicate Eq. (6.17) (divided by two) demonstrating the value of the number of produced zero-mode gravitons without coupling to the KK modes. comprises all the features of the spectrum even quantita- tively reasonably well [cf. dashed lines in Fig. 25] N outn,k,• ≃ 0.2 ωoutn,k ys for mn ≫ mc . (5.6) The 1/ys-dependence is compelling. It follows imme- diately from the considerations on the energy and the scaling behavior discussed above [cf. Figs. 16 and 17]. For completeness we now write 1/ωoutn,k instead of the KK mass mn only, since what matters is the total energy of a mode. Throughout this section this has not been important since we considered only k ≪ 1 such that ωoutn,k becomes independent of k for large masses mn ≫ k [cf. Fig. 19]. E. Short wavelengths k ≫ 1 For short wave lengths k ≫ 1 (short compared to the AdS-curvature scale L set to one in the simulations) a completely new and very interesting effect appears. The behavior of the four-dimensional graviton mode changes drastically. We find that the zero mode now couples to the KK gravitons and no longer evolves virtually independently of the KK modes, in contrast to the behavior for long wavelengths. In Fig. 26 we show the evolution of the zero-mode graviton number N0,k,•(t) and final KK-graviton spectra N outn,k,• for ys = 3, maximal brane velocity vb = 0.1 and three-momenta k = 10 and 30. One observes that the evolution of the four-dimensional graviton depends on the number of KK modes nmax taken into account, i.e. the zero mode couples to the KK gravitons. For -100 -50 0 50 100 time t 0 10 20 30 40 0.02/(k-1.8) analytical FIG. 27: 4D-graviton number N0,k,•(t) for k = 3, 5, 10, 20 and 30 with ys = 3 and maximal brane velocity vb = 0.1. The small plot shows the final graviton spectrum N out 0,k,• together with a fit to the inverse law a/(k + b) [dashed line] and the analytical fitting formula Eq. (6.23) [solid line]. For k = 10 and 30 the corresponding KK-graviton spectra are shown in Fig. 26. k = 10 the first 60 KK modes have to be included in the simulation in order to obtain a numerically stable result for the zero mode. In the case of k = 30 one already needs nmax ≃ 100 in order to achieve numerical stability for the zero mode. Figure 27 displays the time-evolution of the number of produced zero-mode gravitons N0,k,•(t) for ys = 3 and vb = 0.1. For large k the production of massless gravitons takes place only at the bounce since these short wavelength modes are sub-horizon right after the bounce. Corresponding KK-particle spectra for k = 10, 30 are depicted in Figs. 26 and 28. The insert in Fig. 27 shows the resulting final four-dimensional graviton spectrum N out0,k,•, which is very well fitted by an inverse power law N out0,k,• = 0.02/(k − 1.8) 11. Consequently, for k ≫ 1 the zero-mode particle number N out0,k,• declines like 1/k only, in contrast to the 1/k2 behavior found for k ≪ 1. The dependence of N out0,k,• on the maximal brane velocity vb also changes. In Fig. 28 we show N0,k,•(t) together with the corresponding KK-graviton spectra for ys = 3, k = 5 and 10 in each case for different vb. Using nmax = 60 KK modes in the simulations guarantees numerical stability for the zero mode. The velocity dependence of N out0,k,• is not given by a simple power law as it is the case for k ≪ 1. This is not very surprising 11 The momenta k = 5, 10, 20, 30 and 40 have been used to obtain the fit. Fitting the spectrum for k = 20, 30 and 40 to a power law gives N out 0,k,• ∝ k−1.1. -100 -50 0 50 100 time t -100 -50 0 50 100 time t 1 10 100 KK mass m 1 10 100 KK mass m = 0.03 = 0.05 = 0.1 = 0.03 = 0.05 = 0.1 = 0.3 = 0.1 = 0.03 = 0.03 = 0.1 = 0.3 FIG. 28: Zero-mode particle number N0,k,•(t) and corre- sponding final KK-particle spectraN outn,k,• for ys = 3, k = 5, 10 and different maximal brane velocities vb. nmax = 60 guaran- tees numerically stable solutions for the zero mode. since now the zero mode couples strongly to the KK modes [cf. Fig. 26]. For k = 10, for example, one finds N out0,k,• ∝ v1.4b if vb <∼ 0.1. As in the long wavelengths case, the zero-mode particle number does not depend on the position of the static brane ys even though the zero mode now couples to the KK modes. This is demonstrated in Fig. 29 where the evolution of the zero-mode particle number N0,k,•(t) and the corresponding KK-graviton spectra with k = 10, vb = 0.1 for the two values ys = 3 and 10 are shown. One needs nmax = 60 for ys = 30 in order to obtain a stable result for the zero mode which is not sufficient in the case ys = 10. Only for nmax ≃ 120 the zero-mode solution approaches the stable result which is identical to the result obtained for ys = 3. What is important is not the number of the KK modes the four-dimensional graviton couples to, but rather a particular mass mzm ≃ k. The zero mode couples to all KK modes of masses below mzm no matter how many KK modes are lighter. Recall that the value of ys just determines how many KK modes belong to a given mass interval ∆m since, roughly, mn ≃ nπ/ys. The KK-spectra for k ≥ 1 show the same scaling behavior as demonstrated for long wavelengths in Figs. 16 and 17. The production of four-dimensional gravitons of short wavelengths takes place on the expense of the KK modes. In Fig. 30 we show the numerical results for the final KK-particle spectra with vb = 0.1, ys = 3 and k = 3, 5, 10 and 30 obtained for different coupling combinations. These spectra should be compared with those shown in Fig. 23 for the long wavelengths case. For k >∼ 10 the number of the produced lightest KK gravi- tons is smaller in the full coupling case compared to the situation where only the KK-intermode coupling is taken -100 -50 0 50 100 time t -100 -50 0 50 100 time t 1 10 50 KK mass m 1 10 50 KK mass m FIG. 29: Zero-mode particle numberN0,k•(t) and correspond- ing KK-graviton spectra for k = 10, vb = 0.1 and 2nd brane positions ys = 3 and 10. 1 10 100 KK-mass m 1 10 100 KK-mass m k=3 k=5 k=10 k=30 FIG. 30: Final KK-particle spectra N outn,k,• for vb = 0.1, ys = 3 and k = 3, 5, 10 and 30 and different couplings. Circles cor- respond to the full coupling case, squares indicate the results if Mij = Mii = 0, i.e. no KK-intermode couplings and dia- monds correspond to Mi0 = 0, i.e. no coupling of KK modes to the zero mode. into account. In case k = 30, for instance, the numbers of produced gravitons for the first four KK modes are smaller for the full coupling case. This indicates that the lightest KK modes couple strongly to the zero mode. Their evolution is damped and graviton production in those modes is suppressed. The production of zero-mode gravitons on the other hand is enhanced compared to the long wavelengths case. For short wavelengths, the evolution of the KK modes therefore contributes to the production of zero-mode gravitons. This may be interpreted as creation of zero-mode gravitons out of KK-mode vacuum fluctuations. As in the long wavelengths case, the KK-particle spectrum becomes independent of k if mn ≫ k and 10 20 100 200 frequency ω k=40, n k=30, n k=20, n k=10, n k=5, n FIG. 31: Final KK-particle spectra N outn,k,• for vb = 0.1, ys = 3 and k = 5, 10, 20, 30 and 40. The dashed lines in- dicate Eq. (6.35) for k = 10, 20, 30 and 40. For k ≥ 20, the simple analytical expression (6.35) agrees quite well with the numerical results. the evolution of the KK modes is dominated by the KK-intermode coupling. This is visible in Fig. 30 for k = 3 and 5. Also the bend in the spectrum when the KK-intermode coupling starts to dominate is observable. For k = 10 and 30 this regime with mn ≫ k is not reached. As we have shown before, in the regime mn ≫ k the KK-particle spectrum behaves as 1/ωoutn,k which will dominate the energy density of produced KK gravitons. If 1 ≪ mn <∼ k, however, the zero mode couples to the KK modes and the KK-graviton spectrum does not decay like 1/ωoutn,k. This is demonstrated in Fig. 31 where the number of produced final state gravitons N outn,k,• is plotted as function of their frequency ωoutn,k for parameters vb = 0.1, ys = 3 and k = 5, 10, 20, 30 and 40. While for k = 5 the KK-intermode coupling dominates for large masses [cf. Fig. 30] leading to a bending over in the spectrum and eventually to an 1/ωoutn,k-decay, the spectra for k = 20, 30 and 40 show a different behavior. All the modes are still coupled to the zero mode leading to a power-law decrease ∝ 1/(ωoutn,k)α with α ≃ 2. The case k = 10 corresponds to an intermediate regime. Also shown is the simple analytical expression given in Eq. (6.35) which describes the spectra reasonably well for large k (dashed line). The KK-particle spectra in the region 1 ≪ mn <∼ k will also contribute to energy density since the cutoff scale is the same for the integration over k and the summation over the KK-tower (see Section VID below). 1 2 3 4 5 10 20 30 100 Kaluza-Klein mass m =0 (kink) =0.005 =0.015 =0.05 = 2.2 x 10 exp(-0.1315 m FIG. 32: KK-particle spectrum for ys = 3, vb = 0.1 and k = 0.1 for the bouncing as well as smooth motions with ts = 0.005, 0.015, and 0.05 to demonstrate the influence of the bounce. nmax = 60 KK modes have been taken into account in the simulations and the result for the kink motion is shown as well. F. A smooth transition Let us finally investigate how the KK-graviton spec- trum changes when the kink-motion (2.18) is replaced by the smooth motion (5.1). In Fig. 32 we show the numeri- cal results for the final KK-graviton spectrum for ys = 3, vb = 0.1 and k = 0.1 for the smooth motion (5.1) with ts = 0.05, 0.015 and 0.005. nmax = 60 modes have been taken into account in the simulation and the results are compared to the spectrum obtained with the kink-motion (2.18). The parameter ts defines the scale Ls ≃ 2ts at which the kink is smoothed, i.e. Ls corresponds to the width of the transition from contraction to expansion. The numerical results reveal that KK gravitons of masses smaller than ms ≃ 1/Ls are not affected, but the pro- duction of KK particles of masses larger than ms is exponentially suppressed. This is in particular evident for ts = 0.05 where the particle spectrum for masses mn > 10 has been fitted to a exponential decrease. Going to smaller values of ts, the suppression of KK-mode pro- duction sets in for larger masses. For the example with ts = 0.005 the KK-particle spectrum is identical to the one obtained with the kink-motion within the depicted mass range. In this case the exponential suppression of particle production sets in only for masses mn > 100. Note that the exponential decay of the spectrum for the smooth transition from contraction to expansions also shows that no additional spurious effects due to the dis- continuities in the velocity when switching the brane dy- namics on and off occur. Consequently, tin and tout are appropriately chosen. VI. ANALYTICAL CALCULATIONS AND ESTIMATES A. The zero mode: long wavelengths k ≪ 1/L The numerical simulations show that the evolution of the zero mode at large wavelengths is not affected by the KK modes. To find an analytical approximation to the numerical result for the zero mode, we neglect all the couplings of the KK modes to the zero mode by setting Mij = 0 ∀ i, j and keeping M00 only. Then only the evo- lution equation for ǫ 0 ≡ δα0 ǫ is important; it decouples and reduces to ǫ̈+ [k2 + V(t)]ǫ = 0 , (6.1) with “potential” V = Ṁ00 −M200 . (6.2) The corresponding vacuum initial conditions are [cf. Eqs. (3.21), (3.22); here we do not consider the unim- portant phase] ǫ = 1 , lim ǫ̇ = −ik. (6.3) A brief calculation using the expression for M00 (cf. Ap- pendix B) leads to V = y y2s − y2b 3y2b − 2y2s y2s − y2b (6.4) = − y y2s − y2b y2s − y2b . (6.5) If one assumes that the static brane is much further away from the Cauchy horizon than the physical brane, ys ≫ yb, it is simply V = −H2 − Ḣ , (6.6) and one recovers Eq. (2.50). For the particular scale factor (2.17) one obtains H = ȧ sgn(t) |t|+ tb and (6.7) Ḣ = 2δ(t) (|t|+ tb)2 (6.8) such that Ḣ +H2 = 2δ(t) . (6.9) The δ-function in the last equation models the bounce. Without the bounce, i.e. for an eternally radiation dom- inated dynamics, one has V = 0 and the evolution equa- tion for ǫ would be trivial. With the bounce, the potential is just a delta-function potential with “height” propor- tional to −2√vb/L V = − δ(t) , (6.10) where vb is given in Eq (2.20). Equation (6.1) with poten- tial (6.10) can be considered as a Schrödinger equation with δ-function potential. Its solution is a classical text- book problem. Since the approximated potential V vanishes for all t < 0 one has, with the initial condition (6.3), ǫ(t) = e−ikt , t < 0 . (6.11) Assuming continuity of ǫ through t = 0 and integrating the differential equation over a small interval t ∈ [0−, 0+] around t = 0 gives (6.12) = ǫ̇(0+)− ǫ̇(0−)− ǫ(0) . (6.13) The jump of the derivative ǫ̇ at t = 0 leads to parti- cle creation. Using ǫ(0+) = ǫ(0) = ǫ(0−) and ǫ̇(0+) = ǫ̇(0−) + ǫ(0) as initial conditions for the solution for t > 0, one obtains ǫ(t) = Ae−ikt +Beikt , t > 0 (6.14) A = 1 + i , B = −i . (6.15) The Bogoliubov coefficient B00 after the bounce is then given by B00(t ≥ 0) = e−ikt 1 + i ǫ(t)− i ǫ̇(t) (6.16) where we have used that M00 = −H if ys ≫ yb. At this point the importance of the coupling matrix M00 becomes obvious. Even though the solution ǫ to the dif- ferential equation (6.1) is a plane wave right after the bounce, |B00(t)|2 is not a constant due to the motion of the brane itself. Only once the mode is inside the hori- zon, i.e. H/k ≪ 1, |B00(t)|2 is constant and the number of generated final state gravitons (for both polarizations) is given by N out0,k = 2|B00(kt ≫ 1)|2 = 2 |ǫ|2 + |ǫ̇| (kL)2 (6.17) where we have used that the Wronskian of ǫ, ǫ∗ is 2ik. As illustrated in Fig. 9 the expression (6.17) is indeed in excellent agreement with the (full) numerical results, not only in its k-dependence but also the amplitude agrees without any fudge factor. The evolution of the four- dimensional graviton mode and the associated genera- tion of massless gravitons with momentum k < 1/L can therefore be understood analytically. Note that the approximation employed here is only valid if y2s − yb(0)2 ≫ yb(0)2. In the opposite limit, if ∆y ≡ ys − yb(0) ≪ yb(0) one can also derive an analytical ap- proximation along the same lines. For k ≤ 1/∆y one obtains instead of Eq. (6.17) N out0,k = 2(k∆y)2 , (6.18) if ∆y ≡ ys − yb(0) ≪ yb(0) , k∆y <∼ 1 . In order to calculate the energy density, we have to take into account that the approximation of an exactly ra- diation dominated Universe with an instant transition breaks down on small scales. We assume this break down to occur at the string scale Ls, much smaller than L [cf. Eqs. (2.14),(2.15)]. Ls is the true width of the transition from collapse to expansion, which we have set to zero in the treatment. Modes with mode numbers k ≫ (2π)/Ls will not ’feel’ the potential and are not generated. We therefore choose kmax = (2π)/Ls as the cutoff scale. Then, with Eq (4.21), one obtains for the energy density 2 π2a4 ∫ 2π/Ls dkk3N0,k . (6.19) For small wave numbers, k < 1/L, we can use the above analytical result for the zero-mode particle number. However, as the numerical simulations have revealed, as soon as k >∼ 1/L, the coupling of the four-dimensional graviton to the KK modes becomes important and for large wave numbers N out0,k decays only like 1/k. Hence the integral (6.19) is entirely dominated by the upper cutoff. The contributions from long wavelengths to the energy density are negligible. For the power spectrum, on the other hand, we are interested in cosmologically large scales, 1/k ≃ several Mpc or more, but not in short wavelengths kL ≫ 1 dominating the energy density. Inserting the expression for the number of produced long wavelength gravitons (6.17) into (4.11), the gravity wave power spectrum at late times becomes P0(k) = (2π)3 for kt ≫ 1. (6.20) This is the asymptotic power spectrum, when ǫ starts oscillating, hence inside the Hubble horizon, kt ≫ 1. On super Hubble scales, kt ≪ 1 when the asymptotic out-state of the zero mode is not yet reached, one may use Eq. (4.10) with R0,k(t) = |ǫ(t)|2 − 1 ≃ 4vba . (6.21) For the ≃ sign we assume t ≫ L and t ≫ tb so that one may neglect terms of order t/L in comparison to√ vb(t/L) 2. We have also approximated a = (t+ tb)/L ≃ t/L. Inserting this in Eq. (4.8) yields P0(k) = 2 , kt ≪ 1 . (6.22) Both expressions (6.20) and (6.22) are in very good agreement with the corresponding numerical results, see Figs. 9, 10 and 11. B. The zero mode: short wavelengths k ≫ 1/L As we have demonstrated with the numerical analysis, as soon as k >∼ 1/L, the coupling of the zero mode to the KK modes becomes important, and for large wave numbers N out0,k,• ∝ 1/k. We obtain a good asymptotic behavior for the four-dimensional graviton spectrum if we set N out0,k,• ≃ 5(kL) . (6.23) This function and Eq. (6.17) (divided by two for one po- larization) meet at kL = 5. Even though the approxi- mation is not good in the intermediate regime it is very reasonable for large k [cf. Fig. 27]. Inserting this approximation into Eq (6.19) for the energy density, one finds that the integral is dominated entirely by the upper cutoff, i.e. by the blue, high energy modes: . (6.24) The power spectrum associated with the short wave- lengths k ≫ 1/L is not of interest since the gravity wave spectrum is measured on cosmologically large scales only, k ≪ 1/L. C. Light Kaluza-Klein modes and long wavelengths k ≪ 1/L The numerics indicates that light (mn < 1) long wave- length KK modes become excited mainly due to their coupling to the zero mode. Let us take only this cou- pling into account and neglect also the time-dependence of the frequency, setting ωn,k(t) ≡ ωoutn,k = ωinn,k since it plays an inferior role as shown by the numerics. The Bogoliubov coefficients are then determined by the equations ξ̇n,k + iω n,kξn,k = 2ωoutn,k Sn(t; k) (6.25) η̇n,k − iωoutn,kηn,k = − 2ωoutn,k Sn(t; k) (6.26) with the “source” Sn(t; k) = (ξ0 − η0)Mn0 . (6.27) We have defined ξn,k ≡ ξ(0)n,k, ηn,k ≡ η n,k, ξ0 ≡ ξ 0,k, and η0 ≡ η(0)0,k. This source is known, since the evolution of the four-dimensional graviton is know. From the result for ǫ above and the definition of ξ0 and η0 in terms of ǫ and ǫ̇ one obtains ξ0 − η0 = −ik + 1|t|+ tb e−itk , t < 0 (6.28) ξ0 − η0 = 2 1− iktb k2tb(t+ tb) e−itk k2tb(t+ tb) eitk , t > 0 .(6.29) Furthermore, if ys ≫ yb, one has [cf. Eq. (B3)] Mn0 = 2 Y1(mnys)2 Y1(mnyb)2 − Y1(mnys)2 . (6.30) Assuming ysmn ≫ 1 and ybmn ≪ 1 one can expand the Bessel functions and arrives at Mn0 ≃ ẏb = − πmnL2 L sgn(t) (|t|+ tb)2 To determine the number of created final state gravi- tons we only need to calculate ηn,k [cf. Eq. (3.32) with ∆+n,k(|t| → ∞) = 1 and ∆ n,k(|t| → ∞) = 0], N outn,k,• = |B0n,k(tout)|2 = ωoutn,k |ηn,k|2 (6.31) The vacuum initial conditions require limt→−∞ ηn,k = 0 so that ηn,k is given by the particular solution ηn,k(t) = ωoutn,k ′; k)e−it ′ωoutn,kdt′ , (6.32) and therefore N outn,k,• = 4ωoutn,k Sn(t; k)e −itωout n,kdt (6.33) where the integration range has been extended from −∞ to +∞ since the source is very localized around the bounce. This integral can be solved exactly. A some- what lengthy but straight forward calculation gives N outn,k,• = πm5nL 2ωoutn,kkys ∣∣∣2iRe +k)tbE1(i(ω n,k + k)tb) +(ktb) −1ei(ω n,k−k)tbE1(i(ω n,k − k)tb) −ei(ω +k)tbE1(i(ω n,k + k)tb) . (6.34) Here E1 is the exponential integral, E1(z) ≡∫∞ t−1e−tdt . This function is holomorphic in the com- plex plane with a cut along the negative real axis, and the above expression is therefore well defined. Note that this expression does not give rise to a simple dependence of N outn,k on the velocity vb = (L/tb)2. In the preceding section we have seen that, within its range of validity, Eq. (6.34) is in excellent agreement with the numerical results (cf., for instance, Figs. 12 and 13). As already mentioned before, this excellent agreement between the numerics and the analytical approximation demonstrates that the numerical results are not contam- inated by any spurious effects. D. Kaluza-Klein modes: asymptotic behavior and energy density The numerical simulations show that the asymptotic KK-graviton spectra (i.e. for masses mn ≫ 1) decay like 1/ωoutn,k if mn ≫ k and like 1/ωoutn,k with α ≃ 2 if mn <∼ k. The corresponding energy density on the brane is given by the summation of Eq. (4.23) over all KK modes up to the cutoff. Since the mass mn is simply the momentum into the extra dimension, it is plausible to choose the same cutoff scale for both, the k-integral and the summation over the KK modes, namely 2π/Ls. The main contribution to the four-dimensional particle density and energy density comes from mn ∼ 2π/Ls and k ∼ 2π/Ls, i.e. the blue end of the spectrum. The large-frequency behavior of the final KK-spectrum can be approximated by N outn,k,• ≃ 0.2v2b   ωoutn,k if 1/L <∼ k <∼ mn 2(α−1)/2 (ωoutn,k) if mn <∼ k <∼ 2π/Ls (6.35) with α ≃ 2 which is particularly good for large k. Both expression match at mn = k and are indicated in Figures 25 and 31 as dashed lines. Given the complicated coupling structure of the problem and the multitude of features visible in the particle spectra, these compact expressions describe the numerical results reasonable well for all parameters. The deviation from the numer- ical results is at most a factor of two. This accuracy is sufficient in order to obtain a useful expression for the energy density from which bounds on the involved energy scales can be derived. The energy density on the brane associated with the KK gravitons is given by [cf. Eq. (4.23)] ρKK ≃ πa6ys dkk2 N outn,k,• ωoutn,k mn . (6.36) Splitting the momentum integration into two integrations from 0 to mn and mn to the cutoff 2π/Ls, and replacing the sum over the KK masses by an integral one obtains ρKK ≃ C(α) π5v2b . (6.37) The power α in Eq. (6.35) enters the final result for the energy density only through the pre-factor C(α) which is of order unity. VII. DISCUSSION The numerical simulations have revealed many inter- esting effects related to the interplay between the evolu- tion of the four-dimensional graviton and the KK modes. All features observed in the numerical results have been interpreted entirely on physical grounds and many of them are supported by analytical calculations and ar- guments. Having summarized the results for the power spectrum and energy densities in the preceding section, we are now in the position to discuss the significance of these findings for brane cosmology. A. The zero mode For the zero-mode power-spectrum we have found that P0(k) = k2 if kt ≪ 1 (La)−2 if kt ≫ 1 . (7.1) Therefore, the gravity wave spectrum on large, super Hubble scales is blue with spectral tilt nT = 2 , (7.2) a common feature of ekpyrotic and pre-big-bang models. The amplitude of perturbations on scales at which fluctuations of the Cosmic Microwave Background (CMB) are observed is of the order of (H0/mPl) 2, i.e. very suppressed on scales relevant for the anisotropies of the CMB. The fluctuations induced by these Casimir gravitons are much too small to leave any observable imprint on the CMB. For the zero-mode energy density at late times, kt ≫ 1, we have obtained [cf Eq. (6.24)] ρh0 ≃ . (7.3) In this section we denote the energy density of the zero mode by ρh0 in order not to confuse it with the 12 Note that even the transition from the summation over the KK- tower to an integration according to (4.33) “eats up” the 1/ys term in (6.36), the final energy density (6.37) depends on ys since it explicitly enters the particle number. present density of the Universe. Recall that Ls is the scale at which our kinky approximation (2.17) of the scale factor breaks down, i.e. the width of the bounce. If this width is taken to zero, the energy density of gravitons is very blue and diverges. This is not so surprising, since the kink in a(t) leads to the generation of gravitons of arbitrary high energies. However, as the numerical simulations have shown, when we smooth the kink at some scale Ls, the production of modes with energies larger than ≃ 1/Ls is exponentially suppressed [cf. Fig. 32]. This justifies the introduction of Ls as a cutoff scale. In the following we shall determine the density pa- rameter of the generated gravitons today and compare it to the Nucleosynthesis bound. For this we need the quantities ab given in Eq (2.20) and Here ab is the minimal scale factor andHb is the maximal Hubble parameter, i.e. the Hubble parameter right after the bounce. (Recall that in the low energy approximation t = η.) During the radiation era, curvature and/or a cos- mological constant can be neglected so that the density ρrad = a−4 = . (7.4) In order to determine the density parameter of the gen- erated gravitons today, i.e., at t = t0, we use Ωh0 = ρh0(t0) ρcrit(t0) ρh0(t0) ρrad(t0) ρrad(t0) ρcrit(t0) ρh0(t0) ρrad(t0) Ωrad. (7.5) The second factor Ωrad is the present radiation density parameter. For the factor ρh0/ρrad, which is time inde- pendent since both ρh0 and ρrad scale like 1/a 4, we insert the above results and obtain Ωh0 = Ωrad = Ωrad (7.6) Ωrad . (7.7) The nucleosynthesis bound [14] requests that Ωh0 <∼ 0.1Ωrad , (7.8) which translates into the relation (LPl/Ls) (L/Ls) <∼ 0.1 (7.9) which, at first sight, relates the different scales involved. But since we have chosen the cutoff scale Ls to be the higher-dimensional fundamental scale (string scale), Equation (7.9) reduces to vb <∼ 0.2 (7.10) by virtue of Equation (2.15). All one has to require to be consistent with the nucleosynthesis bound is a small brane velocity which justifies the low energy approach. In all, we conclude that the model is not severely con- strained by the zero mode. This result itself is remark- able. If there would be no coupling of the zero mode to the KK modes for small wavelengths the number of produced high energy zero-mode gravitons would behave as ∝ k−2 as it is the case for long wavelengths. The production of high energy zero-mode gravitons from KK gravitons enhances the total energy density by a factor of about L/Ls. Without this enhancement, the nucleosyn- thesis bound would not lead to any meaningful constraint and would not even require vb < 1. B. The KK modes As derived above, the energy density of KK gravitons on the brane is dominated by the high energy gravitons and can be approximated by [cf. Eq. (6.37)] ρKK ≃ π5v2b . (7.11) Let us evaluate the constraint induced from the require- ment that the KK-energy density on the brane be smaller than the radiation density ρKK(t) < ρrad(t) at all times. If this is not satisfied, back-reaction cannot be neglected and our results are no longer valid. Clearly, at early times this condition is more stringent than at late times since ρKK decays faster then ρrad. Inserting the value of the scale factor directly after the bounce where the produc- tion of KK gravitons takes place, a−2b = vb, one finds, using again the RS fine tuning condition (2.15), ≃ 100 v3b . (7.12) If we use the largest value for the brane velocity vb ad- mitted by the nucleosynthesis bound vb ≃ 0.2 and re- quire that ρKK/ρrad be (much) smaller than one for back- reaction effects to be negligible, we obtain the very strin- gent condition . (7.13) Let us first discuss the largest allowed value for L ≃ 0.1mm. The RS-fine tuning condition (2.15) then deter- mines Ls = (LL 1/3 ≃ 10−22 mm ≃ 1/(106 TeV). In this case the brane tension is T = 6κ4/κ25 = 6L2Pl/L6s = 6/(LL3s) ∼ (10TeV)4. Furthermore, we have (L/Ls)2 ≃ 1042 so that ys > L(L/Ls) 2 ≃ 1041mm ≃ 3 × 1015Mpc, which is about 12 orders of magnitude larger than the present Hubble scale. Also, since yb(t) ≪ L in the low energy regime, and ys ≫ L according to the inequality (7.13), the physical brane and the static brane are very far apart at all times. Note that the distance between the physical and the static brane is dy = L log(ys/yb) >∼ L ≫ Ls . This situation is probably not very realistic. Some high energy, stringy effects are needed to provoke the bounce and one expects these to be relevant only when the branes are sufficiently close, i.e. at a distance of order Ls. But in this case the constraint (7.13) will be violated which implies that back-reaction will be relevant. On the other hand, if one wants that ys ≃ L and back-reaction to be unimportant, then Eq. (7.12) implies that the bounce velocity has to be exceedingly small, vb <∼ 10−15. A way out of this conclusion is to assume that the brane distance at the bounce, ∆y = ys − yb(0), becomes of the order of the cutoff Ls or smaller. Then the pro- duction of KK gravitons is suppressed. However, then the approximation (6.18) has to be used to determine the energy density of zero-mode gravitons which then becomes ρh0 ≃ (Ls∆y) Setting ∆y ≃ Ls, the nucleosynthesis bound, ρh0 <∼ 0.1ρrad, then yields the much more stringent limit on the brane velocity, v2b < . (7.14) One might hope to find a way out of these conclusions by allowing the bounce to happen in the high energy regime. But then vb ≃ 1 and the nucleosynthesis bound is violated since too many zero-mode gravitons are pro- duced. Even if one disregards this limit for a moment, saying that the calculation presented here only applies in the low energy regime, vb ≪ 1, the modification coming from the high energy regime are not expected to allevi- ate the bounds. In the high energy regime one may of course have yb(t) ≫ L and therefore the physical brane can approach the static brane arbitrarily closely without the latter having to violate (7.13). Those results suggest that even in the scenario of a bounce at low energies, the back reaction from KK gravitons has to be taken into account. But this does not need to exclude the model. VIII. CONCLUSIONS We have studied the evolution of tensor perturbations in braneworld cosmology using the techniques developed for the standard dynamical Casimir effect. A model consisting of a moving and a fixed 3-brane embedded in a five-dimensional static AdS bulk has been considered. Applying the dynamical Casimir effect formulation to the study of tensor perturbations in braneworld cosmology represents an interesting alternative to other approaches existing in the literature so far and provides a new perspective on the problem. The explicit use of coupling matrices allows us to obtain detailed information about the effects of the intermode couplings generated by the time-dependent boundary conditions, i.e. the brane motion. Based on the expansion of the tensor perturbations in instantaneous eigenfunctions, we have introduced a consistent quantum mechanical formulation of graviton production by a moving brane. Observable quantities like the power spectrum and energy density can be directly deduced from quantum mechanical expectation values, in particular the number of gravitons created from vacuum fluctuations. The most surprising and at the same time most interesting fact which this approach has revealed is that the energy density of the massive gravitons decays like 1/a6 with the expansion of the Universe. This is a direct consequence of the localization of gravity: five-dimensional aspects of it, like the KK gravitons, become less and less ’visible’ on the brane with the expansion of the Universe. The 1/a6-scaling behavior remains valid also when the fixed brane is sent off to infinity and one ends up with a single braneworld in AdS, like in the original RS II scenario. Consequently, KK gravitons on a brane moving through an AdS bulk cannot play the role of dark matter. As an explicit example, we have studied graviton production in a generic, ekpyrotic-inspired model of two branes bouncing at low energies, assuming that the energy density on the moving brane is dominated by a radiation component. The numerical results have revealed a multitude of interesting effects. For long wavelengths kL ≪ 1 the zero mode evolves virtually independently of the KK modes. zero-mode gravitons are generated by the self coupling of the zero mode to the moving brane. For the number of produced massless gravitons we have found the simple analytical expression 2vb/(kL). These long wavelength modes are the once which are of interest for the gravitational wave power spectrum. As one expects for an ekpyrotic scenario, the power spectrum is blue on super-horizon scales with spectral tilt nT = 2. Hence, the spectrum of these Casimir gravitons has much too little power on large scales to affect the fluctuations of the cosmic microwave background. The situation changes completely for short wavelengths kL ≫ 1. In this wavelength range, the evolution of the zero mode couples strongly to the KK modes. Produc- tion of zero-mode gravitons takes place on the expense of KK-graviton production. The numerical simulation have revealed that the number of produced short-wavelength massless gravitons is given by 2vb/(5kL). It decays only like 1/k instead of the 1/k2-behavior found for long wavelengths. These short wavelength gravitons dominate the energy density. Comparing the energy density with the nucleosynthesis bound and taking the cutoff scale to be the string scale Ls, we have shown that the model is not constrained by the zero mode. As long as vb <∼ 0.2, i.e. a low energy bounce, the nucleosynthesis bound is not violated. More stringent bounds on the model come from the KK modes. Their energy density is dominated by the high energy modes which are produced due to the kink which models the transition from contraction to expansion. Imposing the reasonable requirement that the energy density of the KK modes on the brane be (much) smaller than the radiation density at all times in order for back reaction effects to be negligible, has led to two cases. On the one hand, allowing the largest values for the AdS curvature scale L ≃ 0.1mm and the bounce velocity vb ≃ 0.2, back reaction can only be neglected if the fixed brane is very far away from the physical brane ys ∼ 1041mm. As we have argued, this is not very realistic since some high energy, stringy effects provoking the bounce are expected to be relevant only when the branes are sufficiently close, i.e. ys ∼ Ls. On the other hand, by only requiring that ys ≃ L ≫ Ls, the bounce velocity has already to be exceedingly small, vb <∼ 10−15, for back reaction to be unimportant. Therefore, one of the main conclusions to take away from this work is that back reaction of massive gravitons has to be taken into account for a realistic bounce. Many of the results presented here are based on numerical calculations. However, since the used ap- proach provides the possibility to artificially switch on and off the mode couplings, we were able identify the primary sources driving the time evolution of the perturbations in different wavelength and KK mass ranges. This has allowed us to understand many of the features observed in the numerical results on analytical grounds. On the other hand, it is fair to say that most of the presented results rely on the low energy approach, i.e. on the approximation of the junction condition (generalized Neumann boundary condition) by a Neumann boundary condition. Even though we have given arguments for the goodness of this approximation, it has eventually to be confirmed by calculations which take the exact boundary condition into account. This is the subject of future work. Acknowledgment We thank Cyril Cartier who participated in the early stages of this work and Kazuya Koyama and David Lan- glois for discussions. We are grateful for the use of the ’Myrinet’-cluster of Geneva University on which most of the quite intensive numerical computations have been performed. This work is supported by the Swiss National Science Foundation. APPENDIX A: VARIATION OF THE ACTION Let us consider the variation of the action (2.27) with respect to h•. It is sufficient to study the action for a fixed wave number k and polarization • Sh•(k) = yb(t) |∂th•|2 − |∂yh•|2 − k2|h•|2 and we omit the normalization factor L3/κ5 as well as the factor two related to Z2 symmetry. The variation of (A1) reads δSh•(k) = yb(t) (∂th•)(∂tδh •) (A2) −(∂yh•)(∂yδh∗•)− k2h•δh∗• + h.c. . Here, T denotes a time interval within the variation is performed and it is assumed in the following that the variation vanishes at the boundaries of the time interval T . Performing partial integrations and demanding that the variation of the action vanishes leads to 0 = (A3) yb(t) − ∂2t h• + y3 − k2h• [(v∂t + ∂y)h•] δh •|yb(t) − (∂yh•)δh with v = dyb(t)/dt. The first term in curly brackets is the wave operator (2.24). In order for h• to satisfy the free wave equation (perturbation equation) (2.24) the term in curly brackets in the second integral has to vanish. Allowing for an evolution of h• on the branes, i.e. in general δh•|brane 6= 0, enforces the boundary conditions (v∂t + ∂y)h•|yb(t) = 0 and ∂yh•|ys = 0 , (A4) hence, the junction condition (2.26). Consequently, any other boundary conditions than (A4) are not compati- ble with the free perturbation equation (2.24) under the influence of a moving brane (provided δh• 6= 0 at the branes). APPENDIX B: COUPLING MATRICES The use of several identities of Bessel functions leads M00 = ŷb y2s − y2b , (B1) M0j = 0 , (B2) Mi0 = φ0 = ŷb y2s − y2b , (B3) Mii = m̂i , (B4) Mij = M ij +M ij (B5) MAij = (ŷb + m̂i)yb 2m2iNiNj m2j −m2i × (B6) × [ys C2(mjys)J1(miys)− yb C2(mjyb)J1(miyb)] where J1(mi y) = [J2(miyb)Y1(miy)− Y2(miyb)J1(miy)] MNij = NiNjmim̂i dyy2C1(miy)C2(mjy). (B8) This integral has to be solved numerically. Note that, because of the boundary conditions, one has the identity dyy2C1(miy)C2(mjy) = − dyy2C1(miy)C0(mjy). Furthermore, one can simplify J1(mi yb) = πmiyb , J1(mi ys) = πmiyb Y1(miys) Y1(miyb) (B10) where the limiting value has to be taken for the last term whenever Y1(miyb) = Y1(miys) = 0. APPENDIX C: ON POWER SPECTRUM AND ENERGY DENSITY CALCULATION 1. Instantaneous vacuum In Section III the in - out state approach to particle creation has been presented. The definitions of the in - and out- vacuum states Eq. (3.9) are unique and the particle concept is well defined and meaningful. If we interpret tout as a continuous time variable t, we can write the Bogoliubov transformation Eq. (3.24) âα,k,•(t) = Aβα,k(t)âinβ,k,• + B∗βα,k(t)â β,−k,• where at any time we have introduced a set of operators {âα,k•(t), â†α,k,•(t)}. Vacuum states defined at any time can be associated with these operators via âα,k,•(t)|0, t〉 = 0 ∀ α,k • . (C2) Similar to Eq. (3.11) a ”particle number” can be intro- duced through Nα,k(t) = 〈0, in|â†α,k•(t)âα,k,•(t)|0, in〉 |Bβα,k(t)|2 . (C3) We shall denote |0, t〉 as the instantaneous vacuum state and the quantity Nα,k(t) as instantaneous particle num- ber 13. However, even if we call it ”particle number” and plot it in section V for illustrative reasons, we consider only the particle definitions for the initial and final state (asymptotic regions) outlined in section III as physically meaningful. 2. Power spectrum In order to calculate the power spectrum Eq. (4.7) we need to evaluate the expectation value (t, yb,k)ĥ (t, yb,k ′)〉in = (C4) φα(t, yb)φα′ (t, yb)〈q̂α,k,•(t)q̂†α′,k′,•(t)〉in where we have introduced the shortcut 〈...〉in = 〈0, in|...|0, in〉. Using the expansion (3.15) of q̂α′,k′,•(t) in initial state operators and complex functions ǫ α,k(t) one finds 〈q̂α,k,•(t)q̂†α′,k′,•(t)〉in = α,k(t) ǫ α′,k (t) 2ωinβ,k δ(3)(k− k′). From the initial conditions (3.21) it follows that the sum in (C4) diverges at t = tin. This divergence is related to the usual normal ordering problem and can be removed by a subtraction scheme. However, in order to obtain a well defined power spectrum at all times, it is not suffi- cient just to subtract the term (1/2)(δαα′/ω α,k)δ (3)(k − ′) which corresponds to 〈q̂α,k,•(tin)q̂†α′,k′,•(tin)〉in in the above expression. In order to identify all terms contained in the power spectrum we use the instantaneous particle concept which allows us to treat the Bogoliubov coeffi- cients (3.25) and (3.26) as continuous functions of time. First we express the complex functions ǫ α,k in (C5) in terms of Aγα,k(t) and Bγα,k(t). This is of course equiv- alent to calculating the expectation value (C5) using [cf. Eq.(3.7)] q̂α,k,•(t) = 2ωα,k(t) âα,k,•(t)Θα,k(t) α,−k,•(t)Θ α,k(t) and the Bogoliubov transformation Eq. (C1). The result consists of terms involving the Bogoliubov coefficients and the factor (1/2)(δαα′/ωα,k(t))δ (3)(k − k′), leading potentially to a divergence at all times. This term cor- responds to 〈0, t|q̂α,k,•(t)q̂†α′,k′,•(t)|0, t〉, and is related to 13 It could be interpreted as the number of particles which would have been created if the motion of the boundary (the brane) stops at time t. the normal ordering problem (zero-point energy) with re- spect to the instantaneous vacuum state |0, t〉. It can be removed by the subtraction scheme 〈q̂α,k,•(t)q̂†α′,k′,•(t)〉in,phys (C7) = 〈q̂α,k,•(t)q̂†α′,k′,•(t)〉in − 〈0, t|q̂α,k,•(t)q̂ α′,k′,•(t)|0, t〉 where we use the subscript “phys” to denote the physi- cally meaningful expectation value. Inserting this expectation value into (C4), and using Eq. (4.2), we find 〈ĥ•(t, yb,k)ĥ•(t, yb,k′)〉in (C8) Rα,k(t)Y2α(a)δ(3)(k− k′) with Rα,k(t) defined in Eq. (4.9). The function ONα,k appearing in Eq. (4.9) is explicitely given by ONα,k = 2ℜ Θ2α,kAβα,kB∗βα,k +Θα,k α′ 6=α ωα′,k Yα′ (a) Yα(a) Θ∗α′,kB∗βαBβα′ +Θα′,kAβαB∗βα′ and Oǫα,k appearing in Eq. (4.10) reads Oǫα,k = β,α′ 6=α Yα′(a) Yα(a) ωinβ,k . (C10) 3. Energy density In order to calculate the energy density we need to evaluate the expectation value 〈 ˙̂hij(t,x, yb) ˙̂hij(t,x, yb)〉in. Using (2.22) and the relation e•ij(−k) = (e•ij(k))∗ we ob- 〈 ˙̂hij(t,x, yb) ˙̂hij(t,x, yb)〉in = (2π)3/2 (2π)3/2 (C11) × 〈 ˙̂h (t, yb,k) ′(t, yb,k ′)〉inei(k−k ′)·xe•ij(k) ′ ij(k′) By means of the expansion (3.17) the expectation value 〈 ˙̂h (t, yb,k) ′(t, yb,k ′)〉in becomes 〈 ˙̂h (t, yb,k) ′(t, yb,k ′)〉in (C12) 〈p̂α,k,•(t)p̂†α′,k′,•′(t)〉inφα(t, yb)φα′ (t, yb). From the definition of p̂α,k,•(t) in Eq. (3.18) it is clear that this expectation value will in general contain terms proportional to the coupling matrix and its square when expressed in terms of ǫ α,k. However, we are interested in the expectation value at late times only when the brane moves very slowly such that the mode couplings go to zero and a physical meaningful particle definition can be given. In this case we can set 〈p̂α,k,•(t)p̂†α′,k′,•′(t)〉in = ˙̂qα,k,•(t) ˙̂q α′,k′,•′(t) .(C13) Calculating this expectation value by using Eq. (3.15) leads to an expression which, as for the power spec- trum calculation before, has a divergent part related to the zero-point energy of the instantaneous vacuum state (normal ordering problem). We remove this part by a subtraction scheme similar to Eq (C7). The final result reads 〈 ˙̂qα,k,•(t) ˙̂q†α′,k′,•′(t)〉in,phys (C14) α,k(t)ǫ̇ α′,k′(t)√ ωinβ,kω − ωα,k(t)δαα′ ′δ(3)(k− k′). Inserting this result into Eq. (C12), splitting the summa- tions in sums over α = α′ and α 6= α′ and neglecting the oscillating α 6= α′ contributions (averaging over several oscillations), leads to 〈 ˙̂h (t, yb,k) ′(t, yb,k ′)〉in (C15) Kα,k(t)Y2α(a)δ••′δ(3)(k− k′) where the function Kα,k(t) is given by Kα,k(t) = |ǫ̇(β)α,k(t)|2 ωinβ,k − ωα,k(t) = ωα,k(t)Nα,k(t) , (C16) and we have made use of Eq. (4.2). The relation be- tween β |ǫ̇ α,k(t)|2/ωinβ,k and the number of created par- ticles can easily be established. Using this expression in Eq. (C11) leads eventually to 〈 ˙̂hij(t,x, yb) ˙̂hij(t,x, yb)〉in (C17) (2π)3 Kα,k(t)Y2α(a) where we have used that the polarization tensors satisfy e•ij(k) e• ij(k) = 2. (C18) The final expression for the energy density Eq. (4.18) is then obtained by exploiting that κ5/L = κ4. APPENDIX D: NUMERICS In order to calculate the number of produced gravitons the system of coupled differential equations (3.34) and (3.35) is solved numerically. The complex functions ξ α,k are decomposed into their real and imaginary parts: α,k = u α,k + iv α,k , η α,k = x α,k + iy α,k. (D1) The system of coupled differential equations can then be written in the form (cf. Eq. (A2) of [16]) k (t) = Wk(t)X k (t) (D2) where 0,k ...u nmax,k 0,k ...x nmax,k 0,k ...v nmax,k 0,k ...y nmax,k The matrix Wk(t) is given by Eq. (A4) of [16] but here indices start at zero. The number of produced gravitons can be calculated directly from the solutions to this system using Eqs. (3.28) and (3.32). Note that for a given truncation parameter nmax the above system of size 4(nmax + 1) × 4(nmax + 1) has to be solved nmax + 1 - times, each time with different initial conditions (3.38). The main difficulty in the numerical simulations is that most of the entries of the matrix Wk(t) [Eq. (A4) of [16]] are not known analytically. This is due to the fact that Eq. (2.40) which determines the time-dependent KK masses mi(t) does not have an (exact) analytical solution. Only the 00-component of the coupling matrix Mαβ is known analytically. We therefore have to deter- mine the time-dependent KK-spectrum {mi(t)}nmaxi=1 by solving Eq. (2.40) numerically. In addition, also the part MNij [Eq. (B8)] has to be calculated numerically since the integral over the particular combination of Bessel functions can not be found analytically. We numerically evaluate the KK-spectrum and the integral MNij for discrete time-values ti and use spline routines to assemble Wk(t). The system (D2) can then be solved using standard routines. We chose the distribution of the ti’s in a non-uniform way. A more dense mesh close to the bounce and a less dense mesh at early and late times. The independence of the numerical results on the distribution of the ti’s is checked. In order to implement the bounce as realistic as possible, we do not spline the KK-spectrum very close to the bounce but re-calculate it numerically at every time t needed in the differential equation solver. This minimizes possible artificial effects caused by using a spline in the direct vicinity of the bounce. The same was done for MNij but we found that splining MNij when propagating through the bounce does not affect the numerical results. Routines provided by the GNU Scientific Library (GSL) [59] have been employed. Different routines for root finding and integration have been compared. The code has been parallelized (MPI) in order to deal with the 1 10 100 KK-mass m 1 10 100 KK-mass m FIG. 33: Comparison of the final KK-graviton spectrum n,k,• with the expression dn,k(tout) describing to what accu- racy the diagonal part of the Bogoliubov relation (D4) is sat- isfied. Left panel: ys = 3, k = 0.1, vb = 0.03 and nmax = 100 [cf. Fig. 25]. Right panel: ys = 3, k = 30, vb = 0.1 and nmax = 100 [cf. Fig. 26]. intensive numerical computations. The accuracy of the numerical simulations can be assessed by checking the validity of the Bogoliubov relations Aβα,k(t)A∗βγ,k(t)− B∗βα,k(t)Bβγ,k(t) = δαγ (D4) Aβα,k(t)B∗βγ,k(t)− B∗βα,k(t)Aβγ,k(t) = 0. (D5) In the following we demonstrate the accuracy of the nu- merical simulations by considering the diagonal part of (D4). The deviation of the quantity dα,k(t) = 1− |Aβα,k(t)|2 − |Bβα,k(t)|2 from zero gives a measure for the accuracy of the numerical result. We consider this quantity at final times tout and compare it with the corresponding final particle spectrum. In Fig. 33 we compare the final KK- graviton spectrum N outn,k,• with the expression dn,k(tout) for two different cases. This shows that the accuracy of the numerical simulations is very good. Even if the expectation value for the particle number is only of order 10−7 to 10−6, the deviation of dn,k(tout) from zero is at least one order of magnitude smaller. This demonstrates the reliability of our numerical simulations and that we can trust the numerical results presented in this work. APPENDIX E: DYNAMICAL CASIMIR EFFECT FOR A UNIFORM MOTION We consider a real massless scalar field on a time- dependent interval [0, y(t)]. The time evolution of its mode functions are described by a system of differential equations like (2.49) where the specific form of Mαβ de- pends on the particular boundary condition the field is subject to. In [15, 17] a method has been introduced to study particle creation due to the motion of the boundary y(t) (i.e. the dynamical Casimir effect) fully numerically. We refer the reader to these publications for further de- tails. If the boundary undergoes a uniform motion y(t) = 1+vt (in units of some reference length) it was shown in [57, 58] that the total number of created scalar particles diverges, caused by the discontinuities in the velocity at the begin- ning and the end of the motion. In particular, for Dirich- let boundary conditions (no zero mode), it was found in [58] that 〈0, in|N̂outn |0, in〉 ∝ v2/n if n > 6 and v ≪ 1. Thereby in- and out- vacuum states are defined like in the present work and the frequency of a mode function is given by ωn = π n , n = 1, 2, ... . In Figure 34 we show spectra of created scalar particles obtained numer- ically with the method of [17] for this particular case. 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0704.0791

Spectral analysis of Swift long GRBs with known redshift

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 6 August 2021 (MN LATEX style file v2.2) Spectral analysis of Swift long GRBs with known redshift J. I. Cabrera1⋆, C. Firmani1,2, V. Avila–Reese1, G. Ghirlanda2, G. Ghisellini2 and L. Nava2,3 1Instituto de Astronomı́a, Universidad Nacional Autónoma de México, A.P. 70–264, 04510, México, D.F. 2Osservatorio Astronomico di Brera, via E.Bianchi 46, I–23807 Merate, Italy 3Universitá degli studi dell’Insubria, Dipartimento di Fisica e Matematica, via Valleggio 11, I–22100 Como, Italy 6 August 2021 ABSTRACT We study the spectral and energetics properties of 47 long–duration gamma–ray bursts (GRBs) with known redshift, all of them detected by the Swift satellite. Due to the narrow energy range (15–150 keV) of the Swift–BAT detector, the spectral fitting is reliable only for fitting models with 2 or 3 parameters. As high uncertainty and cor- relation among the errors is expected, a careful analysis of the errors is necessary. We fit both the power law (PL, 2 parameters) and cut–off power law (CPL, 3 pa- rameters) models to the time–integrated spectra of the 47 bursts, and we present the corresponding parameters, their uncertainties, and the correlations among the uncer- tainties. The CPL model is reliable only for 29 bursts for which we estimate the νfν peak energy Epk. For these GRBs, we calculate the energy fluence and the rest–frame isotropic–equivalent radiated energy, Eγ,iso, as well as the propagated uncertainties and correlations among them. We explore the distribution of our homogeneous sample of GRBs on the rest–frame diagram E′pk vs Eγ,iso. We confirm a significant correlation between these two quantities (the “Amati” relation) and we verify that, within the un- certainty limits, no outliers are present. We also fit the spectra to a Band model with the high energy power law index frozen to −2.3, obtaining a rather good agreement with the “Amati” relation of non–Swift GRBs. Key words: gamma rays: bursts — gamma rays: sources 1 INTRODUCTION The Swift Mission (Gehrels et al. 2004) was designed mainly to rapidly detect, locate, and observe gamma–ray bursts (GRBs). After more than two years of operation, the Swift satellite has observed approximately 180 GRBs, for which more than 50 events have known redshifts reported in the Mission homepage1. In this paper we present results from an homogeneous spectral analysis that we have carried out to the Swift long–duration GRBs with known z with the main aim of estimating the time–averaged spectral parame- ters of the prompt emission of these bursts, as well as related global quantities like the prompt energy fluence, Sγ , and the rest–frame isotropic–equivalent radiated energy, Eγ,iso. Unfortunately, the spectral coverage of the Swift γ–ray detector, the Burst Alert Telescope (BAT, Barthelmy et al. 2005), is very narrow. The recommendation is to limit the spectral analysis to channels between 15 keV and 195 keV, though the Swift–BAT’s team suggests a more conservative ⋆ E–mail: [email protected] 1 http://swift.gsfc.nasa.gov/ upper limit of 150 keV because, above this energy, the cal- ibration is not yet sufficiently reliable. Thus, the BAT en- ergy range is not broad enough to allow the spectra to be fitted unambiguously by the usual GRB broken power–law function of 4 parameters. Therefore, the inference of γ–ray spectral parameters and global quantities related to these parameters for the Swift GRBs is not an easy task. A main concern of this task is the determination of the uncertainties of the spectral parameters and their propaga- tion into the composite quantities. Due to the narrow energy range of Swift BAT we know a priori that the errors in the fitted spectral parameters will be large and correlated among them. Therefore, the appropriate handling of errors is cru- cial in order to make useful the spectral data generated by BAT. The effort is justified by the great value that an homo- geneous sample has, where the spectral information for all the events is obtained with the same detector and analyzed with the same techniques and methods. This is the case of the growing–in–number sample of Swift GRBs. The characterization of the time–averaged photon spectra of GRBs with known redshift is important for several reasons. On one hand, this characterization offers c© 0000 RAS http://arxiv.org/abs/0704.0791v3 2 Cabrera et al. clues for understanding the radiation and particle accel- eration mechanisms at work during the prompt phase of the bursts (see for recent reviews e.g., Zhang & Mészáros 2004; Piran 2005; Mészáros 2006; Zhang 2007). On the other hand, the spectral parameters and the global quantities inferred from them (Sγ , Eγ,iso, etc.) are among the main properties that characterize GRBs. The study of the rest–frame correlations among these and other properties (e.g,. Amati et al. 2002; Atteia 2003; Ghirlanda, Ghisellini & Lazzati 2004a; Yonetoku et al. 2004; Liang & Zhang 2005; Firmani et al. 2006a) is cur- rently allowing to learn key aspects of the nature of GRBs (e.g., Thompson 2006; Thompson, Rees & Mészáros 2007; Ramirez–Ruiz & Granot 2006). Furthermore, those corre- lations that are tight enough can be used to standardize the energetics of GRBs, making possible to apply GRBs for constructing the Hubble diagram up to unprecedentedly high redshifts (Ghirlanda et al. 2004b, 2006; Dai et al. 2004; Firmani et al. 2005, 2006b, 2007; Liang & Zhang 2005; Xu, Dai & Liang 2005; Schaefer 2007). These tight corre- lations, for a given cosmological model, can be used also to estimate the (pseudo)redshifts of GRBs with not mea- sured redshifts, mainly from the extensive CGRO–BATSE database2 (e.g., Lloyd-Ronning, Fryer & Ramirez-Ruiz 2002; Atteia 2003). Thus, by combining the distribution in redshift of a large sample of GRBs with the observed flux distribution, inferences on the luminosity function and formation rate of GRBs can be obtained (Firmani et al. 2004; Guetta et al. 2005). The outline of this paper is as follows. In §2 we ex- plain the main steps of the spectral analysis carried out by us: selection of the sample (§§2.1), spectral deconvolution (fit) procedure and fitting models (§§2.2), and error analysis (§§2.3). The results of the spectral analysis are given in §3, along with the estimates of Sγ and Eγ,iso as well as some correlations among the obtained quantities. The summary and conclusions of the paper are presented in §4. In the Ap- pendix the deconvolved spectra of the 47 GRBs studied here are plotted. We adopt the concordance ΛCDM cosmology with ΩM = 0.3, ΩΛ = 0.7, and H0 = 70 km s −1Mpc−1. 2 SPECTRAL ANALYSIS The first step in our spectral analysis is the selection of the Swift sub-sample of GRBs to be studied (§§2.1). Then we proceed to the spectral deconvolution with XSPEC and de- termination of the GRB spectral parameters for two– and three–parameter photon models (§§2.2). By using these pa- rameters, the energy fluence and isotropic–equivalent energy are calculated. Finally, we construct the error confidence el- lipses for the spectral parameters and Eγ,iso (§§2.3). The BAT is a large aperture γ−ray telescope. It consists of a (2.4×1.2)m coded aperture mask supported one meter 2 In a recent paper, Li (2007a) showed that due both to a red- shift degeneracy in the so–called Amati correlation (Amati et al. 2002) and to its high scatter, is not possible to infer reliable (pseudo)redshifts from this correlation. above the 5200 cm2 area detector plane. The detector con- tains 32,768 individual cadmium-zinc-telluride detector ele- ments, each one of (4×4×2)mm size. The effective efficiency of BAT starts at approximately 15 keV, attains a broad maximum at approximately 30–100 keV and then falls off, attaining at 195 keV the same level as at 15 keV. Due to the 1 mm thickness of the lead tiles, the BAT coded mask starts to be transparent around 150 keV. We reduce the BAT’s data using the analysis techniques provided by the BAT’s instrument team3. 2.1 The sample We analyse the time–integrated spectra of all available Swift GRBs with known redshift z. The list of events was taken from the daily updated compilation of J. Greiner4 for the period January 2005 to January 2007. The corre- sponding file spectra were loaded from the legacy NASA database5. The total number of events in this list was 55. We exclude from the list the short–duration bursts (T90 < 2s; GRB050509, GRB050724, and GRB061217) as well as those events with incomplete observational informa- tion (GRB050730, GRB060124, GRB060218, GRB060505 and GRB060512), leaving us with a preliminary list of 47 long GRBs with measured z. The list of 47 events is given in Table 1. The first four columns of Table 1 give respectively the burst name, the class (see below), the redshift z, and the time Tintthat we use to integrate the spectrum. In principle T90 could be used as an integration time. Nevertheless two problems arise in this case. The first one has to do with the systematic under- estimate of the fluence, hence of Eγ,iso. The second problem may be more important; the elimination of the (early and late) wings in the light curve can influence the estimates of the spectral parameters. For example the elimination of the late wing may harden artificially the spectrum. For these reasons we have fixed the integration time by a visual in- spection of the light curve as the period where the signal is clearly identifiable in the background noise. A modest overestimate of Tint does not influence on the values of the integrated parameters, though increases the noise. We would like to remark that Tint is not a good estimate of the burst duration because its uncertainty is related to the identification of faint (some time extended) wings. 2.2 Spectral fits It is well known that the GRB photon spectra are in general well described, in the energy range of ∼ 10 keV to a few MeV, by the so–called Band function (Band et al. 1993), which is a two smoothly connected power laws: N(E) = N0 100 keV E 6 E0(α− β) (1) E0(α− β) 100 keV ](α−β) exp(β − α) 100 keV 3 http://heasarc.gsfc.nasa.gov/docs/swift/analysis/threads 4 http://www.mpe.mpg.de/ jcg/grbgen.html 5 ftp://legacy.gsfc.nasa.gov/swift/data/obs c© 0000 RAS, MNRAS 000, 000–000 Spectral analysis of Swift GRBs with known z 3 GRB Class z Tint LgN0[sd] α[sd] e1 n1 e2 n2 χ sec 50 keV (1) (2) (3) (4) (5) (6) (7) (8) (9) 050126 A 1.29 40.0 -2.656[0.026] -1.317[0.101] 0.026(-1.00, 0.01) 0.106(-0.01,-1.00) 1.399 050223 A 0.5915 30.0 -2.652[0.028] -1.837[0.106] 0.025(-0.99, 0.13) 0.109(-0.13,-0.99) 0.902 050315 D 1.949 129.0 -2.567[0.013] -2.101[0.055] 0.012(-0.99, 0.11) 0.056(-0.11,-0.99) 0.889 050318 A 1.44 34.5 -2.375[0.014] -1.968[0.057] 0.012(-0.99, 0.13) 0.060(-0.13,-0.99) 0.936 050319 D 3.24 190.0 -3.100[0.034] -2.045[0.118] 0.026(-0.98, 0.17) 0.119(-0.17,-0.98) 0.687 050401 B 2.9 43.0 -1.667[0.012] -1.472[0.046] 0.011(-1.00,-0.04) 0.047( 0.04,-1.00) 1.060 050416 D 0.6535 5.0 -2.225[0.055] -3.119[0.155] 0.026(-0.95, 0.30) 0.182(-0.30,-0.95) 0.971 050505 A 4.27 70.0 -2.418[0.018] -1.415[0.071] 0.018(-1.00,-0.03) 0.073( 0.03,-1.00) 0.789 050525 A 0.606 15.0 -0.932[0.003] -1.769[0.015] 0.003(-1.00,-0.09) 0.015( 0.09,-1.00) 2.934 050603 B 2.821 16.0 -1.347[0.010] -1.159[0.043] 0.010(-1.00,-0.05) 0.043( 0.05,-1.00) 0.913 050803 A 0.422 165.0 -2.868[0.022] -1.526[0.085] 0.021(-1.00, 0.04) 0.088(-0.04,-1.00) 0.882 050814 A 5.3 95.0 -2.750[0.030] -1.798[0.108] 0.027(-0.99, 0.12) 0.117(-0.12,-0.99) 1.036 050820 C 2.61 32.0 -2.461[0.016] -1.689[0.067] 0.015(-1.00, 0.09) 0.067(-0.09,-1.00) 0.988 050824 D 0.83 38.0 -3.102[0.081] -2.743[0.219] 0.036(-0.95, 0.30) 0.241(-0.30,-0.95) 1.0157 050904 C 6.29 207.0 -2.603[0.009] -1.205[0.041] 0.009(-1.00,-0.04) 0.041( 0.04,-1.00) 1.070 050908 A 3.344 24.0 -2.685[0.031] -1.947[0.107] 0.025(-0.99, 0.17) 0.114(-0.17,-0.99) 0.969 050922C B 2.198 10.0 -1.761[0.009] -1.371[0.038] 0.009(-1.00,-0.02) 0.038( 0.02,-1.00) 0.900 051016B D 0.9364 6.0 -2.509[0.048] -2.331[0.145] 0.030(-0.97, 0.24) 0.158(-0.24,-0.97) 1.344 051109 A 2.346 43.0 -2.052[0.033] -1.626[0.124] 0.032(-1.00, 0.05) 0.129(-0.05,-1.00) 0.819 051111 B 1.55 45.0 -2.080[0.008] -1.305[0.034] 0.008(-1.00,-0.04) 0.035( 0.04,-1.00) 0.764 051221 C 0.547 3.1 -1.422[0.008] -1.397[0.037] 0.008(-1.00,-0.03) 0.037( 0.03,-1.00) 0.996 060115 A 3.53 128.0 -2.851[0.018] -1.771[0.071] 0.016(-0.99, 0.11) 0.073(-0.11,-0.99) 0.875 060206 A 4.048 14.0 -2.184[0.011] -1.693[0.047] 0.011(-1.00, 0.06) 0.049(-0.06,-1.00) 1.061 060210 A 3.91 278.0 -2.579[0.015] -1.544[0.059] 0.015(-1.00,-0.01) 0.060( 0.01,-1.00) 0.928 060223 A 4.41 14.8 -2.320[0.020] -1.775[0.078] 0.019(-0.99, 0.11) 0.082(-0.11,-0.99) 1.009 060418 C 1.489 80.0 -1.993[0.006] -1.634[0.030] 0.006(-1.00,-0.01) 0.030( 0.01,-1.00) 0.815 060502A B 1.51 34.0 -2.157[0.011] -1.431[0.045] 0.011(-1.00,-0.01) 0.046( 0.01,-1.00) 1.031 060510B A 4.9 289.0 -2.839[0.011] -1.778[0.048] 0.010(-1.00, 0.07) 0.049(-0.07,-1.00) 0.643 060522 A 5.11 80.0 -2.829[0.024] -1.578[0.093] 0.023(-1.00, 0.08) 0.098(-0.08,-1.00) 1.188 060526 C 3.21 320 -3.387[0.035] -1.924[0.140] 0.028(-0.99, 0.17) 0.132(-0.17,-0.99) 0.967 060604 D 2.68 67.0 -3.356[0.085] -2.040[0.283] 0.060(-0.98, 0.21) 0.290(-0.21,-0.98) 0.804 060605 A 3.711 37.0 -2.810[0.026] -1.474[0.105] 0.025(-1.00, 0.07) 0.109(-0.07,-1.00) 0.912 060607 A 3.082 127.0 -2.686[0.011] -1.436[0.046] 0.010( 1.00, 0.02) 0.047(-0.02,-1.00) 0.947 060614 C 0.125 154.0 -1.877[0.005] -1.936[0.024] 0.004(-1.00, 0.01) 0.024( 0.01,-1.00) 1.072 060707 A 3.425 81.0 -2.652[0.022] -1.667[0.082] 0.021(-1.00, 0.06) 0.087(-0.06,-1.00) 1.116 060714 A 2.71 135.0 -2.614[0.016] -1.935[0.063] 0.013(-0.99, 0.12) 0.064(-0.12, 0.99) 1.161 060729 C 0.54 134.0 -2.658[0.021] -1.870[0.083] 0.019(-0.99, 0.12) 0.084(-0.12,-0.99) 0.863 060904B A 0.703 187.0 -3.059[0.022] -1.628[0.090] 0.021(-1.00, 0.10) 0.090(-0.10,-1.00) 0.855 060906 A 3.686 63.0 -2.386[0.020] -2.006[0.075] 0.017(-0.99, 0.14) 0.078(-0.14,-0.99) 1.114 060908 A 2.43 30.0 -2.040[0.009] -1.339[0.039] 0.009(-1.00,-0.01) 0.040( 0.01,-1.00) 0.864 060926 D 3.2 13.0 -2.724[0.044] -2.460[0.146] 0.026(-0.97, 0.24) 0.151(-0.24,-0.97) 0.972 060927 A 5.6 32.0 -2.388[0.013] -1.654[0.052] 0.012(-1.00, 0.07) 0.055(-0.07,-1.00) 1.158 061007 C 1.261 85.0 -1.334[0.004] -1.002[0.016] 0.003(-0.99,-0.13) 0.016( 0.13,-0.99) 0.523 061110B C 3.44 147.0 -3.055[0.021] -1.069[0.091] 0.021(-1.00,-0.05) 0.091( 0.05,-1.00) 1.175 061121 C 1.314 125.0 -1.936[0.004] -1.410[0.018] 0.004(-1.00,-0.08) 0.018( 0.08,-1.00) 0.450 061222B A 3.355 57.0 -2.367[0.023] -1.962[0.084] 0.020(-0.99, 0.14) 0.089(-0.14,-0.99) 1.280 070110 C 2.352 103.0 -2.806[0.016] -1.581[0.702] 0.016(-1.00, 0.06) 0.070(-0.06,-1.00) 1.007 Table 1. The sample of Swift long GRBs with known z analysed here (47 events). The z and duration (Tint) of the burst as well as the results from the PL photon model fit are reported. Second column refers to the class group assigned to the burst according to our spectral analysis (see text). The best fit values of the parameters and their one–dimensional standard deviations (sd) are given in columns 5 and 6; e1,2 and n1,2 in columns 7 and 8 are the semi-axis lengths and principal axes of the error ellipse, respectively. The reduced χ reported in the last column (dof=58). E > E0(α− β) where α and β are the photon indices of the low and high energy power laws, respectively, and E0 is the e−folding (break) energy related to the peak energy in the νfν [or E2f(E)] spectrum by Epk = E0× (2+α). Note that Epk is well defined for α > −2 and β < −2. The normalization N0 is in photons s−1 cm−2 keV−1. The spectral range of BAT is narrower than that one of BATSE and other previous detectors. In the most recent update of the BAT–team homepage6 it is written that due to varying threshold levels in individual detectors, channels below 15 keV should not be used for spectral analysis. Like- 6 http://swift.gsfc.nasa.gov/docs/swift/analysis/bat-digest.html c© 0000 RAS, MNRAS 000, 000–000 4 Cabrera et al. wise, channels above 150 keV are unreliable due to a lack of calibration data at those energies. Then, we are limited to the narrow energy band of 15–150 keV for the spectral anal- ysis of Swift bursts. This implies that model fitting with the 4–parameter Band function will not be reliable for most of these burst. Therefore, we proceed to fit the observed pho- ton spectra to both the power law (PL) and cutoff power law (CPL) models. The former has only two parameters, the normalization, N0, and the photon index, α, and is given by: N(E) = N0 50 keV The CPL model implies one more parameter, an energy e−folding spectral break, E0, related to Epk in the same way as in the Band model (see above). The CPL model is defined as: N(E) = N0 50 keV It is easy to verify that the CPL model (named also the Compton –COMP– model) is the Band model with β → As we will see later, the complex nature of the errors due to the small energy range of the spectra makes it more convenient to use the logarithms of N0 and E0 (or Epk) rather than the linear quantities. This choice reduces the asymmetry of the errors and of the confidence levels (CL) contour shapes. Therefore, we have introduced into XSPEC new models corresponding to Eqs. (2) and (3) (and also Eq. (1); see §§3.3) to carry out the fits with LogN0 and LogEpk. The spectral analysis was carried out by using the hea- soft6.1.2 public software7. The first step in our analysis is to define the background–subtracted light curve from the cor- responding data file. The photon counts at each channel are taken in the time interval Tint where the main light curve is clearly above the background noise. The spectral file was corrected by position and systematic errors. Then, the pho- ton counts at each channel is convolved with a response ma- trix, build up for a given event with the calibration matrix8, in order to obtain the time–integrated energy spectrum. As mentioned above, the BAT detector is well calibrated only in the 15–150 keV energy range. Therefore, we used the 60 BAT channels in this range, with the default binning. The spectral fit to the different photon models mentioned above was performed with the heasoft package XPSEC (version 11.3.1). For the GRBs for which the (CPL) Epk may be esti- mated, we calculate the fluence, Sγ , corresponding to the observed spectral band (15–150 keV) only with the aim of completeness. In addition the rest isotropic–equivalent en- ergy, Eγ,iso, is calculated from the observer bolometric flu- ence SγB corresponding to the spectral range 1–10000 keV at rest: Eγ,iso = 4πSγBdL(z) (1 + z) , (4) where dL(z) is the cosmology–dependent luminosity dis- tance. 7 http://heasarc.gsfc.nasa.gov/docs/software/lheasoft/download.html 8 We use the January 2007 matrix calibration provided by the Swift–BAT team. 2.3 Errors from the fit and error propagation A main concern of the spectral deconvolution of BAT spec- tra is the expected large uncertainty and correlations among the fitted spectral parameters. Consequently, a careful inter- pretation of the errors is crucial to make usable and reliable the spectral analysis of Swift GRBs. First, as mentioned above, we have carried out the fits using LogN0 (PL model), and LogN0 and LogEpk (CPL model), instead of the corresponding linear values. The er- rors in the linear parameters are very asymmetric and corre- lated among them in a very complex way; the corresponding regions at a given CL, as plotted with the XSPEC com- mand steppar in the plane of two parameters, have shapes strongly deviating from ellipses. Instead, the errors and the CL regions when the fits are carried out with logarithmic quantities, tend to be more symmetric and ellipse–shaped, respectively. This facilitates the handling of the errors to calculate composite quantities and correlations among the GRB parameters. The fit command of XSPEC, generates ellipsoidal con- fidence intervals that approximate the complex regions of joint variation of the parameters. These ellipsoids are char- acterized by the semi–axis lengths, ei (they are the square root of the variances), where i is for each axis, and the cor- responding principal axes (unitary vectors ni). We use these data to calculate the covariance matrix, Cij , and the ellip- tical boundary of the desired CL region in the subspace of interest (see e.g., Press et al. 1999). The diagonal elements of the matrix, Cii, are the variances of each model param- eter (two for the PL model and three for the CPL model), while the off–diagonal elements (covariances) show the cor- relations among the parameters. The error propagation from the spectral quantities (LogN0, α, LogEpk) to the composite quantities LogE (being E′pk the peak energy at rest E pk = Epk× [1+z)]) and LogEγ,iso is performed in two steps. First, we have ex- tracted sets of spectral quantities by a Monte Carlo method according to their elliptical CL. Then, for each set we have calculated the corresponding composite quantities as well as their covariance matrix averaging on the overall extrac- tions. From this last covariance matrix the elliptical CL of the composite quantities is obtained. 3 RESULTS In Table 1 we present the results from the spectral fit to the PL model for the whole sample of 47 long GRBs. Columns 1 to 4 give the burst name, the class group of the burst, z, and Tint. Columns 5 and 6 give the best fit values of the normalization, LogN0, and the photon index, α, with the corresponding one–dimensional symmetric standard devia- tions in brackets, as obtained with the XSPEC error com- mand. In columns 7 and 8 we report the square root of the variances, ei, together with the unitary vectors ni (principal axes), which provide respectively the semi–axis lenghts and directions of each axis (i = 1, 2) of the joint error ellipse. The reduced χ2 is reported in column 9. As it may be appreciated from Table 1, the best–fit α is smaller than −2 for 7 bursts out of the 47 (GRB 060906 is marginal and it will be recovered by the CPL fitting). Most c© 0000 RAS, MNRAS 000, 000–000 Spectral analysis of Swift GRBs with known z 5 Figure 1. Error CL contours projected in the plane of the pair of variables LogEpk–(-α) for each one of the bursts in samples A and B (CPL model). The irregular contours (black lines in the electronic version) were calculated with the XSPEC command steppar, while the elliptical contours (red lines) were constructed from the variances and principal axes given by the XSPEC command fit. Inner (thick line) and outer (thin line) contours in each case are for ∆χ2 = 1 and 2.3, respectively (see text). The dotted line indicates the level of 150 keV. c© 0000 RAS, MNRAS 000, 000–000 6 Cabrera et al. GRB LgN0[sd] α[sd] LgEpk[sd] e1 n1 e2 n2 e3 n3 χ 50 keV keV (1) (2) (3) (4) (5) (6) (7) (8) 050126 -2.38[0.21] -0.75[0.44] 2.07[0.19] 0.022 (-0.83,-0.17,-0.53) 0.519 ( 0.39,-0.85,-0.34) 0.092 ( 0.39,0.49,-0.78) 1.38 050223 -2.47[0.22] -1.50[0.42] 1.83[0.21] 0.022 (-0.89,-0.36,-0.29) 0.506 (-0.44, 0.83, 0.34) 0.140 (-0.12,-0.43, 0.90) 0.90 050318 -2.02[0.14] -1.32[0.26] 1.67[0.05] 0.010 (-0.77,-0.40,-0.49) 0.289 ( 0.46,-0.89,-0.02) 0.051 ( 0.43, 0.24,-0.87) 0.80 050401 -1.56[0.08] -1.23[0.19] 2.27[0.23] 0.011 (-0.96,-0.11,-0.25) 0.306 (-0.27, 0.61, 0.74) 0.056 (-0.07,-0.78, 0.62) 1.05 050505 -2.20[0.14] -0.95[0.31] 2.10[0.16] 0.016 (-0.87,-0.16,-0.46) 0.366 ( 0.37,-0.84,-0.41) 0.070 ( 0.32, 0.53,-0.79) 0.75 050525 -0.62[0.03] -0.98[0.07] 1.91[0.01] 0.002 (-0.73,-0.20,-0.65) 0.012 ( 0.59, 0.30,-0.75) 0.077 (-0.35, 0.93, 0.10) 0.34 050603 -1.27[0.07] -0.97[0.17] 2.51[0.31] 0.010 (-0.98,-0.04,-0.21) 0.358 (-0.20, 0.46, 0.86) 0.049 (-0.06,-0.89, 0.46) 0.90 050803 -2.60[0.18] -0.99[0.37] 1.99[0.15] 0.018 (-0.85,-0.23,-0.48) 0.425 ( 0.41,-0.86,-0.32) 0.081 ( 0.34, 0.47,-0.82) 0.85 050814 -2.10[0.30] -0.58[0.56] 1.73[0.59] 0.017 (-0.60,-0.27,-0.75) 0.0.75 ( 0.65, 0.38,-0.66) 0.628 (-0.46, 0.88, 0.05) 0.93 050908 -2.29[0.27] -1.26[0.48] 1.65[0.08] 0.019 (-0.76,-0.41,-0.51) 0.550 ( 0.49,-0.87,-0.02) 0.093 ( 0.43, 0.26,-0.86) 0.94 050922C -1.66[0.07] -1.16[0.15] 2.35[0.22] 0.009 (-0.97,-0.09,-0.22) 0.274 (-0.24, 0.54, 0.81) 0.045 (-0.04,-0.84, 0.54) 0.87 051109 -1.75[0.27] -1.04[0.55] 1.92[0.17] 0.026 (-0.82,-0.26,-0.52) 0.628 ( 0.42,-0.88,-0.22) 0.112 ( 0.40, 0.40,-0.83) 0.81 051111 -2.02[0.06] -1.17[0.14] 2.54[0.34] 0.008 (-0.99,-0.04,-0.15) 0.374 (-0.15, 0.35, 0.92) 0.040 (-0.01,-0.93, 0.36) 0.76 060115 -2.51[0.16] -1.13[0.32] 1.80[0.08] 0.013 (-0.80,-0.33,-0.51) 0.361 ( 0.45,-0.88,-0.15) 0.063 ( 0.40, 0.35,-0.85) 0.80 060206 -1.90[0.10] -1.11[0.21] 1.89[0.06] 0.009 (-0.83,-0.28,-0.48) 0.237 ( 0.42,-0.88,-0.22) 0.044 ( 0.37, 0.39,-0.85) 0.91 060210 -2.38[0.11] -1.12[0.26] 2.07[0.14] 0.013 (-0.89,-0.19,-0.41) 0.307 (-0.37, 0.82, 0.43) 0.060 (-0.25,-0.53, 0.81) 1.06 060223 -2.04[0.18] -1.16[0.35] 1.80[0.09] 0.015 (-0.80,-0.33,-0.50) 0.400 ( 0.45,-0.88,-0.15) 0.072 ( 0.39, 0.34,-0.85) 0.96 060502A -2.05[0.08] -1.19[0.18] 2.27[0.22] 0.010 (-0.96,-0.12,-0.25) 0.296 (-0.27, 0.60, 0.75) 0.054 (-0.06,-0.79, 0.61) 1.01 060510B -2.72[0.09] -1.53[0.19] 1.99[0.17] 0.009 (-0.94,-0.27,-0.22) 0.267 (-0.35, 0.70, 0.62) 0.069 ( 0.01,-0.66, 0.75) 0.62 060522 -2.37[0.23] -0.70[0.44] 1.85[0.08] 0.016 (-0.71,-0.26,-0.66) 0.071 ( 0.55, 0.39,-0.74) 0.501 (-0.45, 0.88, 0.13) 1.13 060605 -2.57[0.22] -1.00[0.44] 2.01[0.23] 0.022 (-0.87,-0.24,-0.43) 0.539 (-0.41, 0.82, 0.39) 0.103 (-0.26,-0.51, 0.82) 0.90 060607 -2.52[0.09] -1.09[0.19] 2.15[0.15] 0.010 (-0.93,-0.17,-0.34) 0.251 (-0.35, 0.74, 0.57) 0.050 (-0.15,-0.65, 0.75) 0.89 060707 -2.18[0.19] -0.73[0.40] 1.84[0.06] 0.015 (-0.68,-0.25,-0.69) 0.062 ( 0.59, 0.37,-0.72) 0.443 (-0.43, 0.90, 0.10) 1.00 060714 -2.52[0.12] -1.77[0.24] 1.80[0.20] 0.012 (-0.91,-0.38,-0.15) 0.304 (-0.40, 0.74, 0.54) 0.138 ( 0.09,-0.55, 0.83) 1.17 060904B -2.77[0.19] -1.07[0.37] 1.90[0.13] 0.017 (-0.84,-0.29,-0.47) 0.432 (-0.44, 0.86, 0.26) 0.081 (-0.33,-0.43, 0.84) 0.82 060906 -2.16[0.16] -1.60[0.31] 1.65[0.09] 0.014 (-0.83,-0.45,-0.33) 0.346 (-0.46, 0.89,-0.05) 0.094 (-0.31,-0.11, 0.94) 1.09 060908 -1.84[0.08] -0.90[0.17] 2.15[0.10] 0.008 (-0.89,-0.15,-0.42) 0.208 ( 0.36,-0.81,-0.47) 0.039 ( 0.27, 0.57,-0.78) 0.74 060927 -2.02[0.12] -0.93[0.24] 1.86[0.06] 0.009 (-0.77,-0.27,-0.58) 0.043 ( 0.47, 0.37,-0.80) 0.272 (-0.44, 0.89, 0.16) 0.98 061222B -2.03[0.20] -1.30[0.37] 1.67[0.06] 0.015 (-0.76,-0.40,-0.51) 0.422 (-0.47, 0.88, 0.01) 0.075 (-0.44,-0.24, 0.86) 1.30 Table 2. Spectral fit results using the CPL photon model for the class A and B GRBs. The best fit values of the parameters and their one–dimensional standard deviations (sd) are given in columns 2–4; e1,2,3 and n1,2,3 in columns 5–7 are the semi-axis lengths and principal axes of the error ellipsoid, respectively. The reduced χ2 is reported in the last column (dof=57). likely, the νfν spectra of these GRBs peak at energies below 15 keV. They could be related to the X–ray rich events and X–ray Flashes discovered by BeppoSAX and HETE–2 (Kip- pen et al. 2001; Sakamoto et al. 2005). We have also fitted the 47 GRB spectra to the CPL model and found that for 18 bursts the fit produces too large uncertainties and/or unphysical solutions. For these 18 cases the PL model is definitively more reliable than the CPL one. According to the results from our spectral fits (judged by the joint 68% CL contour errors), we classify the 47 Swift GRBs analysed here in four classes: Class A – The CPL–model fit is acceptable and Epk is within the 15–150 keV range. Class B – The CPL–model fit is acceptable, but Epk > 150 keV. Class C – the CPL–model fit gives unreliable results, while the PL–model fit is acceptable and gives α> −2, sug- gesting that Epk is at energies > 150 keV. Class D – The CPL–model fit gives unreliable results, while the PL–model fit is acceptable and gives α< −2, sug- gesting that Epk is smaller than 15 keV. The class of each GRB is indicated in the second col- umn of Table 1. This classification can be easily understood after a visual inspection of the plots shown in Appendix and figures 1 and 2. In Figs. A1, A2, and A3 of the Appendix we present the deconvolved time–integrated νfν [or E 2f(E)] observed spectra of the 47 Swift GRBs analysed here. The CPL model was used for the spectral deconvolution. The error bars show the observations while the continuous lines show the fitted curves. The spectra of all the events that we classify in groups A and B show some curvature, evi- dence of a peak in the E2f(E) distribution. Nevertheless, in most of the cases the formal F−test analysis says that the spectral fits from the PL model to the CPL one (which introduces one more parameter, Epk) do not improve sig- nificantly. Therefore, one expects large uncertainties in the determination of Epk for most of the events analyzed here. This is why it is important to carry out a carefully statistical analysis of the errors and to adequately propagate them for calculating composite quantities and inferring correlations. Figures 1 and 2 show different contours of the CLs pro- jected in the plane of the pair of variables LogEpk vs. (−α) for each burst in groups A and B (CPL model). The irregu- lar CL contours (black lines in the electronic version) were calculated with the XSPEC command steppar, while the el- liptical CLs (red lines) were constructed from the joint vari- ances and principal axes given by the XSPEC command fit (see §§2.3; columns 5, 6, and 7 in Table 2 below). Thick lines show the CLs for ∆χ2 = 1, while thin lines show the CLs for ∆χ2 = 2.3. For ∆χ2 = 1, the projections of the CL regions in each axis enclose ≈ 68% of the one–dimensional probabil- ity for the given parameter (one standard deviation), while c© 0000 RAS, MNRAS 000, 000–000 Spectral analysis of Swift GRBs with known z 7 Figure 2. Continues Fig. 1 for ∆χ2 = 2.3, the plotted region encompasses ≈ 68% of the probability for the joint variation of the two parameters (one standard deviation) (e.g., Press et al. 1999). As can be appreciated, the ellipses are a reasonable description of the CL regions in most cases, with the great advantage that they are handled analytically. We see from Figs. 1 and 2 that for many of the GRBs, the LogEpk and α spectral parameters are correlated (in- clined, thin ellipses). A similar situation arises for the other combination of parameters. Therefore, for this kind of data, a simple statistical quantification of the errors, such as the one–dimensional standard deviation, is not enough. This is why we have worked out the full covariance matrix calcu- lated from the error ellipse. We use it to propagate errors in the calculation of E′pk and Eγ,iso (§§2.3). This is a crucial c© 0000 RAS, MNRAS 000, 000–000 8 Cabrera et al. GRB Sγ [s.d] LgE [sd] LgE′ [sd] e1 n1 e2 n2 10−7erg cm2 1050erg s−1 keV (1) (2) (3) (4) (5) (6) 050126 8.55[1.82] 1.70[0.18] 2.42[0.18] 0.062 (0.73,-0.69) 0.247 ( 0.69, 0.73) 050223 6.14[0.83] 1.00[0.20] 2.04[0.21] 0.113 (0.72,-0.69) 0.266 ( 0.69, 0.72) 050318 12.90[0.54] 2.07[0.11] 2.06[0.04] 0.107 (1.00, 0.03) 0.044 (-0.03, 1.00) 050401 89.41[7.24] 3.51[0.14] 2.86[0.23] 0.042 (0.86,-0.51) 0.261 ( 0.51, 0.86) 050505 25.79[3.06] 3.13[0.12] 2.82[0.16] 0.040 (0.79,-0.62) 0.195 ( 0.62, 0.79) 050525 158.20[1.59] 2.34[0.01] 2.12[0.01] 0.018 (0.79, 0.61) 0.008 (-0.61, 0.79) 050603 74.60[8.18] 3.55[0.22] 3.10[0.30] 0.061 (0.81,-0.59) 0.372 ( 0.59, 0.81) 050803 20.83[2.57] 1.13[0.14] 2.14[0.15] 0.044 (0.73,-0.68) 0.199 ( 0.68, 0.73) 050814 14.60[1.16] 2.94[0.11] 2.53[0.06] 0.114 (0.93, 0.36) 0.047 (-0.36, 0.93) 050908 4.36[0.46] 2.25[0.16] 2.29[0.08] 0.156 (1.00, 0.06) 0.079 (-0.06, 1.00) 050922C 16.97[0.94] 2.63[0.14] 2.86[0.22] 0.034 (0.85,-0.53) 0.257 ( 0.53, 0.85) 051109 35.27[5.85] 2.80[0.19] 2.44[0.16] 0.238 (0.77, 0.64) 0.077 (-0.64, 0.77) 051111 37.32[3.59] 2.80[0.22] 2.96[0.34] 0.056 (0.84,-0.54) 0.399 ( 0.54, 0.84) 060115 16.04[1.07] 2.78[0.10] 2.46[0.07] 0.121 (0.84, 0.54) 0.044 (-0.54, 0.84) 060206 8.29[0.35] 2.59[0.06] 2.59[0.06] 0.086 (0.72, 0.70) 0.024 (-0.70, 0.72) 060210 69.24[3.74] 3.52[0.10] 2.76[0.14] 0.030 (0.81,-0.59) 0.170 ( 0.59, 0.81) 060223 6.54[0.52] 2.56[0.12] 2.53[0.08] 0.139 (0.86, 0.50) 0.052 (-0.50, 0.86) 060502A 22.87[1.66] 2.42[0.14] 2.67[0.22] 0.039 (0.85,-0.53) 0.259 ( 0.53, 0.85) 060510B 38.64[2.88] 3.53[0.13] 2.76[0.17] 0.046 (0.80,-0.59) 0.207 ( 0.59, 0.80) 060522 10.49[1.04] 2.78[0.09] 2.63[0.08] 0.120 (0.74, 0.67) 0.036 (-0.67, 0.74) 060605 5.33[1.12] 2.31[0.20] 2.69[0.22] 0.074 (0.74,-0.67) 0.289 ( 0.67, 0.74) 060607 24.91[1.58] 2.94[0.10] 2.76[0.15] 0.026 (0.83,-0.55) 0.173 ( 0.55, 0.83) 060707 16.26[1.13] 2.70[0.08] 2.48[0.06] 0.097 (0.81, 0.59) 0.036 (-0.59, 0.81) 060714 30.58[3.96] 3.11[0.15] 2.37[0.20] 0.111 (0.85,-0.53) 0.219 ( 0.53, 0.85) 060904B 14.78[1.91] 1.44[0.13] 2.13[0.13] 0.174 (0.71, 0.71) 0.047 (-0.71, 0.71) 060906 23.82[1.92] 3.16[0.14] 2.32[0.09] 0.142 (0.98,-0.20) 0.087 ( 0.20, 0.98) 060908 26.56[1.11] 2.78[0.07] 2.68[0.10] 0.016 (0.81,-0.59) 0.123 ( 0.59, 0.81) 060927 11.66[0.46] 2.92[0.06] 2.68[0.06] 0.077 (0.72, 0.70) 0.022 (-0.70, 0.72) 061222B 22.35[1.23] 2.97[0.14] 2.31[0.06] 0.137 (1.00, 0.01) 0.064 (-0.01, 1.00) Table 3. Calculated fluence and rest–frame isotropic and peak energies for our GRB groups A and B. The corresponding one–dimensional standard deviations are given between square brackets. The last 2 columns give the estimated semi–axis lengths (e1,2) and principal axes (n1,2) of the error ellipse associated to the pair of quantities LogE and LogEγ,iso. Note that the errors in most of the cases are correlated significantly (see also the plotted ellipses in Figs. 4 and 5). step when using the data, for instance, for establishing cor- relations among the spectral or composite quantities (§§3.3). The CPL–model fittings for GRB groups A and B are presented in Table 2. Column 1 gives the burst names. Columns 2, 3, and 4 report the best fit values of LogN0, α, and LogEpk with their corresponding one–dimensional standard deviations, as calculated with the XSPEC error command. In columns 5, 6, and 7 we report the lengths ei of each semi–axis together with the unitary vectors ni (principal axes), which provide the directions of each axis (i = 1, 2, 3). The reduced χ2 is reported in column 8 (dof = Finally, in Table 3 we report the calculated values for Sγ , LogEγ,iso, and LogE pk , as well as the corresponding propagated errors for the samples A and B, using the CPL– model fit results (see §2.3 for the used error propagation procedure). Column 1 gives the burst name. Columns 2 and 3 give respectively the values of Sγ (in the observed range 15–150 keV) and of Eγ,iso (extrapolated to the rest–frame energy range 1–10000 keV) corresponding to the best fit spectral parameters. Column 4 gives the value of the rest– frame peak energy, E′pk. The one–dimensional standard de- viationss of Sγ , Eγ,iso, and E pk are given within square brackets. However, due to the high correlation among the errors the simple standard deviation is not enough to char- acterize statistically the errors. Thus, in columns 5 and 6 we report the error ellipse elements (semi–axis lenghts, ei, and principal axes ni, i = 1, 2) associated to the pair of quantities LogE′pk and LogEγ,iso. It is important to remark that the correlation between the uncertainties of LogE′pk and LogEγ,iso is not entirely due to the correlation among LogN0, α, and LogEpk. For instance an increase in LogEpk alone is able to produce an increase in both LogE′pk and LogEγ,iso. This effect will turn out to be helpful in the next section, where we will study how LogE′pk and LogEγ,iso do correlate in the entire GRB sample. The median and dispersion (quartiles) of the best– fit CPL parameter α for the 24 + 5 classes A+B GRBs are −1.12+0.15 −0.14, which is in agreement with the re- sults from BATSE bright–GRB (Kaneko et al. 2006) and HETE–2 GRB (Barraud et al. 2003; Sakamoto et al. 2005) time–integrated spectra. Our α distribution is in particular similar to the one from HETE–2, with only ∼ 25% of the bursts having α . −1.2. The median and quartiles of the LogEpk [keV] parameter are 1.91 +0.19 −0.11 . Unfortunately, due to the low energy limit of the Swift–BAT, the sample anal- ysed here covers only the ∼ 15% fraction of bursts in the (CPL–model) Epk distribution inferred for BATSE bright bursts (Kaneko et al. 2006). The distribution of E′pk (at rest) is much broader than c© 0000 RAS, MNRAS 000, 000–000 Spectral analysis of Swift GRBs with known z 9 Figure 3. Top panel: the observed Epk as derived by the CPL fit as a function of the spectral index αCPL of the same fit. Bottom panel: Epk (as derived by the CPL fit) as a function of the spectral index αPL of the single power law fit. The correlation of Epk with αPL is induced by Epk entering in the Swift energy band (see text), and cannot be used to infer Epk when knowing only αPL. the one of Epk (the median and quartiles of LogE pk[keV] are 2.54+0.22 −0.21). This is expected due to the broad distribution in redshfits of the Swift bursts analysed here. For our 24 + 5 classes A+B GRBs, the z distribution has a broad maximum at z ∼ 3 − 4; the median and quartiles are 3.08+0.82 −1.5 . The median and quartiles of LogEγ,iso [10 50 erg] for our 24 + 5 classes A+B GRBs are 2.78+0.30 −0.44 . For the 8 long–duration Swift GRBs in common with the compilation presented in Amati (2006b), the values of E′pk reported here and in that paper are similar, but those of Eγ,iso are systematically larger in Amati (2006b). The latter difference is probably due to the fact that Amati (2006b) did not use for most of those GRBs the spectral information from the Swift satel- lite but from other space missions (Konus, HETE–2), which allowed to find a fitting with the Band model; as will be discussed in §§3.3, this implies typically a larger Eγ,iso with respect to the case a CPL model is used. 3.1 Bursts detected also by other instruments Among the bursts listed in Tables 2 and 3 there are four GRBs which have been detected by other instruments be- sides Swift. Three of them have been detected by Konus– Wind (GRB 050401, 050603 and 051109A) and one by HETE–2 (GRB 050922C). For GRB 050401 the Konus–Wind results reported by Golenetskii et al. (2005a) concern the spectral parameters for the two peaks displayed by its light curve, fitted with a Band model in the energy range [20–2000] keV. The first peak had α = −1.15 ± 0.16, β = −2.65 ± 0.31 and Epk = 132 ± 16 keV, while the second peak is described by α = −0.83 ± 0.21, β = −2.37 ± 0.14 and Epk = 119 ± 26 keV. Our analysis with the Swift data (see Tab. 3) for the time integrated spectrum and a CPL model yielded α = −1.23± 0.19 and Epk = 186 −76 keV. Within the (relatively large) errors, the results are consistent. The Konus–Wind data of GRB 050603 have been fitted by Golenetskii et al. (2005b) with a Band model in the [20– 3000] keV energy range, with α = −0.79±0.06, β = −2.15± 0.09 and Epk = 349 ± 28 keV. Our analysis with a CPL model gives α = −1.23± 0.19 and Epk = 323 −164 keV. The derived low energy spectral index is somewhat softer, while the value of Epk is consistent. GRB 050922C was detected by HETE–2 with the FRE- GATE instruments in the 7–30, 7–80, and 30–400 keV bands. It was fitted with a CPL law model by Crew et al. (2005) with α = −0.83+0.23 −0.26 and Epk = 130.5 +50.9 −26.8 keV, to be com- pared with our values of α = −1.16±0.15 and Epk = 224 keV. Again, within the errors, there is consistency. GRB 051109, detected by Konus–Wind and fitted with a CPL model by Golenetskii et al. (2005c) in the [20–500] keV energy range, showed α = −1.25+0.44 −0.59 and Epk = 161+224 −58 keV. This should be compared with our α = −1.04± 0.55 and Epk = 83 −27 keV. Even in this case there is con- sistency, within the relatively large errors. 3.2 Peak energy vs spectral index In Fig. 3 we show Epk, derived with the CPL fit, as a func- tion of the spectral index αCPL of the CPL fit (top panel) and as a function of αPL, the index derived when fitting with a single power law. As can be seen, although there is no relation between Epk and αCPL, a correlation appears when fitting with a single power law. This correlation, how- ever, has no physical meaning, since it is the result of the attempt of the single power law to account for the data point above the peak, with smaller flux (in νfν ). As Epk decreases, a larger fraction of data lies above the peak, in- ducing the single power law to steepen. This also means that this effect vanishes when Epk is outside the Swift en- ergy range, therefore if we do not know Epk, we cannot use this correlation to infer it. Furthermore, note the difference in the derived values of the spectral indices. As expected for these bursts for which we could derive Epk, the index αPL is systematically softer than αCPL. Given the obvious result of this subsection, we have considered unnecessary to show the upper panel of Fig. 3 with the 68% CL error ellipses. A complete treatment of the errors in this sense confirms the previous result. 3.3 Correlation between E′pk and Eγ,iso In Fig. 4, E′pk is plotted vs Eγ,iso for the class A (red–line ellipses) and class B (green–line ellipses) Swift GRBs. The ellipses correspond to the joint 68% CL error regions calcu- lated as explained in §§2.3. The dotted straight lines encom- pass the 3σ scatter of the updated Amati correlation pre- sented in Ghirlanda (2007) for 49 GRBs; the central thin solid straight line is the best fit for that sample. A visual inspection shows that the Swift data analyzed in this paper are indeed correlated in the E′pk–Eγ,iso diagram. Taking into account the uncertainties, no GRB may be classified as outlier. Eight GRBs show their central point above the c© 0000 RAS, MNRAS 000, 000–000 10 Cabrera et al. Figure 4. Correlation between E′ and Eγ,iso (rest frame) as obtained from spectral fits using the CPL model for the class A (red ellipses) and B (green ellipses) bursts. The ellipses correspond to the joint 68% CL error regions of each data point. The dotted lines encompass the 3σ scatter of the updated Amati correlation presented in Ghirlanda (2007) for 49 GRBs; the thin solid line is the best fit for that sample. The thick solid line is the best fit that we find for the Swift data presented here taking into account the correlated errors, and the dot–dashed line is for the 1σ uncertainty, computed in the barycentre of the data points. 3σ strip, but taking into account the 68% CL contours as well as the fact that the spectral fitting was performed us- ing a CPL model instead of the Band one (see below), make such GRBs compatible with the 3σ strip. Notice that the 5 (marginal) class B events are not outliers. Therefore, we may include them for a fit along with the 24 class A events. The thick solid line in Fig. 4 is the best fit that we find for the 29 class A and B GRBs. The linear fit has been performed by an iterative minimum χ2 method, where the coordinate origin of the data is shifted to the barycentre, and the uncertainty component of each datapoint is calculated from the corresponding error ellipse in the orthogonal direc- tion of the straight fitted line. The dot–dashed line curves describe the standard error of the mean values of the fit parameters computed in the barycentre of the data points. The best fit in logarithmic quantities is: = (2.25±0.01)+(0.54±0.02)Log Eγ,iso 1052.44erg We point out the weight of GRB050525 in determining the best fit due to its very–reduced spectral parameter uncer- tainties. We have also carried out the linear fit for only the 24 class A GRBs: = (2.23±0.01)+(0.54±0.02)Log Eγ,iso 1052.42erg As expected, the fit is very similar to the one that includes the 5 class B GRBs. Thus, a correlation between E′pk and Eγ,iso is confirmed for 29 long GRBs observed by the Swift satellite and ho- mogeneously analyzed by us. The values of the power–law index (α ∼ 0.54) and normalization (K ∼ 100 keV for Eγ,iso given in units of 10 52erg, E′pk = KEγ,iso α) corre- sponding to the best fit, lie in the ranges of values found with different previous samples (see for a review of previ- ous studies in Amati 2006a). It is encouraging that with the Swift data, as analysed here, one obtains an E′pk–Eγ,iso c© 0000 RAS, MNRAS 000, 000–000 Spectral analysis of Swift GRBs with known z 11 correlation fully consistent with the one obtained for GRBs observed with other satellite detectors (the so–called Am- ati relation; Amati et al. 2002, see also Amati 2006a and Ghirlanda 2007 for the most recent updates), which had a much broader energy (spectral) range than BAT. The er- rors in the fit parameters reported in eqs. (5) and (6) are at the 68% CL. These errors are somewhat larger than those reported in Ghirlanda (2007) and Amati (2006a; note that therein the errors are at the 90% CL). In general, the scatter around the correlation obtained here is within the range of different previous determinations of the Amati correlation. The value of the reduced χ2 for the fit with the 29 class A+B GRBs is χ2ν = 379/27 = 14 close to the values χ2ν = 493/41 = 12 and χ ν = 530/47 = 11.3 obtained for larger samples (43 and 49 GRBs) by Ghirlanda et al. (2006) and Ghirlanda (2007), respectively. We have also estimated the variance of our fit, s2 (see e.g., Bevington & Robinson 1992, Chapter 11), by taking into account the joint variance of the data, i.e. the elliptical CL contours. For the fit using the 24+5 GRBs, we find s = 0.13 (dof=27). Note that for calculating the deviations, we use orthogonal distances to the best–fitting line rather than dis- tances projected in the LogE′pk (Y) axis. An estimate of the vertical 1σ spread of the data around the best–fitting model would be σ LogE′pk = 0.15, in agreement with previous esti- mates (see e.g., Amati 2006a). The virtue of the Swift sample analysed here is its homo- geneity, in the sense of observations, reduction, and analysis. Previous samples used to infer the E′pk–Eγ,iso correlation were constructed from different detectors, with an informa- tion collected from different authors, who used different data processing and assumptions. This heterogeneity in the data should introduce a spurious scatter. In spite that the data used here have large observational errors, the dispersion of our fit lies within the range of fits reported previously. This is namely due to the homogeneity of the sample and be- cause, having taken into account the whole error ellipses associated to the data points, it results that the major–axis orientation of most of these ellipses is close to the direction of the correlation line. Examining in more detail Fig. 4, one sees that the nor- malization of the correlation found here [eq. (5)] is slightly higher (by ≈ 0.1 dex at the sample median value of Eγ,iso) than the previously established one. This is indeed expected. Due to the narrow energy range of BAT, we had to force the GRB spectra to be fitted by the CPL model, while most spectra could have been better described by the Band model, if higher energies had been detected. It is known that when fitting the CPL model to a spectrum whose shape is ac- tually described by the Band model, then (i) Epk is larger than it would otherwise be, and (ii) the low–energy power– law index α is more negative than it would otherwise be (Band et al. 1993; Kaneko et al. 2006). On the other hand, (iii) in the CPL model the νfν spectrum is cut–off exponen- tially after the break energy E0, while in the Band model the spectrum decreases as a power law. Item (i) implies di- rectly that Epk is likely overestimated when the CPL model is used in the spectral analysis, and items (ii) and (iii) im- ply that Eγ,iso, which is calculated in the rest–frame energy range of 1–10000 keV by extrapolating the fitted spectrum, is likely underestimated. In order to explore in a statistical sense the effect in the E′pk–Eγ,iso correlation of using the CPL model instead of the Band one for the spectral fitting, we carried out the following experiment. The time–averaged spectra of the 29 long GRBs that had a reasonable CPL–model fit (groups A and B) were fitted with XSPEC to a Band model, but with the high–energy power law index β frozen to the typical value of −2.3. As in the case of CPL, we have modified the Band model in order to perform the fit for the logarithms of the normalization and Epk. Then, we followed the same procedures described in §2 for handling the errors, taking into account the correlation among them. Fig. 5 shows the final data points and their 68% CL ellipses in the Epk–Eγ,iso diagram. As expected, the normalization of the E′pk–Eγ,iso correlation decreases (with respect to Fig. 4), now being almost equal to the one of the updated Amati relation given in Ghirlanda (2007) for the range of Eγ,iso values studied here. The best fit in logarithmic quantities is: = (2.23±0.01)+(0.53±0.04)Log Eγ,iso 1052.61erg 4 SUMMARY AND CONCLUSIONS We have analysed homogeneously the time–integrated spec- tra of a relatively large sample of long–duration GRBs with known redshift obtained by a same detector, the Swift–BAT instrument. We have carried out spectral fits to 47 GRBs by using the two–parameter PL and three–parameter CPL photon models. Due to the narrow spectral range of BAT, 15–150 keV, only two– or three–parameter models can give reliable fits. However, even in these cases, large uncertain- ties are expected in the fitted parameters, making the study of the (highly correlated) errors important. We have per- formed Monte Carlo simulations in order to propagate the errors and calculate composite quantities such as E′pk and Eγ,iso. The main results and conclusions are as follow: • For 29 GRBs (classes A+B) out of the 47 bursts anal- ysed here, the CPL model, which identifies a peak energy Epk in the νfν spectrum, gives reliable fits, though in 5 cases (class B), the best fit value for Epk lies beyond the upper limit (150 keV) of the BAT energy band. The un- certainties in the fitted parameters are in most cases highly correlated among them. • The spectra of the remaining 18 bursts (classes C+D) are well fitted by the simple PL model. For 11 bursts (class C) of the 47 GRBs, the best PL–fit value of the photon index α is greater than −2, suggesting that Epk > 150 keV. The other 7 bursts (class D) show a photon index α lower than −2 which implies a decreasing power–law νfν spectrum in the analysed energy range. In these cases, the peak energy could lie in the X–ray range. • The fluence, Sγ , and rest–frame isotropic equivalent energy, Eγ,iso, as well as their uncertainties, were calculated for the 29 (classes A+B) GRBs with reliable fits to the CPL model. This is currently the largest homogeneous sample of long GRBs with determined spectral parameters and Eγ,iso. The mean values of the rest peak energyE′pk and Eγ,iso in the sample are ∼ 350 keV and ∼ 4× 1052 erg, respectively. • The Epk as inferred from the CPL fit and αPL as in- ferred from the PL fit do correlate. However, this correlation c© 0000 RAS, MNRAS 000, 000–000 12 Cabrera et al. Figure 5. Same as Fig. 4 but for E′ and Eγ,iso inferred from spectral fits using the Band model with β frozen to −2.3. Note that using the Band model the correlation shifts to the higher (lower) Eγ,iso (E ) side with respect to the case when the CPL model is used. is not physical, but is the result of the attempt of the sin- gle PL to account for the data points above the peak, with smaller flux. As Epk decreases, a larger fraction of data lies above the peak, inducing the single power law to steepen. Therefore, the correlation should not be used to infer Epk when knowing only αPL. Indeed, we show that Epk and αCPL do not correlate. • A correlation between E′pk and Eγ,iso (the ’Amati’ relation) is confirmed for our sample [Eq. (5)]. This correla- tion and its scatter are consistent with the ones established previously for non–Swift bursts, showing that the E′pk–Eγ,iso correlation is hardly an artifact of selection effects. The zero– point of our correlation (in Log–Log) is larger by ∼ 0.1 dex at the sample median value of Eγ,iso than the latest updated “Amati” correlation (Ghirlanda 2007). This difference is ex- pected due to the use of a CPL model for describing the observed spectra instead of the Band model. • When the Band model with the high–energy photon index frozen to β = −2.3 is fitted to the spectra of our 29 (classes A+B) GRBs, the zero–point of the resultant E′pk– Eγ,iso correlation becomes smaller than in the case of the CPL model. For the Band model, the obtained E′pk–Eγ,iso correlation is given by Eq. (7), which is virtually indistinct from the ’Amati’ correlation established previously for un- even observable data-sets from different satellites. Although in this work we were able to process ho- mogeneously the spectra of a significant fraction of the Swift GRBs with known redshift, the sample is still lim- ited for conclusive results regarding the physical meaning of the correlations among the burst observable properties. During the refereeing process of this work, appeared a pa- per by Li (2007b), where the author finds evidence of sig- nificant change in the ’Amati’ correlation parameters with redshift for the heterogeneous sample of 48 GRBs compiled by Amati (2006a). The number of GRBs with known red- shifts and full spectral information obtained homogeneously should increase in order to attain more conclusive results. c© 0000 RAS, MNRAS 000, 000–000 Spectral analysis of Swift GRBs with known z 13 ACKNOWLEDGMENTS JIC and VA-R gratefully acknowledge the hospitality ex- tended by Osservatorio Astronomico di Brera in Merate dur- ing research stays. JIC is supported by a CONACyT (Mex- ico) fellowship. We are grateful to the anonymous Referee for his/her comments, which helped to correct some bugs in the processed data and to improve the manuscript. This work was supported by the PAPIIT–UNAM grant IN107706-3 to VA-R, and by an italian 2005 PRIN–INAF grant. The au- thors acknowledge the use of the Swift publicly available data as well as the public XSPEC software package. We thank J. Benda for grammar corrections to the manuscript. REFERENCES Amati, L., et al. 2002, A&A, 390, 81 Amati, L. 2006a, MNRAS, 372, 233 Amati, L. 2006b, in proceedings of ”SWIFT and GRBs: Unveiling the Relativistic Universe”, preprint (astro-ph/0611189) Atteia, J.-L. 2003, A&A, 407, L1 Band, D., et al. 1993, ApJ, 413, 281 Barraud, C. et al. 2003, A&A, 400, 1021 Barthelmy, S. D. et al. 2005, Spa. Sci. Rev. 120, 143 Bevington, P. R. & Robinson, D. 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A1, A2, and A3, the CPL model was used to fit the observed spectra. c© 0000 RAS, MNRAS 000, 000–000 http://arxiv.org/abs/astro-ph/0611189 http://arxiv.org/abs/astro-ph/0702212 http://arxiv.org/abs/astro-ph/0608379 http://arxiv.org/abs/astro-ph/0608282 14 Cabrera et al. Figure A1. Time–averaged νfν spectra (red error bars) of the Swift GRBs with known redshift included in our sample. Continuous (green) line: best fit curve with the CPL photon model. The spectral flux of each burst was averaged over its integration time, Tint. c© 0000 RAS, MNRAS 000, 000–000 Spectral analysis of Swift GRBs with known z 15 Figure A2. Continues Fig. A1. c© 0000 RAS, MNRAS 000, 000–000 16 Cabrera et al. Figure A3. Continues Fig. A1. c© 0000 RAS, MNRAS 000, 000–000 Introduction Spectral analysis The sample Spectral fits Errors from the fit and error propagation Results Bursts detected also by other instruments Peak energy vs spectral index Correlation between Epk and E,iso Summary and Conclusions The observed spectra

0704.0792

The Relationship Between Molecular Gas Tracers and Kennicutt-Schmidt Laws

Accepted for publication in the Astrophysical Journal, July 16, 2007 Preprint typeset using LATEX style emulateapj v. 08/22/09 THE RELATIONSHIP BETWEEN MOLECULAR GAS TRACERS AND KENNICUTT-SCHMIDT LAWS Mark R. Krumholz and Todd A. Thompson Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544 Accepted for publication in the Astrophysical Journal, July 16, 2007 ABSTRACT We provide a model for how Kennicutt-Schmidt (KS) laws, which describe the correlation between star formation rate and gas surface or volume density, depend on the molecular line chosen to trace the gas. We show that, for lines that can be excited at low temperatures, the KS law depends on how the line critical density compares to the median density in a galaxy’s star-forming molecular clouds. High critical density lines trace regions with similar physical properties across galaxy types, and this produces a linear correlation between line luminosity and star formation rate. Low critical density lines probe regions whose properties vary across galaxies, leading to a star formation rate that varies superlinearly with line luminosity. We show that a simple model in which molecular clouds are treated as isothermal and homogenous can quantitatively reproduce the observed correlations between galactic luminosities in far infrared and in the CO(1 → 0) and HCN(1 → 0) lines, and naturally explains why these correlations have different slopes. We predict that IR-line luminosity correlations should change slope for galaxies in which the median density is close to the line critical density. This prediction may be tested by observations of lines such as HCO+(1 → 0) with intermediate critical densities, or by HCN(1 → 0) observations of intensely star-forming high redshift galaxies with very high densities. Recent observations by Gao et al. hint at just such a change in slope. We argue that deviations from linearity in the HCN(1 → 0)−IR correlation at high luminosity are consistent with the assumption of a constant star formation efficiency. Subject headings: ISM: clouds — ISM: molecules — stars: formation — galaxies: ISM — radio lines: 1. INTRODUCTION Schmidt (1959, 1963) first proposed that the rate at which a gas forms stars might follow a simple power law correlation of the form ρ̇∗ ∝ ρNg , where ρ̇∗ is the star formation rate per unit volume, ρg is the gas den- sity, and N is generally taken to be in the range 1 − 2. In the decades since, observations have revealed two strong correlations that appear to be evidence for this hypothesis. First, galaxy surveys reveal that the in- frared luminosity of a galaxy, which traces the star for- mation rate, varies with its luminosity in the CO(1 → 0) line, which traces the total mass of molecular gas, as LFIR ∝ L1.4−1.6CO (Gao & Solomon 2004a,b; Greve et al. 2005; Riechers et al. 2006a). Kennicutt (1998a,b) iden- tified the closely-related correlation between gas surface density Σg and star formation rate surface density Σ̇∗, Σ̇∗ ∝ Σ1.4±0.15g , a relation that has come to be known as the Kennicutt Law. Since over the bulk of the dy- namic range of Kennicutt’s data galaxies are predomi- nantly molecular, this is effectively a correlation between molecular gas, as traced by CO(1 → 0) line emission, and star formation. Spatially resolved observations of galaxies confirm that, at least for molecule-rich galax- ies where resolved CO(1 → 0) observations are possible, star formation is more closely coupled with gas traced by CO(1 → 0) than with atomic gas (Wong & Blitz 2002; Heyer et al. 2004; Komugi et al. 2005; Kennicutt et al. 2007) Electronic address: [email protected], [email protected] 1 Hubble Fellow 2 Lyman Spitzer Jr. Fellow Second, Gao & Solomon (2004a,b) find that there is a strong correlation between the IR luminosity of galax- ies and emission in the HCN(1 → 0) line, which mea- sures the mass at densities significantly greater than that probed by CO(1 → 0). However, they find that their correlation, which covers nearly three decades in total galactic star formation rate, is linear: LFIR ∝ LHCN. Wu et al. (2005) show that this correlation ex- tends down to individual star-forming clumps of gas in the Milky Way, provided that their infrared luminosities are >∼ 104.5 L⊙. Interestingly, however, Gao et al. (2007) find a deviation from linearity in the IR-HCN correlation for a sample of intensely star-forming high redshift galax- ies. These sources show small but significant excesses of infrared emission for their observed HCN emission. The difference in power law indices between the LFIR− LCO and LFIR −LHCN correlations is statistically signif- icant, and, on its face, puzzling. An index near N = 1.5 seems natural if one supposes that a roughly constant fraction of the gas present in molecular clouds will be converted into stars each free-fall time. In this case one expects ρ̇∗ ∝ ρ1.5g (Madore 1977; Elmegreen 1994). If gas scale heights do not vary strongly from galaxy to galaxy, this implies Σ̇∗ ∝ Σ1.5g as well, which is consis- tent with the observed Kennicutt law. More generally, since the dynamical timescale in a marginally Toomre- stable (Q ≈ 1; see Martin & Kennicutt 2001) galactic disk is of order Ω−1 ∝ (Gρg)−1/2, where Ω is the angular frequency of the disk, an index close to N = 1.5 is ex- pected if star formation is regulated by any phenomenon that converts a fixed fraction of the gas into stars on this time scale (Elmegreen 2002). http://arxiv.org/abs/0704.0792v2 mailto:[email protected], [email protected] On the other hand, Wu et al. (2005) suggest a simple interpretation of the linear IR-HCN correlation. They ar- gue that the individual HCN-emitting molecular clumps that they identify in the Milky Way represent a funda- mental unit of star formation. The linear correlation between star formation rate and HCN luminosity across galaxies arises because a measurement of the HCN lu- minosity for a galaxy simply counts the number of such structures present within it, each of which forms stars at some roughly fixed rate regardless of its galactic environ- ment. However, in this interpretation it is unclear why the structures traced by HCN(1 → 0) emission should form stars at the same rate in any galaxy. After all, one could equally well argue that molecular clouds traced by CO(1 → 0) are fundamental units of star formation, but the non-linear IR-CO correlation clearly shows that these objects do not form stars at a fixed rate per unit mass. Moreover, the evidence presented by Gao et al. (2007) that the linear IR-HCN correlation varies in extremely luminous high redshift galaxies suggests that the rela- tionship between HCN emission and star-formation may be somewhat more complex. In this paper we attempt to explain the origin of the difference in slope between the CO and HCN correla- tions with star formation rate, and more generally to give a theoretical framework for understanding how cor- relations between star formation rate and line luminosity, which we generically refer to as Kennicutt-Schmidt (KS) laws, depend on the tracer used to define them. Our cen- tral argument is conceptually quite simple, and in some sense represents a combination of the intuitive arguments for CO and HCN given above. Consider an observation of a galaxy in a molecular tracer with critical density ncrit, which essentially mea- sures the mass of gas at densities of ncrit or more, i.e. the gas that is dense enough for that particular transition to be excited. In galaxies where the median density of the molecular gas is significantly larger than ncrit, this means that the observation will detect the majority of the gas, and the bulk of the emission will come from gas whose density is near the median density. Since the gas den- sity will vary from galaxy to galaxy, the star formation rate per unit gas mass will vary as roughly ρ1.5g , with one factor of ρg coming from the amount of gas available for star formation, and an additional factor of ρ0.5g coming from the dependence of the free-fall or dynamical time on the density. On the other hand, in galaxies where the median gas density is small compared to the critical density for the chosen transition, observations will pick out only high density peaks. Since the density in these peaks is set by ncrit, and not by the conditions in the galaxy, these peaks are at essentially the same density in any galaxy where they are observed, and the corresponding free-fall times in these regions are constant as well. As a result, the star formation rate per unit mass of gas traced by that line is approximately the same in every galaxy, because the corresponding free-fall time is the same in every galaxy. In the rest of this paper, we give a quantitative version of this intuitive argument, and then discuss its conse- quences. In § 2 we develop a simple formalism to com- pute the star formation rate and the molecular line lu- minosity of galaxies, and in § 3 we use this formalism to predict the correlation between star formation rate and luminosity. We show that our predictions provide a very good fit for a variety of observations, and make predic- tions for future observations. We discuss the implications of our work and its limitations in § 4, and summarize our conclusions is § 5. 2. STAR FORMATION RATES AND LINE LUMINOSITIES 2.1. Cloud Properties Consider a galaxy in which the star-forming molecu- lar clouds have a volume-averaged mean molecular hy- drogen number density n = ρg/µH2 , where ρg is the volume-averaged mass density of the molecular clouds in the galaxy and µH2 = 3.9×10−24 g is the mean mass per hydrogen molecule for a gas of standard cosmic composi- tion. Observations indicate that n varies by two to three decades over the galaxies for which the Kennicutt and Gao & Solomon correlations are measured, from n ≈ 50 cm−3 in normal spirals like the Milky Way (McKee 1999) up to n ≈ 104 cm−3 in the strongest starburst systems in the local universe (e.g. Downes & Solomon 1998). There is strong evidence that densities in molec- ular clouds follow a lognormal probability distribution function (PDF; see reviews by Mac Low & Klessen 2004 and Elmegreen & Scalo 2004) d lnx (lnx− lnx)2 , (1) where x = n/n is the molecular hydrogen number den- sity n relative to the average density, σ is the width of the lognormal, and lnx = −σ2/2. For this distribution the median density is nmed = n exp(σ 2/2). Numerical experiments show that for supersonic isothermal turbu- lence σ2 ≈ ln 1 + 3M2/4 , where M is the 1D Mach number of the turbulence (Nordlund & Padoan 1999; Ostriker et al. 1999; Padoan & Nordlund 2002). Mach numbers in star-forming molecular clouds range from M ∼ 30 (McKee 1999) in normal spirals to M ∼ 100 in strong starbursts (Downes & Solomon 1998), imply- ing that median densities in molecular clouds range from ∼ 103 cm−3 in normal spirals to ∼ 106 cm−3 in star- bursts. Star forming clouds within a galaxy are approx- imately isothermal, except very near strong sources of stellar radiation, so we assume a fixed temperature T for the clouds. Observationally, T ranges from roughly 10 K in normal spirals (McKee 1999) up to as much about 50 K in strong starbursts (Downes & Solomon 1998; Gao & Solomon 2004b). 2.2. Star Formation Rates First let us ask how quickly stars form in such a medium. Krumholz & McKee (2005) give a model for star formation regulated by supersonic turbulence in which a population of molecular clouds of total mass Mcl form stars at a rate Ṁ∗ = SFRffMcl/tff(n), where tff(n) is the free-fall time evaluated at the mean density and SFRff is a number of order 10 −2 that depends weakly on M. We therefore estimate the star formation rate per unit volume as a function of the mean density given by ρ̇∗ ≈ SFRff 32Gµ3H2n . (2) Molecular Gas and Kennicutt-Schmidt Laws 3 We adopt the Krumholz & McKee result SFRff ≈ 0.014(M/100)−0.32 for clouds with a fiducial virial ra- tio of αvir = 1.3. Alternately, Krumholz & Tan (2007) point out that observed correlations between the star formation rate and the luminosity in different density tracers imply that over a 3− 4 decade range in density n, Ṁ∗ ≈ 10−2 Mcl(> n) tff(n) , (3) where Mcl(> n) is the mass of gas with a density of n or higher, and Mcl = Mcl(> 0). For a given choice of n this provides an alternative estimate of the star for- mation rate which is purely empirical, and independent of any particular theoretical model. However, the differ- ence between the star formation rates predicted by (2) and (3) is small. For gas with a lognormal PDF, Mcl(> n) = 1 + erf −2 lnx+ σ2 23/2σ , (4) and using this to evaluate equation (3) indicates that, for Mach numbers in the observed range, the two prescrip- tions (2) and (3) give about the same star formation rate over a very broad range in x. For example, at M = 30 the two estimates agree to within a factor of 3 for den- sities in the range 0.2 < x < 4 × 104. Given the scatter inherent in observational estimates of the star formation rate, a factor of 3 difference is not particularly signifi- cant, so it matters little which prescription we adopt. In practice, we will use equation (2). 2.3. Line Luminosities Now we must compute the luminosity of molecular line emission from the galaxy. Even for a cloud that is not in local thermodynamic equilibrium (LTE), for optically thin emission this calculation is straightforward. How- ever, the molecular lines used most often in galaxy sur- veys are generally optically thick. To handle the effect of finite optical depth on molecule level populations and line luminosities, we adopt an escape probability approxima- tion and treat clouds as homogeneous spheres. This is not fully consistent with our assumption that clouds have lognormal density PDFs, since the escape probability for- malism assumes a uniform level population throughout the cloud, and the essence of our argument in this paper turns on how the level population varies with density. However, this approach gives us an approximate way of incorporating the optical thickness of star-forming clouds into our model, the only alternative to which for turbu- lent media is full numerical simulation (e.g. Juvela et al. 2001). We therefore proceed by treating clouds as ho- mogeneous in order to determine their escape probabili- ties, and we then relax the assumption of homogeneity, while keeping the escape probabilities fixed, in order to determine level populations and cloud luminosities as a function of density. Consider a cloud of radius R in statistical equilibrium but not necessarily in LTE. In the escape probability approximation, the fraction fi of molecules of species S in state i is given implicitly by the linear system (nqji + βjiAji) fj = (nqij + βijAij)  fi (5) fi=1, (6) where qij is the collision rate for transitions from state i to state j, Aij is the Einstein spontaneous emission coefficient for this transition, βij is the cloud-averaged escape probability for photons emitted in this transition, the sums are over all quantum states, and we understand that Aij = 0 for i ≤ j and qij = 0 for i = j. Equations (5) and (6) allow us to compute the level populations fi for given values of βij . To completely specify the system, we must add an additional consis- tency condition relating the values of βij to the level populations. For a homogeneous spherical cloud, the es- cape probability for a given line is related to the optical depth from the center to the edge of the cloud τij by (B. Draine, 2007, private communication) βij ≈ 1 + 0.5τij , (7) where τij is computed at the central frequency of the line. In turn, the optical depth is related to the level populations by τij = 4(2π)3/2Mcs nX(S)fjR , (8) where λij is the wavelength of transition i → j, gi and gj are the statistical weights of states i and j, cs is the isothermal sound speed of the gas, and X(S) is the abundance of molecules of species S. Note that this equation implicitly assumes that the cloud has a uni- form Maxwellian velocity distribution with 1D disper- sion Mcs, consistent with our treatment of the clouds as homogeneous spheres. One additional complication is that we do not directly know cloud radii for most exter- nal galaxies, where observations cannot resolve individ- ual molecular clouds. However, we often can diagnose the optical depths of transitions by comparing line ratios of molecular isotopomers of different abundances. We therefore take τ10, the optical depth of the transition be- tween the first excited state and the ground state, as known. For a given level population this fixes the value of R. We solve equations (5)–(8) using Newton-Raphson it- eration. In this procedure, we guess an initial set of es- cape probabilities βij , and solve the linear system (5) and (6) to find the corresponding initial level populations fi. We then compute the optical depths τij from equation (8). The guessed escape probabilities βij and the corre- sponding optical depths τij generally will not satisfy the consistency condition (7), so we then iterate over βij val- ues using a Newton-Raphson approach, seeking βij for which the level populations give optical depths τij such that all elements of the matrix βij − 1/(1 + 0.5τij) are equal to zero within some specified tolerance. We use the LTE level populations and escape probabilities for our initial guess, so that the iteration converges rapidly when the system is close to LTE. Once we have determined the escape probabilities βij , we compute the luminosity by holding the βij values fixed but allowing the level populations to vary with density, then integrating over the PDF. Thus, the total luminos- TABLE 1 Model Parameters Parameter Normal galaxy Intermediate Starburst Reference T 10 20 50 1–4 M 30 50 80 1–4 X(CO) 2× 10−4 4× 10−4 8× 10−4 5 X(HCO+) 2× 10−9 4× 10−9 8× 10−9 6, 7 X(HCN) 1× 10−8 2× 10−8 4× 10−8 6–8 τCO(1→0) 10 20 40 9 τHCO+(1→0) 0.5 1.0 2.0 6, 7 τHCN(1→0) 0.5 1.0 2.0 6, 7 OPR 0.25 0.25 0.25 10 Note. — OPR = H2 ortho- to para-ratio. References: 1 – Solomon et al. (1987), 2 – Gao & Solomon (2004b), 3 – Downes & Solomon (1998), 4 – Wu et al. (2005), 5 – Black (2000), 6 – Nguyen et al. (1992), 7 – Wild et al. (1992), 8 – Lahuis & van Dishoeck (2000), 9 – Combes (1991), 10 – Neufeld et al. (2006) ity per unit volume in a particular line is Lij = X(S)βijAijhνij d lnx d lnx, (9) where νij is the line frequency, fi is an implicit function of n given by the solution to equations (5) and (6), and we assume that the abundance X(S) is independent of n. The line luminosity per unit mass is Lij/(µH2n). An IDL code that implements this calcu- lation is available for public download from http://www.astro.princeton.edu/∼krumholz/ astron- omy.html. 3. CORRELATIONS AND KENNICUTT-SCHMIDT LAWS 3.1. Lines and Parameters Using the formalism of § 2, we can now predict the correlation between the star formation rate and the luminosity of a galaxy in molecular lines. We make these predictions for three representative molecular lines: CO(1 → 0), HCO+(1 → 0), and HCN(1 → 0). For the first and last of these transitions, there are exten- sive observational surveys. We select HCO+(1 → 0) in addition to these two because there is some obser- vational data for it, and because its critical density of ncrit = βHCO+4.6 × 104 cm−3 makes it intermediate between CO(1 → 0), with ncrit = βCO560 cm−3, and HCN(1 → 0), with ncrit = βHCN2.8 × 105 cm−3.3 Here βS is the escape probability for the 1 → 0 transition of species S. These critical densities are for T = 20 K. All molecular data are taken from the Leiden Atomic and Molecular Database4 (Schöier et al. 2005). We make our calculations for three sets of fiducial pa- rameters which we summarize in Table 1. The three sets correspond roughly to typical conditions in normal disk galaxies like the Milky Way, to starburst galaxies like Arp 220, and to a case intermediate between the two. We have selected parameters for each case to roughly model the systematic variation of ISM parameters as one moves 3 Note that our critical density for HCN(1 → 0) is somewhat larger than the value quoted by Gao & Solomon (2004a,b), proba- bly because their calculation is based on somewhat different as- sumptions about how to extrapolate from calculated rate coef- ficients for HCN collisions with He to collisions with H2. See Schöier et al. (2005) for details. 4 http://www.strw.leidenuniv.nl/∼moldata/ from normal disk galaxies to starbursts. Thus, we vary the ISM temperature from 10− 50 K and the molecular cloud Mach number from 30−80 as we move from Milky Way-like molecular clouds to temperatures and Mach numbers typical of starbursts (e.g. Downes & Solomon 1998). Similarly, starbursts, which preferentially occur at galactic centers, have systematically larger metallici- ties than galaxies like the Milky Way (e.g. Zaritsky et al. 1994; Yao et al. 2003; Netzer et al. 2005). To explore this effect, we use abundances and 1 → 0 optical depths are twice and four times as large for our intermediate and starburst models, respectively, as for our normal galaxy model. 3.2. Kennicutt-Schmidt Laws We first plot, in Figure 1, the quantities L−1[dL(< n)/d lnn] (solid lines) and M−1[dM(< n)/d lnn] (dot- ted lines) as a function of density n for galaxies with mean densities n = 102, 103, and 104 cm−3, for the tracers CO(1 → 0), HCO+(1 → 0), and HCN(1 → 0), and for the Mach number and temperature corre- sponding to our intermediate case in Table 1. Here L(< n) and M(< n) are the luminosity and mass per unit volume contributed by gas of density n or less, i.e. L(< n) = X(S)βijAijhνij ∫ lnn fin(dp/d lnn)d lnn, M(< n) = ∫ lnn µH2n(dp/d lnn)d lnn, L = L(< ∞), and M = M(< ∞). Physically, L−1[dL(< n)/d lnn] and M−1[dM(< n)/d lnn] represent the fractional contribu- tion to the total line luminosity and the total mass that comes from each unit interval in the logarithm of den- sity. The plot shows what density range provides the dominant contribution to the line luminosity in differ- ent lines and for galaxies of differing mean densities, and how the gas contributing light compares to the gas con- tributing mass. Because the mass distribution is entirely specified by n and M, the dotted lines are the same in each of the three panels. Additionally, because of our choice M = 50 (Table 1), the median density (the den- sity corresponding to the peak in M−1[dM(< n)/d lnn]) is nmed ≈ 43n. In each panel, the critical density for each molecule is identified by a vertical dashed line. The top panel clearly shows that for the CO line, the light and the mass track one another very closely, even at the lowest densities. Thus, because nmed > ncrit, the http://www.astro.princeton.edu/~krumholz/ Molecular Gas and Kennicutt-Schmidt Laws 5 Fig. 1.— Fractional contribution to the total luminosity L−1[dL(< n)/d lnn] (solid lines) and mass M−1[dM(< n)/d lnn] (dotted lines) versus density n for the lines CO(1 → 0) (top panel), HCO+(1 → 0) (middle panel), and HCN(1 → 0) (bottom panel). The three curves show the cases n = 102 cm−3, 103 cm−3, and 104 cm−3, from leftmost to rightmost. We also show the critical density of each molecule, corrected for radiative trapping (dashed vertical lines). These calcula- tions use the parameters for the intermediate case listed in Table 1. solid lines move in lock-step with the dashed lines as n increases. In contrast, for HCN most of the luminosity comes from densities near the critical density regardless of the mass distribution. For the lowest n this means that the line luminosity is entirely dominated by the high den- sity tail of the mass distribution. As the median density nmed varies by a factor of 100 (from 4.3× 103− 4.3× 105 cm−3), the peak of L−1[dL(< n)/d lnn] moves by just a factor of a few in n. The HCO+ line is intermedi- ate between CO and HCN. For n = 102 cm−3 and 103 cm−3, nmed <∼ ncrit, and as with HCN most of the emis- sion comes from near the critical density. For n = 104 cm−3, nmed > ncrit, and the light starts to follow the mass, in a pattern similar to that for CO. Although Fig- ure 1 shows only the intermediate case, the normal galaxy and starburst cases give qualitatively identical results. This confirms the intuitive argument given in § 1: high critical density transitions trace regions of similar den- sity in every galaxy, while low critical density transitions trace regions whose density is close to the median density. Now consider how the luminosity in a given line corre- lates with the star formation rate in galaxies of varying mean densities. For a given n, we can compute the vol- ume density of star formation from equation (2) and the line luminosity density from (9). To facilitate compari- son with observations, rather than considering the total line luminosity, we use the quantity L′ (Solomon et al. 1997), which is related to the luminosity L by 8πkBν2 L, (10) converted to the units K km s−1 pc2. Similarly, we can estimate the far infrared luminosity from the star formation rate. There is a tight correla- tion between far-IR emission and star formation, par- ticularly for dense, dusty galaxies like those that make up most of the dynamic range of the Kennicutt (1998a) sample. To the extent that most or all of the light from young stars is re-processed by dust before escaping the galaxy, the bolometric luminosity integrated over the wavelength range 8− 1000 µm, which we define as LFIR, simply provides a calorimetric measurement of the total energy output by young stars, and is therefore an excel- lent tracer of recent star formation (Sanders & Mirabel 1996; Rowan-Robinson et al. 1997; Kennicutt 1998a,b; Hirashita et al. 2003; Bell 2003; Iglesias-Páramo et al. 2006). We therefore estimate the FIR luminosity from the star formation rate via LFIR = ǫṀ∗c 2, (11) where ǫ is an IMF-dependent constant. For consistency with Kennicutt (1998a,b), we take ǫ = 3.8× 10−4. To be precise and to facilitate comparison with observations, we adopt the Sanders & Mirabel (1996) definition of LFIR as a weighted sum of the luminosity in the 60 and 100 µm IRAS bands. This definition of the infrared luminos- ity generically underestimates the total infrared luminos- ity [8 − 1000]µm by a factor of 1.5 − 2 (Calzetti et al. 2000; Dale et al. 2001; Bell 2003). However, we use the ǫ value appropriate for LFIR rather than for the total IR luminosity because some of the observations to which we wish to compare our model (see § 3.3) provide only LFIR. Note that this choice for the connection between the star formation rate and the infrared luminosity is not fully consistent with our choice of the gas temperature for the three sets of parameters — normal, intermediate, and starburst — listed in Table 1, an issue we discuss in more detail in § 4.3. We plot the ratio of star formation rate to line lumi- nosity, and infrared luminosity to line luminosity, as a function of n in Figure 2. First consider the top panel, which shows all three lines computed for the interme- diate case. This again confirms our intuitive argument. Since the luminosity per unit volume in the CO line is roughly proportional to the mass density, and the star formation rate / IR luminosity is proportional to mass density to the 1.5 power, the ratios Ṁ∗/L ′ and LFIR/L vary roughly as n0.5. A powerlaw fit to the data over the range shown in Figure 2 gives an index of 0.57. In con- trast, the ratio of star formation density to HCN luminos- ity density is nearly constant for galaxies with n < 103 cm−3, and varies quite weakly with n up to densities of 104 cm−3, values found in the densest starbursts. A powerlaw fit from 10 cm−3 to 104 cm−3 gives an index of 0.17; from 10 cm−3 and 103 cm−3, the best fit powerlaw index is 0.08. As in Figure 1, the slope of the Ṁ∗/L curve for HCO+ represents an intermediate case, with a roughly constant ratio of Ṁ∗/L ′ and LFIR/L ′ at low n, rising to a slope comparable to that for CO at high values of n. Now consider the bottom three panels in Figure 2. Each panel shows the ratio of star formation rate and infrared luminosity to line luminosity for a single line, computed for each of the three galaxy models. The most Fig. 2.— Ratio of star formation rate or infrared luminosity to line luminosity, as a function of mean density n. In the top panel we show the lines CO(1 → 0) (solid line), HCO+(1 → 0) (dot- dashed line), and HCN(1 → 0) (dashed line) for the intermediate case in Table 1. In the next three panels we show the CO(1 → 0), HCO+(1 → 0), and HCN(1 → 0) lines for the normal galaxy case (dot-dashed line), intermediate case (solid line), and starburst case (dashed line). important point to take from these plots is that the choice of galaxy model has little effect in most cases. The largest differences are for HCN, where at n = 10 cm−3 the IR to line ratio predicted for the intermediate case differs from the normal galaxy case by a factor of 6.1, and from the starburst case by a factor of 4.1. This variation comes primarily from changes in the Mach number and the op- tical depth between models. The higher Mach number of the starburst model significantly increases the amount of mass in the high overdensity tail of the probability dis- tribution, while the higher optical depth lowers the ef- fective critical density. Both of these effect increase the amount of mass dense enough to emit in HCN(1 → 0) and reduce Ṁ∗/L ′. At higher mean densities these effects become less important and the models converge, so that by n = 104 cm−3 the range in Ṁ∗/L ′ from the normal to the starburst case is only a factor of 3.5. Most importantly, our central conclusion that Ṁ∗/L HCN is roughly constant across galaxies, while Ṁ∗/L CO rises as roughly [L 0.5, still holds when we consider how conditions vary across galaxies. Galaxies with low mean densities n are generally closest to the normal galaxy case, while those with high mean densi- ties should be closest to the starburst case, and this sys- tematic variation in galaxy properties with n still leaves Ṁ∗/L ′ relatively flat for HCN, and varying with a slope close to 0.5 for CO. From the normal galaxy case at n = 10 cm−3 to the starburst case at n = 104 cm−3, the value of Ṁ∗/L ′ varies by more than a factor of 50 for the CO(1 → 0), but by less than a factor of 3 for the HCN(1 → 0). 3.3. Comparison with Observations The calculations illustrated in Figure 2 demonstrate the basic argument that one expects a roughly constant star formation rate per unit line luminosity for high den- sity tracers (e.g., HCN), and a star formation rate per unit luminosity that rises like luminosity to the ∼ 0.5 for low density tracers (e.g., CO). However, in large sur- veys one cannot always determine the mean density in a galaxy, which would be required to construct an ob- servational analog to Figure 2. Instead, we can use our calculated dependence of star formation rate and line luminosity on density to compare to observations as fol- lows. Equation (9) gives the total molecular line lumi- nosity per unit volume and equation (2) gives the star formation rate, which we convert to an IR luminosity via equation (11). For fixed assumed volume of molecu- lar star-forming gas (Vmol) we can then predict the ex- pected correlations between L′ in a given molecular line and LFIR. The three panels of Figure 3 show our results for LFIR as a function of L CO, L HCN, and L for the intermediate model (see Table 1) and for several values of Vmol. Figure 4 shows how are results vary as a func- tion of the assumed T and M. There, for fixed Vmol, we compare our predictions for the intermediate model with the normal and starburst models. In both figures we compare our models to data culled from the literature. From the work of Gao & Solomon (2004a,b), Greve et al. (2005, their Fig. 7), Riechers et al. (2006b, their Fig. 5), and Gao et al. (2007), as well as the theoretical arguments in the preceding sections, we expect a strong, but not linear, correlation between the CO luminosity and the star formation rate — as measured by LFIR — with the approximate form LFIR ∝ L CO . The left panel of Figure 3 shows the CO data, the approximate correlation expected (solid line segment; offset from the data for clarity) and the theoretical prediction (solid lines) for a total volume of molecular gas of Vmol = 10 7, 108, and 109 pc3. Because at fixed LFIR, galaxies exhibit a dispersion in Vmol we expect there to be intrinsic scatter in this correlation, roughly bracketed by the range of Vmol plotted. The middle and right panels of Figure 3 show the same prediction for L′ and L′HCN. In these cases, because the molecular line luminosity is nearly linearly propor- tional to LFIR, the dependence on Vmol is much weaker than for L′CO. However, systematic changes or differences in the fiducial parameters for the calculation (see Table 1) introduce uncertainty and scatter into the correlation. Figure 4 assesses this dependence. It compares the pre- dictions of our model for normal (dot-dashed lines), inter- mediate (solid lines), and starburst (dashed lines) galax- ies, as defined in Table 1, for fixed Vmol = 10 8 pc3. Our simple model reproduces the data rather well, and it pre- dicts that generically there may be more intrinsic scatter in the L′CO−LFIR correlation than in either L HCN−LFIR or L′ − LFIR. Molecular Gas and Kennicutt-Schmidt Laws 7 Fig. 3.— LFIR (L⊙) versus L (1 → 0) (left panel), L′ (1 → 0) (middle panel), and L′ (1 → 0) (K km s−1 pc2; right panel). The lines in each panel derive from the model presented in this paper with a constant total volume of molecular material of 107, 108, and 109 pc3 (lowest to highest). The thick solid line segment shows power-law slopes to guide the eye. Data in the left and right panels are from Gao & Solomon (2004a,b) (circles) and Gao et al. (2007) (open squares for detections, arrows for upper limits). The middle panel combines data from Nguyen et al. (1992) (small circles with lines), Graciá-Carpio et al. (2006) (big circles), and Riechers et al. (2006b) (open square; using the Gao et al. (2007) FIR luminosity and magnification factor). For all data, LFIR is defined based on a weighted sum of the galaxy luminosity in the 60 and 100 µm IRAS bands, as described by Sanders & Mirabel (1996). For the Nguyen et al. (1992) data, the uncertainties in LHCO+ indicated by the lines arise because Nguyen et al. provide both HCN(1 → 0) and HCO +(1 → 0) intensities, but the values for L′ derived from their work generally fall a factor of 2 − 3 below the L′ from Gao & Solomon (2004a,b) for the same systems. This is probably because Nguyen et al. use a single beam pointing rather than integrating fully over extended sources, and therefore miss some of the flux. We therefore show two values of L′ , connected by a line, for each Nguyen et al. data point: a smaller value calculated directly from the data listed in their Table 2, and a larger value obtained by multiplying the L′HCN value of Gao & Solomon for that galaxy by the ratio IHCO+/IHCN measured by Nguyen et al. If this ratio is constant over the source, this estimate should correctly account for the flux outside the beam in the Nguyen et al. HCO+ observation. Note that in both the middle and right panels of Fig- ures 3 and 4, one expects a turn upward in the corre- lation at high LFIR, a deviation from linearity. This follows from the fact that in our model, at fixed Vmol, systems with higher LFIR have higher average densities. At sufficiently high LFIR we thus expect L HCN−LFIR and −LFIR to steepen, in analogy with the L′CO−LFIR correlation. The data points with very high LFIR in Fig- ures 3 and 4, which might be used to test this prediction of our model, are gravitationally lensed, at high redshift, and contaminated by bright AGN. It is therefore un- clear if the deviation from linearity implied particularly by the upper limits in L′HCN in the right panels of Fig- ures 3 and 4 is a result of enhanced LFIR, caused by the AGN emission (Carilli et al. 2005), or is instead a result of less molecular line emission per unit star formation, as our model implies (Fig. 2). Gao et al. (2007) note, however, that in the three systems for which the contri- bution from the AGN has been estimated (F10214+4724, D. Downes & P. Solomon 2007, in preparation; Clover- leaf, Weiß et al. 2003; APM 08279+5255, Weiß et al. 2005, 2007) the corrections are only significant for APM 08279+5255. This suggests that the data are so far con- sistent with our interpretation, but clearly much more data at high LFIR — or, more precisely for our purposes, at high density — is required to test our predictions. We discuss the issue of AGN contamination further in § 4.3. As a final note, the data so far do support the utility of HCO+ as a useful tracer of dense gas. Papadopoulos (2007) has argued against the utility of HCO+ as a faith- ful tracer of mass in starbursts on the basis that, since it is an ion, its abundance is strongly dependent on the free- electron abundance and might therefore vary strongly between galaxies with different ionizing radiation back- grounds. We cannot rule out this possibility given the limited data set available, but we see no strong evi- dence in favor of it from the data shown in Figures 3 and 4. As we have argued, HCO+(1 → 0) is particu- larly useful because its critical density is between that of CO(1 → 0) and HCN(1 → 0) and, thus, as Fig- ure 3 and 4 show, the correlation between LFIR and L′HCO+ should steepen from linear to super-linear over the range of galaxies presented in the CO panels. A care- ful, large-scale HCO+(1 → 0) survey similar to the work of Gao & Solomon (2004a) on HCN(1 → 0) should reveal these trends. Lines with similarly low excitation temper- atures and intermediate critical densities like CS(1 → 0) should behave analogously. 4. DISCUSSION 4.1. Implications for Kennicutt-Schmidt Laws and Star Formation Efficiencies Our results suggest that KS laws in different tracers naturally fall into two regimes, although there is a broad range of molecular tracers that are intermediate between the two extremes. Tracers for which the critical density is small compared to the median density in a galaxy repre- sent one limit. In these tracers, the light faithfully follows the mass, so the KS law measures a relationship between total mass and star formation. In any model in which star formation occurs at a roughly constant rate per dy- namical time, this must produce a KS law in which the star formation rate rises with density to a power of near 1.5, and the ratio of star formation to luminosity rises as density to the 0.5 power. In terms of surface rather than volume densities, this implies Σ̇∗ ∝ Σ −1/2. If we further add the observation that the scale heights h of the star-forming molecular layers of galaxies are roughly constant across galaxy types, one form of the observed Kennicutt (1998a,b) star formation law follows immedi- ately (Elmegreen 2002). Moreover, in a galactic disk, Fig. 4.— The same as Figure 3, but with constant Vmol = 10 8 pc3, and for the model parameters corresponding to “starburst” (dashed), “intermediate” (solid), and “normal” (dot-dashed) (see Table 1). Therefore, the middle solid line in each panel of Figure 3 is the same as the solid line in each panel in this Figure. h ∝ Σg/n and n ∝ Ω2/Q (e.g. Thompson et al. 2005); since in star-forming disks the Toomre-Q is about unity (Martin & Kennicutt 2001), substituting for h immedi- ately gives Σ̇∗ ∝ ΣΩ, the alternate form of the Kennicutt (1998a,b) law. The other limit is tracers for which the critical density is large compared to the median galactic density. These tracers pick out a particular density independent of the mean or median density in the galaxy, and thus all the regions they identify have the same dynamical time re- gardless of galactic environment. In this case the star formation rate will simply be proportional to the total mass of the observed regions, yielding a constant ratio of star formation rate to mass, as is observed for HCN in the local universe (Fig. 3, right panel; Gao & Solomon 2004a,b; Wu et al. 2005). We predict that there should be a transition between linear and super-linear scaling of LFIR with line luminos- ity at the point where galaxies transition from median densities that are smaller than the line critical density to median densities larger than the critical density. The HCO+(1 → 0) line, and other lines with similar critical densities, e.g. CS(1 → 0) and SO(1 → 0), should show this behavior for galaxies in the local universe. The ob- served correlation between LHCO+ and LFIR appears to be consistent with our prediction, although at present the data are not of sufficient quality to distinguish between a break and a single powerlaw relation. There are hints that the very highest luminosity star-forming galaxies, which all reside at high redshift and may well reach ISM densities not found in any local systems, show such a break in the IR-HCN correlation. One important point to emphasize in this analysis is that we have been able to explain the observed correla- tions between line and infrared luminosities, and hence between gas masses at various densities and star for- mation rates, without resorting to the hypothesis that the star formation process is fundamentally different in galaxies of different properties. Although uncertainties in both our model and the observations do not preclude an order-unity change in the star formation efficiency or SFRff as a function of LFIR, there is currently no evi- dence for such a change in the data, contrary to claims made by, e.g. Graciá-Carpio et al. (2006). In fact, all of the observational trends are predicted by our sim- ple model with constant star formation efficiency. This is consistent with other lines of evidence that the frac- tion of mass at a given density that turns into stars is roughly 1% per free-fall time independent of density (Krumholz & Tan 2007). 4.2. Does Star Formation Have a Fundamental Size or Density Scale? Based on the linear correlation between HCN(1 → 0) luminosity and star formation rate, seen both in ex- ternal galaxies and in individual molecular clumps in the Milky Way, Gao & Solomon (2004a,b) and Wu et al. (2005) propose that HCN(1 → 0) emission traces a fun- damental unit of star formation. They explain the linear IR-HCN correlation as a product of this; in their model, HCN luminosity correlates linearly with star formation rate because HCN luminosity simply counts the number of such units. Based on our analysis, we argue that this hypothesis is only partially correct. We concur with Gao & Solomon and Wu et al. that the HCN(1 → 0) luminosity of a galaxy does simply reflect the mass of gas that is dense enough to excite the HCN(1 → 0) line. However, our analysis shows that this does not necessarily imply that this density represents a special density in the star for- mation process, or that objects traced by HCN(1 → 0) represent a physically distinct class. We show that a linear correlation between star formation rate and line luminosity is expected for any line with a critical density comparable to or larger than the median molecular cloud density in the galaxies used to define the correlation. It is possible that HCN(1 → 0)-emitting regions represent a physically distinct scale of star formation as Wu et al. propose, but one can explain the linear IR-HCN correla- tion equally well if they are just part of the same contin- uous medium as the regions traced by CO(1 → 0) and by other transitions. Even the star-forming clouds them- selves may simply be parts of a continuous distribution of ISM structures occupying the entire galaxy, as argued by Wada & Norman (2007). In this case there need be no special density scales other than the mean and me- dian densities for the star-forming clouds on their largest scales, and the density at which star formation becomes rapid, converting the mass into stars in of order a free- Molecular Gas and Kennicutt-Schmidt Laws 9 fall time. This transition scale is unknown, but must be considerably larger than the density traced by HCN (Krumholz & Tan 2007). 4.3. Limitations and Cautions 4.3.1. Self-Consistency As mentioned in § 3.2, our approach of leaving the gas temperature T and Mach number M as free parame- ters is not entirely consistent with our calculation of the IR luminosity, since the IR luminosity and temperature are of course related. In principle, with a model of how the energy output from stars heats the dust and gas, to- gether with a structural model connecting the energy and momentum output from stars to the generation of turbu- lence, it should be possible to self-consistently compute both the gas temperature and the Mach number from the volumetric star formation rate (see, e.g., Thompson et al. 2005). Such a model would return T andM as a function of n and possibly other galaxy properties, while simulta- neously predicting a set of Kennicutt-Schmidt laws. If the line luminosity depended strongly on T or M, or if one required knowledge of the temperature to com- pute the infrared luminosity of a galaxy, we would would have no alternative to constructing such a model if we wished to explain the observed IR-line luminosity cor- relation. However, we can avoid this by relying on the observationally-calibrated star formation-IR correlation, and because, as we show in Figures 2, 3, and 4, the line luminosity varies quite weakly over a reasonable range of T and M for our chosen lines. For this reason, any model for computing T and M as a function of galaxy proper- ties, if it were consistent with observations, would not significantly alter the IR-line luminosity correlation we derive. This is true, however, only for lines that require low temperatures to excite. As we discuss in § 4.3.2, lines that require higher temperatures to excite do de- pend sensitively on the temparature in the galaxy, and a model capable of predicting the IR-line luminosity cor- relation for these lines must also include a calculation of the temperature structure of the galaxy. 4.3.2. Isothermality Our assumption of isothermality means that our anal- ysis will only apply to molecular lines for which the tem- perature Tup corresponding to the upper state energy is < 10 K, low enough to be excited even in the coolest molecular clouds in normal spiral galaxies. The reason for this is that at temperatures larger than Tup, the lu- minosity in a line generally varies at most linearly with the temperature. As the similarity between the results with our different galaxy models illustrates, changing the temperature within the range of ∼ 10 − 50 K produces only a factor of a few change in the luminosity of the lines we have studied. In contrast, line luminosity re- sponds exponentially to temperature changes when the temperature is below the value corresponding to the up- per state’s energy. This means that lines sensitive to high temperatures pick out primarily the regions that are warm enough for the line to be excited. Density has only a secondary effect. The emission will therefore re- flect the temperature distribution in star-forming clouds more than the density distribution, an effect that our isothermal assumption precludes us from treating. KS laws in high temperature tracers are likely to find lin- ear relationships between star formation rate and mass regardless of the critical density of the molecule in ques- tion because they will simply be correlating the mass of dust warmed to >∼ 100 Kelvin, which is essentially what is measured by LFIR, with the mass of gas warmed to temperatures above Tup. However, our model will not apply to these lines, and for this reason we do not attempt to compare to observations using higher tran- sitions of CO (3 → 2, 4 → 3, 5 → 4, 6 → 5, and 7 → 6, which have Tup = 33, 55, 83, 116, and 154 K, respectively; Greve et al. 2005, Solomon & Vanden Bout 2005), CS(5 → 4) (Tup = 35 K; Plume et al. 1997), or other high temperature tracers. 4.3.3. Molecular Abundances We have not considered density-dependent variations in molecule abundances. One potential source of vari- ation in molecular abundance is freeze-out onto grain surfaces at high densities and low temperatures (e.g. Tafalla et al. 2004a,b). Chemodynamical models suggest that freeze-out is not likely to become significant for ei- ther carbonaceous or nitrogenous species until densities n >∼ 106 cm−3 (Flower et al. 2006), but may become se- vere at higher densities, so whether depletion is signif- icant depends on what fraction of the total luminosity would be contributed by gas of this density or higher were there no freeze-out. Figure 1 suggests that freeze- out is likely to modify the total galactic luminosity of CO, HCO+, and HCN fairly little even at a mean ISM density of n = 104 cm−3, but may have significant effects for galaxies of larger mean densities or for lines for which the critical densities is comparable to the freeze-out den- sity. If freeze-out is significant, our conclusions will be modified. 4.3.4. Atomic Gas In the simple model developed here, we have neglected the role of atomic gas entirely. Whether the density or surface density of atomic gas plays a role in controlling the star formation rate is subject to debate on both ob- servational and theoretical grounds (Kennicutt 1998a,b; Wong & Blitz 2002; Heyer et al. 2004; Komugi et al. 2005; Krumholz & McKee 2005; Kennicutt et al. 2007), so it is unclear how much a limitation this omission re- ally is. We can say with confidence that in molecule-rich galaxies, which provide almost all the dynamic range of both the Kennicutt (1998a,b) correlation and the cor- relations illustrated in Figures 3 and 4, the atomic gas plays almost no role simply because there is so little of it. Thus, our predictions should be quite robust, except perhaps at the very low luminosity ends of Figures 3 and 4.3.5. AGN Contributions A final point is not so much a limitation of our work as a cautionary note about comparing our model with ob- servations. We have included in our model IR luminosity only from star formation, and molecular line luminosity only from molecules in cold star-forming clouds. How- ever, an AGN may make a significant contribution to a galaxy’s luminosity in the far infrared by direct heating of dust grains, and in molecular lines via an X-ray dissocia- tion region. Indeed, several of the systems with the high- est IR luminosities in Figures 3 and 4 are contaminated by AGN. As noted in §3.3, this complicates an assessment of our prediction of an up-turn in the L′HCN − LFIR and L′HCO+ − LFIR correlations at high luminosity. This de- viation from linearity at high gas density (at fixed Vmol, high LFIR) is an essential prediction of our model, but testing it relies on a careful separation of the contribu- tion of the AGN to both the IR and line luminosities (e.g. Maloney et al. 1996). In fact, Carilli et al. (2005) discuss the possibility that the AGN’s contribution to the IR lu- minosity in these systems causes them to be above the local linear L′HCN−LFIR correlation. Such a contamina- tion would mimic the prediction of our model. However, Gao et al. (2007) argue that the sub-millimeter galaxies in their sample are not AGN dominated and that just one of three quasars in their sample (APM 08279+5255) has a large AGN IR component. See Gao et al. (2007) for more discussion. For these reasons we contend that although our model is consistent with the existing data, the current evidence for a break in the L′HCN−LFIR cor- relation should be viewed with caution and more data in high density/luminosity systems is clearly required to understand the role of AGN contamination in shaping the correlation. 5. CONCLUSIONS We provide a simple model for understanding how Kennicutt-Schmidt laws, which relate the star formation rate to the mass or surface density of gas as inferred from some particular line, depend on the line chosen to define the correlation. We show that for a turbulent medium the luminosity per unit volume in a given line, provided that line can be excited at temperatures lower than the mean temperature in a galaxy’s molecular clouds, in- creases faster than linearly with the density for molecules with critical densities larger than the median gas density. The star formation rate also rises super-linearly with the gas density, and the combination of these two effects pro- duces a close to linear correlation between star formation rate and line luminosity. In contrast, the line luminosity rises only linearly with density for lines with low critical densities, producing a correlation between star formation rate and line luminosity that is super-linear. Based on this analysis, we construct a model for the correlation between a galaxy’s infrared luminosity and its luminosity in a particular molecular line. Our model is extremely simple, in that it relies on an observationally- calibrated IR-star formation rate correlation, it treats molecular clouds as having homogenous density and ve- locity distributions, temperatures, and chemical compo- sitions, and it only very crudely accounts for variations in molecular cloud properties across galaxies. Despite these approximations, the model naturally explains why some observed correlations between infrared luminosity and line luminosity in galaxies are linear, and some are super-linear. Using it, we are able to compute quantita- tively the correlation between infrared and HCN(1 → 0) line luminosity, and between IR and CO(1 → 0) line lu- minosity. We show that our model provides a very good fit to observations in these lines, and we are able to make similar predictions for any molecular line that can be ex- cited at low temperatures, as we demonstrate for the example of HCO+(1 → 0). Moreover, we are able to ex- plain the observed data without recourse to the hypoth- esis that the star formation process is somehow different, either more or less efficient, in different types of galaxies or for media of different densities. Instead, our model is able to explain the observed correlations using a simple, universal star formation law. One strong prediction of our model is that there should be a break from linear to non-linear scaling in the HCN- IR correlation at very high IR luminosity, and a similar break in the HCO+-IR correlation at somewhat lower lu- minosity. The data for HCO+ are consistent with this prediction but do not yet strongly favor a break over pure powerlaw behavior. However, there is some preliminary evidence for a break in the IR-HCN correlation in high redshift galaxies more luminous than any found in the local universe, although with these high redshift obser- vations it is difficult to rule out the alternative explana- tion of the break as arising due to a progressively rising AGN contribution to the IR luminosity (see §3.3 and §4.3.5). Future galaxy surveys both in the local universe and at high redshift may be used to test our predictions for HCO+(1 → 0), HCN(1 → 0), and other molecular lines. We thank L. Blitz, B. Draine, A. Leroy, E. Rosolowsky, and A. Socrates for helpful discussions, N. Evans and the anonymous referee for useful comments on the manuscript, and R. Kennicutt for kindly providing a preprint of his submitted paper. We thank Y. Gao for providing LFIR for the systems used in Figures 3 and 4. 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0704.0793

Friedmann Equations and Thermodynamics of Apparent Horizons

Friedmann Equations and Thermodynamics of Apparent Horizons Yungui Gong1, 2, ∗ and Anzhong Wang2, † College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China GCAP-CASPER, Department of Physics, Baylor University, Waco, TX 76798, USA With the help of a masslike function which has dimension of energy and equals to the Misner-Sharp mass at the apparent horizon, we show that the first law of thermodynamics of the apparent horizon dE = TAdSA can be derived from the Friedmann equation in various theories of gravity, including the Einstein, Lovelock, nonlinear, and scalar-tensor theories. This result strongly suggests that the relationship between the first law of thermodynamics of the apparent horizon and the Friedmann equation is not just a simple coincidence, but rather a more profound physical connection. PACS numbers: 98.80.-k,04.20.Cv,04.70.Dy The derivation of the thermodynamic laws of black holes from the classical Einstein equation suggests a deep connection between gravitation and thermodynamics [1]. The discovery of the quantum Hawking radiation [2] and black hole entropy which is proportional to the area of the event horizon of the black hole [3] further supports this connection and the thermodynamic (physical) inter- pretation of geometric quantities. The interesting rela- tion between thermodynamics and gravitation became manifest when Jacobson derived Einstein equation from the first law of thermodynamics by assuming the propor- tionality of the entropy and the horizon area for all local acceleration horizons [4]. In cosmology, like in black holes, for the cosmologi- cal model with a cosmological constant (called de Sit- ter space), there also exist Hawking temperature and en- tropy associated with the cosmological event horizon, and thermodynamic laws of the cosmological event horizon [5]. In de Sitter space, the event horizon coincides with the apparent horizon (AH). For more general cosmologi- cal models, the event horizon may not exist, but the AH always exists, so it is possible to have Hawking tempera- ture and entropy associated with the AH. The connection between the first law of thermodynamics of the AH and the Friedmann equation was shown in [6]. Now, we must ask if this interesting relation between gravitation and thermodynamics exists in more general theories of grav- ity, like Brans-Dicke (BD) theory and nonlinear gravi- tational theory. In [7], the gravitational field equations for the nonlinear theory of gravity were derived from the first law of thermodynamics by adding some nonequi- librium corrections. In this Letter, we show that equi- librium thermodynamics indeed exists for more general theories of gravity, provided that a new masslike function is introduced. To show our claim, we begin by reviewing the ther- modynamics of the AH with the use of the Misner- Sharp (MS) mass in Einstein and BD theories of gravity, whereby we find the equilibrium thermodynamics fails to hold for the BD theory. The Einstein equation can be rewritten as the mass formulas with the help of the MS mass M. The energy flow through the AH dE is related with the MS mass. Since the MS mass M, the Hawking temperature TA, and the entropy SA of the AH are geo- metric quantities, the first law of thermodynamics of the AH can be thought of as a geometric relation. Therefore, we expect the geometric relation to hold in other gravi- tational theory if it holds in Einstein theory. To achieve this, we replace the MS mass M by a masslike function M which equals to the MS mass M at the AH, we then show that the connection between the first law of ther- modynamics of the AH and the gravitational equations holds in scalar-tensor and nonlinear theories of gravity without adding nonequilibrium correction. For a spherically symmetric space-time with the metric ds2 = gabdx adxb+ r̃2dΩ2, using the MS mass M = r̃(1− gabr̃,ar̃,b)/2G [8], the a − b components of the Einstein equation give the mass formulas [9, 10] M,a = 4πr̃2(T ba − δ aT )r̃,b, (1) where the unit spherical metric is given by dΩ2 = dθ2 + sin2 θdϕ2 and T = T aa . From now on, all the indices are raised and lowered by the metric gab and the covariant derivative is with respect to gab. The AH is r̃A = arA = (H 2 + k/a2)−1/2. (2) At the AH, the MS mass M = 4πr̃3Aρ/3, which can be interpreted as the total energy inside the AH. Now we use the (approximate) generator ka = (1, −Hr) of the AH, which is null at the horizon, to project the mass formulas. Since kar̃,a = 0, at the AH we find that − dE = −ka∇aMdt = d(r̃A)/G = TAdSA, (3) where the horizon temperature is TA = 1/(2πr̃A) and the horizon entropy is SA = πr̃ A/G. On the other hand, us- ing the mass formulae (1), we get the energy flow through the AH − dE = −ka∇aMdt = −4πr̃2T ba r̃,bk = 4πr̃3AH(ρ+ p)dt. (4) Therefore, the Friedmann equation gives rise to the first law of thermodynamics −dE = TAdSA of the AH. From http://arxiv.org/abs/0704.0793v2 the above definitions, we see that the relation −dE = TAdSA is a geometrical relation which depends on the only assumption of the Robertson-Walker metric. To connect the geometrical quantity dE with the energy flow through the AH, we need to use the Friedmann equa- tions. Therefore, for any gravitational theory, if we can write the gravitational field equation as Gµν = 8πGTµν and regard the right-hand side as the effective energy- momentum tensor, then we find the energy flow through the AH, whereby we derive the first law of thermodynam- ics of the AH −dE = TAdSA. For example, in the Jordan frame of the scalar-tensor theory of gravity, if we take the right-hand side of gravitational field equation as the total effective energy-momentum tensor, then the Friedmann equation can be regarded as a thermodynamic identity at the AH [11]. The connection between the first law of thermodynam- ics and the Friedmann equation at the AH was also found for gravity with Gauss-Bonnet term, the Lovelock theory of gravity [6], and the braneworld cosmology [12]. For a general static spherically symmetric and stationary ax- isymmetric space-times, it was shown that Einstein equa- tion at the horizon give rise to the first law of thermody- namics [13, 14]. For the Lovelock gravity, the interpreta- tion of gravitational field equation as a thermodynamic identity was proposed in [15]. Alternatively, the mass formulae (1) can be written as the so-called unified first law ∇aM = AΨa + W∇aV [16, 17], where W = (ρ − p)/2 and Ψa = T ba r̃,b + Wr̃,a. Projecting the unified first law along the direc- tion tangent to the AH (or trapping horizon in Hay- ward’s terminology), the first law of thermodynamics dM = TdS + WdV can be derived, where the hori- zon temperature and entropy are given, respectively, by T = ✷r̃/(4π) and S = A/(4G). Based on this result, the connection between the Friedmann equation and the first law of thermodynamics of the AH with the work term was widely discussed for Einstein gravity, Love- lock’s gravity, the scalar-tensor theory of gravity, the nonlinear theory of gravity, and the braneworld scenario [18, 19, 20, 21, 22, 23]. This connection between the Friedmann equation and the first law of thermodynamics of the AH suggests the unique role of the AH in thermodynamics of cosmology. This may be used to probe the property of dark energy [10, 24]. For example, if we assume that the temperature of the dark components is T = bTA, then use the relation T = (ρ+p)/s = (ρ+p)a3/σ, we find that the total energy density of the dark components is given by ρ = ρΛ + ρ0 , (5) where ρ0 = σ 2b2Ga−60 /(6π), ρΛ = 3Λ/(8πG) is the en- ergy density of the cosmological constant, σ is the con- stant comoving entropy density, and s is the physical en- tropy density. The right-hand side of the above equation contains three different terms, which correspond to, re- spectively, the cosmological constant, the stiff fluid, and the pressureless matter. However, the coefficients of these terms are not all independent. In fact, the current obser- vational constraints tell us that the stiff fluid is negligibly small, for which we must assume ρ0 ≪ 1. This in turn im- plies that the pressureless matter given by the last term is also negligibly small. So the pressureless matter in the last term cannot account for dark matter. In other words, the dark matter must not be in equilibrium with the AH. For the BD theory [25] L = − φR + ωgµν ∂µφ∂νφ , (6) the BD scalar φ plays the role of the gravitational con- stant. The MS mass is [26] M = φr̃(1− gabr̃,ar̃,b)/2. (7) At the AH, M = φr̃A/2. The horizon entropy is SA = πr̃2Aφ, so we get TAdSA = r̃Adφ/2 + φdr̃A. (8) On the other hand, we have − dE = −ka∇aMdt = −r̃Adφ/2 + φdr̃A. (9) Comparing Eqs. (8) with (9), we find that the equilibrium thermodynamics −dE = TAdSA fails to hold for the BD theory. Similarly, it can be shown that −dE = TAdSA does not hold in the nonlinear and scalar-tensor theories of gravity. It is exactly because of this that it was argued nonequilibrium treatment might be needed. As mentioned above, the mass, temperature and en- tropy of the AH are all geometrical quantities, and the first law of thermodynamics of the AH can be regarded as a geometric relation. Now, the important question is whether a mass function exists that serves as the bridge between the Friedmann equation and the first law of ther- modynamics of the AH without nonequilibrium correc- tion. In the following, we show that the answer is affir- mative. It has exactly the dimension of energy, and is equal to the MS mass at the AH. To distinguish it with the MS mass, we call it the masslike function. To show our above claim, let us write the a − b com- ponents of the Einstein equation as M,a = −4πr̃2(T ba − δ aT )r̃,b + r̃,a, (10) where the mass-like function M is defined as (1 + gabr̃,ar̃,b). (11) At the AH, gabr̃,ar̃,b = 0 and the masslike function M = r̃A/2G, which is equal to the MS mass. For the Robertson-Walker metric we have gtt = −1, grr = a2/(1 − kr2) and r̃ = ar. Then, the mass formulas (10) yield the Friedmann equations ρ, (12) (ρ+ 3p). (13) Combining Eqs. (12) and (13), we can derive the energy conservation law ρ̇ + 3H(ρ + p) = 0. Thus, the mass formulas (10) give rise to the full set of the cosmological equations. At the AH, the masslike function M = 4πr̃3Aρ/3, which is the total energy inside the AH. The energy flow is dE = ka∇aMdt = d(r̃A)/G = TAdSA. (14) On the other hand, using the mass formulas (10), we get the energy flow through the AH dE = ka∇aMdt = −4πr̃2T ba r̃,bk = 4πr̃3AH(ρ+ p)dt. (15) Therefore, the Friedmann equation gives rise to the first law of thermodynamics dE = TAdSA of the AH. While this result is the same as that obtained by using the MS mass, we show below that the equilibrium thermodynam- ics can be derived for BD and nonlinear gravities by using our newly defined masslike function, although it cannot be done by using the MS mass, as shown above. For the BD theory, the mass-like function is defined as M ≡ φr̃(1 + gabr̃,ar̃,b)/2. (16) At the AH, it reduces to the MS mass, M = M = φr̃A/2. The a− b components of the gravitational field equation become M,a =− 4πr̃2(T ba − δ aT )r̃,b + 2πr̃ + (φr̃),a ω + 2 r̃2φ,aφ,br̃ r̃2r̃,aφ,bφ ;b − r̃r̃,ar̃,bφ;b r̃2φ;abr̃ r̃2✷r̃φ,a − φ,a✷φ. Applying to the Robertson-Walker metric, the above equation gives the Friedmann equations , (18) (ρ+ 3p)− . (19) The mass formulas (17) or Eqs. (18) and (19) are not sufficient to describe the full dynamics of the BD cosmol- ogy. In the BD cosmology, we also need the equation of motion of the BD scalar field φ in addition to Eqs. (18) and (19), which is given by φ̈+ 3Hφ̇ = 3 + 2ω (ρ− 3p). (20) From the definition of the masslike function (16), at the AH we find dE = M,ak adt = r̃Adφ/2 + φdr̃A = TAdSA, (21) where the entropy now is SA = πr̃ Aφ. Using the mass formulas (17), we get the energy flow through the AH 3 + 2ω r̃3AH [(ω + 2)ρ+ ωp] + r̃3AH − 2r̃3AH r̃Aφ̇, where we used Eq. (20) in deriving the above equation. From Eqs. (18)-(20), the right-hand side of Eq. (22) can be written as 1 r̃Aφ̇+ φ ˙̃rA. Therefore, we see that in BD theory, the first law of thermodynamics of the AH dE = TAdSA can be derived from the Friedmann equation. The thermodynamic prescription can be easily ex- tended to general scalar-tensor theory of gravity with the Lagrangian L = f(φ)R − gµν∂µφ∂νφ/2− V (φ). (23) In this case, f(φ) plays the role of the gravitational con- stant, so now we can define the mass-like function as M ≡ f(φ)r̃ 1 + gabr̃,ar̃,b /2, (24) and the horizon entropy as SA = πr̃ Af(φ). Then, using these definitions, we can show that dE = M,ak adt = TAdSA. For the nonlinear theory of gravity f(R), we can define the masslike function as f ′(R)r̃ 1 + gabr̃,ar̃,b , (25) and the horizon entropy SA = πr̃ ′(R), where f ′(R) = df/dR. Again, it is easy to show that dE = M,ak adt = TAdSA. Therefore, the thermodynamics of the AH holds for both the general scalar-tensor theory of gravity and the nonlinear theory of gravity. Now we show how to derive the first law of ther- modynamics of the AH from the Friedmann equation in the Lovelock gravity. The Lovelock Lagrangian is n=0 cnLn [27], where Ln = 2 µ1ν1···µnνn α1β1···αnβn Rα1β1µ1ν1 · · ·R Using the Robertson-Walker metric, we obtain the Fried- mann equations in N + 1 dimensional space-time N(N − 1) ρ, (26) )i−1 ( N − 1 (ρ+ p), where ĉ0 = c0/[N(N − 1)], ĉ1 = 1 and ĉi = ci j=3(N + 1−j) for i > 1. The masslike function can now be defined N(N − 1)ΩN r̃N 2r̃−2i − = ΩN r̃ A ρ, (28) where ΩN is the volume of unit N -dimensional sphere and the last equality is evaluated at the AH. Note that although the geometric form is different, the masslike function at the AH has the same value as that in Ein- stein theory of gravity, which is the total energy inside the AH. The entropy of the AH is i(N − 1) N − 2i+ 1 ĉir̃ N+1−2i A . (29) From Eqs. (28) and (29), we can easily check that dE = adt = TAdSA holds with the horizon temperature TA = 1/(2πr̃A). Using the Friedmann Eqs. (26) and (27), we find the energy flow through the AH is dE = NΩNHr̃ A (ρ+p), which is the same as that in Einstein’s gravity. By properly defining the masslike function in each the- ory of gravity, we find that the corresponding Friedmann equations can be written in the form dE = TAdSA of the first law of thermodynamics at the AH. In other words, the thermodynamic description of the gravita- tional dynamics is manifest through the mass formulas. Therefore, the gravitational dynamics can be considered as the thermodynamic identity dE = TAdSA. This is true for a variety of theories of gravity, including the Einstein, Lovelock, nonlinear, and scalar-tensor theories. This non-trivial connection between the thermodynamics of the AH and the Friedmann equation may represent a generic connection, and it suggests the unique role that the AH can play in the thermodynamics of cosmology. Such a thermodynamic description of the AH can also be used to probe other physical systems and properties, such as the nature of dark energy and the thermodynamics of black holes in each of these theories. Finally, we would like to note that, although the newly defined masslike function reduces to the MS mass at the AH, the corresponding energy flows passing through the horizon are different. This explains why our masslike function gives rise to the first law of thermodynamics in various theories of gravity, while the MS mass does not. Because of the masslike function, the energy mo- mentum tensor includes the contribution of gravitational fields such as BD scalars, or curvature scalars in non- linear theory of gravity, in addition to the matter fields. This treatment allows a reinterpretation of the nonequi- librium correction introduced in [7]. The studies of other properties of the newly-defined masslike function, includ- ing the physical and geometrical difference between the MS mass and it are important and should be reported somewhere else. Y.G. Gong is supported by NNSFC under Grants No. 10447008 and 10605042, CMEC under Grant No. KJ060502, and SRF for ROCS, State Education Min- istry. A. Wang’s work was partially supported by a VPR fund from Baylor University. ∗ [email protected] † anzhong˙[email protected] [1] J.M. Bardeen, B. Carter and S.W. Hawking, Commun. 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0704.0794

Constraints on the Interactions between Dark Matter and Baryons from the X-ray Quantum Calorimetry Experiment

Constraints on the Interactions between Dark Matter and Baryons from the X-ray Quantum Calorimetry Experiment Adrienne L. Erickcek1,2, Paul J. Steinhardt2,3, Dan McCammon4, and Patrick C. McGuire5 Division of Physics, Mathematics, & Astronomy, California Institute of Technology, Mail Code 103-33, Pasadena, CA 91125, USA Department of Physics, Princeton University, Princeton, NJ 08544, USA Princeton Center for Theoretical Physics, Princeton University, Princeton, NJ 08544, USA Department of Physics, University of Wisconsin, Madison, WI 53706, USA and McDonnell Center for the Space Sciences, Washington University, St. Louis, M0 63130, USA Although the rocket-based X-ray Quantum Calorimetry (XQC) experiment was designed for X- ray spectroscopy, the minimal shielding of its calorimeters, its low atmospheric overburden, and its low-threshold detectors make it among the most sensitive instruments for detecting or constraining strong interactions between dark matter particles and baryons. We use Monte Carlo simulations to obtain the precise limits the XQC experiment places on spin-independent interactions between dark matter and baryons, improving upon earlier analytical estimates. We find that the XQC experiment rules out a wide range of nucleon-scattering cross sections centered around one barn for dark matter particles with masses between 0.01 and 105 GeV. Our analysis also provides new constraints on cases where only a fraction of the dark matter strongly interacts with baryons. PACS numbers: 95.35.+d, 12.60.-i, 29.40.Vj I. INTRODUCTION From Vera Rubin’s discovery that the rotation curves of galaxies remain level to radii much greater than pre- dicted by Keplerian dynamics [1] to the Wilkinson Mi- crowave Anisotropy Probe (WMAP) measurement of the cosmic microwave background (CMB) temperature anisotropy power spectrum [2], observations indicate that the luminous matter we see is only a fraction of the mass in the Universe. The three-year WMAP CMB anisotropy spectrum is best-fit by a cosmological model with Ωm = 0.241 ± 0.034 and a baryon density that is less than one fifth of the total mass density. The cold collisionless dark matter (CCDM) model has emerged as the predominant paradigm for discussing the missing mass problem. The dark matter is assumed to consist of non-relativistic, non-baryonic, weakly interacting par- ticles, often referred to as Weakly Interacting Massive Particles (WIMPs). Although the CCDM model successfully predicts ob- served features of large-scale structure at scales greater than one megaparsec [3], there are indications that it may fail to match observations on smaller scales. Numerical simulations of CCDM halos [4, 5, 6, 7, 8, 9, 10, 11, 12] imply that CCDM halos have a density profile that in- creases sharply at small radii (ρ ∼ r−1.2 according to Ref. [12]). These predictions conflict with lensing ob- servations of clusters [13, 14] that indicate the presence of constant-density cores. X-ray observations of clusters have found cores in some clusters, although density cusps have also been observed [15, 16, 17]. On smaller scales, observations of dwarf and low-surface-brightness galaxies [18, 19, 20, 21, 22, 23, 24] indicate that these dark mat- ter halos have constant-density cores with lower densities than predicted by numerical simulations. Observations also indicate that cores are predominant in spiral galaxies as well, including the Milky Way [25, 26, 27]. Numerical simulations of CCDM halos also predict more satellite halos than are observed in the Local Group [28, 29] and fossil groups [30]. Astrophysical explanations for the discord between the density profiles predicted by CCDM simulations and ob- servations have been proposed: for instance, dynamical friction may transform density cusps into cores in the inner regions of clusters [31], and the triaxiality of galac- tic halos may mask the true nature of their inner den- sity profiles [32]. There are also models of substructure formation that explain the observed paucity of satellite halos [33, 34, 35, 36]. Another possible explanation for the apparent failure of the CCDM model to describe the observed features of dark matter halos is that dark matter particles scat- ter strongly off one another. The discrepancies between observations and the CCDM model are alleviated if one introduces a dark matter self-interaction that is compa- rable in strength to the interaction cross section between neutral baryons [37, 38]: = 8× 10−25 − 1× 10−23cm2 GeV−1, (1) where σDD is the cross section for scattering between dark matter particles and mdm is the mass of the dark mat- ter particle. Numerical simulations have shown that in- troducing dark matter self-interactions within this range reduces the central slope of the halo density profile and reduces the central densities of halo cores, in addition to destroying the extra substructure [39, 40]. The numerical coincidence between this dark mat- ter self-interaction cross section and the known strong- interaction cross section for neutron-neutron or neutron- proton scattering has reinvigorated interest in the pos- sibility that dark matter interacts with itself and with http://arxiv.org/abs/0704.0794v2 baryons through the strong nuclear force. We refer to dark matter of this type as “strongly interacting dark matter” where “strong” refers specifically to the strong nuclear force. Strongly interacting dark matter candi- dates include the dibaryon [41, 42], the Q-ball [43], and O-helium [44]. Surprisingly, the possibility that the dark matter may be strongly interacting is not ruled out. While there are numerous experiments searching for WIMPs, they are largely insensitive to dark matter that interacts strongly with baryons. The reason is that WIMP searches are typ- ically conducted at or below ground level based on the fact that WIMPs can easily penetrate the atmosphere or the Earth, whereas strongly interacting dark matter is multiply scattered and thermalized by the time it reaches ground level and its thermal kinetic energy is too small to produce detectable collisions with baryons in WIMP detectors. Consequently, there are few experiments ca- pable of detecting strongly interacting dark matter di- rectly. Starkman et al. [45] summarized the constraints on strongly interacting dark matter from experiments prior to 1990, and these constraints were later refined [38, 46, 47, 48]. The strength of dark matter interactions with baryons may also be constrained by galactic dynam- ics [45], cosmic rays [45, 49], Big Bang nucleosynthesis (BBN) [49], the CMB [50], and large-scale structure [50]. The X-ray Quantum Calorimetry (XQC) project launched a rocket-mounted micro-calorimeter array in 1999 [51]. At altitudes above 165 km, the XQC detector collected data for a little less than two minutes. Although its primary purpose was X-ray spectroscopy, the limited amount of shielding in front of the calorimeters and the low atmospheric overburden makes the XQC experiment a sensitive detector of strongly interacting dark matter. In this article, we present a new numerical analysis of the constraints on spin-independent interactions be- tween dark matter particles and baryons from the XQC experiment using Monte Carlo simulations of dark mat- ter particles interacting with the XQC detector and the atmosphere above it. Our work is a significant improve- ment upon the earlier analytic estimates presented by some of us in Refs. [38, 48] because it accurately models the dark matter particle’s interactions with the atmo- sphere and the XQC instrument. Our calculation here also supersedes the analytic estimate by Zaharijas and Farrar [52] because they only considered a small por- tion of the XQC data and did not include multiple scat- tering events nor the overburden of the XQC detector. We restrict our analysis to spin-independent interactions because the XQC calorimeters are not highly sensitive to spin-dependent interactions. Only a small fraction of the target nuclei in the calorimeters have non-zero spin; consequently, the bound on spin-dependent interactions between baryons and dark matter from the XQC experi- ment is about four orders of magnitude weaker than the bound on spin-independent interactions [52]. This article is organized as follows. In Section II we summarize the specifications of the XQC detector. We then review dark matter detection theory in Section III. This Section includes a discussion of coherent versus in- coherent scattering and how we account for the loss of coherence in our analysis. A complete description of our analysis follows in IV, and our results are presented in Section V. Finally, in Section VI, we summarize our findings and compare the constraints to strongly inter- acting dark matter from the XQC experiment to those from other experiments. II. THE XQC EXPERIMENT Calorimetry is the use of temperature deviations to measure changes in the internal energy of a material. By drastically reducing the specific heat of the absorbing material, the use of cryogenics in calorimetry allows the absorbing object to have a macroscopic volume and still be sensitive to minute changes in energy. These detectors are sensitive enough to register the energy deposited by a single photon or particle and gave birth to the technique of “quantum calorimetry,” the thermal measurement of energy quanta. The quantum calorimetry experiment [51] we use to constrain interactions between dark matter particles and baryons is the second rocket-born experiment in the XQC (X-ray Quantum Calorimetry) Project, a joint under- taking of the University of Wisconsin and the Goddard Space Flight Center [53, 54]. It launched on March 28, 1999 and collected about 100 seconds of data at alti- tudes between 165 and 225 km above the Earth’s surface. The detector consisted of thirty-four quantum calorime- ters operating at a temperature of 0.06 K; for detailed information on the XQC detector functions, please refer to Refs. [51, 54]. These detectors were separated from the exterior of the rocket by five thin filter panes [51]. The small atmospheric overburden at this altitude and the minimal amount of shielding in front of the calorime- ters makes this experiment a promising probe of strongly interacting dark matter. The absorbers in the XQC calorimeters are composed of a thin film of HgTe (0.96 µm thick) deposited on a sili- con (Si) substrate that is 14 µm thick. The absorbers rest on silicon spacers and silicon pixel bodies. Figs. 1 and 2 show side and top views of the detectors with the di- mensions of each layer. Temperature changes in all four components are measured by the calorimeter’s internal thermometer. The calorimeters report the average tem- perature over an integration time of 7 ms in order to reduce the effect of random temperature fluctuations on the measurement. Multiple scatterings by a dark matter particle will register as a single event because the time it takes the dark matter particle to make its way through the calorimeter is small compared to the integration time. The detector array consists of two rows of detec- tors, with seventeen active calorimeters and one inactive calorimeter in each row, and is located at the bottom of a conical detector chamber. Within a 32-degree angle Si Substrate 0.5 mm x 2.0 mm 14 µm 12 µm HgTe Absorber 0.96µm Si Pixel Body 0.25 mm x 1.0 mm Si Spacer 0.245 mm x 0.245 mm FIG. 1: A vertical cross section of an XQC calorimeter. The relative thicknesses of the layers are drawn to scale, as are their relative lengths, but the two scales are not the same. To facilitate the display of the layers, the vertical dimension has been stretched relative to the horizontal dimension. 2.0 mm 1.0 mm Absorber Panel (HgTe on Si) Pixel Body (Si) Spacer (Si) 0.245 mm x 0.245 mm FIG. 2: A top view of an XQC calorimeter. The absorber is the top layer and underneath it lies the spacer, followed by the pixel body. These dimensions are drawn to scale. from the detector normal, the incoming particles only pass through the aforementioned filters. The five filters are located 2 mm, 6 mm, 9 mm, 11 mm and 28 mm above the detectors. Each filter consists of a thin layer of alu- minum (150 Å) supported on a parylene (CH) substrate (1380 Å). The pressure inside the chamber is less than 10−6 Torr. At this level of evacuation, a dark matter particle with a mass of 106 GeV and a baryon interac- tion cross section of 106 barns, would have less than a 20% chance of colliding with an air atom in the cham- ber. Therefore, we assume that the chamber is a perfect vacuum in our analysis. While the atmospheric pressure at the altitudes at which the detector operated is about 10−8 times the at- mospheric pressure at sea level, the atmospheric over- burden of the XQC detector is still sufficient to scat- ter incoming strongly interacting dark matter particles. Simulating a dark matter particle’s path through the atmosphere requires number-density profiles for all the molecules in the atmosphere. These profiles were ob- tained using the MSIS-E-90 model1 for the time (1999 1 Available at http://modelweb.gsfc.nasa.gov/models/msis.html FIG. 3: The points depict the MSIS-E-90 density profiles for the seven most prevalent constituents of the atmosphere above the XQC detector, and the lines show the piecewise exponen- tial fits used in our analysis. March 28 9:00 UT) and location (White Sands Missile Range, New Mexico) of the XQC rocket launch. During the data collection period, the average altitude of the XQC rocket was 201.747 km. At this altitude and above, the primary constituents of the atmosphere are molecular and atomic oxygen, molecular and atomic ni- trogen, helium, atomic hydrogen, and argon. The MSIS- E-90 model provides tables of the number densities of each of these seven chemical species. In our analysis, computational efficiency demanded that we fit analytic functions to these data. We found exponential fits for the density profiles in three altitude ranges: 200-300 km, 300-500 km and 500-1000 km. The error in the proba- bility of a collision between a dark matter particle and an element of the atmosphere introduced by using these fits instead of the original data is 0.02%. Fig. 3 shows the number density profiles provided by the MSIS-E-90 model and the exponential fits used to model the data. The XQC detector collected data for a total of 150 seconds. During these 150 seconds of activity, the thirty- four individual calorimeters were not all operational at all times. Furthermore, events that could not be ac- curately measured by the calorimeters and events at- tributed to cosmic rays hitting the base of the detec- tor array were removed from the XQC spectrum, and these cuts also contribute to the dead time of the sys- tem. Specifically, events that arrived too close together for the calorimeters to accurately measure distinct ener- gies were discarded. This criterion removed 12% of the observed events and the resulting loss of sensitivity was http://modelweb.gsfc.nasa.gov/models/msis.html FIG. 4: Top panel: The XQC energy spectrum from 0 - 4 keV in 5 eV bins. This spectrum does not have non-linearity corrections applied (see Ref. [51]), so the calibration lines at 3312 eV and 3590 eV appear slightly below their actual en- ergies. The cluster of counts to the left of each calibration peak result from X-rays passing through the HgTe layer and being absorbed in the Si components where up to 12% of the energy may then be trapped in metastable states. Bottom panel: The XQC energy spectrum from 0 - 2.5 keV in 5 eV bins. This spectrum, combined with the over-saturation rate of 0.6 events per second with energies greater than 4000 eV, was used in our analysis. included in the dead time of the calorimeters. When a cosmic ray penetrates the silicon base of the detector ar- ray, the resulting temperature increase is expected to reg- ister as multiple, nearly simultaneous, low-energy events on nearby calorimeters. To remove these events from the spectrum, we cut out events that were part of either a pair of events in adjacent detectors or a trio of events in any of the detectors that arrived within 3 ms of each other and had energies less than 2.5 keV. This procedure was expected to remove more than 97% of the events that resulted from cosmic rays hitting the base of the array. Nearly all of the events attributed to heating from cosmic rays had energies less than 300 eV, and a high fraction of the observed low-energy events were included in this cut. For example, seventeen of the observed twenty-four events with energies less than 100 eV were removed. The expected loss of sensitivity due to events being falsely attributed to cosmic rays was included in the calculated dead time of the calorimeters. Once all the dead time is accounted for, the 150 seconds of data collection is equiv- alent to 100.7 seconds of observation with all thirty-four calorimeters operational. The XQC calorimeters are capable of detecting energy deposits that exceed 20 eV, but full sensitivity is not reached until the energy surpasses 36 eV, and for approx- imately half of the detection time, the detector’s lower threshold was set to 120 eV. The calorimeters cannot re- solve energies above 4 keV, and the 2.5-4 keV spectrum is dominated by the detector’s interior calibration source: a ring of 2µCi 41Ca that generates Kα and Kβ lines at 3312 eV and 3590 eV, respectively. We refer the reader to Ref. [51] for a complete discussion of the calibration of the detector. These limitations restrict the useful portion of the XQC spectrum to 0.03-2.5 keV. This spectrum is shown in Fig. 4, along with the full spectrum from 0-4 keV. The XQC field of view was centered on a region of the sky known to have an enhanced X-ray background in the 100-300 eV range, possibly due to hot gas in the halo, and this surge in counts can be seen in Fig. 4. In addition to the information present in this spectrum, we know that the XQC detector observed an average over- saturation event rate of 0.6 per second. This corresponds to a total of 60 events that deposited more than 4000 eV in a calorimeter. In Section IVB, we describe how we use the observed spectrum between 29 eV and 2500 eV and the integrated over-saturation rate to constrain the total cross section for elastic scattering between dark matter particles and nucleons. III. DETECTING DARK MATTER A. Incidence of dark matter particles The expected flux of dark matter particles into the de- tector depends on the density of the dark matter halo in the Solar System. Unfortunately, the local dark matter density is unknown and the range of theoretical predic- tions is wide. By constructing numerous models of our galaxy with various dark matter density profiles and halo characteristics, rejecting those models that contradict ob- servations, and finding the distribution of local dark mat- ter densities in the remaining viable models, Ref. [55] pre- dicted that the local dark matter density is between 0.3 and 0.7 GeV cm−3 assuming that the dark matter halo is flattened, and the predicted local density decreases as the halo is taken to be more spherical. Another approach [56] used numerical simulations of galaxies similar to our own to find the dark matter density profile and then fit the profile parameters to Galactic observations, predict- ing a mean local dark matter density between 0.18 GeV cm−3 and 0.30 GeV cm−3. Given that it lies in the in- tersection of these two ranges, we use the standard value of 0.3 GeV cm−3 for the local dark matter density in our primary analysis. This assumption ignores the possi- ble presence of dark matter streams or minihalos, which do occur in numerical simulations [56] and could lead to local deviations from the mean dark matter density. We also assume that the velocities of the dark matter particles with respect to the halo are isotropic and have a bounded Maxwellian distribution: the probability that a particle has a velocity within a differential volume in velocity-space centered around a given velocity ~v is P (~v) = d3~v if v ≤ vesc, 0 if v > vesc. where v0 is the dispersion velocity of the halo, vesc is the Galactic escape velocity at the Sun’s position, and k is a normalization factor [57]: k = (πv20) . (3) Numerical simulations indicate that dark matter particle velocities may not have an isotropic Maxwellian distri- bution [56]. Ref. [58] examines how assuming a more complicated velocity distribution would alter the flux of dark matter particles into an Earth-based detector. Given the flat rotation curve of the spiral disk at the Sun’s radius and beyond and assuming a spherical halo, the local dispersion speed v0 is the maximum rotational velocity of the Galaxy vc [59]. Reported values for the rotational speed include 222±20 km s−1 [60], 228±19 km s−1 [61], 184± 8 km s−1 [62] and 230 ± 30 km s−1 [63]. Recent measurements of the Galaxy’s angular velocity have yielded values of Ωgal = 28 ± 2 km s−1 kpc−1 [64] and 32.8 ± 2 km s−1 kpc−1 [65]. If the Sun is located 8.0 kpc from the Galactic center, these angular velocities correspond to tangential velocities 224± 16 km s−1 and 262 ± 16 km s−1 respectively. We adopt vc = 220 ± 30 km s−1 as a centrally conservative value for the Galaxy’s circular velocity at the Sun’s location. The final parameter we need to obtain the dark mat- ter’s velocity distribution is the escape velocity in the So- lar vicinity. The largest observed stellar velocity at the Sun’s radius in the Milky Way is 475 km s−1, which es- tablishes a lower bound for the local escape velocity [66]. Ref. [67] used the radial motion of Carney-Latham stars to determine that the escape velocity is between 450 and 650 km s−1 to 90% confidence, and Ref. [68] obtained a 90% confidence interval of 498 to 608 km s−1 from ob- servations of high-velocity stars. A kinematic derivation of the escape velocity [59] gives v2esc = 2v 1 + ln , (4) where R0 is the distance from the Sun to the center of the Galaxy, and Rgal is radius of the Galaxy. Observations of other galaxies suggest that our galaxy extends to about 100 kpc [59], and observations of Galactic satellites indi- cate that the Galaxy’s flat rotation curve extends to at least 110 kpc [63]. The commonly accepted value for the Solar radius is R0 = 8.0 kpc [69]. Recent measurements include R0 = 7.9± 0.3 kpc [70] and R0 = 8.01± 0.44 kpc [71], and a compilation of measurements over the past decade [71] yields an average value of R0 = 7.80 ± 0.33 kpc. To estimate the escape velocity, we use 100 kpc as a conservative estimate of the Galactic radius and the standard value R0 = 8.0 kpc. These parameters, com- bined with vc = 220 km s −1, predict an escape velocity of 584 km s−1, which falls near the middle of the ranges proposed in Refs. [67, 68]. The isotropic Maxwellian velocity distribution given by Eq. (2) specifies the dark matter particles’ motion relative to the halo. However, we are interested in their motion relative to the XQC detector: ~vobserved = ~vdm − ~vdetector where the latter two velocities are measured with respect to the halo. The velocity of the detector with respect to the halo has three components: the velocity of the Sun relative to halo, the velocity of the Earth with respect to the Sun, and the velocity of the detector with respect to the Earth. When discussing these velocities, it is useful to de- fine a Galactic Cartesian coordinate system. In Galactic coordinates, the Sun is located at the origin, and the xy-plane is defined by the Galactic disk. The x-axis points toward the center of the Galaxy, and the y-axis points in the direction of the Sun’s tangential velocity as it revolves around the Galactic center. The z-axis points toward the north Galactic pole and is antiparal- lel to the angular momentum of the rotating disk. The motion of the Sun through the halo has two compo- nents. First, there is the Sun’s rotational velocity as it orbits the Galactic center: vc in the y direction. Sec- ond, there is the motion of the Sun relative to the spiral disk [72]: ~v⊙ = (10.00 ± 0.36, 5.25 ± 0.62, 7.17 ± 0.38) km s−1 in Galactic Cartesian coordinates. When the Earth’s motion through the Solar System during its an- nual orbit of the Sun is expressed in Galactic coor- dinates [57], the resulting velocity at the time of the XQC experiment (7.3 days after the vernal equinox) is ~vEarth = (29.14, 5.330,−3.597) km s−1. The final consideration is the velocity of the detector relative to the Earth. The maximum velocity attained by the XQC rocket was less than 1.2 km s−1. This velocity is insignificant compared to the motion of the Sun relative to the halo. Moreover, the XQC detector collected data while the rocket rose and while it fell, and the average velocity of the rocket was only 0.104 km s−1. Therefore, we neglect the motion of the rocket in the calculation of the dark matter wind. Combining the motion of the Sun and the Earth then gives the total velocity of the XQC detector with respect to the halo during the experiment in Galactic Cartesian coordinates: ~vdetector = (39.14 ± 0.36, 230.5 ± 30, 3.573 ± 0.38) km s−1. Subtracting the velocity vector of the detector relative to the halo from the velocity vector of the dark matter relative to the halo gives the dark matter’s velocity relative to the detector in Galactic coordinates. However, we want the dark matter particles’ velocities in the coordinate frame defined by the detector, where the z-axis is the field-of-view vector. The XQC field of view was centered on l = 90◦, b = +60◦ in Galactic latitude and longitude [51], so the rotation from Galactic coordinates to detector coordinates may be described as a clockwise 30◦ rotation of the z-axis around the x-axis, which is taken to be the same in both coordinate systems. B. Dark Matter Interactions Calorimetry measures the kinetic energy transferred from the dark matter to the absorbing material without regard for the specific mechanism of the scattering or any other interactions. Consequently, the dark matter detection rate for a calorimeter depends only on the mass of the dark matter particle and the total cross section for elastic scattering between the dark matter particle and an atomic nucleus of mass number A, which is proportional to the cross section for dark matter interactions with a single nucleon (σDn). The calorimeter measures the recoil energy of the target nucleus (mass mT), Erec = 2mTmdm (mT +mdm)2 (1 − cos θCM), (5) where mdm and vdm are dark matter particle’s mass and velocity prior to the collision in rest frame of the target nucleus and θCM is the scattering angle in the center-of- mass frame. If the momentum transferred to the nucleus, q2 = 2mTErec, is small enough that the corresponding de Broglie wavelength is larger than the radius R of the nucleus (qR ≪ ~), then the scattering is coherent. In co- herent scattering, the scattering amplitudes for each in- dividual component in the conglomerate body are added prior to the calculation of the cross section, so the total cross section is proportional to the square of the mass number of the target nucleus. Including kinematic fac- tors [45, 73], the cross section for coherent scattering off a nucleus is given by σcoh(A) = A mred(DM,Nuc) mred(DM, n) σDn, (6) where mred(DM,Nuc) is the reduced mass of the nucleus and the dark matter particle, mred(DM, n) is the reduced mass of a nucleon and the dark matter particle, and A is the mass number of the nucleus. Coherent scattering is isotropic in the center-of-mass frame of the collision. Dark matter particles may be massive and fast-moving enough that the scattering is not completely coherent when the target nucleus is large [74]. When the scat- tering is incoherent, the dark matter particle “sees” the internal structure of the nucleus, and the cross section for scattering is reduced by a “form factor,” which is a func- tion of the momentum transferred to the nucleus during the collision (q) and the nuclear radius (R): F 2(q, R). (7) Since q depends on the recoil energy, which in turn de- pends on the scattering angle, incoherent scattering is not isotropic. In this discussion of coherence, we have neglected the possible effects of the dark matter particle’s internal structure by assuming that σDn is independent of recoil energy. If the dark matter particle is not point-like then σDn decreases as the recoil momentum increases due to a loss of coherence within the dark matter particle. Inco- herence within the dark matter particle has observational consequences [75], but these effects depend on the size of the dark matter particle. To avoid restricting ourselves to a particular dark matter model, we assume that the dark matter particle is small enough that nucleon scat- tering is always coherent; when we discuss incoherence, we are referring to the effects of the nucleus’s internal structure. According to the Born approximation, the form fac- tor for nuclear scattering defined in Eq. (7) is the Fourier transform of the nuclear ground-state mass den- sity [57, 76]. The most common choice for the form factor [74, 77] is F 2(q, R) = exp[−(qRrms)2/(3~2)], where Rrms is the root-mean-square radius of the nucleus. For a solid sphere, R2rms = (3/5)R 2, so this form factor is equivalent to the form factor used in Ref. [52]. This form factor is an accurate approximation of the Fourier transform of a solid sphere for (qR)/~ ∼< 2, but it grossly underesti- mates the reduction in σ for larger values of q [57]. The maximum speed of a dark matter particle with respect to the XQC detector is ∼ 800 km s−1 (escape velocity + detector velocity), and at that speed, the maximum possible value of qR/~ for a collision with a Hg nucleus (A = 200) is nearly ten for a 100 GeV dark matter parti- cle, and the maximum possible value of qR/~ increases as the mass of the dark matter particle increases. Clearly, this approximation is not appropriate for a large portion of the dark matter parameter space probed by the XQC experiment. Furthermore, a solid sphere is not a very realistic model of the nucleus. A more accurate model of the nuclear mass density is ρ(r) = d3r′ρ0(r ′)ρ1(r− r′), where ρ0 is constant inside a radius R20 = R 2 − 5s2 and zero beyond that radius and ρ1 = exp[−r2/(2s2)], where s is a “skin thickness” for the nucleus [78]. The resulting form factor F (q, R) = 3 sin(qR/~)− (qR/~) cos(qR/~) (qR/~)3 × exp (qs/~)2 . (8) We follow Ref. [57] in setting the parameters in Eq. (8): s = 0.9 fm and R2 = [(1.23A1/3 − 0.6)2 + 0.631π2 − 5s2] fm2, (9) where A is the mass number of the target nucleus. Despite its simple analytic form, the form factor given by Eq. (8) is computationally costly to evaluate repeat- edly. We use an approximation: F 2 = 0.9 fm if qR 9(0.81) (qR/~)4 0.9 fm if qR The low-q approximation combines the standard approx- imation for the solid sphere with the factor accounting for the skin depth of the nucleus. The high-q approxi- mation was derived from the asymptotic form of the first spherical Bessel function and normalized so that the to- tal cross section is as close as possible to the exact result. The error in the total cross section due to the use of the approximation is less than 1% for nearly all dark mat- ter masses; the sole exception is mdm ∼ 10 − 100 GeV, and even then the error is less than 5%. Unless other- wise noted, we use this approximation for the form factor throughout this analysis. We also assume that the dark matter particle does not interact with nuclei in any way other than elastic scattering. IV. ANALYSIS OF XQC CONSTRAINTS To obtain an accurate description of the XQC experi- ment’s ability to detect strongly interacting dark matter particles, we turned to Monte Carlo simulations. The Monte Carlo code we wrote to analyze the XQC experi- ment simulates a dark matter particle’s journey through the atmosphere to the XQC detector chamber, its path through the detector chamber to a calorimeter, and its in- teraction with the sensitive components of the calorime- ter. This latter portion of the code also records how much energy the particle deposits in the calorimeter through scattering. The results of several such simulations for the same set of dark matter properties may be used to pre- dict the likelihood that a given dark matter particle will deposit a particular amount of energy into the calorime- ter. These probabilities of various energy deposits predict the recoil-energy spectrum the XQC detector would ob- serve if the dark matter particles have a given mass and nucleon-scattering cross section. This simulated spec- trum may then be compared to the XQC data to find which dark matter parameters are excluded by the XQC experiment. A. Generating Simulated Energy-Recoil Spectra The basic subroutine in our Monte Carlo algorithm is the step procedure. The step procedure begins with a particle with a certain velocity vector and position in a given material and moves the particle a certain dis- tance in the material, returning its new position and ve- locity. The step procedure also determines whether or not a scattering event occurred during the particle’s trek and updates the velocity accordingly. The number of ex- pected collisions in a step of length l through a material with target number density n is n× σtot × l, where σtot is the total scattering cross section obtained by integrat- ing Eq. (7) over the scattering angle, or equivalently, the recoil momentum q: σtot = q2max F 2(q, R) dq2, (11) where qmax is the maximum possible recoil momentum. The step length l is chosen so that it is at most a tenth of the mean free path through the material, so the num- ber of expected collisions is less than one and represents the probability of a collision. After each step, a ran- dom number between zero and one is generated using the “Mersenne Twister” (MT) algorithm [79] and if that random number is less than the probability of a collision, the particle’s energy and trajectory are updated. First, a recoil momentum is selected according to the probability distribution P (q2) = F 2(q, R)σcoh/(q maxσtot), where the exact form factor is used for qR/~ > 2 so that the oscilla- tory nature of the form factor is not lost. The recoil mo- mentum determines the recoil energy and the scattering angle in the center-of-mass frame through Eq. (5). The scattering is axisymmetric around the scattering axis, so the azimuthal angle is assigned a random value between 0 and 2π. The scattering angles are used to update the particle’s trajectory, and its speed is decreased in accor- dance with the kinetic energy transferred to the target nucleus. The step subroutine repeats until the particle exits the simulation, or its kinetic energy falls below 0.1 eV, or the energy deposited in the calorimeter exceeds the saturation point of 4000 eV. Our simulation treats the atmosphere as a 4.6×4.6 cm square column with periodic boundary conditions, the bottom face of which covers the top of the conical detec- tor chamber described in Section II. This implementation assumes that for every particle that exits one side of the column, there is a particle that enters the column from the opposite side with the same velocity. The infinite extent of the atmosphere and its translational invariance makes this assumption reasonable. The atmosphere col- umn extends to an altitude 1000 km; increasing the at- mosphere height beyond 1000 km has a negligible effect on the total number of collisions in the atmosphere. The simulation begins with a dark matter particle at the top of the atmosphere column at a random initial position on the 4.6×4.6 cm square. Its initial velocity with re- spect to the dark matter halo is selected according to the isotropic Maxwellian velocity distribution function given by Eq. (2), and then the velocity relative to the detector is found via the procedure described in Section IIIA. The dark matter particle’s path from the top of the at- mosphere to the detector is modeled using the step pro- cedure described above. The simulation of the particle’s interaction with the atmosphere ends if the particle’s al- titude exceeds 1000 km or if the particle falls below the height of the XQC rocket. We use the time-averaged alti- tude (201.747 km) as the constant altitude of the rocket. We made this simplification because it allows us to ig- nore the periodic inactivity of each calorimeter and treat the detector as thirty-four calorimeters that are active for 100.7 seconds. When the dark matter particle hits the rocket, its path through the five filter layers is also mod- eled using the step procedure, as is its path through the calorimeters. In addition to being smaller than the mean free path, the step length is chosen so that the particle’s position relative to the boundaries of the detectors is ac- curately modeled. The simulation ends when the dark matter particle’s random-walk trajectory takes it out of the detector chamber. As mentioned in Section II, the calorimeter detects the sum of all the recoil energies if the dark matter particle is scattered multiple times. When the dark matter particle is unlikely to experience more than one collision in the calorimeter, this simula- tion is far more detailed than is required to accurately predict the energy deposited by the dark matter parti- cle. This is the case for the lightest (mdm ≤ 102 GeV) and weakest-interacting (σDn ≤ 10−26 cm2) dark mat- ter particles that the XQC calorimeters are capable of detecting. Since the lightest dark matter particles are also the most numerous, many Monte Carlo trials are required to sample all the possible outcomes of a dark matter particle’s encounter with the detector. The sim- ulation described above is too computationally intensive to run that many trials, so we used a faster and simpler simulation to model the interactions of these dark matter particles. This simulation assumes that the particle will experience at most one collision in the atmosphere and at most one collision in each filter layer and each layer of the calorimeter. The simulation ends if the probability of two scattering events in either the atmosphere or any of the filter layers exceeds 0.1. Instead of tracking the dark matter particle’s path through the atmosphere, the total overburden for the atmosphere is used to determine the probability that the dark matter particle scatters in the atmosphere, and the particle only reaches the detec- tor if its velocity vector points toward the detector after the one allowed scattering event. Also, instead of the small step lengths required to accurately model the ran- dom walk of a strongly interacting particle, each layer is crossed with a single step. These simplifications reduce the runtime of the simulation by a factor of 100, making it possible to run 1010 trials in less than one day. B. Comparing the Simulations to the XQC Data In order to compare the probability spectra produced by our Monte Carlo routine to the results of the XQC experiment, we must multiply the probabilities by the number of dark matter particles that are encountered by the initial surface of the Monte Carlo routine. When the initial velocity of the dark matter particle is chosen, the initial velocity may point toward or away from the detec- tor; in the latter case, the trial ends immediately. Con- FIG. 5: Simulated event spectra for dark matter particles with masses of 1, 10 and 100 GeV and a total nucleon-scattering cross section of 10−27.3 cm2. In addition to the events de- picted in these spectra, the simulations predict 1300 ± 160 events with energies greater than 4000 eV when mdm = 10 GeV and 10, 000 ± 1200 such events when mdm = 100 GeV. The histogram represents the XQC observations. sequently, the Monte Carlo probability that the particle deposits no energy in the calorimeter already includes the probability that the dark matter particle does not have a halo trajectory that takes it into the atmosphere. There- fore, the probabilities resulting from the Monte Carlo routine should be multiplied by the number of particles in the volume swept out by the initial 4.6×4.6 cm2 square surface during the 100.7f(E) seconds of observation time, where f is the fraction of the observing time that the XQC detector was sensitive to deposits of energy E. For energies between 36 and 88 eV, f is 0.5083, and the value of f increases to one over energies between 88 and 128 eV. The detector was also slightly sensitive to lower energies: between 29 and 35 eV, f increases from 0.3815 to 0.5083. The normal of the initial surface points along the detec- tor’s field of view, and the surface moves with the detec- tor; using the detector velocity given in Section IIIA, the number of dark matter particles encountered by the ini- tial surface is Ndm = f × (ρdm/mdm)× [(2.5± 0.3)× 1010 cm3], where ρdm is the local dark matter density. The simulated event spectra produced by our Monte Carlo routine indicate that particles with masses less than 1 GeV very rarely deposit more than 100 eV inside the XQC calorimeters. Conversely, particles with masses greater than 100 GeV nearly always deposit more than 4000 eV when they interact with the XQC calorimeters, so constraints on σDn for these mdm values arise from the FIG. 6: Simulated event spectra for 10-GeV dark mat- ter particles with total nucleon-scattering cross sections of 10−21.6, 10−27.3 and 10−28.3 cm2. In addition to the events depicted in these spectra, the simulations predict 140 ± 37 events with energies greater than 4000 eV when σDn = 10 −21.6 cm2, 1300 ± 160 such events when σDn = 10 −27.3 cm2, and 120±15 such events when σDn = 10 −28.3 cm2. The histogram represents the XQC observations. over-saturation (E ≥ 4000 eV) event rate. Fig. 5 shows simulated spectra for three mdm values that lie between these two extremes, along with a histogram that depicts the XQC observations. Given an initial velocity of 300 km s−1 relative to the XQC detector, a 1-GeV particle can only deposit up to 66 eV in a single collision with an Si nucleus, so the spectrum for these particles is confined to very low energies. Meanwhile, a 10-GeV particle and a 100-GeV particle with the same initial velocity can de- posit up to 900 eV and 44,000 eV, respectively, in a single collision with an Hg nucleus. In fact, ignoring any loss of coherence, all recoil energies between 0 and 44,000 eV are equally likely during a collision between an Hg nu- cleus and a 100-GeV dark matter particle. That’s why the mdm = 100 GeV spectrum in Fig. 5 is flat below 2500 eV and why the simulations predict 10,000 events with energies greater than 4000 eV for this value of mdm and Fig. 6 shows how changing the total cross section for elastic scattering off a nucleon affects the simulated spec- tra generated by our Monte Carlo routine for a single dark matter particle mass (mdm = 10 GeV). We see that increasing σDn from 10 −28.3 cm2 to 10−27.3 cm2 increases all of the counts by a factor of ten but leaves the ba- sic shape of the spectrum unchanged. For much larger values of σDn, however, the particle loses a considerable Energy Range (eV) Counts Energy Range (eV) Counts 29 - 36 0 945 - 1100 31 36 - 128 11 1100 - 1310 30 128 - 300 129 1310 - 1500 29 300 - 540 80 1500 -1810 32 540 - 700 90 1810 - 2505 15 700 - 800 32 ≥ 4000 60 800 - 945 48 TABLE I: The binned XQC results used for comparison with our Monte Carlo simulations. amount of its energy while traveling through the atmo- sphere. Consequently, high-energy recoil events become less frequent, as shown by the spectrum for σDn = 10 −21.6 cm2. For larger values of σDn, too much energy is lost in the atmosphere for the particle to be detectable by the XQC experiment. When comparing the simulated measurements to the XQC data, we group the events into the thirteen energy bins given in Table I. We generally use large bins be- cause it reduces the fractional error in the probabilities generated by our Monte Carlo routine by increasing the probability of each bin: δpi/pi = 1/ pit, where t is the number of trials and pi is the probability of an energy deposit in the ith bin. Given that the number of trials is limited by runtime constraints, increasing the bin size is often the only way to obtain bin probabilities with δpi/pi values much less than one. When choosing our binning scheme, we attempted to maximize bin size while pre- serving as many features of the observed spectrum as possible. We also grouped all energies for which f 6= 1 into two bins; we ignore the variation in f within these bins and set f = 0.3815 in the lowest-energy bin and f = 0.5083 in the next-to-lowest bin. Unfortunately, we do not know the number of X-ray events in any of the bins listed in Table I. We considered using a model to subtract off the X-ray background but, given any model’s questionable accuracy, we decided not to use it in our analysis. Our ignorance of the X-ray background forces us to treat the number of observed counts in each bin as an upper limit on the number of dark matter events in that energy range. Consequently, we define a parameter X2 that measures the extent of the discrepancy between the simulated results for a given mdm and σDn and the XQC observations while ignoring bins in which the observed event count exceeds the pre- dicted contribution from dark matter: i=# of Bins (Ei − Ui)2 with Ui < Ei , (12) where Ei = Ndm × pi is the number of counts in the ith bin predicted by the Monte Carlo simulation and Ui is the number of observed counts in the same bin. We use a second Monte Carlo routine to determine how likely it is that a set of observations would give a value of X2 as large or larger than the one derived from the XQC data given a mean signal described by the set of Ei derived from the simulation Monte Carlo. In the comparison Monte Carlo, a trial begins by gen- erating a new set of Ei by sampling the error distribu- tions of Ndm and pi. The distribution of Ndm values is assumed to be Gaussian with the mean and standard de- viation given above. The probability pi is derived from pi× t events in the simulation Monte Carlo (recall that t is the number of trials), so a new value for pi is generated by sampling a Poisson distribution with a mean of pi × t and dividing the resulting number by t. Once a new set of Ei has been found, the routine generates a simulated number of observed counts for each bin according to a Poisson distribution with a mean of Ei. The value of the X2 parameter for the new Ei and Ui is computed and compared to the value for the original Ei and the XQC observations, X2XQC. The number of trials needed to ac- curately measure the probability P(X) thatX2 ≥ X2XQC is determined by requiring that the variation in the mean value of X2 over ten Monte Carlo simulations does not exceed (100-C)%, where C% is the desired confidence level and that the range P(X)±(5× the variation in P(X)) does not contain (100− C)/100. V. RESULTS AND DISCUSSION The XQC experiment rules out the enclosed region in (mdm, σDn) parameter space shown in Fig. 7. The over- burden from the atmosphere and the filtering layers as- sures that there will be a limit to how strongly a dark matter particle can interact with baryons and still reach the XQC calorimeters; this overburden is responsible for the top edge of the exclusion region. Conversely, if σDn is too small, the dark matter particles will pass through the calorimeters without interacting. The low-energy thresh- old of the XQC calorimeters places a lower bound on the excluded dark matter particle masses; ifmdm is too small, then the recoil energies are undetectable. On the other side of the mass range, the XQC detector is not sensitive to mdm ∼> 105 GeV because the number density of such massive dark matter particles is too small for the XQC experiment to detect. The exclusion region shown in Fig. 7 has a complicated shape, but its features are readily explicable. As mdm in- creases, the range of excluded σDn values shifts to lower values and then moves up again. The downward shift for mdm between 0.1 GeV and 100 GeV is due to the effects of coherent nuclear scattering. Since σcoh increases with in- creasing mdm for fixed σDn, a 100-GeV particle interacts more strongly in the atmosphere and in the detector than a 1-GeV particle with the same σDn. Consequently, both the upper and lower boundaries of the excluded region decrease with increasing mass for mdm ∼< 100 GeV. The scattering of dark matter particles with larger masses is incoherent, and the form factor discussed in Section III B FIG. 7: The region of dark matter parameter space excluded by the XQC experiment; σDn is the total cross section for scat- tering off a nucleon and mdm is the mass of the dark matter particle. This exclusion region follows from the assumption that the local dark matter density is 0.3 GeV cm−3 and that all of the dark matter shares the same value of σDn. causes σtot to decrease as mass increases for fixed σDn. Moreover, particles that are more massive than the target nuclei have straighter trajectories than lighter dark mat- ter particles due to smaller scattering angles in the de- tector rest frame. The loss of coherence also contributes because incoherent scattering makes small scattering an- gles more probable. A straight trajectory is shorter than a random walk, so the more massive particles interact less in the atmosphere and the detector than the more easily-deflected lighter particles. Due to both of these effects, the upper and lower boundaries of the exclusion region increase with increasingmdm for mdm ∼> 100 GeV. The lower left corner of the exclusion region also has two interesting features. First, the lower bound on the excluded value of σDn decreases sharply as mdm increases from 0.1 GeV to 0.5 GeV. A dark matter particle with the maximum possible velocity with respect to the detector (800 km s−1) must have a mass greater than 0.24 GeV to be capable of depositing 29 eV in the calorimeter in a single collision. Lighter particles are only detectable if they scatter multiple times inside the calorimeter, and multiple scatters require a higher value of σDn. Since their analysis does not allow multiple collisions, the XQC exclusion region found in Ref. [52] does not extend to masses lower than 0.3 GeV for any value of σDn. Second, there is a kink in the lower boundary at mdm = 10 GeV; the constraint on σDn is not as strong for this mass. The FIG. 8: The region of dark matter parameter space excluded to 90% confidence by the XQC experiment for several values of the local density of dark matter with a total nucleon scattering cross section σDn and mass mdm. The four densities shown are 0.3 GeV cm−3 (solid line), 0.15 GeV cm−3 (long dashed line), 0.075 GeV cm−3 (short dashed line) and 0.03 GeV cm−3 (dotted line). simulated spectra produced by our Monte Carlo routine for mdm = 10 GeV and σDn ∼< 10−25 cm2 reveal that the particle is most likely to deposit between 100 and 600 eV, as exemplified by the spectra depicted in Fig. 6. The background in this energy range is very high, so the XQC constraints are not as strict at these energies. Altering the local density of dark matter that strongly interacts with baryons changes the exclusion region. Fig. 8 shows the 90%-confidence exclusion regions for four values of the local density of dark matter particles with strong baryon interactions: 0.3 GeV cm−3 (solid line), 0.15 GeV cm−3 (long dashed line), 0.075 GeV cm−3 (short dashed line) and 0.03 GeV cm−3 (dotted line). These different local densities could arise due to varia- tions in the local dark matter density due to mini-halos or streams. They also describe models where the dark matter does not consist of a single particle species and the dark matter that strongly interacts with baryons is a fraction fd of the local dark matter. In that case, the four exclusion regions in Fig. 8 correspond to fd =1, 0.5, 0.25, and 0.1. Fig. 8 indicates that the top and left boundaries of the XQC exclusion region are not highly sensitive to the dark matter density. In particular, the upper left corner of the exclusion region (0.01 ≤ mdm ≤ 0.1 GeV) is nearly un- affected by lowering the dark matter density. This con- sistency indicates our Monte Carlo-generated exclusion region is smaller than the true exclusion region in this corner. If the dark matter is light (mdm ∼< 0.1 GeV), then the number of dark matter particles encountered by the XQC detector is very large (Ndm ∼> 7 × 1010). As previously mentioned, the upper left corner of the XQC exclusion region results from multiple scattering events, so the simpler version of our Monte Carlo code described in Section IVA is not applicable. Consequently, it is not possible to run more than 109 trials in a week, so each scattering event in the simulation corresponds to more than one scattering event in the detector for all the den- sities shown in Fig. 8. Therefore, decreasing the density does not change the result. If it were possible to run 1011 trials, then the upper left corner of the exclusion region would expand and differences between the different den- sity contours would emerge. Since the upper left corner of the XQC exclusion region is already ruled out by astro- physical constraints (see Fig. 9), we have not invested in the computational time necessary to expand this corner. The upper boundary of the exclusion region is also not greatly affected by decreasing the particle density, even when Ndm is small enough that the Monte Carlo routine is capable of running more than Ndm trials (mdm ≥ 100 GeV). This robustness indicates that the overburden of the XQC experiment effectively prevents all dark matter particles with σDn values greater than the upper bound of the exclusion region from reaching the detector, so it does not matter how many particles are encountered. Finally, as discussed previously, the lower portion of the exclusion region’s left boundary (σDn ≤ 10−23 cm2) is set by the energy threshold for detection and is therefore independent of Ndm. Examining the features of the excluded region allows us to predict how the region may be expanded by a fu- ture XQC-like experiment. Decreasing the overburden by either increasing the rocket’s altitude or reducing the fil- tering will push the top boundary of the excluded region upwards. Decreasing the energy detection threshold will extend the excluded region to lower masses. It may also extend the exclusion region to higher values of σDn for all masses since strongly interacting particles lose much of their energy in the atmosphere and arrive at the calorime- ter with too little energy to produce a detectable sig- nal. Increasing the size or number of calorimeters would increase the sensitivity and extend the excluded region to lower values of σDn. Finally, increasing the observa- tion time would increase Ndm, and that would extend the right and bottom boundaries of the excluded region. VI. CONCLUSION The X-ray Quantum Calorimetry (XQC) experiment is a powerful detector of dark matter that interacts strongly with baryons due to its high altitude and minimal shield- ing. The XQC measurements rule out a large range of hitherto unconstrained dark matter masses and scatter- FIG. 9: Plot of the scattering cross section for dark matter particles and nucleons (σDn) versus dark matter particle mass (mdm) showing the new XQC limits along with other current experimental limits. The red XQC exclusion region is the same as shown in Fig. 7, and the other experiments are discussed in the text. The dark gray region shows the maximal range of dark matter self-interaction cross section consistent with the strongly self-interacting dark matter model of structure formation [37, 38]. The square marks the value of the scattering cross section for neutron-nucleon interactions. ing cross sections. The excluded range was first derived in Refs. [38, 48] based on rough analytic estimates. In this paper, we have improved upon these results using detailed Monte Carlo simulations to predict how a dark matter particle of a given mass and cross section for nu- cleon scattering would interact with the XQC calorime- ters. Unlike Ref. [52], our analysis includes the atmo- sphere and the shielding of the detector, so our result in- cludes the upper limit on excluded σDn values, which had not yet been accurately determined. Our simulation also models the internal geometry of the XQC detector and the random walk of particles through it, which is not pos- sible using the analytical approaches of Refs. [38, 48, 52]. The resulting exclusion region is significantly different than its analytical predecessors. When multiple scatter- ings are included, the XQC experiment is sensitive to dark matter particles with masses below 0.3 GeV and cross sections for nucleon scattering between 10−24 and 10−20 cm2. Unlike Ref. [52], we find that the XQC ex- clusion region does not include σDn < 10 −29 cm2 for dark matter masses less than 10 GeV. Ref. [52] obtained a more restrictive upper bound because they assumed a specific X-ray background while we treat all events as po- tential dark matter interactions. At higher masses, the lower boundary of our exclusion region is much higher than in Refs. [38, 48] because they over-estimated the XQC sensitivity by assuming coherent scattering. It also appears that Refs. [38, 48] underestimated the atmo- spheric and shielding overburden for the XQC detector because our exclusion region does not extend to values of σDn as large as those included in their exclusion region. We also assume a lower local dark matter density than Refs. [38, 48] (0.3 instead of 0.4 GeV cm−3), so some of the shrinkage of the exclusion region may be attributed to the reduction in the assumed number density of dark matter particles. Fig. 9 shows how the XQC exclusion region depicted in Fig. 7 complements the exclusion regions from other ex- periments that are sensitive to similar values of σDn and mdm. For a summary of some of the other experimen- tal constraints as of 1994, see Ref. [46]. The constraints to σDn from Pioneer 11 [80], Skylab [81], and IMP7/8 [82] were interpreted by Refs. [38, 46, 48]. There have been two balloon-borne searches for dark matter, the IMAX experiment [46, 47] and the Rich, Rocchia & Spiro (RRS) [83] experiment. Although underground detectors are designed to detect WIMPs, DAMA [84, 85] does ex- clude σDn values within the range of interest, and relevant constraints may be derived from Edelweiss (EDEL) and CDMS [86, 87]. All of the exclusion regions shown in Fig. 9 were de- rived assuming that all the dark matter is strongly in- teracting. A local dark matter density of 0.4 GeV cm−3 was assumed in the analysis of the exclusion regions from Pioneer 11, Skylab and the RRS experiment, while all the other exclusion regions were derived assuming a lo- cal dark matter density of 0.3 GeV cm−3. Furthermore, the derivations of all the shown exclusion regions other than the XQC region and the EDEL+CDMS region as- sume that the scattering between dark matter particles and nuclei is coherent. Therefore, these exclusion regions are likely too broad because they over-estimate the cross section for nuclear scattering. A comparison of the XQC exclusion region reported in Refs. [38, 48] and our exclu- sion region indicates that assuming coherent scattering extends the exclusion region for mdm ≥ 1000 GeV to σDn values that are roughly A× smaller than the lower boundary of our exclusion region, where A is the mass number of the largest target nucleus. Fig. 9 also shows the bound on σDn from the CMB and large-scale structure (LSS) obtained when one as- sumes prior knowledge of the Hubble constant H0 and the cosmic baryon fraction (from BBN) [50]. This bound is nominally stronger than the bound from disk stabil- ity [45], but it is less direct in that it requires combining different measurements and depends on the cosmologi- cal model; consequently, we show both bounds in Fig. 9. Measurements of primordial element abundances give an upper limit of σDn/mdm ∼< 4 × 10−16 cm2 GeV−1 [49]. Since this upper bound lies well beyond the upper bound from disk stability, we do not include it in Fig. 9. We also do not display the constraints from cosmic rays [49] because they are derived from inelastic interactions that are model-dependent. As shown in Fig. 9, the XQC experiment rules out a wide region of (mdm, σDn) parameter space that was not probed by prior dark matter searches. Of partic- ular interest is the darkly shaded range of σDn values that corresponds to the maximal range of dark matter self-interaction cross sections consistent with the strongly self-interacting dark matter model of structure formation [37, 38]. If the dark matter consists of exotic hadrons whose interactions with nucleons are comparable to their self-interactions, then σDn for these particles would lie in or near the darkly shaded region in Fig. 9. Previous esti- mates of the XQC exclusion region [38, 48] indicated that the XQC experiment rules out all the darkly shaded σDn values for 1 ∼< mdm ∼< 104 GeV. Our analysis reveals that this is not the case; portions of the darkly shaded region for mdm ∼> 20 GeV are not excluded by the XQC ex- periment, although they are ruled out by observations of LSS and the CMB. The mass-σ combination correspond- ing to nucleon-neutron scattering (the square in Fig. 9) lies within the exclusion region of the XQC experiment, and the only portion of the darkly shaded region that is unconstrained corresponds to dark matter masses smaller than 0.25 GeV. It is important to note, however, that the cross section for dark matter self-interactions need not be comparable to the cross section for nucleon scattering; σDn could dif- fer by a few orders of magnitude from the self-interaction cross section (as is the case for Q-balls). Furthermore, no interactions with baryons are required for self-interacting dark matter to resolve the tension between the collision- less dark matter model and observations of small-scale structure. Another XQC detector is scheduled to launch in the upcoming year. This experiment will have twice the ob- serving time of the XQC experiment used in this analy- sis. As discussed in Section V, increasing the observing time will extend the exclusion region to higher masses and weaker interactions. The future XQC experiment will also have a lower energy threshold (15 eV) and will maintain sensitivity to all energies above this threshold throughout the run. The increased sensitivity to low en- ergies will shift the lower (σDn ≤ 10−23 cm2) left bound- ary of the exclusion region to lower masses. A lower en- ergy threshold of 15 eV will make the experiment sensi- tive to single recoil events involving dark matter particles more massive than 0.17 GeV, as discussed in Section V. Clearly, the next-generation XQC experiment will be an even more powerful probe of interactions between dark matter particles and baryons than its predecessor. Acknowledgments A. L. E. would like to thank Robert Lupton and Michael Ramsey-Musolf for useful discussions. D. 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0704.0795

Heavy-Light Semileptonic Decays in Staggered Chiral Perturbation Theory

CU-TP-1177 Heavy-Light Semileptonic Decays in Staggered Chiral Perturbation Theory C. Aubin Physics Department, Columbia University, New York, NY 10027 C. Bernard Department of Physics, Washington University, St. Louis, MO 63130 Abstract We calculate the form factors for the semileptonic decays of heavy-light pseudoscalar mesons in partially quenched staggered chiral perturbation theory (SχPT), working to leading order in 1/mQ, where mQ is the heavy quark mass. We take the light meson in the final state to be a pseudoscalar corresponding to the exact chiral symmetry of staggered quarks. The treatment assumes the validity of the standard prescription for representing the staggered “fourth root trick” within SχPT by insertions of factors of 1/4 for each sea quark loop. Our calculation is based on an existing partially quenched continuum chiral perturbation theory calculation with degenerate sea quarks by Bećirević, Prelovsek and Zupan, which we generalize to the staggered (and non- degenerate) case. As a by-product, we obtain the continuum partially quenched results with non-degenerate sea quarks. We analyze the effects of non-leading chiral terms, and find a relation among the coefficients governing the analytic valence mass dependence at this order. Our results are useful in analyzing lattice computations of form factors B → π and D → K when the light quarks are simulated with the staggered action. PACS numbers: 12.39.Fe,12.39.Hg, 11.30.Rd, 12.38.Gc http://arxiv.org/abs/0704.0795v2 I. INTRODUCTION Extraction of the CKM matrix elements |Vub| and |Vcs| from the experimentally measured semileptonic decay rates for B → πℓν andD → Kℓν requires reliable theoretical calculations of the corresponding hadronic matrix elements. Recently, there has been significant progress in computing these matrix elements on the lattice, with good control of the systematic uncertainties [1, 2, 3, 4]. Since computation time increases as a high power of the inverse quark mass, the light (u, d) quark masses used in the simulations are heavier than in nature, and a chiral extrapolation is necessary to obtain physical results. To keep systematic errors small, the simulated u, d masses should be well into the chiral regime, giving pion masses ∼300MeV or lighter. Such masses in lattice calculations of leptonic and semileptonic heavy- light decays are accessible with staggered quarks [5, 6, 7, 8, 9, 10]. The trade-off for this benefit is the fact that staggered quarks do not fully remove the species doubling that occurs for lattice fermions; for every flavor of lattice quark, there are four “tastes,” which are related in the continuum by an SU(4) symmetry (or an SU(4)L×SU(4)R symmetry in the massless case). The taste symmetry is broken at non-zero lattice spacing a by terms of order a2. The breaking of taste symmetry on the lattice implies that one must take into account taste-violations in the chiral extrapolations, leading to a joint extrapolation in both the quark masses and the lattice spacing. Staggered chiral perturbation theory (SχPT) [11, 12, 13] allows us to make such extrapolations systematic. For quantities with heavy quarks, one must also incorporate Heavy Quark Effective Theory (HQET) [14, 15, 16, 17, 18] into SχPT. This has been done in Ref. [19], and then applied to leptonic heavy-light decays. Here, we extend the analysis of Ref. [19] to the semileptonic case. In addition to the practical implications of taste symmetry violations for chiral extrap- olations, the violations lead to a potentially more serious theoretical concern. Simulations such as Refs. [5, 6, 7, 8, 9, 10, 20] take the fourth root of the staggered quark determinant [21] in an attempt to obtain a single taste per quark flavor in the continuum limit. Were the taste symmetry exact at finite lattice spacing, the fourth root prescription would obviously accomplish the desired goal, since it would be equivalent to using a local Dirac operator obtained by projecting the staggered operator onto a single-taste subspace. Because the taste symmetry is broken, however, the fourth root is necessarily a nonlocal operation at non-zero lattice spacing [22]. The question of whether the rooted theory is in the correct universality class therefore becomes nontrivial. Nevertheless, there are strong theoretical arguments [22, 23, 24, 25, 26] in the interacting theory, as well as free-theory and numerical evidence [27] that the fourth-root trick is valid, i.e., that it produces QCD in the continuum limit. The current paper does not actually need to assume that the rooting procedure itself is valid.1 Instead, like previous SχPT calculations for the rooted theory [12, 13, 19, 28], it requires a narrower assumption: that the rooting can be represented at the chiral level by multiplying each sea quark loop by a factor of 1/4. This can be accomplished by a quark flow analysis [29], or, more systematically, by use of the replica trick [30]. In Ref. [24], it was shown that the correctness of this representation of the fourth root in SχPT follows in turn from certain — in our opinion, rather plausible — assumptions. As such, we assume here that this representation is valid. Fitting lattice quantities to SχPT formulae (as in Refs. [10, 20]) provides an additional empirical test of the validity of this representation. The main purpose of the current paper is to find SχPT expressions for the form factors of the semileptonic decay B → Pℓν, where P is some light pseudoscalar meson, which we will refer to generically as a “pion.” We consider first the partially quenched case, and obtain the full QCD results afterward by taking the limit where valence masses equal the sea masses. The B is a heavy-light meson made up of a b heavy quark and a valence light quark spectator of flavor x; we use the notation Bx when confusion as to the identity of the light spectator could arise. The P meson (more precisely Pxy) is composed of two light valence quarks, of flavor x and y. For simplicity we consider only the case where the outgoing pion is (flavor) charged; in other words x 6= y. The flavor structure of the weak current responsible for the decay is ȳγµb. In our calculation, we take the heavy quark mass mQ to be large compared to ΛQCD and work to leading order in the 1/mQ expansion. Our analysis also applies when the heavy quark is a c (i.e., to D mesons), but we use B to denote the heavy-light meson to stress the fact that only lowest order terms HQET are kept. For D mesons, of course, the higher order terms omitted here would be more important than for B mesons. Discretization errors coming from the heavy quark are not included in the current calcu- lations. We assume that such errors will be estimated independently, using HQET as the 1 Of course, were the fourth root trick to prove invalid, the motivation for the current work would be lost. effective-theory description of the lattice heavy quark [31]. It is expected that the errors from staggered quark taste-violations, which are considered here, are significantly more important at most currently accessible lattice spacings than the heavy-quark errors [4]. However, since taste-violations decrease rapidly2 when the lattice spacing is reduced, this may change in the not too distant future. In any case, the precise quantification of the total discretization error will always require simulation at several lattice spacings. An additional practical constraint on the current calculation is that amQ must not be too large compared to unity. When amQ ≫ 1, the effects of the heavy quark doublers would need to be included in the chiral theory, and the analysis would become prohibitively complicated. A detailed discussion of this and other issues involved in incorporating heavy quarks into SχPT appears in Ref. [19]. The calculations of interest here have been performed in continuum partially quenched chiral perturbation theory (PQχPT) by Bećirević, Prelovsek and Zupan [33] for Nsea de- generate sea quarks. In this paper we show how one can generalize the PQχPT formulae to the corresponding SχPT formulae, thereby avoiding the necessity of recomputing all the diagrams from scratch. Some results from the current work, as well as a brief discussion of how to generalize PQχPT to SχPT, appear in Ref. [34]. In addition, our results have already been used in chiral fits to lattice data in Refs. [8, 9]. A related calculation for the B → D∗ and B → D semileptonic form factors has been presented by Laiho and Van de Water [28]. The outline of this paper is as follows: We first include a brief description of heavy-light SχPT in Sec. II. In Sec. III, we discuss the procedure for generalizing PQχPT to SχPT, using the heavy-light form factors as examples, although the procedure can be used for many other quantities in SχPT. Using this procedure and starting from Ref. [33], we write down, in Sec. IV, the one-loop SχPT results for the semileptonic form factors. The partially quenched staggered case with non-degenerate sea quarks, as well as its continuum limit, is presented in Sec. IVA. In that section, we also discuss a method for treating — in a way that appears to be practical for fitting lattice data — some spurious singularities which arise in the calculations. Section IVB considers full-QCD special cases of the results from 2 Taste violations with improved staggered fermions go like α2 a2. See Fig. 1 in Ref. [32] for a test of this relation. Sec. IVA; while Sec. IVC discusses the analytic contributions to the form factors at this order. In Sec. V we add in the effects of a finite spatial lattice volume. Sec. VI presents our conclusions. We include three appendices: Appendix A gives expressions for the SχPT propagators and vertices, as well as the corresponding continuum versions. Appendix B lists the integrals used in the form factor calculations; while Appendix C collects necessary wavefunction renormalization factors that were calculated in Refs. [13, 19]. II. HEAVY-LIGHT STAGGERED CHIRAL PERTURBATION THEORY References [14, 15, 16, 17, 18] show how to incorporate heavy-light mesons into continuum χPT; the extension to SχPT appears in Ref. [19]. Here we review the key features needed for our calculations. The heavy-light vector (B∗µa) and pseudoscalar (Ba) mesons are combined in the field 1 + v/ µa + iγ5Ba , (1) which destroys a meson. Here v is the meson velocity, and a is the “flavor-taste” index of the light quark in the meson. For n flavors of light quarks, a can take on 4n values. Later, we will write a as separate flavor (x) and taste (α) indices, a→ (x, α), and ultimately drop the taste index, since the quantities we calculate will have trivial dependence on the light quark taste. The conjugate field Ha creates mesons: Ha ≡ γ0H†aγ0 = µa + iγ5B ] 1 + v/ . (2) As mentioned in the introduction, we use B to denote generic heavy-light mesons to empha- size that we are working to leading order in 1/mQ. Under SU(2) heavy-quark spin symmetry, the heavy-light field transforms as H → SH , H → HS† , (3) with S ∈ SU(2), while under the SU(4n)L × SU(4n)R chiral symmetry, H → HU† , H → UH , (4) with U ∈ SU(4n) defined below. We keep the flavor and taste indices implicit here. The light mesons are combined in a Hermitian field Φ(x). For n staggered flavors, Φ is a 4n× 4n matrix given by: U π+ K+ · · · π− D K0 · · · K− K̄0 S · · · . . . . (5) We show the n = 3 portion of Φ explicitly. Each entry in Eq. (5) is a 4×4 matrix, written in terms of the 16 Hermitian taste generators TΞ as, for example, U = Ξ=1 UΞTΞ. The component fields of the flavor-neutral elements (UΞ, DΞ, . . . ) are real; the other (flavor- charged) fields (π+Ξ , K Ξ, . . . ) are complex. The TΞ are TΞ = {ξ5, iξµ5, iξµν(µ < ν), ξµ, ξI} , (6) with ξµ the taste matrices corresponding to the Dirac gamma matrices, and ξI ≡ I the 4×4 identity matrix. We define ξµ5 ≡ ξµξ5, and ξµν ≡ (1/2)[ξµ, ξν]. The mass matrix is the 4n× 4n matrix muI 0 0 · · · 0 mdI 0 · · · 0 0 msI · · · . . . , (7) where the portion shown is again for the n = 3 case. From Φ one constructs the unitary chiral field Σ = exp[iΦ/f ], with f the tree-level pion decay constant. In our normalization, f ∼ fπ ∼= 131 MeV. Terms involving the heavy-lights are conveniently written using use σ ≡ Σ = exp[iΦ/2f ]. These fields transform trivially under the SU(2) spin symmetry, while under SU(4n)L × SU(4n)R we have Σ→ LΣR† , Σ† → RΣ†L† , (8) σ → LσU† = UσR† , σ† → Rσ†U† = Uσ†L† , (9) with global transformations L ∈ SU(4n)L and R ∈ SU(4n)R. The transformation U, defined by Eq. (9), is is a function of Φ and therefore of the coordinates. It is convenient to define objects involving the σ field that transform only with U and †. The two possibilities with a single derivative are σ†∂µσ + σ∂µσ , (10) σ†∂µσ − σ∂µσ† . (11) Vµ transforms like a vector field under the SU(4n)L×SU(4n)R chiral symmetry and, when combined with the derivative, can form a covariant derivative acting on the heavy-light field or its conjugate: Dµ)a = Hb Dbaµ ≡ ∂µHa + iHbVbaµ , DµH)a = Dabµ Hb ≡ ∂µHa − iVabµ Hb , (12) with implicit sums over repeated indices. The covariant derivatives and Aµ transform under the chiral symmetry as Dµ → (H DµH → U( DµH) , Aµ → UAµU† . (13) The combined symmetry group of the theory includes Euclidean rotations (or Lorentz symmetry), translations, heavy-quark spin, flavor-taste chiral symmetries, and the discrete symmetries C, P , and T . Many of these symmetries are violated by lattice artifacts and/or light quark masses. Violations to a given order are encoded as spurions in the Symanzik action. From these spurions, the heavy-light and light-light fields, derivatives, the heavy quark 4-velocity vµ, and the light quark gamma matrix γµ, we can construct the chiral Lagrangian and relevant currents order by order. Reference [19] finds the lowest order heavy-chiral Lagrangian and left-handed current, as well as higher order corrections. We need primarily the lowest order results here. For convenience, we write the Lagrangian in Minkowski space, so that we can make contact with the continuum literature. We write the leading order (LO) chiral Lagrangian as LLO = Lpion + LHL (14) where Lpion is the standard SχPT Lagrangian [13] for the light-light mesons, and LHL is the contribution of the heavy-lights. We have3 Lpion = Tr(∂µΣ∂ µΣ†) + µf 2Tr(MΣ+MΣ†) (UI +DI + SI + . . .) 2 − a2V , (15) −V = C1Tr(ξ(n)5 Σξ †) + C3 [Tr(ξ(n)ν Σξ ν Σ) + h.c.] [Tr(ξ ν5 Σξ 5ν Σ) + h.c.] + C6 Tr(ξ(n)µν Σξ + C2V [Tr(ξ(n)ν Σ)Tr(ξ ν Σ) + h.c.] + C2A [Tr(ξ ν5 Σ)Tr(ξ 5ν Σ) + h.c.] + C5V Tr(ξ(n)ν Σ)Tr(ξ †) + C5A ν5 Σ)Tr(ξ †) , (16) LHL = −iTr(HHv· D) + gπ Tr(HHγ µγ5Aµ) . (17) Here Tr denotes a trace over flavor-taste indices and, where relevant, Dirac indices. The product HH is treated as a matrix in flavor-taste space: (HH)ab ≡ HaHb. The covariant derivative D acts only on the field immediately preceding it. For convenience, we work with diagonal fields (U , D, . . . ) and leave the anomaly (m20) term explicit in Eq. (15). We can take m20 →∞ and go to the physical basis (π0, η, . . . ) at the end of the calculation [37]. To calculate semileptonic form factors, we need the chiral representative of the left-handed current which destroys a heavy-light meson of flavor-taste b. At LO this takes the form γµ (1− γ5)H σ†λ(b) , (18) where λ(b) is a constant vector that fixes the flavor-taste: (λ(b))c = δbc, and trD is a trace on Dirac indices only. The power counting is a little complicated in the heavy-light case, since many scales are available. Let mq be a generic light quark mass, and let m π ∝ mq be the corresponding “pion” mass, with p its 4-momentum. Further, take k as the heavy-light meson’s residual momentum. Then our power counting assumes k2 ∼ p2 ∼ m2π ∼ mq ∼ a2, where appropriate powers of the chiral scale or ΛQCD are implicit. The leading heavy-light chiral Lagrangian 3 There is a missing minus sign in Eq. (35) of Ref. [19]. LHL is O(k), the leading light-light Lagrangian Lpion is O(p2, mq, a2), and the leading heavy- light current j LO is O(1). Only these leading terms are relevant to the calculation of non- analytic “chiral logarithms” at first non-trivial order, which give O(mq, a2) corrections to leading expressions for semileptonic form factors. In principle, finding the corresponding analytic corrections requires complete knowledge of the next-order terms in the Lagrangian and current. However, since the form factors depend only on the the valence and sea quark masses, a2, and the pion energy in the rest frame of the B (namely v·p), the form of these corrections is rather simple and is easily determined by the symmetries. The large number of chiral parameters that can appear in higher-order terms in the Lagrangian and the current collapse down into relatively few free parameters in the form factors. Unless one wants to write these free parameters in terms of the chiral parameters, complete knowledge of the higher-order terms in the Lagrangian and current is often unnecessary. However, one does need to know enough about the higher- order terms to check for the possibility of relations among the free parameters that multiply different quantities or that appear in different form factors. At the order we work here, there is one relation among the various parameters that determine the linear dependence of the two form factors on the valence masses. In order to be sure that this relation is valid, we need to know all terms at next order that can contribute such linear dependence. Fortunately, all such terms are known. For the light-light Lagrangian, Eq. (15), the relevant terms are the standard O(p4 ∼ m2q) terms in the continuum [38]. All terms of O(mqa2, a4), which are special to SχPT, are also available [39]. For the heavy-light La- grangian and current, Ref. [19] lists all terms which are higher order than Eqs. (17) and (18) by a factor of mq (most important here) or a 2. Reference [19] does not attempt a complete catalog of the terms which are higher than Eqs. (17) and (18) by one or two powers of k, i.e., having one or two derivative insertions. However, a sufficient number of representative terms of this type are listed to see that the corresponding free parameters in the form factors are all independent. We discuss the determination of the analytic terms further in Sec. IVC. III. GENERALIZING CONTINUUM PQχPT TO SχPT We wish to compute the decay Bx → Pxy in SχPT, where x and y are (light) flavor labels. The taste of the light quarks in B, P and the current also needs to be specified. We take the Pxy to be a “Goldstone pion” with taste ξ5. Let the light quark in the B have taste α (α = 1, . . . , 4); in flavor-taste notation the light quark has index a ↔ xα. The current, Eq. (18), has flavor-taste b ↔ yβ. Despite the existence of taste violations at non-zero lattice spacing, the amplitude turns out to be proportional to (ξ5/2)αβ, with a proportionality factor that is independent of the tastes α, β. We will often keep this rather trivial taste-dependence implicit. In Ref. [33], Bećirević, et al. have calculated the form factors for B → π and B → K transitions in continuum PQχPT. They assume degenerate sea-quark masses, but leave Nsea, the number of sea quarks, arbitrary. As we explain below, the Nsea dependence is a marker for the underlying quark flow [29] within the meson diagrams. Once we have separated the meson diagrams into their contributions from various the quark flow diagrams, we can easily generalize the continuum PQχPT results to the staggered case, without actually having to calculate any SχPT diagrams. To check our method, however, we have also computed many of the diagrams directly in SχPT; the results agree. The key feature that makes possible the generalization of continuum PQχPT results to SχPT results is the taste-invariance of the leading-order Lagrangian for the heavy-light mesons [19]. This means that the continuum vertices and propagators involving heavy- light mesons are trivially generalized to the staggered case: flavor indices (which can take Nsea values if they describe sea quarks) simply become flavor-taste indices (taking 4Nsea sea-quark values). In one-loop diagrams, taste violations arise only from the light meson (“pion”) propagators. Propagators and vertices for the staggered and continuum cases are listed Appendix A. Looking at the expressions in Appendix B of Ref. [33], we see that there are two types of terms that can contribute to each diagram for Bx → Pxy: a term proportional to Nsea, and a term proportional to 1/Nsea. This is the same behavior that appears, for example, in light-light [35] or heavy-light [36] PQχPT decay constants. The term which is proportional to Nsea comes solely from connected quark-level diagrams, an example of which is shown in Fig. 1 (where (a) is the meson-level diagram and (b) is the quark-level diagram).4 The appearance of the quark loop accounts for the factor of 4 By definition, the pion propagator in Fig. 1(a) is connected; the version with a disconnected propagator is shown in Fig. 2(a). Nsea. In detail, using Eqs. (A15) and (A16), the loop integrand is proportional to the connected contraction ΦijΦji′ , where the index j is repeated because the heavy- light propagator conserves flavor. Equation (A18) then implies that the sum over j produces a factor of Nsea when the sea quarks are degenerate. In the non-degenerate case, there is no factor of Nsea but simply a sum over the sea-quark flavor of the virtual valence-sea pion. In the staggered case, the internal heavy propagators, Eqs. (A1) and (A2), as well as the vertices coupling heavy-light mesons to pions (e.g., Eq. (A3)), preserve both flavor and taste. Therefore Fig. 1 is now simply proportional to ΦabΦba′ Φiα,jβΦjβ,i′α′ where we have replaced the flavor-taste indices (a, b, . . . ) with separate flavor (i, j, . . . ) and taste (α, β, . . . ) indices. From Eq. (A8), the loop integrand is then proportional to iδii′δαα′ p2 −m2ij,Ξ + iǫ , (19) where the δαα′ factor shows that, despite the existence of taste violations, the loop preserves the taste of the light quark in the heavy-light meson and is independent of that taste. The overall factor of the SχPT diagram must be such as to reproduce the continuum result in the a → 0 limit. Since pions come in 16 tastes, the sum over pion tastes Ξ in Eq. (19) must come with a factor of 1/16 compared to the continuum expression. To see this explicitly, note first that there are two factors of 1/2 relative to the continuum coming from the vertices (compare Eqs. (A3) and (A16)), due to the non-standard normalization of the taste generators, Eq. (6). An additional factor of 1/4 comes from the SχPT procedure for taking into account the fourth root of the staggered determinant: This is a diagram with a single sea quark loop. Finally, we need to consider how such a diagram depends on the tastes α and β of the heavy-light meson and the current. Since the taste indices flow trivially through the heavy- light lines and vertices, and, as we have seen, through the loops, the taste dependence is simply (ξ5/2)αβ, where the ξ5 comes from the outgoing light meson. The factor of 1/2 is due to the normalization of the taste generators. The net result is that terms with factors of Nsea in the continuum calculation of Ref. [33] are converted to SχPT by the rule: NseaF(m2M)→ (ξ5)αβ F(m2fz,Ξ) (20) where the sum over f is over the sea quark flavors, z is the valence flavor flowing through the loop (either x or y), mM is the common mass of the Nsea mesons made up of a z valence quark and the degenerate sea quarks, and F is some function of the pion masses. (For heavy-light quantities, F is often also a function of the pion energy in the heavy-light rest frame.) The masses of pions of various tastes and flavors (mfz,Ξ) are given in Eq. (A7). The terms that are proportional to 1/Nsea are more subtle. They arise from diagrams with disconnected pion propagators. The simplest example is shown at the meson level in Fig. 2(a) and at the quark level in Fig. 2(b). The continuum form of the disconnected propagator is given in Eq. (A21). Using the continuum values δ′ = m20/3 and m η ≈ Nseam20/3, we see that the disconnected propagator produces an overall factor of 1/Nsea as m0 → ∞. Equation (A21) can then be written as a sum of residues times poles, where the residues can be rather complicated when the sea masses are non-degenerate (see Appendix B). Thus, the final answer after integration amounts to something of the form R̂jF̃(m2j) , (21) where F̃ is again a general function resulting from the loop integral, R̂j is the residue of the pole at q2 = m2j , and j ranges over the flavor-neutral mesons involved: the sea mesons, π0, η, . . . , and the “external” mesons in the disconnected propagator, called ii and i ′i′ in Eq. (A21). When mi′i′ = mii, there is a double pole, and Eq. (21) should be replaced by ∂m2ii R̂jF̃(m2j ) , (22) where the sum over j now does not include i′i′. When the sea quarks are degenerate, the residues simplify considerably. However, by comparing the general forms in Eqs. (21) and (22) to the rather simple terms in Ref. [33], it is easy to move backwards from the degenerate case and determine the form of the expressions for non-degenerate sea quarks. The flavor structure in the staggered case is identical to that in the continuum: Flavor remains a good quantum number, so meson propagators in both cases can only be discon- nected if they are flavor neutral. Because of taste violations, however, disconnected hairpin diagrams can contribute to mesons propagators with three different tastes (singlet, vector, and axial vector) at this order in SχPT. These three hairpin contributions are quite similar to each other, but there are a few important differences: • The strength of the hairpin, δ′Ξ, depends on the taste Ξ — see Eq. (A9). • In the taste-singlet case, as in the continuum, the hairpin (4m20/3) comes from the anomaly and makes the flavor-singlet meson heavy. Decoupling the η′I by taking m20 → ∞ is therefore a good approximation, and we do it throughout, giving rise to an overall factor of 1/Nsea. But in the taste-vector and taste-axial-vector cases, the hairpins are not particularly large; indeed they are taste-violating effects that vanish like a2 (up to logarithms) as a→ 0. So we cannot decouple the corresponding mesons, η′V and η A, in the taste-vector and taste-axial-vector channels. • The taste matrices associated with the vector and axial-vector mesons, ξµ and ξµ5, anticommute with the ξ5 coming from the outgoing Goldstone pion. Therefore the vector and axial hairpin contributions will have an opposite sign from the singlet (and continuum) contribution if the ξ5 needs to be pushed past a ξµ or ξµ5 to contract with the external pion state. Figure 3 shows the tree-level diagrams that contribute to the form factors, while Figs. 4 and 5 show all the non-vanishing one-loop diagrams. As a first example of the treatment of diagrams with disconnected meson propagators, consider Fig. 5(b). It is not hard to see that this diagram has only a disconnected contribution, shown as a quark-flow diagram in Fig. 6. A connected contribution would require the contraction of the external light quark fields x and y, which make up the outgoing pion. That is impossible since we have chosen x 6= y.5 In our notation, the result from Ref. [33] for this diagram in the continuum partially quenched case with Nsea degenerate sea quarks is: (4πf)2 m2Y −m2U m2Y −m2X J sub1 (mY , v·p)− m2X −m2U m2Y −m2X J sub1 (mX , v·p) , (23) where mU is the mass of any of the mesons made up of a sea quark and a sea anti-quark, mX and mY are the masses of the flavor-neutral mesons made up of xx̄ and yȳ quarks, respectively, and the function J sub1 , defined below in Eq. (51), is the result of the momentum integral. 5 A similar argument will be given in more detail below when discussing Figs. 7 and 8. The ratios of mass differences in Eq. (23) can be recognized as the residue functions (see Appendix B) for the various poles. For example, (m2Y −m2U)/(m2Y −m2X) is the residue for the pole at q2 = m2Y . These residues are rather simple in this case because of the degeneracy of the sea quarks. To generalize Eq. (23) to the completely non-degenerate case, we simply need to replace the residues by their general expressions. ForNsea non-degenerate sea quarks, Eq. (23) is replaced by (4πf)2 [Nsea+1,Nsea] 1 (mj , v·p) , (24) where the Minkowski residues R̂ [n,k] j are defined in Eq. (B2), and the sum over j is over the Nsea+1 mesons that make up the denominator masses in the disconnected propagator after m20 → ∞. (See Eq. (A21) and the discussion following it.) We leave implicit, for now, the arguments to the residues in Eq. (24); we will be more explicit in the final results below. In addition, we will ultimately express everything in terms of Euclidean-space residues R [n,k] Eq. (B4), simply because those are what have been defined and used previously [13, 19]. Cases with double poles present no additional problems, since Ref. [33] shows these explicitly as derivatives with respect to squared masses of the results of single-pole integrals. We will therefore simply get derivatives of the usual residues, as in Eq. (B3). As discussed above, we will need the expression before the m20 →∞ limit is taken in order to generalize the result to the disconnected taste-vector and axial-vector cases. Equation (A21) and the fact that m2η′ ≈ Nseam20/3 for large m0 allow us to rewrite Eq. (24) as (4πf)2 [Nsea+2,Nsea] 1 (mj , v·p) . (25) The sum over j now includes the η′. The sign difference between Eqs. (24) and (25) comes from the sign of the mass term in the Minkowski-space η′ propagator. Generalizing Eq. (25) to the staggered case is then straightforward. For the taste-singlet hairpin contributions, we simply replace each continuum pion mass by the mass of the corresponding taste-singlet pion. In other words, we just let mj → mj,I in Eq. (25). Note that, after the staggered fourth root is properly taken into account, the taste-singlet η′ mass goes like Nseam 0/3 for large m0, as it does in the continuum, so one could reverse the process that led to Eq. (25) and use instead Eq. (24) or even Eq. (23) (for degenerate sea-quarks), with mj → mj,I in both cases. Just as for diagrams with connected pion propagators (see Eq. (20)), there is also a trivial overall factor of (ξ5)αβ/2, where α and β are the tastes of the heavy-light meson and the current, respectively, and the ξ5 is due to the pseudoscalar (Goldstone) taste of the outgoing pion. For the taste-vector and axial-vector disconnected contributions, a little more work is required. We first note that the factor of m20/3 in Eq. (25) is simply δ Ξ/4 with Ξ = I, the strength of the taste-singlet hairpin, Eq. (A9).6 For the other tastes we then replace δ′Ξ by the appropriate hairpin strength from Eq. (A9) and also replace the pion masses: mj → mj,Ξ. In addition, there is an overall sign change for this diagram in going from the singlet to the vector or axial-vector tastes. This comes from the fact that the outgoing pion line in Fig. 5(b) lies between the two ends of the disconnected propagator. Using Eq. (A13) and the Feynman rules for the heavy-light propagators and vertices in Appendix A, one sees that the diagram with a taste-Ξ disconnected propagator goes like TΞ ξ5 TΞ . This leads to a positive sign for Ξ = I but a negative sign for tastes that anticommute with ξ5. Finally, the fact that there are four degenerate taste-vector (or axial-vector) pions at this order leads to an additional overall factor of four. When we attempt to apply the same procedure to the other diagrams in Figs. 4 and 5, we find a further complication in diagrams Fig. 4(a), Fig. 4(b), and Fig. 5(c), where the external pion and one or more internal pions emerge from the same vertex. The problem is that the ordering of the taste matrices at the vertex is not determined by the meson-level diagram (i.e., each diagram can correspond to several orderings), so we do not immediately know the relative sign of taste-vector and axial-vector contributions relative to the singlet contribution. Nevertheless, a quark-flow analysis allows us to identify appropriate “flags” that signal which terms in Ref. [33] come from which orderings at the vertex. As an example of the procedure in this case, consider Fig. 5(c). The corresponding quark flow diagrams with disconnected pion propagators are shown Fig. 7. In Fig. 7(a), the outgoing pion lies between the two ends of the disconnected propagator. This produces a change in sign of the taste-vector and axial-vector hairpin contributions relative to the taste-singlet one, just as for Fig. 5(b). In Fig. 7(b), on the other hand, the outgoing pion is emitted outside the disconnected propagator, and all the hairpin contributions have the 6 The factor of 1/4 just comes from the different conventional normalization of the generators in the con- tinuum and staggered cases; see Appendix A for further discussion of normalization issues. same sign. The same is true of the reflected version of Fig. 7(b), which has the outgoing pion emerging from the other side of the vertex. Fortunately, Figs. 7(a) and (b) are distinguished by their flavor structure, even in the continuum. In Fig. 7(a), the two “external” mesons in the disconnected propagator have different flavors: The one on the left is an X meson (an xx̄ bound state); while the one on the right is a Y meson (a yȳ bound state). In Fig. 7(b), both external mesons in the disconnected propagator are Y mesons. Similarly, the reflected version of Fig. 7(b) has two X mesons in the disconnected propagator. This flavor structure is immediately apparent in the results of Ref. [33]. The parts of Fig. 5(c) that come from the quark flow of Fig. 7(a) are proportional to the function called H1, which depends on the masses mX and mY (in our notation), as well as the sea-meson mass. The parts of Fig. 5(c) that come from the quark flow of Fig. 7(b) (or its reflected version) are proportional to the function called G1, which depends only on the mass mY (or mX) and the sea-meson mass. To generalize the results of Ref. [33] to the staggered case, we thus can use the method outlined above, and simply include an extra minus sign for those taste-vector and axial-vector hairpin contributions proportional to H1 (relative to the taste-singlet contributions), but not for those proportional to G1. This approach also works for the other problematic diagrams, Figs. 4(a) and (b). The reader may wonder why the complication associated with ordering the taste matrices at the vertices does not occur when the internal pion propagator is connected, but only in the disconnected case. Figure 8 shows possible quark-flow diagrams for Fig. 5(c) with a connected pion propagator. Figure 8(a) cannot occur in our case because we have assumed that x, the light flavor of the heavy-light meson, is different from y, the light flavor of the weak current.7 The same reasoning is what allows us to rule out any connected contributions to Fig. 5(b), as mentioned above. Thus all contributions with connected propagators are of the type shown in Fig. 8(b), or its reflected version, and these never have a sign difference between terms with different internal pion tastes. We note that one can reproduce the SχPT results for light-light [13] and heavy-light [19] mesons by starting from the continuum PQχPT in Refs. [35] or [36], respectively, and following the procedure described above. The computations are in fact slightly more difficult in those cases than in the one at hand, because Refs. [35] and [36] do not explicitly separate 7 Equivalently, we have assumed that the outgoing pion is flavor charged. double-pole from single-pole contributions. It is therefore takes a little work to express the answers from those references in the form of our residue functions, which is the necessary first step before generalizing to the staggered case. IV. FORM FACTORS FOR B → P DECAY The standard form factor decomposition for the matrix element between a Bx meson and a Pxy meson is 〈Pxy(p)|ȳγµb|Bx(pB)〉 = (pB + p)µ − qµ m2Bx −m m2Bx −m qµF0(q 2) , (26) where q = pB − p is the momentum transfer. We are suppressing taste indices everywhere, but emphasize that the light pseudoscalar Pxy is assumed to be the Goldstone meson (taste ξ5). In the heavy quark limit, it is more convenient to write this in terms of form factors which are independent of the heavy meson mass 〈Pxy(p)|ȳγµb|Bx(v)〉HQET = [pµ − (v·p)vµ] fp(v·p) + vµfv(v·p) , (27) where v is the four-velocity of the heavy quark, and v·p is the energy of the pion in the heavy meson rest frame. Recall that the QCD heavy meson state and the HQET heavy meson state are related by |B(pB)〉QCD = mB|B(v)〉HQET . (28) The form factors fp and and fv are often called f⊥ and f‖, respectively. As discussed in Sec. III, the taste indices are left implicit in Eqs. (26) and (27), as are the trivial overall factors of (ξ5/2)αβ in the matrix elements. The tree-level diagrams for Bx → Pxy are shown in Fig. 3. Figure 3(a) is the tree-level “point” contribution to fv, while Fig. 3(b) is the tree-level “pole” contribution to fp. We f treev (v·p) = , f treep (v·p) = v·p+∆∗ , (29) where ∆∗ = mB∗ − mB is the mass difference of the vector and pseudoscalar heavy-light meson masses at leading order in the chiral expansion, i.e., neglecting all effects of light- quark masses. As in Refs. [33, 40], we drop this splitting inside loops, but keep it in the internal B∗ line in the tree-level diagram Fig. 3(b). This forces the tree-level pole in fp to be atmB∗ , the physical point. Dropping ∆ ∗ inside loops is consistent at leading order in HQET, which is the order to which we are working. It would also be consistent, parametrically, to drop ∆∗ everywhere. But this would not be convenient, since the mB∗ pole is physically important for fp. The non-zero diagrams that correct the form factors to one loop are shown in Fig. 4 for fv and Fig. 5 for fp. Table I lists the correspondences between these diagrams and those of Ref. [33]. A number of other diagrams, which can arise in principle, vanish identically due to the transverse nature of the B∗ propagator, Eq. (A2); these additional diagrams can be found in Ref. [33]. We do not indicate hairpin vertices explicitly in Figs. 4 and 5; the internal pion propagators in these diagrams may be either connected or disconnected. Before generalizing the results in Ref. [33] to SχPT, we discuss a subtle issue that affects Fig. 5(a). If the splitting ∆∗ is dropped on internal B∗ lines in loop diagrams, as is done in Ref. [33], this diagram has a spurious singularity (a double pole) at v · p = 0, the edge of the physical region. The singularity arises from the presence of the two B∗ lines that are not inside the loop integral and therefore can be on mass shell in the absence of B∗-B splitting. Including ∆∗ on all such internal “on-shell” B∗ lines (i.e., lines not inside the loops themselves), as is done in Ref. [40], at least pushes the unnatural double pole out of the physical region. We will follow this prescription for including the splitting, but take it one step further. The loop in Fig. 5(a) is a self-energy correction on the internal B∗ line. The double pole results from not iterating the self-energy and summing the geometric series. We will follow the more natural course and sum the series; doing so restores a standard single-pole singularity. There is a further one-loop contribution that can naturally be included in Fig. 5(a). The corresponding tree-level graph, Fig. 3(b), gets two kinds of corrections that are not shown in Fig. 5. One comes simply from the wavefunction renormalizations on the external pion and B lines; we include those terms explicitly below. The second contribution arises from the one-loop shift in the external meson mass. Since this mass shift depends on the flavor of the light quark in the Bx, namely x, we call it δMx = Σx(v ·k = 0), with Σx(v ·k) the self-energy for Bx or B x. Note that Σx is the same for both the Bx and the B x, since the splitting ∆∗ is dropped inside loops. When the external Bx line in Fig. 3(b) is put on mass-shell at one loop, the denominator of the internal B∗y propagator changes from −2(v·p + ∆ ∗) to −2(v·p+∆∗ − δMx). It is convenient to define ∆∗yx as the full splitting between a B∗y and a yx ≡MB∗y −MBx = ∆ + δMy − δMx (30) The internal B∗y propagator now becomes −2(v·p+∆ yx − δMy). The contribution from the mass shift may then be combined with the tree-level and Fig. 5(a) contributions to give: f selfp (v·p) = v·p+∆∗yx +D(v·p) , (31) D(v·p) ≡ Σy(v·p)− Σy(0) , (32) where the subtraction in D comes from the effect of putting the Bx on mass shell, via Eq. (30). The main difference between the approach taken to the spurious singularity of Fig. 5(a) and that of Bećirević et al. [33] is that they work to first order in the self-energy in the corresponding diagram (their diagram (7)). Expanding Eq. (31), we find that D is related in the continuum limit to what Ref. [33] calls δf D(v·p) = −v·p δf (7)p . (33) Thus we can find the staggered D(v·p) simply by applying the methods of Sec. III to δf (7)p . We can now write down the expressions for the form factors for Bx → Pxy decay. For the point form factor, fv, we have fBx→Pxyv = f 1 + δfBx→Pxyv + c xmx + c ymy + c sea(mu +md +ms) + cv1(v·p) + cv2(v·p)2 + cvaa2 , (34) where f treev is given by Eq. (29), and the analytic coefficients c y, . . . arise from next-to- leading order (NLO) terms in the heavy-light chiral Lagrangian (see Sec. IVC). The non- analytic pieces, which come from the diagrams shown in Fig. 4 as well as the wavefunction renormalizations, are included in δf Bx→Pxy δfBx→Pxyv = δf v + δf δZBx + δZPxy . (35) The wavefunction renormalization terms, δZBx and δZPxy , have been calculated previously [13, 19] in SχPT and are listed in Appendix C. For the fp form factor, we write fBx→Pxyp = f p + f̃ δfBx→Pxyp + c xmx + c ymy + c sea(mu +md +ms) 1(v·p) + c 2(v·p)2 + cpaa2 . (36) where f selfp is defined in Eq. (31), and f̃ treep (v·p) ≡ v·p+∆∗xy . (37) Non-analytic contributions are summarized in the function D(v·p) in f selfp , Eq. (32), and Bx→Pxy p , which comes from Figs. 5(b)-(d) and wavefunction renormalizations. Explicitly, δfBx→Pxyp = δf p + δf p + δf δZBx + δZPxy . (38) For simplicity, we do not include the superscript Bx → Pxy on the individual diagrams in Eqs. (35) and (38). Using f̃ treep , which includes the full B y–Bx splitting ∆ yx, rather than f p , Eq. (29), changes Eq. (36) only by higher-order terms. However, it is convenient to keep the same splitting in both f selfp and the other terms in Eq. (36). Note that it is also consistent at this order to use the alternative form fBx→Pxyp = f 1 + δfBx→Pxyp + c xmx + c ymy + c sea(mu +md +ms) 1(v·p) + c 2(v·p)2 + cpaa2 , (39) The analytic terms in fv and fp are not all independent. As mentioned in Sec. II, there is one relation among the terms that control the valence mass dependence: cpx + c x = c y + c y (40) We show that this relation follows from the higher order terms in the Lagrangian and current in Sec. IVC. All other NLO parameters in Eqs. (34) and (36) are independent. A. Form factors for 3-flavor partially quenched SχPT First we display the results for the individual diagrams shown in Figs. 4 and 5 for the fully non-degenerate case with three dynamical flavors (the “1+1+1” case). This means that we have already taken into account the transition from 4 to 1 tastes per flavor. Indeed, our method of generalizing the partially quenched continuum expressions to the staggered case automatically includes this adjustment. We detail below the minor changes needed to obtain 2+1 results from those in the 1+1+1 case. We first define sets of masses which appear in the numerators and denominators of the disconnected propagators with taste labels implicit (see Appendix B): µ(3) = {m2U , m2D, m2S} , (41) M(3,x) = {m2X , m2π0 , m2η} , (42) M(4,x) = {m2X , m2π0 , m2η, m2η′} , (43) M(4,xy) = {m2X , m2Y , m2π0, m2η} , (44) M(5,xy) = {m2X , m2Y , m2π0, m2η, m2η′} . (45) For the mass sets (42) and (43), there are also corresponding sets with x→ y and X → Y . When we show explicit taste subscripts such as I or V on the mass sets µ orM, it means that all the masses in the set have that taste. The functions that appear in the form factors are8 I1(m) = m , (46) I2(m,∆) = −2∆2 ln − 4∆2F + 2∆2 , (47) J1(m,∆) = −m2 + 2 (∆2 −m2)F m2 , (48) F (x) = 1− x2 tanh−1 1− x2 , 0 ≤ x ≤ 1 x2 − 1 tan−1 x2 − 1 , x ≥ 1 . The main difference in these formulae with those of Bećirević et al. [33] is that they keep the divergence pieces, while we have renormalized as in Refs. [13, 19]. To convert to our form, replace the MS scale µ in Ref. [33] with the chiral scale Λ and set their quantity ∆̄ to zero, where ∆̄ ≡ 2 − γ + ln(4π) + 1 , (50) 8 For ease of comparison to Ref. [33], we use I1(m) instead of ℓ(m 2) (as in Refs. [13, 19]) for the chiral logarithm. with d the number of dimensions. F (x) is only needed for positive x; so we use the simpler form given in Ref. [40], rather than the more general version worked out in Ref. [41] and quoted in Ref. [33]. We do not list the function J2, which appears in the integral J µν of Eq. (B8) but does not enter the final answers. We also define a “subtracted” J1 function by J sub1 (m,∆) ≡ J1(m,∆)− . (51) The subtraction term cancels the singularity when ∆→ 0. The function J sub1 enters naturally in the expression for the self energy correction D(v·p) because of the the subtraction in Eq. (32). It also turns out to arise from the integral in Fig. 5(b) — see Eq. (26) in Ref. [40]. For the point corrections in the 1+1+1 case, we have δf4(a)v )Bx→Pxy 1+1+1 2(4πf)2 I1(myf,Ξ) + 2I2(myf,Ξ, v·p) j∈M(4,xy) [4,3] M(4,xy)I ;µ [I1(mj,I) + 2I2(mj,I , v·p)] ∂m2Y,I j∈M(3,y) [3,3] M(3,y)I ;µ [I1(mj,I) + 2I2(mj,I , v·p)] + a2δ′V ∂m2Y,V j∈M(4,y) [4,3] M(4,y)V ;µ [I1(mj,V ) + 2I2(mj,V , v·p)] j∈M(5,xy) [5,3] M(5,xy)V ;µ [I1(mj,V ) + 2I2(mj,V , v·p)] + [V → A] , (52) δf4(b)v )Bx→Pxy 1+1+1 = − 1 6(4πf)2 [I1(mxf,Ξ) + I1(myf,Ξ)] ∂m2Y,I j∈M(3,y) [3,3] M(3,y)I ;µ I1(mj,I) ∂m2X,I j∈M(3,x) [3,3] M(3,x)I ;µ I1(mj,I) j∈M(4,xy) [4,3] M(4,xy)I ;µ I1(mj,I) + a2δ′V ∂m2Y,V j∈M(4,y) [4,3] M(4,y)V ;µ I1(mj,V ) ∂m2X,V j∈M(4,x) [4,3] M(4,x)V ;µ I1(mj,V ) j∈M(5,xy) [5,3] M(5,xy)V ;µ I1(mj,V ) + [V → A] . (53) Those that correct the pole form factors are Bx→Pxy 1+1+1 = − 3g2πv·p (4πf)2 J sub1 (myf,Ξ, v·p) j∈M(3,y) ∂m2Y,I [3,3] M(3,y)I ;µ J sub1 (mj,I , v·p) + a2δ′V j∈M(4,y) ∂m2Y,V [4,3] M(4,y)V ;µ J sub1 (mj,V , v·p) + [V → A] , (54) δf5(b)p )Bx→Pxy 1+1+1 (4πf)2 j∈M(4,xy) [4,3] M(4,xy)I ;µ J sub1 (mj,I , v·p) +a2δ′V j∈M(5,xy) [5,3] M(5,xy)V ;µ J sub1 (mj,V , v·p) + [V → A] ,(55) δf5(c)p )Bx→Pxy 1+1+1 6(4πf)2 [I1(mxf,Ξ) + I1(myf,Ξ)] j∈M(3,y) ∂m2Y,I [3,3] M(3,y)I ;µ I1(mj,I) j∈M(3,x) ∂m2X,I [3,3] M(3,x)I ;µ I1(mj,I) j∈M(4,xy) [4,3] M(4,xy)I ;µ I1(mj,I) + a2δ′V j∈M(4,y) ∂m2Y,V [4,3] M(4,y)V ;µ I1(mj,V ) j∈M(4,x) ∂m2X,V [4,3] M(4,x)V ;µ I1(mj,V ) j∈M(5,xy) [5,3] M(5,xy)V ;µ I1(mj,V ) + [V → A] , (56) δf5(d)p )Bx→Pxy 1+1+1 = − 1 2(4πf)2 I1(myf,Ξ) j∈M(3,y) ∂m2Y,I [3,3] M(3,y)I ;µ I1(mj,I) +a2δ′V j∈M(4,y) ∂m2Y,V [4,3] M(4,y)V ;µ I1(mj,V ) + [V → A] .(57) In Eqs. (52) through (57), the explicit factors of 1/3 in front of terms involving the taste- singlet (I) mesons come from the factors of 1/Nsea in Ref. [33]. To get the full corrections for both fv and fp, we need to add in the wavefunction renor- malizations, given in Appendix C in Eqs. (35) and (38). Putting these together with the analytic terms and (for fp) theD term, Eqs. (34) and (36) give the complete NLO expressions for the form factors in SχPT. The above 1+1+1 results are expressed in terms of the Euclidean residue functions R [n,k] Eq. (B4). In the 2+1 case, there is a cancellation in the residues between the contribution of the U or D in the numerator and that of the π0 in the denominator. Thus, to obtain the 2+1 from the 1+1+1 case, one must simply reduce by one all superscripts on the residues, i.e., R[n,k] → R[n−1,k−1], and remove mπ0 and (say) mD from the mass sets: µ(3) → {m2U , m2S} , (58) M(3,x) → {m2X , m2η} , (59) M(4,x) → {m2X , m2η, m2η′} , (60) M(4,xy) → {m2X , m2Y , m2η} , (61) M(5,xy) → {m2X , m2Y , m2η, m2η′} . (62) We also write here the expressions for three non-degenerate dynamical flavors in contin- uum PQχPT, which to our knowledge do not appear in the literature. These expressions can be obtained either by returning to Ref. [33] and using the residue functions to generalize to the non-degenerate case, or simply by taking the continuum limit of the above equations. Either way, the results for fv are δf4(a),contv )Bx→Pxy 2(4πf)2 I1(myf) + 2I2(myf , v·p) j∈M(4,xy) [4,3] M(4,xy);µ(3) [I1(mj) + 2I2(mj , v·p)] j∈M(3,y) [3,3] M(3,y);µ(3) [I1(mj) + 2I2(mj , v·p)] δf4(b),contv )Bx→Pxy = − 1 6(4πf)2 [I1(mxf) + I1(myf )] j∈M(3,y) [3,3] M(3,y);µ(3) I1(mj) j∈M(3,x) [3,3] M(3,x);µ(3) I1(mj) j∈M(4,xy) [4,3] M(4,xy);µ(3) I1(mj) , (63) while those for fp are Dcont )Bx→Pxy = − 3g (4πf)2 J sub1 (myf , v·p) j∈M(3,y) [3,3] M(3,y);µ(3) J sub1 (mj , v·p) δf5(b),contp )Bx→Pxy (4πf)2 j∈M(4,xy) [4,3] M(4,xy);µ(3) J sub1 (mj , v·p) δf5(c),contp )Bx→Pxy 6(4πf)2 [I1(mxf) + I1(myf )] j∈M(3,y) [3,3] M(3,y);µ(3) I1(mj) j∈M(3,x) [3,3] M(3,x);µ(3) I1(mj) j∈M(4,xy) [4,3] M(4,xy);µ(3) I1(mj) δf5(d),contp )Bx→Pxy 2(4πf)2 I1(myf ) j∈M(3,y) [3,3] M(3,y);µ(3) I1(mj) . (64) Corresponding continuum-limit results for the wave-function renormalizations are given in Appendix C. B. Full QCD Results Adding together the complete results for the “full QCD” case is straightforward. For simplicity, we specialize to case mu = md (i.e., 2 + 1). For B → π, the complete corrections (including wave-function renormalization) are: DB→π = − 3g (4πf)2 2J sub1 (mπ,Ξ, v·p) + J sub1 (mK,Ξ, v·p) J sub1 (mπ,I , v·p) + J sub1 (mη,I , v·p) j∈{π,η,η′} (−a2δ′V )R [3,1] j ({mπ,V , mη,V , mη′,V }; {mS,V }) J sub1 (mj,V , v·p) V → A , (65) δfB→πp = (4πf)2 1 + 3g2π [2I1(mπ,Ξ) + I1(mK,Ξ)] 1 (mπ,I , v·p) + 1 (mη,I , v·p) + 1 + 3g2π 3I1(mπ,I)− I1(mη,I) j∈{π,η,η′} a2δ′VR [3,1] j ({mπ,V , mη,V , mη′,V }; {mS,V }) 1 (mj,V , v·p) + 1 + 3g2π I1(mj,V ) + [V → A] , (66) δfB→πv = (4πf)2 1− 3g2π [2I1(mπ,Ξ) + I1(mK,Ξ)] + 2I2(mπ,Ξ, v·p) + I2(mK,Ξ, v·p) 1 + 3g2π I1(mπ,I)− I1(mη,I) j∈{π,η,η′} a2δ′VR [3,1] j ({mπ,V , mη,V , mη′,V }; {mS,V }) 3(g2π − 1) I1(mj,V )− 2I2(mj,V , v·p) + [V → A] . (67) For B → K,9 we have DB→K = − 3g2π(v·p) (4πf)2 2J sub1 (mK,Ξ, v·p) + J sub1 (mS,Ξ, v·p) J sub1 (mη,I , v·p)− J sub1 (mS,I , v·p) j∈{S,η,η′} (−a2δ′V )R [3,1] j ({mS,V , mη,V , mη′,V }; {mπ,V })J sub1 (mj,V , v·p) V → A , (68) δfB→Kp = (4πf)2 −2 + 3g I1(mK,Ξ)− I1(mS,Ξ)− 3g2πI1(mπ,Ξ) 1 (mη,I , v·p) + I1(mπ,I)− 4 + 3g2π I1(mη,I) + I1(mS,I) + a2δ′V −m2η,V J sub1 (mη,V , v·p)− J sub1 (mη′,V , v·p) j∈{π,η,η′} [3,1] j ({mπ,V , mη,V , mη′,V }; {mS,V }) I1(mj,V ) j∈{S,η,η′} [3,1] j ({mS,V , mη,V , mη′,V }; {mπ,V }) I1(mj,V ) + [V → A] , (69) 9 The transition B → K occurs through penguin diagrams; D → K is a standard semileptonic decay due to the current in Eq. (18). We keep the notation B → K however to stress that we are working to lowest order in the heavy quark mass. δfB→Kv = (4πf)2 2− 3g2π I1(mK,Ξ)− 3g2πI1(mπ,Ξ) + I1(mS,Ξ) + 2I2(mK,Ξ, v·p) + I2(mS,Ξ, v·p) I1(mS,I) + I1(mπ,I) + 8− 3g2π I1(mη,I) + I2(mη,I , v·p)− I2(mS,I , v·p) + a2δ′V I1(mη′,V )− I1(mη,V ) + I2(mη′,V , v·p)− I2(mη,V , v·p) −m2η,V j∈{S,η,η′} [3,1] j ({mS,V , mη,V , mη′,V }; {mπ,V }) I1(mj,V ) + I2(mj,V , v·p) j∈{π,η,η′} [3,1] j ({mπ,V , mη,V , mη′,V }; {mS,V }) I1(mj,V ) + [V → A] . (70) C. Analytic terms From the power counting discussed in Sec. II, as well as interchange symmetry among the sea quark masses, the form factors at the order we are working can only depend only on the valence quark masses mx and my, the sum of the sea quark masses mu +md +ms, the pion momentum (through v·p), and the lattice spacing, a. The last must appear quadratically, since the errors of the staggered action are O(a2). Recall that we do not include any discretization errors coming from the heavy quark in our effective theory. Thus we expect to have the analytic terms shown in Eqs. (34) and (36) with coefficients i and c i . (Here i = {x, y, sea, 1, 2, a}.) We then can examine, one by one, the known NLO terms in the Lagrangian and current to check for the existence of relations among the c and/or cvi . As soon as a sufficient number of terms are checked to ensure that the parameters are independent, we are done. It is therefore not necessary in all cases to have a complete catalog of NLO terms. Unless otherwise indicated, all NLO terms discussed in this section come from Ref. [19]. Note first of all that we do not need to include explicitly the effects of mass- renormalization terms in the NLO heavy-light Lagrangian, such as 2λ1Tr + 2λ′1Tr , (71) where we define M± = 1 σMσ ± σ†Mσ† . (72) The effect of the terms in Eq. (71) is absorbed into the B∗y -Bx mass difference ∆ yx, Eq. (30), just like the one-loop contribution to the mass. Corresponding O(a2) term in the Lagrangian, which can be obtained by replacing M+ above by various taste-violating operators, can likewise be ignored here. We now consider the discretization corrections parametrized by cpa and c a. There are a large number of O(a2) corrections to the Lagrangian and the current that can contribute to these coefficients, so it is not surprising that they are independent. For example, consider the following terms in the NLO heavy-light Lagrangian cA3,k Tr HHγµγ5{Aµ,OA,+k } , (73) where the OA,+k are various taste-violating operators, similar to those in Eq. (16) above. These terms do not contribute to cva, but only to c a, though corrections to the B-B vertex in Fig. 3(b). On the other hand, there are many terms that contribute both to cva and to cpa. An example is the following correction to the current rA1,k trD (γ µ(1− γ5)H)OA,+k σ †λ(b) , (74) which contributes equally to cva and c a. Additional examples are provided by those terms with two derivatives in the O(mqa2) pion Lagrangian [39], which correct both coefficients though their effect on the pion wave-function renormalization. We consider the v·p and (v·p)2 terms next, namely cv1, cv2, c 1, and c 2. This is a case where a complete catalog of Lagrangian and current corrections does not exist. However, it is easy to find corrections that contribute only to fv or only to fp. As in the previous case, corrections to the B-B∗-π vertex in Fig. 3(b) only affect fp at the order we are working. Thus, (v · →DHH −HHv · ←D) γµγ5Aµ contributes to c 1 only; while HHγµγ5(v · D )2Aµ contributes to c 2 only. Similarly, only fv is affected, though Fig. 3(a), by any correction to the current whose expansion in terms of pion fields starts at linear order (i.e., corrections of schematic form H( iΦ + · · · ), with · · · denoting higher order terms in Φ). Thus, γµ (1−γ5)H v · A σ†λ(b) (77) contributes to cv1 only; while γµ (1−γ5)H v · →Dv · A σ†λ(b) (78) contributes to cv2 only. Since there is at least one Lagrangian or current term that contributes to each of cv1, c 1, and c 2 exclusively, these coefficients are independent. The argument for the independence of the sea-quark mass terms, i.e., the coefficients cvsea and cpsea, is similar. The Lagrangian correction HHγµγ5A Tr(M+) (79) contributes to cpsea only; while the current correction ρ2 trD (γ µ(1− γ5)H)σ†λ(b)Tr(M+) (80) contributes equally to both cpsea and c sea. These two observations are enough to guarantee that cvsea and c sea are independent. We now turn to the coefficients that control the valence quark mass dependence of the form factors: cvx, c x, and c y. At first glance, it would seem unlikely that there could be any constraint among these parameters, since there are seven terms in the Lagrangian and current in Ref. [19] that could generate valence mass dependence.10 However, three of these terms are immediately eliminated, either because they could only contribute to flavor- neutral pions (with x = y), or because they produce no fewer than two pions. There are then two remaining corrections to the heavy-light Lagrangian, ik1Tr HHv·←DM+ − v·→DHHM+ + k3Tr HHγµγ5{Aµ,M+} and two corrections to the current, ρ1 trD (γ µ(1− γ5)H)M+σ†λ(b) + ρ3 trD (γµ(1− γ5)H)M−σ†λ(b) (82) 10 There are additional terms involving Tr(M+), as in Eqs. (79) and (80), that only give sea quark mass dependence at this order. The k3 term in Eq. (81) contributes only to fp, through the B-B ∗-π vertex. However, because of the anticommutator, its contribution is proportional to mx+my, so it gives equal contributions to cpx and c y. Similarly, because the one-pion term in M− is proportional to ΦM +MΦ, the ρ3 term contributes equally to cvx and cvy (but not at all to cpx and cpy). Further, sinceM+ creates only even number of pions, we can replace it byM in Eq. (82). The ρ1 term can then easily be seen to contribute equally to c y and c x, since the current needs to annihilate a B∗y in the fp case and a Bx in the fv case. The contributions of the k1 term in Eq. (81) are the most non-trivial. It contributes to both cvx and c x through wave function renormalization on the external Bx line, but it also contributes to cpy through an insertion on the internal B y line in Fig. 3(b). However, since wave-function renormalization effects on external lines go like Z, the contributions of this term to both cvx and c x are exactly half of its contribution to c y. Thus, all four terms in Eqs. (81) and (82) are consistent with the relation given in Eq. (40). We still need to worry about valence mass dependence generated by the standard O(p4) pion Lagrangian [38] through wave function renormalization of the external pion. Such contributions do exist (from L5), but the x ↔ y symmetry of the pion guarantees they are proportional to mx +my in both fp and fv, and hence do not violate Eq. (40). A consistency check of the relation, Eq. (40), as well as of the claimed independence of the other analytic terms, can be performed by considering the change in the chiral logarithms in Eqs. (52) through (57) and Eqs. (C1) and (C2) under a change in chiral scale. To simplify the calculation, it is very convenient to use the conditions obeyed by sums of residues, which are given in Eq. (38) of the second paper in Ref. [13]. We find that such a scale change can be absorbed by parameters that obey Eq. (40) but are otherwise independent. In the continuum limit, cpsea and c sea remain independent, as do c 1, and c 2. We disagree on these points with Ref. [33], which found cpsea = c sea, and did not consider analytic terms giving v·p dependence. The difference can be traced to the inclusion here of the effects of the complete set of NLO mass-dependent terms, as well as a sufficient number of higher derivative terms (Eqs. (75) through (78)). In particular, the independence of cpsea and c sea can be traced to the existence of the Lagrangian correction, Eq. (79), which was not considered in Ref. [33]. On the other hand, the relation among the valence mass coefficients, Eq. (40), is obeyed by the expressions for these coefficients found in Ref. [33]. This occurs because the contributions of the terms proportional to k3 and ρ3 in Eqs. (81) and (82), which were not considered in Ref. [33], are proportional to mx +my and automatically obey Eq. (40). Note, finally, that the relation in Eq. (40) is almost certain to be violated at next order in HQET. This is because the contributions from operators like the k1 term in Eq. (81) will affect the B and the B∗ differently at O(1/mQ), destroying the cancellation that made Eq. (40) possible. V. FINITE VOLUME EFFECTS In a finite volume, we must replace the integrals in Eqs. (B5) through (B8) by discrete momentum sums. We assume that the time direction is large enough to be considered infinite (this is the case in MILC simulations), and that each of the spatial lengths has (dimensionful) size L. The correction to Eq. (B5) is given explicitly in Ref. [12]. In finite volume, we need only make the replacement I1(m)→ I fv1 (m) = I1(m) +m2δ1(mL) . (83) Here δ1 is a sum over modified Bessel functions δ1(mL) = ~r 6=0 K1(rmL) , (84) where ~r is a 3-vector with integer components, and r ≡ |~r |. Arndt and Lin [42] have worked out the finite volume correction to Eq. (B6). In our notation, the function I2(m,∆) is replaced by its finite volume form, I 2 (m,∆), I2(m,∆)→ I fv2 (m,∆) = I2(m,∆) + δI2(m,∆, L) , (85) where the correction δI2(m,∆, L) is given simply in terms of the function JFV(m,∆, L) defined in Eq. (44) of Ref. [42]:11 δI2(m,∆, L) = −(4π)2∆ JFV(m,∆, L) JFV(m,∆, L) ≡ ~r 6=0 ωq(ωq +∆) sin(qrL) , (86) 11 We have added the L argument to JFV for consistency with our notation with ωq = q2 +m2. The asymptotic form of JFV(m,∆, L) for large mL is useful for practical applications, where typically mL > 3, and often mL > 4 [8, 9]. Arndt and Lin have found [42]: JFV(m,∆, L) = ~r 6=0 e−rmLA , (87) A = e(z2) [1− Erf(z)] + 1− Erf(z) 1− Erf(z) (rmL)3 , (88) where . (89) Computing higher orders in the 1/(mL) expansion is possible if greater precision is needed. Since the functions I1(m) and I2(m,∆) arise from the integral Iµ3 (m,∆) in Eq. (B7), as well as from Eqs. (B5) and (B6), which serve to define them, it is necessary to check that the finite volume corrections coming from Eq. (B7) are just those given by Eqs. (83) and (85) above. This is easily seen to be true in the rest frame of the heavy quark, in which we are working. It is a consequence of the facts that: (1) in the rest frame, only the µ = 0 component of Iµ3 (m,∆) is non-zero, and (2) the integral over dq0 is unaffected by finite volume, since we assume large time-extent of the lattices. The finite volume integral then splits into I fv1 (m) and I 2 (m,∆) pieces, just as in infinite volume. Finally, we have to examine the finite volume corrections to the integral J µν , Eq. (B8). Since the function J2(m,∆) does not enter our final results, we need only evaluate J ≡ (gνµ − vνvµ)J µν = (2π)4 i(gνµ − vνvµ)qµqν (v·q −∆+ iǫ)(q2 −m2 + iǫ) (2π)4 (v·q −∆+ iǫ)(q2 −m2 + iǫ) (4π)2 J1(m,∆) , (90) where q is the spatial 3-vector part of qµ. In the last line, the arrow refers to the fact that the function J1 arises after regularization and renormalization of the integral. A useful regulator in the present context is given by the insertion of a factor of exp(−ωq/Λ0), where Λ0 is a cutoff. After performing the contour integral over q (2π)3 2ωq(ωq +∆) (2π)3 (2π)3 (2π)3 ∆2 −m2 2ωq(ωq +∆) . (91) The first term is a pure divergence with no m or ∆ dependence. It is thus the same in finite volume or infinite volume [12]. The correction to the middle term is proportional to the correction to I1, since the same integral appears after performing the q 0 integration in Eq. (B5). Similarly, the integral in the third term is proportional to that arising from the q0 integration in Eq. (B6), and the correction is therefore already known. We have J1(m,∆)→ J fv1 (m,∆) = J1(m,∆) + δJ1(m,∆, L) , (92) where δJ1(m,∆, L) = m2 −∆2 δI2(m,∆, L)− δ1(mL) (93) The correction to J sub1 , Eq. (51), is δJ sub1 (m,∆, L) = δJ1(m,∆, L) + 16π2m2 JFV(m, 0, L) , (94) where JFV(m, 0, L) is the same as Eq. (87) with A = 1. With the expressions in this section, it is straightforward to incorporate the corrections to I1, I2, and J1 numerically into fits to finite-volume lattice data. VI. CONCLUSIONS We have presented the NLO expressions in partially quenched SχPT for the form factors associated with B → Pxy semileptonic decays, for both infinite and finite volume. Using a quark flow analysis, we have obtained these results by generalizing the NLO PQχPT expressions calculated in the continuum in Ref. [33]. The main subtlety in applying this technique is due to the appearance of taste matrices inside the Feynman diagrams, since non-trivial signs can arise from the anticommutation relations of the taste generators. We have shown that these signs can be accounted for by a careful analysis of the relevant quark flow diagrams. The SχPT expressions are generally necessary for performing chiral fits to lattice simu- lations where staggered light quarks are used. For simpler quantities than the form factors, SχPT has been seen to be essential [10, 20] in order to get reliable extrapolations both to the continuum limit and to the physical quark mass values. For form factors, the lattice data in Ref. [9] was not yet sufficiently precise for the SχPT expressions to be required (over continuum forms) for acceptable fits. However, we expect that the forms derived here will become more and more important as the lattice data improves. Our results are valid to lowest order in HQET; in general, we neglect 1/mB corrections. We do however include the B∗-B splitting ∆∗ on internal B∗ lines that are not in loops. This prescription allows the form factor fp to have the physical m B pole structure. Our treatment of the B∗-B splitting is similar, but not identical, to that of Refs. [33, 40]. Unlike those authors, we iterate self-energy contributions, namely Fig. 5(a) and the effect of the one-loop mass shift of the B, to all orders. This seems to us to be a natural choice, and also makes the one-loop corrections better behaved. Indeed, with the values of light quark masses and momenta typically used in staggered simulations [8, 9], the one-loop B mass shift can dominate other one-loop corrections, so summing such self-energy contributions to all orders seems entirely appropriate. The final answers are then expressed in terms of the splitting ∆∗yx ≡ MB∗y −MBx . In fitting lattice data, we suggest using the actual lattice values of this mass difference (at the simulated light quark mass values and lattice spacings), rather than applying a one-loop formula for the mass shifts. Our primary results for the staggered, partially quenched case with three non-degenerate sea quarks are found in Sec. IVA. The form factor fv (also known as f‖) is given by Eq. (34) in terms of quantities defined in Eqs. (35), (52) and (53), as well as the wave function renormalization factors δZPxy and δZBx that are listed in Eqs. (C1) and (C2) of Appendix C. Similarly, the form factor fp (also known as f⊥), is given by Eq. (36) in terms of quantities defined in Eqs. (31), (37), (38), and (54) through (57), as well as the wave function renormalization factors. We have also found a single relation, Eq. (40), among the parameters that control the analytic valence mass dependence. While this relation is also satisfied by the parameters written down in Ref. [33], it is important to know that it persists even in the presence of the complete NLO forms of the Lagrangian and current. Appropriate limits of our expressions can be taken for various relevant cases, including the case of full (unquenched) staggered QCD (in Sec. IVB) and the case of continuum PQχPT with non-degenerate sea quark masses [Eqs. (63), (64), (C3), and (C4)]. Despite the fact that the latter are continuum results, they have not, to our knowledge, appeared in the literature before. Finally, our expressions can be corrected for finite volume effects using the results of Sec. V. ACKNOWLEDGMENTS We thank J. Bailey, B. Grinstein, A. Kronfeld, P. Mackenzie, S. Sharpe and our colleagues in the MILC collaboration for helpful discussions. We also are grateful to D. Lin for dis- cussions on finite volume corrections and for sharing with us the Mathematica code used to make the expansions in Ref. [42]. This work was partially supported by the U.S. Department of Energy under grant numbers DE-FG02-91ER40628 and DE-FG02-92ER40699. APPENDIX A: FEYNMAN RULES In this appendix we list the SχPT propagators and (some of) the vertices in Minkowski space [19], as well as the corresponding continuum versions. In SχPT, the propagators for the heavy-light mesons are (k) = 2(v·k + iǫ) , (A1) B∗µaB (k) = −iδab(gµν − vµvν) 2(v·k −∆∗ + iǫ) . (A2) Here a, b indicate the flavor-taste of the light quarks, and ∆∗ is the B∗-B splitting in the chiral limit, which we often neglect since we work to leading order in HQET. The BB∗π vertex is: B†a B µb −B∗†µa Bb ∂µΦba , (A3) where repeated indices are summed. Other needed vertices come from the expansion of the LO current, Eq. (18). We have: LO = κB δac − ΦabΦbc + · · · + κvµBa Φac + · · · , (A4) where repeated indices are again summed and · · · represents terms involving higher numbers of pions, as well as contributions from the axial vector part of the current, which are not relevant to the form factors. If desired, each flavor-taste index can be replaced by a pair of indices representing flavor and taste separately. We use Latin indices in the middle of the alphabet (i, j, . . . ) as pure flavor indices, which take on the values 1, 2, . . . , Nsea in full QCD. Greek indices at the beginning of the alphabet (α, β, γ, . . . ) are used for quark taste indices, running from 1 to 4. Thus we can replace a→ iα and write, for example, (k) = iδijδαβ 2(v·k + iǫ) . (A5) As in Refs. [13, 19], pion propagators are treated most easily by dividing them into connected and disconnected pieces, where the disconnected parts come from insertion (and iteration) of the hairpin vertices. The connected propagators are ΦΞijΦ (p) = iδii′δjj′δΞΞ′ p2 −m2ij,Ξ + iǫ , (A6) where Ξ is one of the 16 meson tastes [as defined after Eq. (5)], and mij,Ξ is the tree-level mass of a taste-Ξ meson composed of quarks of flavor i and j: m2ij,Ξ = µ(mi +mj) + a 2∆Ξ. (A7) Here ∆Ξ is the taste splitting, which can be expressed in terms of C1, C3, C4 and C6 in Eq. (16) [13]. There is a residual SO(4) taste symmetry [11] at this order, implying that the mesons within a given taste multiplet (P , V , T , A, or I) are degenerate in mass. We therefore usually use the multiplet label to represent the splittings. Since the heavy-light propagators are most simply written with flavor-taste indices, as in Eqs. (A1) and (A2), it is convenient to rewrite Eq. (A6) in flavor-taste notation also: ΦabΦb′a′ (p) ≡ Φiα,jβΦj′β′,i′α′ (p) = iδii′δjj′T p2 −m2ij,Ξ + iǫ , (A8) where TΞ are the 16 taste generators, Eq. (6). For flavor-charged pions (i 6= j), the complete propagators are just the connected propa- gators in Eq. (A6) or (A8). However, for flavor-neutral pions (i = j), there are disconnected contributions coming from one or more hairpin insertions. At LO, these appear only for taste singlet, vector, or axial-vector pions. Denoting the Minkowski hairpin vertices as −iδ′Ξ, we have [13]: δ′Ξ = a2δ′V , TΞ ∈ {ξµ} (taste vector); a2δ′A, TΞ ∈ {ξµ5} (taste axial-vector); 4m20/3, TΞ = ξI (taste singlet); 0, TΞ ∈ {ξµν , ξ5} (taste tensor or pseudoscalar) δ′V (A) ≡ (C2V (A) − C5V (A)) . (A10) The disconnected pion propagator is then ΦΞijΦ (p) = δijδj′i′δΞΞ′DΞii,i′i′ , (A11) where [13] DΞii,i′i′ = −iδ′Ξ (p2 −m2ii,Ξ) (p2 −m2 i′i′,Ξ) (p2 −m2U,Ξ)(p2 −m2D,Ξ)(p2 −m2S,Ξ) (p2 −m2 )(p2 −m2η,Ξ)(p2 −m2η′,Ξ) . (A12) For concreteness we have assumed that there are three sea-quark flavors: u, d, and s; the generalization to Nsea flavors is immediate. Here mU,Ξ ≡ muu,Ξ is the mass of a taste-Ξ pion made from a u and a ū quark, neglecting hairpin mixing (and similarly for mD,Ξ and mS,Ξ), mπ0,Ξ, mη,Ξ, and mη′,Ξ are the mass eigenvalues after mixing is included, and the iǫ terms have been left implicit. When specifying the particular member of a taste multiplet appearing in the disconnected propagator is unnecessary, we abuse this notation slightly following Eq. (A7) and refer to DVii,i′i′ , DAii,i′i′ , or DIii,i′i′. In flavor-taste notation we have: ΦabΦb′a′ (p) ≡ Φiα,jβΦj′β′,i′α′ (p) = δijδj′i′ TΞαβT β′α′DΞii,i′i′ (A13) For comparison, we now describe the continuum versions of the Feynman rules [18]. Since taste violations do not appear in LHL, Eq. (17), the continuum-theory version of Eqs. (A1) and (A2) are unchanged except that flavor-taste indices are replaced by pure flavor indices (i, j): (k) = 2(v·k + iǫ) [continuum], (A14) B∗µiB (k) = −iδij(gµν − vµvν) 2(v·k + iǫ) [continuum]. (A15) Similarly, the continuum BB∗π [18] and current vertices are identical to those in SχPT, aside from the redefinition of the indices and a factor of 2 for each Φab field due to the non-standard normalization of the generators in the SχPT case, Eq. (6). The continuum version of Eq. (A3) is µi Bj − B ∂µΦji [continuum]; (A16) while the continuum version of Eq. (A4) is LO = κB δℓk − ΦℓiΦik + · · · + κvµBℓ Φℓk + · · · [continuum]. (A17) Because of taste-violations in the SχPT pion sector, the differences between the propaga- tors Eqs. (A6), (A11) and (A12) and their continuum versions are slightly less trivial. The continuum connected propagator is ΦijΦj′i′ (p) = iδii′δjj′ p2 −m2ij + iǫ [continuum], (A18) m2ij = µ(mi +mj) [continuum]. (A19) The continuum disconnected propagator is ΦijΦj′i′ (p) = δijδj′i′Dii,i′i′ [continuum], (A20) where [13] Dii,i′i′ = −iδ′ (p2 −m2ii) (p2 −m2 (p2 −m2U )(p2 −m2D)(p2 −m2S) (p2 −m2 )(p2 −m2η)(p2 −m2η′) [continuum], (A21) with now δ′ = m20/3. Note the difference in normalization between δ′ and the SχPT taste-singlet hairpin, δ′I , Eq. (A9). This arises from the fact that m20/3 is defined to be the strength of the hairpin vertex when one has a single species of quark on each side of the vertex [12]. In the staggered case, each normalized taste-singlet field is made out of four species (tastes), for example φI = 1 (φ11+φ22+φ33+φ44), where φ is flavor neutral, and only taste indices are shown. In the disconnected propagator of two such fields, there are 16 terms, and a factor of (1/2)2 from the normalization, so there is an overall factor of 4 relative to a single-species disconnected propagator, such as that of φ11 with φ22. At one loop, the “external” fields in this propagator are always valence fields, so the normalization issue has nothing directly to do with the fourth root trick for staggered sea quarks. (The normalization is in fact compensated by the extra factors of 2 in the continuum vertices.) The rooting does however affect the η′I mass that appears in denominator of Eq. (A21), which comes from iterations of the hairpin and therefore involves sea quarks. The end result is that m2η′,I ≈ Nseam20/3 (for large m0), rather than ≈ 4Nseam20/3, the value in the unrooted theory [13]. In the continuum, we also have m2η′ ≈ Nseam20/3. APPENDIX B: INTEGRALS Here we collect the integrals needed in evaluating the diagrams for the semileptonic form factors [19, 33]. The disconnected propagators can be written as a sum of single or double poles using the (Euclidean) residue functions introduced in Ref. [13] or their Minkowski-space versions. We define {m} ≡ {m1, m2, . . . , mn} as the set of masses that appear in the denominator of Eq. (A12), and {µ} ≡ {µ1, µ2, . . . , µk} as the numerator set of masses. Then, for n > k and all masses distinct, we have: I [n,k] ({m};{µ}) ≡ i=1(q 2 − µ2i ) j=1(q 2 −m2j + iǫ) [n,k] j ({m};{µ}) q2 −m2j + iǫ , (B1) where the Minkowski space residues R̂ [n,k] j are given by [n,k] j ({m};{µ}) ≡ i=1(m j − µ2i ) r 6=j(m j −m2r) . (B2) If there is one double pole term for q2 = m2ℓ (where mℓ ∈ {m}), then I [n,k]dp (mℓ; {m};{µ}) ≡ i=1(q 2 − µ2i ) (q2 −m2 + iǫ) j=1(q 2 −m2j + iǫ) [n,k] j ({m};{µ}) q2 −m2j + iǫ . (B3) In the end we want to write the results in terms of the Euclidean-space residues R [n,k] because they are ones we have used previously [13, 19]. In Euclidean space the sign of each factor in Eq. (B2) is changed. We therefore have [n,k] j ({m};{µ}) ≡ i=1(µ i −m2j ) r 6=j(m r −m2j) = (−1)n+k−1R̂[n,k]j ({m};{µ}) . (B4) The integrals needed for the form factors are ([17, 33]) I1 = µ4−d (2π)d q2 −m2 + iǫ (4π)2 I1(m) , (B5) I2 = µ4−d (2π)d (v·q −∆+ iǫ)(q2 −m2 + iǫ) (4π)2 I2(m,∆) , (B6) Iµ3 = µ4−d (2π)d (v·q −∆+ iǫ)(q2 −m2 + iǫ) (4π)2 [I2(m,∆) + I1(m)] , (B7) J µν = µ4−d (2π)d iqµqν (v·q −∆+ iǫ)(q2 −m2 + iǫ) (4π)2 [J1(m,∆)g µν + J2(m,∆)v µvν ] , where the arrows represent the fact that the r.h.s. of these expressions have already been renormalized (unlike the corresponding equations in Ref. [33]). APPENDIX C: WAVEFUNCTION RENORMALIZATION FACTORS The one loop chiral corrections to the wave function renormalization factors ZB and ZP are are [13, 19] δZPxy = 3(4πf)2 [I1 (mxf,Ξ) + I1 (myf,Ξ)] j∈M(3,x) ∂m2X,I [3,3] M(3,x)I ;µ I1(mj,I) j∈M(3,y) ∂m2Y,I [3,3] M(3,y)I ;µ I1(mj,I) j∈M(4,xy) [4,3] M(4,xy)I ;µ I1(mj,I) + a2δ′V j∈M(4,x) ∂m2X,V [4,3] M(4,x)V ;µ I1(mj,V ) j∈M(4,y) ∂m2Y,V [4,3] M(4,y)V ;µ I1(mj,V ) j∈M(5,xy) [5,3] M(5,xy)V ;µ I1(mj,V ) V → A , (C1) δZBx = −3g2π (4πf)2 I1(mxf,Ξ) j∈M(3,x) ∂m2X,I [3,3] M(3,x)I ;µ I1(mj,I) + a2δ′V j∈M(4,x) ∂m2X,V [4,3] M(4,x)V ;µ I1(mj,V ) + [V → A] , (C2) where f runs over the sea quarks (u, d, s). For the continuum result in partially quenched χPT, we can simply set a = 0 and ignore taste splittings. In the 1+1+1 case, we get δZcontPxy = 3(4πf)2 [I1 (mxf) + I1 (myf )] j∈M(3,x) [3,3] M(3,x);µ(3) I1(mj) j∈M(3,y) [3,3] M(3,y);µ(3) I1(mj) j∈M(4,xy) [4,3] M(4,xy);µ(3) I1(mj) , (C3) δZcontBx = −3g2π (4πf)2 I1(mxf) j∈M(3,x) [3,3] M(3,x);µ(3) I1(mj) Returning to a 6= 0, and taking the valence quark masses to be mx = my = mu = md, we have the 2+1 full QCD pion result in SχPT: δZπ = 3(4πf)2 [4I1(mπ,Ξ) + 2I1(mK,Ξ)] +(−4a2δ′V ) (m2SV −m (m2ηV −m2πV )(m −m2πV ) I1(mπV ) + (m2SV −m (m2πV −m2ηV )(m −m2ηV ) I1(mηV ) (m2SV −m (m2πV −m )(m2ηV −m I1(mη′ V → A . (C5) Taking the valence quark masses to be mx = mu = md and my = ms gives the 2+1 full QCD kaon result: δZK = 3(4πf)2 (2I1(mπ,Ξ) + 3I1(mK,Ξ) + I1(mS,Ξ)) I1(mπI ) + I1(mηI )− I1(mSI ) +(−a2δ′V ) (m2SV +m − 2m2ηV ) (m2πV −m2ηV )(m −m2ηV )(m −m2ηV ) I1(mηV ) (m2SV +m − 2m2 (m2πV −m )(m2SV −m )(m2ηV −m I1(mη′ m2SV −m (m2ηV −m2πV )(m −m2πV ) I1(mπV ) + m2πV −m (m2ηV −m −m2SV ) I1(mSV ) V → A . (C6) Setting mx = mu = md in Eq. 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[33] (left column) and this paper (right column). Ref. [33] This work (4) Fig. 4(a) (7) Fig. 5(a) (9) Fig. 5(b) (12) Fig. 5(c) (13) Fig. 5(d) (14) Fig. 4(b) (a) (b) FIG. 1: Example of a connected one-loop form factor diagram at (a) the meson level and (b) the quark level. For the meson diagram, the double line is a heavy-light meson while the single line is a pion. For the quark-level diagram, the solid line is a heavy quark and the dashed line is a light quark. The internal sea quark loop is required by the (quark-flow) connected pion propagator; purely valence diagrams are only possible with a disconnected pion propagator. Therefore this diagram gives rise to a factor of Nsea in the degenerate case. (a) (b) FIG. 2: Example of a disconnected one-loop form factor diagram at (a) the meson level and (b) the quark level. The cross in the meson diagram represents the two-point interactions in χPT, and is represented by the “hairpin” in the quark-level diagram. There are no factors of Nsea but instead factors of 1/Nsea coming from the decoupling of the η′. (a) (b) FIG. 3: Tree level diagrams for (a) fv and (b) fp. The double line is the heavy-light meson and the single line is the pion. (a) (b) FIG. 4: One-loop fv diagrams. The internal light meson lines may in general be connected or disconnected: possible hairpin insertions are not shown explicitly. FIG. 5: One-loop fp diagrams. The internal light meson lines may in general be connected or disconnected: possible hairpin insertions are not shown explicitly. FIG. 6: The quark-flow diagram for Fig. 5(b), omitting the heavy quark line for clarity. The mesons in the loop are X and Y mesons, flavor-neutral mesons made up of x and y quarks. Note that even though only a single hairpin insertion is shown explicitly, the figure should be interpreted as representing all diagrams with one or more hairpins. (a) (b) FIG. 7: Possible quark-flow diagrams for Fig. 5(c) with a disconnected meson propagator in the loop. The solid rectangle encloses the 5-point vertex of Fig. 5(c). The heavy quark line has been omitted for clarity. A “reflected” version of diagram (b), with the outgoing pion on the other side of the vertex, is also possible. (a) (b) FIG. 8: Possible quark-flow diagrams for Fig. 5(c) with a connected meson propagator in the loop. The solid rectangle encloses the 5-point vertex of Fig. 5(c). The heavy quark line has been omitted for clarity. Since we have assumed that x and y are different flavors, diagram (a) cannot occur in our case. Diagram (b) can occur, as can a “reflected” version with the outgoing pion on the other side of the vertex. Introduction Heavy-Light Staggered Chiral Perturbation Theory Generalizing Continuum PQ0.4exPT to S0.4exPT Form Factors for BP Decay Form factors for 3-flavor partially quenched S0.4exPT Full QCD Results Analytic terms Finite Volume Effects Conclusions Feynman Rules Integrals Wavefunction Renormalization Factors References

0704.0796

A density tensor hierarchy for open system dynamics: retrieving the noise

arXiv:0704.0796v4 [quant-ph] 16 Jul 2007 April, 2007 A density tensor hierarchy for open system dynamics: retrieving the noise Stephen L. Adler Institute for Advanced Study Princeton, NJ 08540 Send correspondence to: Stephen L. Adler Institute for Advanced Study Einstein Drive, Princeton, NJ 08540 Phone 609-734-8051; FAX 609-924-8399; email [email protected] http://arxiv.org/abs/0704.0796v4 ABSTRACT We develop a density tensor hierarchy for open system dynamics, that recovers information about fluctuations (or “noise”) lost in passing to the re- duced density matrix. For the case of fluctuations arising from a classical probability distribution, the hierarchy is formed from expectations of prod- ucts of pure state density matrix elements, and can be compactly summarized by a simple generating function. For the case of quantum fluctuations arising when a quantum system interacts with a quantum environment in an overall pure state, the corresponding hierarchy is defined as the environmental trace of products of system matrix elements of the full density matrix. Whereas all members of the classical noise hierarchy are system observables, only the lowest member of the quantum noise hierarchy is directly experimentally measurable. The unit trace and idempotence properties of the pure state density matrix imply descent relations for the tensor hierarchies, that relate the order n tensor, under contraction of appropriate pairs of tensor indices, to the order n− 1 tensor. As examples to illustrate the classical probability distribution formalism, we consider a spatially isotropic ensemble of spin- 1/2 pure states, a quantum system evolving by an Itô stochastic Schrödinger equation, and a quantum system evolving by a jump process Schrödinger equation. As examples to illustrate the corresponding trace formalism in the quantum fluctuation case, we consider the tensor hierarchies for collisional Brownian motion of an infinite mass Brownian particle, and for the the weak coupling Born-Markov master equation. In different specializations, the lat- ter gives the hierarchies generalizing the quantum optical master equation and the Caldeira–Leggett master equation. As a further application of the density tensor, we contrast stochastic Schrödinger equations that reduce and that do not reduce the state vector, and discuss why a quantum system cou- pled to a quantum environment behaves like the latter. The descent relations for our various examples are checked in a series of Appendices. 1. Introduction Increasing attention is being paid to the dynamics of open quantum systems, that is, to quantum systems acted on by an environment. Such systems are of interest for studies of dissipative phenomena, decoherence, backgrounds to quantum computers and to precision measurements, and theories of quantum measurement. A principal tool in studying open quantum systems is the reduced density matrix, obtained from the pure state density matrix by tracing over environment degrees of freedom, or in stochastic models where the environ- ment is represented by a noise term in the Schrödinger equation, by averaging over the noise. As is well-known, this transition from the pure state density matrix to the reduced density matrix is not one-to-one, since information about the total system is lost. For example, in stochastic models, there is known to be a continuum of different unravelings, or pure state density matrix stochastic evolutions, that yield the same master equation for the reduced density matrix. The question that we investigate here is the extent to which one can form objects that refer only to the basis vectors of the system Hilbert space, but that nonetheless recapture information that is lost in passing to the reduced density matrix. In the first part of this paper (Sections 2 through 5), we discuss classical noise arising from fluctuations defined by classical probability distributions. In the second part (Sections 6 through 9), we give an analogous discussion of quantum noise, which appears in the physically important case of a quantum system coupled to a quantum environment in an overall pure state. We also give an extension, making contact with the discussion of the first part, to the case in which the overall system is in a mixed state superposition of pure states. The final section contains a discussion of quantum measurements that relates the material in the first and second parts. For the case of classical probability distributions, a relevant discussion appears in Chapter 5 of the book The Theory of Open Quantum Systems by Breuer and Petruccione [1], following up on earlier papers by those authors [2], by Wiseman [3] and by Mølmer, Castin, and Dalibard [4]. In simplified form, Breuer and Petruccione introduce an ensemble of pure state vectors |ψα〉, each drawn from the same system Hilbert space HS , with each vector assumed to occur in the ensemble with probability wα, wα = 1. Measurement of a general self-adjoint operator R for a system prepared in |ψα〉 typically gives a range of values, the mean of which given by 〈ψα|R|ψα〉. The mean or expectation over the ensemble of pure state vectors is then given by wα〈ψα|R|ψα〉 = TrρR , (1a) with ρ the mixed state or reduced density matrix defined by wα|ψα〉〈ψα| . (1b) Breuer and Petruccione point out that there are three variances that are relevant. The variance of measurements of R over all pure states in the ensemble is given by Var(R) = Trρ(R − TrρR)2 = TrρR2 − (TrρR)2 . (2a) This can be written as the sum of two non-negative terms, Var(R) = Var1(R) + Var2(R) , (2b) with Var1(R) the ensemble average of the variances of R within each pure state of the ensemble, Var1(R) = wα[〈ψα|R 2|ψα〉 − 〈ψα|R|ψα〉 2] , (2c) and with Var2(R) the variance of the pure state means of R over the ensemble, Var2(R) = wα〈ψα|R|ψα〉 2 − [ wα〈ψα|R|ψα〉] 2 . (2d) Thus, Var1(R) is an ensemble average of the quantum variances of R, while Var2(R) is a measure of the spread of the average values of R resulting from the statistical properties of the ensemble. As Breuer and Petruccione note, neither of the subsidiary variances Var1,2 can be expressed as the density matrix expectation of some self-adjoint operator. Our aim in the first part of this paper is to extend the formalism of ref [1] by utilizing a density tensor hierarchy, which captures the statistical information that is lost in forming the reduced density matrix of Eq. (1b). A density tensor, defined as an ensemble average of density matrices, was first introduced by Mielnik [5], and was applied to discussions of density functions on the space of quantum states and their application to thermalization of quantum systems by Brody and Hughston [6]. These papers, in addition to introducing the concept of a density tensor which is developed further here, also contain the important result that in the case of a continuum probability distribution, the density tensor hierarchy gives all of the information needed to reconstruct the probability function wα. In particular, the variances Var1,2 for any observable, and more general statistical properties of the ensemble as well, can be expressed as contractions of density tensor matrix elements with appropriate matrix elements of the observable(s) of interest. The basic construction of the density tensor hierarchy corresponding to a classical probability distribution {wα} is given in Sec. 2. Here we generalize the reduced density matrix of Eq. (1b) to a density tensor, formed by taking a product of pure state density matrix elements, and averaging over the ensemble of pure states. When the ψα are independent of α, this tensor reduces to an n-fold product of reduced density matrices, and so the difference between the density tensor and this product is a measure of the statistical fluctuations in the ensemble. In the generic case of non-trivial dependence of ψα on α, there are some general statements that can be made. First of all, the order n density tensor is a symmetric tensor in its pair indices, and it can be considered as a matrix operator acting on the n-fold tensor product of the system Hilbert space HS with itself. The symmetry of the density tensor allows construction of a generating function that on expansion gives the density tensors of all orders. Additionally, as a consequence of the unit trace and idempotence conditions obeyed by the pure state density matrix, the density tensor hierarchy satisfies a system of descent equations, relating the order n tensor to the order n−1 tensor when any row index is contracted with any column index. We show that the variances Var1,2 defined by Breuer and Petruccione can be expressed in terms of appropriate contractions of density tensor elements with operator matrix elements. In subsequent sections we develop some concrete applications of the general formal- ism for classical probability distributions. In Sec. 3, we consider an isotropic ensemble of spin-1/2 pure state density matrices, construct the density tensors through order 3, ver- ify the descent equations, and calculate the generating function. In Sec. 4 we apply the formalism to a quantum system evolving under the influence of noise as described by a stochastic Schrödinger equation, with the ensemble defined as the set of all histories of an initial quantum state under the influence of the noise. Assuming white noise described by the Itô calculus, we give the dynamics of the general density tensor in terms of the general unraveling of the Lindblad equation constructed by Wiseman and Diósi [7], and show that the order two and higher density tensors distinguish between inequivalent unravelings that give the same reduced density matrix (i.e., the same order one density tensor). In Sec. 5 we develop an analogous formalism for the case of jump (piecewise deterministic process) unravelings of the Lindblad equation. We turn next to an analysis of a quantum system coupled to a quantum environment, rather than to an external classical noise source. Here, one is confronted with the problem of discussing the system dissipation associated with the system-environment interaction within a single overall pure state of system plus environment (or in a thermal state that is a weighted average of such pure states). Typically, in master equation derivations, the system- environment interaction1 H has vanishing expectation in the environment, but its square H2 does not have a vanishing expectation, because the environment is not in an eigenstate of H . The associated variance is then a measure of quantum fluctuations associated with the environment state, and is the source of quantum “noise” driving the system dissipation. Our aim in the second part of this paper is to generalize the formalism of the first part to recapture information about this noise that is lost in the passage to the system reduced density matrix. We do this in Sec. 6 by defining a density tensor hierarchy as the trace over the environment of a product of environment operators constructed as the system matrix elements of the total density matrix. Unlike the classical noise construction, which uses only the system density matrix, the construction in the quantum noise case requires knowledge of the full system plus environment density matrix, and so (except for the order one case) does not give a system observable. It is nonetheless computable in any theory of the system plus environment, and is of theoretical, rather than empirical, interest. Because the environment 1 What we call H is usually denoted by HI in the open systems literature. To avoid confusion, all other Hamiltonians will carry subscripts, e.g., HS and HE for the system and environment Hamiltonians, HTOT for the total Hamiltonian, etc. operators entering the construction are non-commutative, this hierarchy is no longer totally symmetric in its system index pairs, but by the cyclic permutation property of the trace, it is symmetric under cyclic permutation of the system index pairs. Also, because the system trace of these environment operators gives only the reduced environment density matrix, rather than unity, there is in general no descent equation associated with taking this trace. However, when indices of adjacent system operators are contracted, one gets the square of the overall density matrix, and so there remains a set of descent relations connecting the order (n) tensor to the order (n− 1) tensor. Finally, in the case of thermal (or other mixed) overall states, we define the appropriate tensor as a weighted sum of pure state tensors, in analogy with the definition of Sec. 2. In subsequent sections, we give applications of the trace hierarchy formalism to several classic problems discussed in the theory of quantum master equations. In Sec. 7 we consider the quantum Brownian motion (and resulting decoherence) of a massive Brownian particle in interaction with an independent particle bath of scatterers. In Sec. 8 we discuss the tensor hierarchy corresponding to the weak coupling Born–Markov master equation, and it specialization to the quantum optical master equation. Finally in Sec. 9, we give an analogous discussion for the Caldeira–Leggett model of a particle in interaction with a system of environmental oscillators. We conclude with a discussion that bridges the considerations of the classical noise and the quantum noise cases. In Sec. 10, we contrast two different Itô stochastic Schrödinger equations, both of which have the same Lindblad, but only one of which leads to state vector reduction. We relate this to the fact that the equation giving the time derivative of the stochastic expectation of operator variances involves the order two density tensor, which differs for the two cases. We discuss the analogous equation for the time dependence of the variance of a “pointer operator” in the case of a quantum system coupled to a quantum environment, and show why this does not lead to state vector reduction. Thus we see no mechanism for quantum “noise” in a closed quantum system plus environment to provide a resolution of the quantum measurement problem. 2. The density tensor for classical noise and its kinematical properties We proceed to establish our notation and to define the density tensor hierarchy in the classical noise case. We denote the pure state density matrix formed from the unit normalized state |ψα〉 by ρα, with ρα = |ψα〉〈ψα| , (3a) and its general matrix element between states |i〉 and |j〉 of HS by ρα;ij ≡ 〈i|ρα|j〉 . (3b) The unit trace condition on ρα states that Trρα = 〈ψα|ψα〉 = 1 , (3c) and the idempotence condition on ρα states that ρ2α = |ψα〉〈ψα||ψα〉〈ψα| = |ψα〉〈ψα| = ρα . (3d) We now define the order n density tensor by i1j1,i2j2,...,injn wαρα;i1j1ρα;i2j2 ...ρα;injn = E[ρα;i1j1ρα;i2j2 ...ρα;injn ] , (4a) with E[Fα] a shorthand for E[Fα] = wαFα . (4b) Since wαρα;ij = wα〈i|ρα|j〉 , (5a) we see that this is just the |i〉 to |j〉 matrix element of the reduced density matrix ρ defined in Eq. (1b), ij = 〈i|ρ|j〉 , (5b) and so the density tensor of Eq. (4a) is a natural generalization of the usual reduced density matrix. When the states |ψα〉 are independent of the label α, the definition of Eq. (4a) simplifies to i1j1,i2j2,...,injn = ρi1j1ρi2j2 ...ρinjn , (5c) and so the difference between Eq. (4a) and a product of reduced density matrix elements is a reflection of the statistical structure of the ensemble. Since the factors within the expectation E[...] on the right of Eq. (4a) are just ordinary complex numbers, the density tensor is symmetric under interchange of any index pair iljl with any other index pair imjm. Consequently, we can define a generating function for the density tensor by G[aij ] = E[e ρα;ijaij ] = ai1j1 ...ainjn i1j1,...,injn , (5d) where repeated indices i, j are summed. It will often be convenient to abbreviate ρα;ijaij by ρα · a, so that the generating function becomes in this notation G[a] = E[e ρα ·a]. Although the density tensor for n > 1 is not an operator on HS , it clearly has the structure of an operator on the n-fold tensor product HS ⊗ HS ⊗ ... ⊗ HS . Motivated by this, we will often find it convenient to write the definition of Eq. (4a) as ρ(n) = E , (5e) with each factor ρα;ℓ acting on a distinct factor Hilbert space HS;ℓ in the tensor product ℓ=1 HS;ℓ. One can pass easily back and forth from this notation to one in which the system matrix indices are displayed explicitly. Let us consider next the result of contracting any row index il with any column index jk. There are two basic cases: (i) one can contract a row index il with its corresponding column index jl, and (ii) one can contract a row index il with a column index jk with k 6= l. Since the density tensor is symmetric in its index pairs, it suffices to consider only one example of each case, since all others can be obtained by permutation. For the contraction of i1 with j1 we find δi1j1ρ i1j1,i2j2,...,injn = E[(Trρ)ρα;i2j2 ...ρα;injn ] = E[ρα;i2j2 ...ρα;injn ] = ρ (n−1) i2j2,...,injn , (6a) where we have used the unit trace condition of Eq. (3c). For the contraction of j1 with i2, we find δj1i2ρ i1j1,i2j2,...,injn = E[(ρ2)α;i1j2 ...ρα;injn ] = E[ρα;i1j2ρα;i3j3 ...ρα;injn ] = ρ (n−1) i1j2,i3j3,...,injn , (6b) where now we have used the idempotence condition of Eq. (3d). As an illustration of how this works when all possible index pair contractions are considered, we give the complete set of contractions reducing the second order density tensor to a first order density tensor, δi1j1ρ i1j1,i2j2 δi2j2ρ i1j1,i2j2 δj1i2ρ i1j1,i2j2 δj2i1ρ i1j1,i2j2 Referring to the generating function of Eq. (5d), the general descent equations can be sum- marized compactly by the two identities, ∂G[aij ] =E[(Trρα)e ρα;ijaij ] = G[aij ] , ∂2G[aij ] ∂amr∂apq =E[ρmrρrqe ρα;ijaij ] = E[ρmqe ρα;ijaij ] = ∂G[aij ] When the density matrix ρ used to define the density tensor is a mixed state density matrix, the trace descent relation of Eq. (6a) is unchanged, while the indempotency relation of Eq. (6b) relates the contraction an order (n) tensor to an order (n− 1) tensor in which one factor ρ is replaced by ρ2; this is not a member of the original hierarchy, but still gives a useful relation for checking calculations. To conclude this section, let us return to the variances introduced by Breuer and Petruccione. In terms of the order one and order two density tensors, we evidently have Var1(R) =ρ (R2)j1i1 − ρ i1j1,i2j2 Rj1i1Rj2i2 , Var2(R) =ρ i1j1,i2j2 Rj1i1Rj2i2 − (ρ Rj1i1) Var(R) =ρ (R2)j1i1 − (ρ Rj1i1) with Rji = 〈j|R|i〉. Clearly, other statistical properties of the ensemble are readily expressed in terms of the density tensor hierarchy. For example, the ensemble average of the product of the expectations of two different operators R and S is given by wα〈ψα|R|ψα〉〈ψα|S|ψα〉 = ρ i1j1,i2j2 Rj1i1Sj2i2 , (8b) which can be used, together with information obtained from ρ(1), to calculate the covariance and correlation of R and S. 3. Isotropic spin-1/2 ensemble As a simple example of the density tensor formalism, let us follow Breuer and Petruc- cione [1] and consider the case of an isotropic spin-1/2 ensemble. Let ~v be a vector in three dimensions, and consider the ensemble of spin-1/2 pure state density matrices ρ(~v) = (1 + ~v · ~σ) , (9a) with ~σ = (σ1, σ2, σ3) the standard Pauli matrices, and with a uniform probability distribution of ~v over the unit sphere |~v | = 1 specified by w(~v ) = δ(|~v | − 1) . (9b) (Clearly, ~v has the same significance as the label α used in the preceding section.) Defining E[P (~v )] = d3vw(~v )P (~v ) , (10a) a standard calculation gives E[1] = 1 , E[vsvt] = δst , ... , (10b) with all averages of odd powers of ~v vanishing. From Eq. (9a), we have ρ(~v )ij = (δij + vrσ ij) , (11a) and the general density tensor over this ensemble is defined by i1j1,...,injn = E[ρ(~v )i1j1 ...ρ(~v )injn ] . (11b) From Eq. (10b), the first three tensors in this hierarchy are now easily found to be δi1j1 , i1j1,i2j2 δi1j1δi2j2 + ~σi1j1 · ~σi2j2 i1j1,i2j2,i3j3 δi1j1δi2j2δi3j3 + (δi1j1~σi2j2 · ~σi3j3 + δi2j2~σi1j1 · ~σi3j3 + δi3j3~σi1j1 · ~σi2j2) Using the relations Tr~σ = 0 and (~σ 2)ij = 3δij , it is now easy to verify that the descent relations of Eqs. (6a) and (6b) are satisfied by Eq. (12). For the isotropic spin-1/2 ensemble, the generating function of Eq. (5d) becomes G[aij ] = E[e ρ(~v)ijaij ] , (13) with ρ(~v )ij given by Eq. (11a). Defining the vector ~A by ~σijaij , (14a) a simple calculation gives G[aij ] = exp( sinh | ~A | | ~A | = exp( Tra)[1 + ( ~A 2)2 + ...] , (14b) from which one can read off the values of the low order density tensors given in Eq. (12). The verification of the descent relations of Eq. (7b) for the generating function of Eq. (14b) is given in Appendix A. 4. Itô stochastic Schrödinger equation We consider next a state vector |ψ〉 with a time evolution described by a stochas- tic Schrödinger equation, which is a frequently used model approximation to open system dynamics. In this case the state vector and the corresponding pure state density matrix ρ = |ψ〉〈ψ| are implicit functions of the noise, which takes a different sequence of values for each history of the system. In the notation of Sec. 2, the different histories are labeled by the subscript α, and the expectation of Eq. (4b) is an average over all possible histories. It is customary, however, in discussing stochastic Schrödinger equations to omit the subscript α, treating the history dependence of ρ as understood. So in this context, the definition of Eq. (4a) becomes i1j1,...,injn = E[ρi1j1 ...ρinjn ] , (15) with E[...] the stochastic expectation, and the generating function G[aij ] takes the same form as given in Eq. (5d) but with the subscript α omitted. Our aim in this section is to derive an equation of motion for the generating function, which on expansion yields equations of motion for all density tensors ρ(n), taking as input the general pure state density matrix evolution constructed by Wiseman and Diósi [7], that corresponds to a given Lindblad form [8,9] for the time evolution of the reduced density matrix ρ(1) = E[ρ]. We begin by recapitulating the results of ref [7]. The most general evolution of a density matrix ρ that preserves Trρ = 1 and obeys the complete positivity condition is the Lindblad form dρ = dtLρ , (16a) Lρ ≡ −i[HTOT, ρ] + ckρc ck, ρ} , (16b) with {, } denoting the anticommutator, and with the repeated index k summed. The set of Lindblad operators ck describes the effects on the system of the reservoir or environment that is modeled by an external classical noise. Wiseman and Diósi show that the most general evolution of the pure state density matrix ρ for which E[dρ] reduces to Eqs. (16a) and (16b) takes the form dρ = dtLρ+ |dφ〉〈ψ|+ |ψ〉〈dφ| . (17a) Here |dφ〉 is a state vector that is a pure noise term, so that E[|dφ〉] = 0 , (17b) that is orthogonal to |ψ〉, so that 〈ψ|dφ〉 = 0 , (17c) and that obeys |dφ〉〈dφ| = dtW . (17d) The operator W is the Diósi transition rate operator [5] given by W =Lρ− {ρ,Lρ}+ ρTr(ρLρ) =(ck − 〈ck〉)ρ(ck − 〈ck〉) , (18) where 〈ck〉 is a shorthand for the quantum state expectation 〈ψ|ck|ψ〉 = Trρck. Although |dφ〉〈dφ| is completely fixed, Wiseman and Diósi show that |dφ〉|dφ〉 is free, with different choices for this and different phase choices for the ck corresponding to different pure state evolutions (or “unravelings”) that yield the same evolution of Eqs. (16a) and (16b) for the reduced density matrix ρ. Wiseman and Diósi further show that |dφ〉 can be parameterized by complex Wiener processes by writing |dφ〉 = (ck − 〈ck〉)|ψ〉dξ k , (19a) E[dξk] = E[dξ k] = 0 (19b) and with dξj(t)dξ k(t) =dtδjk dξj(t)dξk(t) =dtujk , (19c) where ukj = ujk is a set of arbitrary complex numbers subject to the condition that the norm of the complex matrix u ≡ [ujk] be less than or equal to 1. (See Eqs. (4.10) and (4.11) of ref. [7].) In terms of this parameterization of |dφ〉, the pure state evolution of Eq. (17a) takes the form dρ = dtLρ+ (ck − 〈ck〉)ρdξ k + ρ(ck − 〈ck〉) †dξk , (19d) and the corresponding stochastic Schrödinger equation for the wave function is [7] d|ψ〉 =− iHψdt|ψ〉+ (ck − 〈ck〉)dξ k|ψ〉 , −iHψ =− iHTOT − kck − 2〈ck〉 ∗ck + 〈ck〉 ∗〈ck〉 (19e) We proceed now to use pure state evolution of Eq. (19d) to calculate the evolution equation for the generating function G[aij ] = E[exp(ρijaij)] . (20a) To calculate the differential of Eq. (20a), we use the Itô stochastic calculus rule for the differential of a function f(w) of a stochastic variable w, df(w) = dwf ′(w) + (dw)2f ′′(w) . (20b) Applying this to Eq. (20a), we get dG[aij ] = E[(dρmramr + dρmramrdρpqapq) exp(ρijaij)] . (20c) Substituting Eq. (19d) for dρ, and using Eqs. (19a-c), together with the Itô calculus rule E[dwf(w)] = 0, we get dG[aij ] = dtE amr(Lρ)mr + amrapqCmr,pq exp(ρijaij) , (21a) with the coefficient of the quadratic term in aij given by Cmr,pq =Cpq,mr = dρmrdρpq =〈m|(ck − 〈ck〉)ρ|r〉〈p|ρ(ck − 〈ck〉) +〈m|ρ(ck − 〈ck〉) †|r〉〈p|(ck − 〈ck〉)ρ|q〉 +〈m|(ck − 〈ck〉)ρ|r〉〈p|(cℓ − 〈cℓ〉)ρ|q〉u +〈m|ρ(ck − 〈ck〉) †|r〉〈p|ρ(cℓ − 〈cℓ〉) †|q〉ukℓ (21b) This expression can be rearranged by using the identity, valid for general operators A,B, general states |r〉, |m〉, and general pure state (idempotent) density matrix ρ, ρA|r〉〈m|Bρ = ρ〈m|BρA|r〉 , (22a) giving an alternative result for Cmr,pq Cmr,pq =Wmqρpr +Wprρmq +[(ck − 〈ck〉)ρ]mqu kℓ[(cℓ − 〈cℓ〉)ρ]pr +[ρ(ck − 〈ck〉) †]prukℓ[ρ(cℓ − 〈cℓ〉) †]mq , (22b) where we have used Eq. (18) defining the operator W , and where we use the subscript notation of Eq. (3b) for matrix elements, so that in general Amr = 〈m|A|r〉. From the evolution equation of Eqs. (21a,b) and (22b) for the generating function, by expansion in powers of a we can read off the evolution equation for the general density tensor of order n. Employing now the condensed notation of Eq. (5e), in which matrix indices are not indicated explicitly, we have dρ(n) =dtE[ (ρ1...ρn)ℓ(Lρ)ℓ ℓ<m=1 (ρ1...ρn)ℓmCℓm] . (23a) Here (ρ1...ρn)ℓ denotes the product j=1 ρj with the factor ρℓ omitted, and similarly, (ρ1...ρn)ℓm denotes the product ρj with the factors ρℓ and ρm omitted. 2 The coefficient Cℓm is given by Cℓm = Cmℓ =[(ck − 〈ck〉)ρ]ℓ[ρ(ck − 〈ck〉) +[ρ(ck − 〈ck〉) †]ℓ[(ck − 〈ck〉)ρ]m +[(ck − 〈ck〉)ρ]ℓ[(ck̄ − 〈ck̄〉)ρ]mu +[ρ(ck − 〈ck〉) †]ℓ[ρ(ck̄ − 〈ck̄〉) †]mukk̄ , (23b) which corresponds in an obvious way to Eq. (21b) when matrix elements are written explicitly between states 〈m| and |r〉 in the Hilbert space labeled by ℓ, and between states 〈p| and |q〉 in the Hilbert space labeled by m. (No relation is implied between the m used as a state label, and the m used as a Hilbert space label.) Since Cℓm in Eq. (23a), which depends through the terms involving ukk̄ on the choice of unraveling, is multiplied by two powers of a, it does not contribute to the evolution equation for the reduced density matrix ρ(1). So as expected, the reduced density matrix evolution is given solely by the Lindblad term and is independent of the choice of unraveling. Higher density tensors ρ(n), with n ≥ 2, have evolution equations that receive contributions from Cℓm, and so contain information that distinguishes between different unravelings of the Lindblad evolution. As a simple illustration of how the tensors ρ(n) for n ≥ 2 distinguish between different unravelings, let us consider the case of real noise, dξk = dξ k, for which ujk = δjk, and with 2 For n = 1, (ρ1)1 = 1 and (ρ1)ℓm = 0, while for n = 2, (ρ1ρ2)12 = 1. a single Lindblad c1, which we choose as either c1 = A or c1 = iA, with A a self-adjoint operator. Both choices of c1 lead to the same Lindblad, since L is invariant under rephasing of ck, but through the ukk̄ terms they lead to different expressions for Cmr,pq. When ck = A, we find from Eq. (21b) Cmr,pq =〈m|{A− 〈A〉, ρ}|r〉〈p|{A− 〈A〉, ρ}|q〉 =〈m|[ρ, [ρ, A]]|r〉〈p|[ρ, [ρ, A]]|q〉 , (24a) while when ck = iA, we have instead Cmr,pq = −〈m|[A, ρ]|r〉〈p|[A, ρ]|q〉 . (24b) We will return to this example in Sec. 10. Using the expression of Eq. (21a) for the time evolution of the generating function, the descent equations of Eq. (7b) can be verified; this calculation is carried out in Appendix 5. Jump process Schrödinger equation As our next density tensor application we consider the jump process (piecewise de- terministic process, or PDP) Schrödinger equation, given by d|ψ〉 = Adt|ψ〉+BkdNk|ψ〉 , (25a) where a sum over k is understood, with A and the Bk general (non-self-adjoint) operators, and with the dNk independent discrete random variables obeying dNjdNk = δjkdNk , dNjdt = 0 . (25b) Straightforward calculation shows that this process preserves the norm of |ψ〉 and the pure state condition ρ2 = ρ = |ψ〉〈ψ|, provided that A and B obey the restrictions 〈A+ A†〉 =0, 〈Bk +B kBk〉 =0 , (25c) with no summation over k on the second line, which must hold individually for each value of k. Corresponding to Eq. (25a), the density matrix obeys the evolution equation dρ =(Aρ+ ρA†)dt+QkdNk , Qk =Bkρ+ ρB +BkρB (25d) with a sum over k understood in the dNk term on the first line, but no sum over k understood in the second line. Let now E|ψ〉[...] denote an expectation conditioned on the current value of the wave function being |ψ〉, and E[...] be the expectation value over the entire history of the jump process (which leads to an ensemble of different current values of the wave function). We wish to find restrictions on A, Bk, and on E|ψ〉[dNk] ≡ vkdt , (26a) such that the expectation of dρ takes the Lindblad form of Eq. (16b), that is, E[dρ] =dtLρ Lρ =− i[HTOT, ρ] + ckρc kck, ρ} . (26b) Making the Ansatz ck −Kk − 1 , (27a) withKk constants (this Ansatz includes both the standard quantum jump equation (Kk = 0), and the orthogonal jump equation (Kk = 〈ck〉), as special cases; see Schack and Brun [10] for a concise review), some calculation shows that the conditions of Eqs. (25c) and (26b) are satisfied if we choose vk =〈(ck −Kk) †(ck −Kk)〉 , A =− iHTOT − kck + kck〉+ ckK (〈ck〉K k + 〈ck〉 ∗Kk) . (27b) Let us now define the order n density tensor for the jump models by ρ(n) = E[ ρℓ] , (28a) where we use the condensed notation of Eq. (5e). For the differential of this, we find dρ(n) =E[ (ρ1...ρn)ℓdρℓ + ℓ<m=1 (ρ1...ρn)ℓmdρℓdρm ℓ<m<p=1 (ρ1...ρn)ℓmpdρℓdρmdρp + ... + dρ1dρ2dρ3...dρn−1dρn] , (28b) where all powers of dρ must be retained because dN2k = dNk. Using the conditional proba- bility formula p(|ψ〉∩dNk) = p(dNk| |ψ〉)p(|ψ〉), we get the conditional expectation formula, valid for an arbitrary function F of the state |ψ〉, E[F (|ψ〉)dNk] = E[F (|ψ〉)E|ψ〉[dNk]] = E[F (|ψ〉)vk] . (29a) Using this equation to evaluate the higher order terms in Eq. (28b), together with Eq. (26b) for the leading term, we get dρ(n) =dtE[ (ρ1...ρn)ℓ(Lρ)ℓ ℓ<m=1 (ρ1...ρn)ℓmvk(Qk)ℓ(Qk)m ℓ<m<p=1 (ρ1...ρn)ℓmpvk(Qk)ℓ(Qk)m(Qk)p + ...+ vk(Qk)1(Qk)2(Qk)3...(Qk)n−1(Qk)n] , (29b) with a sum over k in each term containing vk. Writing the corresponding generating function in compact notation as G[a] = E[ea·ρ] , (30a) the evolution equation for G is given , with the k sum now indicated explicitly, by dG[a] =E[ea·dρea·ρ]− E[ea·ρ] (a · dρ)p )ea·ρ] =dtE[(a · Lρ+ (a ·Qk) )ea·ρ] . (30b) From Eq. (30b), and the identities which follow, after some algebra, from Eqs. (16b), (25d), (27a), and (27b) {ρ,Lρ} =Lρ− {ρ,Qk} =Qk −Q (30c) one can prove that Eq. (30b) obeys the descent equations, as shown in Appendix C. 6. The density tensor for quantum noise and its kinematical properties Let us now consider a closed quantum system, consisting of a system S interacting with an environment E . In such a situation, one does not have a classical probability distri- bution wα and fluctuations associated with this probability distribution. Instead, one deals with the system plus environment as the only pure state that is given, with the fluctuations that are averaged over in deriving the master equation coming from quantum fluctuations associated with the system-environment interaction. Weighted averages of the sort that we have used in our definition of Eq. (4a) appear only when the total state is a mixture of pure states, such as a thermal state, but in this case, important system quantum fluctuations still occur in each pure state component of this mixture. In order to describe this more general situation, we shall have to generalize our definition of a density tensor hierarchy. To achieve this, we initially suppose the overall system plus environment to have the pure state density matrix ρ. We denote the system basis states by |i〉 , as well as |j〉, and denote the environment basis states by |ea〉, a = 1, 2, ..... A general density matrix element has the form 〈e1i|ρ|e2j〉, and the standard reduced density matrix, with the environment traced out, is defined by ij = (TrEρ)ij = 〈ei|ρ|ej〉 . (31) In order to recapture fluctuations that are averaged over in the trace in Eq. (31), we define the density tensor ρ(n) by i1j1,i2j2,...,injn e1,e2,...,en 〈e1i1|ρ|e2j1〉〈e2i2|ρ|e3j2〉...〈en−1in−1|ρ|enjn−1〉〈enin|ρ|e1jn〉 =TrEρi1j1ρi2j2 ...ρinjn . (32a) Here we have defined ρiℓjℓ as the matrix, labeled by the system state labels iℓ, jℓ, acting on the environment Hilbert space HE according to (ρiℓjℓ)e1e2 = 〈e1|ρiℓjℓ |e2〉 = 〈e1iℓ|ρ|e2jℓ〉 . (32b) The density tensor ρ(n) is again an operator on a tensor product of system Hilbert spaces ℓ=1 HS;ℓ. Thus, in a condensed notation analogous to that of Eq. (5e), we can also write Eq. (32a) as ρ(n) = TrEρ1ρ2....ρn , (32c) where ρℓ is an operator acting on HE ⊗HS;ℓ. We have avoided using a product notation ℓ=1 in Eq. (32c) because the factors ρiℓjℓ in Eq. (32a) and ρℓ in Eq. (32c) are different operators on the environment for each ℓ and thus do not commute. Hence the density tensor is not symmetric under permutation of its pair indices iℓjℓ, but it is symmetric under cyclical permutation of the indices, as a result of the cyclic symmetry of the trace. For n = 2, cyclic symmetry is equivalent to symmetry under pair index interchange, and for n = 3, using the identity TrABC = Tr ([A,B]C + {A,B}C) , (33) cyclic symmetry is equivalent to the statement that the density tensor ρ(3) can be written as the sum of two tensors ρ(3) = ρ(3S) + ρ(3A), with ρ(3S) completely symmetric, and ρ(3A) completely antisymmetric, under pair index interchange. Also because the density tensor is not totally symmetric in its pair indices, we cannot introduce a generating function by imitating Eq. (5d) Similarly, because of factor non-commutativity, the density tensor satisfies only a subset of the descent equations of Eqs. (6a), (6b), and (7b). Contraction with δiℓjℓ does not lead to a descent condition, since δiℓjℓρiℓjℓ is not unity, but rather TrSρ, the reduced density matrix that acts on the environment when the system is traced out. Contraction of a general jk with a general iℓ for k 6= ℓ gives nothing useful, since in general non-commuting factors stand between ρk and ρℓ. However, when a column index jk is contracted with the adjacent row index ik+1, the two density matrices to which they are attached are linked to form the product ρ2 = ρ, and so we get the descent relation of Eq. (6b), and others related to it by cyclic permutation symmetry, δj1i2ρ i1j1,i2j2,...,injn (n−1) i1j2,i3j3,...,injn . (34) As noted before, even when ρ2 6= ρ, the descent relation corresponding to Eq. (34) is still useful for checking calculations. Since we cannot define a generating function as in Eq. (5d), in the quantum noise case we do not have analogs of the descent equations in the form of Eq. (7b); when verifying the descent equations in the various cases considered below, we will work directly from Eq. (34). We will also consider a more general definition of the density tensor, corresponding to the case in which the system plus environment is in a mixed state composed of pure states ρα with weights wα. Typically, α refers to an eigenvalue of a conserved quantum number of the total system, such as the energy; when the environment is considered in the independent particle approximation, with the system back reaction on the environment neglected, α then can refer to the energies and momenta of each environmental particle. In this case we define the density tensor by i1j1,i2j2,...,injn α;i1j1,i2j2,...,injn , (35a) α;i1j1,i2j2,...,injn = TrEρα;i1j1ρα;i2j2 ...ρα;injn . (35b) This definition gives information about both the quantum noise or fluctuations contained within each ρα, and the classical noise or fluctuations associated with the probability distri- bution wα. Note that in the mixed state case one could also define a density tensor that is a direct analog of the classical noise definition of Sec. 2, by (n);CL i1j1,i2j2,...,injn TrEρα;iℓjℓ (36) which would give information only about the classical noise fluctuations associated with the probability distribution wα. In the examples computed in the following sections, where a weak coupling approximation is made, the definition of Eq. (36) typically contains no more information than could be gotten from a product of n reduced density matrix factors, each of the form wαTrEρα;iℓjℓ . As already noted, the density tensor ρ(n) is not measurable by any operation on the system Hilbert space. Its construction requires knowledge of the full system plus envi- ronment density matrix, which is not experimentally accessible for complex environments. Nonetheless ρ(n) is computable in any theory of the system-environment interaction, and we believe it to be of conceptual and theoretical interest, even if not of direct empirical relevance. We close out this section by noting that in the quantum noise case, there is no analog of Eq.(8a), which relates the positive semidefinite variations Var1,2 to the density tensor ρ in the classical noise case. The closest analog we find to the fluctuation formulas of Eq. (8a) involves the n = 3 density tensor. The reason for this is that whereas E[1] = 1, the trace over the environment of unity is the dimension of the environmental Hilbert space; to get a unit trace over the environment we must include a factor of ρE ≡ TrSρ, the reduced density matrix for the environment. This pushes up the order of the density tensor involved from 2 to 3. Specifically, let AS be an operator acting on HS , but which acts as the unit operator on HE . In place of the expectations used in the classical noise discussion of Eqs. (1a) through (2d), in the quantum noise case of system plus environment we consider the expression AE ≡ TrSρAS , which is an operator on the environmental Hilbert space. The trace of this operator over the environment is TrEAE = TrETrSρAS = TrS(TrEρ)AS = TrSρ (1)AS , giving the expectation of the operator AS when the environment is not observed. On the other hand, the expectation of this operator formed from the environmental reduced density matrix is TrEρEAE . The mean squared fluctuation of this operator over the environment is positive semidefinite, and is given by TrEρE(AE − TrEρEAE) 2 = TrEρEA E − (TrEρEAE) 2 , (37a) where we have used the fact that TrEρE = Trρ = 1. Reexpressing Eq. (37a) entirely in terms of the pure state density matrix ρ, we have TrETrSρ(TrSρAS) 2 − (TrETrSρTrSρAS) =δj1i1ASj2i2ASj3i3ρ i1j1,i2j2,i3j3 − (δj1i1ASj2i2ρ i1j1,i2j2 (37b) where we have used the fact that the right hand side of Eq. (37b) involves only the symmetric part of the order 3 density tensor. Thus, as noted above, where a n = 2 density tensor appears in Eq. (8a), a n = 3 density tensor appears in Eq. (37b), and where a n = 1 density tensor appears in Eq. (8a), a n = 2 density tensor appears in Eq. (37b). 7. Collisional Brownian Motion As our first application of Eqs. (32a-c) and Eqs. (35a,b), we consider the collisional Brownian motion of a massive Brownian particle immersed in a bath of scattering particles. We work in the approximation of neglecting recoil of the Brownian particle, and of treating the bath as a collection of free particles of massm. We consider the pure state density matrix corresponding to definite momenta {~ki} of the bath particles, calculate the corresponding order n density tensor defined by Eqs. (32a-c), and then average over the thermal distribution of the bath particles as in Eqs. (35a,b). Thus the initial density matrix for the total system, corresponding to the factor ρℓ in Eq. (33d), is ρTOTℓ = ρℓρE , (38a) with ρℓ the initial density matrix of the Brownian particle, characterized by its coordinate matrix elements 〈~Rℓ|ρℓ|~R ℓ〉, and with ρE the product density matrix for the bath particles, |~ki〉〈~ki| . (38b) Since the bath particle scatterings are all independent, we focus on the effect of the scattering of a single bath particle, of initial momentum ~k, on the Brownian particle, which we take to be in a superposition of position eigenstates. Thus the initial state of the Brownian particle and the bath particle that we are considering is |I〉 = ~R 〉|~k 〉 , (39a) corresponding to an initial state density matrix ρI =|I〉〈I| |~R 〉|~k〉〈~k|〈~R′| . (39b) The corresponding Brownian particle matrix element of ρI , which is still an operator on the bath particle state, takes the form 〈~R|ρI |~R ′〉 = ρ(~R, ~R′)|~k〉〈~k| , (39c) ρ(~R, ~R′) = c~Rc . (39d) Asymptotically, the effect of the scattering is to replace the initial state |I〉 by |F 〉 = S|I〉, with S the scattering matrix. Substituting Eq. (39a), and using translation invariance to relate the scattering matrix S with the Brownian particle at a general coordinate, to the scattering matrix S0 with the Brownian particle at the origin, we get [11] |F 〉 =S|I〉 = c~RS| ~R 〉|~k 〉 ~R 〉e−i ~kOP·~RS0e i~kOP·~R|~k 〉 , (40a) with ~kOP the momentum operator for the bath particle. The corresponding final density matrix is then ρF =|F 〉〈F | |~R 〉e−i ~kOP·~RS0e i~kOP·~R|~k 〉〈~k|e−i ~kOP·~R i~kOP·~R 〈~R′| , (40b) and the Brownian particle matrix element of ρF , which is again an operator acting on the bath particle, is 〈~R|ρF |~R ′〉 = ρ(~R, ~R′)e−i ~kOP·~RS0e i~kOP·~R|~k〉〈~k|e−i ~kOP·~R i~kOP·~R . (40c) Substituting this expression into Eq. (33d), we get ~R1 ~R ,..., ~Rn ~R′n;F ρ(~Rℓ, ~R ~k|e−i ~kOP·~R i~kOP·~R ~kOP·~Rℓ+1S0e i~kOP·~Rℓ+1 |~k〉 ρ(~Rℓ, ~R i~kOP·(~R −~Rℓ+1)S0|~k〉e i~k·(~Rℓ+1−~R (40d) with ~Rn+1 = ~R1. The matrix element appearing in the final line of Eq. (40d) is one that is familiar from the standard calculation of the reduced density matrix (that is, ρ ~R, ~R′ ) for collisional decoherence [12]. Writing 〈~k|S i~kOP·(~R −~Rℓ+1)S0|~k〉e i~k·(~Rℓ+1−~R ) = 1 + f(~Rℓ+1 − ~R ℓ) , (41a) with f proportional to the square of the scattering amplitude, the product of matrix elements in Eq. (40d) can be written, to second order accuracy in the scattering amplitude, as [1 + f(~Rℓ+1 − ~R ℓ)] ≃ 1 + f(~Rℓ+1 − ~R ℓ) . (41b) We also note that Eq. (39c), when substituted into Eq. (32c), implies that the value of ρ(n) before the scattering is ~R1 ~R ,..., ~Rn ~R′n;I ρ(~Rℓ, ~R ℓ) . (41c) Thus when the approximation of Eq. (41b) is substituted into Eq. (40d), we get ~R1 ~R ,..., ~Rn ~R′n;F ~R1 ~R ,..., ~Rn ~R′n;I f(~Rℓ+1 − ~R ~R1 ~R ,..., ~Rn ~R′n;I . (41d) At this point our work is essentially finished, since the remaining steps are identical to the standard calculation [11,12,13] proceeding from the n = 1 case of Eq. (41d), and the structure of Eq. (41d) makes it clear how to generalize the standard result for ρ(1) to the case of general ρ(n). In brief, the standard procedure is to multiply the right hand side of Eq. (41d) by the number of scattering particles, which combines with a normalizing factor of the inverse volume to give an overall factor of N , the scattering particle density. The effect of the thermal distribution µ(~k) of momenta ~k is taken into account by including an integral d~kµ(~k), in accordance with the mixed state procedure of Eq. (35a). Finally, expressing the S matrix in terms of the scattering amplitude f(~k′, ~k), and noting that the squared delta function for energy conservation gives an overall factor of the elapsed time, Eq. (41d) becomes, in the limit of small elapsed time, a formula for the time derivative of ρ(n). For the n = 1 case, the standard answer obtained this way is ∂ρ(1)(t)~R~R′ = −F (~R − ~R′)ρ(1)(t)~R~R′ , (42a) F (~R) = N d~kµ(~k) 1− ei( ~k−n̂|~k|)·~R) |f(n̂|~k|, ~k)|2 , (42b) where n̂ is a unit vector which gives the direction of the scattered particle momentum ~k′ = n̂|~k|. To compare Eq. (42b) with the n = 1 case of Eq. (41d), we replace ~R by ~R2 = ~R1 and ~R ′ by ~R′1. Then we see that the generalization to n ≥ 1 is given by ∂ρ(n)(t)~R1 ~R′1,..., ~Rn ~R′n F (~Rℓ+1 − ~R (n)(t)~R1 ~R′1,..., ~Rn ~R′n . (42c) This is our final result for collisional Brownian motion, giving the evolution equation obeyed by the order n density tensor . We see that it has the generic symmetries expected in the quantum noise case: although not totally symmetric in its pair indices, ρ(n) is symmetric under cyclic permutation of these indices. As additional checks, we see that for n = 2 the factor involving F is F (~R2 − ~R 1) + F ( ~R1 − ~R 2) , (43a) which is symmetric under the interchange 1 ↔ 2, while for n = 3 we have F (~R2 − ~R 1) + F ( ~R3 − ~R 2) + F ( ~R1 − ~R 3) = F S + FA , F S = [F (~R2 − ~R 1) + F ( ~R3 − ~R 2) + F ( ~R1 − ~R +F (~R1 − ~R 2) + F ( ~R3 − ~R 1) + F ( ~R2 − ~R 3)] , [F (~R2 − ~R 1) + F ( ~R3 − ~R 2) + F ( ~R1 − ~R −F (~R1 − ~R 2)− F ( ~R3 − ~R 1)− F ( ~R2 − ~R 3)] , (43b) with F S symmetric, and FA antisymmetric, under any of the pair interchanges 1 ↔ 2, or 1 ↔ 3, or 2 ↔ 3. Checking the descent equations is easy. Setting ~R′1 = ~R2, the term F (~R2− ~R 1) in Eq. (42c) vanishes, so that on integrating over ~R′1 one is left on the right hand side with a sum F (~R1 − ~R n) + F ( ~R3 − ~R 2) + ... that does not involve ~R′1, times d~R′1ρ (n)(t)~R1 ~R′1, ~R ,..., ~Rn ~R′n , (43c) and so the descent equation for ρ(n)(t) then implies the descent equation for its time deriva- tive. 8. The weak coupling Born-Markov approximation and the quantum optical master equation for the density tensor We turn next to the density tensor extension of the standard weak coupling Born- Markov approximation, that is used to give a master equation for the reduced density matrix ρ(1) for a system S interacting with an environment E . We assume a total system plus envi- ronment Hamiltonian HTOT = HE +HS +H , with HE and HS respectively the environment and system Hamiltonians, and with H the system-environment interaction Hamiltonian. (We omit the customary subscript I on the interaction Hamiltonian to avoid a proliferation of subscripts.) We shall work in this section in interaction picture, in which the operators carry the time dependence associated with HE and HS . Thus the interaction Hamiltonian carries a time dependence H(t), and the density matrix obeys the equation of motion dρ(t) = −i[H(t), ρ(t)] (44a) which can be integrated to give ρ(t) = ρ(0)− i ds[H(s), ρ(s)] . (44b) Substituting Eq. (44b) back into Eq. (44a) gives the additional evolution equation dρ(t) = −i[H(t), ρ(0)]− ds[H(t), [H(s), ρ(s)]] . (44c) One then notes that up to an error of order H3, the time argument of the factor ρ(s) in the double commutator term is irrelevant, so this factor can be approximated as ρ(t), giving dρ(t) = −i[H(t), ρ(0)]− ds[H(t), [H(s), ρ(t)]] , (44d) which is used as the starting point for the standard master equation derivation. Our first step is to derive a suitable extension of Eq. (44d) for the product ρ1ρ2...ρn that appears in Eq. (32c). By the chain rule, we have d(ρ1ρ2...ρn) ρ2...ρn + ρ1... ...ρn + ρ1.... . (45a) For each undifferentiated factor on the right of Eq. (45a) we substitute Eq. (44b), and for each time derivative factor we substitute Eq. (44c), with appropriate subscripts added. Let us now organize the terms obtained this way according to the number of factors of H that appear. Since Eq. (44c) contains at least one factor of H , there are no terms in Eq. (45a) with no factors of H . The general term in Eq. (45a) with one factor of H comes from the term in Eq. (44c) with one factor of H , multiplied by the product of the terms from Eq. (44b) with no factors of H , giving [H1(t), ρ1(0)]ρ2(0)...ρn(0)+ρ1(0)[H2(t), ρ2(0)]...ρn(0)+ ...+ρ1(0)ρ2(0)...[Hn(t), ρn(0)] (45b) The terms in Eq. (45a) with two factors of H are of two types: (1) the quadratic term in H on the right of Eq. (44c) times factors of ρ(0), and (2) the linear term in H on the right of Eq. (44c), multiplied by one factor of the linear term on the right of Eq. (44b), times factors of ρ(0). We now note that up to an error of order H3, in terms that already contain two factors of H we can replace all factors ρ(0) or ρ(s) by the corresponding ρ(t), since the differences ρ(t) − ρ(s) and ρ(t) − ρ(0) are all of order H . Collecting everything, we get the following formula, which gives the needed extension of Eq. (44d), d(ρ1ρ2...ρn) ρ1(0)...ρℓ−1(0)[Hℓ(t), ρℓ(0)]ρℓ+1(0)...ρn(0) ρ1(t)...ρℓ−1(t) ds[Hℓ(t), [Hℓ(s), ρℓ(t)]]ρℓ+1(t)...ρn(t) {ρ1(t)...ρℓ−1(t)[Hℓ(t), ρℓ(t)]ρℓ+1(t)...ρm−1(t) ds[Hm(s), ρm(t)]ρm+1(t)...ρn(t) + ρ1(t)...ρℓ−1(t) ds[Hℓ(s), ρℓ(t)]ρℓ+1(t)...ρm−1(t)[Hm(t), ρm(t)]ρm+1(t)...ρn(t)}+O(H (45c) Taking the overall TrE of this expression then gives a formula for the time evolution of ρ (n)(t) as defined by Eq.(32c). We now make two standard assumptions. First of all, we assume at that at the initial time t = 0, the density matrix factorizes so that ρ(0) = ρEρS , with ρE and ρS respectively density matrices for the environment and the system which commute with one another, and with ρE a pure state density matrix obeying ρ E = ρE . Secondly, we assume that 〈H〉E = TrEρEH = 0, that is, we take the interaction Hamiltonian to have a vanishing expectation in the initial environmental state. As a result of these two assumptions, the environmental trace of the first term on the right hand side of Eq. (45c) vanishes, since TrEρ1(0)...ρℓ−1(0)[Hℓ(t), ρℓ(0)]ρℓ+1(0)...ρn(0) =ρS1...ρSℓ−1[(TrEρEHℓ(t)), ρSℓ]ρSℓ+1...ρSn = 0 . (46a) The remaining terms in Eq. (45c) all have two factors of H . Since ρ(t) and ρ(0) differ by one power of H , in these terms, up to an error of order H3, we can replace all factors ρ(t) by the factorized approximation ρ(t) ≃ ρ(0) = ρEρS = ρETrEρ(0) ≃ ρETrEρ(t) = ρEρ (1)(t) . (46b) With these simplifications, and remembering that system operator factors ρ with different index values ℓ act on different Hilbert spaces HS;ℓ and so commute, Eq. (45c) becomes an extended version of the Redfield equation, dρ(n)(t)/dt = − 1 (t)...ρ n (t))ℓTrEρ ds[Hℓ(t), [Hℓ(s), ρ (t)ρE ]] 1 (t)...ρ n (t))ℓmTrE n−(m−ℓ)−1 × {[Hℓ(t), ρ (t)ρE ]ρ m−ℓ−1 E [Hm(s), ρ m (t)ρE ] + [Hℓ(s), ρ (t)ρE ]ρ m−ℓ−1 E [Hm(t), ρ m (t)ρE ]} . (46c) This is converted to the Born-Markov equation by setting s→ t− s, and then extending the upper limit of the s integration from t to ∞, giving dρ(n)(t)/dt = − 1 (t)...ρ n (t))ℓTrEρ ds[Hℓ(t), [Hℓ(t− s), ρ (t)ρE ]] 1 (t)...ρ n (t))ℓmTrE n−(m−ℓ)−1 × {[Hℓ(t), ρ (t)ρE ]ρ m−ℓ−1 E [Hm(t− s), ρ m (t)ρE ] + [Hℓ(t− s), ρ (t)ρE ]ρ m−ℓ−1 E [Hm(t), ρ m (t)ρE ]} . (46d) We now note that Eq. (46d) can be further simplified, by taking account of the fact that whenever an H factor is sandwiched between factors of ρE it vanishes, since ρEHρE = ρE〈H〉E = 0. This eliminates all terms in the sum over ℓ,m that are not adjacent in a cyclic sense, i.e., that do not either have m = ℓ + 1, ℓ = 1, ..., n − 1, or ℓ = 1, m = n. The latter, by use of the cyclic properties of the trace, can be rearranged to give the ℓ = n term of the former set. We thus get a simplified set of Born-Markov equations. For n = 1, we get the usual starting point for the Born-Markov master equation derivation, dρ(1)(t)/dt = −TrE ds[H(t)H(t− s)ρ(1)(t)ρE + ρ (1)(t)ρEH(t− s)H(t) −H(t)ρ(1)(t)ρEH(t− s)−H(t− s)ρ (1)(t)ρEH(t)] , (47a) and for n ≥ 2, with the subscript n+ 1 identified with 1, dρ(n)(t)/dt = −TrEρE × {(ρ 1 (t)...ρ n (t))ℓ[Hℓ(t)Hℓ(t− s)ρ (t) + ρ (t)Hℓ(t− s)Hℓ(t)] 1 (t)...ρ n (t))ℓℓ+1[ρ (t)Hℓ(t)Hℓ+1(t− s)ρ ℓ+1(t) + ρ (t)Hℓ(t− s)Hℓ+1(t)ρ ℓ+1(t)]} . (47b) At this point it is useful to check (and we have done so) that the descent equations are satisfied by Eqs. (47a) and (47b). The remainder of the derivation follows closely the standard master equation deriva- tion, in the rotating wave approximation, that proceeds from Eq. (47a), so we will only give a sketch. For further details, and in particular a discussion of the physical justification for the approximations involved, see Sec. 3.3 of ref [1] and also ref [13]. One assumes that Hℓ(t) has the form Hℓ(t) = eiωtA ℓα(ω)Bα(t) , (48a) with A ℓα acting only in the system Hilbert space HS;ℓ and with Bα acting only in the environment Hilbert space HE , and with the Hermiticity properties A (ω) = Aℓα(−ω) and B†α(t) = Bα(t). Since Eqs. (47a,b) are quadratic in H , one uses Eq. (48a) twice; for each Hk(t− s) (regardless of the value of the index k) one writes Hk(t− s) = e−iω(t−s)Akβ(ω)Bβ(t− s) , (48b) and for each Hk(t) (again regardless of the value of k) one writes Hk(t) = (ω′)B†α(t) . (48c) The rotating wave approximation then consists of neglecting terms in the double sum with ω′ 6= ω, so that only the diagonal terms ω′ = ω are left. From the trace over the environment, and the integral over s, one gets correlators of the form dseiωs〈B†α(t)Bβ(t− s)〉E ≡ Γαβ(ω) , dseiωs〈Bβ(t− s)B α(t)〉E = Γαβ(−ω) (49a) where in the second line we have used the definition of the first line and the adjointness properties of the integrand. It is also customary to decompose the reservoir correlation function Γαβ into self-adjoint and anti-self-adjoint parts, according to Γαβ(ω) = γαβ(ω) + iSαβ(ω) . (49b) Proceeding in this fashion, after some algebra one gets the final result, which can be written as an equation for all n ≥ 1 by including a δn1 to take account of the special nature of the n = 1 equation, dρ(n)(t)/dt = 1 (t)...ρ n (t))ℓi[ρ Sαβ(ω)A (ω)Aℓβ(ω)] γαβ(ω) 1 (t)...ρ n (t))ℓ δn1Aℓβ(ω)ρ (ω)Aℓβ(ω), ρ 1 (t)...ρ n (t))ℓℓ+1ρ (ω)Aℓ+1β(ω)ρ ℓ+1(t) (50a) Despite the fact the the n = 1 and n ≥ 2 density tensors have a different structure, the descent equations are satisfied by Eq. (50a), as verified in Appendix D. Finally, we note that Eq. (50a) is readily converted to the quantum optical master equation and its density tensor generalizations, by taking α to be a three-vector index, so that Aα becomes ~A, which is related to the dipole operator by Eq. (3.182) of ref [1]. Also, one takes Sαβ(ω) = δαβS(ω), with S(ω) given by Eq. (3.205) of ref [1], and γαβ(ω) = (4ω3/3)[1 + N(ω)]δαβ , with N(ω) = 1/(e βω − 1) the photon number operator. One gets in this way the density tensor generalization of the quantum optical master equation, dρ(n)(t)/dt = 1 (t)...ρ n (t))ℓi[ρ S(ω) ~A (ω) · ~Aℓ(ω)] (4ω3/3)[1 +N(ω)] 1 (t)...ρ n (t))ℓ δn1 ~Aℓ(ω) · ρ ℓ (t) ℓ (ω)− ℓ (ω) · ~Aℓ(ω), ρ ℓ (t)} 1 (t)...ρ n (t))ℓℓ+1ρ (t) ~A (ω) · ~Aℓ+1(ω)ρ ℓ+1(t) (50b) which is our final result of this section. 9. The Caldeira–Leggett model master equation for the density tensor The Caldeira–Leggett model [14] describes the damping of the one-dimensional mo- tion of a Brownian particle of mass m, moving in a potential V (x), and interacting with an environment consisting of harmonic oscillators with masses mo and frequencies ωo, and annihilation operator bo. The interaction Hamiltonian is assumed to be a linear coupling H = −xB, with κoxo = κo(bo + b o)/(2moωo) 2 (51a) a weighted sum of the harmonic oscillator coordinates. A counter-term formally of order H2, Hc = x 2moωo ≡ x2C , (51b) is included in the calculation, so that the total Hamiltonian is HTOT = HE +HS +H +Hc , (52a) with HE and HS respectively the oscillator and particle Hamiltonians, obo + + V (x) . (52b) Our aim will be to get a description of the effect on the particle motion of the couplings to the oscillator environment, in the high temperature limit. Our derivation of the density tensor generalization of the high temperature master equation closely follows that of Sec. 3.6 of ref [1], to which the reader is referred for a discussion of the physical motivation of the approximations involved. Since the environmental expectation of the interaction Hamiltonian H vanishes, we can proceed directly from the simplified Born-Markov equation of Eqs. (47a) and (47b). The first step is to transform the density matrix ρ(t) back to Schrödinger picture; it is easy to see that the effect of this is to replace H(t) by H = H(0), to replace H(t − s) by H(−s) (with H(−s) still in the interaction picture), and to change d/dt to D/dt, defined by Dρ(n)(t)/dt = dρ(n)(t)/dt+ i TrEρ1(t)...ρℓ−1(t)[p ℓ/(2m) + V (xℓ), ρℓ(t)]ρℓ+1(t)...ρn(t) . (53a) It is also necessary to explicitly include commutators arising from the counter term, which is easy since this term is treated as being already quadratic in H . For the analog of Eq. (47a) for the special case n = 1, we find Dρ(1)(t)/dt = −i[Hc, ρ (1)(t)]− TrE ds[HH(−s)ρ(1)(t)ρE + ρ (1)(t)ρEH(−s)H −Hρ(1)(t)ρEH(−s)−H(−s)ρ (1)(t)ρEH ] , (53b) and for the analog of Eq. (47b) for n ≥ 2, we have dρ(n)(t)/dt =− i 1 (t)...ρ n (t))ℓ[Hcℓ, ρ ℓ (t)] −TrEρE 1 (t)...ρ n (t))ℓ[HℓHℓ(−s)ρ (t) + ρ (t)Hℓ(−s)Hℓ] 1 (t)...ρ n (t))ℓℓ+1[ρ (t)HℓHℓ+1(−s)ρ ℓ+1(t) + ρ (t)Hℓ(−s)Hℓ+1ρ ℓ+1(t)]} . (53c) We next note that Hℓ = −xℓ(0)B(0) , Hℓ(−s) = −xℓ(−s)B(−s) , (54a) where, using the assumption that the system evolution is slow compared to the oscillator time scale, we approximate xℓ(−s) by its free particle dynamics, xℓ(−s) ≃ xℓ − s . (54b) Since the right hand sides of Eqs. (53b,c) are quadratic in H , the operator B giving the coupling to the oscillators appears, after the environmental trace is taken, only through the correlators D(s) ≡i〈[B(0), B(−s)]〉E , D1(s) ≡〈{B(0), B(−s)}〉E , (55a) so that we have 〈B(0)B(−s)〉E = [D1(s)− iD(s)] , 〈B(−s)B(0)〉E = [D1(s) + iD(s)] . (55b) These correlators appear in the following integrals, which are evaluated or approximated in Sec. 3.6.2 of ref [1], dsD(s) =2C , dsD1(s) =4mγkBT , dssD(s) =2mγ , dssD1(s) =4mγkBT/Ω ≃ 0 , (55c) with C the constant defined by the counter term of Eq. (51b), with γ a constant determined by the harmonic oscillator spectral density, with kB and T respectively the Boltzmann con- stant and environment temperature, and with Ω a frequency cutoff. For a spectral density J(ω) with a Lorentz-Drude cutoff function, one has J(ω) = 2moωo δ(ω − ωo) = Ω2 + ω2 . (55d) This completes the specification of the calculation; the rest is just the algebra of assembling all the pieces, and so we pass directly to the result. For n = 1, we get the Caldeira–Leggett master equation, Dρ(1)(t)/dt = −iγ[x, {p, ρ(1)(t)}]− 2mγkBT [x, [x, ρ (1)(t)]] . (56a) For the density tensors with n ≥ 2, we correspondingly get Dρ(n)/dt = 1 (t)...ρ n (t))ℓ −2mγkBT{x ℓ , ρ ℓ (t)}+ iγ ℓ (t)pℓxℓ − xℓpℓρ ℓ (t) 1 (t)...ρ n (t))ℓℓ+1[4mγkBTρ (t)xℓxℓ+1ρ ℓ+1(t) + iγρ (t)(xℓpℓ+1 − pℓxℓ+1)ρ ℓ+1(t)] . (56b) We also note that the term proportional to γ on the first line of Eq. (56b) can be written in the alternative form, (t)pℓxℓ − xℓpℓρ (t) + (t), {xℓ, pℓ}] . (56c) Equations (56a) and (56b) are our final results for the Caldeira–Leggett model. As was the case for the master equations derived in the preceding section, despite the differences between the structure of the n = 1 and the n ≥ 2 equations, the descent equations are satisfied, as verified in Appendix E. 10. An application to state vector reduction We turn now to considerations that bridge the discussions given above in the classical and quantum noise cases. We begin with an analysis of two Itô stochastic Schrödinger equations, d|ψ〉 = − (A− 〈A〉)2dt|ψ〉+ (A− 〈A〉)dWt|ψ〉 , (57a) d|ψ〉 = − A2dt|ψ〉+ iAdWt|ψ〉 , (57b) with dWt a real Brownian noise obeying dW t = dt, and where we have dropped the Hamil- tonian term. These lead to the respective density matrix evolution equations dρ = − [A, [A, ρ]]dt+ [ρ, [ρ, A]]dWt , (58a) dρ = − [A, [A, ρ]]dt+ i[A, ρ]dWt , (58b) which correspond to the same Lindblad type evolution equation for the expectation E[ρ], dE[ρ] = LE[ρ]dt , Lρ = − [A, [A, ρ]] . (58c) Let us now consider the effect of the stochastic evolutions of Eqs. (58a,b,c) on the expectation of the variance V = Var(A) of the operator A, V =TrρA2 − (TrρA)2 , E[V ] =TrE[ρ]A2 −E[ρi1j1ρi2j2 ]Aj1i1Aj2i2 =Trρ(1)A2 − ρ i1j1,i2j2 Aj1i1Aj2i2 , (59a) where in the final line we have used the density tensor definition of Eq. (15). For the time evolution of E[V ] we have dE[V ]/dt =Tr(LE[ρ])A2 − dρ i1j1,i2j2 Aj1i1Aj2i2 =Tr(LE[ρ])A2 − 2E[ρi1j1(Lρ)i2j2 ]Aj1i1Aj2i2 − E[Ci1j1,i2j2 ]Aj1i1Aj2i2 , (59b) where we have used Eq. (58c) in the first line and Eq. (23a) in the second line. Since the cyclic property of the trace implies that Tr[A, [A, ρ]]A = 0 , the terms in Eq. (59b) involving the Lindblad L all vanish, and so the time derivative of E[V ] comes entirely form the final term, dE[V ]/dt = −E[Ci1j1,i2j2 ]Aj1i1Aj2i2 , (59c) and thus is determined by the evolution equation for the second order density tensor. This is why the state vector evolutions of Eqs. (57a) and (57b), or equivalently the density matrix evolutions of Eqs. (58a) and (58b), lead to very different results for the evolution of the variance of the operator A. The tensor Ci1j1,i2j2 corresponding to Eqs. (57b) and (58b) is given in Eq. (24b), and since the cyclic property of the trace implies that Tr[A, ρ]A = 0, one has dE[V ]/dt = 0 for this evolution. On the other hand, the tensor Ci1j1,i2j2 corresponding to Eqs. (57a) and (58a) is given in Eq. (24a), and through Eq. (59c) implies that dE[V ]/dt = −E[(Tr[ρ, [ρ, A]]A)2] = −E[(Tr([ρ, A])2)2] , (59d) which is negative definite. Starting from Eq. (59d), some simple inequalities imply that the stochastic evolution of Eqs. (57a) and (58a) drives the variance of A to zero as t→ ∞, and hence reduces the state vector to an eigenstate of A, as discussed in detail in refs [15]. Let us now consider a quantum system S, consisting of a microscopic system coupled to a macroscopic measuring apparatus, interacting with a quantum environment E , with the totality forming a closed system. A general result [16], using just the linearity of quantum mechanics, shows that state vector reduction cannot occur in this case. To understand this result through an analysis similar to that just given for Eqs. (57a,b), let us consider the behavior of the variance of a system operator A which is a good “pointer observable”. By definition, a system operator commutes with the environment Hamiltonian HE , and since the system in this case includes the apparatus and so is macroscopic, the pointer observable also obeys [17] [A,H ] = 0, with H the system-environment interaction Hamiltonian. Let us now write the density matrix evolution in Schrödinger picture, dρ/dt = −i[HTOT, ρ] = −i[HS +HE +H, ρ] , (60a) with HS the system Hamiltonian. We consider the system evolution after a brief interaction has entangled the apparatus states with the microscopic subsystem quantum states that are to be distinguished by the pointer reading. For the time evolution of the variance of the pointer observable A, we have dV/dt = Tr(dρ/dt)A2 − 2(TrρA)(Tr(dρ/dt)A) , (60b) which substituting Eq. (60a), and using the cyclic property of the trace and the fact that A commutes with both HE and H , simplifies to dV/dt = iTrρ[HS , A 2]− 2i(TrρA)(Trρ[HS , A]) . (60c) This can be further simplified by using the definition of the reduced density matrix ρ(1) = TrEρ, together with the fact that the commutators in Eq. (60c) involve only system operators, giving dV/dt = iTrSρ (1)[HS , A 2]− 2i(TrSρ (1)A)(TrSρ (1)[HS , A]) . (60d) We see that, unlike the Itô equation case discussed above, the time derivative of V here is determined by ρ(1), rather than by ρ(2). Let us now take the pointer observable to be a pointer center of mass coordinate A = X , in which case, once the entanglement of the pointer with the microsystem being measured has been established, the relevant part of the system Hamiltonian HS is P 2/(2M), with P the total momentum operator for the pointer of macroscopic mass M . Evaluating the commutators, and writing TrSρ (1)O = 〈O〉, we see that dV/dt =(1/M)〈{P,X}〉 − (2/M)〈X〉〈P 〉 =(1/M)〈{X − 〈X〉, P − 〈P 〉}〉 . (61a) By the Schwartz inequality, the right hand side of Eq. (61a) is bounded by (2/M)〈(X − 〈X〉)2〉 2 〈(P − 〈P 〉)2〉 2 = (2/M)∆X∆P . (61b) Let us now determine the minimum value of the bound of Eq. (61b) that is compatible with the parameters of a feasible measurement. Since the uncertainty principle implies that ∆X∆P ≥ 1/2, we get a least upper bound on Eq. (61b) by substituting ∆X∆P ∼ 1/2. This shows that |dV/dt| can be made as small as ∼ 1/M , which since M is macroscopic, can be made essentially arbitrarily small.3 Hence the variance of the pointer variable A stays essentially constant, and is not forced to reduce to zero in the course of the measurement. We conclude, in agreement with the arguments of [16], that a quantum apparatus interacting with a quantum environment does not act like the stochastic equation of Eq. (57a) in terms of reducing the state vector. Although a quantum environment acts on a quantum system with a form of “noise”, our analysis of the density tensor hierarchy in the classical and 3 Restoring factors of Planck’s constant, |dV/dt| can be as small as h̄/M , for which the reduction time dt is at least of order MdV/h̄. For M ∼ 1024mproton and dV ∼ (1cm) this gives dt ∼ M(1cm)2/h̄ ∼ 1027s ∼ 1010 times the age of the universe. Note that our argument places no restriction on the mean pointer momentum 〈P 〉 that establishes the time needed to attain one or the other of the measurement outcomes X starting from the initial pointer position. quantum noise cases shows that structures with different kinematical symmetries,4 different dynamical evolutions, and different implications for the measurement process are involved. As a result, the quantum noise in a closed quantum system does not mimic the action of the classical noise in objective reduction models, and cannot be invoked to give a resolution of the quantum measurement problem within the framework of unmodified quantum mechanics. Acknowledgments I wish to thank Angelo Bassi, Todd Brun, Lajos Diósi, Larry Horwitz, and Lane Hughston for instructive conversations over a number of years that helped motivate this study, and Francesco Petruccione for the gift some years ago of a copy of ref [1]. This work was supported in part by the Department of Energy under Grant #DE–FG02–90ER40542. 4 The dissimilarities between the symmetries of the classical noise and quantum noise hierarchies are least for the order two density tensor. In the order two case, cyclic symmetry is equivalent to full permutation symmetry, and so the index symmetry properties are the same in the classical and quantum noise cases, and as a consequence the descent equations in the quantum noise case correspond to the idempotence descent equations in the classical noise case. Only the classical noise descent equation implied by the unit trace condition has no precise quantum noise counterpart: in the classical case, one has δi1j1ρ i1j1,i2j2 = δi1j1E[ρi1j1ρi2j2 ] = E[ρi2j2 ] = ρ whereas in the quantum noise case one instead has δi1j1ρ i1j1,i2j2 = δi1j1TrEρi1j1ρi2j2 = TrEρEρi2j2 6= TrEρi2j2 = ρ with ρE = TrSρ the reduced density matrix of the environment with the system traced out. Appendix A: Descent equations for the isotropic spin-1/2 ensemble Let us write the generating function of Eq. (14b) as G[aij ] =fg , f(x) = sinh x 2 , x = ~A 2 , Tra . Then, we find δmrG+ gf ′ ~A · ~σmr , ∂amr∂apq δpqgf ′ ~A · ~σmr +gf ′′ ~A · ~σmr ~A · ~σpq + gf ′~σmr · ~σpq . Here the primes denote derivatives of f with respect to x, and in this notation f obeys the second order differential equation xf ′′ + f ′ = f . (A3) Contracting the first expression in Eq. (A2) with δmr , and using the tracelessness of the Pauli matrices, gives the first equation in Eq. (7b). Contracting the second expression in Eq. (A2) with δrp, and using the differential equation of Eq. (A3) together with the Pauli matrix identities (~σ 2)mq = 3δmq and σ iσj = δij+ iǫijkσk, which implies ~A ·~σmp ~A ·~σpq = ~A 2δmq , gives the second equation in Eq. (7b). Appendix B: Descent equations for the Itô stochastic Schrödinger equation We wish here to verify that dG[a] = dtE amr(Lρ)mr + amrapqCmr,pq obeys the descent equations of Eq. (7b). Since δmr(Lρ)mr = δmrCmr,pq = δpqCmr,pq = 0 , (B2a) we have ∂dG[a] = dtE amr(Lρ)mr + amrapqCmr,pq (Trρ)eρ·a = dG[a] , (B2b) giving the first identity in Eq. (7b). Next we calculate ∂dG[a] (Lρ)mq + auv(Cmq,uv + Cuv,mq) auv(Lρ)uv + auvarsCuv,rs (B3a) while for the contraction of the second variation we have (with indices m, q implicit on the right hand side) ∂2dG[a] ∂amr∂arq = dtE S1 + S2 + auv(T1uv + T2uv) , (B3b) (Cmr,rq + Crq,mr) , S2 ={Lρ, ρ}mq , T1uv = [(Cmr,uv + Cuv,mr)ρrq + ρmr(Cuv,rq + Crq,uv)] , T2uv = (Lρ)uv + arsCuv,rs ρmq , (B3c) We see immediately that auvT2uv gives all of the second line of Eq. (B3a). From Eqs. (16b) and (18) we find {Lρ, ρ}mq = (Lρ)mq − [(ck − 〈ck〉)ρ(ck − 〈ck〉) †]mq − ρmq〈(ck − 〈ck〉) †(ck − 〈ck〉)〉 , (B4a) while from Eq. (21b) we have (Cmr,rq + Crq,mr) = [(ck − 〈ck〉)ρ(ck − 〈ck〉) †]mq + ρmq〈(ck − 〈ck〉) †(ck − 〈ck〉)〉 . (B4b) Hence S1 + S2 = (Lρ)mq, giving the Lρ part of the first line of Eq. (B3a). Finally, again using Eq. (21b) we find that [(Cmr,uv + Cuv,mr)ρrq + ρmr(Cuv,rq + Crq,uv)] = (Cmq,uv + Cuv,mq) , (B4c) and so auvT1uv gives the remainder of the first line of Eq. (B3a), completing the check of the descent equations. Appendix C: Descent equations for the jump Schrödinger equation We verify here that dG[a] = dtE (a · Lρ+ (a ·Qk) )ea·ρ (C1a) obeys the descent equations of Eq. (7b). Since Tr(Lρ) = 0 and TrQk = 〈Bk+B Bk〉 = 0, we have ∂dG[a] = dtE (a · Lρ+ (a ·Qk) )(Trρ)ea·ρ , (C1b) checking the first line of Eq. (7b). Next we calculate the first variation of G, ∂dG[a] (Lρ)mq + (a ·Qk) (p− 1)! (Qk)mq +(a · Lρ+ (a ·Qk) )ρmqe (C2a) and the contracted second variation, ∂2dG[a] ∂amr∂arq = dtE (S1 + S2 + S3 + S4)e , (C2b) (a ·Qk) (p− 2)! (Q2k)mq , S2 ={Lρ, ρ}mq , (a ·Qk) (p− 1)! {Qk, ρ}mq , a · Lρ+ (a ·Qk) ρmq . (C2c) We see immediately that S4 gives all of the second line of Eq. (C2a). From Eq. (30c), which we rewrite here, {Lρ, ρ} =Lρ− {Qk, ρ} =Qk −Q (C3a) we see that the Lρ part of S2 and the Qk part of {Qk, ρ} in S3 give the first line of Eq. (C2a). To complete the verification, we must show that S1 cancels against the remainder of S2+S3, which is (a ·Qk) (p− 1)! (Q2k)mq . (C3b) But separating off the p = 2 term of S1, and making the change of variable p → p + 1 in the remaining sum, we see that S1 is exactly the negative of Eq. (C3b), completing the argument. Appendix D: Descent equations for the Born-Markov master equation We wish here to verify that Eq. (50a) obeys the descent equations of Eq. (34). We separate the verification into two parts, first checking the descent from n = 2 to n = 1, and then checking the descent from general n > 2 to n − 1. For the n = 2 density tensor time derivative, writing out all terms in Eq. (50a) explicitly, and using the fact that since operators labeled with subscripts 2 and 1 act on different Hilbert spaces, the order in which they are written is irrelevant, we have dρ(2)(t)/dt =i[ρ 1 (t), Sαβ(ω)A 1α(ω)A1β(ω)]ρ 2 (t) 1 (t)i[ρ 2 (t), Sαβ(ω)A 2α(ω)A2β(ω)] γαβ(ω) 1α(ω)A1β(ω), ρ 1 (t)}ρ 2 (t) γαβ(ω)ρ 1 (t) 2α(ω)A2β(ω), ρ 2 (t)} γαβ(ω)[ρ 1 (t)A 1α(ω)A2β(ω)ρ 2 (t) + A1β(ω)ρ 1 (t)ρ 2 (t)A 2α(ω)] . (D1a) Contracting the column index associated with the subscript 1 with the row index associated with the subscript 2, and dropping the subscripts since all operators now act in the same Hilbert space, we get dρ(2)(t)/dt→i[ρ(1)(t), Sαβ(ω)A α(ω)Aβ(ω)]ρ (1)(t) +ρ(1)(t)i[ρ(1)(t), Sαβ(ω)A α(ω)Aβ(ω)] γαβ(ω) {A†α(ω)Aβ(ω), ρ (1)(t)}ρ(1)(t) γαβ(ω)ρ (1)(t) {A†α(ω)Aβ(ω), ρ (1)(t)} γαβ(ω)[ρ (1)(t)A†α(ω)Aβ(ω)ρ (1)(t) + Aβ(ω)(ρ (1)(t))2A†α(ω)] =i[(ρ(1)(t))2, Sαβ(ω)A α(ω)Aβ(ω)] γαβ(ω) Aβ(ω)(ρ (1)(t))2A†α(ω)− {(ρ(1)(t))2, A†α(ω)Aβ(ω)} (D1b) which has the structure of dρ(1)(t)/dt and so verifies the 2 → 1 descent. To verify the n→ n− 1 descent we make some simplifications in notation. We omit all superscripts (1), since this leads to no ambiguities, as well as all time arguments (t) and all frequency arguments (ω). We also abbreviate Sαβ(ω)A (ω)Aℓβ(ω) , γαβ(ω)A (ω)Aℓβ(ω) . (D2a) Our general strategy is to split the sum ℓ=1 containing (ρ1...ρn)ℓ into ℓ=2 plus the ℓ = 1 and the ℓ = n terms, and to split the sum ℓ=1 containing (ρ1...ρn)ℓℓ+1 into ℓ=2 plus the ℓ = 1, ℓ = n− 1, and ℓ = n terms. For the part of dρ(n)/dt involving Lℓ, we have (ρ1...ρn−1)ℓρni[ρℓ, Lℓ] + (ρ2...ρn)i[ρ1, L1] + (ρ1...ρn−1)i[ρn, Ln] , (D2b) which on contracting the column index associated with the subscript n with the row index associated with the subscript 1, and relabeling all quantities that had subscript n with subscript 1, since they act now in the same Hilbert space, gives (ρ21ρ2...ρn−1)ℓi[ρℓ, Lℓ] + ρ2...ρn−1i(ρ1[ρ1, L1] + [ρ1, L1]ρ1) (ρ21ρ2...ρn−1)ℓi[ρℓ, Lℓ] + ρ2...ρn−1i[ρ 1, L1] , (D2c) which has the correct structure for the corresponding part of dρ(n−1)/dt, with ρ1 replaced by ρ21. The remainder of dρ (n)/dt is (ρ1...ρn)ℓ {Mℓ, ρℓ} − (ρ2...ρn) {M1, ρ1} − (ρ1...ρn−1) {Mn, ρn} (ρ1...ρn)ℓℓ+1ρℓA Aℓ+1βρℓ+1 + ρ3...ρnρ1A 1αA2βρ2 + ρ1...ρn−2ρn−1A n−1αAnβρn + ρ2...ρn−1ρnA nαA1βρ1 (D3a) Again, contracting the column index associated with the subscript n with the row index associated with the subscript 1, and relabeling all quantities that had subscript n with subscript 1, since they act now in the same Hilbert space, gives (ρ21...ρn−1)ℓ {Mℓ, ρℓ} − (ρ2...ρn−1) ρ1M1ρ1(∗) + {M1, ρ (ρ21...ρn−1)ℓℓ+1ρℓA Aℓ+1βρℓ+1 + (ρ3...ρn−1)ρ 1αA2βρ2 + ρ2...ρn−2ρn−1A n−1αA1βρ 1 + ρ2...ρn−1ρ1A 1αA1βρ1(∗) (D3b) which on canceling the terms marked with (∗) gives the corresponding part of dρ(n−1)/dt, with ρ1 replaced by ρ 1. This completes the verification of the n→ n− 1 descent. Appendix E: Descent equations for the Caldeira–Leggett model We verify here that Eqs. (56a) and (56b) obey the descent equations of Eq. (34). As in the preceding appendix, we simplify the notation by omitting all superscripts (1) and all time arguments (t). We first verify the n = 2 to n = 1 descent. For the n = 2 case of Eq. (56b), we have Dρ(2)/dt =ρ2[−2mγkBT (x 1ρ1 + ρ1x 1) + iγ(ρ1p1x1 − x1p1ρ1)] +ρ1[−2mγkBT (x 2ρ2 + ρ2x 2) + iγ(ρ2p2x2 − x2p2ρ2)] +4mγkBT (ρ1x1x2ρ2 + ρ2x2x1ρ1) + iγ[ρ1(x1p2 − p1x2)ρ2 + ρ2(x2p1 − p2x1)ρ1] . (E1A) Contracting the column index associated with the subscript 1 with the row index associated with the subscript 2, and dropping subscripts since all operators now act in the same Hilbert space, we get Dρ(2)/dt→− 2mγkBT (x 2ρ2 + ρx2ρ) + iγ(ρpxρ− xpρ2) − 2mγkBT (ρx 2ρ+ ρ2x2) + iγ(ρ2px− ρxpρ) +4mγkBT (ρx 2ρ+ xρ2x) + iγ[ρ(xp− px)ρ+ pρ2x− xρ2p] . (E1B) We see that the terms that have an operator sandwiched between two factors of ρ cancel, leaving only terms involving ρ2, which have the form of Eq. (56a) with ρ replaced by ρ2. To check the n > 2 to n−1 descent, we split the sums that occur in the same manner as in Appendix D. We thus write Eq. (56b) in the form Dρ(n)/dt = (ρ1...ρn)ℓ[−2mγkBT{x ℓ , ρℓ}+ iγ(ρℓpℓxℓ − xℓpℓρℓ)] +ρ2...ρn[−2mγkBT{x 1, ρ1}+ iγ(ρ1p1x1 − x1p1ρ1)] +ρ1...ρn−1[−2mγkBT{x n, ρn}+ iγ(ρnpnxn − xnpnρn)] (ρ1...ρn)ℓℓ+1[4mγkBTρℓxℓxℓ+1ρℓ+1 + iγρℓ(xℓpℓ+1 − pℓxℓ+1)ρℓ+1] +ρ3...ρn[4mγkBTρ1x1x2ρ2 + iγρ1(x1p2 − p1x2)ρ2] +ρ1...ρn−2[4mγkBTρn−1xn−1xnρn + iγρn−1(xn−1pn − pn−1xn)ρn] +ρ2...ρn−1[4mγkBTρnxnx1ρ1 + iγρn(xnp1 − pnx1)ρ1] . We now contract the column index associated with the subscript n with the row index associated with the subscript 1, and relabel all quantities that had subscript n with subscript 1, since they act now in the same Hilbert space. As is readily seen by inspection of Eq. (E2), this gives Eq. (56b) with n replaced by n−1 and with ρ1 replaced by ρ 1, together with terms of the wrong structure, that grouped together give (4−2−2)ρ2...ρn−1mγkBTρ1x 1ρ1 = 0 and (1 − 1)ρ2...ρn−1iγρ1(x1p1 − p1x1)ρ1 = 0, which thus vanish. This completes the verification of the descent equation for Eq. (56b). References [1] Breuer H-P and Petruccione F (2002) The Theory of Open Quantum Systems (Oxford: Oxford University Press) [2] Breuer H-P and Petruccione F (1996) Phys. Rev. A 54 1146 [3] Wiseman H M (1993) Phys. Rev. A 47 5180 [4] Mølmer K, Castin Y and Dalibard J (1993) J. Opt. Soc. Am. B 10 524 [5] Mielnik, B (1974) Commun. Math. Phys. 37 221; see especially p 240. I wish to thank Lane Hughston for bringing this reference, and ref [6] as well, to my attention. [6] Brody, D C and Hughston, L P (1999) J. Math. Phys. 40 12, Eqs. (31) and (32); Brody, D C and Hughston, L P (1999) Proc. Roy. Soc. A 455 1683, Sec. 2(e); Brody, D C and Hughston, L P (2000) J. Math. Phys. 41, 2586, Eq. (9) and subsequent discussion. [7] Wiseman H M and Diósi L (2001) Chem. Phys. 268 91. See also Diósi L (1986) Phys. Lett. A 114 451 for the transition rate operator. [8] Lindblad G (1976) Commun. Math. Phys. 48 119 [9] Gorini V, Kossakowski A and Sudarshan E C G (1976) J. Math. Phys. 17 821 [10] Schack R and Brun T A (1997) Comp. Phys. Commun. 102 210 [11] Gallis M R and Fleming G N (1990) Phys. Rev. A 42 38 [12] Diósi L (1995) Europhys. Lett. 30 63; Dodd P J and Halliwell J J (2003) Phys. Rev. D 67 105018; Hornberger K and Sipe J E (2003) Phys. Rev. A 68 012105; Adler S L (2006) J. Phys. A: Math. Gen. 39 14067 [13] Hornberger K (2006) Introduction to decoherence theory, arXiv: quant-ph/0612118 [14] Caldeira A O and Leggett A J (1983) Physica A 121 587 [15] Ghirardi G C, Pearle P and Rimini A (1990) Phys. Rev. A 42 78; Hughston L P (1996) Proc. Roy. Soc. A 452 953; Adler S L and Horwitz L P (2000) J. Math. Phys. 41 2485; Adler S L, Brody D C, Brun T A and Hughston L P (2001) J Phys. A: Math. Gen. 34 8795; Adler S L (2004) Quantum Theory as an Emergent Phenomenon (Cambridge UK: http://arxiv.org/abs/quant-ph/0612118 Cambridge University Press) Sec. 6.2 [16] Bassi A and Ghirardi G C Phys. Lett. A 275 373 [17] Zurek W H (1981) Phys. Rev. D 24 1516; Schlosshauer M (2004) Rev. Mod. Phys. 75 1267, p. 1280 [18] For reviews of stochastic reduction models, see Bassi A and Ghirardi G C (2003) Phys. Reports 379 257; Pearle P (1999) Collapse models, in Open Systems and Measurements in Relativistic Quantum Field Theory, Lecture Notes in Physics 526, Breuer H-P and Petruc- cione F eds. (Berlin: Springer-Verlag)

0704.0797

Scalar self-force on eccentric geodesics in Schwarzschild spacetime: a time-domain computation

Scalar self-force on eccentric geodesics in Schwarzschild spacetime: A time-domain computation Roland Haas Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G 2W1 (Dated: April 3, 2007) We calculate the self-force acting on a particle with scalar charge moving on a generic geodesic around a Schwarzschild black hole. This calculation requires an accurate computation of the retarded scalar field produced by the moving charge; this is done numerically with the help of a fourth-order convergent finite-difference scheme formulated in the time domain. The calculation also requires a regularization procedure, because the retarded field is singular on the particle’s world line; this is handled mode-by-mode via the mode-sum regularization scheme first introduced by Barack and Ori. This paper presents the numerical method, various numerical tests, and a sample of results for mildly eccentric orbits as well as “zoom-whirl” orbits. PACS numbers: 04.25.-g, 04.40.-b, 41.60.-m, 45.50.-j, 02.60.Cb, 02.70.Bf I. INTRODUCTION The inspiral and capture of solar-mass compact objects by supermassive black holes is one of the most promis- ing and interesting sources of gravitational radiation to be detected by the future space-based gravitational-wave antenna LISA [1]. For these extreme mass-ratio inspirals, one can treat the compact object as a point mass and de- scribe its influence on the spacetime perturbatively. Go- ing beyond the test mass limit, its motion is no longer along a geodesic of the unperturbed spacetime of the cen- tral black hole; it is a geodesic of the perturbed space- time created by the presence of the moving body. When viewed from the unperturbed spacetime, the small body is said to move under the influence of its gravitational self-force. The self-force induces radiative losses of energy and angular momentum, which will eventually drive the object into the black hole. To describe the motion of the body, including its inspiral toward the black hole, we seek to evaluate the self-force and calculate its effect on the motion. One way of doing this uses the mode-sum reg- ularization procedure introduced by Barack and Ori [2]. (For a comprehensive introduction of the problem, see the special issue of Classical and Quantum Gravity [3].) In this paper, in an effort to build expertise to calculate the gravitational self-force, we retreat to the technically simpler problem of a point particle of mass m endowed with a scalar charge q orbiting a Schwarzschild black hole of mass M . Following up on a previous paper [4], we implement the numerical part of the regularization pro- cedure for generic orbits with a time-domain integration of the scalar-wave equation. A. The problem Our goal is to calculate the regularized self-force acting on a scalar point charge in orbit around a Schwarzschild black hole. In analogy with the gravitational case, where in a first-order (in m/M) perturbative calculation the particle moves on a geodesic of the background space- time, we take the orbit of the particle to be a geodesic and calculate the self-force as a vector field on this geodesic. We start by writing the Schwarzschild metric using the tortoise coordinate r∗ = r + 2M ln ds2 = f −dt2 + dr∗2 + r2dΩ2, (1.1) where f = 1− 2M , dΩ2 = dθ2 + sin2 θdφ2 is the metric on a two-sphere, and t, r, θ and φ are the usual Schwarzschild coordinates. Our task is to solve the scalar wave equation gαβ∇α∇βΦ(x) = −4πµ(x), (1.2) µ(x) = q δ4(x, z(τ))dτ , (1.3) where ∇α is the covariant derivative compatible with the metric gαβ , Φ(x) is the scalar field created by a scalar charge q which moves along a world line γ : τ 7→ z(τ) parametrized by proper time τ . The source term µ(x) appearing on the right-hand side is written in terms of a scalarized four-dimensional Dirac δ-function δ4(x, x ′) := δ(x0 − x′0)δ(x1 − x′1)δ(x2 − x′2)δ(x3 − x′3)/ − det(gαβ). Because of the singularity in the source term, the re- tarded solution to Eq. (1.2) is singular on the world line, and the näıve expression for the self-force, Fα(τ) = q∇αΦ(z(τ)), (1.4) must be regularized. Following DeWitt and Brehme [5], Mino, Sasaki, Tanaka [6], Quinn and Wald [7], Quinn [8] carried out this regularization for the electromagnetic, scalar and gravitational radiation reaction. In later work, Detweiler and Whiting [9] introduced a very useful de- composition of the retarded solution of Eq. (1.2) in terms of a singular part ΦS and a regular remainder ΦR: Φ = ΦS +ΦR. (1.5) ΦR is regular and differentiable at the position of the par- ticle, satisfies the homogeneous wave equation associated http://arxiv.org/abs/0704.0797v2 with Eq. (1.2), and is solely responsible for the self-force acting on the particle. ΦS , on the other hand, satisfies Eq. (1.2), is just as singular at the particle’s position as the retarded solution, and produces no force on the par- ticle. Rearranging Eq. (1.5) and differentiating once, we can write the regularized self-force as Fα := q∇αΦR = q ∇αΦ−∇αΦS . (1.6) In a previous paper [4], we described our implemen- tation of the regularization procedure to find a mode- sum representation of ∇αΦS along a generic geodesic of the Schwarzschild spacetime. Schematically, we intro- duce a tetrad eα and decompose the tetrad components Φ(µ) := e ∇αΦ of the field gradient in terms of ordinary scalar spherical harmonics Yℓm: Φ(µ)(t, r, θ, φ) = Φℓm(µ)(t, r)Yℓm(θ, φ). (1.7) Each mode Φℓm (t, r) is finite at the position of the par- ticle, but their sum diverges on the world line. In [4], we derive analytic expressions for the mode-sum decompo- sition of ΦS(µ), ΦS(µ) =q ΦS(µ),ℓ (1.8) ΦS(µ),ℓ = A(µ) +B(µ) + (ℓ− 1 )(ℓ + 3 + · · · , (1.9) where the coefficients A(µ), B(µ), C(µ), and D(µ) are in- dependent of ℓ; they are listed in Appendix B for conve- nience. As each mode of Φ is finite, it is straightforward to compute the modes of the retarded solution using nu- merical methods, and we will describe how this was done in Sec. IV. We use the numerical solutions in Eq. (1.6) to calculate the regularized self-force, regularizing mode- by-mode: ΦR(µ) = Φ(µ),ℓ − ΦS(µ),ℓ , (1.10) where Φ(µ),ℓ := Yℓm (no summation over ℓ im- plied). For numerical purposes it is convenient to define ψℓm Φ(x) = ℓm, (1.11) where Yℓm are the usual scalar spherical harmonics. Af- ter substituting in Eq. (1.2), this yields a reduced wave equation for the multipole moments ψℓm: −∂2tψℓm + ∂2r∗ψℓm − Vℓψℓm = − 4πq Ȳℓm(π/2, φ0)δ(r ∗ − r∗0), (1.12) where Vℓ = f ℓ (ℓ+ 1) . (1.13) An overbar denotes complex conjugation, E = −ut is the particle’s conserved energy per unit mass, and uα = dz is its four velocity. Quantities bearing a subscript “0” are evaluated at the particle’s position; they are functions of τ that are obtained by solving the geodesic equation uβ∇βuα = 0 (1.14) in the background spacetime. Without loss of general- ity, we have confined the motion of the particle to the equatorial plane θ = π Once we have numerically solved Eq. (1.12), we ex- tract numerical estimates for ψℓm, ∂tψℓm and ∂r∗ψℓm, which can then be used to find Φℓm, ∂tΦℓm and ∂rΦℓm. These—together with the translation table displayed in Eqs. (1.23)–(1.26) of [4], reproduced in Appendix A— allow us to find the tetrad components Φ(µ)ℓm with re- spect to the tetrad defined by Eqs. (1.18)–(1.21) of [4]. Eventually we regularize the multipole coefficients Φ(µ)ℓ = Φ(µ)ℓm(t0, r0)Y ℓm(π/2, φ0) (1.15) using Eq. (1.10); this involves the regularization param- eters listed in Eqs. (1.30)–(1.45) of [4], which are repro- duced in Appendix B. B. Organization of this paper In Sec. II we introduce the main ideas behind the discretization scheme used in the numerical simulation. Sec. III describes the choices we make in order to handle the problems of specifying initial data and proper bound- ary conditions. The next section—Sec. IV—provides de- tails on the concrete implementation of the ideas put forth in Secs. II and III. In Sec. V we describe the tests we performed in order to validate our implementation of the numerical method. Sec. VI finally presents sample results for a small number of representative simulations. C. Future work This work, which deals with a scalar charge moving in the Schwarzschild spacetime, is not intended to produce physically or astrophysically interesting results. Instead, its goal is to help us evaluate the merits of several strate- gies that could be used to tackle the more interesting (and difficult) problems of electromagnetism and gravity. One future project we are currently exploring is to ap- ply the formalism developed so far to the electromagnetic self-force acting on an electric charge. Beyond the tech- nical complication of having to deal with a vector field instead of a single scalar quantity, we are also faced with the reality of having to impose a gauge (in our case: the Lorenz gauge) and to eliminate (or at least control) gauge violations in the numerical simulation. The first step, namely, the calculation of the regularization parameters A(µ), B(µ), C(µ), and D(µ) for the self-force, is currently underway. Also underway is the calculation of the regu- larization parameters for he gravitational self-force. Another project is the implementation of a scheme to use the calculated self-force to update the orbital pa- rameters of a particle on its inspiral toward the black hole. The standard proposed approach to this problem in the past has been to calculate the self-force on a set of geodesics which are momentarily tangent to the par- ticle’s trajectory. The self-force calculated in this way is then used to update the orbital elements. This “after the fact” calculation of the motion requires one to build (in advance) a large database of self-force values for the anticipated set of orbital parameters that the particle’s trajectory will assume during its inspiral. Alternatively, and conceptually more simply, the self-force could be cal- culated self-consistently along the real, accelerated tra- jectory. Such an approach requires changes in the expres- sions of the regularization parameters, which so far have been derived only for geodesic orbits. We are currently investigating the merits of such an approach. II. NUMERICAL METHOD In this section we describe the algorithm used to inte- grate the reduced wave equation [Eq. (1.12)] numerically. For the most part we use the fourth-order algorithm in- troduced by Lousto [10], with some modifications to suit our needs. We choose to implement a fourth-order con- vergent code because second-order convergence for the potential Φ, while much easier to achieve, would guaran- tee only first-order convergence for ∇αΦ, the quantity in which we are ultimately interested. With a fourth-order convergent code we can expect to achieve third-order con- vergence for ∇αΦ, which is required for an accurate es- timation of the self-force. Numerical experiments, how- ever, show that in practice we do achieve fourth-order convergence for the derivatives of Φ, a fortunate outcome that we exploit but cannot explain. From now on, we will suppress the subscripts ℓ and m on Vℓ and ψℓm for convenience of notation. The wave equation consists of three parts: the wave-operator term (∂2r∗ − ∂2t )ψ and the potential term V ψ on the left-hand side, and the source term on the right-hand side of the equation. Of these, the wave operator turns out to be easiest to handle, and the source term does not create a substantial difficulty. The term involving the potential V turns out to be the most difficult one to handle. Following Lousto we introduce a staggered grid with step sizes ∆t = 1 ∆r∗ ≡ h, which follows the characteris- tic lines of the wave operator in Schwarzschild spacetime; see Fig. 1 for a sketch of a typical grid cell. The basic idea behind the method is to integrate the wave equation over a unit cell of the grid, which nicely deals with the Dirac-δ source term on the right-hand side. To this end, we introduce the Eddington-Finkelstein null coordinates v = t + r∗ and u = t − r∗ and use them as integration variables. A. Differential operator Rewriting the wave operator in terms of u and v, we find −∂2t + ∂2r∗ = −4∂u∂v, which allows us to evaluate the integral involving the wave operator exactly. We find −4∂u∂vψ du dv =− 4[ψ(t+ h, r∗) + ψ(t− h, r∗) − ψ(t, r∗ − h)− ψ(t, r∗ + h)]. (2.1) B. Source term If we integrate over a cell traversed by the particle, then the source term on the right-hand side of the equation will have a non-zero contribution. Writing the source term as G(t, r∗)δ(r∗ − r∗0(t)) with G(t, r∗) = −4πq f Ȳℓm(π/2, φ0), (2.2) we find Gδ(r∗ − r∗0(t)) du dv =− f0(t) r0(t) × Ȳℓm(π/2, φ0(t)) dt, (2.3) where t1 and t2 are the times at which the particle enters and leaves the cell, respectively. While we do not have an analytic expression for the trajectory of the particle (except when the particle follows a circular orbit), we can numerically integrate the first-order ordinary differential equations that govern the particle’s motion to a precision that is much higher than that of the partial differential equation governing ψ. In this sense we treat the integral over the source term as exact. To evaluate the integral we adopt a four-point Gauss-Legendre scheme, which has an error of order h8. C. Potential term The most problematic term—from the point of view of implementing an approximation of sufficiently high or- der in h—turns out to be the term V ψ in Eq. (1.12). Since this term does not contain a δ-function, we have to approximate the double integral V ψ du dv (2.4) t0 − h t0 + h t0 − 2h 0 − 2h r 0 + 2h 0 − 3h r 0 + 3hr 0 + hr 0 − h 13 34 87 9 10 FIG. 1: Points used to calculate the integral over the potential term for vacuum cells. Grid points are indicated by blue cir- cles while red cross-hairs indicate points in between two grid points. We calculate field values at points that do not lie on the grid by employing the second-order algorithm described in [10]. up to terms of order h6 for a generic cell in order to achieve an overall O(h4) convergence of the scheme. Here we have to treat cells traversed by the particle (“sourced” cells) differently from the generic (“vacuum”) cells. While much of the algorithm can be transferred from the vacuum cells to the sourced cells, some modifica- tions are required. We will describe each case separately in the following subsections. 1. Vacuum case To implement Lousto’s algorithm to evolve the field across the vacuum cells, we use a double Simpson rule to compute the integral Eq. (2.4). We introduce the nota- g(t, r∗) = V (r∗)ψ(t, r∗) (2.5) and label our points in the same manner (see Fig. 1) as in [10]: g du dv = [g1 + g2 + g3 + g4 + 4(g12+ g24 + g34 + g13) + 16g0] +O(h 6). (2.6) Here, for example, g1 is the value of g at the grid point labeled 1, and g12 is the value of g at the off-grid point labeled 12, etc. Deviating from Lousto’s algorithm, we choose to calculate g0 using an expression different from that derived in [10]. Unlike Lousto’s approach, our ex- pression exclusively involves points that are within the past light cone of the current cell. We find 8V4 ψ4 + 8V1 ψ1 + 8V2 ψ2 − 4V6 ψ6 − 4V5 ψ5 + V10 ψ10 + V7 ψ7 − V9 ψ9 − V8 ψ8 +O(h4). (2.7) In order to evaluate the term in parentheses in Eq. (2.6), we again use a variant of the equations given in [10]. Lousto’s equations (33) and (34), g13 + g12 =V (r 0 − h/2) (ψ1 + ψ0) V (r∗0 − h/2) +O(h4), (2.8) g24 + g34 =V (r 0 + h/2) (ψ0 + ψ4) V (r∗0 + h/2) +O(h4) (2.9) contain isolated occurrences of ψ0, the value of the field at the central point. Since Eq. (2.7) only allows us to find g0 = V0ψ0, finding ψ0 would involve a division by V0, which will be numerically unstable very close to the event horizon where V0 ≈ 0. Instead we choose to express the potential term appearing in the square brackets as a Taylor series around r∗0 . This allows us to eliminate the isolated occurrences of ψ0, and we find g13+g12 + g24 + g34 = 2V (r V (r∗0) + V (r∗0 − h/2)ψ1 V (r∗0 − h/2) + V (r∗0 + h/2)ψ4 V (r∗0 + h/2) V (r∗0 − h/2)− 2V (r∗0) + V (r∗0 + h/2) (ψ1 + ψ4) +O(h 4). (2.10) Because of the factor in Eq. (2.6), this allows us to reach the required O(h6) convergence for a generic vacuum cell. This—given that there is a number of order N = 1/h2 of such cells—yields the desired overall O(h4) convergence of the full algorithm, at the end of the N steps required to finish the simulation. 2. Sourced cells For vacuum cells, the algorithm described above is the complete algorithm used to evolve the field forward in time. For cells traversed by the particle, however, we have to reconsider the assumptions used in deriving Eqs. (2.7) and (2.10). When deriving Eq. (2.10) we have employed the second-order evolution algorithm (see [10]), in which the single step equation ψ3 =− ψ2 + (ψ1 + ψ4) (2.11) O(h3)O(h4)O(h5) O(h5)O(h4) traje tory t0 + h t0 − h t0 − 2h FIG. 2: Cells affected by the passage of the particle, showing the reduced order of the single step equation is accurate only to O(h3) for cells traversed by the parti- cle. For these cells, therefore, the error term in Eq. (2.10) is O(h3) instead of O(h4). As there is a number of or- der N ′ = 1/h of cells that are traversed by the parti- cle in a simulation run, the overall error—after including factor in Eq. (2.6)—is of order h4. We can therefore afford this reduction of the convergence order in Eq. (2.10) Equation (2.7), however, is accurate only to O(h) for cells traversed by the particle. Again taking the factor into account, this renders the overall algorithm O(h2). Figure 2 shows the cells affected by the particle’s traversal and the reduced order of the single step equa- tion for each cell. Cells whose convergence order is O(h5) or higher do not need modifications, since there is only a number N ′ = 1/h of such cells in the simulation. We are therefore concerned about cells neighboring the particle’s trajectory and those traversed by the particle. a. Cells neighboring the particle These cells are not traversed by the particle, but the particle might have traversed cells in their past light-cone, which are used in the calculation of g0 in Eq. (2.7). For these cells, we use a one-dimensional Taylor expansion of g(t, r∗) within the current time-slice t = t0, 5V (r∗0 − h)ψ(t0, r∗0 − h) + 15V (r∗0 − 3h)ψ(t0, r∗0 − 3h) − 5V (r∗0 − 5h)ψ(t0, r∗0 − 5h) + V (r∗0 − 7h)ψ(t0, r∗0 − 7h) +O(h4) (2.12) for the cell on the left-hand side, and 5V (r∗0 + h)ψ(t0, r 0 + h) + 15V (r∗0 + 3h)ψ(t0, r 0 + 3h) − 5V (r∗0 + 5h)ψ(t0, r∗0 + 5h) + V (r∗0 + 7h)ψ(t0, r 0 + 7h) +O(h4) (2.13) for the cell on the right-hand side, where (t0, r 0) is the center of the cell traversed by the particle. Both of these are more accurate than is strictly necessary; we would t0 − h t0 + h 0 + 2hr 0 − 2h r 0 − h r 0 + h (3a) (3b) (2a) (2b) FIG. 3: Typical cell traversal of the particle. We split the domain into sub-parts indicated by the dotted line based on the time the particle enters (at t1) and leaves (at t2) the cell. The integral over each sub-part is evaluated using an iterated two-by-two point Gauss-Legendre rule. need error terms of order h3 to achieve the desired over- all O(h4) convergence of the algorithm. Keeping the ex- tra terms, however, improves the numerical convergence slightly. b. Cell traversed by the particle We choose not to implement a fully explicit algorithm to handle cells tra- versed by the particle, because this would increase the complexity of the algorithm by a significant factor. In- stead we use an iterative approach to evolve the field using the integrated wave equation −4(ψ3+ψ2 − ψ1 − ψ4)− V ψ du dv = − 8πq f0(t) r0(t) Ȳℓm(π/2, φ0(t)) dt. (2.14) In this equation the integral involving the source term can be evaluated to any desired accuracy at the begin- ning of the iteration, because the motion of the particle is determined by a simple system of ordinary differential equations, which are easily integrated with reliable nu- merical methods. It remains to evaluate the integral over the potential term, which we do iteratively. Schemati- cally the method works as follows: • Make an initial guess for ψ3 using the second-order algorithm. This guess is correct up to terms of O(h3). • Match a second-order piecewise interpolation poly- nomial to the six points that make up the past light- cone of the future grid point, including the future point itself. • Use this approximation for ψ to numerically calcu- V ψ du dv, using two-by-two point Gauss-Legendre rules for the six sub-parts indicated in Fig. 3. • Update the future value of the field and repeat the process until the iteration has converged to a re- quired degree of accuracy. trajectory FIG. 4: Numerical domain evolved during the simulation. We impose an inner boundary condition close to the black whole where we can implement it easily to the accuracy of the un- derlying floating point format. Far away from the black hole, we evolve the full domain of dependence of the initial data domain without imposing boundary conditions. III. INITIAL VALUES AND BOUNDARY CONDITIONS As is typical for numerical simulations, we have to pay careful attention to specifying initial data and appropri- ate boundary conditions. These aspects of the numerical method are highly non-trivial problems in full numerical relativity, but they can be solved or circumvented with moderate effort in the present work. A. Initial data In this work we use a characteristic grid consisting of points lying on characteristic lines of the wave operator to evolve ψ forward in time. As such, we need to specify characteristic initial data on the lines u = u0 and v = v0 shown in Fig. 4. We choose not to worry about specifying “correct” initial data, but instead arbitrarily choose ψ to vanish on u = u0 and v = v0: ψ(u = u0, v) = ψ(u, v = v0) = 0. (3.1) This is equivalent to adding spurious initial waves in the form of a homogeneous solution of Eq. (1.12) to the cor- rect solution. This produces an initial wave burst that moves away from the particle with the speed of light, and quickly leaves the numerical domain. Any remain- ing tails of the spurious initial data decay as t−(2ℓ+2) as shown in [11] and become negligible after a short time. We conclude that the influence of the initial-wave con- tent on the self-force becomes negligible after a time of the order of the light-crossing time of the particle’s orbit. B. Boundary conditions On the analytical side we would like to impose ingoing boundary conditions at the event horizon r∗ → −∞ and outgoing boundary conditions at spatial infinity r∗ → ∞, r∗→−∞ ∂uψ =0, lim ∂vψ =0. (3.2) Because of the finite resources available to a computer we can only simulate a finite region of the spacetime, and are faced with the reality of implementing boundary conditions at finite values of r∗. Two solutions to this problem present themselves: 1. choose the numerical domain to be the domain of dependence of the initial data surface. Since the effect of the boundary condition can only propagate forward in time with at most the speed of light, this effectively hides any influence of the boundary. This is what we choose to do in order to deal with the outer boundary condition. 2. implement boundary conditions sufficiently “far out” so that numerically there is no difference be- tween imposing the boundary condition there or at infinity. Since the boundary conditions depend on the vanishing of the potential V (r) appearing in the wave equation, this will happen once 1−2M/r ≈ 0. Near the horizon r ≈ 2M(1 + exp(r∗/2M)), so this will happen—to numerical accuracy—for mod- estly large (negative) values of r∗ ≈ −73M . We choose to implement the ingoing waves condition ∂uψℓm = 0 there. IV. IMPLEMENTATION Making more precise the ideas developed in the pre- ceding sections, we implement the following numerical scheme. A. Particle motion Following Darwin [12] we introduce the dimensionless semi-latus rectum p and the eccentricity e such that for a bound orbit around a Schwarzschild black hole of mass 1 + e , r2 = (4.1) are the radial positions of the periastron and apastron, respectively. Energy per unit mass and angular momen- tum per unit mass are then given by (p− 2− 2e)(p− 2 + 2e) p (p− 3− e2) , L2 = p− 3− e2 (4.2) Together with these definitions it is useful to introduce an orbital parameter χ such that along the trajectory of the particle, r(χ) = 1 + e cosχ , (4.3) where χ is single-valued along the orbit. We can then write down first-order differential equations for χ(t) and the azimuthal angle φ(t) of the particle, (p− 2− 2e cosχ)(1 + e cosχ)(1 + e cosχ) (Mp2) p− 6− 2e cosχ (p− 2− 2e)(p− 2 + 2e) , (4.4) (p− 2− 2e cosχ)(1 + e cosχ)2 p3/2M (p− 2− 2e)(p− 2 + 2e) . (4.5) We use the embedded Runge-Kutta-Fehlberg (4, 5) algo- rithm provided by the GNU Scientific Library routine gsl odeiv step rkf45 and an adaptive step-size control to evolve the position of the particle forward in time. Intermediate values of the particle’s position are found using a Hermite interpolation of the nearest available cal- culated positions. B. Initial data We do not specify initial data. The field is set to zero on the initial characteristic slices, u = u0 and v = v0. C. Boundary conditions We adjust the outer boundary of the numerical do- main at each time-step so that we cover the domain of dependence of the initial characteristic surfaces and the particle’s world line. The resulting numerical domain was already shown in Fig. 4. Near the event horizon, at r∗ ≈ −73M , we implement an ingoing-wave boundary condition by imposing ψ(t+ h, r∗) = ψ(t, r∗ − h). (4.6) This allows us to drastically reduce the number of cells in the numerical domain, and consequently the running time of the simulation. D. Evolution in vacuum Cells not traversed by the particle are evolved using Eqs. (2.1), (2.6) – (2.10). Explicitly written out, we use ψ3 = −ψ2 (V0 + V1) + V0 (V0 + V1) (V0 + V4) + V0 (V0 + V4) (g12 + g24 + g34 + g13 + 4g0), (4.7) where g0 is given by Eq. (2.7) and the sum g12 + g24 + g34 + g13 is given by Eq. (2.10). E. Cells next to the particle Vacuum cells close to the current position of the parti- cle require a different approach to calculate g0, since the cells in their past light cone could have been traversed by the particle. We use Eqs. (2.12) and (2.13) to find g0 in this case. Other than this modification, the same algorithm as for generic vacuum cells is used. F. Cells traversed by the particle We evolve cells traversed by the particle using the it- erative algorithm described in Sec. II C 2. Here ψ3 =− ψ1 + ψ2 + ψ4 − V ψ du dv f0(t) r0(t) Ȳℓm(π/2, π0(t)) dt, (4.8) where the initial guess for the iterative evolution of∫∫ V ψ du dv is obtained using the second order algo- rithm of Lousto and Price [13], ψ3 =− ψ1 + × [ψ2 + ψ4] f0(t) r0(t) Ȳℓm(π/2, π0(t)) dt. (4.9) Successive iterations use a four-point Gauss-Legendre rule to evaluate the integral of V ψ; this requires a second- order polynomial interpolation of the current field values as described in Appendix C. G. Extraction of the field data at the particle In order to extract the value of the field and its first derivatives at the position of the particle, we again use a polynomial interpolation at the points surrounding the particle’s position. Using a fourth-order polynomial, as described in Appendix C, we can estimate ψ, ∂tψt, and ∂r∗ψ at the position of the particle up to errors of order h4. As was briefly mentioned in Sec. II, we would expect an error term of order h3 for ∂tψt and ∂r∗ψ. The O(h accuracy we actually achieve by using a fourth-order (in- stead of a third-order) piecewise polynomial shows up clearly in a regression plot such as Fig. 7. H. Regularization of the mode sum We use the calculated multipole moments ψℓm to con- struct the multipole moments Φℓm, and first derivatives ∂tΦℓm and ∂rΦℓm, of the scalar field. These, in turn, are used to calculate the tetrad components Φ(0)ℓm, Φ(+)ℓm, Φ(−)ℓm, and Φ(3)ℓm of the field gradient according to Eqs. (1.23)–(1.26) of [4], which are reproduced in Ap- pendix A. These multipoles then give rise to the multi- pole coefficients of the retarded field, Φ(µ)ℓ(t, r, θ, φ) = Φ(µ)ℓm(t, r)Yℓm(θ, φ), (4.10) which are subjected to the regularization procedure de- scribed by Eq. (1.29) of [4], ΦR(µ)(t, r0, π/2, φ0) = lim Φ(µ)ℓ(t, r0 +∆, π/2, φ0) (ℓ+ 1/2)A(µ) +B(µ) (ℓ + 1/2) (ℓ− 1/2)(ℓ+ 3/2) + · · · , (4.11) using the regularization parameters A(µ), B(µ), C(µ), and D(µ) tabulated in Appendix B. Finally we reconstruct the vector components of the field gradient using Eqs. (1.47)–(1.48) of [4], ΦRt = (0), (4.12) ΦRr = ΦR(+)e −iφ0 +ΦR(−)e , (4.13) ΦRθ = −r0ΦR(3), (4.14) ΦRφ = − ΦR(+)e −iφ0 − ΦR(−)e , (4.15) and calculate the self-force Fα = qΦ α . (4.16) We recall the discussion in Sec. I A concerning the def- inition of ΦR, its connection to the self-force acting on the particle, and its regularity at the particle’s position. V. NUMERICAL TESTS In this section we present the tests we have performed to validate our numerical evolution code. First, in order to check the fourth-order convergence rate of the code, we perform regression runs with increasing resolution for both a vacuum test case, where we seeded the evolution with a Gaussian wave packet, and a case where a particle is present. As a second test, we compute the regularized self-force for several different combinations of orbital el- ements p and e and check that the multipole coefficients decay with ℓ as expected. This provides a very sensi- tive check on the overall implementation of the numerical scheme, as well as the analytical calculations that lead to the regularization parameters. Finally, we calculate the self-force for a particle on a circular orbit and show that it agrees with the results presented in [4, 14]. A. Convergence tests: Vacuum As a first test of the validity of our numerical code we estimate the convergence order by removing the particle and performing regression runs for several resolutions. We use a Gaussian wave packet as initial data, ψ(u = u0, v) = exp(−[v − vp]2/[2σ2]), (5.1) ψ(u, v = v0) = 0, (5.2) where vp = 75M and σ = 10M , v0 = −u0 = 6M + 2M ln 2, and we extract the field values at r∗ = 20M . Several such runs were performed, with varying resolu- tion of 2, 4, 8, 16, and 32 grid points per M . Figure 5 shows ψ(2h)−ψ(h) rescaled by appropriate powers of 2, so that in the case of fourth-order convergence the curves would lie on top of each other. As can be seen from the plots, they do, and the vacuum portion of the code is indeed fourth-order convergent. B. Convergence tests: Particle While the convergence test described in section VA clearly shows that the desired convergence is achieved for vacuum evolution, it does not test the parts of the code that are used in the integration of the inhomoge- neous wave equation. To test these we perform a second set of regression runs, this time using a non-zero charge q. We extract the field at the position of the particle, thus also testing the implementation of the extraction algorithm described in section IVG. For this test we choose the ℓ = 6, m = 4 mode of the field generated by a particle on a mildly eccentric geodesic orbit with p = 7, e = 0.3. As shown in Fig. 6 the convergence is still of fourth order, but the two curves no longer lie precisely on top of each other at all times. The region before t ≈ 100M is dominated by the initial wave burst and therefore does not scale as expected, yielding two very different curves. In the region 300M . t . 400M the two curves lie on top of each other, as expected for a fourth-order convergent algorithm. In the region between t ≈ 200M and t ≈ 300M , however, the dashed curves -8.0e-07 -6.0e-07 -4.0e-07 -2.0e-07 0.0e+00 2.0e-07 4.0e-07 6.0e-07 8.0e-07 0 20 40 60 80 100 120 δ16-8 δ32-16 FIG. 5: Convergence test of the numerical algorithm in the vacuum case. We show differences between simulations using different step sizes h = 0.5M (ψ2), h = 0.25M (ψ4), h = 0.125M (ψ8), h = 0.0625M (ψ16), and h = 0.03125M (ψ32). Displayed are the rescaled differences δ4−2 = ψ4 −ψ2, δ8−4 = 24(ψ8 − ψ4), δ16−8 = 4 4(ψ8 − ψ4), and δ32−16 = 8 4(ψ8 − ψ4) for the real part of the ℓ = 2, m = 2 mode at r∗ ≈ 20M . The maximum value of the field itself is of the order of 0.1, so that the errors in the field values are roughly five orders of magnitude smaller than the field values themselves. We can see that the convergence is in fact of fourth-order, as the curves lie nearly on top of each other, with only the lowest resolution curve δ4−2 deviating slightly. have slightly smaller amplitudes than the solid one, indi- cating an order of convergence different from (but close to) four. To explain this behavior we have to examine the terms that contribute significantly to the error in the simula- tion. The numerical error is almost completely domi- nated by that of the approximation of the potential term∫∫ V ψ du dv in the integrated wave equation. For vac- uum cells the error in this approximation scales as h6, where h is the step size. For cells traversed by the parti- cle, on the other hand, the approximation error depends also on the difference t2−t1 of the times at which the par- ticle enters and leaves the cell. This difference is bounded by h but does not necessarily scale as h. For example, if a particle enters a cell at its very left, then scaling h by 1 would not change t2 − t1 at all, thus leading to a scaling behavior that differs from expectation. To investigate this further we conducted test runs of the simulation for a particle on a circular orbit at r = 6M . In order to observe the expected scaling behav- ior, we have to make sure that the particle passes through the tips of the cell it traverses. When this is the case, then t2 − t1 ≡ h and a plot similar to the one shown in Fig. 6 shows the proper scaling behavior. As a further test we artificially reduced the convergence order of the vacuum algorithm to two by implementing the second-order algo- rithm described in [10]. By keeping the algorithm that deals with sourced cells unchanged, we reduced the rela- -4e-05 -2e-05 2e-05 4e-05 100 200 300 400 500 δ16-8 FIG. 6: Convergence test of the numerical algorithm in the sourced case. We show differences between simulations using different step sizes of 4 (ψ4), 8 (ψ8), 16 (ψ16), and 32 (ψ32) cells per M . Displayed are the rescaled differences δ8−4 = ψ8 −ψ4, etc. (see caption of Fig. 5 for definitions) of the field values at the position of the particle for a simulation with ℓ = 6, m = 4 and p = 7, e = 0.3. We see that the convergence is approximately fourth-order. -4e-05 -2e-05 2e-05 4e-05 100 200 300 400 500 δ32-16 δ16-8 FIG. 7: Convergence test of the numerical algorithm in the sourced case. We show differences between ∂rΦ for simula- tions using different step sizes of 4 (Φr,4), 8 (Φr,8), 16 (Φr,16), and 32 (Φr,32) cells per M . Displayed are the rescaled differ- ences δ8−4 = Φr,8 − Φr,4 etc. of the values at the position of the particle for a simulation with ℓ = 6, m = 4 and p = 7, e = 0.3. Although there is much noise caused by the piece- wise polynomials used to extract the data, we can see that the convergence is approximately fourth-order. tive impact on the numerical error. This, too, allows us to recover the expected (second-order) convergence. Fig- ures 8 and 9 illustrate the effects of the measures taken to control the convergence behavior. -3.000e-05 -2.000e-05 -1.000e-05 0.000e+00 1.000e-05 2.000e-05 3.000e-05 200 210 220 230 240 250 260 270 280 δ64-32 δ32-16 δ16-8 -3.000e-05 -2.000e-05 -1.000e-05 0.000e+00 1.000e-05 2.000e-05 3.000e-05 200 210 220 230 240 250 260 270 280 FIG. 8: Behavior of convergence tests for a particle in circular orbit at r = 6M . We show differences between simulations of the ℓ = 2, m = 2 multipole moment using different step sizes of 2 (ψ2), 4 (ψ4), 8 (ψ8), 16 (ψ16), 32 (ψ32) and 64 (ψ64) cells per M . Displayed are the real part of the rescaled differences δ4−2 = (ψ4 −ψ2) etc. of the field values at the position of the particle, defined as in Fig. 5. The values have been rescaled so that—for fourth order convergence—the curves should all coincide. The upper panel corresponds to a set of simulations where the particle traverses the cells away from their tips. The curves do not coincide perfectly with each other, seem- ingly indicating a failure of the convergence. The lower panel was obtained in a simulation where the particle was carefully positioned so as to pass through the tips of each cell it tra- verses. This set of simulations passes the convergence test more convincingly. C. High-ℓ behavior of the multipole coefficients Inspection of Eq. (4.11) reveals that a plot of Φ(µ)ℓ as a function of ℓ (for a selected value of t) should display a linear growth in ℓ for large ℓ. Removing the A(µ) term should produce a constant curve, removing the B(µ) term (given that C(µ) = 0) should produce a curve that decays as ℓ−2, and finally, removing the D(µ) term should pro- duce a curve that decays as ℓ−4. It is a powerful test of the numerical methods to check whether these expecta- tions are borne out by the numerical data. Fig. 10 plots the remainders as obtained from our numerical simula- tion, demonstrating the expected behavior. It displays, on a logarithmic scale, the absolute value of ReΦR(+)ℓ, the real part of the (+) component of the self-force. The orbit is eccentric (p = 7.2, e = 0.5), and all components of the self-force require regularization. The first curve (in trian- gles) shows the unregularized multipole coefficients that increase linearly in ℓ, as confirmed by fitting a straight line to the data. The second curve (in squares) shows par- tially regularized coefficients, obtained after the removal of (ℓ + 1/2)A(µ); this clearly approaches a constant for large values of ℓ. The curve made up of diamonds shows the behavior after removal of B(µ); because C(µ) = 0, it decays as ℓ−2, a behavior that is confirmed by a fit to -8.000e-05 -4.000e-05 0.000e+00 4.000e-05 8.000e-05 200 210 220 230 240 250 260 270 280 -8.000e-05 -4.000e-05 0.000e+00 4.000e-05 8.000e-05 200 210 220 230 240 250 260 270 280 -8.000e-05 -4.000e-05 0.000e+00 4.000e-05 8.000e-05 200 210 220 230 240 250 260 270 280 δ64-32 δ32-16 δ16-8 FIG. 9: Behavior of convergence tests for a particle in circular orbit at r = 6M . We show differences between simulations of the ℓ = 2, m = 2 multipole moment using different step sizes of 8 (ψ8), 16 (ψ16), 32 (ψ32), and 64 (ψ64) cells per M . Displayed are the real part of the rescaled differences δ16−8 = ψ16−ψ8 etc. of the field values at the position of the particle, defined as in Fig. 5. The values have been rescaled so that— for second order convergence—the curves should all coincide. The upper two panels correspond to simulations where the second order algorithm was used throughout. For the topmost one, care was taken to ensure that the particle passes through the tip of each cell it traverses, while in the middle one no such precaution was taken. Clearly the curves in the middle panel do not coincide with each other, indicating a failure of the second-order convergence of the code. The lower panel was obtained in a simulation using the mixed-order algorithm described in the text. While the curves still do not coincide precisely, the observed behavior is much closer to the expected one than for the purely second order algorithm. the ℓ ≥ 5 part of the curve. Finally, after removal of D(µ)/[(ℓ − 12 ) (ℓ + )] the terms of the sum decrease in magnitude as ℓ−4 for large values of ℓ, as derived in [15]. Each one of the last two curves would result in a con- verging sum, but the convergence is much faster after subtracting the D(µ) terms. We thereby gain more than 2 orders of magnitude in the accuracy of the estimated Figure 10 provides a sensitive test of the implemen- tation of both the numerical and analytical parts of the calculation. Small mistakes in either one will cause the difference in Eq. (4.11) to have a vastly different behav- 0 2 4 6 8 10 12 14 ReΦ(+) ReΦ(+)-A ReΦ(+)-A-B ReΦ(+)-A-B-D FIG. 10: Multipole coefficients of the dimensionless self-force ReΦR(+) for a particle on an eccentric orbit (p = 7.2, e = 0.5). The coefficients are extracted at t = 500M along the trajectory shown in Fig. 12. The plots show several stages of the regularization procedure, with a closer description of the curves to be found in the text. 0 2 4 6 8 10 12 14 16 18 20 FIG. 11: Multipole coefficients of ΦR(0) for a particle on a circu- lar orbit. Note that ΦR(0)ℓ is linked to Φ t via Φ The multipole coefficients decay exponentially with ℓ until ℓ ≈ 16, at which point numerical errors start to dominate. D. Self-force on a circular orbit For the case of a circular orbit, the regularization pa- rameters A(0), B(0), and D(0) all vanish identically, so that the (0) (or alternatively the t) component of the self-force does not require regularization. Figure 11 thus shows only one curve, with the magnitude of the multi- pole coefficients decaying exponentially with increasing As a final test, in Table I we compare our result for the self-force on a particle in a circular orbit at r = 6M to those obtained in [4, 14] using a frequency-domain code. For a circular orbit, a calculation in the frequency domain TABLE I: Results for the self-force on a scalar particle with scalar charge q on a circular orbit at r0 = 6M . The first column lists the results as calculated in this work us- ing time-domain numerical methods, while the second and third columns list the results as calculated in [4, 14] using frequency-domain methods. For the t and φ components the number of digits is limited by numerical roundoff error. For the r component the number of digits is limited by the trun- cation error of the sum of multipole coefficients. This work: Previous work: Diaz-Rivera time-domain frequency-domain [4] et. al. [14] ΦRt 3.60339 × 10 −4 3.60907254 × 10−4 1.6767 × 10−4 1.67730 × 10−4 1.6772834 × 10−4 ΦRφ −5.30424 × 10 −3 −5.30423170 × 10−3 is more efficient, and we expect the results of [4, 14] to be much more accurate than our own results. This fact is reflected in the number of regularization coefficients we can reliably extract from the numerical data, before being limited by the accuracy of the numerical method: the frequency-domain calculation found usable multipole coefficients up to ℓ = 20, whereas our data for ΦR(0)ℓ is dominated by noise by the time ℓ reaches 16. Figure 11 shows this behavior. E. Accuracy of the numerical method Several figures of merit can be used to estimate the accuracy of numerical values for the self-force. An estimate for the truncation error arising from cut- ting short the summation in Eq. (4.11) at some ℓmax can be calculated by considering the behavior of the remain- ing terms for large ℓ. Detweiler et. al. [15] showed that the remaining terms scale as ℓ−4 for large ℓ. They find the functional form of the terms to be EP3/2 (2ℓ− 3)(2ℓ− 1)(2ℓ+ 3)(2ℓ+ 5) , (5.3) where P3/2 = 36 2. We fit a function of this form to the tail end of a plot of the multipole coefficients to find the coefficient E in Eq. (5.3). Extrapolating to ℓ → ∞ we find that the truncation error is ℓ=ℓmax [Eq. (5.3)] (5.4) 2Eℓmax (2ℓmax + 3)(2ℓmax + 1)(2ℓmax − 1)(2ℓmax − 3) (5.5) where ℓmax is the value at which we cut the summation short. For all but the special case of the (0) component for a circular orbit, for which all regularization parame- ters vanish identically, we use this approach to calculate an estimate for the truncation error. A second source of error lies in the numerical calcula- tion of the retarded solution to the wave equation. This error depends on the step size h used to evolve the field forward in time. For a numerical scheme of a given con- vergence order, we can estimate this discretization error by extrapolating the differences of simulations using dif- ferent step sizes down to h = 0. This is what was done in the graphs shown in Sec. VB. We display results for mildly eccentric orbits. A high eccentricity causes ∂rΦ (displayed in Fig. 7) to be plagued by high frequency noise produced by effects similar to those described in Sec. VB. This makes it impossible to reliably estimate the discretization error for these orbits. We do not expect this to be very different from the errors for mildly eccentric orbits. Finally we compare our final results for the self-force Fα to “reference values”. For circular orbits, frequency- domain calculations are much more accurate than our time-domain computations. We thus compare our results to the results obtained in [4]. Table II lists typical values for the various errors listed above. error estimation mildly eccentric orbit truncation error (M Φ(+)) ≈ 2× 10 discretization error (M ∂rΦℓm) ≈ 10 comparison with reference values circular orbit Ft 0.2% Fr 0.04% Fφ 2× 10 TABLE II: Estimated values for the various errors in the com- ponents of the self-force as described in the text. We show the truncation and discretization errors for a mildly eccentric orbit and the total error for a circular orbit. The truncation error is calculated using a plot similar to the one shown in Fig. 16. The discretization error is estimated using a plot similar to that in Fig. 7 for the ℓ = 2, m = 2 mode, and the total error is estimated as the difference between our values and those of [4]. We use p = 7.2 , e = 0.5 for the mildly eccentric orbit. Note that we use the tetrad component Φ(+) for the truncation error and the vector component ∂rΦ for the discretization error. Both are related by the translation table Eqs. (A6) – (A9), we expect corresponding errors to be comparable for Φ(+) and ∂rΦ. VI. SAMPLE RESULTS In this section we describe some results of our numer- ical calculation. A. Mildly eccentric orbit We choose a particle on an eccentric orbit with p = 7.2, e = 0.5 which starts at r = pM/(1−e2), halfway between 15 10 5 0 5 10 trajectory for p=7.2, e=0.5 FIG. 12: Trajectory of a particle with p = 7.2, e = 0.5. The cross-hair indicates the point where the data for Fig. 10 was extracted. -0.014 -0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0.002 100 200 300 400 500 600 700 800 900 1000 time/M FIG. 13: Regularized dimensionless self-force M and M Fφ on a particle on an eccentric orbit with p = 7.2, e = 0.5. periastron and apastron. The field is evolved for 1000M with a resolution of 16 grid points per M , both in the t and r∗ directions, for ℓ = 0. Higher values of ℓ (and thus m) require a corresponding increase in the number of grid points used to achieve the same fractional accuracy. Multipole coefficients for 0 ≤ ℓ ≤ 15 are calculated and used to reconstruct the regularized self-force Fα along the geodesic. Figure 13 shows the result of the calcula- tion. For the choice of parameters used to calculate the force shown in Fig. 13, the error bars corresponding to the truncation error (which are already much larger than than the discretization error) would be of the order of the line thickness and have not been drawn. Already for this small eccentricity, we see that the self- force is most important when the particle is closest to the black hole (ie. for 200M . t . 400M and 600M . t . 60 50 40 30 20 10 0 10 20 30 40 50 trajectory for p=7.8001, e=0.9 FIG. 14: Trajectory of a particle on a zoom-whirl orbit with p = 7.8001, e = 0.9. The cross-hairs indicate the positions where the data shown in Fig. 16 and 17 was extracted. 800M); the self-force acting on the particle is very small once the particle has moved away to r ≈ 15M . B. Zoom-whirl orbit Highly eccentric orbits are of most interest as sources of gravitational radiation. For nearly parabolic orbits with e . 1 and p & 6+2e, a particle revolves around the black hole a number of times, moving on a nearly circu- lar trajectory close to the event horizon (“whirl phase”), before moving away from the black hole (“zoom phase”). During the whirl phase the particle is in the strong field region of the black hole, emitting copious amounts of radiation. Figures 14 and 15 show the trajectory of a particle and the force on such an orbit with p = 7.8001, e = 0.9. Even more so than for the mildly eccentric orbit discussed in Sec. VIA, the self-force (and thus the amount of radiation produced) is much larger while the particle is close to the black hole than when it zooms out. Defining energy E per unit mass and angular momen- tum L per unit mass in the usual way, E = − uα, L = uα, (6.1) and following eg. the treatment of Wald [16], Ap- pendix C, it is easy to see that the rates of change Ė and L̇ (per unit proper time) are directly related to com- ponents of the acceleration aα (and therefore force) ex- perienced by the particle via Ė = −at, L̇ = aφ. (6.2) The self-force shown in Fig. 15 therefore confirms our näıve expectation that the self-force should decrease both the energy and angular momentum of the particle as ra- diation is emitted. -0.025 -0.02 -0.015 -0.01 -0.005 0.005 500 1000 1500 2000 2500 3000 time/M -0.002 -0.001 0.001 0.002 0.003 0.004 2000 2050 2100 2150 2200 FIG. 15: Self-force acting on a particle. Shown is the dimen- sionless self-force M Fr and Fφ on a zoom-whirl orbit with p = 7.8001, e = 0.9. The inset shows a magni- fied view of the self-force when the particle is about to enter the whirl phase. No error bars showing an estimate error are shown, as the errors shown eg. in Table II are to small to show up on the graph. Notice that the self-force is essentially zero during the zoom phase 500M . t . 2000M and reaches a constant value very quickly after the particle enters into the whirl phase. It is instructive to have a closer look at the force acting on the particle when it is within the zoom phase, and also when it is moving around the black hole on the nearly cir- cular orbit of the whirl phase. In Fig. 16 and Fig. 17 we show plots of Φ(0)ℓ vs. ℓ after the removal of the A(µ), B(µ), and D(µ) terms. While the particle is still zooming in toward the black hole, Φ(0)ℓ behaves exactly as for the mildly eccentric orbit described in Sec. VIA over the full range of ℓ plotted; ie. the magnitude of each term scales as ℓ0, ℓ−2 and ℓ−4, after removal of the A(µ), B(µ), and D(µ) terms respectively. Close to the black hole, on the other hand, the particle moves along a nearly circular tra- jectory. If the orbit were perfectly circular for all times, ie. ṙ ≡ 0, then the (0) component would not require reg- ularization at all, and the multipole coefficients would decay exponentially, resulting in a straight line on the semi-logarithmic plot shown in Fig. 17. As the real orbit is not precisely circular, curves eventually deviate from a straight line. Removal of the A(µ) term is required almost immediately (beginning with ℓ ≈ 3), while the D(µ) term starts to become important only after ℓ ≈ 11. This shows that there is a smooth transition from the self-force on a circular orbit, which does not require regularization for the t and φ components, to that of a generic orbit, for which all components of the self-force require regulariza- tion. 0 2 4 6 8 10 12 14 Φ(0)-A Φ(0)-A-B Φ(0)-A-B-D FIG. 16: Multipole coefficients of M ReΦR(0) for a particle on a zoom-whirl orbit (p = 7.8001, e = 0.9). The coefficients are extracted at t = 2000M as the particle is about to enter the whirl phase. As ṙ is non-zero, all components of the self-force require regularization and we see that the dependence of the multipole coefficients on ℓ is as predicted by Eq. 1.9. After the removal of the regularization parameters A(µ), B(µ), and D(µ) the remainder is proportional to ℓ0, ℓ−2 and ℓ−4 respectively. 0 2 4 6 8 10 12 14 Φ(0)-A Φ(0)-A-B Φ(0)-A-B-D FIG. 17: Multipole coefficients of ReΦR(0) for a particle on a zoom-whirl orbit (p = 7.8001, e = 0.9). The coefficients are extracted at t = 2150M while the particle is in the whirl phase. The orbit is nearly circular at this time, causing the dependence on ℓ after removal of the regularization parame- ters to approximate that of a true circular orbit. Acknowledgments We thank Eric Poisson and Eran Rosenthal for useful discussions and suggestions. This work was supported by the Natural Sciences and Engineering Council of Canada. APPENDIX A: TRANSLATION TABLES We quote the results of [4] for the translation table be- tween the modes Φℓm and the tetrad components Φ(µ)ℓm with respect to the pseudo-Cartesian basis eα(0) = , 0, 0, 0 , (A1) eα(1) = f sin θ cosφ, cos θ cosφ,− sinφ r sin θ , (A2) eα(2) = f sin θ sinφ, cos θ sinφ, r sin θ , (A3) eα(3) = f cos θ,−1 sin θ, 0 , (A4) and the complex combinations eα := eα ± ieα eα(±) = f sin θe±iφ, cos θe±iφ, ±ie±iφ r sin θ . (A5) With these, the spherical-harmonic modes Φ(µ)ℓm(t, r) are given in terms of Φℓm(t, r) by Φ(0)ℓm = Φℓm, (A6) Φ(+)ℓm =− (ℓ+m− 1)(ℓ +m) (2ℓ− 1)(2ℓ+ 1) − ℓ− 1 Φℓ−1,m−1 (ℓ−m+ 1)(ℓ −m+ 2) (2ℓ+ 1)(2ℓ+ 3) Φℓ+1,m−1, (A7) Φ(−)ℓm = (ℓ −m− 1)(ℓ−m) (2ℓ− 1)(2ℓ+ 1) − ℓ− 1 Φℓ−1,m+1 (ℓ+m+ 1)(ℓ +m+ 2) (2ℓ+ 1)(2ℓ+ 3) Φℓ+1,m+1, (A8) Φ(3)ℓm = (ℓ −m)(ℓ+m) (2ℓ− 1)(2ℓ+ 1) Φℓ−1,m (ℓ−m+ 1)(ℓ +m+ 1) (2ℓ+ 1)(2ℓ+ 3) Φℓ+1,m. (A9) APPENDIX B: REGULARIZATION PARAMETERS For completeness we list the regularization parameters as calculated in [4]. Quantities bearing a subscript “0” are evaluated at the particle’s position. A(0) = 0 + L sign(∆), (B1) A(+) = −eiφ0 0 + L sign(∆), (B2) A(3) = 0, (B3) where f0 := 1 − 2M/r0 and sign(∆) is equal to +1 if ∆ > 0 and to −1 if ∆ < 0. We have, in addition, A(−) = Ā(+), A(1) = Re[A(+)], and A(2) = Im[A(+)]. We also use B(0) = − Er0ṙ0√ 0 + L 2)3/2 E + Er0ṙ0 0 + L 2)3/2 B(+) = e Bc(+) − iB , (B5) Bc(+) = 0 + L 2)3/2 r20 + L 0 + L 2)3/2 f0 − 1 r20 + L K, (B6) Bs(+) = − f0)ṙ0 r20 + L E + (2− f0)ṙ0 r20 + L B(3) = 0. (B8) In addition, B(−) = B̄(+), B(1) = Re[B(+)] = cosφ0 + B sinφ0, and B(2) = Im[B(+)] = Bc(+) sinφ0 −B (+) cosφ0. Here, the rescaled elliptic integrals E and K are defined E := 2 ∫ π/2 (1− k sin2 ψ)1/2 dψ = F ; 1; k K := 2 ∫ π/2 (1− k sin2 ψ)−1/2 dψ = F ; 1; k (B10) in which k := L2/(r20 + L We also use C(µ) = 0 (B11) D(0) = − Er30(r 0 − L2)ṙ30 0 + L 2)7/2 E(r70 + 30Mr 0 − 7L2r50 + 114ML2r40 + 104ML4r20 + 36ML6)ṙ0 16r40 0 + L 2)5/2 Er30(5r 0 − 3L2)ṙ30 0 + L 2)7/2 E(r50 + 16Mr 0 − 3L2r30 + 42ML2r20 + 18ML4)ṙ0 16r20 0 + L 2)5/2 K, (B12) D(+) = e Dc(+) − iD , (B13) Dc(+) = r30(r 0 − L2)ṙ40 0 + L 2)7/2 − r0ṙ 4(r20 + L 2)3/2 (3r70 + 6Mr 0 − L2r50 + 31ML2r40 + 26ML4r20 + 9ML6)ṙ20 0 + L 2)5/2 (3r70 + 8Mr 0 + L 2r50 + 26ML 2r40 + 22ML 4r20 + 8ML 16r60(r 0 + L 2)3/2 0 + 2Mr 0 + 4ML r20 + L 0 − 3L2)ṙ40 0 + L 2)7/2 8(r20 + L 2)3/2 − (7r 0 + 12Mr 0 − L2r30 + 46ML2r20 + 18ML4)ṙ20 16r20 0 + L 2)5/2 − (7r 0 + 6Mr 0 + 6L 2r30 + 12ML 2r20 + 4ML 16r40(r 0 + L 2)3/2 r20 + L K, (B14) Ds(+) = r20(r 0 − 7L2)( f0 − 2)ṙ30 0 + L 2)5/2 − (2r 0 +Mr 0 + 5L 2r50 + 10ML 2r40 + 29ML 4r20 + 14ML 6)ṙ0 8r50L(r 0 + L 2)3/2 (r50 −Mr40 + 4L2r30 − 5ML2r20 + 2ML4)ṙ0 4r30L 0 + L 2)3/2 r20(r 0 − 3L2)( f0 − 2)ṙ30 0 + L 2)5/2 (4r50 + 2Mr 0 + 7L 2r30 + 10ML 2r20 + 14ML 4)ṙ0 16r30L(r 0 + L 2)3/2 − (2r 0 − 2Mr20 + 5L2r0 − 8ML2)ṙ0 0 + L 2)3/2 K, (B15) D(3) = 0. (B16) And finally, D(−) = D̄(+), D(1) = Re[D(+)] = cosφ0 + D sinφ0, and D(2) = Im[D(+)] = Dc(+) sinφ0 −D (+) cosφ0. APPENDIX C: PIECEWISE POLYNOMIALS In two places in the numerical simulation we introduce piecewise polynomials to approximate the scalar field ψℓm across the world line, where it is continuous but not dif- ferentiable. By a piecewise polynomial we mean a poly- nomial of the form p(t, r∗) =   n,m=0 unvm if r∗(u, v) > r∗0 n,m=0 unvm if r∗(u, v) < r∗0 , (C1) where u = t−r∗, v = t+r∗ are characteristic coordinates, r∗0 is the position of the particle at the time t(u, v), and N is the order of the polynomial, which for our purposes is N = 4 or less. The two sets of coefficients cnm and c′nm are not independent of each other, but are linked via jump conditions that can be derived from the wave equation [Eq. (1.12)]. To do so, we rewrite the wave equation in the characteristic coordinates u and v and reintroduce the integral over the world line on the right- hand side, −4∂u∂vψ − V ψ = Ŝ(τ)δ(u − up)δ(v − vp) dτ , (C2) where Ŝ(τ) = −8πq Ȳℓm(π/2,φp(τ)) rp(τ) is the source term and quantities bearing a subscript p are evaluated on the world line at proper time τ . Here and in the following we use the notation [∂nu∂ v ψ] = lim [∂nu∂ v ψ(t0, r 0 + ǫ)− ∂nu∂mv ψ(t0, r∗0 − ǫ)] to denote the jump in ∂nu∂ v ψ across the world line. First, we notice that the source term does not contain any derivatives of the Dirac δ-function, causing the solution ψ to be continuous. This means that the zeroth-order jump vanishes: [ψ] = 0. Our task is then to find the re- maining jump conditions at a point (t0, r 0) for n,m ≤ 4. Alternatively, instead of crossing the world line along a line t = t0 = const we can also choose to cross along lines of u = u0 = const or v = v0 = const, noting that for a line of constant v the coordinate u runs from u0+ ǫ to u0 − ǫ to cross from the left to the right of the world line. Figure 18 provides a clearer description of the paths taken. (u0 − ǫ, v0) (u0, v0 + ǫ)(u0 + ǫ, v0) (u0, v0 − ǫ) (t0, r 0) = (u0, v0) FIG. 18: Paths taken in the calculation of the jump condi- tions. (u0, v0) denotes an arbitrary but fixed point along the world line γ. The wave equation is integrated along the lines of constant u or v indicated in the sketch. Note that in order to move from the domain on the left to the domain on the right, u has to run from u0 + ǫ to u0 − ǫ. Where appropriate we label quantities connected to the domain on the left by a subscript “−” and quantities connected to the domain on the right by “+”. In order to find the jump [∂uψ] we integrate the wave equation along the line u = u0 from v0 − ǫ to v0 + ǫ ∫ v0+ǫ ∂u∂vψdv − ∫ v0+ǫ V ψdv = Ŝ(τ)δ(u0 − up) ∫ v0+ǫ δ(v − vp)dv dτ , which, after involving ∫ v0+ǫ δ(v − vp)dv = θ(vp − v0 + ǫ)θ(v0 − vp + ǫ) and δ(g(x)) = δ(x− x0)/ |g′(x0)|, yields [∂uψ] = − E − ṙ0 Ŝ(τ0), (C5) where the overdot denotes differentiation with respect to proper time τ . Similarly, after first taking a derivative of the wave equation with respect to v and integrating from u0+ ǫ to u0 − ǫ, we obtain ∫ u0−ǫ vψdu− ∫ u0−ǫ V ψdu = Ŝ(τ) ∫ u0−ǫ δ(u− up)du δ′(v0 − vp)dτ . We find E + ṙ0 E + ṙp Ŝ(τ) |τ=τ0 . (C7) Systematically repeating this procedure we find expres- sions for the jumps in all the derivatives that are purely in the u or v direction. Table III lists these results. Jump [ψ] =0 [∂uψ] =− bS(τ0), [∂vψ] = bS(τ0) bS(τ ) |τ=τ0 bS(τ ) |τ=τ0 V ξ0ξ̄ 0 [∂uψ]− bS(τ ) |τ=τ0 V ξ̄0ξ 0 [∂vψ] + bS(τ ) |τ=τ0 |τ=τ0 + 3ξ0ξ̄ 0 ∂uV + ξ 0 ∂vV [∂uψ] + bS(τ ) |τ=τ0 |τ=τ0 + 3ξ̄0ξ 0 ∂vV + ξ̄ 0 ∂uV [∂vψ]− bS(τ ) |τ=τ0 TABLE III: Jump conditions for the derivatives purely in the u or v directions. ṙ and r̈ are the particle’s radial velocity and acceleration, respectively. They are obtained from the equa- tion of motion for the particle. ξ̄ := E−ṙ and ξ := E+ṙ introduced for notational convenience. Quantities bearing a subscript p are evaluated on the particle’s world line, while quantities bearing a subscript 0 are evaluated at the parti- cle’s current position. Derivatives of V with respect to either u or v are evaluated as ∂uV = − f∂rV and ∂vV = f∂rV , respectively. conditions for derivatives involving both u and v are ob- tained directly from the wave equation [Eq. (C2)]. We see that [∂u∂vψ] = 0, (C8) and taking an additional derivative with respect to u on both sides reveals that ∂2u∂vψ V [∂uψ] . (C9) Systematically repeating this procedure we can find jump conditions for each of the mixed derivatives by evaluating ∂n+1u ∂ [∂nu∂ v (V ψ)] , (C10) where n,m ≥ 0 and derivatives of V with respect to either u or v are evaluated as ∂uV = − 12f∂rV and ∂vV = f∂rV , respectively. The results of Table III and Eq. (C10) allow us to express the coefficients of the left-hand polynomial in Eq. (C1) in terms of the jump conditions and the co- efficients of the right-hand side: c′nm = cnm − [∂nu∂mv ψ] . (C11) For N = 4 this leaves us with 25 unknown coefficients cnm which can be uniquely determined by demanding that the polynomial match the value of the field on the 25 grid points surrounding the particle. When we are interested in integrating the polynomial, as in the case of the potential term in the fourth-order algorithm, we do not need all these terms. Instead, in order to calculate e.g. the integral V ψ du dv up to terms of order h5, as is needed to achieve overall O(h4) convergence, it is sufficient to include only terms such that n+m ≤ 2, thus reducing the number of unknown coefficients to 6. In this case Eq. (C1) becomes p(t, r∗) = m+n≤2 unvm if r∗(u, v) > r∗0 m+n≤2 unvm if r∗(u, v) < r∗0 . (C12) The six coefficients can then be determined by matching the polynomial to the field values at the six grid points which lie within the past light cone of the grid point whose field value we want to calculate. [1] The LISA web site is located at http://lisa.jpl.nasa.gov/. [2] L. Barack and A. Ori, Phys. Rev. D 61, 061502 (2000), gr-qc/9912010. [3] C. Lousto, Class. Quantum Grav. 22, S543 (2005). [4] R. Haas and E. Poisson, Phys. Rev. D 74, 044009 (pages 29) (2006), gr-qc/0605077, URL http://link.aps.org/abstract/PRD/v74/e044009 . [5] B. S. DeWitt and R. W. Brehme, Annals of Physics 9, 220 (1960). [6] Y. Mino, M. Sasaki, and T. Tanaka, Phys. Rev. D 55, 3457 (1997), gr-qc/9606018. [7] T. C. Quinn and R. M. Wald, Phys. Rev. D 56, 3381 (1997), gr-qc/9610053. [8] T. C. Quinn, Phys. Rev. D 62, 064029 (2000), gr- qc/0005030. [9] S. Detweiler and B. F. Whiting, Phys. Rev. D 67, 024025 (2003), gr-qc/0202086. [10] C. O. Lousto, Class. Quant. Grav. 22, S543 (2005), gr- qc/0503001. [11] R. H. Price, Phys. Rev. D 5, 2419 (1972). [12] C. G. Darwin, Proc. R. Soc. A 249, 180 (1959). http://lisa.jpl.nasa.gov/ http://link.aps.org/abstract/PRD/v74/e044009 [13] C. O. Lousto and R. H. Price, Phys. Rev. D 56, 6439 (1997), gr-qc/9705071. [14] L. M. Diaz-Rivera, E. Messaritaki, B. F. Whit- ing, and S. Detweiler, Physical Review D (Par- ticles, Fields, Gravitation, and Cosmology) 70, 124018 (pages 14) (2004), gr-qc/0410011, URL http://link.aps.org/abstract/PRD/v70/e124018. [15] S. Detweiler, E. Messaritaki, and B. F. Whiting, Phys. Rev. D 67, 104016 (2003), gr-qc/0205079. [16] R. M. Wald, General relativity (University of Chicago Press, Chicago, 1984), ISBN 0226870324. http://link.aps.org/abstract/PRD/v70/e124018

0704.0798

Nonimmersions of RP^n implied by tmf, revisited

arXiv:0704.0798v1 [math.AT] 5 Apr 2007 NONIMMERSIONS OF RP n IMPLIED BY tmf, REVISITED DONALD M. DAVIS AND MARK MAHOWALD Abstract. In a 2002 paper, the authors and Bruner used the new spectrum tmf to obtain some new nonimmersions of real projective spaces. In this note, we complete/correct two oversights in that paper. The first is to note that in that paper a general nonimmersion result was stated which yielded new nonimmersions for RPn with n as small as 48, and yet it was stated there that the first new result occurred when n = 1536. Here we give a simple proof of those overlooked results. Secondly, we fill in a gap in the proof of the 2002 paper. There it was claimed that an axial map f must satisfy f∗(X) = X1+X2. We realized recently that this is not clear. However, here we show that it is true up multiplication by a unit in the appropriate ring, and so we retrieve all the nonimmersion results claimed in [6]. Finally, we present a complete determination of tmf8∗(RP∞ × RP∞) and tmf∗(CP∞ × CP∞) in positive dimensions. 1. Introduction In [6], the authors and Bruner described a proof of the following theorem, along with some additional nonimmersion results. Theorem 1.1. ([6, 1.1]) Assume that M is divisible by the smallest 2-power greater than or equal to h. • If α(M) = 4h − 1, then P 8M+8h+2 cannot be immersed in ( 6⊆) 16M−8h+10. • If α(M) = 4h− 2, then P 8M+8h 6⊆ R16M−8h+12. Here and throughout, α(M) denotes the number of 1’s in the binary expansion of M , and P n denotes real projective space. Date: April 5, 2007. 2000 Mathematics Subject Classification. 57R42, 55N20. Key words and phrases. immersion, projective space, elliptic cohomology. We thank Steve Wilson for causing us to take a look at these matters. http://arxiv.org/abs/0704.0798v1 2 DONALD M. DAVIS AND MARK MAHOWALD In [6], the theorem is followed by a comment that this is new provided α(M) ≥ 6, i.e., h ≥ 2, and the first new result occurs for P 1536. In this note, we point out that 1.1 is valid when h = 1, and these results are new when M is even, including new nonimmersions of P n for n as small as 56. A remark in [6, p.66] that the nonimmersions when h = 1 were implied by earlier work of the authors was incorrect. Letting h = 1 in 1.1, we have the following result. Corollary 1.2. a. If α(M) = 3, then P 8M+10 6⊆ R16M+2. b. If α(M) = 2, then P 8M+8 6⊆ R16M+4. Part (a) is new when M is even. It is 2 better than the previous best result, proved in [4], and the nonembedding result that it implies is also new, 1 better than the previous best, proved in [3]. In [7], a table of known nonimmersions, immersions, nonembeddings, and embeddings of P n is presented, arranged according to n = 2i+d with 0 ≤ d < 2i and d < 64. Part (a) enters the table with a new result for d = 58, applying first to P 122. If M is even, 1.2.b is new, 1 better than the previous best result, of [12], and the nonembedding result implied is also new. It enters [7] at d = 24 and 40, with a new result for P n with n as small as 56. The result of 1.2.b with M = 2i + 1 was also proved very recently by Kitchloo and Wilson in [15]. This result for P 2 k+16, 2 better than the previous result of [4] and also new as a nonembedding, enters [7] at d = 16, and applies for n as small as 48. In Section 2, we present a self-contained proof of Corollary 1.2. The primary reason for doing this, which amounts to a reproof of part of [6, 1.1], is that the proof of the general case in [6] requires some extremely elaborate arguments and calculations. Our proof here, which is just for the case h = 1, is much more comprehensible. The proof in [6] contained an oversight which we shall correct here. The argument there was that an immersion of RP n in Rn+k implies existence of an axial map P n× f−→ Pm+k for an appropriate value of m, and obtains a contradiction for certain n, m, and k by consideration of tmf∗(f). Here tmf is the spectrum of topological modular forms, which was discussed in [6]. A class X ∈ tmf8(P n) was described, along with X1 = X × 1 and X2 = 1 × X in tmf8(P n × Pm). It was asserted that f ∗(X) = X1+X2, and a contradiction obtained by showing that, for certain values of NONIMMERSIONS IMPLIED BY TMF, REVISITED 3 the parameters, we might have Xℓ = 0 but (X1+X2) ℓ 6= 0. We recently realized that it is conceivable that f ∗(X) might contain other terms coming from tmf8(P n ∧ Pm). In Section 3 (see Theorem 3.7) we perform a complete calculation of tmf∗(P∞×P∞) in positive gradings divisible by 8, and in Section 4 we use it to show that effectively f ∗(X) = u(X1 + X2), where u is a unit in tmf ∗(P∞ × P∞), which enables us to retrieve all the nonimmersions of [6]. In Section 5, we compute tmf∗(CP∞ × CP∞) in positive gradings. The original purpose of doing this was, prior to our obtaining the argument of Section 4, to see whether we might mimic the argument of [2] and [8] to conclude that if f is an axial map, then f ∗(X) might necessarily equal u(X1 − X2), where u is a unit in tmf∗(CP × CP ). This approach to retrieving the nonimmersions of [6] did not yield the desired result, but the later approach given in Section 4 did. Nevertheless the nice result for tmf∗(CP∞ × CP∞) obtained in Theorem 5.19 should be of independent interest. 2. Proof of Corollary 1.2 We begin by proving 1.2.a. The following standard reduction goes back at least to [14]. If P 8M+10 ⊆ R16M+2, then gd((2L+3 − 8M − 11)ξ8M+10) ≤ 8M − 8, hence this bundle has (2L+3−16M −3) linearly independent sections, and thus there is an axial P 8M+10 × P 2L+3−16M−4 f−→P 2L+3−8M−12. The bundle here is the stable normal bundle, L is a sufficiently large integer, and gd refers to geometric dimension. Let X , X1, and X2 be elements of tmf 8(−) described in [6] and also in Section 1. In Section 4, we will show that we may assume that f ∗(X) = X1+X2, as was done in [6], since this is true up to multiplication by a unit. Since tmf2 L+3−8M−8(P 2 L+3−8M−12) = 0, we have 0 = f ∗(0) = f ∗(X2 L−M−1) = (X1+X2) 2L−M−1 ∈ tmf2L+3−8M−8(P 8M+10×P 2L+3−16M−4). Expanding, we obtain 2L−M−1 XM+11 X 2L−2M−2 2L−M−1 XM1 X 2L−2M−1 2 as the only terms which are possibly nonzero. Next we note that, with all u’s representing odd integers, 2L−M−1 = 2α(M)−ν(M+1)u2 = 2 3−ν(M+1)u2, 4 DONALD M. DAVIS AND MARK MAHOWALD where we have used α(M) = 3 at the last step. Here and throughout, ν(2eu) = e. Similarly, 2L−M−1 = 2α(M)u4 = 2 3u4. Thus an immersion implies that in L+3−8M−8(P 8M+10 × P 2L+3−16M−4), we have 23−ν(M+1)u2X 2L−2M−2 2 + 2 2L−2M−1 2 = 0. (2.1) We recall [6, 2.6], which states that there is an equivalence of spectra P k+8b+8 ∧ tmf ≃ Σ8P kb ∧ tmf. Combining this with duality, we obtain tmf 8M+8(P 8M+10) ≈ tmf−1(P−3) ≈ Z/8, and so 8XM+11 X2 L−2M−2 2 = 0. Here and throughout, Pn = P RP∞/RP n−1. Similarly tmf2 L+3−16M−8(P 2 L+3−16M−4) ≈ tmf7(P3) ≈ Z/16, and hence 16XM1 X 2L−2M−1 2 = 0. Duality also implies L+3−8M−8(P 8M+10 × P 2L+3−16M−4) ≈ tmf14(P−3 ∧ P3). Calculations such as E2(tmf∗(P−3∧P3)), the E2-term of the Adams spectral sequence (ASS), were made by Bruner’s minimal-resolution computer programs in our work on [6]. This one is in a small enough range to actually do by hand. The result is given in Diagram 2.2. Diagram 2.2. E2(tmf∗(P−3 ∧ P3)), ∗ ≤ 15 0 3 7 11 15 r r r r r� r r r The Z/8 ⊕ Z/16 arising from filtration 0 in grading 14 in 2.2 is not hit by a differential from the class in (15, 0) because, as explained in the last paragraph of page 54 of [6], the class in (15, 0) corresponds to an easily-constructed nontrivial map. The monomials XM+11 X 2L−2M−2 2 and X 2L−2M−1 2 are detected in mod-2 cohomology, NONIMMERSIONS IMPLIED BY TMF, REVISITED 5 and so their duals emanate from filtration 0. We saw in the previous paragraph that 8 and 16, respectively, annihilate these monomials, and hence also their duals. Since the chart shows that the subgroup of tmf14(P−3∧P3) generated by classes of filtration 0 is Z/8 ⊕ Z/16, we conclude that 8 and 16, respectively, are the precise orders of the monomials. In particular, the order of XM1 X 2L−2M−1 2 is 16, and hence the class in (2.1) is nonzero since it has a term 8uXM1 X 2L−2M−1 2 , and so (2.1) contradicts the hypothesized immersion. Part b of 1.2 is proved similarly. If P 8M+8 immerses in R16M+4, then there is an axial map P 8M+8 × P 2L+3−16M−6 f−→P 2L+3−8M−10, and hence, up to odd multiples, 22−ν(M+1)XM+11 X 2L−2M−2 2 + 2 2XM1 X 2L−2M−1 2 (2.3) = 0 ∈ tmf2L+3−8M−8(P 8M+8 ∧ P 2L+3−16M−6), since α(M) = 2. We have tmf8M+8(P 8M+8) ≈ tmf−1(P−1) ≈ Z/2, and L+3−16M−8(P 2 L+3−16M−6) ≈ tmf−1(P−3) ≈ Z/8. Thus the two monomials in (2.3) have order at most 2 and 8, respectively. On the other hand, the group in (2.3) is isomorphic to tmf6(P−1 ∧ P−3). A minimal resolution calculation easier than the one in Diagram 2.2 shows that tmf6(P−1∧P−3) has Z/2⊕Z/8 emanating from filtration 0 (and another Z/2⊕Z/8 in higher filtration). The monomials of (2.3) are generated in filtration 0, and since the above upper bound for their orders equals the order of the subgroup generated by filtration-0 classes, we conclude that the orders of the monomials in (2.3) are precisely 2 and 8, respectively, and so the term 4XM1 X 2L−2M−1 2 in (2.3) is nonzero, contradicting the immersion. 3. tmf-cohomology of P∞ × P∞ In this section, we compute tmf∗(P∞) and tmf8∗(P∞ × P∞) in positive gradings. These will be used in the next section in studying the axial class in tmf-cohomology. There is an element c4 ∈ π8(tmf) which reduces to v41 ∈ π8(bo); it has Adams filtration 4. It acts on tmf∗(X) with degree −8. Recall also that π∗(bo) = bo∗ is as depicted in 5.1. We denote bo∗ = bo−∗. We use P1 and P ∞ interchangeably. 6 DONALD M. DAVIS AND MARK MAHOWALD Theorem 3.1. There is an element X ∈ tmf8(P1) of Adams filtration 0, described in [6], such that, in positive dimensions divisible by 8, tmf∗(P1) is isomorphic as an algebra over Z(2)[c4] to Z(2)[c4][X ]. In particular, each tmf 8i(P1) with i > 0 is a free abelian group with basis {cj4X i+j : j ≥ 0}. There is a class L ∈ t0(P1) such that • tmf0(P1) is a free abelian group with basis {L, cj4Xj : j ≥ 1}, • L2 = 2L and LX = 2X. Moreover, in positive dimensions tmf∗(P1) is isomorphic as a graded abelian group to bo∗[X ], and is depicted in Diagram 3.6. Remark 3.2. A complete description of tmf∗(P1) as a graded abelian group could probably be obtained using the analysis in the proof which follows, together with the computation of the E2-term of the ASS converging to tmf∗(P−1), which was given in [10]. However, this is quite complicated and unnecessary for this paper, and so will be omitted. Proof. We begin with the structure as graded abelian group. There are isomorphisms tmf∗(P1) ≈ lim← tmf ∗(P n1 ) ≈ lim← tmf−∗−1(P −n−1) = tmf−∗−1(P (3.3) Since H∗(tmf;Z2) ≈ A//A2, there is a spectral sequence converging to tmf∗(X) with E2(X) = ExtA2(H ∗X,Z2). Here A2 is the subalgebra of the mod 2 Steenrod algebra A generated by Sq1, Sq2, and Sq4. Also Z2 = Z/2. We compute E2(P −∞) from the exact sequence → Es−1,t2 (P∞−1)→ E −∞)→ E q∗−→ Es,t2 (P∞−1)→ . (3.4) It was proved in [17] that ExtA2(P −∞,Z2) ≈ ExtA1(Σ Z2,Z2). Here we have initiated a notation that Pmn := H ∗(Pmn ). A complete calculation of ExtA2(P −1,Z2) was performed in [10], but all we need here are the first few groups. We can now form a chart for E2(P −∞) from (3.4), as in Diagram 3.5, where ◦ indicate elements of ExtA2(P −1,Z2) suitably positioned, and lines of negative slope correspond to cases of q∗ 6= 0 in (3.4). NONIMMERSIONS IMPLIED BY TMF, REVISITED 7 Diagram 3.5. tmf∗(P −∞), −17 ≤ ∗ ≤ 2 −17 −9 −1 · · · ✻ ✻ ✻ r r r r r r ✻ ✻ ✻ r r r r r r r r r r r r r r r Dualizing, we obtain Diagram 3.6 for the desired tmf∗(P∞1 ). Diagram 3.6. tmf∗(P∞1 ), ∗ ≥ −2 0 8 16 ✻ ✻ ✻✻ ✻ ✻ r r r r r r · · · Naming of the generators X i is clear since X has filtration 0. The free action of c4 is also clear. The class L is (up to sign) the composite P1 λ−→ S0 → tmf, where λ is the well-known Kahn-Priddy map. Thus L is the image of a class L̂ ∈ π0(P1). Lin’s theorem ([16]) says that π0(P1) ≈ Z∧2 , generated by L̂. Since π0(P1)→ ko0(P1) is an isomorphism, and, since (1 − ξ)2 = 2(1 − ξ) for a generator (1 − ξ) of ko0(P1), we obtain L̂2 = 2L̂, and hence also for L. We chose the generator to be (1 − ξ) rather than (ξ − 1) to avoid minus signs later in the paper. 8 DONALD M. DAVIS AND MARK MAHOWALD To prove the claim about LX , first note that, by the structure of tmf8(P1), we must have LX = p(c4X)X for some polynomial p. Multiply both sides by L and apply the result about L2 to get 2LX = p(c4X)LX , hence 2p = p 2, from which we conclude p = 2. In tmf∗(P1 × P1), for i = 1, 2, let Li and Xi denote the classes L and X in the ith factor. Note that there is an isomorphism as tmf∗-modules, but not as rings, tmf∗(P1 × P1) ≈ tmf∗(P1 ∧ P1)⊕ tmf∗(P1 × ∗)⊕ tmf∗(∗ × P1). Theorem 3.7. In positive dimensions divisible by 8, tmf∗(P1 ∧ P1) is isomorphic as a graded abelian group to a free abelian group on monomials X i1X 2 with i, j > 0 direct sum with a free Z[c4]-module with basis {L1X i2, X i1L2 : i ≥ 1}. The product and Z[c4]-module structure is determined from 3.1 and c4(X1X2) = (c4X1)X2 = X1(c4X2) = 4(L1X 1 L2), for certain integers γi with γ0 divisible by 8. The proof of this theorem involves a number of subsidiary results. They and it occupy the remainder of this section. We will use duality and exact sequences similar to (3.4). But to get started, we need ExtA2(P ⊗ P,Z2). Here we have begun to abbreviate P := P∞−∞. We begin with a simple lemma. Throughout this section, x1 and x2 denote nonzero elements coming from the factors in H 1(RP × RP ;Z2). Lemma 3.8. ([9]) There is a split short exact sequence of A-modules 0→ Z2 ⊗P→ P⊗P→ (P/Z2)⊗P→ 0. Proof. The Z2 is, of course, the subgroup generated by x 0, which is an A-submodule. A splitting morphism P⊗P g−→Z2 ⊗P is defined by g(xi1 ⊗ x 2) = x 1 ⊗ x 2 . This is A-linear since g(Sqk(xi1⊗x 2)) = x01⊗x i+j+k x01⊗x i+j+k 2 = Sq k g(xi1⊗x The following result is more substantial. We will prove it at the end of this section. NONIMMERSIONS IMPLIED BY TMF, REVISITED 9 Proposition 3.9. There is a short exact sequence of A2-modules 0→ C → (P/Z2)⊗P→ B → 0, where C has a filtration with Fp(C)/Fp−1(C) ≈ Σ8pA2/ Sq2, p ∈ Z, and B has a filtration with Fp(B)/Fp−1(B) ≈ Z copies Σ4p−2A2/ Sq 1, p ∈ Z. The generator of Fp(C)/Fp−1(C) is x 2 ; a basis over Z2 for C is {x21xi+22 +x41xi2, x41xi2+x81xi−42 , i ∈ Z}∪{x11xi−12 +x21xi−22 , i 6≡ 0 (8)}∪{x11x 2 , p ∈ Z}. A minimal set of generators as an A2-module for the filtration quotients of B is {x8i−11 x 2 : i, j ∈ Z}. Corollary 3.10. A chart for Ext (P ⊗ P,Z2) in 8p − 3 ≤ t − s ≤ 8p + 4 is as suggested in Diagram 3.11, for all integers p. The big batch of towers in each grading ≡ 2 (4) represents an infinite family of towers. The pattern of the other classes is repeated with vertical period 4. Thus, for example, in 8p−1 there is an infinite tower emanating from filtration 4i for each i ≥ 0. 10 DONALD M. DAVIS AND MARK MAHOWALD Diagram 3.11. Ext (P⊗P,Z2) in 8p− 3 ≤ t− s ≤ 8p+ 4 8p+ −2 0 2 4 ✻✻✻✻✻✻✻✻✻✻✻ ✻✻✻✻✻✻✻✻✻✻✻✻ ✻✻ ✻✻ ✻✻ Proof of Corollary 3.10. We first note that ExtA2(P,Z2) is identical to the left portion of Diagram 3.5 extended periodically in both directions. Also, ExtA2(A2/ Sq 1,Z2) ≈ ExtA0(Z2,Z2) is just an infinite tower, and ExtA2(A2/ Sq 2,Z2) ≈ ExtA1(A1/ Sq2,Z2) is given as in Diagram 3.14. We will show at the end of this proof that ExtA2(C,Z2) ≈ ExtA2(Σ 8pA2/ Sq 2,Z2) (3.12) and similarly ExtA2(B,Z2) ≈ ExtA2(Σ 4p−2A2/ Sq 1,Z2). These would follow by induction on p once you get started, but since p ranges over all integers, that is not automatic. Thus ExtA2(P⊗P,Z2) is formed from ExtA2(P,Z2)⊕ ExtA2(Σ 8pA2/ Sq 2,Z2)⊕ ExtA2(Σ 4p−2A2/ Sq 1,Z2), NONIMMERSIONS IMPLIED BY TMF, REVISITED 11 using the sequences in 3.8 and 3.9. The Ext sequence of 3.8 must split, and there are no possible boundary morphisms in the Ext sequence of 3.9, yielding the claim of the corollary. To prove (3.12), let (s, t) be given, and choose p0 so that 8p0 < t− 23s+ 2. Since the highest degree element in A2 is in degree 23, Ext (Fp0(C),Z2) = 0. Actually a much sharper lower vanishing line can be established, but this is good enough for our purposes. Thus, for this (s, t), (Fp1(C),Z2) ≈ (Σ8p−2A2/ Sq 2) (3.13) for p1 ≤ p0, as both are 0. Let p1 be minimal such that (3.13) does not hold. Then comparison of exact sequences implies that s−1,t (Fp1−1(C),Z2)→ Ext (Fp1(C)/Fp1−1(C),Z2) must be nonzero. But one or the other of these groups is always 0,1 as both charts (Fp1−1(C),Z2) and Ext (Fp1(C)/Fp1−1(C),Z2) are copies of Diagram 3.14 dis- placed by 4 vertical units from one another. Thus (3.13) is true for all p1, and hence (3.12) holds. A similar proof works when C is replaced by B. Diagram 3.14. ExtA2(A2/ Sq 2,Z2) · · · Now we can prove a result which will, after dualizing, yield Theorem 3.7. The groups ExtA1(Z2,Z2) to which it alludes are depicted in 5.1. The content of this result is pictured in Diagram 3.18. Proposition 3.15. In dimensions t− s ≡ 2 mod 4 with t− s ≤ −10, ExtA2(P−2−∞⊗ P−2−∞,Z2) consists of i infinite towers emanating from filtration 0 in dimensions −8i− 1Actually this is not quite true; for one family of elements we need to use h0- naturality. 12 DONALD M. DAVIS AND MARK MAHOWALD 6 and −8i − 10, together with the relevant portion of two copies of ExtA1(Z2,Z2) beginning in filtration 1 in each dimension −8i − 2. The generators of the towers in −8i− 10 correspond to cohomology classes x−91 x−8i−12 , . . . , x−8i−11 x−92 . The generators of the two copies of ExtA1(Z2,Z2) in −8i−2 arise from h0 times classes corresponding to x−11 x 2 and x −8i−1 Proof. Using exact sequences like (3.4) on each factor, we build Ext (P−2−∞⊗P−2−∞,Z2) fromA := Ext (P⊗P,Z2), B := Ext∗−1,∗A2 (P −1⊗P,Z2), C := Ext ∗−1,∗ (P⊗P∞−1,Z2), and D := Ext ∗−2,∗ (P∞−1 ⊗ P∞−1,Z2), with possible d1-differential from A and into D. In the range of concern, t− s ≤ −9, the D-part will not be present, and the part of Diagram 3.11 in dimension 6≡ 2 mod 4 will not be involved in d1. Using [17] for B and C, the relevant part, namely the portion of A in dimension ≡ 2 mod 4, together with B and C, is pictured in Diagram 3.16. Diagram 3.16. Portion of A+B+C ✻✻✻✻✻✻✻✻✻✻ ✻✻✻✻✻✻✻✻✻✻ ✻✻✻✻✻✻✻✻✻✻ −2 2 68p+ rr rrr In dimension 8p−2, the towers in A arise from all cohomology classes x−8i−11 x −8j−1 with i+ j = −p, while in dimension 8p+ 2, they arise from x8i−11 x 2 ∼ x8i+31 x The finite towers in B arise from x4i−11 x 2 with i ≥ 0, and those from C from x8i−11 x 2 with j ≥ 0. The homomorphism Ext0A2(P⊗P,Z2)→ Ext (P∞−1 ⊗P,Z2)⊕ Ext0A2(P⊗P −1,Z2), NONIMMERSIONS IMPLIED BY TMF, REVISITED 13 which is equivalent to the d1-differential mentioned above, sends classes to those with the same name. In dimension ≤ −10, this is surjective, with kernel spanned by classes with both components < −1. In dimension −8i−6 and −8i−10, there will be i such classes. We illustrate by listing the classes in the first few gradings: −14 : x−91 x−52 ∼ x−51 x−92 −18 : x−91 x−92 −22 : x−171 x−52 ∼ x−131 x−92 , x−91 x−132 ∼ x−51 x−172 −26 : x−171 x−92 , x−91 x−172 . These kernel classes yield infinite towers emanating from filtration 0. For each p < 0, the towers arising from x 2 , j ≥ 0, in A combine with those in the p-summand of ExtA1(Σ 8p−1P∞−1,Z2) as in Diagram 3.17 to yield one of the copies of ExtA1(Z2,Z2) arising from filtration 1. An identical picture results when the factors are reversed. Diagram 3.17. Part of ExtA2(P −∞ ⊗P−2−∞,Z2) ✻ ✻ ✻ Putting things together, we obtain that in dimensions less than −8, ExtA2(P−2−∞⊗ P−2−∞,Z2) consists of a chart described in Proposition 3.15 and partially illustrated in Diagram 3.18 together with the classes in Diagram 3.11 which are not part of the infinite sums of towers in dimension ≡ 2 mod 4. 14 DONALD M. DAVIS AND MARK MAHOWALD Diagram 3.18. Illustration of Proposition 3.15 −26 −18 −10 ✻✻✻✻✻✻ ✻ ✻✻✻ ✻✻✻✻ ✻✻ The only possible differentials in the Adams spectral sequence of P−2−∞∧P−2−∞∧ tmf involving the classes in dimensions 8p − 2 with p < 0 are from the towers in 8p − 1 in Diagram 3.11, but these differentials are shown to be 0 as in [6, p.54]. Similarly to (3.3), we have tmf∗(P1 ∧ P1) ≈ tmf−∗−2(P−2−∞ ∧ P−2−∞), and so we obtain a turned-around version of Diagram 3.18, of the same general sort as Diagram 3.6, as a depiction of a relevant portion of tmf∗(P1∧P1), with the labeled columns in Diagram 3.18 corresponding to cohomology gradings 24, 16, and 8. The classes X i1X 2 described in Theorem 3.7 are detected by the S-duals of the classes from which the filtration-0 towers in dimensions 8p− 2 in Diagram 3.18 arise, and so they can be chosen to be the corresponding elements of tmf8∗(P1 ∧P1). Simi- larly the classes L1X 2 and X 1L2 have Adams filtration 1, and so one would anticipate that they represent the duals of the generators of the two towers in dimension 8p− 2 with p < 0 in Diagram 3.18. This seems a bit harder to prove using the Adams spectral sequence; however, the Atiyah-Hirzebruch spectral sequence shows this quite clearly. The class X i1 is detected by H 8i(P1; π0(tmf)), while L is detected by H 1(P1; π1(tmf)). NONIMMERSIONS IMPLIED BY TMF, REVISITED 15 Under the pairing, their product is detected in H8i+1(P1; π1(tmf)), clearly of Adams filtration 1. The last part of Theorem 3.7 deals with the action of c4 on the monomials X Since tmf is a commutative ring spectrum, tmf∗(P1 ∧ P1) is a graded commutative algebra over tmf∗. The action c4(X1X2) must be of the form i≥0 γic 4(L1X as these are the only elements in tmf8(P1∧P1), and the class must be invariant under reversing factors. The divisibility of γ0 by 8 follows since c4 has Adams filtration 4. Having just completed the proof of Theorem 3.7, we conclude this section with the postponed proof of Proposition 3.9. Proof of Proposition 3.9. Let C denote the A2-submodule of (P/Z2) ⊗ P generated by all x11x 2 , p ∈ Z. Note that Sq2(x11x 2 ) = Sq 4 Sq6(x11x 2 ). Thus a basis of A2/ Sq 2 acting on all x11x 2 spans C. The 24 elements in a basis of A/ Sq acting on x11x 2 yield x 2 + x 2 + x 2 + x 2 , x 2 + x 2 , x 2 , x 2 , x 2 , x 2 , x 2 , x 2 , x 2 , x 2 , x 2 , x 2 , x 2 , x 2 , x 2 , x and x41x 2 + x 2 . These classes with second components shifted by all multiples of 8 exactly comprise the basis for C described in the proposition. The procedure to establish the structure of B = ((P/Z2)⊗P)/C is similar but more elaborate. For the 32 elements θ in a basis of A2/ Sq 1, we list θ(x−11 x 2 ) and θ(x Then we show that these, with each component allowed to vary by multiples of 8, together with C, fill out all of (P/Z2)⊗P. It is convenient to let Q denote the quotient of (P/Z2)⊗P by C and all elements θ(x8i−11 x 2 ) and θ(x 2 ). We will show Q = 0. This will complete the proof of Proposition 3.9, implying in particular that Sq1(x8i−11 x 2 ) and Sq 1(x8i−11 x 2 ) are decomposable over A2. A separate calculation is performed for each mod 8 value of the degree. Here we use repeatedly that the A2-action on x i depends only on i mod 8. We illustrate with the case in which degree ≡ 0 mod 8. The other 7 congruences are handled similarly, although some are a bit more complicated. A basis of A2/ Sq 1 in degree ≡ 2 mod 8 acting on x−11 x−12 yields the following elements: x−11 x 2 + x 2 + x 2 , x 2 + x 2 + x 2 + x 2 + x 2 + x 16 DONALD M. DAVIS AND MARK MAHOWALD and x41x 2 + x 2. A basis of A2/ Sq 1 in degree ≡ 6 mod 8 acting on x−11 x32 yields the following elements: x21x 2 + x 2 + x 2 + x 2 + x 2 + x 2 + x 2 + x 2, and x 2 + x 2. Because we allow both components to vary by multiples of 8, we will list just the first component of the ordered pairs. These are considered as relations in Q. Thus the relation R1 below really means that all x8i−11 x 2 + x 2 + x 2 become 0 in Q. R1 : X−1 +X0 +X1, R2 : X2 +X6, R3 : X−1 +X3 +X4 +X5 +X9, R4 : X4 +X12, R5 : X2 +X3 +X4 +X5, R6 : X−1 +X2 +X5, R7 : X4 +X6 +X10 +X12, R8 : X8 +X16. We will use these relations to show that all classes (in degree ≡ 0 mod 8) are 0 in Q. First, R8 implies that all classes X8i are congruent to one another. Since X0 is 0 in the quotient due to P/Z2, we conclude that all classes X8i are 0 in Q. Next, R4 implies that all X8i+4 are congruent to one another. Since X4 +X8 ∈ C, and we have just shown that X8 ≡ 0 in Q, we deduce that all X8i+4 are 0 in Q. Now we use R2 + R7 to see that all X8i+2 + X8i+4 are congruent to one another, then that X2 +X4 ∈ C to deduce all X8i+2 +X8i+4 ≡ 0, and finally the result of the previous sentence to conclude all X8i+2 ≡ 0. Then R2 implies all X8i+6 ≡ 0. Now R1+R3+R5, together with relations previously obtained, implies all X8i+1 are congruent to one another, and since X1 ∈ C, we conclude all X8i+1 ≡ 0. Finally R1 implies X8i−1 ≡ 0, R6 implies X8i+5 ≡ 0, and then R3 implies X8i+3 ≡ 0. 4. Careful treatment of axial class In this section, we fill the gap in the proof in [6] of its Theorem 1.1 by careful consid- eration of the possible “other terms” in the axial class discussed in the Introduction. We show that, at least as far as the monomials cX i1X 2 in its powers are concerned, NONIMMERSIONS IMPLIED BY TMF, REVISITED 17 the axial class equals u(X1+X2), where u is a unit in tmf 0(RP∞×RP∞). Thus the ℓth power of the axial class is nonzero in tmf8ℓ(RP n×RPm) if and only if (X1+X2)ℓ is nonzero there, and the latter is the condition which yielded the nonimmersions of [6, 1.1]. Thus we have a complete proof of [6, 1.1]. If P n × Pm f−→Pm+k is an axial map, then there is a commutative diagram P n × Pm f−−−→ Pm+k P∞ × P∞ g−−−→ P∞, where g is the standard multiplication of P∞, since P∞ = K(Z2, 1). Since X ∈ tmf8(Pm+k) has been chosen to extend over P∞, we obtain that f ∗(X) is the restric- tion of g∗(X). By Theorem 3.7 and the symmetry of g, we must have g∗(X) = X1 +X2 + 4(L1X 1 L2), (4.1) for some integers κi. This is what we call the “axial class.” Then g ∗(Xℓ) equals the ℓth power of (4.1). Using the formulas for L2i , LiXi, and c4(X1X2) in 3.1 and 3.7 and the binomial theorem, this ℓth power can be written in terms of the basis described in 3.7. If some κi’s are nonzero, the coefficients of X 2 in g ∗(Xℓ) will not equal , as was claimed in [6]. We will study this possible deviation carefully. One simplification is to treat L1 and L2 as being just 2. Note that Li acts like 2 when multiplying by Xi, and if, for example, L1 is present without X1, then the terms ci4L1X 2 cannot cancel our X 2-classes because both are separate parts of the basis. You have to carry the terms along, because they might get multiplied by an X1, and then it is as if L1 = 2. We will incorporate this important simplification throughout the remainder of this section. For example, one easily checks that, using L21 = 2L1 and L1X1 = 2X1, we obtain (X1 +X2 + L1X2) 4 = (X1 + 3X2) 4 − 80X42 + 40L1X42 . The exponent of 2 in each monomial of (X1 + 3X2) 4 − 80X42 is the same as that in (X1 +X2) 4, and L1X 2 is a separate basis element. With this simplification, the axial class in (4.1) becomes X1 +X2 + 2 2 ) (4.2) 18 DONALD M. DAVIS AND MARK MAHOWALD for some integers κi. There was another term 2κ0(X1+X2), but it can be incorporated into the leading (X1 +X2). The odd multiple that it can create is not important. From Theorem 3.7, we have c4(X1X2) = 16(X1 +X2) + 2 (4.3) for some integers γk. The 16 comes from γ0 = 8 and Li = 2. Actually we don’t really know that γ0 = 8, even just up to multiplication by a unit, but it is divisible by 8 and the possibility of equality must be allowed for. This gives 2 ) = 16(X 2 ) + 2 i+k+1 j+k+1 (4.4) Here we use that in a graded tmf∗-algebra tmf ∗(X) with even-degree elements, c(xy) = cx · y, for c ∈ tmf∗ and x, y ∈ tmf∗(X). There is an iterative nature to the action of c4 in (4.4), but the leading coefficient 16 enables us to keep track of 2-exponents of leading terms in the iteration. (As observed above, the leading coefficient might be an even multiple of 16, which would make the terms even more highly 2-divisible. We assume the worst, that it equals 16.) We obtain the following key result about the action of c4 on monomials in X1 and X2. Theorem 4.5. There are 2-adic integers Ai such that 24+iAi Remark 4.6. This formula will be evaluated on (i.e. multiplied by) monomialsXk1X One might worry that the negative powers of X1 or X2 in 4.5 will cause nonsensical negative powers in c4X 2. This will, in fact, not occur because the monomials on which we act always have total degree greater than the dimension of either factor. Thus if, after multiplication by c4, a term with negative exponent of Xi appears, then the accompanying X 3−i-term will be 0 for dimensional reasons. Proof of Theorem 4.5. The defining equation (4.3) may be written as, with θ = X1X2 and z = X1/X2, θ = 16(z + z−1) + i(zi+1 + z−(i+1)). (4.7) NONIMMERSIONS IMPLIED BY TMF, REVISITED 19 Let pi = z i + z−i. We will show that 24+iAip2i+1 (4.8) for certain 2-adic integers Ai, which interprets back to the claim of 4.5. Note that pipj = pi+j + p|i−j|, and hence pe11 · · · p k = pΣiei + L, where L is a sum of integer multiples of pj with j < iei and j ≡ iei mod 2. We will ignore for awhile the coefficients γi which occur in (4.7). This is allowable if we agree that when collecting terms, we only make crude estimates about their 2-divisibility. We have θ = 16p1 + 2θp2 + 2θ 2p3 + 2θ 3p4 + · · · = 16p1 + 2p2(16p1 + 2p2(16p1 + · · · ) + 2p3(16p1 + · · · )2 + · · · ) +2p3(16p1 + 2p2(16p1 + · · · ) + · · · )2 + · · · . Note that the only terms that actually get evaluated must end with a 16p1 factor. Now let T1 = 16p1 and, for i ≥ 2, let Ti = 2θi−1pi. Each term in the expansion of θ involves a sequence of choices. First choose Ti for some i ≥ 1, and then if i > 1 choose (i−1) factors Tj , one from each factor of θi−1. For each of these Tj with j > 1, choose j − 1 additional factors, and continue this procedure. This builds a tree, and we don’t get an explicit product term until every branch ends with T1. Each selected factor Tj with j > 1 contributes a factor 2pj. There will also be binomial coefficients and the omitted γi’s occurring as additional factors. For example, Diagram 4.9 illustrates the choices leading to one term in the expan- sion of θ. This yields the term 2p2 · 2p4 · 16p1 · 2p2 · 16p1 · 2p3 · 16p1 · 2p2 · 16p1, which equals 221(p17 +L), where L is a sum of pi with i < 17 and i odd. By induction, one sees in general that the sum of the subscripts emanating from any node, including the subscript of the node itself, is odd. 20 DONALD M. DAVIS AND MARK MAHOWALD Diagram 4.9. A possible choice of terms T2 T4 T2 T1 T2 T1 The important terms are those in which T2 is chosen k times (k ≥ 0) and then T1 is chosen. These give (2p2) kp1 with no binomial coefficient. This term is 2 k+4(p2k+1+L). Note that a term 2k+4p2i+1 with i < k obtained from L will be more 2-divisible than the 2i+4p2i+1 term that was previously obtained. Thus it may be incorporated into the coefficient of that term. All other terms will be more highly 2-divisible than these. For example, the first would arise from choosing T3 then two copies of T1. This would give 2p3 ·24p1 ·24p1 = 29p5+L, and the 29p5 can be combined with the 26p5 obtained from choosing T2 then T2 then T1. Incorporating γi’s may make terms even more divisible, but the claim of (4.8) is only that p2i+1 occurs with coefficient divisible by 2 Now we incorporate 4.5 into (4.2) to obtain the following key result, which we prove at the end of the section. Theorem 4.10. The monomials ciX 2 in the nth power of the axial class in tmf8n(RP∞ × RP∞) are equal to those in the nth power of (X1 +X2) 24+iαi , (4.11) where u is an odd 2-adic integer and αi are 2-adic integers. The factor which accompanies (X1+X2) in (4.11) is a unit in tmf ∗(RP∞×RP∞); we referred to it earlier as u. Indeed, its inverse is a series of the same form, obtained by solving a sequence of equations. This justifies the claim in the first paragraph of this section regarding retrieval of the nonimmersions of [6, 1.1]. We must also observe that restriction to tmf8ℓ(RP n × RPm) of the non-X i1Xℓ−i2 parts of the basis of tmf8ℓ(RP∞ × RP∞) cannot cancel the X i1Xℓ−i2 terms essential for the nonimmersion. This is proved by noting that these elements such as L1X NONIMMERSIONS IMPLIED BY TMF, REVISITED 21 and ci4L1X 2 will restrict to a class of the same name in tmf 8ℓ(RP n × RPm), and will be 0 there for dimensional reasons, since 8ℓ > n. Proof of Theorem 4.10. Let g∗(X) denote the axial class as in (4.1). From (4.2) and 4.5, the difference g∗(X)− (X1 +X2) equals We let z = X1/X2 and pj = z j + z−j as in the proof of 4.5. The summand with i = 2t becomes 2κi(X1 +X2) X t1X 2jAjp2j+1 = 2κi(X1 +X2)(p2t + L)24i k(p2k+i + L). Here k is a sum of j-values taken from the various factors in the ith power. Also, in pj + L, L denotes a combination of pt’s with t < j. Noting (p2t + L)(p2k+i + L) = p2k+2i + L, this becomes 2(X1 +X2)2 k(p2k+2i + L). (4.12) The argument when i = 2t + 1 is similar but slightly more complicated because (X i+11 +X 2 ) is not divisible by (X1 +X2). We obtain X i+11 +X X1X2)2t+1 2jAjp2j+1 For one of the factors of the ith power, say the first, we treat p2j+1 as X1+X2√ (p2j +L). The expression then becomes 2(X1 +X2)pi+12 k(p2k+i−1 + L), where k is obtained as in the previous case. We again obtain (4.12). Thus when g∗(X) − (X1 +X2) is written as (X1 + X2) βjp2j , the coefficient βj satisfies ν(βj) ≥ (j − 1) + 4 + 1. Here the (j − 1) + 4 comes from the case i = 1, k = j− 1 in (4.12), and the extra +1 is the factor 2 which has been present all along. This yields the claim of (4.11). 22 DONALD M. DAVIS AND MARK MAHOWALD 5. tmf-cohomology of CP∞ × CP∞ In [2], [4], and [8], it was noted, first by Astey, that the axial class using BP (or BP 〈2〉) was u(X2 − X1), where u is a unit in BP ∗(P∞ ∧ P∞). In this section, we review that argument and consider the possibility that it might be true when BP is replaced by tmf, which would render the considerations of the previous section unnecessary. To do this, we calculate tmf∗(CP∞) and tmf∗(CP∞×CP∞) in positive dimensions. (See Theorems 5.15 and 5.19.) Although our conclusion will be that Astey’s BP -argument cannot be adapted to tmf, nevertheless these calculations may be of independent interest. We begin by reviewing Astey’s argument. There is a commutative diagram, in which RP = RP∞ and CP = CP∞ dR−−−→ RP × RP mR−−−→ RP dC−−−→ CP × CP CP 1×(−1) CP × CP mC−−−→ CP The generator XR ∈ BP 2(RP ) satisfies XR = h∗(X). We also have that mC ◦ (1 × (−1))◦dC is null-homotopic. The key fact, which will fail for tmf, is BP ∗(CP×CP ) ≈ BP ∗[X1, X2]. The axial class is m∗R(XR). It equals (h× h)∗(1× (−1))∗m∗C(X). But (1× (−1))∗m∗C(X) ∈ ker(d∗C). By the above “key fact,” d∗C is the projection BP ∗[X1, X2]→ BP ∗[X ] in which each Xi 7→ X . The kernel of this projection is the ideal (X2 −X1). To see this, just note that in grading 2n a kernel element must be 2 with ci = 0, and hence is 2 −Xn1 ) = 1(X2 −X1) n−i−1−j Thus (1 × (−1))∗m∗C(X) = (X2 − X1)u for some u ∈ BP ∗(CP × CP ). This u is a unit by consideration of its reduction to H∗(−;Z), as in [2]. Since h∗(u) will then be a unit in BP ∗(RP ×RP ) and h∗(Xi) = XRi, we obtain the claim about the axial class being a unit times XR2 −XR1. NONIMMERSIONS IMPLIED BY TMF, REVISITED 23 In order to see if there is any chance of adapting this to tmf, we compute tmf∗(CP∞) and tmf∗(CP∞ × CP∞) in positive gradings. We begin with the relevant Ext calcu- lations. Let bo = Ext (Z2,Z2). Recall that a chart for this is given as in Diagram 5.1, extended with period (t− s, s) = (8, 4). Diagram 5.1. Ext (Z2,Z2) 0 4 8 · · · Let M10 denote the A2-module 〈1, Sq4, Sq2 Sq4, Sq4 Sq2 Sq4〉. Lemma 5.2. There is an additive isomorphism (M10,Z2) ≈ bo[v2], where v2 ∈ Ext1,7(−). Thus the chart for Ext (M10,Z2) consists of a copy of bo shifted by (t − s, s) = (6i, i) units for each i ≥ 0. Proof. There is a short exact sequence of A2-modules 0→ Σ7M10 → A2//A1 → M10 → 0. This yields a spectral sequence which builds Ext (M10,Z2) from ∗−i,∗−7i (A2//A1,Z2). Since Ext (A2//A1,Z2) ≈ bo, one easily checks that there are no possible differen- tials in this spectral sequence. Let Cmn = H ∗(CPmn ;Z2). 24 DONALD M. DAVIS AND MARK MAHOWALD Theorem 5.3. There is an additive isomorphism (C∞−∞,Z2) ≈ Σ8p−2bo[v2]. Of course Σ applied to a module or an Ext group just means to increase the t-grading by 1. Proof. There is a filtration of C∞−∞ with Fp/Fp−1 ≈ Σ8p−2M10 for p ∈ Z. We have Sq2 ι8p−2 = Sq 4 Sq2 Sq4 ι8p−10. The same argument used in the last paragraph of the proof of Corollary 3.10 works to initiate an inductive proof of the Ext-isomorphism claimed in the theorem. Corollary 5.4. In gradings (t− s) less than −1, (C−2−∞,Z2) ≈ Σ8p−2bo[v2]. Proof. There is an exact sequence → Exts−1,tA2 (C −1,Z2)→ Ext (C−2−∞,Z2)→ Ext (C∞−∞,Z2) q∗−→ Exts,tA2(C −1,Z2)→ . The result is immediate from this and 5.3, since q∗ sends the initial tower in F0/F−1 isomorphically to the initial tower in ExtA2(C −1,Z2). The A-modules C∞1 and Σ 2C−2−∞ are dual. Thus, by [9, Prop 4], (Z2,C 1 ) ≈ Ext (Σ2C−2−∞,Z2). There is a ring structure on Ext (Z2,C 1 ). We deduce the following result, which is pictured in Diagram 5.12. Corollary 5.5. In (t− s) gradings ≤ 0, there is a ring isomorphism (Z2,C 1 ) ≈ bo[v2][X ], where X ∈ Ext0,−8. Proof. We apply the duality isomorphism to 5.4. The multiplicative structure is obtained from the observation that the powers of the class in Ext0,−8 equal the class in Ext0,−8i for each i > 0. NONIMMERSIONS IMPLIED BY TMF, REVISITED 25 The Ext groups computed here are the E2-term of the ASS converging to tmf −∗(CP∞). We will consider the differentials in this spectral sequence after performing the Ext calculation relevant for tmf∗(CP∞ × CP∞). Now we consider C−2−∞⊗C−2−∞. Now x1 and x2 denote elements of H2(CP ;Z2). Let E2 denote the exterior subalgebra generated by the Milnor primitives of grading 1, 3, and 7. Note that A2//E2 has a basis with elements of grading 0, 2, 4, 6, 6, 8, 10, and 12. Finally we note that for any j ≡ −2 mod 8 with j ≤ −10, there is a nontrivial A2-morphism C ρ−→ΣjZ2. Lemma 5.6. Let K = ker(C−2−∞ ⊗C−2−∞ ρ−→C−2−∞ ⊗ Σ−10Z2). Let S denote the set of all classes x8i−21 x 2 with i ≤ −1 and j ≤ −2, together with the classes x8i−21 x 2 with i ≤ −1 and j ≤ −1. Then K is the direct sum of a free A2//E2-module on S with a single relation Sq 4 Sq2 Sq4(x−101 x 2 ) = 0. Proof. Since the generators of E2 have odd grading, A2//E2 acts on any element of these evenly-graded modules. The action of A2//E2 on x 2 yields the additional elements x−21 x 2 , x 2 , x 2 , x x−21 x 2 , and x 2. The action of A2//E2 on x 2 yields the additional elements x01x 2 + x 2, and x 2 + x 2. Each exponent can be decreased by any multiple of 8. One can easily check that in each grading all classes in C−2−∞ ⊗C−2−∞ are obtained exactly once from the described elements in K together with C−2−∞ ⊗ Σ−10Z2. There are four cases, for the four even mod 8 values. We illustrate with the case of grading 4 mod 8. We will just consider the specific value −28, but it will be clear that it generalizes to all gradings ≡ 4 mod 8. Letting Xi denote xi1x−28−i2 , we have: (1) From generators in −28, we obtain just X−10 in K. The class X−18 is in C −∞ ⊗ Σ−10Z2. (2) From generators in −32, we obtain X−8 + X−6, X−16 + X−14, and X−24 +X−22. (3) From generators in −36, we obtain X−8+X−4 and X−16+X−12. (4) From generators in −40, we obtain X−4, X−12 +X−8, X−20 + X−16, and X−24. 26 DONALD M. DAVIS AND MARK MAHOWALD Note in (4) that X0 and X−28 do not appear because each component must be ≤ −4 and the components sum to −28. One easily checks that the 11 classes listed above, including X−18, form a basis for the space spanned by X−4, . . . , X−24, in an orderly fashion that clearly generalizes to any grading ≡ 4 mod 8. A similar argument works in the other three congruences. There are some minor variations in the top few dimensions. Now we dualize. There is a pairing ExtA2(Z2,C 1 )⊗ ExtA2(Z2,C∞1 )→ ExtA2(Z2,C∞1 ⊗C∞1 ). Let Xi denote the class in grading −8 coming from the ith factor. Then we obtain Theorem 5.7. The algebra Ext (Z2,C 1 ⊗ C∞1 ) in gradings ≤ −8 is isomorphic to Z2[X1, X2]〈X1X2, y−12〉 with y2−12 = X21X2 + X1X22 . The monomials of the form X i1X 2y−12 are acted on freely by Z2[v0, v1, v2]. Let Sn denote the Z2-vector space with basis the monomials X i1X 2 , and define a homomorphism ǫ : Sn → Z2 by sending each monomial to 1. Then Z2[v0, v1, v2] acts freely on ker(ǫ), while bo[v2] acts freely on Sn/ ker(ǫ). Thus in dimensions t − s ≤ −8 Ext∗,∗A2 (Z2,C 1 ⊗ C∞1 ) has, for each i > 0, i copies of Σ−8i−4Z2[v0, v1, v2] and i copies of Σ −8i−16Z2[v0, v1, v2], and also one copy of Σ−8i−8bo[v2]. Here Z2[X1, X2]〈X1X2, y−12〉means a free Z2[X1, X2]-module on basis {X1X2, y−12} Proof. The structure as graded abelian group is straightforward from Lemma 5.6, Corollary 5.5, and the duality isomorphism (Z2,C 1 ⊗C∞1 ) ≈ Ext ∗,∗−4 (C−2−∞ ⊗C−2−∞,Z2). We use that ExtA2(A2//E2,Z2) ≈ Z2[v0, v1, v2]. The reason that we only assert the structure in dimension ≤ −8 is due to the Σ−10 in the cokernel part of Lemma 5.6, and that Theorem 5.5 was only valid in dimension ≤ 0. In the range under consideration, the relation on the top class in Lemma 5.6 does not affect Ext. The ring structure in filtration 0 comes from HomA2(Z2,C 1 ⊗C∞1 ) being isomorphic to elements of C∞1 ⊗C∞1 annihilated by Sq2 and Sq4, which has as basis all elements x4i1 ⊗ x 2 and (x 1 ⊗ x 2 )(x 1 ⊗ x22 + x21 + x42). NONIMMERSIONS IMPLIED BY TMF, REVISITED 27 Now we show that Ext 1,−8n+2 (Z2,C 1 ⊗ C∞1 ) = Z2, and h1 times each mono- mial in Ext 0,−8n (Z2,C 1 ⊗ C∞1 ) equals the nonzero element here. An element in 1,−8n+2 (Z2,C 1 ⊗C∞1 ) = Z2 is an equivalence class of morphisms Σ2A2 ⊕ Σ4A2 h−→C∞1 ⊗C∞1 which increase grading by 8n− 2, and yield a trivial composite when preceded by Σ4A2 ⊕ Σ8A2 Sq2 Sq6 0 Sq4 −−−−−−−−−→ Σ2A2 ⊕ Σ4A2. Morphisms h which can be factored as Σ2A2 ⊕ Σ4A2 Sq2,Sq4−−−−→ A2 k−→C∞1 ⊗C∞1 (5.8) are equivalent to 0 in Ext. We illustrate with the case n = 3. There are A2-morphisms increasing grading by 22 sending either Σ2A2 or Σ 4A2 to any one of the following classes: 2 , x 2 , x (5.9) The classes are listed in this order because any two adjacent monomials are equivalent using as k in (5.8) the morphism sending the generator to the indicated classes in succession: 2 , x For example, (Sq2, Sq4)(x11x 2 ) = (x 2 , x 2 ). Thus all classes in (5.9) are equiva- lent to one another. That h1 times any monomialX 2 equals this nonzero element of Ext 1,8n+2 (Z2,C C∞1 ) follows from usual Yoneda product consideration. If 0 ← Z2 ← C0 ← C1 ← is the beginning of a minimal A2-resolution, with C1 = Σ 1A2 ⊕ Σ2A2 ⊕ Σ4A2, then h1X 2 is represented by the composite C1 → C0 → C∞1 ⊗ C∞1 sending ι2 7→ ι 7→ X i1Xn−i2 , and this is equivalent to the element described in the previous paragraph. Here is a schematic way of picturing Theorem 5.7. We first list the generators in grading greater than −32. Then for each of the two types of generators, we list the structure arising from them in the first 10 dimensions. The bo[v2]-structure in the 28 DONALD M. DAVIS AND MARK MAHOWALD left half of Diagram 5.11 arises from one tower in dimensions −24 and −16, while the Z2[v0, v1, v2]-structure in the right half of diagram 5.11 arises from the other towers in Diagram 5.10. Diagram 5.10. Generators of ExtA2(Z2,C 1 ⊗C∞1 ) −28 −24 −20 −16 −12 ✻✻✻ ✻✻ ✻✻ ✻ ✻ Diagram 5.11. Structure on two types of generators 0 010 10 ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻✻ ✻ ✻ Now we consider the differentials in the ASS converging to tmf∗(CP∞) and then for tmf∗(CP∞ ∧ CP∞). The gradings are negated when considered as tmf-cohomology groups. Corollary 5.5 gives the E2-term converging to [Σ ∗CP∞1 , tmf] ≈ tmf −∗(CP∞1 ). We will maintain the homotopy gradings until just before the end. In diagram 5.12, we depict a portion of the E2-term of this ASS in gradings −16 to 1. There are also classes in higher filtration arising from powers of v41 and v2 acting on generators in lower grading. The elements indicated by •’s are involved in differentials, as explained later. NONIMMERSIONS IMPLIED BY TMF, REVISITED 29 Diagram 5.12. A portion of E2 for [Σ ∗CP∞, tmf] −16 −8 0 We will prove the following key result about differentials in this ASS. Theorem 5.13. The nonzero differentials in the ASS converging to [Σ∗CP∞, tmf], ∗ < 1, are given by −2k+1) = hǫ+11 v for ǫ = 0, 1, i, j ≥ 0, k ≥ 1. Here h1, v 1, and v2 have the usual Ext s,t gradings (s, t) = (1, 2), (4, 12), and (1, 7), respectively. Diagram 5.12 pictures the situation for k = 1 and small values of i and j. The elements indicated by •’s are involved in the differentials. The resulting picture is nicer if the filtrations of all classes built on X−2k+1 are increased by 1. There is a nontrivial extension (multiplication by 2) in dimension −6 due to the preceding differential. This is equivalent to the way that bu∗ is formed from bo∗ and Σ 2bo∗. We obtain Diagram 5.14 from Diagram 5.12 after the differentials, extensions, and filtration shift are taken into account. 30 DONALD M. DAVIS AND MARK MAHOWALD Diagram 5.14. Diagram 5.12 after differentials and filtration shift −16 −8 0 The regular sequence of towers in the chart beginning in filtration 1 in dimension −10 is interpreted as vi1v2, i ≥ 0. After negating dimensions to switch to cohomology indexing, we obtain the follow- ing result, which is immediate from 5.13 after the extensions such as just seen are taken into account. Theorem 5.15. In positive gradings, there is an isomorphism of graded abelian groups tmf∗(CP∞1 ) ≈ Z(2)[Z16](bo∗ ⊕ v2Z(2)[v1, v2]). Here Z16 ∈ tmf16(CP∞1 ), and |v1| = −2 and |v2| = −6. Recall that bo∗ = bo−∗ with bo∗ as suggested in 5.1. Much of the ring structure of tmf∗(CP∞1 ) is described in 5.15, since bo∗ and v2Z(2)[v1, v2] are rings, and it is quite clear how to multiply an element in bo∗ by one in v2Z(2)[v1, v2]. Because of the filtration shift that led to the identification of some of the classes in v2Z(2)[v1, v2], we hesitate to make any complete claims about the ring structure. A complete computation of tmf∗(CP∞) was made in [5]. See there especially Theorem 7.1 and Diagram 7.1. At first glance, the two descriptions appear quite different, but they seem to be compatible. NONIMMERSIONS IMPLIED BY TMF, REVISITED 31 Proof of Theorem 5.15. We first prove that there is a nontrivial class in [Σ−16CP, tmf] detected in filtration 0. This is obtained using the virtual bundle 8(H−1)−(H3−H), where H denotes the complex Hopf bundle. Considered as a real bundle θ, this bundle satisfies w2(θ) and p1(θ) = 0. Here we use from [18] that p1 generates the infinite cyclic summand in H4(BSO;Z) and satisfies r∗(p1) = c 1 − 2c2 under BU r−→BSO, and ρ∗(p1) = 2e1 under BSpin ρ−→ BSO, where H4(BSpin;Z) is an infinite cyclic group generated by e1. The total Chern class of 9H −H3 is (1 + x)9(1 + 3x)−1 = 1 + 6x+ 18x2 + · · · , and hence r∗(p1(θ)) = (c1(9H −H3))2 − 2c2(9H −H3) = (6x)2 − 2 · 18x2 = 0. Thus e1(θ) = 0, hence CP ∞ θ−→BSpin → K(Z, 4) is trivial, and so θ lifts to a map CP∞ → BO[8]. Hence its Thom spectrum induces a degree-1 map T (θ) → MO[8]. Since ψ3(H) = H3 −H , by [19] θ is J(2)-equivalent to 8(H − 1), and hence its Thom spectrum is T (8(H − 1)) = Σ−16CP∞8 . Using the Ando-Hopkins-Rezk orientation ([1]) MO[8]→ tmf, we obtain our desired class as the composite Σ−16CP∞1 col−→ Σ−16CP∞8 T (θ)−−→MO[8]→ tmf . (5.16) We will deduce our differentials from the d3-differential E 3 → E 3 in the ASS converging to π∗(tmf). This can be seen in [13, p.537] or [11, Thm 2.2]. See Remark 5.17 for additional explanation. It is not difficult to show that, with M10 as in 5.2, the morphism (Z2,Z2)→ Exts,tA2(M10,Z2) induced by the nontrivial A2-map M10 → Z2 sends the Z2 in Ext7,23A2 (Z2,Z2) which is not part of the infinite tower to h21v We prefer to think about the ASS for tmf∗(Σ 2CP−2−∞), which, as we have noted, is isomorphic to that of [Σ∗CP∞1 , tmf]. The E2-term was described in 5.4. Let S−16 → Σ2CP−2−∞ ∧ tmf correspond to the map in (5.16). Since E2(CP−2−∞ ∧ tmf) in negative dimensions is built from copies of ExtA2(M10,Z2), we deduce from the previous paragraph that h21v 1v2g−16 in the ASS for tmf∗(Σ 2CP−2−∞) must be hit by a d2- or d3-differential, since it is the image of a class hit by a d3. The only possibility is that it be d2 from h1v 1g−8, as indicated by the dotted line in Diagram 5.12. Naturality 32 DONALD M. DAVIS AND MARK MAHOWALD of differentials with respect to h1 and v 1 implies the differentials of 5.13 for ǫ = 0, 1, all i, j = 0, and k = 1. Using the diagonal map of CP∞1 and the multiplication of tmf, powers of (5.16) give similar nontrivial elements in [Σ−16kCP∞1 , tmf] for all k ≥ 1, and by the argument just presented, we establish the differentials of 5.13 for all k (with j = 0 still). The only possible differentials on v2g−16 would be some dr with r > 2 hitting an element which is acted on nontrivially by h1. However h1v2g−16 has become 0 in E3 since it was hit by a d2-differential. Thus a nonzero differential on v2g−16 would contradict naturality of differentials with respect to h1-action. Hence there is a map S−10 → Σ2CP−2−∞ ∧ tmf hitting v2g−16, and the argument of the previous paragraph implies that d2(h1v 1v2g−8) = h 2g−16 and then other related differentials. This now establishes the differentials of 5.13 when j = 1, and sets in motion an inductive argument to establish these differentials for all j ≥ 1. No further differentials in the spectral sequence are possible, by dimensional and h1-naturality considerations. Remark 5.17. The proof of the key d3-differential in the ASS of tmf from the 17- stem to the 16-stem, which was cited above, has not had a thorough proof in the literature. Giambalvo’s original argument was incorrect and his correction merely refers to “a homotopy argument.” The current authors cited Giambalvo’s result in [11] without additional argument. We provide some more detail here regarding this differential. The relevant portion of the ASS of tmf appears in Diagram 5.18. In [13] and [11], this was pictured as the ASS of MO[8], but through dimension 18, ∗(MO[8]),Z2) ≈ Ext∗,∗A2(Z2 ⊕ Σ Z2,Z2). One way of obtaining the differentials from 15 to 14, as in [13], is to note that the [8]- cobordism group of 14-dimensional manifolds is Z2, and so the top two elements must be killed by differentials. It is not difficult to compute in Ext the Massey product formula B = 〈A, h0, h1〉, where A and B are as in Diagram 5.18. This can be seen as v41 times a similar formula between classes in dimensions 6 and 8. Since A is 0 in homotopy, the associated Toda bracket formula says that B must be divisible by η. NONIMMERSIONS IMPLIED BY TMF, REVISITED 33 But only 0 can be divisible by η in dimension 16 here. Thus B must be killed by a differential, and the depicted way is the only way this can happen. Diagram 5.18. Portion of ASS of tmf 14 16 18 The differentials in the ASS converging to tmf∗(CP −∞∧CP−2−∞) are implied by the same considerations that worked for CP−2−∞. The Z2[v0, v1, v2]-parts in Theorem 5.7 cannot support differentials by dimensionality and h1-naturality. For the bo-like part, we prefer thinking about it as [Σ∗+4CP∞1 ∧ CP∞1 , tmf] ≈ tmf−∗−4(CP∞1 ∧ CP∞1 ), where the product structure is more apparent. Let Zn denote the nonzero element of Ext 0,−8n (Z2,C 1 ⊗C∞1 )/ ker(h1). By Theorem 5.7, Zn can be represented by X 2 for any 1 ≤ i < n. If n is even and n ≥ 4, choosing i even, Zn is an infinite cycle because it is an external product of infinite cycles. Hence by the proof of Theorem 5.13, 2Z2k−1) = h 2 Z2k for ǫ = 0, 1, i, j ≥ 0, and k ≥ 2. Finally, X1X2 is an infinite cycle since there is nothing that it can hit. Also, h1v2X1X2 and h 1v2X1X2 are not hit by differentials since Ext (Z2,C 1 ⊗C∞1 ) = 0 by Theorem 5.7. We obtain the following. Theorem 5.19. In grading ≥ 10, there is an isomorphism of graded abelian groups tmf∗(CP∞1 ∧CP∞1 ) ≈ yZ(2)[v1, v2, X1, X2]⊕ In·Z(2)[v1, v2]⊕Z(2)[Z](bo∗⊕v2Z(2)[v1, v2]), 34 DONALD M. DAVIS AND MARK MAHOWALD where |y| = 12, |Xi| = 8, |Z| = 16, |v1| = −2, and |v2| = −6. Here In = ker(Fn ǫ−→Z), where Fn is a free abelian group with basis {X i1Xn−i2 : 1 ≤ i < n}, and ǫ(X i1Xn−i2 ) = 1. Thus In consists of all polynomials of grading n with sum of coefficients equal to 0. We could have extended the description in 5.19 down to grading 8, but the description would have been slightly more complicated, since it would include h1v2Z and h 1v2Z. The motivation for this section was to see if perhaps ker(tmf∗(CP∞ × CP∞) d −→ tmf∗(CP∞)) might be something nice like the I(X1 − X2) which was the case for BP ∗(−). In Theorem 5.19, we described tmf∗(CP∞ ∧ CP∞). To obtain tmf∗(CP∞ × CP∞), we add on two copies of tmf∗(CP∞), which was described in 5.15. Denote by Z1 and Z2 the generators in tmf 16(CP∞ × CP∞). Monomials Z i1Zn−i2 should equal Zn of 5.19 plus perhaps elements of I2n of 5.19. The class y of 5.19 plus perhaps a sum of elements of higher filtration is in ker(d∗) and not in the ideal generated by (Z1−Z2). Thus, as expected, ker(d∗) does not have the nice form that it did for BP ∗(−), and so we cannot use this argument to show that the axial class in tmf∗(RP∞ × RP∞) is u(X1 − X2). However, we showed something like this by a completely different method in Theorem 4.10. We feel that the results obtained in Theorems 5.15 and 5.19 should be of independent interest. References [1] M. Ando, M. J. Hopkins, and C. Rezk, Multiplicative orientations of KO-theory and of the spectrum of topological modular forms, preprint, www.math.uiuc.edu/∼mando/papers/koandtmf.pdf. [2] L. Astey, Geometric dimension of vector bundles over real projective spaces, Quar Jour Math Oxford 31 (1980) 139-155. [3] , A cobordism obstruction to embedding manifolds, Ill Jour Math 31 (1987) 344-350. [4] L. Astey and D. M. Davis, Nonimmersions of real projective spaces implied by BP , Bol Soc Mat Mex 25 (1980) 15-22. [5] T. Bauer, Elliptic cohomology and projective spaces–a computation, preprint. wwwmath.uni-muenster.de/u/tbauer/cpinfty.pdf. [6] R. R. Bruner, D. M. Davis, and M. Mahowald, Nonimmersions of real projec- tive spaces implied by tmf, Contemp Math 293 (2002) 45-68. [7] D. M. Davis, Table of immersions and embeddings of real projective spaces, http://www.lehigh.edu/∼dmd1/immtable. [8] , A strong nonimmersion theorem for real projective spaces, Annals of Math 120 (1984) 517-528. NONIMMERSIONS IMPLIED BY TMF, REVISITED 35 [9] , On the Segal Conjecture for Z2 × Z2, Proc Amer Math Soc 83 (1981) 619-622. [10] D. M. Davis and M. Mahowald, Ext over the subalgebra A2 of the Steenrod algebra for stunted projective spaces, Can Math Soc Conf Proc 2 (1982) 297- [11] , A new spectrum related to 7-connected cobordism, Springer-Verlag Lecture Notes in Math 1370 (1989) 126-134. [12] D. M. Davis and V. Zelov, Some new embeddings and nonimmersions of real projective spaces, Proc Amer Math Soc 128 (2000) 3731-3740. [13] V. Giambalvo, On 〈8〉-cobordism, Ill Jour Math 15 (1971) 533-541. Correction in Ill Jour Math 16 (1972) 704. [14] I. M. James, On the immersion problem for real projective spaces, Bull Amer Math Soc 69 (1963) 231-238. [15] N. Kitchloo and W. S. Wilson, The second real Johnson-Wilson theory and nonimmersions of RPn, preprint. [16] W. H. Lin, On conjectures of Mahowald, Segal, and Sullivan, Math Proc Camb Phil Soc 87 (1980) 449-458. [17] W. H. Lin, D. M. Davis, M. Mahowald, and J. F. Adams, Calculation of Lin’s Ext groups, Math Proc Camb Phil Soc 87 (1980) 459-469. [18] J. Milnor and J. D. Stasheff, Characteristic classes, Princeton Univ Press (1974). [19] D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Annals of Math 100 (1974) 1-79. Lehigh University, Bethlehem, PA 18015, USA E-mail address : [email protected] Northwestern University, Evanston, IL 60208, USA E-mail address : [email protected]

0704.0799

Spin Evolution of Accreting Neutron Stars: Nonlinear Development of the R-mode Instability

Spin Evolution of Accreting Neutron Stars: Nonlinear Development of the R-mode Instability Ruxandra Bondarescu, Saul A. Teukolsky, and Ira Wasserman Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853 The nonlinear saturation of the r-mode instability and its effects on the spin evolution of Low Mass X-ray Binaries (LMXBs) are modeled using the triplet of modes at the lowest parametric instability threshold. We solve numerically the coupled equations for the three mode amplitudes in conjunction with the spin and temperature evolution equations. We observe that very quickly the mode amplitudes settle into quasi-stationary states that change slowly as the temperature and spin of the star evolve. Once these states are reached, the mode amplitudes can be found algebraically and the system of equations is reduced from eight to two equations: spin and temperature evolution. The evolution of the neutron star angular velocity and temperature follow easily calculated trajectories along these sequences of quasi-stationary states. The outcome depends on whether or not the star will reach thermal equilibrium, where the viscous heating by the three modes is equal to the neutrino cooling (H = C curve). If, when the r-mode becomes unstable, the star spins at a frequency below the maximum of the H = C curve, then it will reach a state of thermal equilibrium. It can then either (1) undergo a cyclic evolution with a small cycle size with a frequency change of at most 10%, (2) evolve toward a full equilibrium state in which the accretion torque balances the gravitational radiation emission, or (3) enter a thermogravitational runaway on a very long timescale of ≈ 106 years. If the star does not reach a state of thermal equilibrium, then a faster thermal runaway (timescale of ≈ 100 years) occurs and the r-mode amplitude increases above the second parametric instability threshold. Following this evolution requires more inertial modes to be included. The sources of damping considered are shear viscosity, hyperon bulk viscosity and viscosity within the core-crust boundary layer. We vary proprieties of the star such as the hyperon superfluid transition temperature Tc, the fraction of the star that is above the threshold for direct URCA reactions, and slippage factor, and map the different scenarios we obtain to ranges of these parameters. We focus on Tc & 5 × 10 9 K where nonlinear effects are important. Wagoner [1] has shown that a very low r-mode amplitude arises at smaller Tc. For all our bounded evolutions the r-mode amplitude remains small ∼ 10−5. The spin frequency of accreting neutron stars is limited by boundary layer viscosity to νmax ≈ 800Hz[Sns/(M1.4R6)] 4/11T −2/11 . Fast rotators are allowed for [Sns/(M1.4R6)] 4/11T −2/11 ∼ 1 and we find that in this case the r-mode instability would be active for about 1 in 1000 LMXBs and that only the gravitational waves from LMXBs in the local group of galaxies could be detected by advanced LIGO interferometers. PACS numbers: 04.40.Dg, 04.30.Db, 97.10.Sj, 97.60.Jd I. INTRODUCTION R-modes are oscillations in rotating fluids that are due to the Coriolis effect. They are subject to the classical Chandrashekar-Friedman-Shutz (CFS) instability [2, 3], which is driven by the gravitational radiation backreac- tion force. Andersson [4] and Friedman and Morsink [5] showed that, in the absence of fluid dissipation, r-modes are linearly unstable at all rotation rates. However, in real stars there is a competition between internal viscous dissipation and gravitational driving [6] that depends on the angular velocity Ω and temperature T of the star. Above a critical curve in the Ω−T plane the n = 3,m = 2 mode, referred to as ’the r-mode’ in this work, becomes unstable. At first, an unstable r-mode grows exponen- tially, but soon it may enter a regime where other in- ertial modes that couple to the r-mode become excited and nonlinear effects become important. Roughly speak- ing, nonlinear effects first become significant as the am- plitude passes its first parametric instability threshold, which is very low (∼ 10−5). Modeling and understand- ing the nonlinear effects is crucial in determining (1) the final saturation amplitude of the r-mode and (2) the lim- iting spin frequency that neutron stars can achieve. The r-mode amplitude and the duration of the instability are among the main factors that determine whether the as- sociated gravitational radiation could be detectable by laser interferometers on Earth. The r-mode instability has been proposed as an expla- nation for the sub-breakup spin rates of both Low Mass X-ray Binaries (LMXBs) [7, 8] and young, hot neutron stars [6, 9]. The idea that gravitational radiation could balance accretion was proposed independently by Bild- sten [7] and Andersson et al. [8]. Cook, Shapiro and Teukolsky [10, 11] model the recycling of pulsars to mil- lisecond periods via accretion from a Keplerian disk onto a bare neutron star with M = 1.4M⊙ when Ω = 0. De- pending on the equation of state they found that spin frequencies of between ≈ 670 Hz and 1600 Hz could be achieved before mass shedding or radial instability set in (these calculations predated the realization that the r-mode instability could limit the spin frequency). For comparison, the highest observed spin rate of millisec- ond pulsars is 716 Hz for PSR J1748-2446ad [12, 13]. http://arxiv.org/abs/0704.0799v3 PSR B1937+21, which was discovered in 1982, was the previous fastest known radio pulsar with a spin rate of 642 Hz [14]; that this “speed” record stood for 24 years suggests that neutron stars rotating this fast are rare. Moreover, based on a Bayesian statistical analysis of the spin frequencies of the 11 nuclear-powered millisecond pulsars whose spin periods are known from burst oscil- lations, Chakrabarty et al. [15] claimed a cutoff limit of νmax = 760 Hz (95% confidence); A more recent analy- sis, which added two more pulsars to the sample, found νmax = 730 Hz [16]. At first sight, one might conclude that mass shedding or radial instability sets νmax, and that it is just above the record ν = 716 Hz determined for PSR J1748-2446ad. However, the nuclear equations of state consistent with this picture all have rather large radii ≈ 16 − 17 km for non-rotating 1.4 M⊙ models; see Table 1 in Cook et al. [10]. For these equations of state, the r-mode insta- bility should lead to νmax somewhat below 716 Hz; see Eq. (33) in Sec. V below. Thus, the r-mode instability may prevent recycling by accretion from reaching mass shedding or radial instability. In other words, the de- tection of the 716 Hz rotator is consistent with accretion spin-up mitigated by the r-mode instability only for equa- tions of state for which mass shedding or radial instability would permit even faster rotation. Ultimately, this may be turned into useful constraints on nuclear equations of state. However, at present the uncertainty in the physics of internal dissipation is a significant hindrance in estab- lishing such constraints. Since a physical model to follow the nonlinear phase of the evolution was initially unavailable, Owen et al. [17] proposed a simple one-mode evolution model in which they assumed that nonlinear hydrodynamics effects satu- rate the r-mode amplitude at some arbitrarily fixed value. According to their model, once this maximum allowed amplitude is achieved, the r-mode amplitude remains constant and the star spins down at this fixed ampli- tude (see Eqs. (3.16) and (3.17) in Ref. [17]). They used this model to study the impact of the r-mode instability on the spin evolution of young hot neutron stars assum- ing normal matter. In their calculation they include the effects of shear viscosity and n-p-e bulk viscosity. They found that the star would cool to approximately 109 K and spin down from a frequency close to the Kepler fre- quency to about 100 Hz in a period of ∼ 1 yr [17]. Most subsequent investigations that did not perform direct hydrodynamic simulations used the one-amplitude model of Ref. [17] for studying the r-mode instability. Levin [18] used this model to study the limiting effects of the r-mode instability on the spin evolution of LMXBs, assuming an r-mode saturation amplitude of ∼ 1; he adopted a modified shear viscosity to match the maxi- mum LMXB spin frequency of 330 Hz known in 1999. Levin found that the neutron star followed a cyclic evo- lution in the Ω − T phase plane. The star spins up for several million years until it crosses the r-mode stability curve, whereupon the r-mode becomes unstable and the star is viscously heated for a fraction of a year until the r-mode reaches its saturation amplitude (∼ 1). At this point the spin and r-mode amplitude evolution equations are changed, following the prescription of Ref. [17] to en- sure constant amplitude. The star then spins down by emitting gravitational radiation for another fraction of a year until it crosses the r-mode stability curve again and the instability shuts off. The time period during which the r-mode is unstable was found to be about 10−6 times shorter than the spin-up time, and Levin concluded that it is unlikely that any neutron stars in LMXBs in our galaxy are currently spinning down and emitting gravita- tional radiation. However, following work by Arras et al. [19] showing that nonlinear effects become significant at small r-mode amplitude, Heyl [20] varied the saturation amplitude, and found that the duration of the spin-down depends sensitively on it. He predicted that the unstable phase could be as much as 30% of the cyclic evolution for an r-mode saturation amplitude of α ≈ 10−5, and that this would make some of the fastest spinning LMXBs in our galaxy detectable by interferometers on Earth. Jones [21] and Lindblom and Owen [22] pointed out that if the star contains exotic particles such as hyperons (massive nucleons where an up or down quark is replaced with a strange quark), internal processes could lead to a very high coefficient of bulk viscosity in the cores of neu- tron stars. While this additional high viscosity coefficient could eliminate the instability altogether in newly born neutron stars [21, 22, 23, 24], Nayyar and Owen [24] pro- posed that it would enhance the probability of detection of gravitational radiation from LMXBs by blocking the thermal runaway. The cyclic evolution found by Levin [18] and gener- alized by Heyl [20] arises when shear or boundary layer viscosity dominates the r-mode dissipation. In the evo- lutionary picture of Nayyar and Owen [24], the r-mode first becomes unstable at a temperature where shear and boundary layer viscosity dominate, but the result- ing thermal runaway halts once hyperon bulk viscosity becomes dominant. The key feature behind the runaway is that shear and boundary layer viscosities both decrease with increasing temperature, so the instability speeds up as the star grows hotter. However, if the bulk viscosity is sufficiently large the star can cross the r-mode stabil- ity curve at a point where the viscosity is an increas- ing function of temperature. Such scenarios were stud- ied by Wagoner [1] for hyperon bulk viscosty with low hyperon superfluid transition temperature; similar evo- lution was found for strange stars by Andersson, Jones and Kokkotas [25]. In this picture, the star evolves near the r-mode stability curve until an equilibrium between accretion spin-up and gravitational radiation spin-down is achieved. The value of the r-mode amplitude remains below the lowest instability threshold found by Brink et al. [26, 27, 28] for modes with n < 30, and hence in this regime nonlinear effects may not play a role. Schenk et al. [29] developed a formalism to study the nonlinear interaction of the r-mode with other inertial modes. They assumed a small r-mode amplitude and treated the oscillations of the modes with weakly nonlin- ear perturbation theory via three-mode couplings. This assumption was tested by Arras et al. [19] and Brink et al. [26, 27, 28]. Arras et al. proposed that a turbu- lent cascade would develop in the strong driving regime. They estimated that r-mode amplitude was small and could have values between 10−1 − 10−4. Brink et al. modeled the star as incompressible and calculated the coupling coefficients analytically. They computed the in- teraction of about 5000 modes via approximatively 1.3 million couplings of the 109 possible couplings among the modes with n ≤ 30. The couplings were restricted to mode triplets with a fractional detuning δω/(2Ω) < 0.002 since near-resonances promote modal excitation at very small amplitudes. Brink et al. showed that the nonlinear evolution saturates at a very small amplitude, generally comparable to the lowest parametric instability thresh- old that controls the initiation of energy sharing among the sea of inertial modes. However, Brink et al. did not model accretion spin-up or neutrino cooling in their cal- culation and only included minimal dissipation via shear viscosity. In this paper we begin a more complete study of the saturation of the r-mode instability including accretion spin up and neutrino cooling. We use a simple model in which we parameterize uncertain properties of the star such as the rate at which it cools via neutrino emission and the rate at which the energy in inertial modes dis- sipates via boundary layer effects [30] and bulk viscos- ity. In order to exhibit the variety of possible nonlinear behaviors, we explore a range of models with different neutrino cooling and viscous heating coefficients by vary- ing the free parameters of our model. In particular, we vary: (1) the slippage factor Sns, which regulates the boundary layer viscosity, between 0 and 1 (see for exam- ple [31, 32, 33] for some models of the interaction between the oscillating fluid core and an elastic crust) ; (2) the fraction of the star that is above the density threshold for direct URCA reactions fdU, which is taken to be between 0 (0% of the star cools via direct URCA) and 1 (100% of the star is subjected to direct URCA reactions), and in general depends on the equation of state used; and (3) the hyperon superfluidity temperature Tc, which is believed to be between 109− 1010 K (We use a single, effective Tc rather than modelling its spatial variation.) We focus on Tc & 5×109 K for which nonlinear effects are important. For low Tc . 3 × 109 K, Wagoner [1] showed that the evolution reaches a steady state at amplitudes below the lowest parametric instability threshold found by Brink et al. [28]. It is important to note that all our evolu- tions start on the part of the r-mode stability curve that decreases with temperature and that the bulk viscosity does not play a role in any of our bound evolutions. We include three modes: the r-mode at n = 3 and the two inertial modes at n = 13 and n = 14 that become unstable at the lowest parametric instability threshold found by Brink et al. [28]. We evolve the coupled equa- tions for the three-mode system numerically in conjunc- tion with the spin and temperature evolution equations. The lowest parametric instability threshold provides a physical cutoff for the r-mode amplitude. In all cases we investigate, the growth of the r-mode is initially halted by energy transfer to the two daughter modes. We ob- serve that the mode amplitudes settle into a series of quasi-stationary states within a period of a few years af- ter the spin frequency of the star has increased above the r-mode stability curve. These quasi-stationary states are algebraic solutions of the three-mode amplitude equa- tions (see Eqs. (6)) and change slowly as the spin and the temperature of the star evolve. Using these solutions for the mode amplitudes, one can reduce the eight evo- lution equations (six for the real and imaginary parts of the mode amplitudes, which are complex [29]; one for the spin, and one for the temperature) to two equations gov- erning the rotational frequency and the temperature of the star. Our work can be regarded as a minimal physical model for modeling amplitude saturation realistically. The outcome of the evolution is crucially dependent on whether the star can reach a state of thermal equilib- rium. This can be predicted by finding the curve where the viscous heating by the three modes balances the neu- trino cooling, referred to below as the Heating = Cooling (H = C) curve. TheH = C curve can be calculated prior to carrying out an evolution using the quasi-stationary solutions for the mode amplitudes. If the spin frequency of the star upon becoming unstable is below the peak of the H = C curve, then the star will reach a state of thermal equilibrium. When such a state is reached we find several possible scenarios. The star can: (1) un- dergo a cyclic evolution; (2) reach a true equilibrium in which the accretion torque is balanced by the rate of loss of angular momentum via gravitational radiation; or (3) evolve in thermal equilibrium until it reaches the peak of the H = C curve, which occurs on a timescale of about 106 yr, and subsequently enter a regime of thermal run- away. On the other hand, if the star cannot find a state of thermal equilibrium, then it enters a regime of ther- mogravitational runaway within a few hundred years of crossing the r-mode stability curve. When this happens, the r-mode amplitude increases beyond the second para- metric instability, and more inertial modes would need to be included to correctly model the nonlinear effects. This will be done in a later paper. This paper focuses on showing how nonlinear mode couplings affect the evolution of the temperature and spin frequency of a neutron star once it becomes prone to the r-mode CFS instability. We do this in the context of three mode coupling, which may be sufficient for large enough dissipation. To illustrate the types of behavior that arise, we adopt a very specific model in which the mode fre- quencies and couplings are computed for an incompress- ible star, modes damp via shear viscosity, boundary layer viscosity and hyperon bulk viscosity, and the star cools via a mixture of fast and slow processes. This model in- volves several parameters that are uncertain, and we vary these to find ‘phase diagrams’ in which different generic types of behavior are expected. Moreover, the model it- self is simplified: (1) A more realistic treatment of the modes could include buoyant forces, and also mixtures of superfluids or of superfluid and normal fluid in different regions. (2) Dissipation rates, particularly from bulk vis- cosity, depend on the composition of high density nuclear matter, which could differ from what we assume. Nevertheless, although the quantitative details may differ from what we compute, we believe that many fea- tures of our calculations ought to be robust. More sophis- ticated treatment of the modes of the star will still find a dense set of modes confined to a relatively small range of frequencies. Most importantly, this set will exhibit nu- merous three mode resonances, which is the prerequisite for strong nonlinear effects at small mode amplitudes. Thus, whenever the unstable r-mode can pass its lowest parametric instability threshold, it must start exciting its daughters. Whether or not that occurs depends on the temperature dependence of the dissipation rate of the r- mode; for the models considered here, where bulk viscos- ity is relatively unimportant, soon after the star becomes unstable its r-mode amplitude passes its first paramet- ric instability threshold. Once that happens, the generic types of behavior we find - cycles, steady states, slow and fast runaway - ought to follow suit. The details of when different behaviors arise will depend on the precise features of the stellar model, but the principles we out- line here (parametric instability, quasisteady evolution, competition between heating and cooling) ought to ap- ply quite generally. In Sec. II we describe the evolution equations of the three modes, the angular frequency and the temperature of the neutron star. We first show how the equations of motion for the modes of Schenk et al. couple to the rota- tional frequency of the star in the limit of slow rotation. We then give a short review of the parametric instability threshold and the quasi-stationary solutions of the three- mode system. The thermal and spin evolution of the star is discussed next. This is followed by a description of the driving and damping rates used. Sec. III provides an overview of the results, which includes a discussion of each evolution scenario and of the initial conditions and input physics that lead to each scenario. Sec. IVA discusses cyclic evolution in more detail. An evolution that leads to an equilibrium steady state is presented next in Sec. IVB. The two types of thermal runaway are then discussed in Sec. IVC. The prospects for detecting gravitational radiation for the evolutions in which the three-mode system correctly models the nonlinear effects are considered in Sec. V. We summarize the results in the conclusion. Appendix A sketches a derivation of the equations of motion for the three modes and Appendix B contains a stability analysis of the evolution equations around the thermal equilibrium state. II. EVOLUTION EQUATIONS A. Three mode system: coupling to uniform rotation In this section we review the equations of motion for the three-mode system in the limit of slow rotation. In terms of rotational phase τ for the time variable with dτ = Ω dt Eq. (2.49) of Schenk et al. [29] can be rewritten = iω̃αCα + 2iω̃ακ̃√ CβCγ , (1) = iω̃βCβ − 2iω̃βκ̃√ = iω̃γCγ − 2iω̃γκ̃√ Here the scaled frequency ω̃j is defined to be ω̃j = ωj/Ω, the dissipation rates of the daughter modes are γβ and γγ , γα is the sum of the driving and damping rates of the r- mode γα = γGR−γαv, and the dimensionless coupling is κ̃ = κ/(MR2Ω2). These amplitude variables are complex and can be written in terms of the variables of Ref. [29] as Cj(t) = Ω(t)cj(t) (see Appendix A for a derivation of Eqs. 1). The index j loops over the three modes j = α, β, γ, where α labels the r-mode or parent mode and β and γ label the two daughter modes in the mode triplet. When the daughter mode amplitudes are much smaller than that of the parent mode, one can approximate the parent mode amplitude as constant. Under this assump- tion one performs a linear stability analysis on Eqs. (1) and finds the r-mode amplitude when the two daughter modes become unstable (see Eqs. (B5-B7) of Ref. [28] for a full derivation). This amplitude is the parametric instability threshold |Cα|2 = 4κ̃2ω̃βω̃γΩ 1 + Ω2 γβ + γγ , (2) where the fractional detuning is δω̃ = ω̃α − ω̃β − ω̃γ . Thorough explorations of the phase space of damped three-mode systems were performed by Dimant [34] and Wersinger et al. [35]. For the three modes at the lowest parametric instabil- ity threshold, ω̃α ≈ 0.66, ω̃β ≈ 0.44, ω̃γ ≈ 0.22, κ̃ ≈ 0.19 and |δω̃| ≈ 3.82 × 10−6. Note that ω̃ is twice the w of Brink et al. [26, 27, 28]. Here β labels the mode with n = 13,m = −3 and γ labels the n = 14,m = 1 mode. The amplitude the r-mode has to reach before exciting these two daughter modes is |Cα| ≈ 1.5× 10−5 Ω [28]. We next rescale the rotational phase τ by the fractional detuning as τ̃ = τ |δω̃| and the mode amplitudes by |Cα|0 = |δω̃| ω̃βω̃γ , |Cβ |0 = |δω̃| ω̃αω̃γ , (3) |Cγ |0 = |δω̃| ω̃βω̃α which for the r-mode is, up to a factor of Ω/Ωc, the no-damping limit of the parametric instability thresh- old below which no oscillations will occur. The coupled equations become |δω̃| C̄α + |δω̃|Ω̃ C̄α − C̄βC̄γ , (4) |δω̃| C̄β − |δω̃|Ω̃ C̄β − C̄αC̄ |δω̃| C̄γ − |δω̃|Ω̃ C̄γ − C̄αC̄ with C̄j = Cj/|Cj |0 and γ̃j = γj/Ωc being the newly rescaled amplitudes and dissipation/driving rates, re- spectively. 1. Quasi-Stationary Solution In terms of amplitudes and phase variables Cj = |Cj |eiφj Eqs. (4) can be rewritten as d|C̄α| Ω̃|δw̃| |C̄α| − sinφ|C̄β ||C̄γ | , (5) d|C̄β | = − γ̃β Ω̃|δw̃| |C̄β |+ sinφ|C̄α||C̄γ | d|C̄γ | Ω̃|δw̃| |C̄γ |+ sinφ|C̄α||C̄β | |δω̃| − cosφ |C̄β ||C̄γ | |C̄α| − |C̄α||C̄γ | |C̄β | − |C̄β ||C̄α| |C̄γ | where we have defined the relative phase difference as φ = φα − φβ − φγ . These equations have the stationary solution |C̄α|2 = 4γ̃βγ̃γ Ω̃|δω̃|2 tan2 φ , (6) |C̄β |2 = 4γ̃αγ̃γ Ω̃|δω̃|2 tan2 φ |C̄γ |2 = 4γ̃αγ̃β Ω̃|δω̃|2 tan2 φ tanφ = γ̃β + γ̃γ − γ̃α Ω̃|δω̃| Note that in the limit in which γβ+γγ >> γα the station- ary solution for the r-mode amplitude |Cα| is the same as the parametric instability threshold. B. Temperature and Spin Evolution The spin evolution equation is obtained from conser- vation of total angular momentum J , where J = IΩ + Jphys. (7) Following Eq (K39-K42) of Schenk et al. [29] the physical angular momentum of the perturbation can be written as ΩJphys = C⋆BCA d3xρ[(Ω̂× ξ⋆B) · (Ω̂× ξA) (8) − i (ω̃A + ω̃B) ξ⋆B · (Ω̂× ξA)]. Since the eigenvectors ξA ∝ eimAφ the cross-terms will vanish for modes with different magnetic quantum num- bers m as ei(mA−mB)φdφ = 0 for mA 6= mB. Eq. (8) can be re-written for our triplet of modes as Jphys = MR 2(kαα|Cα|2 + kββ|Cβ |2 + kγγ |Cγ |2), (9) where kαα is defined as kαα = d3xρ[(Ω̂×ξ⋆α) · (Ω̂×ξα)− iω̃αξ⋆α · (Ω̂×ξα)] and similarly for kββ and kγγ . In terms of the scaled vari- ables C̄j = Cj/|Cj |0 (with |Cj |0 defined in Eq. (3)) the angular momentum of the perturbation can be written Jphys = MR2Ωc|δω̃|2 (4k̃)2ω̃αω̃βω̃γ (kαα|C̄α|2ω̃α (11) +kββ|C̄β |2ω̃β + kγγω̃γ |C̄γ |2). We chose the same normalization for the eigenfuctions as Refs. [19, 26, 27, 28, 29] so that at unit amplitude all modes have the same energy ǫα = MR 2Ω2. The energy of a mode α is Eα = MR 2Ω2|cα|2 = MR2Ω|Cα|2. The rotating frame energy is the same as the canonical energy and physical energy [29]. The canonical angular momen- tum and the canonical energy of the perturbation satisfy the general relation Ec = −(ω/m)Jc [3]. Angular momentum is gained because of accretion and lost via gravitational waves emission = 2γGRJc rmode + Ṁ GMR, (12) where Jc rmode = −(mα/ωα)ǫα|cα|2 = −3MR2Ω|cα|2 = −3MR2|Cα|2. Eq. (12) can be rewritten in terms of the scaled variables C̄j as = −6γ̃GR MR2Ωc|δω̃| (4k̃)2ω̃βω̃γ |C̄α|2 + ΩcΩ̃|δω̃| . (13) Thermal energy conservation gives the temperature evo- lution equation C(T ) 2Ejγj +KnṀc 2 − Lν(T ), (14) = 2MR2Ω(γα v|Cα|2 + γβ |Cβ |2 + γγ |Cγ |2) +KnṀc2 − Lν(T ). The three terms on the right hand side of the equa- tion represent viscous heating, nuclear heating and neu- trino cooling. The specific heat is taken to be C(T ) ≈ 1.5 × 1038 T8 erg K−1, where T = T8 × 108 K. Nu- clear heating occurs because of pycnonuclear reactions and neutron emission in the inner crust [36]. At large accretion rates such as that of the brightest LMXBs of Ṁ ≈ 10−8M⊙/yr, the accreted helium and hydrogen burns stably and most of the heat released in the crust is conducted into the core of the neutron star, where neu- trino emission is assumed to regulate the temperature of the star [36, 37]. The nuclear heating constant is taken to be Kn ≈ 1×10−3 [36]. Following Ref. [1], we take the neutrino luminosity to be Lν = LdUT 8RdU(T/Tp) + LmUT 8RmU(T/Tp) (15) + Le−iT 8 + Ln−nT 8 + LCpT where the constants for the modified and direct URCA re- actions are defined by LmU = 1.0×1032 erg sec−1, LdU = fdU × 108LmU [38, 39], and the electron-ion, neutron- neutron neutrino bremsstrahlung and Cooper pairing of neutrons are given by Le−i = 9.1 × 1029 erg sec−1 [36], Ln−n ≈ 0.01LmU, LCp = 8.9 × 1031 erg sec−1 [40]. The fraction of the star fdU that is above the density thresh- old for direct URCA reactions is in general dependent on the equation of state [41] and in this work we treat fdU a free parameter with values between 0 and 1. The proton superfluid reduction factors for the modi- fied and direct URCA reactions are taken from Ref. [39] (see Eqs. (32) and (51) in Ref. [39]): RdU(T/Tp) = 0.2312 + (0.76880)2 + (0.1438v)2 × exp 3.427− (3.427)2 + v2 RmU(T/Tp) = 0.2414 + (0.7586)2 + (0.1318v)2 × exp 5.339− (5.339)2 + (2v)2 where the dimensionless gap amplitude v for the singlet type superfluidity is given by 1.456− 0.157 + 1.764 . (17) Similar to Ref. [1], we use Tp = 5.0× 109 K. In terms of the scaled variables Eq. (14) becomes C(T ) 2MR2Ω2c |δω̃| (4κ̃)2ω̃αω̃βω̃γ (ω̃αγ̃α v|C̄α|2 + ω̃βγ̃β |C̄β |2(18) +ω̃γγ̃γ |C̄γ |2) + KnṀc 2 − Lν(T ) ΩcΩ̃|δω̃| C. Temperature and Spin Evolution with the Mode Amplitudes in Quasi-Stationary States Assuming that the amplitudes evolve through a series of spin- and temperature-dependent steady states, i.e., dCi/dτ̃ ≈ 0, the spin and thermal evolution equations can be rewritten by taking J ≈ IΩ and using Eqs. (6) in Eq. (13). = − 6γ̃GR Ω̃2|δω̃| γ̃β γ̃γ 4k̃2Ĩω̃βω̃γ tan2 φ MR2ĨΩ̃|δω̃| where Ĩ = I/(MR2). The thermal evolution of the sys- tem is given by C(T ) 2MR2Ω2c (4κ̃)2ω̃αω̃βω̃γ γ̃αγ̃β γ̃γ Ω̃|δω̃| ω̃αγ̃α,v + ω̃β (20) +ω̃γ) tan2 φ KnṀc 2 − Lν(T ) ΩcΩ̃|δω̃| By setting the right hand side of the above equation to zero, one can find the Heating = Cooling (H = C) curve. Below, we find that Eqs. (19)-(20) describe the evolu- tion very well throughout the unstable regime. These equations are a minimal physical model for the effects of nonlinear coupling on r-mode evolution. D. Sources of Driving and Dissipation The damping mechanisms are shear viscosity, bound- ary layer viscosity and hyperon bulk viscosity; for modes j = α, β, γ we write γj v(Ω, T ) = γj sh(T ) + γj bl(Ω, T ) + γj hb(Ω, T ). (21) The r-mode is driven by gravitational radiation and damped by these dissipation mechanisms, while the pair of daughter modes (n = 13,m = −3 labeled as β and n = 14,m = 1 labeled as γ) is affected only by the vis- cous damping. Brink et al. [26, 27, 28] determined that this pair of modes is excited at the lowest parametric instability threshold. Their model uses the Bryan [42] modes of an incompressible star, which has the advan- tage that the mode eigenfrequencies (and eigenfunctions) are known analytically. This enables them to find near resonances efficiently. We are using their results, but we include more realistic effects such as bulk viscosity, whose effect vanishes in the incompressible limit (Γ1 → ∞ in Eq. (29)) For our benchmark calculations, we adopt the neutron star model of Owen et al. Ref. [6] (n = 1 polytrope, M = 1.4M⊙, Ωc = 8.4 × 103 rad sec−1 and R = 12.53 km) and use their gravitational driving rate and shear viscous damping rate for the r-mode γGR(Ω) ≃ sec−1, (22) γα sh(T ) ≃ where τsh = 2.56 × 106 sec. (In Sec. V we consider ap- proximate scalings with M and R.) The damping rate due to shear viscosity for the two daughter modes is calculated using the Bryan modes for a star with the same mass and radius γβ sh(T ) ≃ 3.48× 10−4 sec−1 , (23) γγ sh(T ) ≃ 4.52× 10−4 sec−1 The geometric contribution γsh/η of the individual modes increases significantly with the degree n of the mode scal- ing approximatively like n3 for large n (see Eq. (29) of Brink et al. [27] for an analytic fit to the shear damping rates computed for the 5,000 modes in their network), and hence the inertial modes with n = 13 and n = 14 have shear damping rates about three orders of magni- tude larger than that of the r-mode. The damping due to boundary layer viscosity is calcu- lated using Eq. (4) of Ref. [30], γα bl(T,Ω) ≃ 0.009 sec−1 S2ns , (24) γβ bl(T,Ω) ≃ 0.028 sec−1 S2ns γγ bl(T,Ω) ≃ 0.021 sec−1S2ns Analogous to Wagoner [1], we allow the slippage fac- tor Sns to vary. The slippage factor is defined by Refs. [1, 31, 45] to be S2ns = (2S n + S s )/3, with Sn being the fractional difference in velocity of the normal fluid be- tween the crust and the core [31] and Ss the fractional degree of pinning of the vortices in the crust [45]. Note that γβ bl and γγ bl are both greater than 2 × γα bl and can easily be comparable to γGR in the unstable regime. The damping rate due to bulk viscosity produced by out-of-equilibrium hyperon reactions for the r-mode is found by fitting the results of Nayyar and Owen [24]. This rate is taken to have a form similar to that taken by Wagoner [1] γα hb = fhb t−20α τ(T )Ω̃ 1 + (ω̃αΩτ(T ))2 , (25) and for the daughter modes γβ hb = fhb t−20β τ(T )ω̃ 1 + (ω̃βΩτ(T ))2 , (26) and similarly foe γγ hb. The relaxation timescale τ(T ) = Rhb(T/Tc) The reduction factor is taken to be the product of two single-particle reduction factors [23, 24] Rhb single(T/Tc) = a5/4 + b1/2 0.5068− 0.50682 + y2 where a = 1 + 0.3118y2, b = 1 + 2.556y2 and y = 1.0− T/Tc(1.456 − 0.157 Tc/T + 1.764Tc/T ). The constants t1 ≈ 10−4 sec and t0α ≈ 0.00058 sec are found by fitting the results of Ref. [24]. The factor fhb allows for physical uncertainties; we take fhb = 1 throughout the body of the paper since Tc , which enters γj hb ex- ponentially, is also uncertain. For the daughter modes, the dissipation energy due to bulk viscosity is calculated using the modes for the incompressible star. In the slow rotation limit, it is given to leading order in Γ−21 by − ĖB j = ξj · ∇p . (29) This approximation was proposed by Cutler and Lind- blom [43] and adopted by Kokkotas and Stergioulas [44] for the r-mode and by Brink et al. [27] for the inertial modes. The adiabatic index Γ1 is regarded as a parame- ter; we use Γ1 ≈ 2. The damping rate is γj hb = − ĖB j , (30) where ǫ = MR2Ω2 is the mode’s energy in the rotat- ing frame at unit amplitude and j = β, γ. Using this procedure, we calculate t0β ≈ 1.4× 10−5 sec, (31) t0γ ≈ 1.0× 10−5 sec. III. SUMMARY OF RESULTS Fig. 1(a) shows possible evolutionary trajectories of a neutron star in the angular velocity-temperature Ω̃− T8 plane, where T = T8×108 K is the core temperature, and Ω̃ = Ω/Ωc = Ω/ πGρ̄ with ρ̄ the mean density of the neutron star. Fig. 1(b) displays the regions in fdU − Sns in which the trajectories occur. Here fdU represents the fraction of the star that is above the density threshold for direct URCA reactions and Sns is the slippage factor that reduces the relative motion between the crust and the core taking into account the elasticity of the crust [31]. The stability regions are shown at fixed hyperon superfluidity temperature, Tc = 5.0 × 109 K. The initial part of the evolution is similar in all scenarios and can be divided into phases. Phase 0. Spin up below the r-mode stability curve at T8 = T8 in such that nuclear heating balances neutrino cooling. Phase 1. Linear regime. The r-mode amplitude grows exponentially. The phase ends when the r-mode reaches the parametric instability. Phase 2. The triplet coupling leads to quasi-steady mode amplitudes. The star is secularly heated at approximately constant Ω because of viscous dissipation in all three modes. Phase 3. Several trajectories are possible depending on FIG. 1: (a)Typical trajectories for the four observed evolu- tion scenarios are shown in the Ω̃ - T8 phase space, where Ω̃ = Ω/Ωc. The dashed lines (H = C curves) represent the points in the Ω̃ − T8 phase space where the dissipative effects of the heating from the three-modes exactly compen- sate the neutrino cooling for the given set of parameters (Sns, fdU, Tc, ...) of each evolution. (b)The corresponding sta- bility regions for which these scenarios occur are plotted at fixed hyperon superfluidity temperature Tc = 5.0 × 10 while varying fdU and Sns. The position of the initial angu- lar velocity and temperature (Ω̃in, T8 in) with respect to the maximum of this curve determines the stability of the evo- lution. (I) Ω̃in > Ω̃H=C max. Trajectory R1. Fast Runaway Region. After the r-mode becomes unstable the star heats up, does not find a thermal equilibrium state and continues heating up until a thermogravitational runaway occurs. (II) Ω̃in < Ω̃H=C max. The evolutions are either stable or, if there is a runaway, it occurs on timescales comparable to the ac- cretion timescale. The possible trajectories are (1)Trajectory C. Cycle Region. (2) Trajectories S1 and S2. Steady State Region. (3) Trajectory R2. Slow Runaway Region. how the previous phase ends. a. Fast Runaway. The star fails to reach thermal equilibrium when the trajectory passes over the peak of the Heating = Cooling (H = C) curve. This leads to rapid runaway. The daughter modes damp eventually as bulk viscosity becomes important, and the r-mode grows exponentially until the trajectory hits the r-mode stability curve again. This scenario ends as predicted by Nayyar and Owen [24]. However, the r-mode passes its second parametric instability threshold soon after it starts growing again. This requires the inclusion of more modes to follow the evolution, which is the subject of future work. b. The star reaches thermal equilibrium. There are then three possibilities: (i) Cycle. The star cools and spins down slowly, descending the H = C curve until it crosses the r-mode stability curve again. At this point the instability shuts off. The star cools back to T8 in at constant Ω̃ and then the cycle repeats itself. At Tc = 5.0 × 109 K this scenario occurs for values of Sns < 0.50 and large enough values of fdU. However, if Tc is larger, the cycle region in the fdU-Sns phase space increases dramatically (see Fig. 9(a)). Note that our cycles are different from those obtained by Levin [18] in that the spin-down phase does not start when the r-mode amplitude saturates (or in our case when it reaches the parametric insta- bility threshold), but rather when the system reaches thermal equilibrium. The r-mode amplitude does not grow significantly above its first parametric instability threshold, remaining close to ∼ 105 and so the part of the cycle in which the r-mode is unstable also lasts longer than in Ref. [18]. Also, our cycles are narrow. During spin-down the temperature changes by less than 20 % and Ω̃ changes by less than 10% of the initial value. (See Sec. 2 for a detailed example.) (ii) Steady State. For small Sns and large enough fdU (fdU & 5 × 10−5, Sns . 0.04; see Fig. 1(b)) the star evolves towards an Ω̃ equilibrium. The trajectory either ascends or descends the H = C curve (spins up and heats or spins down and cools). The evolution stops when the accretion torque equals the gravitational radiation emission. (iii) Slow Runaway. For small Sns and very small fdU (Sns . 0.03, fdU < 5×10−5) the star ascends the H = C curve until the peak is overcome and subsequently a runaway occurs. The daughter modes eventually damp and the r-mode grows exponentially until it crosses its second parametric instability threshold and more modes need to be included. Bulk viscosity only affects the runaway evolutions; the cyclic and steady state evolutions found here would be the same if there were no hyperon bulk viscosity. For large Tc ∼ 1010, or for models with no hyperons at all, there would be no runaway region (See Fig. 9(a) for an fdU − Sns scenario space with a larger Tc = 6.5× 109 K where the fast runaway region has shrunk dramatically FIG. 2: Two cyclic trajectories in the Ω̃ − T8 plane are dis- played for a star with Tc = 5.0 × 10 9 K and (a) fdU = 0.15 and Sns = 0.10, and (b) fdU = 0.142 and Sns = 0.35, which is close to the border between the stable and unstable region (see Fig. 1(b)). The thick solid line labeled as the Heating = Cool- ing (H = C) curve is the locus of points in this phase space where the neutrino cooling is equal to the viscous heating due to the unstable modes. The other solid line representing the r-mode stability curve is defined by setting the gravitational driving rate equal to the viscous damping rate. The part of the curve that decreases with T8 is dominated by boundary layer and shear viscosity, while the part of the curve that has a positive slope is dominated by hyperon bulk viscosity. In portion a1 → b1 of the trajectory the star heats up at con- stant Ω̃. Part b1 → c1 represents the spin down stage, which occurs when the viscous heating is equal to the neutrino cool- ing. c1 → d1 shows the star cooling back to the initial T8. Segment d1 → a1 displays the accretional spin-up of the star back to the r-mode stability curve. The cycle a2 → d2 pro- ceeds in the same way. This cycle is close to the peak of the H = C curve. Configurations above this peak will run away. and the slow runaway region has disappeared.) IV. POSSIBLE EVOLUTION SCENARIOS In this section we examine examples of the different types of evolution in more detail. We assume Ṁ = 10−8M⊙/yr and Tc = 5.0× 109 K. A. Cyclic Evolution In this sub-section we present the features of typical cyclic trajectories of neutron stars in the angular velocity temperature plane in more detail. We focus on two cases: (C1) Sns = 0.10 and fdU = 0.15 and (C2) Sns = 0.35 and fdU = 0.142. In this scenario the 3-mode system is sufficient to model the nonlinear effects and successfully stops the thermal runaway. The numerical evolution is started once the star reaches the r-mode stability curve. The initial temperature of the star is at the point where nuclear heating equals neutrino cooling in Eq. (18) that is approximately T8 in ≈ 3.29 for both cases. The initial Ω is the angular velocity that corresponds to this tem- perature on the r-mode stability curve, which differs for the different Sns (Ω̃in = 0.183 for C1 and Ω̃in = 0.288 for Figs. 2(a) and (b) display the cyclic evolution for tra- jectories C1 and C2 of Fig. 1(b). In leg a1 → b1 of the trajectory the r-mode and, once the r-mode amplitude increases above the first parametric instability thresh- old, the two daughter modes it excites, viscously heat up the star until point b1 when the neutrino cooling bal- ances the viscous dissipation. This part of the evolu- tion occurs at constant angular velocity over a period of theat−up ≈ 100 yr and a total temperature change (∆T )a1−b1 ≈ 0.80 (≈ 24% of T8 in). The points where the viscous heating compensates the neutrino cooling are represented by the Heating = Cooling (H = C) curve. This is determined by setting Eq. 18 to zero and using the quasi-stationary solutions given by Eq. (6) for the three modes on the right hand side. The star continues to evolve on the H = C curve for part b1 → c1 of the trajectory as it spins down and cools down back to the r- mode stability curve. This spin-down stage lasts a time tspin−downb1−c1 ≈ 23, 000 yr that is much longer than the heat-up period. This timescale is very sensitive to changes in the slippage factor and can reach 106 yr for smaller values of Sns that are close to boundary of the steady state region. The cycle is very narrow in angular velocity with a total angular velocity change of less than 4%, (∆Ω̃)b1−c1 ≈ 0.0066. The temperature also changes by only about 2%, (∆T8)b1−c1 ≈ 0.08 in this spin-down period. Segment c1 → d1 represents the cooling of the star to the initial temperature on a timescale of ∼ 2, 000 yr. In part d1 → a1 the star spins up by accretion at constant temperature back to the original crossing point on the r-mode stability curve. This last part of the tra- jectory is the longest-lasting one, taking ≈ 200, 000 yr at our chosen Ṁ of 10−8M⊙yr −1. The cycle C2 in Fig. 2(b) proceeds in a similar fashion. It is important to note that this configuration is close to the border between the “FAST RUNAWAY” and “CYCLE” regions and there- fore close to the peak of the H = C curve. Configura- tions above this peak (e.g., with the same fdU and higher Sns) will go through a fast runaway. Fig. 3(a) shows the evolution of the three modes in the first few years after the star first reaches the r- mode stability curve. In this region the r-mode is un- stable and initially grows exponentially. Once it has in- creased above the first parametric instability threshold the daughter modes are excited. The oscillations of the three modes display some of the typical dynamics of a driven three-mode system. When the r-mode transfers energy to the daughter modes they increase exponen- tially while the r-mode decreases. Similarly, when daugh- ter modes decrease the r-mode increases. The viscosity damps the oscillations and the r-mode amplitude settles at a value close to the parametric instability threshold. Fig. 3(b) displays the evolution of the r-mode ampli- tude divided by the parametric instability threshold on a longer timescale. It can be seen that the r-mode never grows significantly beyond this first threshold. Fig. 3(c) shows the evolution of the parametric instability thresh- old as a function of time. The threshold increases as the temperature increases and the star is viscously heated by the three modes. When the star spins down in thermal equilibrium, the threshold decreases to a value close to its initial value. B. Steady State Evolution This sub-section focuses on evolutions that lead to a steady equilibrium state in which the rate of accretion of angular momentum is balanced by the rate of loss via gravitational radiation emission. This scenario is re- stricted to stars with small slippage factor (Sns . 0.04, see Fig. 1(b)) and boundary layer viscosity. A typical trajectory of a star that reaches such an equilibrium is shown in Fig. 4. As always, we start the evolution at the point on the r-mode stability curve at which the nuclear heating balances neutrino cooling. Above the r-mode sta- bility curve the gravitational driving rate is greater than the viscous damping rate and the r-mode grows exponen- tially until nonlinear effects become important. In this case, as in the cyclic evolution, the triplet of modes at the lowest parametric instability threshold is sufficient to stop the thermal runaway. The r-mode remains close to the first instability threshold for the length of the evo- lution and after a few oscillations the three modes settle into their quasi-stationary states, which change only sec- ularly as the spin and temperature of the star evolve. The modes heat the star viscously at constant Ω̃ in seg- ment a → b of the trajectory for theat−up ≈ 1, 100 yr. At point b, the star reaches a state of thermal balance. In leg b → c the star continues its evolution in thermal equilibrium and slowly spins up due to accretion until the angular velocity evolution also reaches an equilib- 5.12 5.13 τ [x 10 0.422 0.424 0.426 0.428 0.432 |cα Th| 0.4 0.5 0.6 0.7 0.8 0.9 1 τ [x 10 0 0.5 1 1.5 2 2.5 3 3.5 4 τ [x 10 0.425 0.435 FIG. 3: (a)The amplitudes of the r-mode |Cα| and of the n = 13, m = −3 and n = 14, m = 1 inertial modes |Cβ | and |Cγ | are shown as a function of time for a star that executes a cyclic evolution (same parameters as in Fig. 2). The low- est parametric instability threshold is also displayed. (b)The ratio of the r-mode amplitude to the parametric instability threshold is plotted as a function of time. It can be seen that once the r-mode crosses the parametric instability threshold it remains close to it for the rest of the evolution. (c)The parametric instability threshold is displayed as a function of time. Its value changes as the angular velocity and tempera- ture evolve. FIG. 4: The trajectory of a neutron star in the Ω̃− T8 phase space is shown for a model with Tc = 5.0× 10 9 K, fdU = 0.03 and Sns = 0.03 that reaches an equilibrium steady state. The star spins up until it crosses the r-mode stability curve and the r-mode becomes unstable. The r-mode then quickly grows to the first parametric instability threshold and excites the daughter modes. In leg a → b of the trajectory the star is viscously heated by the mode triplet until the system reaches thermal equilibrium. Segment b → c shows the star contin- uing to heat and spin up in thermal equilibrium until the accretion torque is balanced by the gravitational radiation emission. The r-mode stability curve represents the points in phase space where the viscous driving rate is equal to the gravitational driving rate. The H=C curve is the locus of points where the viscous dissipation due to the mode triplet balances the neutrino cooling. FIG. 5: The (Ω̃, T8) initial values (region delimited by the solid line) that lead to equilibrium steady states and their corresponding final steady state values (region enclosed by the dashed line) are shown. Since both the initial and final values of T8 are low, these evolutions are roughly independent of Tc. rium. The timescale to reach an equilibrium steady state is tsteady ≈ 3.5× 106 yr for this set of parameters. Fig. 5 displays the possible initial values for the angu- lar velocity Ω̃ and temperature T8 of the star that lead to a balancing between the accreted angular momentum and the angular momentum emitted in gravitational waves. The fraction of the star that is above the threshold for direct URCA reactions and the slippage factor are varied within the corresponding “STEADY STATE” region of Fig. 1(b). The final equilibrium values are also displayed and cluster in a narrower region than the initial values. Because viscosity is so small in this regime, the values of Ω also tend to be small. Thus, although an interest- ing physical regime, this case is most likely not relevant to recycling by accretion to create pulsars with spin fre- quencies as large as 716 Hz. Note that a steady state can be achieved when Sns = 0. This is the probable end state of the problem first calculated by Levin [18]. The reason we do not find a cycle at low Sns is twofold: (1) the shear viscosity we are using is lower (shear viscosity in Ref. [18] is amplified by a factor of 244), and (2) the nonlinear couplings keep all mode amplitudes small. C. Thermal Runaway Evolutions We now consider evolutions in which the three-mode system is not sufficient to halt the thermal runaway. We observe two such scenarios. In the first scenario, the star is unable to reach thermal equilibrium. The run- away occurs on a period much shorter than the accretion timescale and so the whole evolution is at approximately constant angular frequency. In the second scenario, the star reaches a state of thermal equilibrium but the spin evolution does not reach a steady state. The star contin- ues to spin up by accretion until it climbs to the peak of the H = C curve, thermal equilibrium fails and a run- away occurs. 1. Fast Runaway A typical trajectory of a star that goes through a rapid thermal runaway is displayed in Fig. 6. This star has Sns = 0.25 and fdU = 0.058. Initially, the growth of the r-mode is halted by the two daughter modes once the lowest parametric instability threshold is crossed, and the three modes settle in the (Ω,T )-dependent quasi- stationary states of Eqs. (6). They viscously heat up the star until hyperon bulk viscosity becomes important for the daughter modes. As the amplitudes of the daughter modes decrease the coupling is no longer strong enough to drain enough energy to stop the growth of the r-mode. The daughter modes are completely damped and the r- mode increases exponentially. The system goes back to the one-mode evolution described by Ref. [24]. Fig. 6(a) and (b) compare both the temperature evolu- tion and the trajectory in the Ω̃−T8 plane of the star for a 0 0.5 1 1.5 2 2.5 3 3.5 τ [x 10 Full Amplitudes-Ω-T Evolution Steady State Evolution 2nd Parametric Instability Threshold 3 3.5 4 4.5 5 5.5 6 6.5 r-mode Stability Curve Full Amplitude-Ω-T Evolution Steady State Evolution 2nd Parametric Instability Threshold FIG. 6: This plot compares the full evolution resulting from solving Eqs. (4),(13),(18) with the reduced Ω − T evolution that assumes the amplitudes go through a series of steady states Eqs. (19)-(20) for a model with Tc = 5.0 × 10 fdU = 0.058 and Sns = 0.25. (a) The temperature is dis- played as a function of time for the two different methods. (b) The angular velocity Ω̃ = Ω/Ωc is shown as a function of temperature. The evolution occurs at constant spin fre- quency. It can be seen that the steady-state amplitude ap- proximation is extremely good. The ‘X’ shows the point at which the r-mode crosses its second lowest parametric insta- bility threshold, where additional dissipation would become operative. simulation solving the full set of equations to a simulation that assumes quasi-stationary solutions for the three am- plitudes and evolves only the angular velocity and tem- perature of the star. It can be seen that the steady state approximation is very good until the thermal runaway occurs. Afterward, the temperature evolution of the re- duced equations is offset slightly from the quasi-steady result and intersects the r-mode instability curve sooner. This evolution is similar to that described by Nayyar and Owen [24]. However, the r-mode crosses its second low- FIG. 7: The trajectory of a neutron star in the Ω̃ − T8 phase space is shown for a model with Tc = 5.0 × 10 fdU = 4.0 × 10 −5 and Sns = 0.02 that goes through a slow thermogravitational runaway. Portion a → b of the trajectory shows the mode triplet heating up the neutron star through boundary layer and shear viscosity until the system reaches thermal equilibrium. Segment b → c represents the accre- tional spin-up of the star in thermal equilibrium. The dotted- dashed line is the locus of points where the viscous dissipation of the mode triplet is equal to the neutrino cooling, and is la- beled as the H = C curve. The star reaches the maximum of this curve and fails to reach an equilibrium between the ac- cretion torque and gravitational emission. It then continues heating at constant angular velocity and crosses its second lowest parametric instability threshold, at which point more modes would need to be included to make the evolution accu- rate. Eventually the star reaches the r-mode stability curve again. est parametric instability much earlier in the evolution (see the ‘X’ in the figure), and at that point more modes need to be included to model the instability accurately. Thus, we cannot be sure that a runaway must occur in this case. We shall return to this issue in a subsequent paper. 2. Slow Runaway In this section we examine evolutions in which the neu- tron star has both a very small slippage factor, Sns . 0.03, and only a small percentage of the star is above the threshold for direct URCA reactions, fdU < 5 × 10−5. A trajectory for this kind of evolution is displayed in Fig. 7. After the star crosses the r-mode stability curve, the r-mode increases beyond the first parametric insta- bility threshold, and its growth is temporarily stopped by energy transfer to the daughter modes. As in the previous scenarios, the star is viscously heated by the mode triplet at constant Ω in part a → b of the trajec- tory on a timescale of about 5, 000 yr. At point b, it FIG. 8: The spin-down timescale is shown as slippage fac- tor Sns and fraction of the star subject to direct URCA fdU for cyclic evolutions are varied for a fixed hyperon critical temperature of Tc = 5.0 × 10 9 K. This timescale dominates the heat-up timescale and hence represents the time the star spends above the r-mode instability curve. It increases as the viscosity is lowered and the star gets closer to the steady state region. reaches thermal equilibrium. In leg b → c of the tra- jectory, the star continues its evolution by ascending the H = C curve and spinning up because of accretion for about 2 × 106 yr without finding an equilibrium state for the angular momentum evolution. Once it reaches the peak of the H = C curve, the cooling is no longer sufficient to stop the temperature from increasing expo- nentially and a thermal runaway occurs. The cross mark ‘X’ on the trajectory shows the point at which the r-mode amplitude crosses its second lowest parametric instabil- ity threshold. At this stage more inertial modes need to be included to model the rest of this evolution correctly. As for the cases that evolve to steady states, these long- timescale runaways tend to occur at low spin rates. V. PROBABILITY OF DETECTION Fig. 8 shows how the time the star spends above the r-mode stability curve changes when Sns and fdU are var- ied. For large enough values of Sns the boundary layer viscosity dominates. In this region of phase space the spin-down timescale can be approximated by tspin−down = dΩ̃ (32) FIG. 9: (a)The stability regions are plotted at fixed hyperon superfluidity temperature Tc = 6.5×10 9 K, while varying fdU and Sns. The steady state region remains roughly the same as in Fig. 1(b), the slow run-away region disappears, and the cycle region increases dramatically while shrinking the fast- runaway region. (b) The spin-down timescale is shown for the cyclic evolutions in part (a). Ĩτ0GR (4κ̃)2ω̃βω̃γ |δω̃|2 |C̄α|2 <Ω̃>6 ≈ 250 yr ∆νkHz <νkHz>7 M1.4R |cthα | where M1.4 = M/(1.4M⊙), R6 = R/(10 6cm), νkHz = ν/1kHz, Ĩ = 0.261 [17], the r-mode am- plitude at its parametric instability threshold |cthα | ≈ |δω̃|/(4κ̃ ω̃βω̃γ) ≈ 1.5×10−5, and C̄α = Ω̃|cα|/|cα|th. This approximation agrees with spin-up timescales ob- tained from our simulations to ∼ 25%. The maximum ν is approximately the same as the ini- tial frequency, and can be determined by equating the driving and damping rate of the r-mode, since it is on the r-mode stability curve νmax ≈ 800Hz M1.4R6 )4/11 . (33) Thus, the spin-down timescale is very sensitive to the slippage factor tspin−down ∝ S−24/11ns (∆νkHz/νkHz). The dependences on fdU and accretion rate Ṁ are much weaker; a rough approximation, obtained by matching direct URCA cooling and nuclear heat- ing, is T8 in ∝ Ṁ1/6f−1/6dU R 1.4 , and νmax ∝ dU Ṁ −1/33R −34/99 1.4 . The gravitational wave amplitude measured at distance d [46, 47] is h ≈ 1.6R τ0GRc Ω̃3|cα| (34) ≈ 3× 10−25 10kpc M1.4R Taking ν ≈ νmax gives h ∝ S12/11ns M −1/33 1.4 R dU Ṁ −1/11. (35) The maximum distance at which sources could be de- tected by Advanced LIGO interferometers, assuming hmin = 10 −27, [46] is dmax ≈ 3Mpc 10−27 M1.4R |cthα | ≈ 1.5Mpc 10−27 S12/11ns M −1/11 1.4 R 21/11 × T−6/118 |cthα | Eqs. (33) and (36) imply that gravitational radiation from the r-mode instability may only be detectable for sources in the Local Group of galaxies. Eq. (33) implies that for accretion to be able to spin up neutron stars to ν & 700 Hz, we must require (Sns/M1.4R6 T8in) 4/11 & 1. Assuming this to be true, dmax . 1-1.5 Mpc. However, tspin−down ≈ 1000 yr at most, making detection unlikely for any given source. Moreover, unless Sns can differ sub- stantially from one neutron star to another, only those with ν given by Eq. (33) can be r-mode unstable. Slower rotators, including almost all LMXBs, are still in their stable spin-up phases. Still more seriously, Fig. 1(b) shows that spin cycles are only possible for Sns . 0.50, assuming Tc ≈ 5.0× 109 K; Eq. (33) then implies ν . 450 Hz. This would restrict detectable gravitational radiation to galactic sources, al- though the duration of the unstable phase could be longer. Within the context of our three mode calculation, Sns > 0.50, which is needed for explaining the fastest pul- sars, would imply fast runaway. There are two possible resolutions to this problem. One is that including addi- tional modes prevents the runaway; we shall investigate this in subsequent papers. The second is that Tc is larger, or that neutron stars do not contain hyperons (e.g., be- cause they are insufficiently dense). Fig. 9(a) shows the same phase plane as Fig. 1(b) but with Tc = 6.5×109 K, and Fig. 9(b) shows the results for tspin−down analogous to Fig. 8. Larger Tc permits spin cycles for higher values of Sns (and hence ν), but the time spent in the unstable regime is shorter. VI. CONCLUSIONS In this paper, we model the nonlinear saturation of un- stable r-modes of accreting neutron stars using the triplet of modes formed from the n = 3,m = 2 r-mode and the the first two near resonant modes that become unstable (n = 13,m = −3 and n = 14,m = 1) by coupling to the r-mode. This is the first treatment of the spin and thermal evolution including the nonlinear saturation of the r-mode instability to provide a physical cutoff by en- ergy transfer to other modes in the system. The model includes neutrino cooling and shear, boundary layer and hyperon bulk viscosity. We allow for some uncertainties in neutron star physics that is not yet understood by varying the superfluid transition temperature, the slip- page factor that regulates the boundary layer viscosity, and the fraction of the star that is above the density threshold for direct URCA reactions. In all our evolu- tions we find that the mode amplitudes quickly settle into a series of quasi-stationary states that can be calcu- lated algebraically, and depend weakly on angular veloc- ity and temperature. The evolution continues along these sequences of quasi-steady states as long as the r-mode is in the unstable regime. The spin and temperature of the neutron star can follow several possible trajectories depending on interior physics. The first part of the evo- lution is the same for all types of trajectories: the star viscously heats up at constant angular velocity. If thermal equilibrium is reached, we find several pos- sible scenarios. The star may follow a cyclic evolution, and spin down and cool in thermal equilibrium until the r-mode enters the stable regime. It subsequently cools at constant Ω until it reaches the initial temperature. At this point the star starts spinning up by accretion until the r-mode becomes unstable again and the cycle is repeated. The time the star spends in the unstable regime is found to vary between a few hundred years (large Sns ∼ 1) and 106 yr (small Sns ∼ 0.05). Our cycles are different from those previously found by Ref. [18] in that our amplitudes remain small, ∼ 10−5, which slows the viscous heating and causes the star to spend more time in the regime where the r-mode instability is active. Furthermore, we find that the star stops heating when it reaches thermal equilibrium and not when the r- mode reaches a maximum value. The cycles we find are narrow with the spin frequency of the star changing less than 10% even in the case of high spin rates ∼ 750 Hz. Other possible trajectories are an evolution toward a full steady state in which the accretion torque balances the gravitational radiation emission, and a very slow thermo- gravitational runaway on a timescale of ∼ 106 yr. These scenarios occur for very low viscosity (Sns . 0.04). Al- though theoretically interesting, they do not allow for very fast rotators of ∼ 700 Hz. Alternatively, if the star does not reach thermal equi- librium, we find that it continues heating up at constant spin frequency until it enters a regime in which the r- mode is no longer unstable. This evolution is similar to that predicted by Nayyar and Owen [24]. However, the r-mode grows above its second parametric instability threshold fairly early in its evolution and at this point more inertial modes should be excited and the three- mode model becomes insufficient. Modeling this scenario accurately is subject of future work. We have focused on cases with Tc & 5× 109 K. These are cases for which the nonlinear effects are substantial. In this regime, hyperon bulk viscosity is not important except for thermal runaways where we expect other mode couplings, ignored here, to play important roles. Fast ro- tation requires large dissipation, as has long been recog- nized [18, 30] and these models can only achieve ν & 700 Hz if boundary layer viscosity is very large. Alterna- tively, at lower Tc . 3 × 109 K, large rotation rates can be achieved at r-mode amplitudes below the first para- metric instability threshold [1]. Nayyar and Owen found that increasing the mass of the star for the same equation of state makes the hyperon bulk viscosity become impor- tant at lower temperatures [24]. Conceivably, there are accreting neutron stars with relatively low masses that have lower central densities and small hyperon popula- tions. These could evolve as detailed here and only spin up to modest frequencies. Hyperons could be more im- portant in more massive neutron stars leading to larger spin rates and very small steady state r-mode amplitude as found by Wagoner [1]. Our models imply small r-mode amplitudes of ∼ 10−5 and therefore gravitational radiation detectable by ad- vanced LIGO interferometers only in the local group of galaxies up to a distance of a few Mpc. The r- mode instability puts a fairly stringent limit on the spin frequencies of accreting neutron stars of νmax ≈ 800Hz[Sns/(M1.4R6)] 4/11T −2/11 8 . In order to allow for fast rotators of & 700 Hz in our models a large bound- ary layer viscosity with (Sns/M1.4R6 T8in) 4/11 ∼ 1 is required. Slippage factors of order ∼ 1 lead to time peri- ods on which the r-mode is unstable with a timescale of at most 1000 yr, which is about 10−3 times shorter than the accretion timescale. This would mean that only about 1 in 1000 LMXBs in the galaxy are possible LIGO sources. However, lower slippage factors lead to a longer duration of the gravitational wave emission, but also lower fre- quencies. We also note that in this model we have con- sidered only very fast accretors with Ṁ ∼ 10−8M⊙yr−1 and most LMXBs in our galaxy accrete at slower rates. Investigations with more accurate nuclear heating models are a subject for future work. Our analysis could be made more realistic in several ways, such as by including the effects of magnetic fields, compressibility, multi-fluid composition [48], superfluid- ity, superconductivity, etc. These features would render the model more realistic, but its generic features ought to persist, since the upshot would still be a dense set of mode frequencies exhibiting three mode resonances and parametric instabilities with low threshold amplitudes. Although the behavior of the star would differ quanti- tatively in a model different from ours in detail, we ex- pect the qualitative behaviors we have found to be ro- bust, as they are well described by quasi-stationary mode evolutions whose slow variations are determined by com- petitions between dissipation and neutrino cooling, and accretion spin-up and gravitational radiation spin-down. In our model, it seems that three mode evolution involv- ing interactions of the r-mode with two daughters at the lowest parametric instability threshold is often sufficient to quench the instability. Our treatment is inadequate to follow what happens when the system runs away; for this, coupling to additional modes is essential. For this regime, a generalization of the work of Brink et al. [26, 27, 28] that includes accretion spin-up, viscous heating and neu- trino cooling would be needed. Such a calculation is formidable even in a “simple” model involving coupled inertial modes of an incompressible star. Acknowledgments It is a pleasure to thank Jeandrew Brink and Éanna Flanagan for useful discussions. RB would especially like to thank Jeandrew for useful discussions, encourage- ment and advice at the beginning of this project, with- out which the project would not have been started. RB is very grateful to Gregory Daues for steady encourage- ment and support for the duration of this project, and also to Gabrielle Allen and Ed Seidel. This research was funded by grants NSF AST-0307273, NSF AST-0606710 and NSF PHY-0354631. APPENDIX A This appendix will sketch the derivation of Eqs. (1) from the Lagrangian density. We follow closely Appendix A in Schenk et al., which contains the derivation of the equations of motion for constant Ω. The Lagrangian density as given by Eq. (A1) in Schenk et al. [29] is L = 1 ξ̇ · ξ̇ + 1 ξ̇ ·B · ξ − 1 ξ ·C · ξ + aext(t) · ξ, (A-1) where the operators B · ξ = 2Ω× ξ and ρ(C · ξ)i = −∇i(Γ1p∇jξj) +∇ip∇jξj + ρ∇iδφ (A-2) − ∇jp∇iξj + ρξj∇j∇iφ+ ρξj∇j∇iφrot with φrot = −(1/2)(Ω × x)2. We are interested in a situation where the uniform angular velocity of the star changes slowly on the timescale of the rotation period itself. In order to remove the time dependence we define the new displacement and time variables , dτ = Ωdt. (A-3) In terms of these new variables the Lagrangian density can be written as L̃ = 1 ξ̃′ · ξ̃′ + 1 ξ̃′ · (B̃ · ξ̃) + ( |ξ̃|2 (A-4) ξ̃ · C̃ · ξ̃ + aext(t) · ξ̃, where the primes denote derivatives with respect to τ , B̃ = Ω−1B and C̃ = Ω−2C. The momentum canonically conjugate to ξ̃ is = ξ̃′ + Ω̂× ξ̃. (A-5) The associated Hamiltonian density is B̃ · ξ̃ |ξ̃|2 + ξ̃ · C̃ · ξ̃ − · ξ̃. (A-6) Hamilton’s equations of motions can be written as ζ̃′ = T · ζ̃ + F(τ), (A-7) where the operator T is T = T0 + T1 with B̃2 − C̃ − 1 Ω)′′√ F(τ) = We assume solutions of the form ζ̃(τ,x) = eiω̃tζ̃(x). Spe- cializing to the case of no forcing term aext = 0 leads to the eigenvalue equation (T0 − iω̃)ζ̃(x) = 0. (A-8) Since the operator T0 is not Hermitian it will have dis- tinct right and left eigenvectors. Similar to Schenk et al. [29] we label the right eigenvectors of T as ζ̃A, and the associated eigenfrequencies as ω̃A = ωA/Ω, and the eigenvalue equation above becomes (T0 − iω̃A)ζ̃A(x) = 0. (A-9) The left eigenvectors χA satisfy 0 − iω̃⋆A)χ̃A = 0, (A-10) where B̃2 − C̃ For simplicity, in this appendix we specialize to the case of no Jordan chains when the set of right eigenvectors forms a complete basis. The orthonormality relation be- tween right and left eigenvectors is χ̃A, ζ̃B d3xρ(x)χ̃ A · ζ̃B = δAB. (A-11) We can expand ζ(τ,x) in this basis as ζ(τ,x) = CA(τ)ζA(x), (A-12) where the coefficients CA are given by the inverse of this mode expansion CA(τ) = χ̃A, ζ̃(τ,x) . (A-13) Using Eqs. (B-2,A-9,A-11) in Eq. (A-7) leads to the equa- tions of motion for the mode amplitudes C′A − iω̃ACA = g(τ) (A-14) + 〈χ̃A, F (τ)〉 , where g(τ) = ( Ω)′′/ Ω. Following Sec. IV of Schenk et al. [29] we replace the externally applied acceleration by the nonlinear acceleration given by Eq. (4.2) of Ref. [29]. The inner product can be written in terms of the displacement variable ξ̃. The left eigenvectors are χ̃A = where τ̃A can be chosen to be proportional to ξ̃A because they satisfy the same matrix equation. τ̃A = −iξ̃A/b̃A, (A-15) which corresponds to Eq. (A-45) in Schenk et al. [29] with the proportionality constant b̃A = Ω −1bA = MR2/ω̃A (also given by Eq. (2.36) of Ref. [29]). The equations of motion for the mode amplitudes be- C′A − iω̃ACA = ig(τ) d3xξ̃⋆A · ξ̃B(A-16) κ̃⋆ABCC where the nonlinear coupling κ̃ABC = κABC/(MR and κABC is explicitly give by Eq. (4.20) of Ref. [29]. The g(τ) integral mixes only modes with mA = mB because of the eimφ dependence of the displacement eigenvectors ξ̃. ( dφei(mA−mB)φ = 0 if mA 6= mB.) So, this term will be zero for our mode triplet. Also, in the case of a single mode triplet there is only one coupling and Eqs. (A-16) take the form of Eqs. (1). APPENDIX B In this appendix we study the behavior of the mode amplitudes and temperature near equilibrium assuming constant angular velocity. We are performing a first order expansion of Eqs. (5) and (18). Similar to Ref. [49], each of the five variables is expanded about its equilibrium (Xj)e as follows Xj(τ̃ ) = {|C̄α|, |C̄β |, |C̄γ |, φ, T8} = (Xj)e[1 + ζj(τ̃ )] (B-1) where the perturbation |ζj | << 1 and j = α, β, γ, T . The expansion leads to a first order differential equation for each ζj (γ̃α)e Ω̃|δω̃| ζα − ζβ − ζγ − ζφ (B-2) (γ̃β)e Ω̃|δω̃| ζα − ζβ + ζγ + (γ̃γ)e Ω̃|δω̃| ζα + ζβ − ζγ + φe tanφe γ̃α + γ̃β + γ̃γ Ω̃|δω̃| −γ̃α − γ̃β + γ̃γ Ω̃|δω̃| −γ̃α + γ̃β − γ̃γ Ω̃|δω̃| (γ̃α − γ̃β − γ̃γ)e Ω̃|δω̃| MR2Ω2c γ̃αγ̃β γ̃γ 2κ̃2ω̃αω̃βω̃γΩ̃|δω̃|C(Te)T8e tanφ2e γ̃α v ζα + ω̃βζβ + ω̃γζγ + T8e + ω̃β + ω̃γ ΩcΩ̃|δω̃|C(Te) where the equilibrium amplitudes |Cj |e have been written in terms of the corresponding driving and damping rates using Eqs. (6). Eq. (B-2) can be written in matrix form = Aijζi. (B-3) Let ζj ∝ exp(λτ̃ ). The determinant ||Aij − λδij || = 0 leads to the eigenvalue equation λ5 + a4λ 4 + a3λ 3 + a2λ 2 + a1λ+ a0 = 0. (B-4) The coefficients aj with j = 0, 4 are a4 = 2 tanφe = γ̃β + γ̃γ − γ̃α Ω̃|δω̃| , (B-5) tanφ2e γ̃2β + γ̃ γ + γ̃ (Ω̃|δω̃|)2 + tanφ2e − 1, γ̃αγ̃β γ̃γ (Ω̃|δω̃|)3 tanφ2e 4γ̃αγ̃β γ̃γ (Ω̃|δω̃|)3 tanφe + tanφ 2MR2Ω2c κ̃2ω̃αω̃βω̃γC(Te) (γ̃αγ̃β γ̃γ) (Ω̃|δω̃|)4 tanφe tanφ2e 4γ̃αγ̃β γ̃γ (Ω̃|δω̃|)3 tanφe Ω̃|δω̃|C(Te) The eigenvalues can be approximated as λ1,2 ≈ − − ǫ± i ǫ2 + w2 , (B-6) λ3,4 ≈ ǫ± iw, λ5 ≈ − where ǫ = (a2 − a3a4)/a4 and w = a1/a3. The system is unstable when a2 − a3a4 > 0 or a0 < 0. The first two eigenvalues will have a negative real part as long as γ̃β + γ̃γ > γ̃α. If the heating compensates the cooling of the star a0 ≈ 0 and becomes negative if the star can not reach thermal equilibrium. The other critical stability condition a2 − a3a4 = 0 can be written as Ω̃|δω̃| [1+Γβ+Γγ−(Γ2β+Γ2γ)−(Γβ−Γγ)2(Γβ+Γγ)] = 0, (B-7) where Γβ = γβ/γα and Γγ = γγ/γα. Note that we have ignored the smaller terms of orderO([γ̃α/(Ω̃|δω̃|)]5). 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0704.0800

Quantum Auctions

Quantum Auctions Tad Hogg HP Labs Palo Alto, CA Pavithra Harsha Cambridge, MA Kay-Yut Chen HP Labs Palo Alto, CA October 28, 2018 Abstract We present a quantum auction protocol using superpositions to rep- resent bids and distributed search to identify the winner(s). Measuring the final quantum state gives the auction outcome while simultane- ously destroying the superposition. Thus non-winning bids are never revealed. Participants can use entanglement to arrange for correla- tions among their bids, with the assurance that this entanglement is not observable by others. The protocol is useful for information hiding applications, such as partnership bidding with allocative externality or concerns about revealing bidding preferences. The protocol applies to a variety of auction types, e.g., first or second price, and to auctions involving either a single item or arbitrary bundles of items (i.e., com- binatorial auctions). We analyze the game-theoretical behavior of the quantum protocol for the simple case of a sealed-bid quantum, and show how a suitably designed adiabatic search reduces the possibilities for bidders to game the auction. This design illustrates how incen- tive rather that computational constraints affect quantum algorithm choices. http://arxiv.org/abs/0704.0800v1 1 Introduction Quantum information processing [23] offers potential improvements in a va- riety of applications. Computational advantages [26, 14] of quantum com- puters with many qubits have received the most attention but are difficult to implement physically. On the other hand, technology for manipulating and communicating just a few qubits could be sufficient to create new economic mechanisms by altering the information security and strategic incentives of the underlying game. Examples of quantum mechanisms include the prisoner’s dilemma [10, 11, 7, 8], coordination [17, 21] and public goods provisioning [3]. In partic- ular, a quantum mechanism can significantly reduce the free-rider problem without a third-party enforcer or repeated interactions, both in theory and practice [2]. In this paper, we examine quantum mechanisms for another economic scenario: resource allocation by auction [28]. While traditional auction mechanisms can efficiently allocate resources in many cases, quantum auc- tion protocols offer improvements in preserving privacy of the losing bids and dealing with scenarios in which bidders care about what other bidders win when multiple items are auctioned. Specifically, using quantum super- positions to represent bids prevents the auctioneer and other bidders from viewing the bids during the auction without disrupting the auction process. Furthermore, the auction result reveals nothing but the winning bid and allocation. The first part of the paper introduces a general quantum auction protocol for various pricing and allocation rules, multiple unit auctions, combinatorial auctions and partnership bids. For simplicity, we focus on the sealed-bid first-price auction. In this auction, each bidder has one opportunity to submit a bid. The winner is the highest bidder, who pays the amount bid for the item. This auction has been well studied both theoretically [28] and experimentally [5, 4], and contrasts with iterative auctions in which bidders can incrementally increase their bids depending on how others bid. If the auction is not well-matched to the bidders preferences, it can intro- duce perverse incentives and result in poor outcomes, such as lost revenue for the seller or economically inefficient allocations where items are not al- located to those who value them most. Thus it is important to examine incentives introduced with a proposed auction design. In particular, our auction protocol involves quantum search, which introduces incentive issues beyond those examined in prior quantum games [11]. A full analysis of incentive issues is complicated, even for classical auc- tions. In this paper we focus on two incentive issues arising from the quan- tum auction protocol. The first incentive issue arises from the possibility of manipulating the search outcome by altering amplitudes associated with different bids. We show how to revise an adiabatic search method to correct this incentive problem, thereby preserving the classical Nash equilibrium. From a quantum algorithm perspective, this construction of the search il- lustrates how incentive issues affect algorithm design, in contrast to the more common concern with computational efficiency in quantum informa- tion processing. Second, the quantum search for the highest bid is probabilistic, i.e., does not always return the highest bid. While the probability of finding the correct answer can be made as high as one wishes by using more iterations of the search, the small residue probability of awarding the item to someone other than the highest bidder may change bidding behavior. As a step toward addressing the effect of probabilistic outcomes, we show that, with sufficient steps in the quantum search, altering choices from those of the corresponding deterministic auction gives at most a small improvement for that bidder. The paper is organized as follows. Sec. 2 describes the quantum auction and the bidding language encoding bids in quantum states. Sec. 3 describes the quantum search method to find the maximum bid. After these sections describing the auction protocol, in Sec. 4 we turn to strategic issues raised by the quantum nature of the auction beyond those in the corresponding classical auctions. Then, in Sec. 5 we give a game theory analysis of some of these strategic possibilities and describe how simple modifications of the quantum search improves the auction outcome, in theory. Sec. 6 generalizes the results to auctions of multiple items, including combinatorial auctions. Sec. 7 describes scenarios for which the quantum protocol offers likely eco- nomic advantages in terms of information security and ability to compactly express complex dependencies among items and bidders. Finally, Sec. 8 sum- marizes the quantum auction protocol and highlights a number of remaining economic questions. 2 Quantum Auction Protocol In our auction protocol, each bidder selects an operator that produces the de- sired bid from a prespecified initial state. The auctioneer repeatedly asks the bidders to apply their individual operators in a distributed implementation of a quantum search to find the winning bid. More specifically, the quantum auction protocol for sealed-bid auctions involves the following steps: 1. Auctioneer announces conventional aspects of the auction: type of auction (e.g., first or second price and any reservation prices), the good(s) for sale, the allowed price granularity (e.g., if bids can specify values to the penny, or only to the dollar), and the criterion used to determine the winner(s), e.g., maximizing revenue for the seller 2. Auctioneer announces how quantum states will be interpreted, i.e., as specifying a price if only one good is for sale, or a combination of price and a set of goods if combinations are for sale; and also announces the initial quantum state. This state uses p qubits for each bidder. Auctioneer announces the quantum search procedure. 3. Each bidder selects an operator on p qubits. Bidders keep their choice of operator private. 4. Auctioneer produces a set of particles implementing p qubits for each bidder, initializing the set to the announced initial state. 5. Auctioneer and bidders perform a distributed search for the winner Fig. 1 illustrates this procedure for two bidders and repeating the steps of the search twice. Realistic search involves a larger number of steps. In contrast with other quantum games, e.g., public goods, that involve just one round of interaction, the search required to identify the winners involves multiple rounds of interaction among the participants. The required number of iterations depends on the search method. In practice, the auctioneer could pick the number of iterations based on prior experience with similar auctions, or from simulating several test cases using valuations randomly drawn from a plausible distribution of values for the auction items. Alternatively, the auctioneer could repeat the procedure several times (possibly with steps from each repetition interleaved in a random order) and use the best result from these repetitions. auctioneer auctioneer auctioneer bidder 1 bidder 2 bidder 1 bidder 2 start measure state announce result Figure 1: Schematic diagram of distributed search procedure, showing re- peated interactions between auctioneer and bidders, in this case two bidders and two steps of the distributed search. number of bidders n number of items in auction m number of qubits per bidder p state of qubits for bidder j ψj state of all qubits Ψ = ψ1 ⊗ . . .⊗ ψn Table 1: Notation for the quantum auction. This auction protocol uses a distributed search so bidders’ operator choices remain private. Specifically, the search operation requiring input from the bidders is applied locally by each bidder, giving the overall opera- U = U1 ⊗ U2 ⊗ . . .⊗ Un (1) where n is the number of bidders and Ui the operator of bidder i. 3 Quantum Auction Implementation A quantum auction requires finding the winning bid and corresponding bid- der. This procedure has two components: the interpretation of the qubits as bids, and the search procedure to find the winner. The following two sub- sections discuss these components in the context of a single-item auction. Sec. 6 generalizes this discussion to multiple items. 3.1 Creation and interpretation of quantum bids We define a bid as the amount a bidder indicates he is willing to pay for the item. An allocation is a list of bids, one from each bidder. The quan- tum auction protocol manipulates superpositions of allocations. We use an allocation rule to indicate how allocations specify a winner and amount paid. Example 1. Consider an auction of one item with three bidders, willing to pay $1, $3 and $10 for the item, respectively. We represent these bids as |$1〉, |$3〉 and |$10〉, and the corresponding allocation as the product of these states, i.e., |$1, $3, $10〉 with the ordering in the allocation understood to correspond to the bidders. A simple allocation rule selects the highest bidder as the winner, who pays the high bid. In this example, this rule results in the third bidder winning, and paying $10 for the item. Each bidder gets p qubits and can only operate on those bits. Thus each bidder has 2p possible bid values, and can create superpositions of these values. A superposition of bids specifies set of distinct bids, with at most one allowed to win. The amplitudes of the superposition affect the likelihood of various outcomes for the auction. For a single-item auction, a bidder will typically have only one bid. As discussed below, more complicated superpositions are useful for information hiding. Specifically, bidder j selects an operator Uj on p qubits to apply to the initial state for that bidder’s qubits ψinit specified by the auctioneer. The resulting state, ψj = Ujψinit, is a superposition of bids, each of the form where b i is bidder j’s bid for the item. The subscript i indicates one of the possible bids that can be specified with p qubits according to the announced interpretation of the bits. We define the subspace used by bidder j as the set of states spanned by the basis eigenvectors in ψj . Only these basis vectors appear in allocations relevant for the search. As bidders apply their operators during the search, the superposition of allocations remains within the subspace of each bidder. In this case, where each bidder applies an operator only to their own qubits, the superposition of allocations is always a factored form, i.e., Ψ = ψ1 ⊗ . . . ⊗ ψn. More generally, groups of bidders could operate jointly on their qubits, entangling their bids in the allocations as discussed in Sec. 7. To exploit information hiding properties of superpositions, the state re- vealed at the end of the search should specify only the bidder who wins the item and the corresponding bid. To achieve this, instead of a direct repre- sentation of bids, we interpret bids formed from the p qubits available to a bidder as containing a special null value, ∅, indicating a bid for nothing. This null bid has additional benefits in multiple item settings, as discussed in Sec. 6 and Sec. 7. Example 2. Consider bidder j with two qubits and the initial state ψinit = |00〉 corresponding to the vector (1, 0, 0, 0), which is interpreted as the null bid. The other bid states are |01〉, |10〉 and |11〉 corresponding to vectors (0,1,0,0), (0,0,1,0) and (0,0,0,1). These three states are interpreted as three bid values in some preannounced way, e.g., $1, $2 and $3, respectively. The operator 1 0 1 0 0 1 0 1 1 0 −1 0 0 1 0 −1 gives the initial state ψj = Ujψinit as (|00〉+|10〉)/ 2 and specifies the search subspace whose basis is the first and third columns of Uj in this example. Thus the possible allocations involve only |00〉 and |10〉 for this bidder, cor- responding to the null bid and a bid of $2, respectively. In the presence of a null bid, we consider an allocation to be a feasible if it contains exactly one bid not equal to ∅. The corresponding allocation rule assigns no winner to infeasible allocations and, for feasible allocations, the winner is the single bidder in the allocation whose bid is not ∅, and he pays the amount bid. This allocation rule corresponds to a first-price single- item auction, except there can be no winner, analogous to the situation in auctions with a reservation price when no bidder exceeds that price. 3.2 Distributed Search The auctioneer must find the best state according to an announced crite- rion, e.g., maximum revenue. Specifically, the auctioneer has a evaluation function F assigning a quality value to each allocation. The function F assigns a lower value to infeasible allocations than to any feasible one. An example is F equal to the revenue produced by the allocation (if feasible) and otherwise is −1. The auctioneer uses quantum search to find the allocation in the subspace selected by the bidders giving the maximum value for F (e.g., a feasible allocation giving the most revenue to the auctioneer). This could be done via repeated uses of a decision-problem quantum search [14, 1] as a subroutine within a search for the minimum threshold value of F giving a solution to the decision problem, e.g., with a classical binary search on threshold values or using results of prior iterations of the decision problem [9]. Alternatively, we could use a method giving the maximum value directly (e.g., adiabatic [12] if run for a sufficiently long time or heuristic methods [15, 16] based on some prior knowledge of the distribution of bidders values). For definiteness, we focus on the adiabatic method. The adiabatic search is conventionally described as searching for the minimum cost state. We use this convention by defining a state’s cost to be the negative of the evaluation function F . The adiabatic search procedure, if run sufficiently slowly, changes the initial superposition into a final super- position in such a way that the amplitude in each initial eigenstate maps to the same amplitude in the corresponding final eigenstate, up to a phase factor (for nondegenerate eigenstates). We refer to this mapping of initial to final eigenstates as a perfect search. In practice, with a finite time for the search, there will be some transfer of amplitude among the eigenstates so the search will not be perfect in the sense defined here. Instead the auc- tion outcome is probabilistic: the auction will not always produce the best outcome when starting from the ground state. For example, an auction in- tending to find the highest bid could sometimes produce the second highest bid instead. Conventionally, the search operations are chosen so the uniform superposition is the lowest cost initial eigenstate. In our case, bidders are free to choose their operators and need not create uniform superpositions. A discrete implementation of adiabatic search consists of the following steps: • The auctioneer selects a number of search steps S and parameter ∆. These need not be announced to the bidders. • The auctioneer initializes the state of all np qubits to Ψinit = ψinit ⊗ . . . ⊗ ψinit = |0, . . . , 0〉, with n factors of ψinit in the product, and ψinit = |0〉 is the initial state for the p qubits for a single bidder. • The auctioneer sends these initialized qubits to the bidders who use their individual operators and then return the qubits to the auctioneer, jointly creating the state Ψ0 = UΨinit (3) • For s = 1, . . . , S, the auctioneer and bidders update the state to Ψs = UD(f)U †P (f)Ψs−1 (4) with f = s/S the fraction of steps completed. The bid operator U and its adjoint U † are performed by sending bits to the bidders as described in Sec. 2. The diagonal matrices D(f) and P (f) are described below. • The auctioneer measures the state ΨS, resulting in specific values for all the bits, from which the winner and prices are determined by the allocation rule described in Sec. 3.1. The diagonal matrix P (f) adjusts the phases of the amplitudes according to the cost associated with each allocation. In particular, using the cost c(x) = −F (x) for allocation |x〉, we have Pxx(f) = exp (−ifc(x)∆) (5) Similarly, the diagonal matrix D(f) adjusts amplitude phases as defined by a function d(x): Dxx(f) = exp (−i(1− f)d(x)∆) (6) The key property of d(x) is assigning the smallest value, e.g., 0, to |0〉, thereby making the first column of U the ground state eigenvector. Aside from this key property, the choice of d(x) is somewhat arbitrary. The con- ventional choice in the adiabatic method uses the Hamming weight of the state, i.e., d(x) equal to the number of 1 bits in the binary representation of x. However, as described in Sec. 5, other choices for d(x) can improve the incentive properties of the auction. The discrete-step implementation of the continuous adiabatic method [12] involves the limits ∆ → 0 and S∆ → ∞, in which case the final state ψS has high probability to be the lowest cost state. In practice, this outcome can often be achieved with considerably fewer steps using a fixed value of ∆, corresponding to a discrete version of the adiabatic method [16]. 4 Strategies with Quantum Operators Ideally, an auction achieves the economic objective of its design (e.g. maxi- mum revenue for the seller). In practice, an auction design may not provide incentives for participants to behave so as to achieve this objective. Usu- ally auction designs are examined under the assumption of self-interested rational participants. In conventional auctions, strategic issues include mis- representation of the true value, collusion among bidders and false name bidding (where a single bidder submits bids under several aliases). Some of these issues can be addressed with suitable auction rules, e.g., second price auctions encourage truthful reporting of values. Developing suitable designs of classical auctions in a wide range of economic contexts remains a challenging problem [28]. Quantum auctions raise strategic issues beyond those of classical auc- tions. In our case, every step of the adiabatic search requires each bidder to perform an operation on their qubits. Ideally, the bidder should use the same operator U for creating ψinit as in every step of the search in Eq. (4). In addition, bidders should include the null bid in their subspaces. In the clas- sical first-price sealed-bid auction, the bidder makes one choice: the amount to bid. In our quantum setting, this choice amounts to selecting the sub- space to use with the quantum search. The remaining freedom to select U , and possibly a different U for each step in the search, are additional choices provided by the quantum auction. Bidders may be tempted to exploit the flexibility of choosing operators in several general ways. First, they could use a subspace not including the null bid. Second, they could use a different operator for creating ψinit than they use in the rest of the search, thereby producing an altered initial amplitude that is not the ground state eigenvector. Third they could change operators during the search. If any such changes give significant probability for low bids to win, bidders would be tempted to make such changes and include a low bid in their subspace, hoping to profit significantly by winning the auction with a low bid. The remainder of this section describes some strategic issues unique to quantum auctions and possible solutions. We further discuss a game theory analysis of some of these issues in Sec. 5. 4.1 Selecting the Subspace The use of the null bid in our protocol raises the strategic issue illustrated in the following example: Example 3. Consider an auction of a single item with two bidders Alice and Bob. Using operators producing uniform amplitudes for the sake of illustration, they ought to apply operators that create (|∅〉+ |bA〉) and 1√ (|∅〉+ |bB〉) respectively, where bA and bB are their desired bids. The initial superposition for all the qubits is the product of these individual superpositions, i.e., Ψ0 is (|∅,∅〉+ |bA,∅〉+ |∅, bB〉+ |bA, bB〉) If bidders use these same operators during the search, the search algorithm finds the highest revenue allocation, i.e., giving the item to the highest bid- der. Suppose instead Bob picks an operator with a one-dimensional subspace, producing an initial state |bB〉 rather than including ∅. The product super- position is then (|∅, bB〉+ |bA, bB〉) Since the search remains in this subspace and the second allocation is infea- sible, the search will return |∅, bB〉 no matter what Alice bids. Thus Bob always wins the item, and can win using the lowest possible bid. This example shows bidders have an incentive to exclude the null set from their subspace. If all bidders make this choice, there will be no feasible allocations in the joint subspace and the auction will always give no winner. For auctions with more than two bidders, selecting subspaces excluding ∅ is a weak Nash Equilibrium for the quantum auction because any other choice by a single bidder still results in no feasible allocations. 4.2 Altering Initial Amplitudes Strategic choices for bidders also arise from the search procedure itself, even when using the correct subspace consisting of ∅ and the desired bid. In particular, the probabilistic outcome of the search means the optimal bid according to the auction criterion (e.g., highest revenue) will not always win. For the adiabatic search method, bidders could try to arrange for especially tiny eigenvalue gaps between the state corresponding to the best outcome and another state allowing them to win with a low bid. A sufficiently small gap could make the number of steps the auctioneer selects insufficient to give the optimal state with high probability and instead give a significant chance of producing the more favorable outcome. However, because the eigenvalues are a complicated function of the operators of all bidders, and individual bidders do not know the choices made by others, it will be difficult for a bidder to determine how to make such especially small gaps and do so in a way that gives a favorable outcome. Nevertheless, even fairly small proba- bilities for not finding the optimal state could alter the strategic behavior of the bidders. A more direct way a bidder can arrange for a low bid to win is by altering the initial state of the adiabatic search to start not in the ground state but in an eigenvector corresponding to one of the first few eigenvalues above the ground state. The adiabatic search takes such eigenvectors, with high probability, to an outcome in which a bid lower than the highest wins. While a single bidder cannot create an arbitrary initial condition, one bidder can ensure that it is not the ground state. For example, a bidder could chose an operator that gives a nonuniform amplitude for the initial state, in particular (|∅〉−|bA〉)/ 2, while using the uniform state (|∅〉+|bA〉)/ 2 as the ground state through the remainder of the search in Eq. (4). This can result in significant probability for a low bid to win, and so a bidder is tempted to deviate from the nominal operator choice. Fig. 2 illustrates this behavior. Instead of starting in the ground state, 0 0.2 0.4 0.6 0.8 1 high bid wins low bid wins infeasible Figure 2: Correspondence between initial basis and the possible allocations for a single item auction with two bidders in the standard adiabatic search. During the search, as f increases from 0 to 1, the eigenvalues of the four states change as shown schematically in the figure. The states for f = 0 correspond to both bidders starting with the ground state, |00〉, the two states obtained if one of the bidders starts with a different superposition, |01〉 and |10〉 (“single-bidder deviation states”), and the state of both bidders starting with different superpositions, |11〉 (“2-bidder deviation state”). the bidder’s choice gives the initial state as a linear combination of the ground state and the single-deviation state for that bidder, denoted as |01〉 or |10〉 for the two bidders in Fig. 2. Here a “single deviation” state is one that a single bidder can create, i.e., by operating on just the qubits available to that bidder. The adiabatic search splits the degeneracy, thereby giving some probability for the lowest bid to win and some probability for an infeasible allocation. More generally, bidder i uses this strategy by selecting two different operators U initi and Ui to use for forming the initial state and during the search, respectively. These choices result in different joint operators, in Eq. (1), used in Eq. (3) and (4). As with selecting a subspace without ∅, if many or all bidders make this choice, the initial state will have significant amplitude in eigenvectors corresponding to large eigenvalues, which produce infeasible outcomes and 0 0.2 0.4 0.6 0.8 1 high bid wins low bid wins infeasible Figure 3: Correspondence between the initial basis and the possible alloca- tions for a single item auction with two bidders in the search with permuted initial eigenvalues. hence a high probability for no winner. Thus with standard adiabatic search, if everyone uses the same operator for both initialization and search, then each bidder is tempted to use a different initialization operator and bid low, gaining a chance to win with a low bid. However, if multiple bidders attempt this, the outcome will most likely be an infeasible state, with no winner. We can address this problem by reordering the eigenvalues given by the d(x) function in Eq. (6) so that any change in initial operator by a single bidder increases probability of infeasible allocation but not the probability of any feasible allocation with a bid lower than the highest bid. This is possible because bidders only have access to their own bits, so can only form initial superpositions from a limited set of basis vectors. Fig. 3 illustrates the resulting situation. We give an analysis of this approach in Sec. 5.2. 4.3 Changing Operator During Search The distributed search of Eq. (4) has each bidder using the same operator for every step of the search. Thus bidders may gain some advantage by altering their operator during the steps of the search. Gradually changing the operator during the search amounts to a different path from initial to final Hamiltonian during the adiabatic search. Thus, provided the auction- eer uses enough steps, such changes will have at most a minor effect on the outcome probabilities unless the bidder can arrange for particularly small eigenvalue gaps among favorable states. Such arrangement is difficult, par- ticularly since the bidder does not know the choices of other bidders and the auctioneer could treat the bits from the bidders in an arbitrary, unan- nounced order. More significant changes in outcome is possible with sudden, large changes in the operator during search. Since the use of bidders operators gradually decreases during the search (i.e., Dxx(f) given in Eq. (6) approaches the identity operator as f approaches 1), the most problematic situation is for an abrupt change in operator at the beginning of the search. After such a change, the adiabatic search continues its gradual change of states, but now instead of starting in the ground state, it will instead have a linear combination of various states obtained by mapping the original basis onto the basis after the change. 5 Quantum Auction Design In this section, we focus on mechanism design to reduce incentive issues arising from the quantum aspects of the auction. We analyze incentive is- sues with the Nash equilibrium (NE) concept commonly used to evaluate auctions [28]. A given set of behaviors for the bidders is an equilibrium if no single bidder can gain an advantage (i.e., higher expected payoff) by switching to another behavior. Specifically, Sec. 5.1 describes an approach to encouraging bidders to include the null set in their bids. In Sec. 5.2 we show that using the ground state eigenvector is a NE provided bidders do not change the operators during the search. Sec. 5.3 then discusses how the auctioneer can discourage bidders from changing operators. Sec. 5.4 de- scribes how the auction can be made symmetric across the different bidders. We focus on single-item auctions in this section, but the ideas extend to quantum combinatorial auctions, as described in Sec. 6. 5.1 Checking for the Null Set One approach to the incentive to exclude the null set, described in Sec. 4.1, is for the auctioneer to perform a second search: for the allocation with the most ∅ values. This search uses the same distributed protocol of Eq. (4) but with separate qubits and a different cost function to define P (f), i.e., setting c(x) to the number of non-∅ values in the allocation x. Interleaving the additional search in a random order within the steps of the search for the winning bid prevents bidders from knowing which search a given step belongs to. So bidders could not consistently select different operators for the two searches. If all bidders include ∅ in their selected subspace, this additional search returns |∅,∅, . . .〉. Any bidder found not to have included ∅ could be ex- cluded from winning the auction. At this point the auctioneer could either announce there is no winner, or restart the auction for the remaining bid- ders without announcing this restart. The adiabatic search has a small but nonzero probability of returning the wrong result, which would then incor- rectly conclude some bidder did not include ∅. As long as the probability of such errors is smaller than the error probability of the search for the winner, these errors should not greatly affect the incentive structure of the mech- anism. Alternatively, the auctioneer could use a search completing with probability one in a finite number of steps, i.e., with different choices of D and P in Eq. (4), the auctioneer could implement Grover’s algorithm [14] to search for the allocation |∅,∅, . . .〉 in the joint subspace of the bidders. Since the auctioneer does not know the size of the subspaces selected by the bidders, the auctioneer would need to try various numbers of steps [1] before concluding |∅,∅, . . .〉 is not in the selected subspaces. Unlike the adiabatic search, failure would only indicate some bidder had not included ∅, but not which one. Thus the auctioneer’s only alternative in this case is to announce the auction has no winner. While this approach removes the immediate benefit of not including the null bid, its affect on broader strategic issues in the full auction is an open question. 5.2 The First-Price Sealed-Bid Auction In this section we examine the incentive structure of the auction with per- muted eigenstates described in Sec. 4.2. We first review how game the- ory applies to auctions. We then consider the quantum auction when the search runs long enough to give successful completion almost always (“per- fect search”). Finally, we consider the more realistic case of search with small, but not negligible, probability for non-optimal outcomes. 5.2.1 A Game Theory Approach to Auctions Game theory is a common approach to evaluating auctions [20, 28]. Con- sider n people bidding for an item, with person i having value vi for the item. Unlike discrete choice games, such as the prisoner’s dilemma, a strat- egy for a private value auction involves a bidding function b(v), mapping a bidder’s value to a corresponding bid. Theoretical analysis of auctions usually involves identifying a NE strategy, if any. This is a strategy for all players such that no bidder gains by changing this strategy given everyone else is using it. This focus on possible changes by a single bidder assumes bidders do not collude. A primary issue for auction behavior is how much participants know about other bidders’ values. Such knowledge can affect the choice of bid. The most popular model of such knowledge is independent private values, where the vi are independently drawn from the same distribution. Each bidder knows his own value, but not the values of other bidders. However, the distribution from which values come is common knowledge, i.e., known to all bidders, each bidder knows the others know this fact, and so on. A final ingredient for the analysis is an assumption of bidders’ goals. For illustration, we use the common assumption that bidders are risk neutral expected utility maximizers, and within the context of the auction, utility is proportional to profit. We illustrate this approach for a first-price sealed-bid auction, in which each bidder submits a single bid without seeing any of the other bids. This corresponds to the auction scenario considered in this paper. The bidder with the highest bid gets the item and pays the amount of his bid. Thus if bidder i bids bi, his profit is vi−bi if he wins the auction and zero otherwise. To avoid possibly losing money, bidders should ensure bi ≤ vi, and bids are required to be nonnegative. In the symmetric case where bidders’ values all come from the same distribution, a NE is a bidding function b(v). A bidder’s expected payoff is (v − b(v))P (b) where v is his value, b is his bidding function and P (b) is the probability of winning if he is using b(v) (which is also the function others use in equilibrium). Let F be the cumulative distribution of values, i.e., probability a value is at most v, and n be the number of bidders. The equilibrium condition leads to a differential equation satisfied by b(v) [28]. As a simple example, when v is uniformly distributed between 0 and 1, F (v) = v and the NE is b(v) = (n − 1)v/n. Thus, in the equilibrium strategy, a bidder bids somewhat less than his value and the bid gets closer to the value when there is more competition, i.e., larger n. If bidders have differing value distributions, a NE involves a set of bidding functions, {bi(v)}. An auction may have multiple equilibria. 5.2.2 Behavior with Perfect Search With perfect search and non-colluding bidders, if bidders use the same opera- tors for every step of the search, including initialization, and pick a subspace with the null bid then the adiabatic search described in Sec. 3.2 finds the highest revenue state. We now show that the auctioneer can choose eigenval- ues for the search so that bidders have no incentive to create an initial state different from the ground state. This choice corresponds to the auctioneer selecting an appropriate function d(x) in Eq. (6). Suppose bidder i uses operator Ui, giving the overall operator U with Eq. (1). Suppose all bidders except bidder 1 use the same operator to create the initial state as they use for the subsequent search. But bidder 1 uses two operators: U init1 to form the initial state and U1 for the search. Thus the initial state produced by bidder 1, ψ1 = U 1 ψinit, i.e., the first column of U init1 , is not necessarily equal to the first column of U1 that bidder 1 uses for the subsequent search. Instead, ψ1 may have contributions from all columns of U1, i.e., αi |i〉 (7) where |i〉 corresponds to column i, ranging from 0 to 2p−1, of U1. Combining with the initial state of all other bidders, Eq. (3) gives Ψ0 = i αi |i, 0, . . . , 0〉, instead of the initial ground state |0, 0, . . . , 0〉. Significantly, because a bidder can only operate on the p qubits from the auctioneer and not on any of the qubits sent to other bidders, a single bidder can only create a limited set of “single-deviation” initial states. In the case of bidder 1, these states all have the form |i, 0, . . . , 0〉. Similarly, if bidder j is the one using different initial and search operators, the states all have the form |. . . , 0, i, 0, . . .〉, where only the jth position can be nonzero. Thus, among the 2np basis states in the full search space, aside from the correct ground state, only n(2p−1) are possible states some single bidder can create when all other bidders use the same operator for initialization and search. More generally, k bidders can create superpositions of (2p − 1)k basis states in which none of them use the ground state initially, by selecting different operators for initialization and search. Thus there are (2p − 1)k (8) k-deviation states that some set of k bidders can create, while the other n− k bidders use the ground state. Our formulation has n(2p−1) feasible allocations, i.e., situations in which exactly one of the bidders has a non-∅ bid while all other bidders have ∅. To see this, each of the n bidders could have the non-∅ bid, and this bid could have any of 2p − 1 values (since the remaining value for the bidder’s bits represents ∅). The remaining n− 1 bidders have only one choice each, i.e., ∅. Suppose the auctioneer selects d(x) such that d(|0, . . . , 0〉) = 0 is the lowest eigenvalue and d(x) for all single-deviation states x is the largest value, with intermediate values for all other states. Provided the number of infeasible allocations is at least equal to the number of single-deviation states, a perfect search will then map every single-deviation state to an infeasible allocation, resulting in no winner for the auction. This condition amounts to 2np − n(2p − 1) ≥ n(2p − 1) (9) The following claim shows that Eq. (9) always is true in an auction scenario. Claim 1. Eq. (9) is true for all integers n, p ≥ 1 Proof. When p = 1, Eq. (9) reduces to 2n−1 ≥ n, which is true for all n ≥ 1. We prove a stronger condition for p ≥ 2, namely there are enough in- feasible states to handle up to n − 1 bidders deviating. Using Eq. (8), this stronger condition is 2np − n(2p − 1) ≥ (2p − 1) = 2np − 1− (2p − 1)n (10) with the k = 1 term in the sum corresponding to the right-hand side of Eq. (9). Writing x ≡ 2p − 1, Eq. (10) becomes f(x, n) ≡ xn − nx+ 1 ≥ 0. Since p ≥ 2, we have x ≥ 3. For this range of x and for n ≥ 1, f(x, n) is monotonically increasing in both arguments. To see f is monotonic for x, the derivative of f(x, n) with respect to x is n(xn−1 − 1) which is nonnegative since n ≥ 1 and x > 1. Similarly, the derivative with respect to n is x(xn−1 ln(x) − 1) which is at least 3(ln(3) − 1) > 0 since n ≥ 1 and x ≥ 3. Thus for the relevant range of n and x, f(x, n) ≥ f(3, 1) = 1 so Eq. (10) is true for all n ≥ 1 and p ≥ 2. Combining these cases for p = 1 and p ≥ 2 establishes the claim. Using this claim, we demonstrate the permuted eigenvalue choices re- move incentives to alter the initial amplitudes: Theorem 1. If (a) auctioneer chooses eigenvalues as described above, (b) {b∗i (v)}ni=1 is an equilibrium for the first-price classical auction, and (c) bid- ders include the null set as part of their bids and use the same operator in each step in the search except, possibly, for the initial state, then the strategy of using bidding functions {b∗i (v)}ni=1 and the same operator for their initial state as they use in the search is a NE for corresponding quantum auction. Proof. Without loss of generality, suppose only bidder 1 deviates and all the other bidders use {b∗i (v)}ni=2 and the same operator for initialization and search. Then, as described above, the initial state Ψ0 is i αi |i, 0, . . . , 0〉 for some choice of amplitudes αi, with i ranging from 0 to 2 p − 1. A perfect adiabatic search maps each of these states to a corresponding allocation. In particular, with d(|0, . . . , 0〉) having the smallest value of the function d(x), the lowest cost allocation is produced with probability |α0|2. This allocation corresponds to the highest bid winning. Moreover, each |i, 0, . . . , 0〉 with i 6= 0 has the largest value of d(x), and so, because of Eq. (9), maps to an infeasible allocation, giving no winner and hence no value to bidder 1. Hence the expected value for bidder 1 is |α0|2V where V is the value of the expected profit of the corresponding classical auction to bidder 1. Since |α0|2V ≤ V , bidder 1 cannot gain from such a deviation. Furthermore, there is no gain from deviating from the bidding function b∗1(v) since it will only decrease V , because, by assumption, {b∗i (v)}ni=1 is a NE for the corresponding classical auction. Because of Eq. (9), this discussion applies to deviations by any single bidder, not just bidder 1. Thus, using bidding function {b∗i (v)}ni=1 and using the same operator for their initial state as they use in the search is a NE. The stronger condition, Eq. (10), shows that the number of infeasible states is enough to give no winner for any choice of initial amplitudes that up to n − 1 bidders can produce, provided p ≥ 2. Thus if an auctioneer implements a collusion-proof classical auction with the quantum protocol and assigns infeasible states as described then the resulting quantum auction is collusion-proof up to n− 1 bidders for initial amplitude deviations. The choice for d(x) satisfying the above requirements is not unique. As one example, let x be the state index in the full search space, running from 0 to 2np − 1. Consider x as written as a series of n base-2p numbers, |x1, x2, . . . , xn〉. Define d(x) = −r(x) (mod n+ 1) (11) where r(x) is number of nonzero values among x1, x2, ..., xn. The mod oper- ation gives all d(x) values in the range 0 to n. For the initial ground state, x = |0, . . . , 0〉, r(x) = 0 so d(x) = 0, and this is the smallest possible value. Single-deviation states have exactly one of the xi nonzero, giving r(x) = 1 and d(x) = n, the largest possible value. More generally, all k-deviation states have r(x) = k so d(x) = n + 1 − k. This function definition gives values directly from the representation of the state x, so, in particular, the auctioneer can implement it without any knowledge of the subspaces selected by the bidders. The assumption of perfect search is a sufficient but not necessary con- dition for the proof of Theorem 1. The necessary conditions are more com- plicated because we only need that every single bidder deviation maps to a linear combination of infeasible states. Thus mixing among different single- deviation states during search (e.g., due to small eigenvalue gaps among those states), or among states corresponding to two or more bidders deviat- ing, does not affect the proof. 5.2.3 Bounded Number of Search Steps Theorem 1 shows the quantum auction has the same NE as the classical first price auction if the search is perfect and each bidder uses the same operator for every search step of Eq. (4). Since adiabatic search, run for a finite number of steps, is not perfect we examine the effect on the NE of an imperfect search. We show that the NE for perfect search, i.e., bidding as in a classical first price auction and using the same operator initially and during the search, is an ǫ-equilibrium for the auction with imperfect search. Furthermore, ǫ converges to zero as the number of search steps goes to infinity. A strategic profile is an ǫ-equilibrium [24] if for every player, the gains of unilateral defecting to another strategy is at most ǫ. This weaker equilibrium concept is useful in our case because determining how to exploit imperfect search is computationally difficult. Specifically, with the small eigenvalue gaps and degeneracy it is hard to know whether imperfect search benefits a particular bidder. Thus computational cost will likely outweigh the small possible gain. In this situation, an ǫ-equilibrium is a useful generalization of NE. We must prove that for any ǫ there exists an N so that if the search process uses at least N steps, the equilibrium of the game with a perfect search is also an ǫ-equilibrium when using the actual search. To do so, we bound the possible gain from deviation based on prior knowledge of the range of possible bidder values. That is, we assume the distribution of values has a finite upper bound v̄. In our context, one such bound is the maximum bid value expressible by the announced interpretation of each bidders qubits. Theorem 2. If the conditions of Theorem 1 are met, and assuming the pos- sible bidder values are bounded by v̄, for any ǫ > 0, there exists an N so that the NE in the quantum auction with a perfect search, shown in Theorem 1, is also an ǫ-equilibrium of the same auction with an imperfect search using N search steps. Proof. Let ph be the probability of the highest bid wins. Let pinf be the probability of reaching an infeasible state. Then po = 1 − ph − pinf is the probability of a bid other than the highest bid wins. With the adiabatic search, with nonzero eigenvalue gaps, the probability of correctly mapping the initial to final states converges to one as the number of search steps increases. Thus for any δ > 0, there always exists a N where po is at most δ. We define an equilibrium expected payoff function for bidder i with value v as π∗i (v), when all bidders use their equilibrium bidding functions. Without loss of generality, from the perspective of bidder i with value v, the probability of achieving the equilibrium payoff, π∗i (v), if that bidder does not deviate is 1 − δ. Thus the expected payoff of deviating is at most πdeviatei (v) ≤ (1− δ)π∗i (v) + δv̄ because (a) the most any bidder can gain is bounded by v̄, and (b) with probability 1− δ the auction either produces no profit (pinf) or is identical to a classical auction (ph). The expected gain g from deviating is the expected payoff from deviating minus the expected payoff with no deviation, i.e., g = πdeviatei (v) − π∗i (v) ≤ δ(v̄ − π∗i (v)), which in turn is at most δv̄. Thus for any choice of δ, there always exists an N where the maximum deviation benefit is at most δv̄. For any ǫ > 0, using δ = ǫ/v̄ in the above discussion shows there always exists an N where the deviation is at most ǫ. 5.3 Testing for Changed Operators During Search One approach to the incentive issue of changing operators during search, described in Sec. 4.3, is for the auctioneer to test the bidders by randomly inserting additional probe steps in the search. Specifically, suppose at any step of the search the auctioneer, with some probability, decides to check a bidder by sending a new set of qubits in a known state |φ〉, while storing the qubits for the search until a subsequent step. For the test step, the auctioneer sets D or P to the identity operator. The state returned by the bidder is then U ′iU i |φ〉 or U i Ui |φ〉, depending on which part of the search step in Eq. (4) the auctioneer is testing. Without loss of generality, we consider the former case. Ideally, the bidder uses the same operator, so U ′i = Ui and U i is the identity. Suppose the test state is formed from some operator V , randomly selected by the auctioneer, |φ〉 = V |0〉. If U ′iU i is not the identity, the re- turned state has the form α |φ〉+β |φ⊥〉, where |φ⊥〉 is some state orthogonal to |φ〉 and |α|2 + |β|2 = 1. The auctioneer then applies V †, giving α |0〉+ β |a〉 (12) for some value a 6= 0. The auctioneer then measures this state, getting 0 with probability |α|2, indicating the bidder passes the test. Otherwise, the auctioneer observes a different value, indicating the bidder changed the operator. Hence the chance of getting caught depends on how often the auctioneer checks, and how big a change the bidder makes in the operator. Larger operator changes are more likely to be caught. This testing behavior is appropriate as small changes are not likely to have much affect on the search outcome, and instead simply act as an alternate adiabatic path from initial to final states. This technique is especially useful for risk averse bidders since then even a small chance to be caught might be enough to prevent bidders from wanting to change operators. 5.4 Assigning Eigenvalues to Subspaces Quantum search acts on the full space of superpositions of the available qubits, i.e., in our case to all 2np configurations of items and bids. In the auction context, bidders choose operators to restrict the search to a subspace of possible bids, namely the ones they wish to make. Conceptually, the search described above is then restricted to the subspace selected by the bidders. The search can also be viewed as taking place in the full space of 2np configurations. The operator U appearing in the search algorithm is block diagonal (up to a permutation of the basis states), with only the block operating on the selected subspace relevant for the search outcome. This view of the search is that of the auctioneer, who has no prior knowledge of the subspace selected by each bidder. The operator U is not known to any single individual: instead its implementation is distributed among the bidders, with each bidder implementing a part of the overall operator. The auctioneer chooses the eigenvalues for the initial Hamiltonian and the ordering for the qubits assigned to each bidder. These choices, which could change during the search, affect the incentive structure of the auction as described in Sec. 5.2. This section describes how the auctioneer’s choice of d(x) can give the same eigenvalues when restricted to the subspace actually selected for the search. For simplicity, we suppose each bidder uses a 2-dimensional sub- space, consisting of ∅ and the desired bid for the single item. While not essential for the NE results discussed above, uniformity with respect to sub- space choices means bidders are treated uniformly, so convergence of the search is independent of the order in which the auctioneer considers the bidders. 5.4.1 An Example Consider n = 2 bidders, each with p = 2 bits, representing 4 values: ∅ and three bid values 1, 2, 3. A set of 2-bit operators to form a uniform superposition of the form (|∅〉+ |b〉)/ 2 where b is the bid value, 1, 2 or 3, is 1/ 2 times 1 −1 0 0 1 1 0 0 0 0 1 −1 0 0 1 1 1 0 −1 0 0 1 0 −1 1 0 1 0 0 1 0 1 1 0 0 −1 0 1 1 0 0 −1 1 0 1 0 0 1 which we can denote as A1, A2, A3, respectively, with the first columns giving the uniform superposition of the three possible bid values. If the bidders select bids b1, b2, respectively, the overall operator for the search is U = Ab1 ⊗ Ab2 , used in Eq. (4) to perform each step of the search. Thus in this case there are 9 possible subspaces the two bidders can jointly select. Up to a permutation, U is block diagonal with the block containing the nonzero entries of the first column, and hence all the nonzero amplitude during the search, equal to 1 −1 −1 1 1 1 −1 −1 1 −1 1 −1 1 1 1 1 The search using U in the full 4-bit space is thus equivalent to one taking place in the 2-bit subspace selected by the two bidders using this operator The auctioneers’ choice of eigenvalues, i.e., the function d(x) used in Eq. (6) should ensure the uniform superposition within the subspace defined by the two bidders has the lowest value, say 0, and all other eigenstates have larger values. One possibility is the standard choice for the diagonal values d(x) when searching in the full space of 24 states defined by the np = 4 bits, namely the Hamming weight of each state, i.e., the number of 1 bits in its binary representation, ranging from 0 to 4. An alternative approach is picking d(x) so eigenvalues for the four states appearing in V have the same values as they would have with using the Hamming weight for a 2-bit search, ranging from 0 to 2. Doing so requires selecting the eigenvalues to match the corresponding Hamming weights for any choices the bidders make among A1, A2, A3. In this example, each bidder has 2 qubits, so can represent 4 states, which we denote as |0〉 , . . . , |3〉. The states for both bidders are products of these individual states, |0, 0〉 , . . . , |3, 3〉. Examining the 9 possible cases for U , shows a consistent set of choices is d(|x, y〉) equal to the number of nonzero values among x, y. With this d(x), the adiabatic search in the subspace selected by the bidders is identical to the standard adiabatic search for two bits. This choice treats both bidders identically. In this case we see the auctioneer can arrange the adiabatic search to operate symmetrically no matter what choice of subspace each bidder makes (i.e., no matter what value each bidder decides to bid). Thus from the point of view of the bidders, the search, in effect, takes place within the subspace of possible values defined by their bid selections. 5.4.2 General Case For arbitrary numbers of bidders n and bits p, we consider a single-item auction so each bidder would, ideally, pick an operator giving just two terms, with b(j) the bid of bidder j for the single item and no bits needed to specify which item the bidder is interested in. The choice of b(j) corresponds to the bidder picking a 2-dimensional subspace of the 2p possible states. The product of these subspaces gives a subspace S of all np qubits used in the auction. The subspace S has dimension 2n and its states xS can be viewed as strings of n bits. More specifically, we suppose bidder j implements the operator Uj such that the rows and columns corresponding to ∅ and b have nonzero values only for positions ∅ and b(j). That is, the elements of Uj for these two values form a 2× 2 unitary matrix. If the auctioneer knew the subspace S, the eigenvalue function d(x) used in Eq. (6) could be selected to match any desired function dS(xS) of the states in xS ∈ S. Without such knowledge, this is possible only for some choices for dS . Theorem 3. Provided dS(xS) depends only on the Hamming weight of the states xS, a single choice of d(x) in the full space corresponds to dS(xS) in all possible subspaces the bidders could select that include the null set. Proof. Consider the full operator U given by Eq. (1). For the element Ux,y, express the np bits defining the states x and y as sequences of p-bit values, x1, . . . , xn and y1, . . . , yn, respectively, with each xi and yi between 0 and 2p − 1. From Eq. (1), Ux,y = (Ui)xi,yi The matrix U is of size 2np × 2np while each Ui is of size 2p × 2p. Consider the first column of U , i.e., y = 0. Ux,0 is nonzero only for those x such that all the (Ui)xi,0 are nonzero. For this to be the case, each xi is either 0 (corresponding to |∅〉 for that bidder’s superposition) or xi = b(i), i.e., the bid value. Similarly, for all columns with each yi equal to 0 or b These values for x, y are precisely the states in the selected subspace of the bidders, S. For these choices of xi, yi, we can map 0 (i.e., p bits all equal to zero) to the single bit 0, and each b(i) (specified by values for p bits) to the single bit 1. This establishes a one-to-one mapping from states in the full space, of np bits corresponding to the product of bidders’ superpositions, to states in the subspace treated as n-bit vectors. Thus a function dS(xS) applied to the subspace that depends on the Hamming weight, i.e., the number of 1 bits in xS, is the same as a function on the full space depending on the number of nonzero xi values in x = x1, . . . , xn. We must show that a single choice of function d(x) in the full space maps to the desired dS(xS) in any choice of bidder subspaces. To see this is the case, consider any state in the full space x = x1, . . . , xn. Among these xi, suppose h are nonzero, denoted by xa1 , . . . , xah . This state x will appear in all selected subspaces in which bidder aj bids b (aj) = xaj , for j = 1, . . . , h, and the remaining bidders have any choice of bid. That is, x appears in (2p − 1)n−h possible subspaces S. Since x has exactly h nonzero values, in each of these possible subspaces it maps to a state xS with exactly h bits equal to 1, i.e., it has the same Hamming weight, h, in all possible subspaces in which it appears. Thus any choice of function dS(xS) depending only on the Hamming weight of xS will have the same value in all these possible subspaces. This observation allows the auctioneer to select that common value as the value for d(x), consistently giving the desired eigenvalue function for any possible subspace. Since this holds for all values of h, the auctioneer can operate in the full space with identical search behavior no matter what subspace the bidders select. For the auctioneer to operate without knowledge of the actual subspace selected by the bidders and treat bidders identically, we need d(x) to map to the same function on any subspace selected. In this case, the search proceeds exactly as if the auctioneer did know the subspace choices made by the bidders. The theorem gives one type of function for in which this is the case. In particular, Eq. (11) is an example of a function satisfying this theorem. 6 Multiple Items and Combinatorial Auction While the paper focuses on the single item first-price sealed-bid auction, the quantum protocol can apply to multiple items by changing the interpretation of the bids, i.e., the bidding language. Such changes affect the counting of deviation and feasible states, so we must check the validity of Theorem 1. In the single item case, each bidder uses the p qubits to specify the bid amount. With multiple items, the bid must specify both the items of interest and a bid amount for the items. Various bidding languages can encode this information. For multiple items, we divide the p qubits allocated to each bidder into two parts: pitem bits to denote a bundle of items and pprice bits to denote bid value (so p = pitem+pprice). Since qubits are expensive, a succinct represen- tation of items is best. Depending on the type of auction, we have various choices with different efficiency in using bits. For example, the pitem item bits could indicate the item in the bid, allowing pitem qubits to specify up to 2pitem different items. Another case is multiple units of a single item, so pitem could specify how many units a bidder wants (with the understanding the bid is for all those units not a partial amount) so the bits could specify 2pitem different numbers. In the general case, bids are on arbitrary sets of items or bundles, and we represent a bundle with m bits, 1 if the corresponding item is a part of the bundle and 0 otherwise, i.e., m = pitem. We focus on this general case in the remainder of the section. Allowing bids on sets of items is called a combinatorial auction [6]. With multiple items, the bid operator ψj = Ujψinit gives a superposition of bids of the form i , b where b i is bidder j’s bid for a bundle of items I . In this notation, the null bid is |∅, b〉, and the specified amount b is irrelevant so we take it to be zero in the examples. A superposition specifies a set of distinct bids, with at most one allowed to win. Example 4. Consider a combinatorial auction with two items X, Y and integer prices ranging from 0 to 3. With p = 4 bits for each bidder, using 2 bits each to specify item bundles and prices, is sufficient to specify the bids. The full space for a bidder has dimension 2p = 16, consisting of 4 possible item bundle choices and 4 price choices. Suppose a bidder places a bid (|∅, 0〉+ |X, 1〉 + |(X,Y ), 2〉) i.e., a bid of 1 for item X alone, and 2 for the bundle of both items. In this case, the bidder is not interested in item Y by itself. The dimension of the subspace of this bid is 3. Another example is the bid (|∅, 0〉 + |X, 1〉 + |X, 3〉 + |(X,Y ), 4〉) The dimension of the subspace is 4. This superposition has multiple bids on the same item X. This bidding language is both expressive and compact. For instance, a superposition of bundles of items readily expresses exclusive-or preferences, where a bidder wants at most one of the bundles. It is also compact because superpositions allow the bidder to use exactly the same qubits to place no bid (i.e., ∅) and to place all the exponential number of bundles in a combinatorial auction. An allocation, as defined in Sec. 3.1, is a list of bids, one from each bidder. With multiple items, an allocation is feasible if the item sets are pairwise disjoint. As in the single item case, we consider the allocation when all item sets are empty as infeasible. The value of a feasible allocation is the sum total of the bid values of the different bids in the allocation. The number of feasible states is ((n + 1)m − 1)2npprice . This is because we can assign m items among n bidders where all items need not be allocated in (n+1)m ways. The factor n+1 allows for some items to remain unallocated. Since the allocation when all bidders place the null bid is an infeasible state, we subtract 1. Each bidder can specify 2pprice different prices for the bundle giving 2npprice possible choices for n bidders. Note that the number of feasible states for a single item, m = 1, is different from that in Sec. 5.2 because here we have changed the bidding language to represent items also. The null bid in our protocol simplifies the evaluation of allocations for combinatorial auctions. To see this, consider a protocol without the null bid. In a single item case, F (x) for any allocation vector x would be maximum of the bids placed by the different bidders on the item, which is fairly easy to compute. But in the case of multiple items, there could be several allocations for a vector x. For example suppose Alice bids on the set {A,B} and Bob bids on {B,C}. Without the null set then both bids appear in the same state and have to be evaluated by F (x). The possible allocation to the bidders are 1. none to either 2. {A,B} to Alice 3. {B,C} to Bob, and 4. {A,B} to Alice and {B,C} to Bob (which is infeasible) F (x) will have to compute the maximum of the values in all these states. This is computationally complex when there are many items. By contrast, the bidding language with the null bid avoids this combinatorial evaluation within the search function F (x). As in the case of single item auctions, we restrict ourselves to a one-shot sealed bid classical combinatorial auction that we implement in a quantum setting. The total number of states is 2pn and the total number of single bidder deviations states is n(2p − 1). These expressions are the same as the single item case. The condition for all single-deviation states to be mapped to infeasible allocations, resulting in no winner, is 2np − ((n + 1)m − 1)2npprice ≥ n(2p − 1) (13) This condition holds for cases relevant for auctions as seen in the following claim. Claim 2. Eq. (13) is true for all integers m, pprice ≥ 1 and n ≥ 2. Proof. Recall p = m + pprice. We prove a stronger condition for integers n,m ≥ 2, i.e., there exists enough infeasible states to handle joint deviations up to n − 1 bidders. The number of k-bidder deviation states is the same as the single-item case, i.e., Eq. (8). Thus this stronger condition, with the same right-hand side as Eq. (10), is 2np − ((n + 1)m − 1)2npprice ≥ 2np − 1− (2p − 1)n (14) Hence Eq. (14) is true if (2p − 1)n ≥ ((n+ 1)m − 1)2npprice ⇔ 2ppricen(2m − 2−pprice)n ≥ ((n+ 1)m − 1)2npprice ⇔ (2m − 2−pprice)n ≥ (n+ 1)m − 1 Since 2−pprice ≤ 1, Eq. (14) is true if (2m − 1)n ≥ (n+ 1)m − 1 which is true if (2m − 1) m ≥ (n+ 1) Let f(m) ≡ (2m − 1) m and g(n) ≡ (n + 1) n . We establish the required inequality, f(m) ≥ g(n), by showing f(m) is increasing in m when m ≥ 2, g(n) is decreasing in n when n ≥ 2 and noting f(2) = g(2) = Taking the derivative of f(m) with respect to m, we get, (2m − 1) 2m ln(2) 2m − 1 ln(2m − 1) This is positive if and only if 2m − 1 log2(2 m − 1) This is true because log2(2 m − 1) < log2(2m) = m and hence both fractions in the expression are greater than 1. Thus, f(m) is increasing for all m ≥ 2. Taking derivative of g(n) with respect to n, we get, (n+ 1) 1 + n − ln(1 + n) This is negative if and only if ln(1 + n) 1 + n This is true for n ≥ 2. Thus g(n) is decreasing in n for n ≥ 2. Thus we have shown that Eq. (13) is true for n,m ≥ 2. It can be easily checked that Eq. (13), is not true for n = 1 and true when n = 2 and m = 1. Thus, if a classical combinatorial auction has a NE then the correspond- ing quantum auction protocol also has a NE with respect to initial state deviations. Also there is an ǫ-equilibrium of the same auction with an im- perfect search using N search steps. Moreover, the stronger condition of Eq. (14) shows that in auctions with at least two bidders (n > 1), there are enough infeasible states to give no winner for any deviation of initial ampli- tudes that up to n− 1 can produce. Thus no groups, up to size n− 1, can collude to benefit from initial amplitude deviations in the quantum auction. 7 Applications of Quantum Auctions Two properties of quantum information may provide benefits to auctioneers and bidders: the ability to compactly express complicated combinations of preferences via superpositions and entanglement and the destruction of the quantum state upon measurement. This section describes some economic scenarios that could benefit from these properties. As one economic application, quantum auctions provide a natural way to solve the allocative externality problem [18, 25]. In this situation, a bidder’s value for an item depends on the items received by other bidders. For example, consider companies bidding on a big government project requiring multiple companies to work on different parts. Allocative externality refers to the issue that the costs for a company which wins a contract for one part depends on which other companies win other parts. So company A may be willing to bid more aggressively if it knows that company B will work on related parts. Multiple simultaneous auctions for separate parts will not handle these interdependencies and thus will be inefficient. One possible solution is to let companies form partnership bids. That is joint bids that are accepted together or not at all. Quantum information processing allows for a natural way of forming partnership bids via entanglement. With the protocol described in Sec. 6, multiple bids can be entangled so they will either all be accepted together or none will be. Furthermore, quantum auctions may provide more flexibility with respect to information privacy of partnership bids than classical methods. Specifically, with multiple items, groups of bidders could select joint op- erators on their combined qubits, allowing them to express joint constraints (e.g., where they either all win their specified items or none of them do) without any of the other bidders or auctioneer knowing this choice. The bidders do so by creating an entangled state instead of the factored form for their qubits. Thus employing quantum entanglement provides bidders a natural way for expressing any allocative externality. This possibility shows bidding languages based on qubits are highly expressive and compact be- cause bidders can use the same bits to express their individual bids and joint bids via entanglement. Example 5. Alice and Bob could jointly form the state (|∅, 0,∅, 0〉 + |IA, bA, IB , bB〉+ |IC , bC , ID, bD〉) (15) to represent the bidders willing to pay bA and bB for items IA and IB, or to pay bC and bD for items IC and ID, but they are not willing to buy other combinations, such as IA for Alice and ID for Bob. In this scenario, a direct representation of bids, i.e., without a null bid, would not guarantee the joint preferences are satisfied for all entangled bid- ders or none of them. That is, without null bids, the superposition could not express the joint preference through entanglement. A group of k bidders operating jointly on their qubits to form entangled bids could also produce initial amplitudes involving up to k-bidder deviation states. However the discussion with Eq. (14) on multiple item auctions shows our protocol can handle all deviation states a group of up to n−1 bidders can produce, i.e., by mapping them to infeasible outcomes. Thus the additional expressivity used for joint bids does not introduce additional opportunities for collusion to change the outcomes via initial amplitude selection. A second economic application for quantum auctions arises from their privacy guarantee for losing bids. This property is economically useful when bidders have incentives to hide information. An example is a scenario in which companies are bidding for government contracts year after year. A company’s bid usually contains information about its cost structure. If there is reasonable expectation that the losing bids will be revealed, a company may want to bid less aggressively to reduce the amount of information passed to its competition for use in future auctions. This will lead to a less efficient auction than if bidders reveal their true values. In this situation, a privacy guarantee on the losing bids enables bidders to bid with less inhibition. More generally, this privacy issue is only relevant when there are additional interactions between these companies after the auction is concluded, such as future auctions or negotiations where participants may be at a disadvantage if their values are known to others. This strong privacy property is unique to quantum information process- ing. Privacy can be enforced via cryptographic methods for multi-player computation [13], and in an auction can keep losing bids secret [22]. How- ever, the information on the bids, and the key to decrypt them, remains after the auction completes. People who have access to the key may be legally compelled to reveal the information or choose to sell it. So while cryptography can be secured computationally, it cannot guarantee the in- tegrity of the person(s) who have the means to decrypt the information. On the other hand, the quantum method destroys losing bids during the search for the winning one and it is physically impossible to reconstruct the bids after the auction process. Similarly, some of the other properties of quantum auctions, such as correlations for partnership bids, can be pro- vided classically [19]. Moreover, quantum mechanisms are readily simulated classically [27] (as long as they involve at most 20 to 30 qubits). However, these classical approaches lack the information security of quantum states. More study is needed to determine scenarios where the privacy property of the quantum protocol is significant. 8 Discussion This paper describes a quantum protocol for auctions, gives a game theory analysis of some strategic issues the protocol raises and suggests economic scenarios that could benefit from these auctions. These include the privacy of bids and the possibility of addressing allocative externalities. The search used in our protocol can use arbitrary criteria for evaluating allocations, thereby implementing other types of auctions with quantum states. Thus while we focus our attention on the first-price sealed-bid auction, the pro- tocol is more general: it can implement other pricing and allocation rules, as well as multiple-unit-multiple-item auctions with combinatorial bids. For example we can use this protocol in a multiple stage, iterative auction. In fact, the protocol supports general bidding languages. Encoding bids in quantum states raises new game theory issues because the bidders’ strategic choices include specifying amplitudes in the quantum states. The auction is not only probabilistic, but the winning probability is not just a function of the amount bid. Instead a bidder can change the probability of winning by altering the amplitudes of the quantum states encoding his bid. For example, in the context of the first-price sealed-bid auction, the auction does not guarantee the allocation of the item to the highest bidder. We show that the correct design of the protocol can solve a specific version of this incentive problem. The salient design feature is an incentive compatible mechanism so that bidders do not want to cheat, as opposed to an algorithmic secure protocol that prevents bidders from cheating. Thus, our design is an example of a quantum algorithm, in this case adiabatic search, tuned to improve incentive issues rather than the usual focus in quantum information processing on computation or security properties of algorithms. In addition, we show that the Nash equilibrium of the corresponding classical first-price sealed-bid auction is an ǫ-equilibrium of the quantum auction and that ǫ converges to zero when the quantum search associated with the protocol uses an increasing number of steps, under the conditions listed in Theorem 1. This result is with respect to changes in the initial state of the search. It remains to be seen whether other bidder strategies give some unilateral benefit, requiring further adjustments to the auction design. There are multiple directions for future work. First, we plan a series of human subject experiments on whether people can indeed bid effectively in the simple quantum auction scenario described in this paper. As with previous experiments with a quantum public goods mechanism [2], such ex- periments are useful tests of the applicability of game theory in practice, and also suggest useful training and decision support tools. In particular, people’s behavior in a quantum auction could differ from game theory predic- tions that people select a Nash equilibrium based on idealized assumptions of human rationality and full ability to evaluate consequences of strategic choices with uncertainty. Second, we plan to extend studies of quantum auctions to more com- plicated economic scenarios, such as one with allocative externality. Our analysis considers a single auction. An interesting extension is to a series of auctions for similar items. If auctions are repeated, the game theory anal- ysis is more complicated [28]. In particular, privacy concerns become more significant since information revealed by a bidder’s behavior in one auction may benefit other bidders in later auctions. The quantum auction destroys all information about the losing bids. As a result, it is not possible to conduct after-the-fact audits to verify that the auction has been conducted correctly. Is there a way to modify the mechanism to enable audits while preserving some of the privacy guaran- tees? Security is another interesting issue. For example, there may be third parties, aside from the auctioneer and bidders who are interested in inter- cepting and changing bits in transit. Auctioneers may have incentives to detect a bidder’s bid or skew auction results. The question is whether we can build security around the protocol to prevent or at least detect these types of attacks. Similarly, many economics issues surrounding the protocol remain to be resolved. For example, people behave as if they are risk averse in auction situations [5, 4] which can change the predictions of game theory. Another issue arises from the possibility of multiple Nash equilibria. We have only shown that the desirable outcome is an equilibrium. The quantum protocol can also have other equilibria. Since the Nash equilibrium concept alone does not indicate how people select one equilibrium over another, additional study is needed to determine when the desirable outcome is likely to occur. Our protocol makes only limited use of quantum states, in particular encoding bids in the subspace selected by the bidders but not using the amplitudes separately. Thus it would be interesting to examine extensions to the protocol exploiting the wider range of options for bidders. For example, a protocol might use amplitudes of superpositions to indicate a bidder’s probabilistic preferences, say, as in constructing a portfolio of items with various expected values and risks. Such portfolios could be useful if bidders have some uncertainty in their values (e.g., in bidding for oil field exploration rights) rather than the standard private value framework considered in this paper, where bidders know their own values for the items. With uncertain values, probabilistic bids could allow bidders to match their risk preferences along with their value estimates within the auction process. As a final note, the number of qubits necessary to conduct an auction is small compared to the requirement of complex computations such as factor- ing. 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Quantum mechanics helps in searching for a needle in a haystack. Physical Review Letters, 79:325–328, 1997. arxiv.org preprint quant-ph/9706033. [15] Tad Hogg. Quantum search heuristics. Physical Review A, 61:052311, 2000. Preprint at publish.aps.org/eprint/gateway/eplist/aps1999oct19 002. [16] Tad Hogg. Adiabatic quantum computing for random satisfiability problems. Physical Review A, 67:022314, 2003. arxiv.org preprint quant-ph/0206059. http://arxiv.org/abs/quant-ph/0301013 http://arxiv.org/abs/quant-ph/0110122 http://arxiv.org/abs/quant-ph/0104087 http://arxiv.org/abs/quant-ph/9607014 http://arxiv.org/abs/quant-ph/9806088 http://arxiv.org/abs/quant-ph/0004076 http://arxiv.org/abs/quant-ph/9706033 http://arxiv.org/abs/quant-ph/0206059 [17] Bernardo A. Huberman and Tad Hogg. Quantum solution of coordination problems. Quantum Information Processing, 2:421–432, 2003. arxiv.org preprint quant-ph/0306112. [18] Philippe Jehiel and Benny Moldovanu. Allocative and informational external- ities in auctions and related mechanisms. Technical Report SFB/TR 15 142, Free University of Berlin. available at ideas.repec.org/p/trf/wpaper/142.html LOCATION =. [19] David A. Meyer. Quantum communication in games. In S. M. Barnett et al., editors, Quantum Communication, Measurement and Computing, volume 734, pages 36–39. AIP Conference Proceedings, 2004. [20] Paul R. Milgrom and Robert J. Weber. A theory of auctions and competitive bidding. Econometrica, 50:1089–1122, 1982. [21] Pierfrancesco La Mura. Correlated equilibria of classical strategic games with quantum signals. arxiv.org preprint quant-ph/0309033, Sept. 2003. [22] Moni Naor, Benny Pinkas, and Reuben Sumner. Privacy perserving auctions and mechanism design. In Proc. of the ACM Conference on Electronic Com- merce, pages 129–139, NY, 1999. ACM Press. [23] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge Univ. Press, 2000. [24] Roy Radner. Collusive behavior in noncooperative epsilon-equilibria of oligopolies with long but finite lives. J. of Economic Theory, 22:136–154, 1980. [25] Martin Ranger. The generalized ascending proxy auction in the presence of externalities. Technical report, Social Science Research Network, July 2005. available at ssrn.com/abstract=834785. [26] Peter W. Shor. Algorithms for quantum computation: Discrete logarithms and factoring. In S. Goldwasser, editor, Proc. of the 35th Symposium on Founda- tions of Computer Science, pages 124–134, Los Alamitos, CA, November 1994. IEEE Press. [27] S. J. van Enk and R. Pike. Classical rules in quantum games. Physical Review A, 66:024306, 2002. [28] Robert Wilson. Strategic analysis of auctions. In Robert Aumann and Sergiu Hart, editors, Handbook of Game Theory with Economics Applications, vol- ume 1. Elsevier, 1992. Chapter 8. http://arxiv.org/abs/quant-ph/0306112 http://arxiv.org/abs/quant-ph/0309033 Introduction Quantum Auction Protocol Quantum Auction Implementation Creation and interpretation of quantum bids Distributed Search Strategies with Quantum Operators Selecting the Subspace Altering Initial Amplitudes Changing Operator During Search Quantum Auction Design Checking for the Null Set The First-Price Sealed-Bid Auction A Game Theory Approach to Auctions Behavior with Perfect Search Bounded Number of Search Steps Testing for Changed Operators During Search Assigning Eigenvalues to Subspaces An Example General Case Multiple Items and Combinatorial Auction Applications of Quantum Auctions Discussion

0704.0803

Geometric Phase and Superconducting Flux Quantization

Geometric Phase and Superconducting Flux Quantization Geometric Phase and Superconducting Flux Quantization Walter A. Simmons & Sandip S. Pakvasa Department of Physics and Astronomy University of Hawaii at Manoa Honolulu, Hi 96822 Abstract In a ring of s-wave superconducting material the magnetic flux is quantized in units of 0 2 Φ = . It is well known from the theory of Josephson junctions that if the ring is interrupted with a piece of d- wave material, then the flux is quantized in one-half of those units due to a additional phase shift of π . We reinterpret this phenomenon in terms of geometric phase. We consider an idealized hetero-junction superconductor with pure s- wave and pure d-wave electron pairs. We find, for this idealized configuration, that the phase shift of π follows from the discontinuity in the geometric phase and is thus a fundamental consequence of quantum mechanics. Geometric phase has been contained in quantum mechanics since the foundations of the field were set down in the early twentieth century; however, the phase and its importance were not recognized for some time. Pancharatnam1 discovered the classical geometric phase in optics in 1956 and Berry’s important 1987 quantum mechanics paper2 stimulated the rapid development of the field. By 1992, Anandan3, in a review article in Nature, was able to conclude that the phase had been convincingly demonstrated. The first application of geometric phase to Josephson Junctions was carried out by Anandan and Pati in 1997. They showed that the zero voltage tunneling supercurrent is geometric in nature and that it is proportional to the speed of the state vector in projective Hilbert space.4 The appearance of a phase discontinuity of π± arising from geometric phase, under certain circumstances, was shown5 to be a general feature of quantum mechanics in 2003, but has so far found only limited application6. Here we show that a well known phenomenon7,8 in superconductivity, the quantization of magnetic flux in one half of the usual unit9, which is 0 2 Φ = , can be interpreted as an effect of the discontinuity in geometric phase. This phase shift in superconductors has been understood in terms of the physics of the Josephson junctions and the result has been applied to high temperature superconductors10 in order to test the idea that they involve d-wave electron pairs. Our application considers an idealized hetero-junction superconductor with pure s-wave and pure d-wave electron pairs. We find, for this idealized configuration, that the phase shift of π follows from the discontinuity in the geometric phase4 and is thus a fundamental consequence of quantum mechanics. Applications of quantum geometric phase have been made in nearly every branch of physics, from fundamental material science11,12 to quantum computing with superconducting nanocircuits13, as well as in chemistry14, and it has been suggested that phase may become important in biology15. Since the phase has been long present, but not fully recognized, some applications entail reinterpreting known phenomena in terms of the phase. An important and illustrative example is the reinterpretation of the so called Guoy effect in optics as a geometrical phase16, which we will summarize below. An idealized superconducting ring, which consists of a composite of s-wave material and d-wave material, will, in the absence of external electromagnetic fields, exhibit quantization of the magnetic flux in units of one half of the usual unit, 0 2 Φ = . This half-unit quantization of the magnetic flux will occur whenever there is an odd number of phase shifts of magnitude π in the circuit. The theoretical argument6 for the π phase shift in a composite ring was based upon the dynamics of the Josephson junction and on thermodynamics, and has experimental support. We next explain the quantum mechanics of the phase shift and then we shall proceed to reinterpret the s-wave/d-wave superconductor hetro-junction. It has long been known17 that when a light beam converges to a focus, then diverges again, the light experiences a phase shift whose magnitude depends upon the details. For example, for a beam with a Gaussian profile and a very small waistline at the focus, an abrupt phase shift of π occurs for each of the two transverse directions, for a total phase shift of π . This result, which follows from standard classical electrodynamics, has been reinterpreted in terms of geometric phase15. For one transverse direction, the complex curvature of the wave reverses at the focus; the geometric phase, which is directly related to the curvature, changes by π . That optical example of geometric phase is closely analogous to a well-known18 phase flip of π± , which occurs in optics when the polarization of a light beam is rotated from some initial state, through and beyond, another polarization state that is orthogonal to the initial state. The latter example of the geometric phase discontinuity has been shown to occur rather generally in quantum mechanics4. This can be understood by considering the behavior of a complex quantum state vector as it is impelled through a series of states in Hilbert space by an external force. Suppose the initial state is iΨ , the final state is fΨ , and some intermediate state 0Ψ is orthogonal to the initial state, 0 0iΨ Ψ = . In the complex plane, the trajectory is a sequence of projections of each state upon its subsequent state. The trajectory from the initial to the final state passes through the origin (with a positive or negative infinitesimal imaginary part); the phase goes through an inverse-tangent singularity and changes by π± . Finally, turning to superconductivity, we adopt the theoretical framework19, in which superconductivity is viewed as a consequence of the breaking of gauge invariance entailing the formation of Cooper pairs. We consider a ring of material in which the supercurrent is carried by s-wave Cooper pairs. The ring is interrupted by a section of material in which d-wave Cooper pairs carry the supercurrent. The supercurrent passing through the idealized hetro-junction experiences a shift of π± in the geometric phase due to the orthogonality of d-wave and s-wave. Since the orientation of the d- wave relative to the s-wave is not meaningful, we have no sum over dimensions as in the optical beam analogy; a shift of π± is the result of a single inserted section of material and the half-unit magnetic flux quantization follows. Since the 1997 work of Anandan and Pati, it has been known that the zero-voltage current in a tunneling supercurrent arises from the geometry of Hilbert space and is independent of the specific Hamiltonian, (which is a general feature of geometric phase20). Recently, experiments on hetero-junctions have supported the idea that high materials are d-wave superconductors. While we are not discussing here realistic models of high materials here, our results show that a phase shift of π± in s-wave/d-wave hetero-junctions arises from the fundamentals of quantum mechanics. 1 Pancharatnam, S., The Proceedings of the Indian Academy of Sciences, Vol XLIV, No. 5, Sec. A, 247 (1956) in Collected Works of S. Pancharatnam, Oxford University Press, London (175). 2 Berry, M.V. “Quantal Phase Factors Accompanying Adiabatic Changes”, Proc. R. Soc. Lond. A392, 45, (1984). 3 Anandan, J. “The geometric phase”, Nature 360, 307 (1992). 4 Anandan, J. & Pati, A.K., “Geometry of the Josephson effect” Physics Letters A 231, 29 (1997). 5 Mukunda, et al “Bargmann invariants, null phase curves, and a theory of the geometric phase”, Phys. Rev. A 67, 042114 (2003). 6 Simon, R. and Mukunda, N., “Bargmann Invariant and the Geometry of the Guoy Effect”, Phys. Rev. Letters 70, 880 (1993). 7 Bulaevskii, L. N. , Kuzii, V. V. & Sobyanin, A. A. Superconducting system with weak coupling to the current in the ground state. JETP Lett. 25, 290–294 (1977). 8 Tsuei, C.C., and Kirtley, J.R. “Paring symmetry in cuprate superconductors” Rev. Mod. Phys 72, 969 (2000). For more recent results, see Kirtley, et al, Nature 373, 225 (2005) 9 Ashcroft, N.W. & Mermin, N.D., Solid State Physics, Holt, Rinehart and Winston (1976). 10 Hilgenkamp, H., Ariando, Smilde, H.-J. H., Blank, D. H. A., Rijnders, G., Rogalla, H., Kirtley, J. R., and Tsuei, C. C., “Ordering and manipulation of the magnetic moments in large-scale superconducting pi-loop arrays”, Nature 422, 50 (2003). 11 Zak, J. “Berry’s Phase for Energy Bands in Solids”, Phys. Rev. Lett. 62, 2747 (1989). 12 Resta, R. “Manifestations of Berry’s phase in molecules and condensed matter”, J.Phys.Condens. Matter 12, R107 (2000). 13 Falci,G., Fazio, R., Palma, G.M., Siewert, J., and Vedral, V. “Detection of geometric phases in superconducting nanocircuits”, Nature 407, 355 (2000). 14 Mead, C.A. “The geometric phase in molecular systems”, Rev. Mod. Phys. 64, 51 (1992). 15 Kagan, M.L., Kepler, T. B. & Epstein, I.R., “Geometric phase shifts in chemical oscillators”, Nature 349, 506 (1991). 16 Simon, R. and Mukunda, N., “Bargmann Invariant and the Geometry of the Guoy Effect”, Phys. Rev. Letters 70, 880 (1993). 17 Siegman, A.E., Lasers, University Science Books, Mill Valley, California (1986). 18 Bhandari, R. “Polarization of light an the topological phases”, Physics Reports, 281, 1 (1997). 19 Weinberg, S., Quantum Theory of Fields II, Cambridge University Press (1996). 20 Aharonov, Y. & Anandan, J., “Phase Change during a Cyclic Quantum Evolution”, Phys. Rev. Lett 58, 1593 (1987).

0704.0804

Equation-free implementation of statistical moment closures

Equation-Free Implementation of Statistical Moment Closures Francis J. Alexander and Gregory Johnson Los Alamos National Laboratory, P.O.Box 1663, Los Alamos, NM, 87545. Gregory L. Eyink Department of Mathematical Sciences Johns Hopkins University Baltimore, MD 21218 Ioannis G. Kevrekidis Department of Chemical Engineering and PACM Princeton University Princeton, NJ 08544 We present a general numerical scheme for the practical implementation of statistical moment clo- sures suitable for modeling complex, large-scale, nonlinear systems. Building on recently developed equation-free methods, this approach numerically integrates the closure dynamics, the equations of which may not even be available in closed form. Although closure dynamics introduce statistical assumptions of unknown validity, they can have significant computational advantages as they typi- cally have fewer degrees of freedom and may be much less stiff than the original detailed model. The closure method can in principle be applied to a wide class of nonlinear problems, including strongly- coupled systems (either deterministic or stochastic) for which there may be no scale separation. We demonstrate the equation-free approach for implementing entropy-based Eyink-Levermore closures on a nonlinear stochastic partial differential equation. PACS numbers: INTRODUCTION Accurate, fast simulations of complex, large-scale, non- linear systems remain a challenge for computational sci- ence and engineering, despite extraordinary advances in computing power. Examples range from molecular dy- namics simulations of proteins [1], [2] and glasses [3], to stochastic simulations of cellular biochemistry [4, 5], to global-scale, geophysical fluid dynamics [6]. Often for the systems under consideration there is no obvious scale sep- aration, and their many degrees of freedom are strongly coupled. The complex and multiscale nature of these pro- cesses therefore makes them extremely difficult to model numerically. To make matters worse, one is often in- terested not in a single, time-dependent solution of the equations governing these processes, but rather in ensem- bles of solutions consisting of multiple realizations (e.g., sampling noise, initial conditions, and/or uncertain pa- rameters). Often real-time answers are needed (e.g., for control, tracking, filtering). These demands can easily exceed the computational resources available not only now but also for the foreseeable future. In principle, all statistical information for the problem under investigation is contained in solutions to the Liou- ville (if deterministic)/Kolmogorov (if stochastic) equa- tions. These are partial differential equations in a state space of high (possibly infinite) dimension. A straightfor- ward discretization of the Liouville / Kolmogorov equa- tions is therefore impractical. An ensemble approach to solving these equations can be taken; however, quite of- ten, the practical application of the ensemble approach is also problematic. Generating a sufficient number of independent samples for statistical convergence can be a challenge. For some problems, computing even one real- ization may be prohibitive. The traditional approach to making these prob- lems computationally tractable is to replace the Liou- ville/Kolmogorov equation by a (small) set of equations (PDEs or ODEs) for a few, low order statistical moments of its solution. When taking this approach for nonlinear systems, one must make an approximation, a closure, for the dependence of higher order moments on lower order moments. Typically the form of the closure equation is based on expert knowledge, empirical data, and/or phys- ical insight. For example, in the superposition approxi- mation and its extensions [7] for dense liquids and plas- mas, both quantum or classical, one approximates third order moments as functions of second order moments. Moment closure methods of this type have been applied to a number of areas including fluid turbulence (see [8] and references therein, and also the work of Chorin et. al.). Of course, as with any approximation strategy, the quality of the resulting reduced description depends on the approximations made – poor closures lead to poor answers/predictions. In addition to replacing the ensem- ble with a small set of equations for low order moments, these equations are typically easier to solve. They are deterministic and generally far less stiff than the original http://arxiv.org/abs/0704.0804v1 equations. A less exploited variant of this approximation scheme is the probability density function (PDF) based moment- closure approach. For PDF moment closures one makes an ansatz for the system statistics guided by available in- formation (e.g., symmetries). One then uses this ansatz in conjunction with the original dynamical equations to derive moment equations. Such PDF-based closures have been developed for reacting scalars advected by turbu- lence [10], phase-ordering dynamics [11] and a variety of other systems. This approach to moment-closure is a close analogue of the Rayleigh-Ritz method frequently used in solving the quantum-mechanical Schroedinger equation, by exploiting an ansatz for the wave-function. For a formal development of this point of view, see [12]. One of the obstacles to applying moment closures is that often the closure equations are too complicated to write down explicitly, even with the availability of com- puter algebra / symbolic computation systems. This is especially true for large-scale, complex systems, e.g. global climate models. Because of their great complexity, even if one could in principle derive the closure equations analytically, this procedure would be extremely difficult and time-intensive. Moreover, each time a model is up- dated, as climate and ocean models regularly are, the closure equations would have to be rederived. In other cases it may simply be impossible to determine the clo- sure equations analytically. This is especially likely when PDF’s are not Gaussian, which is the case for most use- ful closures. Monte Carlo or other numerical methods may be needed in order to evaluate integrals for the mo- ments [13]. In addition, there may be situations where neither analytic nor numerical/MC integration will yield the closure equations due to the black-box nature of the available numerical simulator such as a compiled numer- ical code with an inaccessible source. Clearly, a need ex- ists for a robust approach to the general closure protocol which circumvents analytical difficulties. We address that need here by combining PDF closures with equation free modeling [14] [15]. The basic premise of the equation-free method is to use an ensemble of short bursts of simulation of the original dynamical system to estimate, on demand, the time-evolution of the the clo- sure equations that we may not explicitly have. The equation-free approach extends the applicability of sta- tistical closures beyond the rare cases where they can be expressed in closed form. This hybrid strategy may be faster than the brute-force solution of a large ensemble of realizations of the dynamical equations since the closure version is generally smoother than the original problem. This paper is organized as follows. In Section 2 we describe the general features of PDF-based moment clo- sures. In Section 3 we explain how to implement the equation-free approach with these closures. We then, in Section 4, apply these ideas for a specific dynamical sys- tem, the stochastic Ginzburg-Landau (GL) equations us- ing a particular PDF-based closure scheme, the entropy method of Eyink and Levermore [16]. We conclude with a discussion of closure quality, computational issues, and the application of our approach to large-scale systems. PDF-BASED MOMENT CLOSURES We consider the very general class of dynamical sys- tems, including maps, formally represented by Ẋ = U(X(t),N(t), t) (1) Xt+1 = Ut(Xt,Nt) (2) where N(t) is a stochastic process with prescribed statis- tics. The stochastic component arises from unknown pa- rameters, random forcing, neglected degrees of freedom and/or random initial conditions. This class includes both deterministic and stochastic systems with discrete and/or continuous states. Queueing systems, molecular dynamics, and stochastic PDEs are just some of the many examples that fall into this category. For concreteness in this paper we restrict ourselves to a special case of equation (2), namely, situations where N(t) is a Markov process (Brownian motion, Poisson pro- cess, etc.) and—more specifically still—Itô stochastic dif- ferential equations of the form: dX = U(X, t)dt + 2S(X, t)dW(t). (3) The deterministic component of the state, X, is gov- erned by the continuously differentiable vector field, U : N × R → RN . For many problems of interest (e.g., cli- mate) U is a highly nonlinear function. The noise com- ponent is modeled by the standard mean 0, covariance matrix I Wiener process, W ∈ RN , possibly modulated by a state-dependent matrix S : RN×R → RN×N . Equa- tion (3) encompasses a wide class of systems including deterministic (S = 0) ones. In many cases one is interested in knowing the low order statistics of equation (3), for example an instanta- neous mean value or possibly multi-point covariance of X. These statistics can be obtained by averaging over an ensemble of stochastic systems, solving equation (3). They can also be obtained via the forward Kolmogorov equation for the probability density function P (X, t): ∂tP = L∗(t)P, (4) where P satisfies the conditions: P (X, t) ≥ 0, and P (X, t) dX = 1, and where L∗ is the generator of the Markov process. In the case of equation (3) this operator takes the form L∗(t)ψ(X) = −∇X·(U(X, t)ψ(X))+∇2X : (D(X, t)ψ(X)). The forward Kolmogorov equation then becomes a Fokker-Planck equation ∂tP +∇X · (UP ) = ∇2X : (DP ) (6) where D(X, t) = S(X, t)S(X, t)T is the nonnegative- definite diffusion matrix arising from the noise term. Un- like the original dynamical equation (3), the forward Kol- morogov equation (FKE) is both linear and determinis- tic. Dealing with it, therefore, has apparent advantages over the original ensemble of stochastic systems simu- lations. The price to pay for these advantages is that the FKE lives in a typically high, potentially infinite- dimensional, space. When equation (3) is a nonlinear PDE, numerical solution to the FKE is usually ruled out. For computational purposes, we would therefore like to reduce the FKE (if possible and useful) to a small system of ordinary differential equations. This reduc- tion should simplify the computation as much as pos- sible while retaining fidelity to the original dynamical processes. The reduction proceeds by taking moments of the FKE with respect to a vector-valued function ξ(X, t) from RN × R+ → RM . The ξ selected should include the relevant variables in the system (slow modes, con- served quantities, etc.). The moments µ(t) of ξ(X, t) are defined by µ(t) = ξ(X, t)P (X, t)dX (7) and give rise to µ̇(t) = ξ̇(X, t)P (X, t)dX, (8) where ξ̇(X, t) = ∂tξ(X, t) + L(t)ξ(X, t) (9) and L is the adjoint of L∗ or the backward Kolmogorov operator. The result (8) can be obtained by averaging over an ensemble of realizations of the stochastic dynam- ics (3). In general, however, (8) is not a closed equa- tion for the moments, µ. One can close this equation by choosing a PDF, P (X, t,µ), which itself is a function of the moments µ. µ̇(t) = V(µ, t) ≡ ξ̇(X, t)P (X, t,µ)dX. (10) Alternatively, one can select a family of probability den- sities P (X, t,α), specified by parameters α = α(µ, t) rather than directly by the moments µ. This is analogous to specifying the temperature in the canonical ensemble as opposed to the average energy. The equivalence of these approaches is guaranteed provided that the param- eters and moments can be determined uniquely from one another. The translation between the parameters and their corresponding moments can be carried out by one of several methods. In some cases one may require Monte Carlo evaluation of the resulting integrals. If the moments and/or parameters are selected judiciously, one hopes that the approximate PDF P (X, t,α(µ)(t)) will be close to the exact solution of the Liouville/Kolmogorov equation (4). The mapping closure approach of Chen et al [10] and the Gaussian mapping method of Yeung et al. [11] are based on this type of parametric PDF closure [19]. In fact, perhaps the most familiar application of the parametric approach is the use of the Rayleigh-Ritz method in quantum me- chanical calculations. This is the essential approach of our paper. EQUATION-FREE COMPUTATION Although we now have obtained a closed moment equa- tion (equation 10), we still need to determine the dynam- ical vector field V. As explained above, this step can be a serious obstacle to the practical implementation of PDF- based moment-closure (PDFMC). A method to calculate V is desirable that (i) does not require a radical revision each time the underlying code or model changes, and (ii) is relatively insensitive to the complexity of the PDFMC. The equation-free approach of Kevrekidis and collabora- tors [14] meets those requirements. It permits one to work with much more sophisticated, physically realistic closures. Equation-free computation is motivated by the simple observation that numerical computations involving the closure equations ultimately do not require closed for- mulae for the closure equations. Instead, one must only be able to sample an ensemble of system states X dis- tributed according to the closure ansatz P (X, t;α) and then evolve each of these via equation (3) for short inter- vals of time. Such sampling and subsequent dynamical evolution would be necessary to calculate the statistics of interest even when not using a closure strategy. It is sufficient to have a (possibly black-box) subroutine avail- able which, given a specific state variable X(t) as input, returns the value of the state X(t + δt) after a short time δt. The ensemble of systems, each of which satis- fies equation (3), is evolved over a time interval δt. The moments/parameters µ or α are determined at the be- ginning and end of this interval and the time derivative µ̇ is estimated from the results of these short ensemble runs. This “coarse timestepper” can be used to estimate locally the right hand side of the closure evolution equa- tions, namely V(µ, t). Coarse projective forward Euler (arguably the simplest of equation-free algorithms) which we will use below il- lustrates the approach succinctly: Starting from a set of coarse-grained initial conditions specified by moments µ(t) we first (a) lift to a consistent fine scale descrip- tion, that is, sample the PDF ansatz P (X, t;α(t)) to generate ensembles of initial conditions X for equation (3) consistent with the set µ(t); (b) starting with these consistent initial conditions we evolve the fine scale de- scription for a (relatively short) time δt; we subsequently restrict back to coarse observables by evaluating the mo- ments µ(t+ δt) as ensemble-averages and (d) use the re- sults to estimate locally the time derivative dµ/dt. This is precisely the right hand-side of the explicitly unavail- able closure, obtained not through a closed form formula, but rather through short, judicious computational exper- iments with the original fine scale dynamics/code. Given this local estimate of the coarse-grained observable time derivatives, we can now exploit the smoothness of their evolution in time (in the form of Taylor series) and take a single long projective forward Euler step: µ(t+∆t) = µ(t) + ∆t µ(t+ δt)− µ(t) . (11) The procedure then repeats itself: lifting, fine scale evo- lution, restriction, estimation, and then (connecting with continuum traditional numerical analysis) a new for- ward Euler step. Beyond coarse projective forward Eu- ler, many other coarse initial-value solvers (e.g. coarse projective Adams-Bashforth, and even implicit coarse solvers) have been implemented; the stability and accu- racy study of such algorithms is progressing [14]. These developments allow us to construct a nonintrusive imple- mentation of PDF moment closures, nonintrusive in the sense that we compute with the closures without explic- itly obtaining them, but rather by intelligently chosen computational experiments with the original, fine-scale problem. There is, however, an obvious objection to the equation-free implementation of moment-closures. Using the same ingredients, one can clearly obtain an estimate of any statistics of interest (for example, the moment- averages µ(t)) without the need of making any closure assumptions whatsoever. This can be done by the much simpler method of direct ensemble averaging. That is, one can sample an ensemble of initial conditions X from any chosen distribution P0(X), evolve each of these real- izations according to the fine-scale dynamics of equation (3), and then evaluate any statistics of interest at time t by averaging over the ensemble of solutions X(t). It would seem that this direct ensemble approach is much more straightforward and accurate than the equation-free implementation of a moment-closure, which introduces additional statistical hypotheses. The response to this important objection is that the fine-scale dynamics (3) is often very stiff for the appli- cations considered, in which the system contains many- degrees-of-freedom interacting on a huge range of length- and time-scales. In contrast, the closure equation (10) is much less stiff, because of statistical-averaging, and its solutions µ(t) are much smoother in time (and space). Thus, to evolve an ensemble of solutions of the fine-scale dynamics (3) from an initial time t0 to a final time t0+T would require O(T/δt) integration steps, where the time- step δt is required to be very small by the intrinsic stiff- ness of the micro-dynamics. In the closure approach, the evolution of the moment equations (10) from time t0 to time t0+T requires only O(T/∆t) integration steps, with (hopefully) ∆t ≫ δt. Each of these closure integration steps by an increment ∆t requires in the equation-free approach just one (or just a few) fine-scale integration step by an increment δt. Thus, there is an over-all savings by a (hopefully) large factor O(∆t/δt). This crude esti- mate is based on a single step coarse projective forward Euler algorithm; clearly, more sophisticated projective integration algorithms can be used. In all of them, however, the computational savings are predicated on the smoothness of the closure equations, and are governed by the ratio of the time that it takes to obtain a good local estimate of dµ/dt from full direct simulation to the time that we can (linearly or even poly- nomially) extrapolate µ(t) in time. It is also worth not- ing that a variety of additional computational tasks, be- yond projective integration (e.g. accelerated fixed point computation) can be performed within the equation-free framework In the next section we show by a concrete example how significant computational economy can be achieved with statistical moment closures implemented in the equation- free framework. A NUMERICAL EXAMPLE We illustrate here the equation-free implementation of moment-closures for a canonical equation of phase- ordering kinetics [17], the stochastic time-dependent Ginzburg-Landau (TDGL) equation in one spatial di- mension. This is written as ∂φ(x, t) = D∆φ(x, t) − V ′(φ(x, t)) + η(x, t) (12) where φ(x, t) represents a local order parameter, e.g. a magnetization. The noise has mean zero and covariance 〈η(x, t)η(x′, t′)〉 = 2kT δ(x−x′)δ(t− t′). The potential V shall be chosen as V (φ) = to represent a single quartic/quadratic well. This stochastic dynamics has an invariant measure which is formally of Hamiltonian form P∗[φ] ∝ exp(−H [φ]/kT ) where H [φ] = D|∇φ(x)|2 + V (φ(x))] dx. (13) The Gibbsian measure P∗[φ] is approached at long times for any random distribution P0[φ] of initial states. One of the simplest dynamical quantities of interest is the bulk magnetization φ(t) = (1/V ) φ(x, t)dx, where V is the total volume. If the initial statistics are space- homogeneous, then the ensemble average µ(t) = 〈φ(t)〉 is also given by µ(t) = 〈φ(x, t)〉 for any space point x. Equa- tion (12) leads to a hierarchy of equations for statistical moments of φ(x, t). For example, the first moment satis- fies the equation ∂〈φ(x, t)〉 = ∆〈φ(x, t)〉 − 〈φ(x, t)〉 − 〈φ3(x, t)〉. (14) The evolution of the mean total magnetization is thus a function of the mean cubic total magnetization. One could write a time evolution equation for 〈φ3〉, but it would involve a higher order term 〈φ5〉, and so on. Each equation contains higher moments and therefore the hi- erarchy does not close. To close the equation for µ(t) we assume a parametric PDF of the form P [φ;α] ∝ exp(−H [φ;α]/kT ) where H [φ;α] = H [φ] + α φ(x) dx is a perturbation of the Hamiltonian (13) by a term pro- portional to the moment variable ξ[φ] = (1/V ) φ(x) dx. This is a special case of a general “entropy-based” clo- sure prescription proposed by Eyink and Levermore [16]. This closure scheme guarantees that α(t) → 0 at long times and therefore the PDF ansatz P [φ;α(t)] relaxes to the correct stationary distribution P∗[φ] of the stochastic process. The determination of the parameter α given the moment µ is here accomplished by Legendre transform α = argmaxα[αµ− F (α)], (15) where the “moment-generating function” F (α) = log〈exp[α φ(x) dx]〉∗ and 〈·〉∗ denotes average with re- spect to the invariant measure P∗[φ]. The numerical op- timization required for the Legendre transform is well- suited to gradient descent algorithms such as the conju- gate gradient method, since (∂/∂α)[αµ− F (α)] = µ− µ(α), where µ(α) = 〈ξ〉α is the average of the moment-function in the PDF ansatz P [φ;α]. In simple cases, F (α) and µ(α) = F ′(α) may be given by closed analytical expres- sions. If not, then both of these averages may be deter- mined together by Monte Carlo sampling techniques. In the numerical calculations below, we discretize equa- tion (12) using a forward Euler-Maruyama stochastic in- tegrator and 3-point stencil for the Laplacian (other dis- cretizations are possible). φ(x, t + δt) = φ(x, t)− δt[φ(x, t) + φ3(x, t)] + (16) (δx)2 [φ(x + δx, t)− 2φ(x, t) + φ(x− δx, t)] + 2kT (δt/δx)N(x, t) where N(x, t) are independent, identically distributed standard normal random variables for each space-time point (x, t). The invariant distribution of the stochastic dynamics space-discretized in this manner has a Gibbsian form ∝ exp(−Hδ/kT ) with discrete Hamiltonian 〈x,x′〉 (φ(x) − φ(x′))2 (17) φ2(x) + φ4(x)] where 〈x, x′〉 are nearest-neighbor pairs. The closure ansatz can be adopted in the consistently discretized form Pδ[φ;α] ∝ exp(−Hδ[φ;α]/kT ) where Hδ[φ;α] = Hδ[φ] + α δxφ(x). In this numerical experiment, we integrate an N = 1000 member ensemble of solutions of equation (17), and measure the ensemble-averaged, global magnetiza- tion µ(t) = 〈φ(t)〉 = (1/V ) 〈φ(x, t)〉 at each time- step. With this we compare the results of the entropy- based closure simulation implemented by the equation- free framework using also an ensemble with N = 1000 samples. In this concrete example, the projective inte- gration scheme works as follows: Suppose we are given the parameter α(t) at time t. The mean µ(t) is first cal- culated from the parametric ensemble at time t by Monte Carlo sampling. Next all N samples are integrated over a short time-step δt to create a time-advanced ensemble. From this ensemble µ(t + δt) is calculated, which yields an estimate of the local time derivative. µ̇app(t) = [µ(t+ δt)− µ(t)]/δt. A large, projective Euler time-step of the moment aver- age is then taken via µ(t+∆t) = µ(t) + ∆t µ̇app(t). The parameter is finally updated by using the Legendre transform inversion to obtain α(t+∆t) from the known value µ(t + ∆t). The cycle may now be repeated to in- tegrate the closure equations by successive time-steps of length ∆t. A critical issue in general application of projective inte- gration is the criterion to determine the projective time- step ∆t. For stiff problems with time-scale separation, the projective time step for stability purposes is of the order of (1/fastest “slow group” eigenvalues), while the “preparatory” simulation time is of the order of (1/slow- est “fast group” eigenvalue). Variants of the approach have been developed for problems with several gaps in their spectrum [18]. Accuracy considerations in real-time projective step selection can, in principle, be dealt with in the traditional way for integrators with adaptive step- size selection and error control: through on-line a poste- riori error estimates. An additional “twist” arises from the error inherent in the estimation of the (unavailable) reduced time derivatives from the ensemble simulations; issues of variance reduction and even on-line hypothe- sis testing (are the data consistent with a local linear model?) must be considered. These are important re- search issues that are currently explored by several re- search groups. Nevertheless, the main factor in computa- tional savings comes from the effective smoothness of the unavailable closed equation: the separation of time scales between the low-order statistics we follow and the higher order statistics whose effect we model (and, eventually, the time scales of the direct simulation of the original model). Figure 1 is a plot comparing Projective Integration with Entropy Closure and direct Ensemble Integration with equation (12) for diffusion constant D = 1000.0 We have selected both the “fine-scale” integration step δt and the “coarse-scale” projective integration step ∆t to be as large as possible, consistent with stability and accuracy. Thus, only steps small enough to avoid numerical blow- ups were considered. Then, values were selected both for δt and for ∆t so that the numerical integrations with those time-steps differed by at most a few percent from fully converged integrations with very small steps. In this manner, the time step required for the Euler-Maruyama integration of (12) was determined to be δt = 0.0004. On the other hand, for projective integration of the closure equation a time step ∆t = 0.01 could be taken. This indicates a gain in time step by a factor of 25, which is also roughly the speed-up in the algorithm or savings in CPU time. The present example is not as stiff as equa- tions that appear in more realistic applications, with a very broad range of length- and time-scales, where even greater computational economies might be expected. In general, the moment-closure results need not agree so well with those of the direct ensemble approach, even when both are converged. In the example presented here, there is good agreement because the closure effectively captures the one-point PDF (see Fig.2). This one-point PDF is the only statistical quantity that enters into Equa- tion (14) as long as the statistics are homogeneous and the Laplacian term vanishes. CONCLUSIONS In this paper, we have described how one can combine recently developed equation-free methods with statisti- cal moment closures to model nonlinear problems. With this method we can numerically integrate complex non- linear systems, for which closure equations may not be available in closed form. In the example presented here the specific entropy-based closure we selected has an H- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 FIG. 1: Mean total field as a function of time. Line (sym- bols): traditional (coarse projective) integration, respectively. See the text for a description of the stepsize selection. FIG. 2: Comparison of the time dependent PDF’s of the local field φ(x, t) for the exact solution (blue) and for the projective integration / closure solution (red). theorem which guarantees relaxation to the equilibrium state of the original dissipative dynamics. However, we stress that the general approach outlined above can be used with a variety of closure methods. The equation-free method has the potential to enhance the flexibility, power, and applications set of the statis- tical moment closure approach. Since little or no an- alytic work is required, the sophistication of statistical moment closures can greatly enhanced beyond Gaussian PDF ansätze. The “practical usefulness” criterion for parametric PDF models that they permit analytical cal- culations is replaced by the criterion that they can be efficiently sampled. We believe that this approach can significantly increase the usefulness of closure methods. In order to model systems like global climate, oceans, and reaction diffusion processes in systems biology, one will have to construct more complex closures. These will likely include higher order moments, correlation functions of the relevant variables, highly non-Gaussian statistics, etc. As the closures become more complex, the lifting step will require more efficient sampling ap- proaches. One will likely have to use nonlocal, acceler- ated sampling methods. One will also likely employ the latest in adaptive time and adaptive mesh methods to optimize performance for large-scale problems. ACKNOWLEDGEMENTS This work, LA-UR-07-2218, was carried out in part at Los Alamos National Laboratory under the auspices of the US National Nuclear Security Administration of the US Department of Energy. It was supported un- der contract number DE-AC52-06NA25396. The work of IGK was partially supported by DARPA and by and US DOE(CMPD). G. Eyink was supported by NSF-ITR grant, DMS-0113649. [1] T. Schlick, R. D. Skeel, A. T. Brunger, L. V. Kale, J. A. Board, Jr. , J. Hermans, and K. Schulten J. Comp. Phys. 151, 9, (1999). [2] M. Karplus and J. A. McCammon, Nature, Structural and Molecular Biology, 9 , 646, (2002). [3] P. G. Debenedetti and F. H. Stillinger, Nature, 410, 259, (2001). [4] D. T. Gillespie, J. Phys. Chem., 81 , 2340, (1977). [5] D. J. Wilkinson, Stochastic Modeling for Systems Biol- ogy, Chapman & Hall / CRC Press, Boca Raton, (2006). [6] A. J. Majda and X. Wang, Nonlinear Dynamics and Sta- tistical Theories for Basic Geophysical Flows, Cambridge Univeristy Press, Cambridge UK, 2006 [7] J. P. Hansen and I. R. MacDonald, Theory of Simple Liquids, Academic, New York, (1986). [8] S. B. Pope, Turbulent Flows, Cambridge University Press, Cambridge, UK, (2000). [9] A. J. Chorin, O. H. Hald, and R. Kupferman, Proceedings of the National Academy of Sciences of the United States of America, 97, 2968, (2000). [10] H. Chen, S. Chen, and R. H. Kraichnan, Phys. Rev. Lett., 63, 2657–2660, 1989. [11] C. Yeung, Y. Oono, and A. Shinozaki Phys. Rev. E, 49, 2693 (1994) [12] G. L. Eyink, Phys. Rev. E 54 (1996) 3419–3435. [13] C.D. Levermore, J. Stat. Phys., 86 (1996), 1021–1065. [14] I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg and K. Theodoropoulos, Comm. Math. Sciences 1(4) pp.715-762 (2003); S. L. Lee and C. W. Gear, J. Comp. App. Math., 201, 258, (2007). [15] I. G. Kevrekidis, C. William Gear and G. Hummer, A.I.Ch.E Journal, 50(7) pp.1346-1354 (2004) [16] G. L. Eyink and C. D. Levermore (preprint) Entropy- Based Closures of Nonlinear Stochastic Dynamics. sub- mitted to ”Communications in Mathematical Sciences” (2006). [17] A. J. Bray, Adv. in Phys., 43, 357, (1994) [18] C. W. Gear and I. G. Kevrekidis, J. Comp. Phys., 187, 95, (2003) [19] In the case of [10] the dynamics is an advection-reaction- diffusion equation for a scalar concentration field X(t) = {θ(x, t) : x ∈ Rd}. The moment functions are the “fine- grained PDF” ξϑ,x[X, t] = δ(θ(x, t) − ϑ), labelled by space point x and scalar value ϑ. The moment average µϑ,x(t) = 〈δ(θ(x, t) − ϑ)〉 is the 1-point PDF p(ϑ;x, t) which gives the distribution of scalar values ϑ at space- time point (x, t). The parametric model P [X;α, t] is the distribution over scalar fields obtained by the ansatz θ(x, t) = X(θ0(x, t),x, t) where θ0(x, t) is a reference ran- dom field of known (Gaussian) statistics and X(·,x, t) : R → R is a “mapping function”. The latter function is the “parameter” αϑ0,x(t) = X(ϑ0,x, t) which determines (and is determined by) the “moment” µϑ,x(t) from the re- lation p(X(ϑ0,x, t);x, t)|∂X/∂ϑ0| = p0(ϑ0,x, t). Here p0 is the 1-point PDF of the reference Gaussian field θ0(x, t). The approach of [11] is similar. The problem is phase- ordering dynamics as given, for example, by our equation (12) and X(t) = {φ(x, t) : x ∈ Rd}. The moment func- tions are the quadratic products ξr[X, t] = φ(r, t)φ(0, t), labelled by the displacement r ∈ Rd and the moment av- erages µr(t) are the spatial correlation function C(r, t). The parametric model P [X;α, t] is the distribution ob- tained by the ansatz φ(x, t) = f(u(x, t)) where u(x, t) is a homogeneous Gaussian random field with mean zero and covariance G(r, t) = 〈u(r, t)u(0, t)〉 and f(z) is the sta- tionary planar interface solution of the TDGL equation (12). In this case, it is the auxiliary correlation function G(r, t) which plays the role of the “parameter” αr(t). It is shown in [11] for various cases how this function may be uniquely related to the “moment” µr(t) = C(r, t).

0704.0805

Opportunistic Relay Selection with Limited Feedback

Opportunistic Relay Selection with Limited Feedback Caleb K. Lo, Robert W. Heath, Jr. and Sriram Vishwanath Wireless Networking and Communications Group Department of Electrical and Computer Engineering The University of Texas at Austin, Austin, Texas 78712-0240 Email: [clo, rheath, sriram]@ece.utexas.edu Abstract— It has been shown that a decentralized relay selec- tion protocol based on opportunistic feedback from the relays yields good throughput performance in dense wireless networks. This selection strategy supports a hybrid-ARQ transmission ap- proach where relays forward parity information to the destina- tion in the event of a decoding error. Such an approach, however, suffers a loss compared to centralized strategies that select relays with the best channel gain to the destination. This paper closes the performance gap by adding another level of channel feedback to the decentralized relay selection problem. It is demonstrated that only one additional bit of feedback is necessary for good throughput performance. The performance impact of varying key parameters such as the number of relays and the channel feedback threshold is discussed. An accompanying bit error rate analysis demonstrates the importance of relay selection. I. INTRODUCTION Message forwarding in multihop networks occurs over in- herently lossy wireless links and coding strategies are needed to meet the network QoS requirements. Hybrid-ARQ is such a coding strategy that is especially effective in dense networks, as intermediate nodes can act as relays, forwarding parity information to the destination. If the destination detects uncor- rectable packet errors and broadcasts a retransmission request to the network, the relays are well-positioned to transmit parity information more reliably than the source. Relay selection techniques have been studied extensively in recent years [1]–[8]. In our previous work on this topic, we proposed a decentralized selection strategy for relay selection in dense mesh networks [14], where decoding relays contend to forward parity information to the destination using rate- compatible punctured convolutional (RCPC) codes [10]. Our random access-based approach, which is based on op- portunistic feedback [9], is distinct from centralized strategies that select the relay with the best instantaneous channel gain to the destination [7], [8]. Such centralized strategies, though, yield better throughput than our decentralized approach. Caleb K. Lo was supported by a Microelectronics and Computer Devel- opment (MCD) Fellowship and a Thrust 2000 Endowed Graduate Fellowship through The University of Texas at Austin. Robert Heath was supported in part by the National Science Foundation under grant CNS-626797, the Office of Naval Research under grant number N00014-05-1-0169, and the DARPA IT-MANET program, Grant W911NF-07-1-0028. Sriram Vishwanath was supported by the National Science Foundation under grants CCF-055274, CCF-0448181, CNS-0615061 and CNS-0626903. To close this performance gap, we propose a refinement of our selection strategy via channel feedback. In our previously proposed approach, if a decoding relay successfully sends a “Hello” message to the source in a minislot, it is declared to be the “winner” for that minislot. The source then randomly chooses a relay among the set of all “winners.” In this paper, we refine the relay selection among the set of all “winners” by biasing the selection towards those relays with channel gains to the destination that are above a particular threshold. For example, if the set of “winners” consists of one relay with a channel gain above the threshold and one relay with a channel gain below the threshold, the relay with a channel gain above the threshold is more likely to be chosen by the source than the other relay. We briefly discuss how our previously proposed relay selec- tion strategy differs from the notion of multiuser diversity [11], [12]. The basic premise behind multiuser diversity is that in a system with many users with independently fading channels, the probability that at least one user will have a “good” channel gain to the transmitter is high. Then, the user with the best channel gain to the transmitter can be serviced, which will yield the maximum throughput. In our setup, the analogous approach would be to always choose the decoding relay that has the best channel gain to the destination to forward parity information. Our decentralized approach, though, allows any decoding relay to have a chance of being selected to forward parity information as long as it sends at least one “Hello” message to the source and wins at least one minislot. II. SYSTEM MODEL Consider the setup in Fig. 1. Each relay operates in a half- duplex mode and is equipped with a single antenna. We use boldface notation for vectors. SNR represents the signal-to- noise ratio. |h|2 denotes the absolute square of h. Q(·) is the standard Q-function, and Pr(X ≤ x) denotes the probability that a realization of the random variable X is at most x. Transmission occurs over a set of time slots {t1, ..., tm} which are of equal duration. We use the ARQ/FEC strategy in [10]. Initially, the source has a k-bit message w that is encoded as an n-bit codeword x. The source chooses code- rates from a RCPC code family, say {R1, R2, ..., Rm} where R1 > R2 > · · · > Rm. http://arxiv.org/abs/0704.0805v1 Tx Rx Relay Relay Relay Relay Relay hg,rht,g Relay Relay Fig. 1. Relay network. Before t1, the source and destination perform RTS/CTS- based handshaking to achieve synchronization. During t1, the source transmits a subset x1 of the bits in x such that x1 forms a codeword from the rate-R1 code. The destination observes yr,1 = ht,rx1 + nr,1 (1) while relay i ∈ {1, 2, ...,Kr} observes yi,1 = ht,ix1 + ni,1. (2) Here, ht,i is a Rayleigh fading coefficient for the channel between the source and node i, while ni,j is additive white Gaussian noise with variance N0 at node i during time slot tj . We assume that all fading coefficients are constant over a time slot and vary from slot to slot; we also assume that fading and additive noise are independent across all nodes. In addition, we assume that all nodes have no prior knowledge of fading coefficients and use training data to learn them. The destination attempts to decode yr,1. If decoding is suc- cessful, the destination broadcasts an ACK message to all of the relays and the source. Otherwise, the destination broadcasts a NACK message; the source now has to select one of the relays to forward additional parity information that will assist the destination in recovering w. III. RELAY SELECTION VIA LIMITED FEEDBACK We briefly review our proposed relay selection strategy in [14]. The framing structure for our relay selection strategy is shown in Fig. 2. We assume in Fig. 2 that the destination sends a NACK after t1 and t2 to trigger the relay contention process. Let Rsel denote the set of relays that can participate in the con- tention process. If relay i ∈ Rsel, then relay i has successfully recovered w and has a channel gain to the destination |hi,r|2 that is above a threshold ηopp. Relay i can determine |hi,r|2 by listening to the destination’s NACK, which is embedded in a packet that contains training data. All relays in Rsel use the same K minislots for feedback to the source. During minislot b, each relay i ∈ Rsel sends a “Hello” message to the source with probability pi. We refer to Source Framing Relay Framing Framing Rate-R1 Listen Listen Transmit Chosen Relay ID Listen Listen Transmit Chosen Relay ID Listen Contention Minislots Contention Minislots Listen Listen Listen Chosen Relay Tx Rate-R2 Other Relays Listen Listen Listen Listen ACK or Listen Listen Listen ACK or Listen Listen Fig. 2. Framing structure for decentralized selection strategy. this approach as a 1-bit strategy, where the “Hello” message is an ID number that has been assigned to each relay. Successful contention occurs during minislot b if exactly one relay i ∈ Rsel sends a “Hello” message. The source then declares that relay as the “winner” for minislot b. After minislot K , the source randomly selects one of the “winners” it; if there are no “winners,” the source will transmit during t2. In this work, we modify the 1-bit strategy by appending a check bit to the “Hello” message; the check bit is set to ‘1’ only if |hi,r|2 > βopp for βopp > ηopp. Again, successful contention occurs during minislot b if exactly one relay i ∈ Rsel sends a “Hello” message. We refer to this approach as a 2-bit strategy. After minislot K , if either all of the “winners” sent a check bit of ’0’, all of the winners sent a check bit of ’1’, or there are no “winners,” the 2-bit strategy reduces to the 1-bit strat- egy. Otherwise, the source will randomly select one of the “winners” it that sent a check bit of ’1’ with probability q > 0.5; one of the “winners” it that sent a check bit of ’0’ is randomly selected with probability 1 − q. Thus, the 2-bit strategy refines the 1-bit strategy by further biasing the selection process in favor of the relays with the best channel gains to the destination. During t2, relay it (or the source) transmits a subset x2 of the bits in x such that x1 ∪ x2 forms a codeword from the rate-R2 code in the RCPC family. The destination combines yr,1 with yr,2 = hit,rx2 + nr,2 (3) and attempts to decode yr,1∪yr,2 based on the rate-R2 code. If unsuccessful decoding occurs again, the destination broadcasts another NACK and the contention process repeats until either the destination successfully recovers w or the rate-Rm code has been used without successful decoding. To compute the dimensionless effective rate Ravg of this strategy, we use [10, equation (16)] Ravg = P + lAV where lAV is the average number of additionally transmitted bits per P information bits. Here, M is the memory of the mother code for the RCPC family. We refer to Ravg as the throughput of this strategy in the rest of this paper. For simulation purposes, we employ the path loss model described in [8]; thus, the received energy at node i is Ei = |hb,i|2Ex (5) = (λc/4πd0) 2(db,i/d0) −µEx (6) where Ex is the energy in the transmitted signal x. Here, λc is the carrier wavelength, d0 is a reference distance, db,i is the distance between transmitting node b and receiving node i, and µ is a path loss exponent. We adopt similar simulation parameters as those in [8]. Here, we employ a carrier frequency fc = 2.4GHz, d0 = 1m, dt,r = 100m and µ = 3, where dt,r is the distance between the source and the destination. We then uniformly distribute Kr = 20 relays in the region between the source and the destination such that each relay i is di,r < dt,r units from the destination. We also use the WiMAX signaling bandwidth, which is roughly 9 MHz [15]; given a noise floor of -204 dB/Hz this yields a noise value N0 = −134 dB. BPSK modu- lation is used for all packet transmissions, and all of the relays and the destination use ML decoding. We use the codes of rates {4/5, 2/3, 4/7, 1/2, 1/3} from the M = 6 RCPC family in [10]. We perform concatenated coding, where the outer code is a (255, 239) Reed-Solomon code with symbols from GF (28); this code can correct at most 8 errors. The mother code for the RCPC family is a rate- 1/3 convolutional code with constraint length 7 and generator polynomial (145 171 133) in octal notation. In this section and in Section IV, we define the average received SNR at the destination as follows. Assume that the source uses a transmit energy of Et(γ) during time slot t1 that yields an average SNR γ at the destination; then, all transmitting nodes will use a transmit energy of Et(γ) during all subsequent transmission cycles. Fig. 3 compares the throughput yielded by the 1-bit and 2-bit strategies. We also plot the throughput yielded by the GPS-based HARBINGER method [8] and by a centralized strategy that always selects the decoding relay with the best instantaneous channel gain to the destination to forward parity information. We have K = 10 minislots. For the 1-bit and 2-bit strategies, we set ηopp = −91dB; we also set βopp = −86dB. We set the feedback probability pi = 0.3 for both strategies. In addition, we set the “winner” selection probability q = 0.75 for the 2-bit strategy. We see that the 2-bit strategy closes the performance gap between the 1-bit strategy and the centralized approach. Thus, using a limited amount of channel feedback improves the performance of our relay selection strategy. We also observe that the 1-bit and 2-bit strategies offer comparable performance to the HARBINGER method. Note that the 2-bit strategy outperforms the HARBINGER method for some values of the received SNR. The intuition behind this result is that the HARBINGER method optimizes the av- erage received SNR at the destination by selecting the closest 2 3 4 5 6 7 8 9 10 average received SNR (dB) RCPC family with M = 6, rates {4/5, 2/3, 4/7, 1/2, 1/3}, d = 100m 1−bit 2−bit best gain HARBINGER Fig. 3. Comparison of 1-bit and 2-bit feedback strategies. decoding relay to the destination. This method, though, does not necessarily select the “best” decoding relay that has the highest instantaneous channel gain to the destination. Also, the inherent randomness of the 1-bit and 2-bit strategies allows for the possibility of choosing the “best” decoding relay. Thus, the HARBINGER method does not necessarily outperform our selection strategies for all received SNR values. IV. PERFORMANCE IMPACT OF VARYING SYSTEM PARAMETERS A joint optimization of all of the key system parameters would enable computation of the maximum throughput yielded by the 1-bit and 2-bit strategies. This optimization, though, is fairly difficult to perform; instead, in this section we provide some insight for system designers by varying some of the key parameters in isolation and illustrating the resulting impact on the throughput. Fig. 4 illustrates the throughput of the 2-bit strategy for various values of the check bit threshold βopp. Here we have Kr = 10 relays and K = 3 minislots. We have an average received SNR at the destination of 8dB. We see that if βopp is close to ηopp, the performance of the 2-bit strategy suffers since the 2-bit strategy essentially reduces to the 1-bit strategy. Also, we see that if βopp is too large, the performance of the 2-bit strategy suffers. This is because the probability of selecting a decoding relay i such that |hi,r|2 > βopp decreases as βopp increases, which causes the 2-bit strategy to reduce to the 1-bit strategy again. Thus, it is apparent that there is an optimal value of βopp for each value of the average received SNR that maximizes the throughput of the 2-bit strategy. Fig. 5 illustrates the throughput of the 1-bit strategy for a varying number of relay nodes. We have K = 3 minislots and an average received SNR of 6dB at the destination. We see that there is an optimal number of relay nodes for which the throughput is maximized. Note that if the number of relay nodes is small, the probability that any of them decode the source message and send a “Hello” message to the source is also small. On the other hand, if the number of relay nodes is large, the probability that at least two relays decode the source message and attempt to send a “Hello” message to the −98 −96 −94 −92 −90 −88 −86 −84 check bit threshold β average received SNR of 8dB, K = 10, K = 3, p = 0.3 Fig. 4. Throughput as a function of check bit threshold. source in each minislot is also large; thus, a collision is likely to occur in each minislot. Fig. 6 also illustrates the effect on the performance of the 1-bit strategy of varying the number of relay nodes. Instead of considering the throughput, though, we consider the bit error rate (BER); we focus on transmission during time slot t2 where the rate-2/3 code from the RCPC family is used. Here we have K = 2 minislots and we set the feedback threshold ηopp = −98dB. Again, we notice that the performance of the 1-bit strategy suffers when the number of relay nodes is either small or large. V. BER ANALYSIS Assume that we employ Viterbi decoding at the relays and at the destination. Recall that P is the puncturing period of the RCPC family. Let Pd be the probability that an incorrect path of weight d is selected by the Viterbi decoder, and let dfree be the free distance of the member of the RCPC family that is currently being used for decoding. Also, let cd be the total number of non-zero information bits on all paths of weight d. From [10, equation (9)] we see that the bit error rate Pb can 4 6 8 10 12 14 16 18 20 0.475 0.485 0.495 0.505 0.515 number of relays K average received SNR of 6dB, K = 3 minislots, p = 0.3 Fig. 5. Throughput as a function of number of relay nodes. 2 4 6 8 10 12 14 16 18 20 0.065 0.075 0.085 0.095 0.105 number of relays K average received SNR of 4dB, K = 2 minislots, p = 0.3 Fig. 6. Bit error rate as a function of number of relay nodes. be upper-bounded as d=dfree cdPd. (7) Let γr denote the received SNR at the destination. Since we are essentially dealing with a binary-input AWGN channel with binary output quantization, we use [13, equation (12.39b)] to see that Pb can be further upper-bounded as d=dfree p(1− p) d=dfree cd · (9) Since g(γr) = Q( 2γr)(1−Q( 2γr)) is a monotonically decreasing function for non-negative γr, we see that Pb mono- tonically decreases for increasing values of the received SNR. This demonstrates the utility of relay selection, as transmission from relay nodes will yield a higher average received SNR at the destination than transmission from the source. To illustrate this point, consider the following simple ex- ample. We have the same simulation parameters as in Section III, except that now we have Kr = 1 relay, K = 1 minislot and a feedback probability pi = 1. We place this relay at a location that is 25 meters from the source and 75 meters from the destination. During time slot t1, the source uses a transmit energy that is 101dB above the noise floor N0, which yields an average received SNR at the destination of γt,r = 0.952dB. We consider the 1-bit strategy here and set ηopp = −91dB. Consider time slot t2, where we assume that the destination did not successfully recover w during t1. Now, if the relay is selected to forward parity information during t2, the average received energy at the destination is 3 · 108 2.4 · 109 10(−134+101)/10 ≈ 1.17 · 10−13. Thus, we have an average received SNR at the destination of γ1,r = Er/N0 ≈ 4.7dB. From [10] we can determine the bit weight enumerating function (WEF) weights cd for the rate-2/3 code from the RCPC family. In particular, we see from [10, Table II(c)] that the only non-zero values of cd are cd = {12, 280, 1140, 5104, 24640, 108512} for d = {6, 7, 8, 9, 10, 11}. Now we substitute these values of cd and d along with γr = γ1,r into (9). We find that the BER Pb is upper-bounded as Pb < 5.42 · 10−4. Since Pr(γr < γ1,r) = 0.368, we want to evaluate the performance of our selection strategy for a wider range of γr. In particular, we find that Pr(γr < 2) = 0.492; if we substitute γr = 2 into (9) we have Pb < 0.0688. On the other hand, assume that the source forwards parity information during t2. If we substitute γr = γt,r into (9), we find that the BER Pb is upper-bounded as Pb < 5.55. Again, since Pr(γr < γt,r) = 0.368, we evaluate the performance of this approach for a wider range of γr. We find that Pr(γr < 0.85) = 0.495; if we substitute γr = 0.85 into (9) we have Pb < 64.7. Thus, it is apparent that relay selection leads to significant gains in BER performance. Since relaying leads to significantly improved BER perfor- mance, we want to determine the probability of relay selection for this example. Here, the relay is selected if it recovers w in t1 and has a channel gain to the destination |h1,r|2 > ηopp. Recall our assumption that all channels in our setup undergo Rayleigh fading. First, the probability that the relay has a sufficiently high channel gain to the destination is e−χ/γ1,rdχ (10) Thus, we only have to consider the probability P1 that the relay recovers w in time slot t1. From [10, equation (20)], the probability Perr that the relay cannot recover w in t1 is Perr < 1− d=dfree where the non-zero values of cd are for the rate-4/5 code from the RCPC family. By using (9), P1 is lower-bounded as P1 = 1− Perr = (1 − Pb)n+M (12) d=dfree cd · (13) )d)n+M In particular, we see from [10, Table II(c)] that the only non- zero values of cd are cd = {24, 376, 3464, 30512, 242734, 1890790} for d = {4, 5, 6, 7, 8, 9}. We have P = 8, n = 2040, M = 6 and the average received SNR at the relay during time slot t1 is γt,1 ≈ 19dB. If we substitute these values of P , n, M and γr = γt,1 into (13) we see that P1 ≈ 1. Again, since Pr(γr < γt,1) = 0.368, we evaluate the performance of our selection strategy for a wider range of γr. In particular, we find that Pr(γr < 5) = 0.0608; if we substitute γr = 5 into (13) we see that P1 > 0.851. Thus, we have a good chance of reaping the benefits of relay selection. VI. CONCLUSION In this paper we presented a strategy for improving the throughput of our previously proposed decentralized relay se- lection protocol. We modified our protocol by using a limited amount of channel feedback to close the performance gap be- tween our protocol and centralized strategies that select the re- lay with the best channel gain to the destination. To understand the performance impact of different system parameters, we presented simulation results and discussed their applicability to system design. We performed a simple BER analysis to further illustrate the gains achieved by relaying. REFERENCES [1] M. Chen, S. Serbetli and A. Yener. Distributed power allocation for parallel relay networks. In Proc. of the IEEE Glob. Telecomm. Conf., 3, St. Louis, MO, November 2005. [2] Z. Lin and E. Erkip. Relay search algorithms for coded cooperative systems. In Proc. of the IEEE Glob. Telecomm. Conf., 3, St. Louis, MO, November 2005. [3] A.K. Sadek, Z. Han and K.J.R. Liu. A distributed relay-assignment algorithm for cooperative communications in wireless networks. In Proc. of the IEEE Intl. Conf. on Commun., 4:1592-1597, Istanbul, Turkey, June 2006. [4] Y. Zhao, R. Adve and T.J. Lim. Improving amplify-and-forward relay networks: optimal power allocation versus selection. In Proc. of the Intl. Symp. on Inform. Theory, 1:1234-1238, Seattle, WA, July 2006. [5] V. Sreng, H. Yanikomeroglu and D.D. Falconer. Relayer selection strategies in cellular networks with peer-to-peer relaying. In Proc. of the IEEE Vehic. Techno. Conf., 3:1949-1953, Orlando, FL, October 2003. [6] A. Bletsas, A. Khisti, D.P. Reed and A. Lippman. A simple cooperative diversity method based on network path selection. IEEE J. Select. Areas Commun., 24(3):659–672, March 2006. [7] J. Luo, R.S. Blum, L.J. Greenstein, L.J. Cimini and A.M. Haimovich. New approaches for cooperative use of multiple antennas in ad hoc wireless networks. In Proc. of the IEEE Vehic. Techno. Conf., 4:2769- 2773, Los Angeles, CA, September 2004. [8] B. Zhao and M.C. Valenti. Practical relay networks: a generalization of hybrid-ARQ. IEEE J. Select. Areas Commun., 23(1):7–18, January 2005. [9] T. Tang and R.W. Heath, Jr. Opportunistic feedback for downlink multiuser diversity. IEEE Comm. Lett., 9(10):948–950, October 2005. [10] J. Hagenauer. Rate-compatible punctured convolutional codes (RCPC codes) and their applications. IEEE Trans. Comm., 36(4):389–400, April 1988. [11] X. Qin and R.A. Berry. Distributed approaches for exploiting multiuser diversity in wireless networks. IEEE Trans. Inform. Theory, 52(2):392– 413, February 2006. [12] P. Viswanath, D.N.C. Tse and R. Laroia. Opportunistic beamforming using dumb antennas. IEEE Trans. Inform. Theory, 48(6):1277–1294, June 2002. [13] S. Lin and D.J. Costello, Jr. Error Control Coding. Pearson Prentice Hall, Upper Saddle River, NJ, 2004. [14] C.K. Lo, R.W. Heath, Jr. and S. Vishwanath. Hybrid-ARQ in multihop networks with opportunistic relay selection. To appear in Proc. of the IEEE Intl. Conf. on Acoustics, Speech and Sig. Proc., Honolulu, HI, April 2007. [15] Wireless MAN Working Group. http://www.wirelessman.org/. http://www.wirelessman.org/ Introduction System Model Relay Selection Via Limited Feedback Performance Impact of Varying System Parameters BER Analysis Conclusion References

0704.0806

The Sloan Digital Sky Survey Quasar Catalog IV. Fifth Data Release

The Sloan Digital Sky Survey Quasar Catalog IV. Fifth Data Release Donald P. Schneider1 Patrick B. Hall2, Gordon T. Richards3,4, Michael A. Strauss5, Daniel E. Vanden Berk1, Scott F. Anderson6, W. N. Brandt1, Xiaohui Fan7, Sebastian Jester8,9, Jim Gray10, James E. Gunn5, Mark U. SubbaRao11, Anirudda R. Thakar3, Chris Stoughton12, Alexander S. Szalay3, Brian Yanny12, Donald G. York13,14, Neta A. Bahcall5, J. Barentine15, Michael R. Blanton16, Howard Brewington15, J. Brinkmann15, Robert J. Brunner17, Francisco J. Castander18, István Csabai19, Joshua A. Frieman20,12,13, Masataka Fukugita21, Michael Harvanek15, David W. Hogg16, Željko Ivezić6, Stephen M. Kent12,13, S. J. Kleinman15, G. R. Knapp5, Richard G. Kron13,12, Jurek Krzesiński22, Daniel C. Long15, Robert H. Lupton5, Atsuko Nitta23, Jeffrey R. Pier24, David H. Saxe25, Yue Shen5, Stephanie A. Snedden15, David H. Weinberg26, Jian Wu1 1Department of Astronomy and Astrophysics, The Pennsylvania State University, 525 Davey Laboratory, University Park, PA 16802. 2 Department of Physics & Astronomy, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada. 3 Department of Physics and Astronomy, The Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218- 2686. 4 Department of Physics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104. 5 Princeton University Observatory, Peyton Hall, Princeton, NJ 08544. 6 Department of Astronomy, University of Washington, Box 351580, Seattle, WA 98195. 7 Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721. 8 School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK. 9 Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany. 10 Microsoft Research, 301 Howard Street, No. 830, San Francisco, CA 94105. 11 University of Chicago and Adler Planetarium and Astronomy Museum, 1300 S. Lake Shore Drive, Chicago, IL 60605. 12 Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510. 13 Department of Astronomy and Astrophysics, The University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637. 14 Enrico Fermi Institute, The University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637. 15 Apache Point Observatory, P.O. Box 59, Sunspot, NM 88349. 16 Department of Physics, New York University, 4 Washington Place, New York, NY 10003. 17 Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801-3080. 18 Institut de Ciències de l’Espai (CSIC-IEEC), Campus UAB, 08193 Bellaterra, Barcelona, Spain. 19 Department of Physics of Complex Systems, Eötvös University, Budapest, Pf. 32, H-1518 Budapest, Hungary. 20 Center for Cosmological Physics, The University of Chicago, 5640 South Ellis Avenue Chicago, IL 60637. 21 Institute for Cosmic Ray Research, University of Tokyo, 5-1-5 Kashiwa, Kashiwa City, Chiba 277-8582, Japan. 22 Mt. Suhora Observatory, Cracow Pedagogical University, ul. Podchorazych 2, 30-084 Cracow, Poland. 23 Gemini Observatory, 670 North A’ohoku Place, Hilo, HI 96720. 24 US Naval Observatory, Flagstaff Station, P.O. Box 1149, Flagstaff, AZ 86002. 25 490 Wilson’s Crossing Road, Auburn, NH 03032. 26 Department of Astronomy, Ohio State University, 140 West 18th Avenue, Columbus, OH 43210-1173. http://arxiv.org/abs/0704.0806v1 – 2 – ABSTRACT We present the fourth edition of the Sloan Digital Sky Survey (SDSS) Quasar Catalog. The catalog contains 77,429 objects; this is an increase of over 30,000 entries since the previous edition. The catalog consists of the objects in the SDSS Fifth Data Release that have luminosities larger than Mi = −22.0 (in a cosmology with H0 = 70 km s −1 Mpc−1, ΩM = 0.3, and ΩΛ = 0.7), have at least one emission line with FWHM larger than 1000 km s−1 or have interesting/complex absorption features, are fainter than i ≈ 15.0, and have highly reliable redshifts. The area covered by the catalog is ≈ 5740 deg2. The quasar redshifts range from 0.08 to 5.41, with a median value of 1.48; the catalog includes 891 quasars at redshifts greater than four, of which 36 are at redshifts greater than five. Approximately half of the catalog quasars have i < 19; nearly all have i < 21. For each object the catalog presents positions accurate to better than 0.2′′ rms per coordinate, five-band (ugriz) CCD-based photometry with typical accuracy of 0.03 mag, and information on the morphology and selection method. The catalog also contains basic radio, near-infrared, and X-ray emission properties of the quasars, when available, from other large-area surveys. The calibrated digital spectra cover the wavelength region 3800–9200 Å at a spectral resolution of ≃ 2000; the spectra can be retrieved from the public database using the information provided in the catalog. The average SDSS colors of quasars as a function of redshift, derived from the catalog entries, are presented in tabular form. Approximately 96% of the objects in the catalog were discovered by the SDSS. Subject headings: catalogs, surveys, quasars:general 1. Introduction This paper describes the Fourth Edition of the Sloan Digital Sky Survey (SDSS; York et al. 2000) Quasar Catalog. Previous versions of the catalog (Schneider et al. 2002, 2003, 2005; hereafter Papers I, II, and III) were published with the SDSS Early Data Release (EDR; Stoughton et al. 2002), the SDSS First Data Release (DR1; Abazajian et al. 2003), and the SDSS Third Data Release (DR3; Abazajian et al. 2005), and contained 3,814, 16,713, and 46,420 quasars, respectively. The current catalog is the entire set of quasars from the SDSS-I Quasar Survey; the SDSS-I was completed on 30 June 2005 and the Fifth Data Release (DR5; Adelman-McCarthy et al. 2007) was made public on 30 June 2006. The catalog contains 77,429 quasars, the vast majority of which were discovered by the SDSS. The SDSS Quasar Survey is continuing via the SDSS-II Legacy Survey, which is is an extension of the SDSS-I. The catalog in the present paper consists of the DR5 objects that have a luminosity larger than Mi = −22.0 (calculated assuming an H0 = 70 km s −1 Mpc−1, ΩM = 0.3, ΩΛ = 0.7 cosmology [Spergel et al. 2006], which will be used throughout this paper), and whose SDSS spectra contain at least one broad emission line (velocity FWHM larger than ≈ 1000 km s−1) or have interesting/complex absorption-line fea- tures. The catalog also has a bright limit of i ≈ 15.0. The quasars range in redshift from 0.08 to 5.41; 78% have redshifts below 2.0. The objects are denoted in the catalog by their DR5 J2000 coordinates; the format for the object name is SDSS Jhhmmss.ss+ddmmss.s. Since the image data used for the astrometric information can change between data releases (e.g., a region with poor seeing that is included in an early release is superseded by a newer observation in good seeing), the coordinates for an object can change at the 0.1′′ to 0.2′′ level; hence – 3 – the designation of a given source can change between data releases. Except on very rare occasions (see §5.1), this change in position is much less than 1′′. When merging SDSS Quasar Catalogs with previous databases one should always use the coordinates, not object names, to identify unique entries. The DR5 catalog does not include classes of Active Galactic Nuclei (AGN) such as Type 2 quasars, Seyfert galaxies, and BL Lacertae objects; studies of these sources in the SDSS can be found in Zakamska et al. (2003) (Type 2), Kauffmann et al. (2003) and Hao et al. (2005) (Seyferts), and Collinge et al. (2005) and Anderson et al. (2007) (BL Lacs). Spectra of the highest redshift SDSS quasars (z > 5.7; e.g., Fan et al. 2003, 2006a) were not acquired as part of the SDSS quasar survey (the objects were identified as candidates in the SDSS imaging data, but the spectra were not obtained with the SDSS spectrographs), so they are not included in the catalog. The observations used to produce the catalog are presented in Section 2; the construction of the catalog and the catalog format are discussed in Sections 3 and 4, respectively. Section 5 presents an overview of the catalog, and a summary is given in Section 6. The catalog is presented in an electronic table in this paper and can also be found at an SDSS public web site.1 2. Observations 2.1. Sloan Digital Sky Survey The Sloan Digital Sky Survey uses a CCD camera (Gunn et al. 1998) on a dedicated 2.5-m telescope (Gunn et al. 2006) at Apache Point Observatory, New Mexico, to obtain images in five broad optical bands (ugriz; Fukugita et al. 1996) over approximately 10,000 deg2 of the high Galactic latitude sky. The sur- vey data-processing software measures the properties of each detected object in the imaging data in all five bands, and determines and applies both astrometric and photometric calibrations (Pier et al., 2003; Lupton et al. 2001; Ivezić et al. 2004). Photometric calibration is provided by simultaneous observations with a 20-inch telescope at the same site (see Hogg et al. 2001, Smith et al. 2002, Stoughton et al. 2002, and Tucker et al. 2006). The SDSS photometric system is based on the AB magnitude scale (Oke & Gunn 1983). The catalog contains photometry from 204 SDSS imaging runs acquired between 19 September 1998 (Run 94) and 13 May 2005 (Run 5326). 2.2. Target Selection The SDSS filter system was designed to identify quasars at redshifts between zero and approximately six; most quasar candidates are selected based on their location in multidimensional SDSS color-space. The Point Spread Function (PSF) magnitudes are used for the quasar target selection, and the selection is based on magnitudes and colors that have been corrected for Galactic extinction (using the maps of Schlegel, Finkbeiner, & Davis 1998). An i magnitude limit of 19.1 is imposed for candidates whose colors indicate a probable redshift of less than≈ 3.0 (selected from the ugri color cube); high-redshift candidates (selected from the griz color cube) are accepted if i < 20.2 and the source is unresolved. The errors on the i measurements are typically 0.02–0.03 and 0.03–0.04 magnitudes at the brighter and fainter limits, respectively. In addition 1http://www.sdss.org/dr5/products/value added/qsocat dr5.html http://www.sdss.org/dr5/products/value$_$added/qsocat$_$dr5.html – 4 – to the multicolor selection, unresolved objects brighter than i = 19.1 that lie within 2.0′′ of a FIRST radio source (Becker, White, & Helfand 1995) are also identified as primary quasar candidates. Target selection also imposes a maximum brightness limit (i ≈ 15.0) on quasar candidates; the spectra of objects that exceed this brightness could contaminate the adjacent spectra on the detectors of the SDSS spectrographs. A detailed description of the quasar selection process and possible biases can be found in Richards et al. (2002a). The primary sample described above was supplemented by quasars that were targeted by the following SDSS spectroscopic target selection algorithms: Galaxy and Luminous Red Galaxy (Strauss et al. 2002 and Eisenstein et al. 2001), X-ray (object near the position of a ROSAT All-Sky Survey [RASS; Voges et al. 1999, 2000] source; see Anderson et al. 2003), Star (point source with a color typical of an interesting class of star), or Serendipity (unusual color or FIRST matches). The SDSS is designed to be complete in the Galaxy, Luminous Red Galaxy and Quasar programs, (in practice various limitations reduce the completeness to about 90%) but no attempt at completeness was made for the other categories. Most of the DR5 quasars that fall below the magnitude limits of the quasar survey were selected by the serendipity algorithm (see §5). While the bulk of the catalog objects targeted as quasars were selected based on the algorithm of Richards et al. (2002a), during the early years of the SDSS the quasar selection software was undergoing constant modification to improve its efficiency. All of the sources in Papers I and II, and some of the Paper III objects, were not identified with the final selection algorithm. Once the final target selection software was installed, the algorithm was applied to the entire SDSS photometric database. Each DR5 quasar has two spectroscopic target selection flags listed in the catalog: BEST, which refers to the final algorithm, and TARGET, which is the target flag used in the actual spectroscopic targeting. There are also two sets of photometric measurements for each quasar: BEST, which refers to the measurements with the latest photometric software on the highest quality data, and TARGET, which are the values used at the time of the spectroscopic target selection. Extreme care must be exercised when constructing statistical samples from this catalog; if one uses the values produced by only the latest version of the selection software, not only must one drop the catalog quasars that were not identified as quasar candidates by the final selection software, one must also account for quasar candidates produced by the final version that were not observed in the SDSS spectroscopic survey (this can occur in regions of sky whose spectroscopic targets were identified by early versions of the selection software). The selection for the UV-excess quasars, which comprise the majority (≈ 80%) of the objects in the DR5 Catalog, has remained reasonably uniform; the changes to the selection algorithm were primarily designed to increase the effectiveness of the identification of 3.0 < z < 3.8 quasars. Extensive discussion of the completeness and efficiency of the selection can be found in Richards et al. (2002a) and Vanden Berk et al. (2005); Richards et al. (2006) describes the process for the construction of statistical SDSS quasar samples (see also Adelman-McCarthy et al. 2007). The survey efficiency (the ratio of quasars to quasar candidates) for the ultraviolet excess-selected candidates, which comprise the bulk of the quasar sample, is about 77%. (The catalog contains information on which objects can be used in a uniform sample; see Section 4.) 2.3. Spectroscopy Spectroscopic targets chosen by the various SDSS selection algorithms (i.e., quasars, galaxies, stars, serendipity) are arranged onto a series of 3◦ diameter circular fields (Blanton et al. 2003). Details of the spectroscopic observations can be found in York et al. (2000), Castander et al. (2001), Stoughton et al. (2002), – 5 – and Paper I. A total of 1458 spectroscopic fields, taken between 5 March 2000 and 14 June 2005, provided the quasars for the DR5 quasar catalog; the locations of the plate centers can be found from the information given by Adelman-McCarthy et al. (2007). The DR5 spectroscopic program attempted to cover, in a well- defined manner, an area of ≈ 5740 deg2. Spectroscopic plate 716 was the first spectroscopic observation that was based on the final version of the quasar target selection algorithm of Richards et al. (2002a); the detailed tiling information in the SDSS database must be consulted to identify those regions of sky targeted with the final selection algorithm (see Richards et al. 2006). The two SDSS double spectrographs produce data covering 3800–9200 Å at a spectral resolution of ≃ 2000. The data, along with the associated calibration frames, are processed by the SDSS spectroscopic pipeline (see Stoughton et al. 2002). The calibrated spectra are classified into various groups (e.g., star, galaxy, quasar), and redshifts are determined by two independent software packages. Objects whose spectra cannot be classified by the software are flagged for visual inspection. Figure 1 shows the calibrated SDSS spectra of four previously unknown catalog quasars representing a range of properties. The processed DR5 spectra have not been corrected for Galactic extinction. 3. Construction of the SDSS DR5 Quasar Catalog The quasars in the catalog were drawn from three sets of SDSS observations: 1) the primary survey area, 2) “Bonus” plates, which are spectroscopic observations of regions near to, but outside of, the primary survey area, and 3) “Special” plates, where the spectroscopic targets were not chosen by the standard SDSS target selection algorithms (e.g., a set of plates to investigate the structure of the Galaxy; see Adelman-McCarthy et al. 2006). The DR5 quasar catalog was constructed, as were the previous editions, in three stages: 1) Creation of a quasar candidate database, 2) Visual examination of the spectra of the quasar candidates, and 3) Application of luminosity and emission-line velocity width criteria. All three tasks were initially done without reference to the material in the previous SDSS Quasar Catalogs, although the results of each task were compared to the Paper III database (e.g., the construction of the quasar database was not viewed as complete until it was understood why any Paper III quasars were not included). 3.1. Creation of the DR5 Quasar Candidate Database This catalog of bona-fide quasars, that have redshifts checked by eye and luminosities and line widths that meet the formal quasar definition, is constructed from a larger “master” table of confirmed quasars and quasar candidates. This master table was created using an SQL query to the public SDSS-DR5 database (i.e., the Catalog Archive Server [CAS]; http://cas.sdss.org/astrodr5/). Two versions of the photometric database exist, which contain the properties of objects when targeted for spectroscopic observations (TARGET) and as determined in the latest processing (BEST). These databases are divided into multiple tables and subtables to facilitate access to only the most relevant data for a particular use. In the case of the quasar catalog construction, we have made use of the PhotoObjAll and SpecObjAll tables, which contain, respectively, the photometric information for all SDSS sources and for all SDSS spectra. In the case of PhotoObjAll, both the TARGET and BEST versions are queried. These tables include duplicate observations of objects and observations of objects that lie outside of the formal SDSS area (as compared to the PhotoObj and SpecObj tables, which include only sources in the formal SDSS area), and are the most complete database files. For http://cas.sdss.org/astrodr5/ – 6 – example, in PhotoObjAll, two (or more) observations of a single object may exist; if so, one is classified as PRIMARY, the other(s) as SECONDARY. This master table contains all objects identified as quasar candidate targets for spectroscopy in either the TARGET or BEST photometric databases. Quasar candidates are those objects which have had one or more of the following flags set by the algorithm described by Richards et al. (2002a): TARGET QSO HIZ OR TARGET QSO CAP OR TARGET QSO SKIRT OR TARGET QSO FIRST CAP OR TARGET QSO FIRST SKIRT ( = 0x0000001F, except for the “special” plates [see Adelman-McCarthy et al. 2006, 2007], where additional care is required in interpreting the flags). Objects flagged as TARGET QSO MAG OUTLIER and TARGET QSO REJECT are not included, as these flags are meant only for diagnostic purposes. (In the CAS documentation and the EDR paper, TARGET QSO MAG OUTLIER is called TARGET QSO FAINT.) Furthermore, the master table includes any objects with spectra that have been classified by the spectroscopic pipeline as quasars (specClass=QSO or HIZ QSO), that have UNKNOWN type, or that have redshifts greater than 0.6. (On rare occasions the spectroscopic pipeline measures the correct redshift for a quasar but classifies the object as a galaxy.) The query was run on the union of the database tables Target..PhotoObjAll, Best..SpecObjAll, and Best..PhotoObjAll. Multiple entries for a given object are retained at this stage. Ten objects in the DR3 Quasar Catalog were missed by this query. One omission was due to an “unmapped” fiber (a spectrum of a quasar was obtained, but because of a failure in the mapping of fiber number to location in the sky, we are no longer certain of the celestial position of the object); the other nine were low-redshift AGN that were not classified as quasars by the spectroscopic pipeline (this result provides an estimate of the incompleteness produced by the query). We were able to identify the information for all ten quasars in the database and add the material to the master table. Four automated cuts were made to the master table database of 329,884 candidates 2: 1) Objects targeted as quasars but whose spectra had not yet been obtained by the closing date of DR5 (124,447 objects), 2) Candidates classified with high confidence as “stars” by the spectroscopic pipeline that had redshifts less than 0.002 (33,653), 3) Objects whose photometric measurements have not been loaded into the CAS (3106) and 4) Multiple spectra (coordinate agreement better than 1.0′′) of the same object (40,007). In cases of duplicate spectra of an object, the “science primary” spectrum is selected (i.e., the spectrum was obtained as part of normal science operations); when there is more than one science primary observation (or when none of the spectra have this flag set), the spectrum with the highest signal-to-noise ratio (S/N) is retained (see Stoughton et al. 2002 for a description of the science primary flag). These actions produced a list of 128,671 unique quasar candidates. 3.2. Visual Examination of the Spectra The SDSS spectra of the remaining quasar candidates were manually inspected by several of the authors (DPS, PBH, GTR, MAS, and SFA); as in previous papers in this series, we found that the spectroscopic 2The master table is known as the QSOConcordanceALL table, which can be found in the SDSS database; see http://cas.sdss.org/astrodr5/en/help/browser/description.asp?n=QsoConcordanceAll&t=U. http://cas.sdss.org/astrodr5/en/help/browser/description.asp?n=QsoConcordanceAll – 7 – pipeline redshifts and classifications of the overwhelming majority of the objects are accurate. Tens of thousands of objects were dropped from the list because they were obviously not quasars (these objects tended to be low S/N stars, unusual stars, and a mix of absorption-line and narrow emission-line objects); this large number of candidates that are not quasars is due to the inclusive nature of our initial database query. Spectra for which redshifts could not be determined (low signal-to-noise ratio or subject to data- processing difficulties) were also removed from the sample. This visual inspection resulted in the revisions of the redshifts of 863 quasars; the changes in the individual redshifts were usually quite substantial, due to the spectroscopic pipeline misidentifying emission lines. An independent determination of the redshifts of 5,865 quasars with redshifts larger than 2.9 in the catalog was performed by Shen et al. (2007). The redshift differences between the two sets of measurements follow a Gaussian distribution (with slightly extended wings), with a mean of 0.002 and a dispersion of 0.01. The catalog contains numerous examples of extreme Broad Absorption Line (BAL) Quasars (see Hall et al. 2002); it is difficult if not impossible to apply the emission-line width criterion for these objects, but they are clearly of interest, have more in common with “typical” quasars than with narrow-emission line galaxies, and have historically been included in quasar catalogs. We have included in the catalog all objects with broad absorption-line spectra that meet the Mi < −22.0 luminosity criterion. 3.3. Luminosity and Line Width Criteria As in Papers II and III, we adopt a luminosity limit of Mi = −22.0. The absolute magnitudes were calculated by correcting the BEST i measurement for Galactic extinction (using the maps of Schlegel, Finkbeiner, & Davis 1998) and assuming that the quasar spectral energy distribution in the ultraviolet- optical can be represented by a power law (fν ∝ ν α), where α = −0.5 (Vanden Berk et al. 2001). (In the 134 cases where BEST photometry was not available, the TARGET measurements were substituted for the absolute magnitude calculation.) This approach ignores the contributions of emission lines and the observed distribution in continuum slopes. Emission lines can contribute several tenths of a magnitude to the k- correction (see Richards et al. 2006), and variations in the continuum slopes can introduce a magnitude or more of error into the calculation of the absolute magnitude, depending upon the redshift. The absolute magnitudes will be particularly uncertain at redshifts near and above five, when the Lyman α emission line (with a typical observed equivalent width of ≈ 400− 500 Å) and strong Lyman α forest absorption enter the i bandpass. Quasars near the Mi = −22.0 luminosity limit are often not enormously brighter in the i-band than the starlight produced by the host galaxy. Although the PSF-based SDSS photometry presented in the catalog are less susceptible to host galaxy contamination than are fixed-aperture measurements, the nucleus of the host galaxy can still contribute appreciably to this measurement for the lowest luminosity entries in the catalog (see Hao et al. 2005). An object of Mi = −22.0 will reach the i = 19.1 “low-redshift” selection limit at a redshift of ≈ 0.4. After visual inspection and application of the luminosity criterion had reduced the number of quasar candidates to under 80,000 objects, the remaining spectra were processed with an automated line-measuring routine. The spectra for objects whose maximum line width was less than 1000 km s−1 were visually examined; if the measurement was deemed to be an accurate reflection of the line (automated routines occasionally have spectacular failures when dealing with complex line profiles), the object was removed from the catalog. – 8 – 4. Catalog Format The DR5 SDSS Quasar Catalog is available in three types of files at the SDSS public web site listed in the introduction: 1) a standard ASCII file with fixed-size columns, 2) a gzipped compressed version of the ASCII file (which is smaller than the uncompressed version by a factor of more than four), and 3) a binary FITS table format. The following description applies to the standard ASCII file. All files contain the same number of columns, but the storage of the numbers differs slightly in the ASCII and FITS formats; the FITS header contains all of the required documentation. Table 1 provides a summary of the information contained in each of the columns in the ASCII catalog. The standard ASCII catalog (Table 2 of this paper) contains information on 77,429 quasars in a 36 MB file. The DR5 format is similar to that of DR3 with a few minor differences. The first 80 lines consist of catalog documentation; this is followed by 77,429 lines containing information on the quasars. There are 74 columns in each line; a summary of the information is given in Table 1 (the documentation in the ASCII catalog header is essentially an expansion of Table 1). At least one space separates all the column entries, and, except for the first and last columns (SDSS designation and the object name if previously known), all entries are reported in either floating point or integer format. Notes on the catalog columns: 1) The DR5 object designation, given by the format SDSS Jhhmmss.ss+ddmmss.s; only the final 18 char- acters are listed in the catalog (i.e., the “SDSS J” for each entry is dropped). The coordinates in the object name follow IAU convention and are truncated, not rounded. 2–3) The J2000 coordinates (Right Ascension and Declination) in decimal degrees. The positions for the vast majority of the objects are accurate to 0.1′′ rms or better in each coordinate; the largest expected errors are 0.2′′ (see Pier et al 2003). The SDSS coordinates are placed in the International Celestial Reference System, primarily through the United States Naval Observatory CCD Astrograph Catalog (Zacharias et al. 2000), and have an rms accuracy of 0.045′′ per coordinate. 4) The quasar redshifts. A total of 863 of the CAS redshifts were revised during our visual inspection. A detailed description of the redshift measurements is given in Section 4.10 of Stoughton et al. (2002). A comparison of 299 quasars observed at multiple epochs by the SDSS (Wilhite et al. 2005) found an rms difference of 0.006 in the measured redshifts for a given object. It is well known that the redshifts of individual broad emission lines in quasars exhibit significant offsets from their systemic redshifts (e.g., Gaskell 1982, Richards et al. 2002b, Shen et al. 2007); the catalog redshifts attempt to correct for this effect in the ensemble average (see Stoughton et al. 2002). 5–14) The DR5 PSF magnitudes and errors (not corrected for Galactic extinction) from BEST photometry for each object in the five SDSS filters. Some of the relevant imaging scans, such as special scans through M31 (see the DR4 and DR5 papers) were never loaded into the CAS, therefore the BEST photometry is not available for them. Thus there are 134 quasars which have entries of “0.000” for their BEST photometric measurements. The effective wavelengths of the u, g, r, i, and z bandpasses are 3541, 4653, 6147, 7461, and 8904 Å, re- spectively (for an α = −0.5 power-law spectral energy distribution using the definition of effective wavelength given in Schneider, Gunn, & Hoessel 1983). The photometric measurements are reported in the natural sys- tem of the SDSS camera, and the magnitudes are normalized to the AB system (Oke & Gunn 1983). The measurements are reported as asinh magnitudes (Lupton, Gunn, & Szalay 1999); see Adelman-McCarthy et – 9 – al. (2007) for additional discussion and references for the accuracy of the photometric measurements. The TARGET PSF photometric measurements are presented in columns 63–72. 15) The Galactic extinction in the u band based on the maps of Schlegel, Finkbeiner, & Davis (1998). For an RV = 3.1 absorbing medium, the extinctions in the SDSS bands can be expressed as Ax = Cx E(B − V ) where x is the filter (ugriz), and values of Cx are 5.155, 3.793, 2.751, 2.086, and 1.479 for ugriz, respectively (Ag, Ar, Ai, and Az are 0.736, 0.534, 0.405, and 0.287 times Au). 16) The logarithm of the Galactic neutral hydrogen column density along the line of sight to the quasar. These values were estimated via interpolation of the 21-cm data from Stark et al. (1992), using the COLDEN software provided by the Chandra X-ray Center. Errors associated with the interpolation are typically expected to be less than ≈ 1× 1020 cm−2 (e.g., see §5 of Elvis, Lockman, & Fassnacht 1994). 17) Radio properties. If there is a source in the FIRST catalog within 2.0′′ of the quasar position, this column contains the FIRST peak flux density at 20 cm encoded as an AB magnitude AB = −2.5 log 3631 Jy (see Ivezić et al. 2002). An entry of “0.000” indicates no match to a FIRST source; an entry of “−1.000” indicates that the object does not lie in the region covered by the final catalog of the FIRST survey. The catalog contains 6226 FIRST matches; 5729 DR5 quasars lie outside of the FIRST area. 18) The S/N of the FIRST source whose flux is given in column 17. 19) Separation between the SDSS and FIRST coordinates (in arc seconds). 20) In cases when the FIRST counterpart to an SDSS source is extended, the FIRST catalog position of the source may differ by more than 2′′ from the optical position. A “1” in column 20 indicates that no matching FIRST source was found within 2′′ of the optical position, but that there is significant detection (larger than 3σ) of FIRST flux at the optical position. This is the case for 2440 SDSS quasars. 21) A “1” in column 21 identifies the 1596 sources with a FIRST match in either columns 17 or 20 that also have at least one FIRST counterpart located between 2.0′′ (the SDSS-FIRST matching radius) and 30′′ of the optical position. Based on the average FIRST source surface density of 90 deg−2, we expect 50–60 of these matches to be chance superpositions. 22) The logarithm of the vignetting-corrected count rate (photons s−1) in the broad energy band (0.1–2.4 keV) in the ROSATAll-Sky Survey Faint Source Catalog (Voges et al. 2000) and the ROSATAll-Sky Survey Bright Source Catalog (Voges et al. 1999). The matching radius was set to 30′′; an entry of “−9.000” in this column indicates no X-ray detection. There are 4133 RASS matches in the DR5 catalog. 23) The S/N of the ROSAT measurement. 24) Separation between the SDSS and ROSAT All-Sky Survey coordinates (in arc seconds). 25–30) The JHK magnitudes and errors from the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) All-Sky Data Release Point Source Catalog (Cutri et al. 2003) using a matching radius of 2.0′′. A – 10 – non-detection by 2MASS is indicated by a “0.000” in these columns. Note that the 2MASS measurements are Vega-based, not AB, magnitudes. The catalog contains 9824 2MASS matches. 31) Separation between the SDSS and 2MASS coordinates (in arc seconds). 32) The absolute magnitude in the i band calculated by correcting for Galactic extinction and assuming H0 = 70 km s −1 Mpc−1, ΩM = 0.3, ΩΛ = 0.7, and a power-law (frequency) continuum index of −0.5. 33) The ∆(g− i) color, which is the difference in the Galactic extinction corrected (g− i) for the quasar and that of the mean of the quasars at that redshift. If ∆(g − i) is not defined for the quasar, which occurs for objects at either z < 0.12 or z > 5.12 the column will contain “−9.000”. See Section 5.2 for a description of this quantity. 34) Morphological information. If the SDSS photometric pipeline classified the image of the quasar as a point source, the catalog entry is 0; if the quasar is extended, the catalog entry is 1. 35) The SDSS SCIENCEPRIMARY flag, which indicates whether the spectrum was taken as a normal science spectrum (SCIENCEPRIMARY= 1) or for another purpose (SCIENCEPRIMARY= 0). The latter category contains Quality Assurance and calibration spectra, or spectra of objects located outside of the nominal survey area. Over 90% of the DR5 entries (69,762 objects) are SCIENCEPRIMARY = 1. 36) This flag provides information on whether the photometric object is designated PRIMARY (1), SECONDARY (2), or FAMILY (3; these are blended objects that have not been deblended). During target selection, only PRIMARY objects are considered (except on occasion for objects located in fields that are not part of the nominal sur- vey area); however, differences between TARGET and BEST photometric pipeline versions make it possible that the BEST photometric object belonging to a spectrum is either not detected at all, or is a non-primary object (see §3.1 above). Over 99% of the catalog entries are PRIMARY; 613 quasars are SECONDARY and 9 are FAMILY. There are 124 quasars with an entry of “0” in this column; each of these is an object that lacks BEST photometry. For statistical analysis, one should use only PRIMARY objects; SECONDARY and FAMILY objects are included in the catalog for the sake of completeness with respect to confirmed quasars. 37) The “uniform selection” flag, either 0 or 1; a “1” indicates that the object was identified as a primary quasar target (37,574 catalog entries) with the final target selection algorithm as given by Richards et al. (2002a). These objects constitute a statistical sample. 38) The 32-bit SDSS target-selection flag from BEST processing (PRIMTARGET; see Table 26 in Stoughton et al. 2002 for details); this is the flag produced by running the selection algorithm of Richards et al. (2002a) on the most recent processing of the image data. The target-selection flag from TARGET processing is found in column 55. 39–45) The spectroscopic target selection breakdown (BEST) for each object. The target selection flag in column 38 is decoded for seven groups: Low-redshift quasar, High-redshift quasar, FIRST, ROSAT, Serendipity, Star, and Galaxy An entry of “1” indicates that the object satisfied the given criterion (see Stoughton et al. 2002). Note that an object can be, and often is, targeted by more than one selection algorithm. The last two columns in Table 3 presents the number of quasars identified by the individual BEST target selection algorithm; the column labeled “Sole” indicates the number of objects that were detected by only one of the seven listed selection algorithms. 46–47) The SDSS Imaging Run number and the Modified Julian Date (MJD) of the photometric observation used in the catalog. The MJD is given as an integer; all observations on a given night have the same integer MJD (and, because of the observatory’s location, the same UT date). For example, imaging run 94 has an – 11 – MJD of 51075; this observation was taken on 1998 September 19 (UT). 48–50) Information about the spectroscopic observation (Modified Julian Date, spectroscopic plate number, and spectroscopic fiber number) used to determine the redshift. These three numbers are unique for each spectrum, and can be used to retrieve the digital spectra from the public SDSS database. 51–54) Additional SDSS processing information: the photometric processing rerun number; the camera column (1–6) containing the image of the object, the field number of the run containing the object, and the object identification number (see Stoughton et al. 2002 for descriptions of these parameters). 55) The 32-bit SDSS target selection flag from the TARGET processing, i.e., the value that was used when the spectroscopic plate was drilled. This may not match the BEST target selection flag because a different versions of the selection algorithm were used, the selection was done with different image data (superior quality data of the field was obtained after the spectroscopic observations were completed), or different processings of the same data were used. Objects with no TARGET flag were either identified as quasars as a result of Quality Assurance observations and/or from special plates with somewhat different targeting criteria (see Adelman-McCarthy 2006). 56–62) The spectroscopic target-selection breakdown (TARGET) for each object; this is the same convention as followed in columns 39–45 for the BEST target-selection flag. 63–72) The DR5 PSF magnitudes and errors (not corrected for Galactic reddening) from TARGET photom- etry. 73) The 64-bit integer that uniquely describes the spectroscopic observation that is listed in the catalog (SpecObjID). 74) Name of object in the NASA/IPAC Extragalactic Database (NED). If there is a source in the NED quasar database within 5.0′′ of the quasar position, the NED object name is given in this column. The NED quasar database contains over 100,000 objects. Occasionally NED will list the SDSS name for objects that were not discovered by the SDSS. 5. Catalog Summary The 77,429 objects in the catalog represent an increase of 31,009 quasars over the Paper III database; of the entries in the new catalog, 74,297 (96.0%) were discovered by the SDSS (with the caveat that NED is not complete). The catalog quasars span a wide range of properties: redshifts from 0.078 to 5.414, 14.94 < i < 22.36 (506 objects have i > 20.5; only 26 have i > 21.0), and−30.27 < Mi < −22.00. The catalog contains 6226, 4133, and 9824 matches to the FIRST, RASS, and 2MASS catalogs, respectively. The RASS and 2MASS catalogs cover essentially all of the DR5 area, but 5729 (7%) of the entries in the DR5 catalog lie outside of the FIRST region. Figure 2 displays the distribution of the DR5 quasars in the i-redshift plane (the 26 objects with i > 21 are not plotted). Objects which NED indicates were previously discovered by investigations other than the SDSS are indicated with open circles. The curved cutoff on the left hand side of the graph is produced by the minimum luminosity criterion (Mi < −22.0). The ridge in the contours at i ≈ 19.1 for redshifts below three reflects the flux limit of the low-redshift sample; essentially all of the large number of z < 3 points with i > 19.1 are quasars selected via criteria other than the primary multicolor sample. – 12 – A histogram of the catalog redshifts is shown in the upper curve in Figure 3. A clear majority of quasars have redshifts below two (the median redshift is 1.48, the mode is ≈ 1.85), but there is a significant tail of objects extending out to redshifts beyond five (zmax = 5.41). The dips in the curve at redshifts of 2.7 and 3.5 arise because the SDSS colors of quasars at these redshifts are similar to the colors of stars; we decided to accept significant incompleteness at these redshifts rather than be overwhelmed by a large number of stellar contaminants in the spectroscopic survey. Improvements in the quasar target selection algorithm since the initial editions of the SDSS Quasar Catalog have increased the efficiency of target selection at redshifts near 3.5 (compare Figure 3 with Paper II’s Figure 4; see Richards et al. 2002a for a discussion of the incompleteness of the SDSS Quasar Survey). This structure in the catalog redshift histogram can be understood by careful modelling of the selection effects (e.g., accounting for emission line effects and using only objects selected in regions whose spectroscopic observations were chosen with the final version of the quasar target selection algorithm; also see Figure 8 in Richards et al. 2006). Repeating the analysis of Richards et al. (2006) for the DR5 sample reveals no structure in the redshift distribution after selection effects have been included (see lower histogram in Figure 3); this is in contrast to the reported redshift structure found in the SDSS quasar survey by Bell & McDiarmid (2006). To construct the lower histogram we have partially removed the effect of host galaxy contamination (by excluding extended objects), limited the sample to a uniform magnitude limit of i < 19.1 (accounting for emission-line effects), and have corrected for the known incompleteness near z ∼ 2.7 and z ∼ 3.5 due to quasar colors lying close to or in the stellar locus. Accounting for selection effects significantly reduces the number of objects as compared with the raw, more heterogeneous catalog, but the smaller, more homogeneous sample is what should be used for statistical analyses. The distribution of the observed i magnitude (not corrected for Galactic extinction) of the quasars is given in Figure 4. The sharp drops in the histogram at i ≈ 19.1 and i ≈ 20.2 are due to the magnitude limits in the low and high-redshift samples, respectively. Figures 5 and 6 display the distribution of the absolute i magnitudes of the catalog quasars. There is a roughly symmetric peak centered at Mi = −26 with a FWHM of approximately one magnitude. The histogram declines sharply at high luminosities (only 1.5% of the objects haveMi < −28.0) and has a gradual decline toward lower luminosities, partially due to host-galaxy contribution. A summary of the spectroscopic selection, for both the TARGET and the BEST algorithms, is given in Table 3. We report seven selection classes in the catalog (columns 39 to 45 for BEST, 56–62 for TARGET). Each selection version has two columns, the number of objects that satisfied a given selection criterion and the number of objects that were identified only by that selection class. About two-thirds of the catalog entries were selected based on the SDSS quasar selection criteria (either a low-redshift or high-redshift candidate, or both). Slightly more than half of the quasars in the catalog are serendipity-flagged candidates, which is also primarily an “unusual color” algorithm; about one-fifth of the catalog was selected by the serendipity criteria alone. Of the 50,093 DR5 quasars that have Galactic-absorption corrected TARGET i magnitudes brighter than 19.1, 48,593 (97.0%) were identified by the TARGET quasar multicolor selection; if one combines TARGET multicolor and FIRST selection (the primary quasar target selection criteria), all but 1015 of the i < 19.1 objects were selected. (The spectra of many of the last category of objects were obtained in observations that were not part of the primary survey.) The numbers are similar if one uses the BEST photometry and selection, although the completeness is not quite as high as with TARGET values. – 13 – 5.1. Discrepancies Between the DR5 and Other Quasar Catalogs The DR3 database is entirely contained in that of DR5, but there are 66 quasars from Paper III (out of 46,420 objects) that do not have a counterpart within 1.0′′ of a DR5 quasar. Three of these “missing” quasars are in the DR5 list; changes in celestial position of 1.1′′, 1.8′′, and 5.3′′ between DR3 and DR5 caused these quasars to be missed with the 1.0′′ matching criterion. The other 63 cases (0.14% of the DR3 total) were individually investigated. Three DR3 objects were dropped because the latest photometry reduced their luminosities below the catalog limit. The remaining 60 objects were removed because 1) the visual examination of the spectrum either convinced us that the object was not a quasar or that the S/N was insufficient to assign a redshift with confidence or 2) The widest line in the latest fit to the spectrum had a FWHM of less than 1000 km s−1. It should be noted that there have been no changes to the DR3 spectra in the DR5 database; the missing objects reflect the inherent uncertainties involved with interpreting objects that either lie near survey cutoffs or have spectra of marginal S/N. There are 40 and 136 DR5 quasars that have redshifts that differ by more than 0.1 from the DR3 and NED values, respectively (there is, of course, considerable overlap in these two groups). In all cases the DR5 measurements are considered more reliable than those presented in previous publications. The 40 objects with |zDR5 − zDR3| > 0.1 are listed in Table 4. 5.2. Quasar Colors It has long been known that the majority of quasars inhabit a restricted range in photometric color, and the large sample size and accurate photometry of the SDSS revealed a relatively tight color-redshift correlation for quasars (Richards et al. 2001). This SDSS color relation, recently presented in Hopkins et al. (2004), has led to considerable success in assigning photometric redshifts to quasars (e.g., Weinstein et al. 2004 and references therein). All photometric measurements used in these analyses have been corrected for Galactic extinction. The dependence of the four standard SDSS colors on redshift for the DR5 quasars is given in Figure 7. The dashed line in each panel is the modal relation for the DR5 quasars; the modal relations are tabulated in Table 5, along with the values for (g− i). The figures show an impressively tight correlation of color with redshift, although the scatter dramatically increases when the Lyman α forest dominates the bluer of the passbands used to form the color. The distribution near the modal curve is roughly symmetric, but there is clearly a significant population of “red” quasars that has no “blue” counterpart. This table is an improvement over previous work in that it is based on a larger sample size (a factor of four increase since this relation was last published) and provides higher redshift resolution (0.01, except near the extrema). As in Hopkins et al. (2004), we compute the mode, rather than the mean or median, as the most representative quantity. However, a formal computation of the mode requires binning the data both in redshift and by color within redshift bins; therefore we estimated the mode from the mean and the median. Typically, the mode is estimated as (3 × median−2 × mean), but we found empirically that (2 × median−mean) appeared to work better for this sample in terms of tracing the modal “ridgeline” with redshift. For each of the DR5 quasars we provide the quantity ∆(g − i), which is defined by ∆(g − i) = (g − i)QSO − 〈(g − i)〉redshift – 14 – where 〈(g − i)〉redshift is the entry in Table 5 for the redshift of the quasar. This “differential color” provides an estimate of the continuum properties of the quasar (values above zero indicate that the object has a redder continuum than the typical quasar at that redshift). 5.3. Bright Quasars Although the spectroscopic survey is limited to objects fainter than i ≈ 15, the SDSS continues to discover a number of “PG-class” (Schmidt & Green 1983) objects. The DR5 catalog contains 81 entries with i < 16.0; 14 of the quasars are not in the NED database or attributed to the SDSS by NED. The spectrum of the brightest post-DR3 discovery, SDSS J165551.37+214601.8 (i = 15.62, z = 0.15), is presented in Figure 1. Three of the SDSS-discovered objects in this catalog have been added since Paper III. 5.4. Luminous Quasars There are 103 catalog quasars with Mi < −29.0 (3C 273 has Mi ≈ −26.6 in our adopted cosmology); 61 were discovered by the SDSS, and 18 are published here for the first time. The redshifts of these quasars lie between 1.3 and 5.0. The most luminous quasar in the catalog is 2MASSI J0745217+473436 (= SDSS J074521.78+473436.2), at Mi = −30.27 and z = 3.22. Spectra of the two most luminous post-DR3 discoveries, with absolute i magnitudes of −29.94 and −29.65, are displayed in the upper two panels of Figure 1. The spectra of both quasars possess a considerable number of absorption features redward of the Lyman α emission line. 5.5. Broad Absorption Line Quasars The SDSS quasar selection algorithm has proven to be effective at finding a wide variety of Broad Absorption Line (BAL) Quasars. An EDR sample of 118 BAL quasars was presented by Tolea, Krolik, & Tsvetanov (2002). There have been two editions of the SDSS BAL Quasar Catalog; the first, associated with Paper I, contained 224 BAL quasars (Reichard et al. 2003); the second was based on the Paper III catalog and presents 4787 BAL quasars (Trump et al. 2006). BAL quasars are usually recognized by the presence of C IV absorption features, which are only visible in SDSS spectra at z > 1.6, thus the frequency of the BAL quasar phenomenon cannot be found from simply taking the ratio of BAL quasars to total number of quasars in the SDSS catalog. The SDSS has discovered a wide variety of extreme BAL quasars (see Hall et al. 2002); the lower right panel in Figure 1 presents the spectrum of an unusual FeLoBAL quasar with strong Balmer absorption (see Hall 2007 for a discussion of this object). 5.6. Quasars with Redshifts Below 0.15 The catalog contains 109 quasars with redshifts below 0.15. All of these objects are of low luminosity (Mi > −24.0, only three have Mi < −23.5) because of the i ≈ 15.0 limit for the spectroscopic sample. About three-quarters of these quasars (83) are extended in the SDSS image data. A total of 40 of the z < 0.15 quasars were found by the SDSS; 21 have been added since Paper III. – 15 – 5.7. High-Redshift (z ≥ 4) Quasars At first light of the SDSS, the most distant known quasar was PC 1247+3406 at redshift of 4.897 (Schneider, Schmidt, and Gunn 1991), which had been discovered seven years earlier. Within a year of operation, the SDSS had discovered quasars with redshifts above five (Fan et al. 1999, 2000); the DR5 catalog contains 60 objects with redshifts greater than that of PC 1247+3406. In recent years the SDSS has identified quasars out to a redshift of 6.4 (Fan et al. 2003, 2006b). Quasars with redshifts larger than ≈ 5.7, however, cannot be found by the SDSS spectroscopic survey because at these redshifts the observed wavelength of the Lyman α emission line is redward of the i band; at this point quasars become single-filter (z) detections. At the typical z-band flux levels for redshift six quasars, there are simply too many “false-positives” to undertake automated targeting. The largest redshift in the DR5 catalog is SDSS J023137.65−072854.5 at z = 5.41, which was originally described by Anderson et al. (2001). The DR5 catalog contains 891 quasars with redshifts larger than four; 36 entries have redshifts above five (11 above z = 5.2), which is more than a factor of two increase since Paper III. The spectra of the 20 highest redshift post-DR3 objects (all with redshifts greater than or equal to 4.99) are displayed in Figure 8. These redshift five spectra display a striking variety of emission line properties, and include an impressive BAL at z = 5.27. We have used archival data from Chandra, ROSAT , and XMM-Newton to check for new X-ray detections of z > 4 quasars with unusual emission-line or absorption-line properties; we do not report all z > 4 X-ray detections here as there are now more than 110 already published.3 We found three remarkable z > 4 X-ray detections: the z = 4.26 BAL quasar SDSS J133529.45+410125.9, the z = 4.11 BAL quasar SDSS J142305.04+240507.8, and the z = 4.50 quasar SDSS J150730.63+553710.8, which shows remarkably strong C iv emission. None of these objects has sufficient counts for detailed X-ray spectral analysis, but we have computed their point-to-point spectral slopes between rest-frame 2500 Å and 2 keV (αox), adopting the assumptions in §2 of Brandt et al. (2002). SDSS J133529.45+410125.9 and SDSS J142305.04+240507.8 were serendipitously detected in archival Chandra ACIS observations and have αox = −2.19 and αox = −1.52, respectively. Comparing these values to the established relation between αox and 2500 Å luminosity (e.g., Steffen et al. 2006), we find that SDSS J133529.45+410125.9 is notably X-ray weak, indicating likely X-ray absorption as is often seen in BAL quasars (e.g., Gallagher et al. 2006) including those at z > 4 (Vignali et al. 2005). In contrast, the level of X-ray emission from SDSS J142305.04+240507.8 is consistent with that from normal, non-BAL quasars; its relatively narrow UV absorption, for a BAL quasar, may indicate a relatively small column density of obscuring material. SDSS J150730.63+553710.8 is weakly detected in a ROSAT PSPC observation and has αox = −1.47; this level of X-ray emission is nominal for a quasar of its luminosity. We have also checked all quasars with z > 5 for new X-ray detections and found none; 21 quasars with z > 5 have previously reported X-ray detections. 5.8. Close Pairs The mechanical constraint that SDSS spectroscopic fibers must be separated by 55′′ on a given plate makes it difficult for the spectroscopic survey to confirm close pairs of quasars. In regions that are covered 3See http://www.astro.psu.edu/users/niel/papers/highz-xray-detected.dat for a list of X-ray detections and references. http://www.astro.psu.edu/users/niel/papers/highz-xray-detected.dat – 16 – by more than one plate, however, it is possible to obtain spectra of both components of a close pair; there are 346 pairs of quasars in the catalog with angular separation less than an arcminute (34 pairs with separations less than 20′′). Most of the pairs are chance superpositions, but there are many sets whose components have similar redshifts, suggesting that the quasars may be physically associated. The typical uncertainty in the measured value of the redshift difference between two quasars is 0.02; the catalog contains 18 quasar pairs with separations of less than an arcminute and with ∆z < 0.02. These pairs, which are excellent candidates for binary quasars, are listed in Table 6. Hennawi et al. (2006) identified over 200 quasar pairs in the SDSS, primarily through spectroscopic observations of SDSS quasar candidates (based on photometric measurements) near known SDSS quasars; statistical arguments based on a correlation-function analysis suggests that most of these pairs are indeed physically associated. 5.9. Morphology The images of 3498 of the DR5 quasars are classified as extended by the SDSS photometric pipeline; 3291 (94%) have redshifts below one (there are nine resolved z > 3.0 quasars). The majority of the large-redshift “re- solved” quasars are probably measurement errors, but this sample may also contain a mix of chance su- perpositions of quasars and foreground objects or possibly some small angle separation gravitational lenses (indeed, several lenses are present in the resolved quasar sample; see Paper II and Oguri et al. 2006). 5.10. Matches with Non-optical Catalogs A total of 6226 catalog objects are FIRST sources (defined by a SDSS-FIRST positional offset of less than 2.0′′). Note that 226 of the objects were selected (with TARGET) solely because they were FIRST matches (all unresolved SDSS sources brighter than i = 19.1 that lie within 2.0′′ of a FIRST source are targeted by the quasar spectroscopic selection algorithm). Extended radio sources may be missed by this matching. The upper left panel in Figure 9 contains a histogram of the angular offsets between the SDSS and FIRST positions; the solid line is the expected distribution assuming a 0.21′′ 1σ Gaussian error in the relative SDSS/FIRST positions (found by fitting the points with a separation less than 1.0′′. The small-angle separations are well-fit to the Rayleigh distribution, but outside of about 0.5′′ there is an obvious excess of observed separations. The number of chance superpositions was estimated by shifting the quasar positions by ±200′′ in declination and matching the new coordinates to the FIRST catalog; only about 0.1% of the reported FIRST matches are false. The large “tail” of this distribution is not likely to be due to measurement errors but probably arises from extended radio emission that may not be precisely centered on the optical image. To recover radio quasars that have offsets of more than 2.0′′, we separately identify all objects with a greater than 3σ detection of FIRST flux at the optical position (2440 sources). For these objects as well as those with a FIRST catalog match within 2′′, we perform a second FIRST catalog search with 30′′ matching radius to identify possible radio lobes associated with the quasar, finding such matches for 1596 sources. Matches with the ROSAT All-Sky Survey Bright and Faint Source Catalogs were made with a maximum allowed positional offset of 30′′; this is the positional coincidence required for the SDSS ROSAT target selection code. The DR5 catalog contains 4133 RASS matches; approximately 1.3% are expected to be false identifications based on an analysis similar to that described in the previous paragraph. The SDSS-RASS offsets for the DR5 sample are presented in the upper right panel of Figure 9; the solid curve, which is the predicted distribution for a 1σ positional error of 11.1′′ (fit using all of the points), matches the data quite – 17 – well. JHK photometric measurements for 9824 DR5 quasars were found by using a matching radius of 2.0′′ in the 2MASS All-Sky Data Release Point Source Catalog. No infrared information was used to select the SDSS spectroscopic targets. The positional offset histogram, given in the lower left panel of Figure 9, is considerably tighter than that for the FIRST matches, although the Rayleigh fit to the separations less than 1.0′′ is virtually identical to the FIRST distribution (1σ of 0.21′′). There are very few 2MASS identifications with offsets between 1′′ and 2′′; virtually all of the infrared matches are correct. 6. Summary The lower right panel in Figure 9 charts the progress of the SDSS Quasar Survey, denoted by the number of spectroscopically-confirmed quasars, over the duration of SDSS-I. Although SDSS-I has now been completed, the SDSS Quasar Survey is continuing under the SDSS-II project. By necessity the SDSS spectroscopy lags the SDSS imaging; at the conclusion of SDSS-I more than 2000 square degrees of SDSS image data in the Northern Galactic Cap lacked spectroscopic coverage (Adelman-McCarthy et al. 2007). A future edition of the SDSS Quasar Catalog will incorporate the observations from SDSS-II and should contain approximately 100,000 quasars. The publication of this catalog marks the completion of the SDSS-I Quasar Survey, and we dedicate this work to the memory of John N. Bahcall. John was the initial co-chair of the SDSS Quasar Working Group, a position he held for nearly a decade. He played a key role in the formation of the SDSS Collaboration and the design of the SDSS Quasar Survey, and was a mentor to many of the members of the Quasar Working Group. We would like to thank Todd Boroson for suggesting several redshift adjustments to some of the DR3 Quasar Catalog redshifts. This work was supported in part by National Science Foundation grants AST-0307582 and AST-0607634 (DPS, DVB, JW), AST-0307384 (XF), and AST-0307409 (MAS), and by NASA LTSA grant NAG5-13035 (WNB, DPS). PBH acknowledges support by NSERC, and GTR was supported in part by a Gordon and Betty Moore Fellowship in Data Intensive Sciences at JHU. XF acknowledges support from an Alfred P. Sloan Fellowship and a David and Lucile Packard Fellowship in Science and Engineering. SJ was supported by the Max-Planck-Gesellschaft (MPI für Astronomie) through an Otto Hahn fellowship. CS was supported by the U.S. Department of Energy under contract DE-AC02-76CH03000. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participat- ing Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Educa- tion Funding Council for England. The SDSS Web site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institu- tions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute of Potsdam, University of Basel, Cambridge University, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle As- trophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of http://www.sdss.org/ – 18 – Washington. 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The spectral resolution of the data ranges from 1800 to 2100; a dichroic splits the beam at 6150 Å. The data have been rebinned to 5 Å pixel−1 for display purposes. The upper two panels display the two most luminous of the newly discovered quasars; both objects have Mi < −29.6. SDSS J165551.37+214601.8 is the brightest (i = 15.62) of the new quasars; SDSS J125942.80+121312.6 is an unusual FeLoBAL quasar with Balmer-line absorption. – 23 – Fig. 2.— The observed i magnitude as a function of redshift for the 77,429 objects in the catalog. Open circles indicate quasars in NED that were recovered but not discovered by the SDSS. The 26 quasars with i > 21 are not plotted. The distribution is represented by a set of linear contours when the density of points in this two-dimensional space causes the points to overlap. The steep gradient at i ≈ 19 is due to the flux limit for the targeted low-redshift part of the survey; the dip in the counts at z ≈ 2.7 arises because of the high incompleteness of the SDSS Quasar Survey at redshifts between 2.5 and 3.0 (also see Figure 3). – 24 – Fig. 3.— The redshift histogram of the catalog quasars. The redshifts range from 0.08 to 5.41; the median redshift of the catalog is 1.48. The redshift bins have a width of 0.05. The dips at redshifts of 2.7 and 3.5 are caused by the reduced efficiency of the selection algorithm at these redshifts. The lower histogram is the redshift distribution of the i < 19.1 sample after correction for selection effects (see Section 5). – 25 – i mag 16 18 20 22 Fig. 4.— The i magnitude (not corrected for Galactic absorption) histogram of the 77,429 catalog quasars. The magnitude bins have a width of 0.108. The sharp drop that occurs at magnitudes slightly fainter than 19 is due to the flux limit for the low-redshift targeted part of the survey. Quasars fainter than the i = 20.2 high-redshift selection limit were found via other selection algorithms, primarily serendipity. The SDSS Quasar survey has a bright limit of i ≈ 15.0 imposed by the need to avoid saturation in the spectroscopic observations. – 26 – Fig. 5.— The absolute i magnitude as a function of redshift for the 77,429 objects in the catalog. Open circles indicate quasars in NED that were recovered but not discovered by the SDSS. The distribution is represented by a set of linear contours when the density of points in this two-dimensional space causes the points to overlap. The steep gradient that runs through the midst of the quasar distribution is produced by the i ≈ 19 flux limit for the targeted low-redshift part of the survey. – 27 – Absolute i Magnitude -30 -28 -26 -24 -22 Fig. 6.— The luminosity distribution of the catalog quasars. The absolute magnitude bins have a width of 0.114. The most luminous quasar in the catalog has Mi ≈ −30.3. In the adopted cosmology 3C 273 has Mi ≈ −26.6. – 28 – Fig. 7.— The quasar color-redshift relation for the DR5 quasars (photometry corrected for Galactic ex- tinction). Contours are used to represent the distribution when the density of points causes the points to overlap. The panels present the four standard SDSS colors; the dashed gray lines are the modal relations presented in Table 5. The influence of emission lines on the colors is readily apparent (in particular Hα in the (i− z) panel). The tightness of the correlations breaks down when the Lyman α forest region dominates the bluer of the two passbands (e.g., above redshifts of 2.2 in the (u− g) relation). – 29 – 005421.42-010921.6 z = 5.09 073103.12+445949.4 z = 5.00 4 084627.84+080051.7 z = 5.03 090245.76+085115.9 z = 5.23 092216.81+265359.0 z = 5.03 111920.64+345248.1 z = 5.01 113246.50+120901.6 z = 5.17 114657.79+403708.6 z = 5.01 115424.73+134145.7 z = 5.01 120207.78+323538.8 z = 5.29 123333.48+062234.2 z = 5.29 133412.56+122020.7 z = 5.13 133728.81+415539.8 z = 5.01 134015.03+392630.7 z = 5.03 134040.24+281328.1 z = 5.34 134141.45+461110.2 z = 5.00 Wavelength (A) 6000 7000 8000 9000 142325.92+130300.6 z = 5.04 Wavelength (A) 6000 7000 8000 9000 4 144350.66+362315.1 z = 5.27 Wavelength (A) 6000 7000 8000 9000 3 162629.19+285857.5 z = 5.02 Wavelength (A) 6000 7000 8000 9000 165902.11+270935.1 z = 5.31 Fig. 8.— SDSS spectra of the 20 new quasars with the highest redshifts (z ≥ 4.99). The spectra have been rebinned to 10 Å pixel−1 for display purposes. The wavelength region below 6000 Å has been removed because of the lack of signal below rest frame wavelengths of 1000 Å in these objects. Five of the quasars have redshifts larger than 5.25. – 30 – FIRST/SDSS Offset (") 0.0 0.5 1.0 1.5 2.0 RASS/SDSS Offset (") 0 10 20 30 2MASS/SDSS Offset (") 0.0 0.5 1.0 1.5 2.0 Modified Julian Date (51600.0 = UTC 2000 Feb 26.0) 51500 52000 52500 53000 53500 Fig. 9.— a) Offsets between the 6226 SDSS and FIRST matches; the matching radius was set to 2.0′′. The smooth curve is the expected distribution for a set of matches if the offsets between the objects are described by a Rayleigh distribution with σ = 0.21′′ (best fit for points with separations of less than 1.0′′). b) Offsets between the 4133 SDSS and RASS FSC/BSC matches; the matching radius was set to 30′′. The smooth curve is the Rayleigh distribution fit (σ = 11.1′′) to all of the points. c) Offsets between the 9824 SDSS and 2MASS matches; the matching radius was set to 2′′. The smooth curve is a Rayleigh distribution with σ = 0.21′′ based on the points with separations smaller than 1.0′′. d) The cumulative number of DR5 quasars as a function of time. The horizontal axis runs from February 2000 to June 2005. The periodic structure in the curve is caused by the yearly summer maintenance schedule. The total number of objects in the catalog is 77,429. – 31 – Table 1. SDSS DR5 Quasar Catalog Format Column Format Description 1 A18 SDSS DR5 Designation hhmmss.ss+ddmmss.s (J2000) 2 F11.6 Right Ascension in decimal degrees (J2000) 3 F11.6 Declination in decimal degrees (J2000) 4 F7.4 Redshift 5 F7.3 BEST PSF u magnitude (not corrected for Galactic extinction) 6 F6.3 Error in BEST PSF u magnitude 7 F7.3 BEST PSF g magnitude (not corrected for Galactic extinction) 8 F6.3 Error in BEST PSF g magnitude 9 F7.3 BEST PSF r magnitude (not corrected for Galactic extinction) 10 F6.3 Error in BEST PSF r magnitude 11 F7.3 BEST PSF i magnitude (not corrected for Galactic extinction) 12 F6.3 Error in BEST PSF i magnitude 13 F7.3 BEST PSF z magnitude (not corrected for Galactic extinction) 14 F6.3 Error in BEST PSF z magnitude 15 F7.3 Galactic extinction in u band 16 F7.3 logNH (logarithm of Galactic H I column density) 17 F7.3 FIRST peak flux density at 20 cm expressed as AB magnitude; 0.0 is no detection, −1.0 source is not in FIRST area 18 F8.3 S/N of FIRST flux density 19 F7.3 SDSS-FIRST separation in arc seconds 20 I3 > 3σ FIRST flux at optical position but no FIRST counterpart within 2′′ (0 or 1) 21 I3 FIRST source located 2′′-30′′ from optical position (0 or 1) 22 F8.3 log RASS full band count rate; −9.0 is no detection 23 F7.3 S/N of RASS count rate 24 F7.3 SDSS-RASS separation in arc seconds 25 F7.3 J magnitude (2MASS); 0.0 indicates no 2MASS detection 26 F6.3 Error in J magnitude (2MASS) 27 F7.3 H magnitude (2MASS); 0.0 indicates no 2MASS detection 28 F6.3 Error in H magnitude (2MASS) 29 F7.3 K magnitude (2MASS); 0.0 indicates no 2MASS detection 30 F6.3 Error in K magnitude (2MASS) 31 F7.3 SDSS-2MASS separation in arc seconds 32 F8.3 Mi (H0 = 70 km s −1 Mpc−1, ΩM = 0.3, ΩΛ = 0.7, αν = −0.5) 33 F7.3 ∆(g − i) = (g − i)− 〈(g − i)〉redshift (Galactic extinction corrected) 34 I3 Morphology flag 0 = point source 1 = extended 35 I3 SDSS SCIENCEPRIMARY flag (0 or 1) 36 I3 SDSS MODE flag (blends, overlapping scans; 1, 2, or 3) 37 I3 Selected with final quasar algorithm (0 or 1) 38 I12 Target Selection Flag (BEST) 39 I3 Low-z Quasar selection flag (0 or 1) 40 I3 High-z Quasar selection flag (0 or 1) 41 I3 FIRST selection flag (0 or 1) – 32 – Table 1—Continued Column Format Description 42 I3 ROSAT selection flag (0 or 1) 43 I3 Serendipity selection flag (0 or 1) 44 I3 Star selection flag (0 or 1) 45 I3 Galaxy selection flag (0 or 1) 46 I6 SDSS Imaging Run Number of photometric measurements 47 I6 Modified Julian Date of imaging observation 48 I6 Modified Julian Date of spectroscopic observation 49 I5 Spectroscopic Plate Number 50 I5 Spectroscopic Fiber Number 51 I4 SDSS Photometric Processing Rerun Number 52 I3 SDSS Camera Column Number 53 I5 SDSS Field Number 54 I5 SDSS Object Number 55 I12 Target Selection Flag (TARGET) 56 I3 Low-z Quasar selection flag (0 or 1) 57 I3 High-z Quasar selection flag (0 or 1) 58 I3 FIRST selection flag (0 or 1) 59 I3 ROSAT selection flag (0 or 1) 60 I3 Serendipity selection flag (0 or 1) 61 I3 Star selection flag (0 or 1) 62 I3 Galaxy selection flag (0 or 1) 63 F7.3 TARGET PSF u magnitude (not corrected for Galactic extinction) 64 F6.3 TARGET Error in PSF u magnitude 65 F7.3 TARGET PSF g magnitude (not corrected for Galactic extinction) 66 F6.3 TARGET Error in PSF g magnitude 67 F7.3 TARGET PSF r magnitude (not corrected for Galactic extinction) 68 F6.3 TARGET Error in PSF r magnitude 69 F7.3 TARGET PSF i magnitude (not corrected for Galactic extinction) 70 F6.3 TARGET Error in PSF i magnitude 71 F7.3 TARGET PSF z magnitude (not corrected for Galactic extinction) 72 F6.3 TARGET Error in PSF z magnitude 73 I21 Spectroscopic Identification flag (64-bit integer) 74 1X, A25 Object Name for previously known quasars “SDSS” designates previously published SDSS object Table 2. The SDSS Quasar Catalog IVa Object (SDSS J) R.A. (deg) Dec (deg) Redshift u g r i z 000006.53+003055.2 0.027228 0.515349 1.8227 20.389 0.066 20.468 0.034 20.332 0.037 20.099 0.041 20.053 0.121 000008.13+001634.6 0.033898 0.276304 1.8365 20.233 0.054 20.200 0.024 19.945 0.032 19.491 0.032 19.191 0.068 000009.26+151754.5 0.038605 15.298476 1.1986 19.921 0.042 19.811 0.036 19.386 0.017 19.165 0.023 19.323 0.069 000009.38+135618.4 0.039088 13.938447 2.2400 19.218 0.026 18.893 0.022 18.445 0.018 18.331 0.024 18.110 0.033 000009.42−102751.9 0.039269 −10.464428 1.8442 19.249 0.036 19.029 0.027 18.980 0.021 18.791 0.018 18.751 0.047 aTable 2 is presented in its entirety in the electronic edition of the Astronomical Journal. A portion is shown here for guidance regarding its form and content. The full catalog contains 74 columns of information on 77,429 quasars. – 34 – Table 3. Spectroscopic Target Selection TARGET TARGET BEST BEST Sole Sole Class Selected Selection Selected Selection Low-z 49010 16422 46460 14444 High-z 16383 5327 16757 4411 FIRST 3501 226 3619 209 ROSAT 4817 380 4918 492 Serendipity 42109 15729 41042 15950 Star 1970 187 820 162 Galaxy 536 99 601 80 – 35 – Table 4. Quasars with |zDR5 − zDR3| > 0.1 SDSS J zDR5 SDSS J zDR5 005508.55−105206.2 1.381 133028.12+600811.7 1.992 013413.55+142900.1 1.195 133951.94+481651.3 0.911 031712.23−075850.3 2.696 134048.37+433359.8 2.069 075052.59+300334.1 3.990 135833.05+634122.6 3.180 075132.75+350535.0 2.077 140012.65+595823.3 2.061 083503.79+322242.0 0.728 140223.63+463604.9 0.925 085339.64+372203.6 1.950 140327.91+613654.2 2.023 090902.73+355334.8 1.638 141230.28+471103.7 2.078 091025.25+365921.3 2.004 142010.28+604722.3 1.345 092415.87+424632.2 0.559 143702.47+613437.0 2.064 093557.85+005528.1 1.301 144939.30+534212.1 1.805 093935.08−000801.1 0.909 151307.26−000559.3 2.030 094326.48+460226.8 2.093 151422.99+481936.3 2.071 100415.17+415802.6 1.977 153257.67+422047.1 1.950 102117.71+623010.1 1.949 160320.97+315248.3 0.727 103039.95+510923.3 1.649 165806.76+611858.9 2.631 103219.66+563456.8 2.017 170929.58+323826.9 1.902 115917.62+100921.5 2.028 205058.45+004709.9 0.932 124345.10+492645.3 1.982 212744.12+005720.3 4.386 131810.57+585416.9 1.900 225246.43+142525.8 4.904 – 36 – Table 5. Quasar Colors as a Function of Redshifta zbin 〈z〉 NQSO (g − i) (u − g) (g − r) (r − i) (i − z) 0.18 0.181 183 0.567 −0.065 0.197 0.379 −0.037 0.21 0.210 290 0.580 0.032 0.223 0.355 −0.034 0.24 0.240 394 0.513 0.000 0.236 0.267 0.115 0.27 0.270 406 0.289 0.055 0.231 0.077 0.397 0.30 0.301 484 0.236 0.067 0.219 0.033 0.472 aTable 5 is presented in its entirety in the electronic edition of the Astro- nomical Journal. A portion is shown here for guidance regarding its form and content. – 37 – Table 6. Candidate Binary Quasars Quasar 1 Quasar 2 z1 z2 ∆θ 001201.87+005259.7 001202.35+005314.0 1.652 1.642 16.0 011757.99+002104.1 011758.83+002021.4 0.612 0.613 44.5 014110.40+003107.1 014111.62+003145.9 1.879 1.882 42.9 024511.93−011317.5 024512.12−011313.9 2.463 2.460 4.5 025813.65−000326.4 025815.54−000334.2 1.316 1.321 29.4 025959.68+004813.6 030000.57+004828.0 0.892 0.900 19.6 074336.85+205512.0 074337.28+205437.1 1.570 1.565 35.5 074759.02+431805.4 074759.66+431811.5 0.501 0.501 9.2 082439.83+235720.3 082440.61+235709.9 0.536 0.536 14.9 085625.63+511137.0 085626.71+511117.8 0.543 0.543 21.8 090923.12+000203.9 090924.01+000211.0 1.884 1.865 15.0 095556.37+061642.4 095559.02+061701.8 1.278 1.273 44.0 110357.71+031808.2 110401.48+031817.5 1.941 1.923 57.3 111610.68+411814.4 111611.73+411821.5 2.980 2.971 13.8 113457.73+084935.2 113459.37+084923.2 1.533 1.525 27.1 121840.47+501543.4 121841.00+501535.8 1.457 1.455 9.1 165501.31+260517.5 165502.02+260516.5 1.881 1.892 9.6 215727.26+001558.4 215728.35+001545.5 2.540 2.553 20.8 aThe quasar pairs were selected by a redshift difference of less than 0.02 and an angular separation less than 60′′. Introduction Observations Sloan Digital Sky Survey Target Selection Spectroscopy Construction of the SDSS DR5 Quasar Catalog Creation of the DR5 Quasar Candidate Database Visual Examination of the Spectra Luminosity and Line Width Criteria Catalog Format Catalog Summary Discrepancies Between the DR5 and Other Quasar Catalogs Quasar Colors Bright Quasars Luminous Quasars Broad Absorption Line Quasars Quasars with Redshifts Below 0.15 High-Redshift (z 4) Quasars Close Pairs Morphology Matches with Non-optical Catalogs Summary

0704.0807

Nuclear forces from chiral effective field theory

Nuclear Forces from Chiral Effective Field Theory∗ R.Machleidt Department of Physics, University of Idaho, Moscow, Idaho, U.S.A. November 20, 2018 Abstract In this lecture series, I present the recent progress in our under- standing of nuclear forces in terms of chiral effective field theory. Contents 1 Introduction and Historical Perspective 3 2 QCD and the Nuclear Force 5 3 Effective Field Theory for Low-Energy QCD 5 3.1 Symmetries of Low-Energy QCD . . . . . . . . . . . . . . . . 6 3.1.1 Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . 6 3.1.2 Explicit Symmetry Breaking . . . . . . . . . . . . . . 9 3.1.3 Spontaneous Symmetry Breaking . . . . . . . . . . . . 9 3.2 Chiral Effective Lagrangians Involving Pions . . . . . . . . . 10 3.3 Nucleon Contact Lagrangians . . . . . . . . . . . . . . . . . . 12 4 Nuclear Forces from EFT: Overview 13 4.1 Chiral Perturbation Theory and Power Counting . . . . . . . 14 4.2 The Hierarchy of Nuclear Forces . . . . . . . . . . . . . . . . 14 ∗Lecture series presented at the DAE-BRNS Workshop on Physics and Astrophysics of Hadrons and Hadronic Matter, Visva Bharati University, Santiniketan, West Bengal, India, November 2006. 5 Two-Nucleon Forces 16 5.1 Pion-Exchange Contributions in ChPT . . . . . . . . . . . . 16 5.1.1 Zeroth Order (LO) . . . . . . . . . . . . . . . . . . . 17 5.1.2 Second Order (NLO) . . . . . . . . . . . . . . . . . . 17 5.1.3 Third Order (NNLO) . . . . . . . . . . . . . . . . . . 19 5.1.4 Fourth Order (N3LO) . . . . . . . . . . . . . . . . . . 20 5.1.5 Iterated One-Pion-Exchange . . . . . . . . . . . . . . 20 5.2 NN Scattering in Peripheral Partial Waves Using the Pertur- bative Amplitude . . . . . . . . . . . . . . . . . . . . . . . . 22 5.3 NN Contact Potentials . . . . . . . . . . . . . . . . . . . . . 28 5.3.1 Zeroth Order . . . . . . . . . . . . . . . . . . . . . . . 29 5.3.2 Second Order . . . . . . . . . . . . . . . . . . . . . . . 30 5.3.3 Fourth Order . . . . . . . . . . . . . . . . . . . . . . . 30 5.4 Constructing a Chiral NN Potential . . . . . . . . . . . . . . 31 5.4.1 Conceptual Questions . . . . . . . . . . . . . . . . . . 31 5.4.2 What Order? . . . . . . . . . . . . . . . . . . . . . . . 33 5.4.3 Charge-Dependence . . . . . . . . . . . . . . . . . . . 34 5.4.4 A Quantitative NN Potential at N3LO . . . . . . . . 36 6 Many-Nucleon Forces 39 6.1 Three-Nucleon Forces . . . . . . . . . . . . . . . . . . . . . . 40 6.2 Four-Nucleon Forces . . . . . . . . . . . . . . . . . . . . . . . 42 7 Conclusions 42 A Fourth Order Two-Pion Exchange Contributions 44 A.1 One-loop diagrams . . . . . . . . . . . . . . . . . . . . . . . . 44 A.1.1 c2i contributions. . . . . . . . . . . . . . . . . . . . . . 44 A.1.2 ci/MN contributions. . . . . . . . . . . . . . . . . . . . 44 A.1.3 1/M2N corrections. . . . . . . . . . . . . . . . . . . . . 45 A.2 Two-loop contributions. . . . . . . . . . . . . . . . . . . . . . 46 B Partial Wave Decomposition of the Fourth Order Contact Potential 48 1 Introduction and Historical Perspective The theory of nuclear forces has a long history (cf. Table 1). Based upon the seminal idea by Yukawa [1], first field-theoretic attempts to derive the nucleon-nucleon (NN) interaction focused on pion-exchange. While the one- pion exchange turned out to be very useful in explaining NN scattering data and the properties of the deuteron [2], multi-pion exchange was beset with serious ambiguities [3, 4]. Thus, the “pion theories” of the 1950s are gen- erally judged as failures—for reasons we understand today: pion dynamics is constrained by chiral symmetry, a crucial point that was unknown in the 1950s. Historically, the experimental discovery of heavy mesons [5] in the early 1960s saved the situation. The one-boson-exchange (OBE) model [6, 7] emerged which is still the most economical and quantitative phenomenol- ogy for describing the nuclear force [8, 9]. The weak point of this model, however, is the scalar-isoscalar “sigma” or “epsilon” boson, for which the empirical evidence remains controversial. Since this boson is associated with the correlated (or resonant) exchange of two pions, a vast theoretical effort that occupied more than a decade was launched to derive the 2π-exchange contribution to the nuclear force, which creates the intermediate range at- traction. For this, dispersion theory as well as field theory were invoked producing the Stony Brook [10], Paris [11, 12], and Bonn [7, 13] potentials. The nuclear force problem appeared to be solved; however, with the discovery of quantum chromodynamics (QCD), all “meson theories” were relegated to models and the attempts to derive the nuclear force started all over again. The problem with a derivation from QCD is that this theory is non- perturbative in the low-energy regime characteristic of nuclear physics, which makes direct solutions impossible. Therefore, during the first round of new attempts, QCD-inspired quark models [14] became popular. These models are able to reproduce qualitatively and, in some cases, semi-quantitatively the gross features of the nuclear force [15, 16]. However, on a critical note, it has been pointed out that these quark-based approaches are nothing but another set of models and, thus, do not represent any fundamental progress. Equally well, one may then stay with the simpler and much more quantita- tive meson models. A major breakthrough occurred when the concept of an effective field theory (EFT) was introduced and applied to low-energy QCD. As outlined by Weinberg in a seminal paper [17], one has to write down the most general Lagrangian consistent with the assumed symmetry principles, particularly Table 1: Seven Decades of Struggle: The Theory of Nuclear Forces 1935 Yukawa: Meson Theory The “Pion Theories” 1950’s One-Pion Exchange: o.k. Multi-Pion Exchange: disaster Many pions ≡ multi-pion resonances: 1960’s σ, ρ, ω, ... The One-Boson-Exchange Model: success Refined meson models, including 1970’s sophisticated 2π exchange contributions (Stony Brook, Paris, Bonn) Nuclear physicists discover 1980’s QCD Quark Cluster Models Nuclear physicists discover EFT 1990’s Weinberg, van Kolck and beyond Back to Pion Theory! But, constrained by Chiral Symmetry: success the (broken) chiral symmetry of QCD. At low energy, the effective degrees of freedom are pions and nucleons rather than quarks and gluons; heavy mesons and nucleon resonances are “integrated out”. So, the circle of his- tory is closing and we are back to Yukawa’s meson theory, except that we have learned to add one important refinement to the theory: broken chiral symmetry is a crucial constraint that generates and controls the dynamics and establishes a clear connection with the underlying theory, QCD. Following the first initiative by Weinberg [18], pioneering work was per- formed by Ordóñez, Ray, and van Kolck [19, 20] who constructed a NN potential in coordinate space based upon chiral perturbation theory at next- to-next-to-leading order. The results were encouraging and many researchers became attracted to the new field [21, 22, 23, 24, 25, 26, 27]. As a conse- quence, nuclear EFT has developed into one of the most popular branches of modern nuclear physics [28, 29]. It is the purpose of these lectures to describe in some detail the recent progress in our understanding of nuclear forces in terms of nuclear EFT. 2 QCD and the Nuclear Force Quantum chromodynamics (QCD) is the theory of strong interactions. It deals with quarks, gluons and their interactions and is part of the Standard Model of Particle Physics. QCD is a non-Abelian gauge field theory with color SU(3) the underlying gauge group. The non-Abelian nature of the theory has dramatic consequences. While the interaction between colored objects is weak at short distances or high momentum transfer (“asymptotic freedom”); it is strong at long distances ( >∼ 1 fm) or low energies, leading to the confinement of quarks into colorless objects, the hadrons. Conse- quently, QCD allows for a perturbative analysis at large energies, whereas it is highly non-perturbative in the low-energy regime. Nuclear physics resides at low energies and the force between nucleons is a residual QCD interac- tion. Therefore, in terms of quarks and gluons, the nuclear force is a very complicated problem. 3 Effective Field Theory for Low-Energy QCD The way out of the dilemma of how to derive the nuclear force from QCD is provided by the effective field theory (EFT) concept. First, one needs to identify the relevant degrees of freedom. For the ground state and the low- energy excitation spectrum of an atomic nucleus as well as for conventional nuclear reactions, quarks and gluons are ineffective degrees of freedom, while nucleons and pions are the appropriate ones. Second; to make sure that this EFT is not just another phenomenology, the EFT must observe all relevant symmetries of the underlying theory. This requirement is based upon a ‘folk theorem’ by Weinberg [17]: If one writes down the most general possible Lagrangian, in- cluding all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, per- turbative unitarity, cluster decomposition, and the assumed sym- metry principles. Thus, the EFT program consists of the following steps: 1. Identify the degrees of freedom relevant at the resolution scale of (low- energy) nuclear physics: nucleons and pions. 2. Identify the relevant symmetries of low-energy QCD and investigate if and how they are broken. 3. Construct the most general Lagrangian consistent with those symme- tries and the symmetry breaking. 4. Design an organizational scheme that can distinguish between more and less important contributions: a low-momentum expansion. 5. Guided by the expansion, calculate Feynman diagrams to the the de- sired accuracy for the problem under consideration. We will now elaborate on these steps, one by one. 3.1 Symmetries of Low-Energy QCD In this section, we will give a brief introduction into (low-energy) QCD, its symmetries and symmetry breaking. A more detailed introduction can be found in the excellent lecture series by Scherer and Schindler [30]. 3.1.1 Chiral Symmetry The QCD Lagrangian reads LQCD = q̄(iγµDµ −M)q − Gµν,aGµνa (1) with the gauge-covariant derivative Dµ = ∂µ + ig Aµ,a (2) and the gluon field strength tensor Gµν,a = ∂µAν,a − ∂νAµ,a − gfabcAµ,bAν,c . (3) In the above, q denotes the quark fields and M the quark mass matrix. Further, g is the strong coupling constant and Aµ,a are the gluon fields. The λa are the Gell-Mann matrices and the fabc the structure constants of the SU(3)color Lie algebra (a, b, c = 1, . . . , 8); summation over repeated indices is always implied. The gluon-gluon term in the last equation arises from the non-Abelian nature of the gauge theory and is the reason for the peculiar features of the color force. On a typical hadronic scale, i.e., on a scale of low-mass hadrons which are not Goldstone bosons, e.g., mρ = 0.78 GeV ≈ 1 GeV; the masses of the up (u), down (d), and—to a certain extend—strange (s) quarks are small [31]: mu = 2± 1 MeV (4) md = 5± 2 MeV (5) ms = 95± 25 MeV (6) It is therefore of interest to discuss the QCD Lagrangian in the limit of vanishing quark masses: L0QCD = q̄iγ µDµq − Gµν,aGµνa . (7) Defining right- and left-handed quark fields, qR = PRq , qL = PLq , (8) (1 + γ5) , PL = (1− γ5) , (9) we can rewrite the Lagrangian as follows: L0QCD = q̄Riγ µDµqR + q̄LiγµDµqL − Gµν,aGµνa . (10) Restricting ourselves now to up and down quarks, we see that L0QCD is invariant under the global unitary transformations 7−→ exp −iΘRi 7−→ exp −iΘLi , (12) where τi (i = 1, 2, 3) are the generators of SU(2)flavor, the usual Pauli spin matrices. The right- and left-handed components of massless quarks do not mix. This is SU(2)R × SU(2)R symmetry, also known as chiral symmetry. Noether’s Theorem implies the existence of six conserved currents; three right-handed currents i = q̄Rγ qR with ∂µR i = 0 (13) and three left-handed currents i = q̄Lγ qL with ∂µL i = 0 . (14) It is useful to consider the following linear combinations; namely, three vec- tor currents i = R i + L i = q̄γ q with ∂µV i = 0 (15) and three axial-vector currents i = R i − L i = q̄γ q with ∂µA i = 0 , (16) which got their names from the fact that they transform as vectors and axial-vectors, respectively. Thus, the chiral SU(2)L × SU(2)R symmetry is equivalent to SU(2)V ×SU(2)A, where the vector and axial-vector transfor- mations are given respectively by 7−→ exp −iΘVi 7−→ exp −iΘAi γ5 . (18) Obviously, the vector transformations are isospin rotations and, therefore, invariance under vector transformations can be identified with isospin sym- metry. There are the six conserved charges, QVi = d3x V 0i = d3x q†(t, ~x) q(t, ~x) with = 0 (19) QAi = d3x A0i = d3x q†(t, ~x)γ5 q(t, ~x) with = 0 , (20) which are also generators of SU(2)V × SU(2)A. 3.1.2 Explicit Symmetry Breaking The mass term −q̄Mq in the QCD Lagrangian Eq. (1) breaks chiral sym- metry explicitly. To better see this, let’s rewrite M, (mu +md) (mu −md) (mu +md) I + (mu −md) τ3 . (23) The first term in the last equation in invariant under SU(2)V (isospin sym- metry) and the second term vanishes for mu = md. Thus, isospin is an exact symmetry if mu = md. However, both terms in Eq. (23) break SU(2)A. Since the up and down quark masses are small as compared to the typical hadronic mass scale of ≈ 1 GeV [cf. Eqs. (4) and (5)], the explicit chiral symmetry breaking due to non-vanishing quark masses is very small. 3.1.3 Spontaneous Symmetry Breaking A (continuous) symmetry is said to be spontaneously broken if a symmetry of the Lagrangian is not realized in the ground state of the system. There is evidence that the chiral symmetry of the QCD Lagrangian is spontaneously broken—for dynamical reasons of nonperturbative origin which are not fully understood at this time. The most plausible evidence comes from the hadron spectrum. From chiral symmetry, one would naively expect the existence of degenerate hadron multiplets of opposite parity, i.e., for any hadron of positive parity one would expect a degenerate hadron state of negative parity and vice versa. However, these “parity doublets” are not observed in nature. For example, take the ρ-meson, a vector meson with negative parity (1−) and mass 776 MeV. There does exist a 1+ meson, the a1, but it has a mass of 1230 MeV and, thus, cannot be perceived as degenerate with the ρ. On the other hand, the ρ meson comes in three charge states (equivalent to three isospin states), the ρ± and the ρ0 with masses that differ by at most a few MeV. In summary, in the QCD ground state (the hadron spectrum) SU(2)V (isospin symmetry) is well observed, while SU(2)A (axial symmetry) is broken. Or, in other words, SU(2)V ×SU(2)A is broken down to SU(2)V . A spontaneously broken global symmetry implies the existence of (mass- less) Goldstone bosons with the quantum numbers of the broken generators. The broken generators are the QAi of Eq. (20) which are pseudoscalar. The Goldstone bosons are identified with the isospin triplet of the (pseudoscalar) pions, which explains why pions are so light. The pion masses are not ex- actly zero because the up and down quark masses are not exactly zero either (explicit symmetry breaking). Thus, pions are a truly remarkable species: they reflect spontaneous as well as explicit symmetry breaking. 3.2 Chiral Effective Lagrangians Involving Pions The next step in our EFT program is to build the most general Lagrangian consistent with the (broken) symmetries discussed above. An elegant formal- ism for the construction of such Lagrangians was developed by Callan, Cole- man, Wess, and Zumino (CCWZ) [32] who worked out the group-theoretical foundations of non-linear realizations of chiral symmetry. The Lagrangians given below are built upon the CCWZ formalism. As discussed, the relevant degrees of freedom are pions (Goldstone bosons) and nucleons. Since the interactions of Goldstone bosons must vanish at zero momentum transfer and in the chiral limit (m→ 0), the low-energy expan- sion of the Lagrangian is arranged in powers of derivatives and pion masses. This is chiral perturbation theory (ChPT). The Lagrangian consists of one part that deals with the interaction among pions, Lππ, and another one that describes the interaction between pions and the nucleon, LπN : Leff = Lππ + LπN (24) Lππ = L(2)ππ + L ππ + . . . (25) LπN = L πN + L πN + L πN + . . . , (26) where the superscript refers to the number of derivatives or pion mass inser- tions (chiral dimension) and the ellipsis stands for terms of higher dimension. The leading order (LO) ππ Lagrangian is given by [33] L(2)ππ = ∂µU∂µU † +m2π(U + U and the LO relativistic πN Lagrangian reads [34] L(1)πN = Ψ̄ iγµDµ −MN + γµγ5uµ Ψ (28) Dµ = ∂µ + Γµ (29) (ξ†∂µξ + ξ∂µξ τ · (π × ∂µπ) + . . . (30) uµ = i(ξ †∂µξ − ξ∂µξ†) = − τ · ∂µπ + . . . (31) U = ξ2 = 1 + τ · π − (τ · π)3 + 8α− 1 π4 + . . . (32) In Eq. (28) the chirally covariant derivative Dµ is applied which introduces the “gauge term” Γµ (also known as chiral connection), a vector current that leads to a coupling of pions with the nucleon. Besides this, the Lagrangian includes a coupling term which involves the axial vector uµ. The SU(2) matrix U = ξ2 collects the Goldstone pion fields. In the above equations, MN denotes the nucleon mass, gA the axial- vector coupling constant, and fπ the pion decay constant. Numerical values will be given later. The coefficient α that appears in Eq. (32) is arbitrary. Therefore, dia- grams with chiral vertices that involve three or four pions must always be grouped together such that the α-dependence drops out (cf. Fig. 4, below). We apply the heavy baryon (HB) formulation of chiral perturbation the- ory [35] in which the relativistic πN Lagrangian is subjected to an expansion in terms of powers of 1/MN (kind of a nonrelativistic expansion), the lowest order of which is L̂(1)πN = N̄ iD0 − ~σ · ~u i∂0 − τ · (π × ∂0π)− τ · (~σ · ~∇)π N + . . . (33) In the relativistic formulation, the nucleon is represented by a four-component Dirac spinor field, Ψ, while in the HB version, the nucleon, N , is a Pauli spinor; in addition, all nucleon fields include Pauli spinors describing the isospin of the nucleon. At dimension two, the relativistic πN Lagrangian reads L(2)πN = ciΨ̄O i Ψ . (34) The various operators O(2)i are given in Ref. [36]. The fundamental rule by which this Lagrangian—as well as all the other ones—are assembled is that they must contain all terms consistent with chiral symmetry and Lorentz invariance (apart from other trivial symmetries) at a given chiral dimension (here: order two). The parameters ci are known as low-energy constants (LECs) and are determined empirically from fits to πN data. The HB projected πN Lagrangian at order two is most conveniently broken up into two pieces, L̂(2)πN = L̂ πN, fix + L̂ πN, ct , (35) L̂(2)πN, fix = N̄ ~D · ~D + i {~σ · ~D, u0} N (36) L̂(2)πN, ct = N̄ 2 c1m π (U + U u20 + c3 uµu ~σ · (~u× ~u) N . (37) Note that L̂(2)πN, fix is created entirely from the HB expansion of the relativistic L(1)πN and thus has no free parameters (“fixed”), while L̂ πN, ct is dominated by the new πN contact terms proportional to the ci parameters, besides some small 1/MN corrections. At dimension three, the relativistic πN Lagrangian can be formally writ- ten as L(3)πN = diΨ̄O i Ψ , (38) with the operators, O(3)i , listed in Refs. [36, 37]; not all 23 terms are of interest here. The new LECs that occur at this order are the di. Similar to the order two case, the HB projected Lagrangian at order three can be broken into two pieces, L̂(3)πN = L̂ πN, fix + L̂ πN, ct , (39) with L̂(3)πN, fix and L̂ πN, ct given in Refs. [36, 37]. 3.3 Nucleon Contact Lagrangians Nucleon contact interactions consist of four nucleon fields (four nucleon legs) and no meson fields. Such terms are needed to renormalize loop integrals, to make results reasonably independent of regulators, and to parametrize the unresolved short-distance contributions to the nuclear force. For more about contact terms, see Sec. 5.3. Because of parity, nucleon contact interactions come only in even num- bers of derivatives, thus, LNN = L NN + L NN + L NN + . . . (40) The lowest order (or leading order) NN Lagrangian has no derivatives and reads [18] L(0)NN = − CSN̄NN̄N − CT N̄~σNN̄~σN , (41) where N is the heavy baryon nucleon field. CS and CT are unknown con- stants which are determined by a fit to the NN data. The second order NN Lagrangian is given by [19] L(2)NN = −C 1[(N̄ ~∇N) 2 + (~∇NN)2]− C ′2(N̄ ~∇N) · (~∇NN) −C ′3N̄N [N̄ ~∇ 2N + ~∇2NN ] −iC ′4[N̄ ~∇N · (~∇N × ~σN) + (~∇N)N · (N̄~σ × ~∇N)] −iC ′5N̄N(~∇N · ~σ × ~∇N)− iC 6(N̄~σN) · (~∇N × ~∇N) −(C ′7δikδjl + C 8δilδkj + C 9δijδkl) ×[N̄σk∂iNN̄σl∂jN + ∂iNσkN∂jNσlN ] −(C ′10δikδjl + C 11δilδkj + C 12δijδkl)N̄σk∂iN∂jNσlN C ′13(δikδjl + δilδkj) +C ′14δijδkl)[∂iNσk∂jN + ∂jNσk∂iN ]N̄σlN . (42) Similar to CS and CT , the C ′i are unknown constants which are fixed in a fit to the NN data. Obviously, these contact Lagrangians blow up quite a bit with increasing order, which why we do not give L(4)NN explicitly here. 4 Nuclear Forces from EFT: Overview In the beginning of Sec. 3, we spelled out the steps we have to take to accomplish our EFT program for the derivation of nuclear forces. So far, we discussed steps one to three. What is left are steps four (low-momentum expansion) and five (Feynman diagrams). In this section, we will say more about the expansion we are using and give an overview of the Feynman diagrams that arise order by order. 4.1 Chiral Perturbation Theory and Power Counting In ChPT, we analyze contributions in terms of powers of small momenta over the large scale: (Q/Λχ)ν , where Q stands for a momentum (nucleon three-momentum or pion four-momentum) or a pion mass and Λχ ≈ 1 GeV is the chiral symmetry breaking scale (hadronic scale). Determining the power ν at which a given diagram contributes has become known as power counting. For a non-iterative contribution involving A nucleons, the power ν is given by ν = −2 + 2A− 2C + 2L+ ∆i , (43) ∆i ≡ di + − 2 , (44) where C denotes the number of separately connected pieces and L the num- ber of loops in the diagram; di is the number of derivatives or pion-mass insertions and ni the number of nucleon fields involved in vertex i; the sum runs over all vertices contained in the diagram under consideration. Note that for an irreducible NN diagram (A = 2), the above formula reduces to ν = 2L+ ∆i (45) The power ν is bounded from below; e.g., for A = 2, ν ≥ 0. This fact is crucial for the power expansion to be of any use. 4.2 The Hierarchy of Nuclear Forces Chiral perturbation theory and power counting imply that nuclear forces emerge as a hierarchy ruled by the power ν, Fig. 1. The NN amplitude is determined by two classes of contributions: con- tact terms and pion-exchange diagrams. There are two contacts of order Q0 [O(Q0)] represented by the four-nucleon graph with a small-dot vertex shown in the first row of Fig. 1. The corresponding graph in the second row, four nucleon legs and a solid square, represents the seven contact terms of O(Q2). Finally, at O(Q4), we have 15 contact contributions represented by a four-nucleon graph with a solid diamond. Now, turning to the pion contributions: At leading order [LO, O(Q0), ν = 0], there is only the well-known static one-pion exchange (1PE), second diagram in the first row of Fig. 1. Two-pion exchange (2PE) starts at next- to-leading order (NLO, ν = 2) and all diagrams of this leading-order two- pion exchange are shown. Further 2PE contributions occur in any higher +... +... +... 2N Force 3N Force 4N Force N LO3 Figure 1: Hierarchy of nuclear forces in ChPT. Solid lines represent nucleons and dashed lines pions. Further explanations are given in the text. order. Of this sub-leading 2PE, we show only two representative diagrams at next-to-next-to-leading order (NNLO) and three diagrams at next-to-next- to-next-to-leading order (N3LO). Finally, there is also three-pion exchange, which shows up for the first time at N3LO (two loops; one representative 3π diagram is included in Fig. 1). At this order, the 3π contribution is negligible [38]. One important advantage of ChPT is that it makes specific predictions also for many-body forces. For a given order of ChPT, two-nucleon forces (2NF), three-nucleon forces (3NF), . . . are generated on the same footing (cf. Fig. 1). At LO, there are no 3NF, and at NLO, all 3NF terms cancel [18, 39]. However, at NNLO and higher orders, well-defined, nonvanishing 3NF occur [39, 40]. Since 3NF show up for the first time at NNLO, they are weak. Four-nucleon forces (4NF) occur first at N3LO and, therefore, they are even weaker. 5 Two-Nucleon Forces In this section, we will elaborate in detail on the two-nucleon force contri- butions of which we have given a rough overview in the previous section. 5.1 Pion-Exchange Contributions in ChPT The effective pion Lagrangians presented in Sec. 3.2 are the crucial ingredi- ents for the evaluation of the pion-exchange contributions to the NN inter- action. We will derive these contributions now order by order. We will state our results in terms of contributions to the momentum- space NN amplitude in the center-of-mass system (CMS), which takes the general form V (~p ′, ~p) = VC + τ 1 · τ 2WC + [VS + τ 1 · τ 2WS ] ~σ1 · ~σ2 + [VLS + τ 1 · τ 2WLS ] −i~S · (~q × ~k) + [VT + τ 1 · τ 2WT ] ~σ1 · ~q ~σ2 · ~q + [VσL + τ 1 · τ 2WσL ] ~σ1 · (~q × ~k ) ~σ2 · (~q × ~k ) , (46) where ~p ′ and ~p denote the final and initial nucleon momenta in the CMS, respectively; moreover, ~q ≡ ~p ′ − ~p is the momentum transfer, ~k ≡ 1 (~p ′ + ~p) the average momentum, ~S ≡ 1 (~σ1 + ~σ2) the total spin, and ~σ1,2 and τ 1,2 are the spin and isospin operators, respectively, of nucleon 1 and 2. For on-energy-shell scattering, Vα and Wα (α = C, S, LS, T, σL) can be expressed as functions of q and k (with q ≡ |~q| and k ≡ |~k|), only. Our formalism is similar to the one used by the Munich group [22, 41, 42] except for two differences: all our momentum space amplitudes differ by an over-all factor of (−1) and our spin-orbit potentials, VLS and WLS , differ by an additional factor of (−2). Our conventions are more in tune with what is commonly used in nuclear physics. In all expressions given below, we will state only the nonpolynomial con- tributions to the NN amplitude. Note, however, that dimensional regular- ization typically generates also polynomial terms. These polynomials are absorbed by the contact interactions to be discussed in a later section and, therefore, they are of no interest here. 5.1.1 Zeroth Order (LO) At order zero [ν = 0, O(Q0), lowest order, leading order, LO], there is only the well-known static one-pion exchange, second diagram in the first row of Fig. 1 which is given by: V1π(~p ′, ~p) = − τ 1 · τ 2 ~σ1 · ~q ~σ2 · ~q q2 +m2π . (48) At first order [ν = 1, O(Q)], there are no pion-exchange contributions (and also no contact terms). 5.1.2 Second Order (NLO) Non-vanishing higher-order graphs start at second order (ν = 2, next-to- leading order, NLO). The most efficient way to evaluate these loop diagrams is to use covariant perturbation theory and dimensional regularization. This is the method applied by the Munich group [22, 41, 42]. One starts with the relativistic versions of the πN Lagrangians (cf. Sec. 3.2) and sets up four-dimensional (covariant) loop integrals. Relativistic vertices and nucleon propagators are then expanded in powers of 1/MN . The divergences that occur in conjunction with the four-dimensional loop integrals are treated by means of dimensional regularization, a prescription which is consistent with chiral symmetry and power counting. The results derived in this way are the same obtained when starting right away with the HB versions of the πN Lagrangians. However, as it turns out, the method used by the Munich group is more efficient in dealing with the rather tedious calculations. Two-pion exchange occurs first at second order, also know as leading- order 2π exchange. The graphs are shown in the first row of Fig. 2. Since a loop creates already ν = 2, the vertices involved at this order can only be from the leading/lowest order Lagrangian L̂(1)πN , Eq. (33), i. e., they carry only one derivative. These vertices are denoted by small dots in Fig. 2. Concerning the box diagram, we should note that we include only the non- iterative part of this diagram which is obtained by subtracting the iter- ated 1PE contribution Eq. (65) or Eq. (66), below, but using M2N/Ep ≈ (N LO)2 (NLO) Figure 2: Two-pion exchange contributions to the NN interaction at order two and three in small momenta. Solid lines represent nucleons and dashed lines pions. Small dots denote vertices from the leading order πN Lagrangian L̂(1)πN , Eq. (33). Large solid dots are vertices proportional to the LECs ci from the second order Lagrangian L̂(2)πN, ct, Eq. (37). Symbols with an open circles are relativistic 1/MN corrections contained in the second order Lagrangian L̂(2)πN , Eqs. (35). Only a few representative examples of 1/MN corrections are shown and not all. M2N/Ep′′ ≈ MN at this order (NLO). Summarizing all contributions from irreducible two-pion exchange at second order, one obtains [22]: WC = − 384π2f4π 4m2π(5g A − 4g A − 1) + q 2(23g4A − 10g A − 1) 48g4Am , (49) VT = − VS = − 3g4AL(q) 64π2f4π , (50) where L(q) ≡ w + q 4m2π + q2 . (52) 5.1.3 Third Order (NNLO) The two-pion exchange diagrams of order three (ν = 3, next-to-next-to- leading order, NNLO) are very similar to the ones of order two, except that they contain one insertion from L̂(2)πN , Eq. (35). The resulting contributions are typically either proportional to one of the low-energy constants ci or they contain a factor 1/MN . Notice that relativistic 1/MN corrections can occur for vertices and nucleon propagators. In Fig. 2, we show in row 2 the diagrams with vertices proportional to ci (large solid dot), Eq. (37), and in row 3 and 4 a few representative graphs with a 1/MN correction (symbols with an open circle). The number of 1/MN correction graphs is large and not all are shown in the figure. Again, the box diagram is corrected for a contribution from the iterated 1PE. If the iterative 2PE of Eq. (65) is used, the expansion of the factor M2N/Ep = MN − p 2/2MN + . . . is applied and the term proportional to (−p2/2MN ) is subtracted from the third order box diagram contribution. Then, one obtains for the full third order contribution [22]: 16πf4π 16MNw2 2m2π(2c1 − c3)− q × w̃2A(q) , (53) 128πMNf4π 3g2Am 4m2π + 2q 2 − g2A(4m π + 3q w̃2A(q) , (54) VT = − 9g4Aw̃ 2A(q) 512πMNf4π , (55) WT = − g2AA(q) 32πf4π (10m2π + 3q , (56) VLS = 3g4Aw̃ 2A(q) 32πMNf4π , (57) WLS = g2A(1− g 32πMNf4π w2A(q) , (58) A(q) ≡ arctan 2m2π + q2 . (60) As discussed in Sec. 5.1.5, below, we prefer the iterative 2PE defined in Eq. (66), which leads to a different NNLO term for the iterative 2PE. This changes the 1/MN terms in the above potentials. The changes are obtained by adding to Eqs. (53)-(56) the following terms: VC = − 256πf4πMN 2 + ω̃4A(q)) (61) 128πf4πMN 2 + ω̃4A(q)) (62) VT = − 512πf4πMN (mπ + ω 2A(q)) (63) WT = − WS = − 256πf4πMN (mπ + ω 2A(q)) (64) 5.1.4 Fourth Order (N3LO) This order, which may also be denoted by next-to-next-to-next-to-leading order (N3LO), is very involved. Three-pion exchange (3PE) occurs for the first time at this order. The 3PE contribution at N3LO has been calculated by the Munich group and found to be negligible [38]. Therefore, we will ignore it. The 2PE contributions at N3LO can be subdivided into two groups, one- loop graphs, Fig. 3, and two-loop diagrams, Fig. 4. Since these contributions are very complicated, we have moved them to Appendix A. 5.1.5 Iterated One-Pion-Exchange Besides all the irreducible 2PE contributions presented above, there is also the reducible 2PE which is generated from iterated 1PE. This “iterative 2PE” is the only 2PE contribution which produces an imaginary part. Thus, one wishes to formulate this contribution such that relativistic elastic uni- tarity is satisfied. There are several ways to achieve this. Kaiser et al. [22] define the iterative 2PE contribution as follows, (KBW) 2π,it (~p ′, ~p) = d3p′′ (2π)3 V1π(~p ′, ~p ′′)V1π(~p ′′, ~p) p2 − p′′2 + i� Q4 (N LO)3 Figure 3: One-loop 2π-exchange contributions to the NN interaction at order four. Basic notation as in Fig. 2. Symbols with a large solid dot and an open circle denote 1/MN corrections of vertices proportional to ci. Symbols with two open circles mark relativistic 1/M2N corrections. Both corrections are part of the third order Lagrangian L̂(3)πN , Eq. (39). Representative examples for all types of one-loop graphs that occur at this order are shown. with V1π given in Eq. (48). Since we adopt the relativistic scheme developed by Blankenbecler and Sugar [43] (BbS) (see beginning of Sec. 5.4), we prefer the following for- mulation which is consistent with the BbS approach (and, of course, with relativistic elastic unitarity): 2π,it (~p ′, ~p) = d3p′′ (2π)3 V1π(~p ′, ~p ′′)V1π(~p ′′, ~p) p2 − p′′2 + i� . (66) The iterative 2PE contribution has to be subtracted from the covariant box diagram, order by order. For this, the expansion M2N/Ep = MN − p2/2MN + . . . is applied in Eq. (65) and M2N/Ep′′ = MN −p ′′2/2MN + . . . in Eq. (66). At NLO, both choices for the iterative 2PE collapse to the same, � � � � � � � � � � � (N LO)3 Figure 4: Two-loop 2π-exchange contributions at order four. Basic notation as in Fig. 2. The oval stands for all one-loop πN graphs some of which are shown in the lower part of the figure. The solid square represents vertices proportional to the LECs di which are introduced by the third order Lagrangian L πN , Eq. (38). More explanations are given in the text. while at NNLO there are obvious differences. 5.2 NN Scattering in Peripheral Partial Waves Using the Perturbative Amplitude After the tedious mathematics of the previous section, it is time for more tangible affairs. The obvious question to address now is: How does the derived NN amplitude compare to empirical information? Since our deriva- tion includes only one- and two-pion exchanges, we are dealing here with the long- and intermediate-range part of the NN interaction. This part of the nuclear force is probed in the peripheral partial waves of NN scattering. Thus, in this section, we will calculate the phase shifts that result from the NN amplitudes presented in the previous section and compare them to the empirical phase shifts as well as to the predictions from conventional meson theory. Besides the irreducible two-pion exchanges derived above, we must also include 1PE and iterated 1PE. In this section [44], which is restricted to just peripheral waves, we will always consider neutron-proton (np) scattering and take the charge- dependence of 1PE due to pion-mass splitting into account, since it is ap- preciable. With the definition V1π(mπ) ≡ − ~σ1 · ~q ~σ2 · ~q q2 +m2π , (67) the charge-dependent 1PE for np scattering is 1π (~p ′, ~p) = −Vπ(mπ0) + (−1) I+1 2Vπ(mπ±) , (68) where I denotes the isospin of the two-nucleon system. We use mπ0 = 134.9766 MeV, mπ± = 139.5702 MeV [31], and 2MpMn Mp +Mn = 938.9182 MeV . (69) Also in the iterative 2PE, we apply the charge-dependent 1PE, i.e., in Eq. (66) we replace V1π with V The perturbative relativistic T-matrix for np scattering in peripheral waves is T (~p ′, ~p) = V (np)1π (~p ′, ~p) + V (EM,np)2π,it (~p ′, ~p) + V2π,irr(~p ′, ~p) , (70) where V2π,irr refers to any or all of the irreducible 2PE contributions pre- sented in Sec. 5.1, depending on the order at which the calculation is con- ducted. In the calculation of the irreducible 2PE, we use the average pion mass mπ = 138.039 MeV and, thus, neglect the charge-dependence due to pion-mass splitting. The charge-dependence that emerges from irreducible 2π exchange was investigated in Ref. [45] and found to be negligible for partial waves with L ≥ 3. For the T -matrix given in Eq. (70), we calculate phase shifts for partial waves with L ≥ 3 and Tlab ≤ 300 MeV. At order four in small momenta, partial waves with L ≥ 3 do not receive any contributions from contact inter- actions and, thus, the non-polynomial pion contributions uniquely predict the F and higher partial waves. We use fπ = 92.4 MeV [31] and gA = 1.29. Via the Goldberger-Treiman relation, gA = gπNN fπ/MN , our value for gA is consistent with g2πNN/4π = 13.63± 0.20 which is obtained from πN and NN analysis [46, 47]. The LECs used in this calculation are shown in Table 2, column “NN periph. Fig. 5”. Note that many determinations of the LECs, ci and d̄i, can be found in the literature. The most reliable way to determine the LECs Table 2: Low-energy constants, LECs, used for a NN potential at N3LO, Sec. 5.4.4, and in the calculation of the peripheral NN phase shifts shown in Fig. 5 (column “NN periph. Fig. 5”). The ci belong to the dimension- two πN Lagrangian, Eq. (37), and are in units of GeV−1, while the d̄i are associated with the dimension-three Lagrangian, Eq. (38), and are in units of GeV−2. The column “πN empirical” shows determinations from πN data. LEC NN potential NN periph. πN at N3LO Fig. 5 empirical c1 –0.81 –0.81 −0.81± 0.15a c2 2.80 3.28 3.28± 0.23b c3 –3.20 –3.40 −4.69± 1.34a c4 5.40 3.40 3.40± 0.04a d̄1 + d̄2 3.06 3.06 3.06± 0.21b d̄3 –3.27 –3.27 −3.27± 0.73b d̄5 0.45 0.45 0.45± 0.42b d̄14 − d̄15 –5.65 –5.65 −5.65± 0.41b aTable 1, Fit 1 of Ref. [48]. bTable 2, Fit 1 of Ref. [37]. from empirical πN information is to extract them from the πN amplitude inside the Mandelstam triangle (unphysical region) which can be constructed with the help of dispersion relations from empirical πN data. This method was used by Büttiker and Meißner [48]. Unfortunately, the values for c2 and all d̄i parameters obtained in Ref. [48] carry uncertainties, so large that the values cannot provide any guidance. Therefore, in Table 2, only c1, c3, and c4 are from Ref. [48], while the other LECs are taken from Ref. [37] where the πN amplitude in the physical region was considered. To establish a link between πN and NN , we apply the values from the above determinations in our calculations of the NN peripheral phase shifts. In general, we use the mean values; the only exception is c3, where we choose a value that is, in terms of magnitude, about one standard deviation below the one from Ref. [48]. With the exception of c3, phase shift predictions do not depend sensitively on variations of the LECs within the quoted uncertainties. In Fig. 5, we show the phase-shift predictions for neutron-proton scat- tering in F waves for laboratory kinetic energies below 300 MeV (for G and H waves, see Ref. [26]). The orders displayed are defined as follows: • Leading order (LO) is just 1PE, Eq. (68). 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) Figure 5: F -wave phase shifts of neutron-proton scattering for laboratory kinetic energies below 300 MeV. We show the predictions from chiral pion exchange at lead- ing order (LO), next-to-leading order (NLO), next-to-next-to-leading order (N2LO), and next-to-next-to-next-to-leading order (N3LO). The solid dots and open circles are the results from the Nijmegen multi-energy np phase shift analysis [49] and the VPI single-energy np analysis SM99 [50], respectively. • Next-to-leading order (NLO) is 1PE, Eq. (68), plus iterated 1PE, Eq. (66), plus the contributions of Sec. 5.1.2 (order two), Eqs. (49) and (50). • Next-to-next-to-leading order (denoted by N2LO in the figures) con- sists of NLO plus the contributions of Sec. 5.1.3 (order three), Eqs. (53)- (58) and (61)-(64). • Next-to-next-to-next-to-leading order (denoted by N3LO in the fig- ures) consists of N2LO plus the contributions of Sec. 5.1.4 (order four), Eqs. (99)-(112) and (115)-(124). It is clearly seen in Fig. 5 that the leading order 2π exchange (NLO) is a rather small contribution, insufficient to explain the empirical facts. In 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) Figure 6: F -wave phase shifts of neutron-proton scattering for laboratory kinetic energies below 300 MeV. We show the results from one-pion-exchange (OPE), and one- plus two-pion exchange as predicted by ChPT at next-to-next-to-next- to-leading order (N3LO) and by the Bonn Full Model [13] (Bonn). Note that the “Bonn” curve does not include the repulsive ω and πρ exchanges of the full model, since this figure serves the purpose to compare just predictions by different mod- els/theories for the π + 2π contribution to the NN interaction. Empirical phase shifts (solid dots and open circles) as in Fig. 5. contrast, the next order (N2LO) is very large, several times NLO. This is due to the ππNN contact interactions proportional to the LECs ci that are introduced by the second order Lagrangian L(2)πN , Eq. (34). These contacts are supposed to simulate the contributions from intermediate ∆-isobars and correlated 2π exchange which are known to be large (see, e. g., Ref. [13]). At N3LO a clearly identifiable trend towards convergence emerges. Ob- viously, 1F3 and 3F4 appear fully converged. However, in 3F2 and 3F3, N3LO differs noticeably from NNLO, but the difference is much smaller than the one between NNLO and NLO. This is what we perceive as a trend towards convergence. In Fig. 6, we conduct a comparison between the predictions from chi- ral one- and two-pion exchange at N3LO and the corresponding predictions from conventional meson theory (curve ‘Bonn’). As representative for con- ventional meson theory, we choose the Bonn meson-exchange model for the NN interaction [13], since it contains a comprehensive and thoughtfully con- structed model for 2π exchange. This 2π model includes box and crossed box diagrams with NN , N∆, and ∆∆ intermediate states as well as di- rect ππ interaction in S- and P -waves (of the ππ system) consistent with empirical information from πN and ππ scattering. Besides this the Bonn model also includes (repulsive) ω-meson exchange and irreducible diagrams of π and ρ exchange (which are also repulsive). However, note that in the phase shift predictions displayed in Fig. 6, the “Bonn” curve includes only the 1π and 2π contributions from the Bonn model; the short-range contri- butions are left out since the purpose of the figure is to compare different models/theories for π + 2π. In all waves shown we see, in general, good agreement between N3LO and Bonn. In 3F2 and 3F3 above 150 MeV and in 3F4 above 250 MeV the chiral model at N3LO is more attractive than the Bonn 2π model. Note, however, that the Bonn model is relativistic and, thus, includes relativistic corrections up to infinite orders. Thus, one may speculate that higher orders in ChPT may create some repulsion, moving the Bonn and the chiral predictions even closer together [51]. The 2π exchange contribution to the NN interaction can also be de- rived from empirical πN and ππ input using dispersion theory, which is based upon unitarity, causality (analyticity), and crossing symmetry. The amplitude NN̄ → ππ is constructed from πN → πN and πN → ππN data using crossing properties and analytic continuation; this amplitude is then ‘squared’ to yield the NN̄ amplitude which is related to NN by cross- ing symmetry [52]. The Paris group [11, 12] pursued this path and calcu- lated NN phase shifts in peripheral partial waves. Naively, the dispersion- theoretic approach is the ideal one, since it is based exclusively on empirical information. Unfortunately, in practice, quite a few uncertainties enter into the approach. First, there are ambiguities in the analytic continuation and, second, the dispersion integrals have to be cut off at a certain momentum to ensure reasonable results. In Ref. [13], a thorough comparison was con- ducted between the predictions by the Bonn model and the Paris approach and it was demonstrated that the Bonn predictions always lie comfortably within the range of uncertainty of the dispersion-theoretic results. There- fore, there is no need to perform a separate comparison of our chiral N3LO predictions with dispersion theory, since it would not add anything that we cannot conclude from Fig. 6. Finally, we need to compare the predictions with the empirical phase shifts. In F waves the N3LO predictions above 200 MeV are, in general, too attractive. Note, however, that this is also true for the predictions by the Bonn π + 2π model. In the full Bonn model, besides π + 2π, (repul- sive) ω and πρ exchanges are included which move the predictions right on top of the data. The exchange of a ω meson or combined πρ exchange are 3π exchanges. Three-pion exchange occurs first at chiral order four. It has be investigated by Kaiser [38] and found to be negligible, at this order. However, 3π exchange at order five appears to be sizable [53] and may have impact on F waves. Besides this, there is the usual short-range phenomenology. In ChPT, this short-range interaction is parametrized in terms of four-nucleon contact terms (since heavy mesons do not have a place in that theory). Contact terms of order four (N3LO) do not contribute to F -waves, but order six does. In summary, the remaining small discrepan- cies between the N3LO predictions and the empirical phase shifts may be straightened out in fifth or sixth order of ChPT. 5.3 NN Contact Potentials In conventional meson theory, the short-range nuclear force is described by the exchange of heavy mesons, notably the ω(782). The qualitative short- distance behavior of the NN potential is obtained by Fourier transform of the propagator of a heavy meson, ei~q·~r m2ω + ~q2 e−mωr . (71) ChPT is an expansion in small momenta Q, too small to resolve struc- tures like a ρ(770) or ω(782) meson, because Q � Λχ ≈ mρ,ω. But the latter relation allows us to expand the propagator of a heavy meson into a power series, m2ω +Q2 −+ . . . , (72) where the ω is representative for any heavy meson of interest. The above expansion suggests that it should be possible to describe the short distance part of the nuclear force simply in terms of powers of Q/mω, which fits in well with our over-all power scheme since Q/mω ≈ Q/Λχ. A second purpose of contact terms is renormalization. Dimensional reg- ularization of the loop integrals of pion-exchanges (cf. Sec. 5.1) typically generates polynomial terms with coefficients that are, in part, infinite or scale dependent. Contact terms pick up infinities and remove scale depen- dence. The partial-wave decomposition of a power Qν has an interesting prop- erty. First note that Q can only be either the momentum transfer between the two interacting nucleons q or the average momentum k [cf. Eq. (47) for their definitions]. In any case, for even ν, Qν = f ν (cos θ) , (73) where fm stands for a polynomial of degree m and θ is the CMS scattering angle. The partial-wave decomposition of Qν for a state of orbital-angular momentum L involves the integral QνPL(cos θ)d cos θ = (cos θ)PL(cos θ)d cos θ , (74) where PL is a Legendre polynomial. Due to the orthogonality of the PL, L = 0 for L > . (75) Consequently, contact terms of order zero contribute only in S-waves, while order-two terms contribute up to P -waves, order-four terms up to D-waves, etc.. We will now present, one by one, the various orders of NN contact terms together with their partial-wave decomposition [54]. Note that, due to parity, only even powers of Q are allowed. 5.3.1 Zeroth Order The contact potential at order zero reads: V (0)(~p′, ~p) = CS + CT ~σ1 · ~σ2 (76) Partial wave decomposition yields: V (0)(1S0) = C̃1S0 = 4π (CS − 3CT ) V (0)(3S1) = C̃3S1 = 4π (CS + CT ) (77) 5.3.2 Second Order The contact potential contribution of order two is given by: V (2)(~p′, ~p) = C1q 2 + C2k 2 + C4k ~σ1 · ~σ2 −i~S · (~q × ~k) + C6(~σ1 · ~q) (~σ2 · ~q) + C7(~σ1 · ~k) (~σ2 · ~k) (78) Second order partial wave contributions: S0) = C1S0(p C2 − 3C3 − C4 − C6 − P0) = C3P0 pp C5 + 2C6 − P1) = C1P1 pp C2 + 2C3 − P1) = C3P1 pp S1) = C3S1(p C2 + C3 + S1 −3 D1) = C3S1−3D1p P2) = C3P2 pp 5.3.3 Fourth Order The contact potential contribution of order four reads: V (4)(~p′, ~p) = D1q 4 +D2k 4 +D3q 2k2 +D4(~q × ~k)2 4 +D6k 4 +D7q 2k2 +D8(~q × ~k)2 ~σ1 · ~σ2 2 +D10k −i~S · (~q × ~k) 2 +D12k (~σ1 · ~q) (~σ2 · ~q) 2 +D14k (~σ1 · ~k) (~σ2 · ~k) + D15 ~σ1 · (~q × ~k) ~σ2 · (~q × ~k) The rather lengthy partial-wave expressions of this order have been relegated to Appendix B. 5.4 Constructing a Chiral NN Potential 5.4.1 Conceptual Questions The two-nucleon system is non-perturbative as evidenced by the presence of a shallow bound state (the deuteron) and large scattering lengths. Wein- berg [18] showed that the strong enhancement of the scattering amplitude arises from purely nucleonic intermediate states. He therefore suggested to use perturbation theory to calculate the NN potential and to apply this potential in a scattering equation to obtain the NN amplitude. We adopt this prescription. Since the irreducible diagrams that make up the potential are calculated using covariant perturbation theory (cf. Sec. 5.1), it is consistent to start from the covariant Bethe-Salpeter (BS) equation [55] describing two-nucleon scattering. In operator notation, the BS equation reads T = V + V G T (81) with T the invariant amplitude for the two-nucleon scattering process, V the sum of all connected two-particle irreducible diagrams, and G the relativistic two-nucleon propagator. The BS equation is equivalent to a set of two equations T = V + V g T (82) V = V + V (G − g)V (83) = V + V1π (G − g)V1π + . . . , (84) where g is a covariant three-dimensional propagator which preserves rela- tivistic elastic unitarity. We choose the propagator g proposed by Blanken- becler and Sugar (BbS) [43] (for more details on relativistic three-dimensional reductions of the BS equation, see Ref. [7]). The ellipsis in Eq. (84) stands for terms of irreducible 3π and higher pion exchanges which we neglect. Note that when we speak of covariance in conjunction with (heavy baryon) ChPT, we are not referring to manifest covariance. Relativity and relativis- tic off-shell effects are accounted for in terms of a Q/MN expansion up to the given order. Thus, Eq. (84) is evaluated in the following way, V ≈ V(on-shell) + V1π G V1π − V1π g V1π , (85) where the pion-exchange content of V(on-shell) is V1π+V ′2π with V1π the on- shell 1PE given in Eq. (48) and V ′2π the irreducible 2π exchanges calculated in Sec. 5.1, but without the box. V1π denotes the relativistic (off-shell) 1PE. Notice that the term (V1π G V1π−V1π g V1π) represents what has been called “the (irreducible part of the) box diagram contribution” in Sec. 5.1 where it was evaluated at various orders. The full chiral NN potential V is given by irreducible pion exchanges Vπ and contact terms Vct, V = Vπ + Vct (86) Vπ = V1π + V2π + . . . , (87) where the ellipsis denotes irreducible 3π and higher pion exchanges which are omitted. Two-pion exchange contributions appear in various orders V2π = V 2π + V 2π + V 2π + . . . (88) as calculated in Sec. 5.1. Contact terms come in even orders, Vct = V ct + V ct + V ct + . . . (89) and were presented in Sec. 5.3. The potential V is calculated at a given order. For example, the potential at NNLO includes 2PE up to V (3)2π and contacts up to V (2)ct . At N 3LO, contributions up to V (4)2π and V ct are included. The potential V satisfies the relativistic BbS equation, Eq. (82). Defining V̂ (~p ′, ~p) ≡ (2π)3 V (~p ′, ~p) T̂ (~p ′, ~p) ≡ (2π)3 T (~p ′, ~p) with Ep ≡ M2N + ~p 2 (the factor 1/(2π)3 is added for convenience), the BbS equation collapses into the usual, nonrelativistic Lippmann-Schwinger (LS) equation, T̂ (~p ′, ~p) = V̂ (~p ′, ~p) + d3p′′ V̂ (~p ′, ~p ′′) p2 − p′′2 + i� T̂ (~p ′′, ~p) . (92) Since V̂ satisfies Eq. (92), it can be used like a usual nonrelativistic potential, and T̂ is the conventional nonrelativistic T-matrix. Iteration of V̂ in the LS equation requires cutting V̂ off for high momenta to avoid infinities, This is consistent with the fact that ChPT is a low- momentum expansion which is valid only for momenta Q � Λχ ≈ 1 GeV. Thus, we multiply V̂ with a regulator function V̂ (~p ′, ~p) 7−→ V̂ (~p ′, ~p) e−(p ′/Λ)2n e−(p/Λ) ≈ V̂ (~p ′, ~p) + . . . with the ‘cutoff parameter’ Λ around 0.5 GeV. Equation (94) provides an indication of the fact that the exponential cutoff does not necessarily affect the given order at which the calculation is conducted. For sufficiently large n, the regulator introduces contributions that are beyond the given order. Assuming a good rate of convergence of the chiral expansion, such orders are small as compared to the given order and, thus, do not affect the accuracy at the given order. In our calculations we use, of course, the full exponential, Eq. (93), and not the expansion. On a similar note, we also do not expand the square-root factors in Eqs. (90-91) because they are kinematical factors which guarantee relativistic elastic unitarity. 5.4.2 What Order? Since in nuclear EFT we are dealing with a perturbative expansion, at some point, we have to raise the question, to what order of ChPT we have to go to obtain the precision we need. To discuss this issue on firm grounds, we show in Table 3 the χ2/datum for the fit of the world np data below 290 MeV for a family of np potentials at NLO and NNLO. The NLO potentials produce the very large χ2/datum between 67 and 105, and the NNLO are between 12 and 27. The rate of improvement from one order to the other is very encouraging, but the quality of the reproduction of the np data at NLO and NNLO is obviously insufficient for reliable predictions. Table 3: χ2/datum for the reproduction of the 1999 np database [56] by families of np potentials at NLO and NNLO constructed by the Juelich group [57]. Tlab bin # of np — Juelich np potentials — (MeV) data NLO NNLO 0–100 1058 4–5 1.4–1.9 100–190 501 77–121 12–32 190–290 843 140–220 25–69 0–290 2402 67–105 12–27 Based upon these facts, it has been pointed out in 2002 by Entem and Machleidt [25, 26] that one has to proceed to N3LO. Consequently, the first N3LO potential was published in 2003 [27]. At N3LO, there are 24 contact terms (24 parameters) which contribute to the partial waves with L ≤ 2 (cf. Sec. 5.3). In Table 4, column ‘Q4/N3LO’, we show how these terms/parameters are distributed over the various partial waves. For comparison, we also show the number of parameters used in the Nijmegen partial wave analysis (PWA93) [49] and in the high-precision CD-Bonn potential [9]. The table reveals that, for S and P waves, the number of parameters used in high-precision phenomenology and in EFT at N3LO are about the same. Thus, the EFT approach provides retroactively a justification for what the phenomenologists of the 1990’s were doing. At NLO and NNLO, the number of parameters is substantially smaller than for PWA93 and CD-Bonn, which explains why these orders are insufficient for a quantitative potential. This fact is also clearly reflected in Fig. 7 where phase shifts are shown for potentials constructed at NLO, NNLO, and N3LO. 5.4.3 Charge-Dependence For an accurate fit of the low-energy pp and np data, charge-dependence is important. We include charge-dependence up to next-to-leading order of the isospin-violation scheme (NLØ, in the notation of Ref. [58]). Thus, we include the pion mass difference in 1PE and the Coulomb potential in pp scattering, which takes care of the LØ contributions. At order NLØ, we have the pion mass difference in 2PE at NLO, πγ exchange [59], and two charge-dependent contact interactions of order Q0 which make possible an accurate fit of the three different 1S0 scattering lengths, app, ann, and anp. Table 4: Number of parameters needed for fitting the np data in phase-shift analysis and by a high-precision NN potential versus the number of NN contact terms of EFT based potentials at different orders. Nijmegen CD-Bonn — Contact Potentials — partial-wave high-precision Q0 Q2 Q4 analysis [49] potential [9] LO NLO/NNLO N3LO 1S0 3 4 1 2 4 3S1 3 4 1 2 4 3S1-3D1 2 2 0 1 3 1P1 3 3 0 1 2 3P0 3 2 0 1 2 3P1 2 2 0 1 2 3P2 3 3 0 1 2 3P2-3F2 2 1 0 0 1 1D2 2 3 0 0 1 3D1 2 1 0 0 1 3D2 2 2 0 0 1 3D3 1 2 0 0 1 3D3-3G3 1 0 0 0 0 1F3 1 1 0 0 0 3F2 1 2 0 0 0 3F3 1 2 0 0 0 3F4 2 1 0 0 0 3F4-3H4 0 0 0 0 0 1G4 1 0 0 0 0 3G3 0 1 0 0 0 3G4 0 1 0 0 0 3G5 0 1 0 0 0 Total 35 38 2 9 24 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) Figure 7: Phase parameters for np scattering as calculated from NN potentials at different orders of ChPT. The dotted line is NLO [57], the dashed NNLO [57], and the solid N3LO [27]. Partial waves with total angular momentum J ≤ 2 are dis- played. Solid dots represent the Nijmegen multienergy np phase shift analysis [49] and open circles are the GWU/VPI single-energy np analysis SM99 [50]. 5.4.4 A Quantitative NN Potential at N3LO NN Scattering. The fitting procedure starts with the peripheral partial waves because they depend on fewer parameters. Partial waves with L ≥ 3 are exclusively determined by 1PE and 2PE because the N3LO contacts contribute to L ≤ 2 only. 1PE and 2PE at N3LO depend on the axial- vector coupling constant, gA (we use gA = 1.29), the pion decay constant, fπ = 92.4 MeV, and eight low-energy constants (LECs) that appear in the dimension-two and dimension-three πN Lagrangians, Eqs. (37) and (38). In the fitting process, we varied three of them, namely, c2, c3, and c4. We found that the other LECs are not very effective in the NN system and, therefore, we kept them at the values determined from πN (cf. Table 2). The most influential constant is c3, which has to be chosen on the low side (slightly more than one standard deviation below its πN determination) for an optimal fit of the NN data. As compared to a calculation that strictly uses the πN values for c2 and c4, our choices for these two LECs lower Table 5: χ2/datum for the reproduction of the 1999 np database [56] by various np potentials. (Numbers in parentheses are the values of cutoff parameters in units of MeV used in the regulators of the chiral potentials.) Tlab bin # of np Idaho Juelich Argonne (MeV) data N3LO [27] N3LO [60] V18 [61] (500–600) (600/700–450/500) 0–100 1058 1.0–1.1 1.0–1.1 0.95 100–190 501 1.1–1.2 1.3–1.8 1.10 190–290 843 1.2–1.4 2.8–20.0 1.11 0–290 2402 1.1–1.3 1.7–7.9 1.04 Table 6: χ2/datum for the reproduction of the 1999 pp database [56] by various pp potentials. Notation as in Fig. 5. Tlab bin # of np Idaho Juelich Argonne (MeV) data N3LO [27] N3LO [60] V18 [61] (500–600) (600/700–450/500) 0–100 795 1.0–1.7 1.0–3.8 1.0 100–190 411 1.5–1.9 3.5–11.6 1.3 190–290 851 1.9–2.7 4.3–44.4 1.8 0–290 2057 1.5–2.1 2.9–22.3 1.4 the 3F2 and 1F3 phase shifts bringing them into closer agreement with the phase shift analysis. The other F waves and the higher partial waves are essentially unaffected by our variations of c2 and c4. Overall, the fit of all J ≥ 3 waves is very good. We turn now to the lower partial waves. Here, the most important fit parameters are the ones associated with the 24 contact terms that contribute to the partial waves with L ≤ 2. In addition, we have two charge-dependent contacts which are used to fit the three different 1S0 scattering lengths, app, ann, and anp. In the optimization procedure, we fit first phase shifts, and then we refine the fit by minimizing the χ2 obtained from a direct comparison with the data. The χ2/datum for the fit of the np data below 290 MeV is shown in Table 5, and the corresponding one for pp is given in Table 6. These tables show that at N3LO a χ2/datum comparable to the high-precision Argonne 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) 0 100 200 300 Lab. Energy (MeV) Figure 8: Neutron-proton phase parameters as described by two potentials at N3LO. The solid curve is calculated from the Idaho N3LO potential [27] while the dashed curve is from the Juelich [60] one. Solid dots and open circles as in Fig. 7. V18 [61] potential can, indeed, be achieved. The “Idaho” N3LO potential [27] produces a χ2/datum = 1.1 for the world np data below 290 MeV which compares well with the χ2/datum = 1.04 by the Argonne potential. In 2005, also the Juelich group produced several N3LO NN potentials [60], the best of which fits the np data with a χ2/datum = 1.7 and the worse with a χ2/datum = 7.9 (see Table 5). While 7.9 is clearly unacceptable for any meaningful application, a χ2/datum of 1.7 is reasonable, although it does not meet the precision standard that few-nucleon physicists established in the 1990’s. Turning to pp, the χ2 for pp data are typically larger than for np be- cause of the higher precision of pp data. Thus, the Argonne V18 produces a χ2/datum = 1.4 for the world pp data below 290 MeV and the best Idaho N3LO pp potential obtains 1.5. The fit by the best Juelich N3LO pp poten- tial results in a χ2/datum = 2.9 which, again, is not quite consistent with the precision standards established in the 1990’s. The worst Juelich N3LO pp potential produces a χ2/datum of 22.3 and is incompatible with reliable predictions. Table 7: Deuteron properties as predicted by various NN potentials are com- pared to empirical information. (Deuteron binding energy Bd, asymptotic S state AS , asymptotic D/S state η, deuteron radius rd, quadrupole moment Q, D-state probability PD; the calculated rd and Q are without meson-exchange current con- tributions and relativistic corrections.) Idaho Juelich N3LO [27] N3LO [60] CD-Bonn[9] AV18[61] Empiricala (500) (550/600) Bd (MeV) 2.224575 2.218279 2.224575 2.224575 2.224575(9) AS (fm−1/2) 0.8843 0.8820 0.8846 0.8850 0.8846(9) η 0.0256 0.0254 0.0256 0.0250 0.0256(4) rd (fm) 1.975 1.977 1.966 1.967 1.97535(85) Q (fm2) 0.275 0.266 0.270 0.270 0.2859(3) PD (%) 4.51 3.28 4.85 5.76 aSee Table XVIII of Ref. [9] for references; the empirical value for rd is from Ref. [62]. Phase shifts of np scattering from the best Idaho (solid line) and Juelich (dashed line) N3LO np potentials are shown in Figure 8. The phase shifts confirm what the corresponding χ2 have already revealed. The Deuteron. The reproduction of the deuteron parameters is shown in Table 7. We present results for two N3LO potentials, namely, Idaho [27] and Juelich [60]. Remarkable are the predictions by the chiral potentials for the deuteron radius which are in good agreement with the latest empirical value obtained by the isotope-shift method [62]. All NN potentials of the past (Table 7 includes two representative examples, namely, CD-Bonn [9] and AV18 [61]) fail to reproduce this very precise new value for the deuteron radius. In Fig. 9, we display the deuteron wave functions derived from the N3LO potentials and compare them with wave functions based upon conventional NN potentials from the recent past. Characteristic differences are notice- able; in particular, the chiral wave functions are shifted towards larger r which explains the larger deuteron radius. 6 Many-Nucleon Forces As noted before, an important advantage of the EFT approach to nuclear forces is that it creates two- and many-nucleon forces on an equal footing. 0 1 2 3 4 5 6 r (fm) Figure 9: Deuteron wave functions: the family of larger curves are S-waves, the smaller ones D-waves. The thick lines represent the wave functions derived from chiral NN potentials at order N3LO (thick solid: Idaho [27], thick dashed: Juelich [60]). The thin dashed, dash-dotted, and dotted lines refer to the wave functions of the CD-Bonn [9], Nijm-I [8], and AV18 [61] potentials, respectively. 6.1 Three-Nucleon Forces The first non-vanishing 3NF terms occur at NNLO and are shown in Fig. 10 (cf. also Fig. 1, row ‘Q3/NNLO’, column ‘3N Force’). There are three di- agrams: the 2PE, 1PE, and 3N-contact interactions [39, 40]. The 2PE 3N-potential is given by V 3NF2PE = i 6=j 6=k (~σi · ~qi)(~σj · ~qj) (q2i +m ijk τ j (95) with ~qi ≡ ~pi′ − ~pi, where ~pi and ~pi′ are the initial and final momenta of nucleon i, respectively, and ijk = δ 4c1m2π ~qi · ~qj �αβγ τ k ~σk · [~qi × ~qj ] . (96) The vertex involved in this 3NF term is the two-derivative ππNN vertex (solid square in Fig. 10) which we encountered already in the 2PE contribu- tion to the NN potential at NNLO. Thus, there are no new parameters and 1 2 3 Figure 10: The three-nucleon force at NNLO (from Ref. [40]). the contribution is fixed by the LECs used in NN . The 1PE contribution is V 3NF1PE = D i 6=j 6=k ~σj · ~qj q2j +m (τ i · τ j)(~σi · ~qj) (97) and, finally, the 3N contact term reads V 3NFct = E j 6=k τ j · τ k . (98) The last two 3NF terms involve two new vertices (that do not appear in the 2N problem), namely, the πNNNN vertex with parameter D and a 6N ver- tex with parameters E. To pin them down, one needs two observables that involve at least three nucleons. In Ref. [40], the triton binding energy and the nd doublet scattering length 2and were used. Alternatively, one may also choose the binding energies of 3H and 4He [63]. Once D and E are fixed, the results for other 3N, 4N, . . . observables are predictions. In Refs. [64, 63], the first calculations of the structure of light nuclei (6Li and 7Li) were re- ported. Recently, the structure of nuclei with A = 10−13 nucleons has been calculated using the ab initio no-core shell model and applying chiral two and three-nucleon forces [65]. The results are very encouraging. Concerning the famous ‘Ay puzzle’, the above 3NF terms yield some improvement of the predicted nd Ay, however, the problem is not solved [40]. Note that the 3NF expressions given in Eqs. (95)-(98) above are the ones that occur at NNLO, and all calculations to date have included only those. Since we have to proceed to N3LO for sufficient accuracy of the 2NF, then consistency requires that we also consider the 3NF at N3LO. The 3NF at N3LO is very involved as can be seen from Fig. 11, but it does not depend on any new parameters. It is presently under construction [66]. So, for the moment, we can only hope that the Ay puzzle may be solved by a complete calculation at N3LO. Figure 11: Three-nucleon force contributions at N3LO (from Ref. [66]). 6.2 Four-Nucleon Forces In ChPT, four-nucleon forces (4NF) appear for the first time at N3LO (ν = 4). Thus, N3LO is the leading order for 4NF. Assuming a good rate of convergence, a contribution of order (Q/Λχ)4 is expected to be rather small. Thus, ChPT predicts 4NF to be essentially insignificant, consistent with ex- perience. Still, nothing is fully proven in physics unless we have performed explicit calculations. Very recently, the first such calculation has been per- formed: The chiral 4NF, Fig. 12, has been applied in a calculation of the 4He binding energy and found to contribute a few 100 keV [68]. It should be noted that this preliminary calculation involves many approximations, but it certainly provides the right order of magnitude of the result, which is indeed very small as compared to the full 4He binding energy of 28.3 MeV. 7 Conclusions The theory of nuclear forces has made great progress since the turn of the millennium. Nucleon-nucleon potentials have been developed that are based on proper theory (EFT for low-energy QCD) and are of high-precision, at the same time. Moreover, the theory generates two- and many-body forces on an equal footing and provides a theoretical explanation for the empirically known fact that 2NF � 3NF � 4NF . . . . (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) Figure 12: The four-nucleon force at N3LO (from Ref. [67]). At N3LO [26, 27], the accuracy can be achieved that is necessary and sufficient for reliable microscopic nuclear structure predictions. First cal- culations applying the N3LO NN potential [27] in the conventional shell model [69, 70], the ab initio no-core shell model [71, 72, 73], the coupled cluster formalism [74, 75, 76, 77, 78], and the unitary-model-operator ap- proach [79] have produced promising results. The 3NF at NNLO is known [39, 40] and has been applied in few-nucleon reactions [40, 80, 81] as well as the structure of light nuclei [64, 63, 65]. How- ever, the famous ‘Ay puzzle’ of nucleon-deuteron scattering is not resolved by the 3NF at NNLO. Thus, one important outstanding issue is the 3NF at N3LO, which is under construction [66]. Another open question that needs to be settled is whether Weinberg power counting, which is applied in all current NN potentials, is consistent. This controversial issue is presently being debated in the literature [82, 83]. Acknowledgements It is a pleasure to thank the organizers of this workshop, particularly, Ananda Santra, for their warm hospitality. I gratefully acknowledge numer- ous discussions with my collaborator D. R. Entem. This work was supported in part by the U.S. National Science Foundation under Grant No. PHY- 0099444. A Fourth Order Two-Pion Exchange Contributions The fourth order 2PE contributions consist of two classes: the one-loop (Fig. 3) and the two-loop diagrams (Fig. 4). A.1 One-loop diagrams This large pool of diagrams can be analyzed in a systematic way by intro- ducing the following well-defined subdivisions. A.1.1 c2i contributions. The only contribution of this kind comes from the football diagram with both vertices proportional to ci (first row of Fig. 3). One obtains [41]: 3L(q) 16π2f4π w2 + c3w̃ 2 − 4c1m2π , (99) WT = − 2L(q) 96π2f4π . (100) A.1.2 ci/MN contributions. This class consists of diagrams with one vertex proportional to ci and one 1/MN correction. A few graphs that are representative for this class are shown in the second row of Fig. 3. Symbols with a large solid dot and an open circle denote 1/MN corrections of vertices proportional to ci. They are part of L̂(3)πN , Eq. (39). The result for this group of diagrams is [41]: VC = − g2A L(q) 32π2MNf4π (c2 − 6c3)q4 + 4(6c1 + c2 − 3c3)q2m2π +6(c2 − 2c3)m4π + 24(2c1 + c3)m , (101) WC = − 2L(q) 192π2MNf4π g2A(8m π + 5q 2) + w2 , (102) WT = − c4L(q) 192π2MNf4π g2A(16m π + 7q 2)− w2 , (103) VLS = 8π2MNf4π w2L(q) , (104) WLS = − c4L(q) 48π2MNf4π g2A(8m π + 5q 2) + w2 . (105) A.1.3 1/M2N corrections. These are relativistic 1/M2N corrections of the leading order 2π exchange diagrams. Typical examples for this large class are shown in row 3–6 of Fig. 3. This time, there is no correction from the iterated 1PE, Eq. (65) or Eq. (66), since the expansion of the factor M2N/Ep does not create a term proportional to 1/M2N . The total result for this class is [42], VC = − 32π2M2Nf 2m8πw −4 + 8m6πw −2 − q4 − 2m4π (106) WC = − 768π2M2Nf q4 + 3m2πq 2 + 3m4π − 6m −k2(8m2π + 5q + 4g4A k2(20m2π + 7q 2 − 16m4πw −2) + 16m8πw +12m6πw −2 − 4m4πq 2w−2 − 5q4 − 6m2πq 2 − 6m4π − 4k2w2 16g4Am , (107) VT = − g4A L(q) 32π2M2Nf q2 +m4πw , (108) WT = − 1536π2M2Nf 7m2π + q2 + 4m4πw − 32g2A m2π + , (109) VLS = g4A L(q) 4π2M2Nf q2 +m4πw , (110) WLS = 256π2M2Nf 16g2A m2π + 4m4πw q2 − 9m2π , (111) VσL = g4A L(q) 32π2M2Nf . (112) A.2 Two-loop contributions. The two-loop contributions are quite intricate. In Fig. 4, we attempt a graphical representation of this class. The gray disk stands for all one-loop πN graphs which are shown in some detail in the lower part of the figure. Not all of the numerous graphs are displayed. Some of the missing ones are obtained by permutation of the vertices along the nucleon line, others by inverting initial and final states. Vertices denoted by a small dot are from the leading order πN Lagrangian L̂(1)πN , Eq. (33), except for the four- pion vertices which are from L(2)ππ , Eq. (27). The solid square represents vertices proportional to the LECs di which are introduced by the third order Lagrangian L(3)πN , Eq. (38). The di vertices occur actually in one- loop NN diagrams, but we list them among the two-loop NN contributions because they are needed to absorb divergences generated by one-loop πN graphs. Using techniques from dispersion theory, Kaiser [41] calculated the imaginary parts of the NN amplitudes, Im Vα(iµ) and Im Wα(iµ), which result from analytic continuation to time-like momentum transfer q = iµ−0+ with µ ≥ 2mπ. From this, the momentum-space amplitudes Vα(q) and Wα(q) are obtained via the subtracted dispersion relations: VC,S(q) = − ImVC,S(iµ) µ5(µ2 + q2) , (113) VT (q) = ImVT (iµ) µ3(µ2 + q2) , (114) and similarly for WC,S,T . In most cases, the dispersion integrals can be solved analytically and the following expressions are obtained [26]: VC(q) = 3g4Aw̃ 2A(q) 1024π2f6π (m2π + 2q 2mπ + w̃ 2A(q) + 4g2Amπw̃ (115) WC(q) = W C (q) +W C (q) , (116) C (q) = 18432π4f6π 192π2f2πw 2g2Aw̃ (g2A − 1)w 6g2Aw̃ 2 − (g2A − 1)w 384π2f2π w̃2(d̄1 + d̄2) + 4m +L(q) 4m2π(1 + 2g A) + q 2(1 + 5g2A) (5 + 13g2A) + 8m π(1 + 2g (117) C (q) = − ImW (b)C (iµ) µ5(µ2 + q2) , (118) where ImW (b)C (iµ) = − 3µ(8πf2π)3 g2A(2m π − µ 2) + 2(g2A − 1)κ − 3κ2x2 + 6κx m2π + κ2x2 ln m2π + κ2x2 µ2 − 2κ2x2 − 2m2π m2π + κ2x2 ; (119) VT (q) = V T (q) + V T (q) VS(q) = − S (q) + V S (q) , (120) T (q) = − S (q) = − 2L(q) 32π2f4π (d̄14 − d̄15) (121) T (q) = − S (q) = ImV (b)T (iµ) µ3(µ2 + q2) , (122) where ImV (b)T (iµ) = − 2g6Aκ µ(8πf2π)3 dx(1− x2) m2π + κ2x2  ; (123) WT (q) = − WS(q) = 2A(q) 2048π2f6π w2A(q) + 2mπ(1 + 2g , (124) where κ ≡ µ2/4−m2π. Note that the analytic solutions hold modulo polynomials. We have checked the importance of those contributions where we could not find an analytic solution and where, therefore, the integrations have to be performed numerically. It turns out that the combined effect on NN phase shifts from C , V T , and V S is smaller than 0.1 deg in F and G waves and smaller than 0.01 deg in H waves, at Tlab = 300 MeV (and less at lower energies). This renders these contributions negligible. Therefore, we omit W (b)C , V and V (b)S in the construction of chiral NN potentials at order N In Eqs. (117) and (121), we use the scale-independent LECs, d̄i, which are obtained by combining the scale-dependent ones, dri (λ), with the chiral logarithm, ln(mπ/λ), or equivalently d̄i = dri (mπ). The scale-dependent LECs, dri (λ), are a consequence of renormalization. For more details about this issue, see Ref. [37]. B Partial Wave Decomposition of the Fourth Or- der Contact Potential The contact potential contribution of order four, Eq. (80), decomposes into partial-waves as follows. V (4)(1S0) = D̂1S0(p ′4 + p4) +D1S0p V (4)(3P0) = D3P0(p ′3p+ p′p3) V (4)(1P1) = D1P1(p ′3p+ p′p3) V (4)(3P1) = D3P1(p ′3p+ p′p3) V (4)(3S1) = D̂3S1(p ′4 + p4) +D3S1p V (4)(3D1) = D3D1p V (4)(3S1 −3 D1) = D̂3S1−3D1p 4 +D3S1−3D1p V (4)(1D2) = D1D2p V (4)(3D2) = D3D2p V (4)(3P2) = D3P2(p ′3p+ p′p3) V (4)(3P2 −3 F2) = D3P2−3F2p V (4)(3D3) = D3D3p ′2p2 (125) The coefficients in the above expressions are given by: D̂1S0 = D1 + D3 − 3D5 − D7 −D11 − D12 − D1S0 = D4 − 10D5 − D7 − 2D8 − D12 − D13 − D14 − D3P0 = − D10 + D11 + D12 − D1P1 = − D2 + 4D5 − D11 − D3P1 = − D10 − 2D11 − D12 + D̂3S1 = D1 + D3 +D5 + D11 + D12 + D3S1 = D12 + D13 + D14 + D3D1 = D10 − D11 + D12 + D13 − D14 − D̂3S1−3D1 = − D11 − D12 − D13 − D3S1−3D1 = − D11 + D12 + D13 − D14 + D1D2 = D12 + D13 − D14 + D3D2 = D10 + D11 − D12 − D13 + D14 + D3P2 = − D10 − D11 + D13 + D3P2−3F2 = D11 − D12 + D13 − D3D3 = D10 − D15 (126) References [1] H. Yukawa, Proc. Phys. Math. Soc. 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B37, 2889 (2006); arXiv:nucl- th/0606017. [69] L. Coraggio et al., Phys. Rev. C 66. 021303 (2002). [70] L. Coraggio et al., Phys. Rev. C 71. 014307 (2005). [71] P. Navrátil and E. Caurier (2004) Phys. Rev. C 69 014311. [72] C. Forssen et al., Phys. Rev. C 71, 044312 (2005). [73] J.P. Vary et al., Eur. Phys. J. A 25 s01, 475 (2005). [74] K. Kowalski et al., Phys. Rev. Lett. 92, 132501 (2004). [75] D.J. Dean and M. Hjorth-Jensen (2004) Phys. Rev. C 69 054320. [76] M. Wloch et al., J. Phys. G 31, S1291 (2005); Phys. Rev. Lett. 94, 21250 (2005). [77] D.J. Dean et al., Nucl. Phys. 752, 299 (2005). [78] J.R. Gour et al., Phys. Rev. C 74, 024310 (2006). [79] S. Fujii, R. Okamato, and K. Suzuki, Phys. Rev. C 69, 034328 (2004). [80] K. Ermisch et al., Phys. Rev. C 71, 064004 (2005). [81] H. Witala, J. Golak, R. Skibinski, W. Glöckle, A. Nogga, E. Epelbaum, H. Kamada, A. Kievsky, and M. Viviani, Phys. Rev. C 73, 044004 (2006). [82] A. Nogga, R. G. E. Timmermans, and U. van Kolck, Phys. Rev. C 72, 054006 (2005). [83] E. Epelbaum, U.-G. Meißner, arXiv:nucl-th/0609037. http://arxiv.org/abs/nucl-th/0606017 http://arxiv.org/abs/nucl-th/0606017 http://arxiv.org/abs/nucl-th/0609037 Introduction and Historical Perspective QCD and the Nuclear Force Effective Field Theory for Low-Energy QCD Symmetries of Low-Energy QCD Chiral Symmetry Explicit Symmetry Breaking Spontaneous Symmetry Breaking Chiral Effective Lagrangians Involving Pions Nucleon Contact Lagrangians Nuclear Forces from EFT: Overview Chiral Perturbation Theory and Power Counting The Hierarchy of Nuclear Forces Two-Nucleon Forces Pion-Exchange Contributions in ChPT Zeroth Order (LO) Second Order (NLO) Third Order (NNLO) Fourth Order (N3LO) Iterated One-Pion-Exchange NN Scattering in Peripheral Partial Waves Using the Perturbative Amplitude NN Contact Potentials Zeroth Order Second Order Fourth Order Constructing a Chiral NN Potential Conceptual Questions What Order? Charge-Dependence A Quantitative NN Potential at N3LO Many-Nucleon Forces Three-Nucleon Forces Four-Nucleon Forces Conclusions Fourth Order Two-Pion Exchange Contributions One-loop diagrams ci2 contributions. ci/MN contributions. 1/MN2 corrections. Two-loop contributions. Partial Wave Decomposition of the Fourth Order Contact Potential

0704.0808

On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields

On a Conjecture of E. M. Stein on the Hilbert Transform on Vector Fields Michael Lacey and Xiaochun Li Michael Lacey, School of Mathematics, Georgia Insti- tute of Technology, Atlanta GA 30332 Xiaochun Li, Department of Mathematics, University of Illinois, Urbana IL 61801 E-mail address : [email protected] http://arxiv.org/abs/0704.0808v3 1991 Mathematics Subject Classification. Primary 42A50, 42B25 The authors are supported in part by NSF grants. M.L. was supported in part by the Guggenheim Foundation. Abstract. Let v be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient condi- tions for the boundedness of the Hilbert transform Hv,ǫ f(x) := p.v. f(x− yv(x)) dy where ǫ is a suitably chosen parameter, determined by the smooth- ness properties of the vector field. It is a conjecture, due to E.M. Stein, that if v is Lipschitz, there is a positive ǫ for which the transform above is bounded on L2. Our principal result gives a sufficient condition in terms of the boundedness of a maximal function associated to v, namely that this new maximal function be bounded on some Lp, for some 1 < p < 2. We show that the maximal function is bounded from L2 to weak L2 for all Lips- chitz vector fields. The relationship between our results and other known sufficient conditions is explored. Contents Preface vii Chapter 1. Overview of Principal Results 1 Chapter 2. Connections to Besicovitch Set and Carleson’s Theorem 7 Besicovitch Set 7 The Kakeya Maximal Function 7 Carleson’s Theorem 8 The Weak L2 Estimate in Theorem 1.15 is Sharp 10 Chapter 3. The Lipschitz Kakeya Maximal Function 11 The Weak L2 Estimate 11 An Obstacle to an Lp estimate, for 1 < p < 2 22 Bourgain’s Geometric Condition 23 Vector Fields that are a Function of One Variable 27 Chapter 4. The L2 Estimate for Hilbert Transform on Lipschitz Vector Fields 31 Definitions and Principle Lemmas 31 Truncation and an Alternate Model Sum 36 Proofs of Lemmata 39 Chapter 5. Almost Orthogonality Between Annuli 63 Application of the Fourier Localization Lemma 63 The Fourier Localization Estimate 79 References 87 Preface This memoir is devoted to a question in planar Harmonic Analysis, a subject which is a circle of problems all related to the Besicovitch set. This anomalous set has zero Lebesgue measure, yet contains a line seg- ment of unit length in each direction of the plane. It is a known, since the 1970’s, that such sets must necessarily have full Hausdorff dimen- sion. The existence of these sets, and the full Hausdorff dimension, are intimately related to other, independently interesting issues [26]. An important tool to study these questions is the so-called Kakeya Maxi- mal Function, in which one computes the maximal average of a function over rectangles of a fixed eccentricity and arbitrary orientation. Most famously, Charles Fefferman showed [10] that the Besicovitch set is the obstacle to the boundedness of the disc multiplier in the plane. But as well, this set is intimately related to finer questions of Bochner-Riesz summability of Fourier series in higher dimensions and space-time regularity of solutions of the wave equation. This memoir concerns one of the finer questions which center around the Besicovitch set in the plane. (There are not so many of these questions, but our purpose here is not to catalog them!) It concerns a certain degenerate Radon transform. Given a vector field v on R2, one considers a Hilbert transform computed in the one dimensional line segment determined by v, namely the Hilbert transform of a function on the plane computed on the line segment {x+ tv(x) | |t| ≤ 1}. The Besicovitch set itself says that choice of v cannot be just mea- surable, for you can choose the vector field to always point into the set. Finer constructions show that one cannot take it to be Hölder continu- ous of any index strictly less than one. Is the sharp condition of Hölder continuity of index one enough? This is the question of E. M. Stein, motivated by an earlier question of A. Zygmund, who asked the same for the question of differentiation of integrals. The answer is not known under any condition of just smoothness of the vector field. Indeed, as is known, and we explain, a positive answer would necessarily imply Carleson’s famous theorem on the con- vergence of Fourier series, [6]. This memoir is concerned with reversing viii PREFACE this implication: Given the striking recent successes related to Car- leson’s Theorem, what can one say about Stein’s Conjecture? In this direction, we introduce a new object into the study, a Lipschitz Kakeya Maximal Function, which is a variant of the more familiar Kakeya Max- imal Function, which links the vector field v to the ‘Besicovitch sets’ associated to the vector field. One averages a function over rectan- gles of arbitrary orientation and—in contrast to the classical setting— arbitrary eccentricity. But, the rectangle must suitably localize the directions in which the vector field points. This Maximal Function ad- mits a favorable estimate on L2, and this is one of the main results of the Memoir. On Stein’s Conjecture, we prove a conditional result: If the Lips- chitz Kakeya Maximal Function associated with v maps is an estimate a little better than our L2 estimate, then the associated Hilbert trans- form is indeed bounded. Thus, the main question left open concerns the behavior of these novel Maximal Functions. While the main result is conditional, it does contain many of the prior results on the subject, and greatly narrows the possible avenues of a resolution of this conjecture. The principal results and conjectures are stated in the Chapter 1; following that we collect some of the background material for this sub- ject, and prove some of the folklore results known about the subject. The remainder of the Memoir is taken up with the proofs of the The- orems stated in the Chapter 1. Acknowledgment. The efforts of a strikingly generous referee has resulted in corrections of arguments, and improvements in presentation throughout this manuscript. We are indebted to that person. Michael T. Lacey and Xiaochun Li CHAPTER 1 Overview of Principal Results We are interested in singular integral operators on functions of two variables, which act by performing a one dimensional transform along a particular line in the plane. The choice of lines is to be variable. Thus, for a measurable map, v from R2 to the unit circle in the plane, that is a vector field, and a Schwartz function f on R2, define Hv,ǫ f(x) := p.v. f(x− yv(x)) dy This is a truncated Hilbert transform performed on the line segment {x+ tv(x) : |t| < 1}. We stress the limit of the truncation in the defi- nition above as it is important to different scale invariant formulations of our questions of interest. This is an example of a Radon transform, one that is degenerate in the sense that we seek results independent of geometric assumptions on the vector field. We are primarily interested in assumptions of smoothness on the vector field. Also relevant is the corresponding maximal function (1.1) Mv,ǫ f := sup 0<t≤ǫ (2t)−1 |f(x− sv(x))| ds The principal conjectures here concern Lipschitz vector fields. Zygmund Conjecture 1.2. Suppose that v is Lipschitz. Then, for all f ∈ L2(R2) we have the pointwise convergence (1.3) lim (2t)−1 f(x− sv(x)) ds = f(x) a. e. More particularly, there is an absolute constant K > 0 so that if ǫ−1 = K‖v‖Lip, we have the weak type estimate (1.4) sup λ|{Mv,ǫ f > λ}|1/2 . ‖f‖2 . The origins of this question go back to the discovery of the Besicov- itch set in the 1920s, and in particular, constructions of this set show that the Conjecture is false under the assumption that v is Hölder con- tinuous for any index strictly less than 1. These constructions, known 2 1. OVERVIEW OF PRINCIPAL RESULTS since the 1920’s, were the inspiration for A. Zygmund to ask if integrals of, say, L2(R2) functions could be differentiated in a Lipschitz choice of directions. That is, for Lipschitz v, and f ∈ L2, is it the case that (2ǫ)−1 f(x− yv(x)) dy = f(x) a.e.(x) These and other matters are reviewed in the next chapter. Much later, E. M. Stein [25] raised the singular integral variant of this conjecture. E.M. Stein Conjecture 1.5. There is an absolute constant K > 0 so that if ǫ−1 = K‖v‖Lip, we have the weak type estimate (1.6) sup λ|{|Hv,ǫ f | > λ}|1/2 . ‖f‖2 . These are very difficult conjectures. Indeed, it is known that if the Stein Conjecture holds for, say, C2 vector fields, then Carleson’s Theorem on the pointwise convergence of Fourier series [6] would follow. This folklore result is recalled in the next Chapter. We will study these questions using modifications of the phase plane analysis associated with Carleson’s Theorem [15–20] and a new tool, which we term a Lipschitz Kakeya Maximal Function. Associated with the Besicovitch set is the Kakeya Maximal Func- tion, a maximal function over all rectangles of a given eccentricity. A key estimate is that the L2 −→ L2,∞ norm of this operator grows logarithmically in the eccentricity, [27, 28]. Associated with a Lipschitz vector field, we define a class of maxi- mal functions taken over rectangles of arbitrary eccentricity, but these rectangles are approximate level sets of the vector field. Perhaps sur- prisingly, these maximal functions admit an L2 bound that is indepen- dent of eccentricity. Let us explain. A rectangle is determined as follows. Fix a choice of unit vectors in the plane (e, e⊥), with e⊥ being the vector e rotated by π/2. Using these vectors as coordinate axes, a rectangle is a product of two intervals R = I × J . We will insist that |I| ≥ |J |, and use the notations (1.7) L(R) = |I|, W(R) = |J | for the length and width respectively of R. The interval of uncertainty of R is the subarc EX(R) of the unit circle in the plane, centered at e, and of length W(R)/ L(R). See Fig- ure 1.1. 1. OVERVIEW OF PRINCIPAL RESULTS 3 0 EX(R) R Figure 1.1. An example eccentricity interval EX(R). The circle on the left has radius one. We now fix a Lipschitz map v of the plane into the unit circle. We only consider rectangles R with (1.8) L(R) ≤ (100‖v‖Lip)−1 . For such a rectangle R, set V(R) = R ∩ v−1(EX(R)). It is essential to impose a restriction of this type on the length of the rectangles, for with out it, one can modify constructions of the Besicovitch set to provide examples which would contradict the main results and conjectures of this work. For 0 < δ < 1, we consider the maximal functions (1.9) Mv,δ f(x) = sup |V(R)|≥δ|R| 1R(x) |f(y)| dy. That is we only form the supremum over rectangles for which the vector field lies in the interval of uncertainty for a fixed positive proportion δ of the rectangle, see Figure 1.2. Weak L2 estimate for the Lipschitz Kakeya Maximal Function 1.10. The maximal function Mδ,v is bounded from L 2(R2) to L2,∞(R2) with norm at most . δ−1/2. That is, for any λ > 0, and f ∈ L2(R2), this inequality holds: (1.11) λ2|{x ∈ R2 : Mδ,v f(x) > λ}| . δ−1‖f‖22 . The norm estimate in particular is independent of the Lipschitz vector field v. A principal Conjecture of this work is: Conjecture 1.12. For some 1 < p < 2, and some finite N and all 0 < δ < 1 and all Lipschitz vector fields v, the maximal function Mδ,v is bounded from Lp(R2) to Lp,∞(R2) with norm at most . δ−N . 4 1. OVERVIEW OF PRINCIPAL RESULTS Figure 1.2. A rectangle, with the vector field pointing in the long direction of the rectangle at three points. We cannot verify this Conjecture, only establishing that the norm of the operator can be controlled by a slowly growing function of ec- centricity. In fact, this conjecture is stronger than what is needed below. Let us modify the definition of the Lipschitz Kakeya Maximal Function, by re- stricting the rectangles that enter into the definition to have an approx- imately fixed width. For 0 < δ < 1, and choice of 0 < w < 1 ‖v‖Lip, parameterizing the width of the rectangles we consider, define (1.13) Mv,δ,w f(x) = sup |V(R)|≥δ|R| w≤W(R)≤2w 1R(x) |f(y)| dy. We can restrict attention to this case as the primary interest below is the Hilbert transform on vector fields applied to functions with fre- quency support in a fixed annulus. By Fourier uncertainty, the width of the fixed annulus is the inverse of the parameter w above. Conjecture 1.14. For some 1 < p < 2, and some finite N and all 0 < δ < 1, all Lipschitz vector fields v and 0 < w < 1 ‖v‖Lip the maximal function Mδ,v,w is bounded from L p(R2) to Lp,∞(R2) with norm at most . δ−N . These conjectures are stated as to be universal over Lipschitz vec- tor fields. On the other hand, we will state conditional results below in which we assume that a given vector field satisfies the Conjecture above, and then derive consequences for the Hilbert transform on vec- tor fields. We also show that e. g. real-analytic vector fields [3] satisfy these conjectures. We turn to the Hilbert transform on vector fields. As it turns out, it is useful to restrict functions in frequency variables to an annulus. Such operators are given by St f(x) = 1/t≤|ξ|≤2/t f̂(ξ) eiξ·x dξ . The relevance in part is explained in part by this result of the authors [15], valid for measurable vector fields. 1. OVERVIEW OF PRINCIPAL RESULTS 5 Theorem 1.15. For any measurable vector field v we have the L2 into λ|{|Hv,∞ ◦ St f | > λ}|1/2 . ‖f‖2 . The inequality holds uniformly in t > 0. It is critical that the Fourier restriction St enters in, for otherwise the Besicovitch set would provide a counterexample, as we indicate in the first section of Chapter 2. This is one point at which the differ- ence between the maximal function and the Hilbert transform is strik- ing. The maximal function variant of the estimate above holds, and is relatively easy to prove, yet the Theorem above contains Carleson’s Theorem on the pointwise convergence of Fourier series as a Corollary. The weak L2 estimate is sharp for measurable vector fields, and so we raise the conjecture Conjecture 1.16. There is a universal constant K for which we have the inequalities (1.17) sup 0<t<‖v‖Lip ‖Hv,ǫ ◦ St‖2→2 <∞ , where ǫ = ‖v‖Lip/K. Modern proofs of the pointwise convergence of Fourier series use the so-called restricted weak type approach, invented by Muscalu, Tao and Thiele in [21]. This method uses refinements of the weak L2 estimates, together with appropriate maximal function estimates, to derive Lp inequalities, for 1 < p < 2. In the case of Theorem 1.15—for which this approach can not possibly work—the appropriate maximal func- tion is the maximal function over all possible line segments. This is the unbounded Kakeya Maximal function for rectangles with zero ec- centricity. One might suspect that in the Lipschitz case, there is a bounded maximal function. This is another motivation for our Lips- chitz Kakeya Maximal Function, and our main Conjecture 1.12. We illustrate how these issues play out in our current setting, with this conditional result, one of the main results of this memoir. Theorem 1.18. Assume that Conjecture 1.14 holds for a choice of Lipschitz vector field v. Then we have the inequalities (1.19) ‖Hv,ǫ ◦ St‖2 . 1 , 0 < t < ‖v‖Lip . Here, ǫ is as in (1.17). Moreover, if the vector field as 1+η derivatives, we have the estimate (1.20) ‖Hv,ǫ‖2 . (1 + log‖v‖C1+η)2 . In this case, ǫ = K/‖v‖C1+η and η > 0. 6 1. OVERVIEW OF PRINCIPAL RESULTS While this is a conditional result, we shall see that it sheds new light on prior results, such as one of Bourgain [3] on real analytic vector fields. See Proposition 3.30, and the discussion of that Proposition. The authors are not aware of any conceptual obstacles to the fol- lowing extension of the Theorem above to be true, namely that one can establish Lp estimates, for all p > 2. As our argument currently stands, we could only prove this result for p sufficiently close to 2, because of our currently crude understanding of the underlying orthogonality ar- guments. Conjecture 1.21. Assume that Conjecture 1.14 holds for a choice of vector field v with 1 + η > 1 derivatives, then we have the inequalities below (1.22) ‖Hv,ǫ‖p . (1 + log‖v‖C1+η)2 , 2 < p <∞ . In this case, ǫ = K/‖v‖C1+η . For a brief remark on what is required to prove this conjecture, see Remark 4.65. The results of Christ, Nagel, Stein and Wainger [7] apply to certain vector fields v. This work is a beautiful culmination of the ‘geometric’ approach to questions concerning the boundedness of Radon trans- forms. Earlier, a positive result for analytic vector fields followed from Nagel, Stein and Wainger [22]. E.M. Stein [25] specifically raised the question of the boundedness of Hv for smooth vector fields v. And the results of D. Phong and Stein [23, 24] also give results about Hv. J. Bourgain [3] considered real–analytic vector field. N. H. Katz [13] has made an interesting contribution to maximal function question. Also see the partial results of Carbery, Seeger, Wainger and Wright [5]. CHAPTER 2 Connections to Besicovitch Set and Carleson’s Theorem Besicovitch Set The Besicovitch set is a compact set that contains a line segment of unit length in each direction in the plane. Anomalous constructions of such sets show that they can have very small measure. Indeed, given ǫ > 0 one can select rectangles R1, . . . , Rn, with disjoint eccentricities, |EX(R)| ≃ n−1, and of unit length, so that |B| ≤ ǫ for B := n=1Rj. On the other hand, letting ej ∈ EX(Rj), one has that the rectangles Rj + ej are essentially disjoint. See Figure 2.1. Call the ‘reach’ of the Besicovitch set Reach := Rj + ej . This set has measure about one. On the Reach, one can define a vector field with points to a line segment contained in the Besicovitch set. Clearly, one has |Hv 1B(x)| ≃ 1 , x ∈ Reach . Further, constructions of this set permit one to take the vector field to be Lipschitz continuous of any index strictly less than one. And conversely, if one considers a Besicovitch set associated to a vector field of sufficiently small Lipschitz norm, of index one, the corresponding Besicovitch set must have large measure. Thus, Lipschitz estimates are critical. The Kakeya Maximal Function The Kakeya maximal function is typically defined as (2.1) MK,ǫ f(x) := sup |EX(R)|≥ǫ 1R(x) |f(y)| dy , ǫ > 0 . One is forced to take ǫ > 0 due to the existence of the Besicovitch set. It is a critical fact that the norm of this operator admits a norm bound on L2 that is logarithmic in ǫ. See Córdoba and Fefferman [8], and 8 2. CONNECTIONS Figure 2.1. A Besicovitch Set on the left, and it’s Reach on the right. Strömberg [27, 28]. Subsequently, there have been several refinements of this observation, we cite only Nets H. Katz [12], Alfonseca, Soria and Vargas [1], and Alfonseca [2]. These papers contain additional references. For the L2 norm, the following is the sharp result. Theorem 2.2. We have the estimate below valid for all 0 < ǫ < 1. ‖MK,ǫ‖2→2 . 1 + log 1/ǫ . The standard example of taking f to be the indicator of a small disk show that the estimate above is sharp, and that the norm grows as an inverse power of ǫ for 1 < p < 2. Carleson’s Theorem We explain the connection between the Hilbert transform on vector fields and Carleson’s Theorem on the pointwise convergence of Fourier series. Since smooth functions have a convergent Fourier expansion, the main point of Carleson’s Theorem is to provide for the control of an appropriate maximal function. We recall that maximal function in this Theorem. Carleson’s Theorem 2.3. For all measurable functions N : R −→ R, the operator below maps L2 into itself. CNf(x) := p.v. eiN(x)y f(x− y)dy The implied operator norm is independent of the choice of measurable N(x). CARLESON’S THEOREM 9 v(x1) N(x1) ξ2 = J σ(ξ1 −N(x1)) Figure 2.2. Deducing Carleson’s Theorem from Stein’s Conjecture. For fixed function f , an appropriate choice of N will give us ∣∣∣p.v. eiNy f(x− y)dy ∣∣∣ . |CNf(x)| . Thus, in the Theorem above we have simply linearized the supremum. Also, we have stated the Theorem with the un-truncated integral. The content of the Theorem is unchanged if we make a truncation of the integral, which we will do below. Let us now show how to deduce this Theorem from an appropriate bound on certain bound on Hilbert transforms on vector fields. (This observation is apparently due to R.Coifman from the 1970’s.) Proposition 2.4. Assume that we have, say, the bound ‖Hv,1‖2→2 . 1 , assuming that ‖v‖C2 ≤ 1. It follows that the Carleson maximal operator is bounded on L2(R). Proof. The Proposition and the proof are only given in their most obvious formulation. Set σ(ξ) = iξy dy . For a C2 function N : R −→ R we deduce that the operator with symbol σ(ξ − N(x)) maps L2(R) into itself with norm that is independent of the C2 norm of the function N(x). A standard limiting argument then permits one to conclude the same bound for all measurable choices of N(x), as is required for the deduction of Carleson’s inequality. This argument is indicated in Figure 2.2. Take the vector field to be v(x1, x2) = (1,−N(x1)/n) where n is chosen much larger than the 10 2. CONNECTIONS C2 norm of the function N(x1). Then, Hv,1 is bounded on L 2(R2) with norm bounded by an absolute constant. The symbol of Hv,1 is σ(ξ1, ξ2) = σ(ξ1 − ξ2N(x1)/n) . The trace of this symbol along the line ξ2 = J defines a symbol of a bounded operator on L2(R). Taking J very large, we obtain a very good approximation to symbol σ(ξ1 − ξ2N(x1)/n), deducing that it maps L2(R) into itself with a bounded constant. Our proof is complete. � The Weak L2 Estimate in Theorem 1.15 is Sharp An example shows that under the assumption that the vector field is measurable, the sharp conclusion is that Hv ◦ S1 maps L2 into L2,∞. And a variant of the approach to Carleson’s theorem by Lacey and Thiele [20] will prove this norm inequality. This method will also show, under only the measurability assumption, that Hv S1 maps L p into itself for p > 2, as is shown by the current authors [15]. The results and techniques of that paper are critical to this one. CHAPTER 3 The Lipschitz Kakeya Maximal Function The Weak L2 Estimate We prove Theorem 1.10, the weak L2 estimate for the maximal function defined in (1.9), by suitably adapting classical covering lemma arguments. The Covering Lemma Conditions. We adopt the covering lemma approach of Córdoba and R. Fefferman [8]. To this end, we regard the choice of vector field v and 0 < δ < 1 as fixed. Let R be any finite collection of rectangles obeying the conditions (1.8) and |V(R)| ≥ δ|R|. We show that R has a decomposition into disjoint collections R′ and R′′ for which these estimates hold. . δ−1 ,(3.1) R∈R′′ ∣∣∣ . (3.2) The first of these conditions is the stronger one, as it bounds the L2 norm squared by the L1 norm; the verification of it will occupy most of the proof. Let us see how to deduce Theorem 1.10. Take λ > 0 and f ∈ L2 which is non negative and of norm one. Set R to be all the rectangles R of prescribed maximum length as given in (1.8), density with respect to the vector field, namely |V(R)| ≥ δ|R|, and f(y) dy ≥ λ|R| . We should verify the weak type inequality (3.3) λ . δ−1/2 . 12 3. LIPSCHITZ KAKEYA Apply the decomposition to R. Observe that . δ−1/2 Here of course we have used (3.1). This implies that . δ−1/2. Therefore clearly (3.3) holds for the collection R′. Concerning the collection R′′, apply (3.2) to see that R∈R′′ . δ−1/2 . This completes our proof of (3.3). The remainder of the proof is devoted to the proof of (3.1) and (3.2). The Covering Lemma Estimates. Construction of R′ and R′′. In the course of the proof, we will need several recursive procedures. The first of these occurs in the selection of R′ and R′′. We will have need of one large constant κ, of the order of say 100, but whose exact value does not concern us. Using this notation hides distracting terms. Let Mκ be a maximal function given as Mκ f(x) = sup |f(y)| dy , sup |f(x+ σω)| dσ Here, Q is the unit square in plane, and Ω is a set of uniformly dis- tributed points on the unit circle of cardinality equal to κ. It follows from the usual weak type bounds that this operator maps L1(R2) into weak L1(R2). To initialize the recursive procedure, set R′ ← ∅ , STOCK← R . THE WEAK L ESTIMATE 13 R′ κR Figure 3.1. The rectangle R′ would have been removed from STOCK upon the selection of R as a member of R′. The main step is this while loop. While STOCK is not empty, select R ∈ STOCK subject to the criteria that first it have a maximal length L(R), and second that it have minimal value of |EX(R)|. Update R′ ←R′ ∪ {R}. Remove R from STOCK. As well, remove any rectangle R′ ∈ STOCK which is also contained in 1κR ≥ κ−1 As the collection R is finite, the while loop will terminate, and at this point we set R′′ def= R−R′. In the course of the argument below, we will refer the order in which rectangles were added to R′. With this construction, it is obvious that (3.3) holds, with a bound that is a function of κ. Yet, κ is an absolute constant, so this depen- dence does not concern us. And so the rest of the proof is devoted to the verification of (3.1). An important aspect of the qualitative nature of the interval of eccentricity is encoded into this algorithm. We will choose κ so large that this is true: Consider two rectangles R and R′ with R ∩ R′ 6= ∅, L(R) ≥ L(R′), W(R) ≥ W(R′), |EX(R)| ≤ |EX(R′)| and EX(R) ⊂ 10EX(R′) then we have (3.4) R′ ⊂ κR . See Figure 3.1. Uniform Estimates. We estimate the left hand side of (3.1). In so doing we expand the square, and seek certain uniform estimates. 14 3. LIPSCHITZ KAKEYA Expanding the square on the left hand side of (3.1), we can estimate l.h.s. of (3.1) ≤ |R|+ 2 (ρ,R)∈P |ρ ∩R| where P consists of all pairs (ρ, R) ∈ R′×R′ such that ρ∩R 6= ∅, and ρ was selected to be a member of R′ before R was. It is then automatic that L(R) ≤ L(ρ). And since the density of all tiles is positive, it follows that dist(EX(ρ),EX(R)) ≤ 2‖v‖LipL(ρ) < 150 . We will split up the collection P into sub-collections {SR : R ∈ R′} and {Tρ : ρ ∈ R′}. For a rectangle R ∈ R′, we take SR to consist of all rectangles ρ such that (a) (ρ, R) ∈ P; and (b) EX(ρ) ⊂ 10EX(R). We assert that (3.5) |R ∩ ρ| ≤ |R|, R ∈ R. This estimate is in fact easily available to us. Since the rectangles ρ ∈ SR were selected to be in R′ before R was, we cannot have the inclusion (3.6) R ⊂ 1κρ > κ Now the rectangle ρ are also longer. Thus, if (3.5) does not hold, we would compute the maximal function of in a direction which is close, within an error of 2π/κ, of being orthog- onal to the long direction of R. In this way, we will contradict (3.6). The second uniform estimate that we need is as follows. For fixed ρ, set Tρ to be the set of all rectangles R such that (a) (ρ, R) ∈ P and (b) EX(ρ) 6⊂ 10EX(R). We assert that (3.7) |R ∩ ρ| . δ−1|ρ|, ρ ∈ R′. This proof of this inequality is more involved, and taken up in the next subsection. Remark 3.8. In the proof of (3.7), it is not necessary that ρ ∈ R′. Writing ρ = Iρ × Jρ, in the coordinate basis e and e⊥, we could take any rectangle of the form I × Jρ. THE WEAK L ESTIMATE 15 . . . Figure 3.2. Proof of Lemma 3.9 These two estimates conclude the proof of (3.1). For any two dis- tinct rectangles ρ, R ∈ P, we will have either ρ ∈ SR or R ∈ Tρ. Thus (3.1) follows by summing (3.5) on R and (3.7) on ρ. The Proof of (3.7). We do not need this Lemma for the proof of (3.7), but this is the most convenient place to prove it. Lemma 3.9. Let S be any finite collection of rectangles with L(R) ≤ 2 L(R′), and with |V(R)| ≥ δ|R| for all R,R′ ∈ S. Then it is the case (3.10) ≤ 2δ−1 . Proof. Fix a point x at which we give an upper bound on the sum above. Let C(x) be any circle centered at x. We shall show that there exists at most one R in S such that V(R) ∩ C(x) 6= ∅. By the assumption |V(R)| ≥ δ|R| this proves the Lemma. We prove this last claim by contradiction of the Lipschitz assump- tion on the vector field v. Assume that there exist at least two rectan- gles R,R′ ∈ S for which the sets V(R) and V(R′) intersect C(x). Thus there exist y and y′ in C(x) such that v(y) ∈ EX(R) and v(y′) ∈ EX(R′). Since v is Lipschitz, we have |v(y)− v(y′)| ≤ ‖v‖Lip|y − y′| ≤ 4‖v‖Lip L(R)|v(y)− v(y′)| , but this is a contradiction to our assumption (1.8). See Figure 3.2. � We fix ρ, and begin by making a decomposition of the collection Tρ. Suppose that the coordinate axes for ρ are given by eρ, associated with the long side of R, and e⊥ρ , with the short side. Write the rectangle as a product of intervals Iρ × J , where |Iρ| = L(ρ). Denote one of the endpoints of J as α. See Figure 3.3. 16 3. LIPSCHITZ KAKEYA Figure 3.3. Notation for the proof of (3.7). For rectangles R ∈ Tρ, let IR denote the orthogonal projection R onto the line segment 2Iρ×{α}. Subsequently, we will consider different subsets of this line segment. The first of these is as follows. For R ∈ Tρ, let VR be the projection of the set V(R) onto 2Iρ × {α}. The angle θ between ρ and R is at most |θ| ≤ 2‖v‖Lip L(ρ) ≤ 150 . It follows that (3.11) 1 L(R) ≤ |IR| ≤ 2 L(R), and δ L(R) . |VR|. A recursive mechanism is used to decompose Tρ. Initialize STOCK← Tρ , U ← ∅ . While STOCK 6= ∅ select R ∈ STOCK of maximal length. Update U ← U ∪ {R}, U(R)← {R′ ∈ STOCK : VR ∩ VR′ 6= ∅}. STOCK← STOCK− U(R). (3.12) When this while loop stops, it is the case that Tρ = R∈U U(R). With this construction, the sets {VR : R ∈ U} are disjoint. By (3.11), we have (3.13) L(R) . δ−1 L(ρ) . The main point, is then to verify the uniform estimate (3.14) R′∈U(R) |R′ ∩ ρ| . L(R) ·W(ρ) , R ∈ U . Note that both estimates immediately imply (3.7). THE WEAK L ESTIMATE 17 Figure 3.4. Proof of Lemma 3.15: The rectangles R,R′ ∈ U(ρ), and so the angles R and R′ form with ρ are nearly the same. Proof of (3.14). There are three important, and more technical, facts to observe about the collections U(R). For any rectangle R′ ∈ U(R), denote its coordinate axes as eR′ and e⊥R′ , associated to the long and short sides of R ′ respectively. Lemma 3.15. For any rectangle R′ ∈ U(R) we have |eR′ − eR| ≤ 12 |eρ− eR| Proof. There are by construction, points x ∈ V(R) and x′ ∈ V(R′) which get projected to the same point on the line segment Iρ × {α}. See Figure 3.4. Observe that |eR′ − eR| ≤ |EX(R′)|+ |EX(R)|+ |v(x′)− v(x)| ≤ |EX(R′)|+ |EX(R)|+ ‖v‖Lip · L(R) · |eρ− eR| ≤ |EX(R′)|+ |EX(R)|+ 1 |eρ− eR| Now, |EX(R)| ≤ 1 |eρ− eR|, else we would have ρ ∈ SR. Likewise, |EX(R′)| ≤ 1 |eR′ − eR|. And this proves the desired inequality. Lemma 3.16. Suppose that there is an interval I ⊂ Iρ such that (3.17) R′∈U(R) L(R′)≥8|I| |R′ ∩ I × J | ≥ |I × J | . Then there is no R′′ ∈ U(R) such that L(R′′) < |I| and R′′∩4I×J 6= ∅. Proof. There is a natural angle θ between the rectangles ρ and R, which we can assume is positive, and is given by |eρ− eR|. Notice that we have θ ≥ 10|EX(R)|, else we would have ρ ∈ SR, which contradicts our construction. 18 3. LIPSCHITZ KAKEYA 4I × J I × J Figure 3.5. Moreover, there is an important consequence of Lemma 3.15: For any R′ ∈ U(R), there is a natural angle θ′ between R′ and ρ. These two angles are close. For our purposes below, these two angles can be regarded as the same. For any R′ ∈ U(R), we will have |κR′ ∩ ρ| |I × J | ≃ κ W(R′) ·W(ρ) θ|I|W(ρ) W(R′) θ · |I| . Recall Mκ is larger than the maximal function over κ uniformly distributed directions. Choose a direction e′ from this set of κ directions that is closest to e⊥ρ . Take a line segment Λ in direction e ′ of length κθ|I|, and the center of Λ is in 4I × J . See Figure 3.5. Then we have |κR′ ∩ Λ| |Λ| ≥ W(R′) θ · |I| Thus by our assumption (3.17), R′∈U(R) |κR′ ∩ Λ| & 1 . That is, any of the lines Λ are contained in the set 1κR′ > κ Clearly our construction does not permit any rectangle R′′ ∈ U(R) contained in this set. To conclude the proof of our Lemma, we seek a contradiction. Suppose that there is an R′′ ∈ U(R) with L(R′′) < |I| and R′′ intersects 2I × J . The range of line segments Λ we can permit THE WEAK L ESTIMATE 19 4I × J Figure 3.6. The proof of Lemma 3.18 is however quite broad. The only possibility permitted to us is that the rectangle R′′ is quite wide. We must have W(R′′) ≥ 1 |Λ| = κ · θ · |I|. This however forces us to have |EX(R′′)| ≥ κ θ. And this implies that ρ ∈ SR′′ , as in (3.5). This is the desired contradiction. Our third and final fact about the collection U(R) is a consequence of Lemma 3.15 and a geometric observation of J.-O. Stromberg [27, Lemma 2, p. 400]. Lemma 3.18. For any interval I ⊂ IR we have the inequality (3.19) R′∈U(R) L(R′)≤|I|≤ κL(R′) |R′ ∩ I × J | ≤ 5|I| ·W(ρ) . Proof. For each point x ∈ 4I × J , consider the square S centered at x of side length equal to κ · |I| · |eR− eρ|. See Figure 3.6. It is Stromberg’s observation that for R′ ∈ U(R) we have |κR′ ∩ I × J | |I × J | ≃ |S ∩ κR′| with the implied constant independent of κ. Indeed, by Lemma 3.15, we have that |κR′ ∩ I × J | |I × J | ≃ κW(R′) |eR− eρ| · |I| ≃ κW(R ′) · |I| · |eR− eρ| (|eR− eρ| · |I|)2 ≃ |S ∩ κR |S| , 20 3. LIPSCHITZ KAKEYA as claimed. Now, assume that (3.19) does not hold and seek a contradiction. Let U ′ ⊂ U(R) denote the collection of rectangles R′ over which the sum is made in (3.19). The rectangles in U ′ were added in some order to the collection R′, and in particular there is a rectangle R0 ∈ U ′ that was the last to be added to U ′. Let U ′′ be the collection U ′−{R0}. We certainly have ∑ R′∈U ′′ |R′′ ∩ I × J | ≥ 4|I × J |. Since we cannot have ρ ∈ SR0 , Stromberg’s observation implies that R′∈U ′′ 1κR′ > κ Here, we rely upon the fact that the maximal function Mκ is larger than the usual maximal function over squares. But this is a contradiction to our construction, and so the proof is complete. � The principal line of reasoning to prove (3.14) can now begin with it’s initial recursive procedure. Initialize C(R′)← R′ ∩ ρ . We are to bound the sum R′∈U(R)|C(R′)|. Initialize a collection of subintervals of IR to be I ← ∅ WHILE there is an interval I ⊂ IR satisfying∑ R′∈V(I) |C(R′) ∩ I × J | ≥ 40|I| ·W(ρ) ,(3.20) V(I) = {R′ ∈ U(R) | |C(R′) ∩ I × J | 6= ∅ , L(R′) ≥ 8|I|} ,(3.21) we take I to be an interval of maximal length with this property, and update I ← I ∪ {I} ; C(R′, I) = C(R′) ∩ I × J , R′ ∈ V(I); C(R′)← C(R′)− I × J, R′ ∈ V(I) . [We remark that this last updating is not needed in the most important special case when all rectangles have the same width. But the case we are considering, rectangles can have variable widths, so that |C(R)| can be much larger than any |I| · |J | that would arise from this algorithm.] Once the WHILE loop stops, we have R′ ∩ ρ = C(R′) ∪ {C(R′, I) | I ∈ I , R′ ∈ V(I)} . THE WEAK L ESTIMATE 21 Here the union is over pairwise disjoint sets. We first consider the collection of sets {C(R′) | R′ ∈ U(R)} that remain after the WHILE loop has finished. Since we must not have R′ ⊂ 1/4κ · ρ, it follows that the minimum value of L(R′) is 1 W(ρ). Thus, if in (3.20), we consider an interval I of length 1 W(ρ), the condition L(R′) ≥ 8|I| in the definition of V(I) in (3.21) is vacuous. Thus, we necessarily have R′∈V(I) |C(R′) ∩ I × J | ≤ 40|I| ·W(ρ) . For if this inequality failed, the WHILE loop would not have stopped. We can partition IR by intervals of length close to W(ρ), showing that we have ∑ R′∈U(R) |C(R′)| . |IR| ·W(ρ) . Turning to the central components of the argument, namely the bound for the terms associated with the intervals in I, consider I ∈ I. The inequality (3.20) and Lemma 3.18 implies that each I ∈ I must have length |I| ≤ κ−1/2|Iρ|. But we choose intervals in I to be of maximal length. Thus, R′∈V(I) |C(R′, I)| ≤ 100 · |I| ·W(ρ) . (3.22) Indeed, suppose this last inequality fails. Let I ⊂ Ĩ ⊂ Iρ be an interval twice as long as I. By Lemma 3.18, we conclude that R′∈V(I) L(R′)≤8|eI| |R′ ∩ Ĩ × J | ≤ 10|I| ·W(ρ) . Notice that we are restricting the sum on the left by the length of |Ĩ|. Therefore, we have the inequalities R′∈V(I) L(R′)>8|eI| |C(R′, I)| ≥ 90 · |I| ·W(ρ) > 40 · |Ĩ| ·W(ρ) . That is, Ĩ would have been selected, contradicting our construction. Lemmas 3.16 and 3.18 place significant restrictions on the collection of intervals I. If we have I 6= I ′ ∈ I with 3 I ∩ 3 I ′ 6= ∅, then we must have e.g. κ|I ′| < |I|, as follows from Lemma 3.18. Moreover, V(I ′) must contain a rectangle R′ with L(R′) < |I|. But this contradicts Lemma 3.16. 22 3. LIPSCHITZ KAKEYA Therefore, we must have |I| . |IR| . L(R). With (3.22), this completes the proof of (3.14). An Obstacle to an Lp estimate, for 1 < p < 2 We address one of the main conjectures of this memoir, namely Conjecture 1.12. Let us first observe Proposition 3.23. We have the estimate below valid for all 0 < w < ‖v‖Lip. λ|{Mv,w f > λ}|2/3 . δ−1/3(1 + logw−1‖v‖Lip)1/3‖f‖3/2 Proof. Let ‖v‖Lip = 1. This just relies upon the fact that with 0 < w < 1 fixed, there are only about log 1/w possible values of L(R). This leads very easily to the following two estimates. Following the earlier argument, consider an arbitrary collection of rectangles R with each R ∈ R satisfying (1.8) and |V(R)| ≥ δ|R|. We can then decompose R into disjoint collections R′ and R′′ for which these estimates hold. . δ−1(log 1/w) ,(3.24) R∈R′′ ∣∣∣ . (3.25) Compare to (3.1) and (3.2). Following the same line of reasoning that was used to prove (3.3), we prove our Proposition. We can devise proofs of smaller bounds on the norm of the maximal function than that given by this proposition. But no argument that we can find avoids the some logarithmic term in the width of the rectangle. Let us illustrate the difficulty in the estimate with an object pointed out to us by Ciprian Demeter. We term it a pocketknife, and it is pictured in Figure 3.7. A pocketknife comes with a handle, namely a rectangle Rhandle that is longer than any other rectangle in the pocketknife. We call a collec- tion of rectangles B a set of blades if these two conditions are met. In the first place, (3.26) Rhandle ∩ R 6= ∅ . BOURGAIN’S GEOMETRIC CONDITION 23 handle blades . . . hinges Figure 3.7. A pocketknife. In the second place, we have angle(R,Rhandle) ≃ angle(R′, Rhandle) , R, R′ ∈ B . Let θ(B) denote the angle between Rhandle and the rectangles in the blade B. We refer to as a hinge a rectangle of dimensions w/θ(B) by w, in the same coordinate system of Rhandle that contains the intersection in (3.26). Now, let B be a collection of blades for the handle Rhandle. Our proof of the weak L2 estimate for the Lipschitz Kakeya Maximal function shows that we can assume ♯B · w2 · θ(B)−1 . |Rhandle| . This is essentially the estimate (3.5). But, to follow the covering lemma approach to the L3/2 estimate for the maximal function, we need to control (♯B)2 · w2 · θ(B)−1 . We can only find control of expressions of this type in terms of some slowly varying function of w−1. Bourgain’s Geometric Condition Jean Bourgain [3] gives a geometric condition on the Lipschitz vec- tor field that is sufficient for the L2 boundedness of the maximal func- tion associated with v. We describe the condition, and show how it immediately proves that the corresponding Lipschitz Kakeya maximal function admits a weak type bound on L1. In particular our Conjec- ture 1.12 holds for these vector fields. To motivate Bourgain’s condition, let us recall the earlier condition considered by Nagel, Stein and Wainger in [22]. This condition imposes 24 3. LIPSCHITZ KAKEYA a restriction on the maximum and minimum curvatures of the integral curves of the vector field through the assumption that supx∈Ω det[∇v(x)v(x), v(x)] infx∈Ω det[∇v(x)v(x), v(x)] Here, Ω is a domain in R2, and one can achieve an upper bound on the norm of the maximal function associated to v, appropriately restricted to Ω, in terms of this ratio. Bourgain’s condition permits the vector field to have integral curves which are flat. Suppose that v is defined on all of R2. Define (3.27) ω(x; t) := |det[v(x+ tv(x)), v(x)]| , |t| ≤ 1 ‖v‖Lip . Assume a uniform estimate of the following type: For absolute con- stants 0 < c, C <∞ and 0 < ǫ0 < 12‖v‖Lip, (3.28) |{|t| ≤ ǫ | ω(x; t) < τ sup |s|≤ǫ ω(x, s)}| ≤ Cτ cǫ , this condition holding for all x ∈ R2, 0 < τ < 1 and 0 < ǫ < ǫ0. The interest in this condition stems from the fact [3] that real- analytic vector fields satisfy it. Also see Remark 3.35. Bourgain proved: Theorem 3.29. Assume that (3.28) holds. Then, the maximal opera- tor Mv,ǫ0 defined in (1.1) maps L 2 into itself. This paper claims that the same methods would prove the bounds ‖Mv,ǫ0‖p . ‖f‖p , 1 < p <∞ . And suggests that similar methods would apply to the localized Hilbert transform with respect to these vector fields. Here, we prove Proposition 3.30. Assume that (3.28) holds. Then, the Lipschitz Kakeya Maximal Functions Mv,δ,w , 0 < δ < 1 , 0 < w < ǫ0 defined in (1.13) satisfy the weak L1 estimate λ|{Mv,δ,w f > λ}| . δ−1(1 + log 1/δ)‖f‖1 . The implied constants depend upon the constants in (3.28). That is, these vector fields easily fall within the scope of our anal- ysis. As a corollary to Theorem 1.18, we see that Hv maps L 2 into itself. BOURGAIN’S GEOMETRIC CONDITION 25 Figure 3.8. Proof of (3.31). Proof. Let us assume that ‖v‖Lip = 1. Fix δ > 0 and 0 < w < ǫ0. Let R be the class of rectangles with L(R) < κ and satisfying |V(R)| ≥ δ|R|. Say that R′ ⊂ R has scales separated by s > 3 iff for R,R′ ∈ R′ the condition 4 L(R) < L(R′) implies that 2s L(R) < L(R′). One sees that R can be decomposed into ≃ s sub-collections with scales separated by s. The fortunate observation is this: Assuming (3.28), and taking s ≃ log 1/δ, any subset R′ ⊂ R with scales separated by s further enjoys this property: If R,R′ ∈ R′ with C EX(R) ∩ C EX(R′) = ∅, with C a fixed constant, then (3.31) L(R) ≃ L(R′) or R ∩R′ = ∅ . Let us see why this is true, arguing by contradiction. Thus we assume that L(R′) ≤ 2−s L(R), R∩R′ 6= ∅ and C EX(R)∩C EX(R′) = ∅. Since the rectangles have an essentially fixed width, it follows that 2|EX(R′)| ≥ |EX(R)|. Fix a line ℓ in the long direction of 2R with |{x ∈ ℓ | v(x) ∈ V(R)}| ≥ δ |ℓ| = δ L(R) . Let x0 be in the set above, x 0 ∈ V (R′) and x′0 is the projection of x′′0 onto the line ℓ. See Figure 3.8. Observe that we can estimate |v(x′′0)− v(x′0)| ≤ 2|v(x0)− v(x′′0)| L(R′)(3.32) Therefore, for C sufficiently large, we have |v(x0)− v(x′0)| ≥ ∣∣|v(x′0)− v(x′′0)| − |v(x′′0)− v(x0)| ≥ |v(x′′0)− v(x0)|(1− 2 L(R′)) ≥ |EX(R′)| provided C is large enough. Now, after a moments thought, one sees that |det[v(x0), v(x′0)]| ≃ angle(v(x0), v(x′0)) . 26 3. LIPSCHITZ KAKEYA Therefore, for any x ∈ ℓ s≤L(R) ω(x; s) & |EX(R′)| . But the vector field satisfies (3.28), which we will apply with τ ≃ EX(R) EX(R′) ≃ L(R It follows that L(R) ≤ |{x ∈ ℓ | ω(x; s) ≤ cτ |EX(R′)|}| ≤ |{x ∈ ℓ | ω(x; s) ≤ τ sup |s|≤ǫ ω(x, s)}| . τ c L(R) . Therefore, we see (δ/2)1/c . L(R′) which is a contradiction to R′ have scales separated by s, and s ≃ 1 + log 1/δ. Let us see how to prove the Proposition now that we have proved (3.31). Take s ≃ log 1/δ, and a finite sub-collection R′ ⊂ R of rectan- gles with scales separated by s. We may take a further subset R′′ ⊂ R′ such that ∥∥∥ R′′∈R′′ . δ−1 ,(3.33) R∈R′−R′′ ∣∣∣ . R∈R′′ |R′′| .(3.34) These are precisely the covering estimates needed to prove the weak L1 estimate claimed in the proposition. But, in choosing R′′ to satisfy (3.33), it is clear that we need only be concerned about rectangles with a fixed length, and the separation in scales are (3.31) will control rectangles of distinct lengths. The procedure that we apply to select R′′ is inductive. Set R′′ ← ∅ , S ← ∅ , STOCK←R′ . WHILE STOCK 6= ∅, select R ∈ STOCK with maximal length, and update R′′ ← R′′ ∪ {R}, as well as STOCK ← STOCK − {R}. In addition, for any R′ ∈ STOCK with R′ ⊂ 4CR, where C ≥ 1 is the VECTOR FIELDS THAT ARE A FUNCTION OF ONE VARIABLE 27 constant that insures that (3.31) holds, remove these rectangles from STOCK and add them to S. Once the WHILE loop stops, we will have STOCK = ∅ and we have our decomposition of R′. By construction, it is clear that (3.34) holds. We need only check that (3.33) holds. Now, consider R,R′ ∈ R′, with two rectangles have their scales separated, thus 2s L(R′) < L(R). If it is the case that R∩R′ 6= ∅ and C EX(R)∩C EX(R′) 6= ∅, then R would been selected to be in R′ first, whence R′ would have been placed in Therefore, C EX(R) ∩ C EX(R′) = ∅, but then (3.31) implies that R ∩ R′ = ∅. Thus, the only contribution to the L∞ norm in (3.33) can come from rectangles of about the same length. But Lemma 3.9 then implies that such rectangles can overlap only about δ−1 times. Our proof is complete. (As the interest in (3.28) is in small values of c, it will be more efficient to use Lemma 3.9 to handle the case of the rectangles having approximately the same length.) � Remark 3.35. To conclude that the Hilbert transform on vector fields is bounded, one could weaken Bourgain’s condition (3.28) to |{|t| ≤ ǫ | ω(x; t) < τ sup |s|≤ǫ ω(x, s)}| ≤ C exp(−(log 1/τ)c)ǫ . This inequality is to hold universally in x ∈ R2, 0 < τ < 1, and 0 < ǫ < ‖v‖Lip. This is of interest for 0 < c < 1. The proof above can be modified to show that the maximal functions Mv,δ,w satisfy the weak L1 inequality, with constant at most . δ−1−1/c. Vector Fields that are a Function of One Variable We specialize to the vector fields that are a function of just one real variable. Assume that the vector field v is of the form (3.36) v(x1, x2) = (v1(x2), v2(x2)) , and for the moment we do not impose the condition that the vector field take values in the unit circle. The point is simply this: If we are interested in transforms where the kernel is not localized, the restriction on the vector field is immaterial. Namely, for any vector field v Hv,∞ f(x) = p.v. f(x− yv(x)) dy = p.v. f(x− yṽ(x)) dy , ṽ(x) = |v(x)| . We return to a theme implicit in the proof of Proposition 2.4. This proof only relies upon vector fields that are only a function of one 28 3. LIPSCHITZ KAKEYA variable. Thus, it is a significant subcase of the Stein Conjecture to verify it for Lipschitz vector fields of just one variable. Indeed, the situation is this. Proposition 3.37. Suppose that a choice of vector field v(x1, x2) = (1, v1(x1)) is just a function of, say, the first coordinate. Then, Hv,∞ maps L2(R2) into itself. Proof. The symbol of Hv,∞ is sgn(ξ1 + ξ2v1(x1)) . For each fixed ξ2, this is a bounded symbol. And in the special case of the L2 estimate, this is enough to conclude the boundedness of the operator. � It is of interest to extend this Theorem in any Lp, for p 6= 2, for some reasonable choice of vector fields. The corresponding questions for the maximal function are also of interest, and here the subject is much more developed. The paper [5] studies the maximal function Mv,∞. They proved the boundedness of this maximal function on Lp, p > 1, assuming that the vector field was of the form v(x) = (1, v2(x)), that D v2 was positive, and increasing, and satisfied a third more technical condition. More recently, [14] has showed that the third condition is not needed. Namely the following is true. Theorem 3.38. Assume that v(x) = (1, v2(x)), and moreover that D v2 ≥ 0 and is monotonically increasing. Then, Mv,∞ is bounded on Lp, for 1 < p <∞. These vector fields present far fewer technical difficulties than a gen- eral Lipschitz vector field, and there are a richer set of proof techniques that one can bring to bear on them, as indicated in part in the proof of Proposition 3.37. The papers [5, 14] cleverly exploit the Plancherel identity (in the independent variable), and other orthogonality consid- erations to prove their results. These considerations are not completely consistent with the domi- nant theme of this monograph, in which the transforms are localized. Nevertheless, it would be interesting to explore methods, possibly mod- ifications of this memoir, that could provide an extension of Proposi- tion 3.37. In this direction, let us state a possible direction of study. The definition of the the sets V(R) for vector fields of magnitude 1 is given as V(R) = R ∩ v−1(EX(R)). For vector fields of arbitrary magnitude, VECTOR FIELDS THAT ARE A FUNCTION OF ONE VARIABLE 29 we define these sets to be V(R) = {x ∈ R | v(x)|v(x)| ∈ EX(R)} . Define a maximal function—an extension of our Lipschitz Kakeya Max- imal Function—by (3.39) M̃v,δf(x) = sup |V(R)|≥δ|R| |R|−1 f(y) dy . In this definition, we require the rectangles to have density δ, but do not restrict their eccentricities, or lengths. Conjecture 3.40. Assume that the vector field is of the form v(x) = (1, v2(x2)), and the derivative D v ≥ 0 and is monotone. Then for all 0 < δ < 1, we have the estimate ‖M̃v,δ‖p . δ−1 , 1 < p <∞ . One can construct examples which show that the L1 to weak L1 norm of the maximal function is not bounded in terms of δ. Indeed, recalling the ‘pocketknife’ examples of Figure 3.7, we comment that one can construct examples of vector fields with these properties, which we describe with the terminology associated with the pocketknife exam- ples. • The width of all rectangles are fixed. And all rectangles have density δ. • The ‘handle’ of the pocketknife has positive angle θ with the x1 axis. • There is ‘hinge’ whose blades have angles which are positive, and greater than θ. The number of blades can be unbounded, as the width of the rectangles decreases to zero. The assumption that the vector field is only a function of x2 then greatly restricts, but does not completely forbid, the existence of addi- tional hinges. So the combinatorics of these vector fields, as expressed in the Lipschitz Kakeya Maximal Function, are not so simple. CHAPTER 4 The L2 Estimate for Hilbert Transform on Lipschitz Vector Fields We prove one of our main conditional results about the Hilbert transform on Lipschitz vector fields, the inequality (1.19) which is the estimate at L2, for functions with frequency support in an annulus, assuming an appropriate estimate for the Lipschitz Kakeya Maximal Function. We begin the proof by setting notation appropriate for phase plane analysis for functions f on the plane supported on an annulus. With this notation, we can define appropriate discrete analogs of the Hilbert transform on vector fields. The Lemmas 4.22 and 4.23 are the combi- natorial analogs of our Theorem 1.15. We then take up the proofs. The main step in the proof is Lemma 4.50 which combines the (standard) orthogonality considerations with the conjectures about the Lipschitz Kakeya Maximal Functions. Definitions and Principle Lemmas Throughout this chapter, κ will denote a fixed small positive con- stant, whose exact value need not concern us. κ of the order of 10−3 would suffice. The following definitions are as in the authors’ previous paper [15]. Definition 4.1. A grid is a collection of intervals G so that for all I, J ∈ G, we have I ∩ J ∈ {∅, I, J}. The dyadic intervals are a grid. A grid G is central iff for all I, J ∈ G, with I ⊂6= J we have 500κ−20I ⊂ J . The reader can find the details on how to construct such a central grid structure in [11]. Let ρ be rotation on T by an angle of π/2. Coordinate axes for R2 are a pair of unit orthogonal vectors (e, e⊥) with ρ e = e⊥. Definition 4.2. We say that ω ⊂ R2 is a rectangle if it is a product of intervals with respect to a choice of axes (e, e⊥) of R 2. We will say that ω is an annular rectangle if ω = (−2l−1, 2l−1)× (a, 2a) for an integer l with 2l < κa, with respect to the axes (e, e⊥). The dimensions of ω are said to be 2l × a. Notice that the face (−2l−1, 2l−1) × a is 32 4. L ESTIMATE FOR Hv es Rs Figure 4.1. The two rectangles ωs and Rs whose prod- uct is a tile. The gray rectangles are other possible loca- tions for the rectangle Rs. tangent to the circle |ξ| = a at the midpoint to the face, (0, a). We say that the scale of ω is scl(ω) := 2l and that the annular parameter of ω is ann(ω) := a. In referring to the coordinate axes of an annular rectangle, we shall always mean (e, e⊥) as above. Annular rectangles will decompose our functions in the frequency variables. But our methods must be sensitive to spatial considerations; it is this and the uncertainty principle that motivate the next definition. Definition 4.3. Two rectangles R and R are said to be dual if they are rectangles with respect to the same basis (e, e⊥), thus R = r1 × r2 and R = r1 × r2 for intervals ri, ri, i = 1, 2. Moreover, 1 ≤ |ri| · |ri| ≤ 4 for i = 1, 2. The product of two dual rectangles we shall refer to as a phase rectangle. The first coordinate of a phase rectangle we think of as a frequency component and the second as a spatial component. We consider collections of phase rectangles AT which satisfy these conditions. For s, s′ ∈ AT we write s = ωs × Rs, and require that ωs is an annular rectangle,(4.4) Rs and ωs are dual,(4.5) The rectangles Rs are from the product of central grids.(4.6) {1000κ−100R | ωs × R ∈ AT } covers R2, for all ωs.(4.7) ann(ωs) = 2 j for some integer j,(4.8) ♯{ωs | scl(s) = scl, ann(s) = ann} ≥ c ,(4.9) scl(s) ≤ κann(s).(4.10) DEFINITIONS AND PRINCIPLE LEMMAS 33 0 ρωs1 Figure 4.2. An annular rectangular ωs, and three as- sociated subintervals of ρωs1, ωs1, and ωs2. We assume that there are auxiliary sets ωs,ωs1,ωs2 ⊂ T associated to s—or more specifically ωs—which satisfy these properties. Ω := {ωs,ωs1,ωs2 | s ∈ AT } is a grid in T,(4.11) ωs1 ∩ ωs2 = ∅, |ωs| ≥ 32(|ωs1|+ |ωs2|+ dist(ωs1,ωs2))(4.12) ωs1 lies clockwise from ωs2 on T,(4.13) |ωs| ≤ K scl(ωs) ann(ωs) ,(4.14) { ξ|ξ| | ξ ∈ ωs} ⊂ ρωs1.(4.15) In the top line, the intervals ωs1 and ωs2 are small subintervals of the unit circle, and we can define their dilate by a factor of 2 in an obvious way. Recall that ρ is the rotation that takes e into e⊥. Thus, eωs ∈ ωs1. See the figures Figure 4.1 and Figure 4.2 for an illustration of these definitions. Note that |ωs| ≥ |ωs1| ≥ scl(ωs)/ann(ωs). Thus, eωs is in ωs1, and ωs serves as ‘the angle of uncertainty associated to Rs.’ Let us be more precise about the geometric information encoded into the angle of uncertainty. Let Rs = rs × rs⊥ be as above. Choose another set of coordinate axes (e′, e′⊥) with e ′ ∈ ωs and let R′ be the product of the intervals rs and rs⊥ in the new coordinate axes. Then K ′ ⊂ Rs ⊂ ′ for an absolute constant K0 > 1. We say that annular tiles are collections AT satisfying the condi- tions (4.4)—(4.15) above. We extend the definition of e⊥, eω⊥, ann(ω) and scl(ω) to annular tiles in the obvious way, using the notation es, es⊥, ann(s) and scl(s). 34 4. L ESTIMATE FOR Hv A phase rectangle will have two distinct functions associated to it. In order to define these functions, set Ty f(x) := f(x− y), y ∈ R2 (Translation operator) Modξ f(x) := e iξ·x f(x), ξ ∈ R2 (Modulation operator) R1×R2 f(x1, x2) := (|R1||R2|)1/p (Dilation operator). In the last display, 0 < p ≤ ∞, and R1 × R2 is a rectangle, and the coordinates (x1, x2) are those of the rectangle. Note that the definition depends only on the side lengths of the rectangle, and not the location. And that it preserves Lp norm. For a function ϕ and tile s ∈ AT set (4.16) ϕs := Modc(ωs)Tc(Rs)D We shall consider ϕ to be a Schwartz function for which ϕ̂ ≥ 0 is supported in a small ball, of radius κ, about the origin in R2, and is identically 1 on another smaller ball around the origin. (Recall that κ is a fixed small constant.) We introduce the tool to decompose the singular integral kernels. In so doing, we consider a class of functions ψt, t > 0, so that Each ψt is supported in frequency in [−θ − κ,−θ + κ].(4.17) |ψt(x)| . CN(1 + |x|)−N , N > 1 .(4.18) In the top line, θ is a fixed positive constant so that the second half of (4.19) is true. Define φs(x) := ϕs(x− yv(x))ψs(y) dy (v(x)) ϕs(x− yv(x))ψs(sy) dy. (4.19) ψs(y) := scl(s)ψscl(s)(scl(s)y).(4.20) An essential feature of this definition is that the support of the integral is contained in the set {v(x) ∈ ωs2}, a fact which can be routinely verified. That is, we can insert the indicator 1 (v(x)) without loss of generality. The set ωs2 serves to localize the vector field, while ωs1 serves to identify the location of ϕs in the frequency coordinate. DEFINITIONS AND PRINCIPLE LEMMAS 35 The model operator we consider acts on a Schwartz functions f , and it is defined by (4.21) Cannf := s∈AT (ann) scl(s)≥‖v‖Lip 〈f, ϕs〉φs. In this display, AT (ann) := {s ∈ AT | ann(s) = ann}, and we have deliberately formulated the operator in a dilation invariant manner. Lemma 4.22. Assume that the vector field is Lipschitz, and satisfies Conjecture 1.14. Then, for all ann ≥ ‖v‖−1Lip, the operator Cann extends to a bounded map from L2 into itself, with norm bounded by an absolute constant. We remind the reader that for 2 < p <∞ the only condition needed for the boundedness of Cann is the measurability of the vector field, a principal result of Lacey and Li [15]. It is of course of great importance to add up the Cann over ann. The method we use for doing this are purely L2 in nature, and lead to the estimate for C := j=1 C2j . Lemma 4.23. Assume that the vector field is of norm at most one in Cα for some α > 1, and satisfies Conjecture 1.14. Then C maps L2 into itself. In addition we have the estimate below, holding for all values of scl. (4.24) ann=−∞ s∈AT (ann) scl(s)=scl 〈f, ϕs〉φs . (1 + log(1 + scl−1‖v‖Cα)). Moreover, these operators are unconditionally convergent in s ∈ AT . These are the principal steps towards the proof of Theorem 1.15. In the course of the proof, we shall not invoke the additional notation needed to account for the unconditional convergence, as it is entirely notational. They can be added in by the reader. Observe that (4.24) is only of interest when scl < ‖v‖Cα. This inequality depends critically on the fact that the kernel sclψ(scly) has mean zero. Without this assumption, this inequality is certainly false. The proof of Theorem 1.15 from these two lemmas is an argument in which one averages over translations, dilations and rotations of grids. The specifics of the approach are very close to the corresponding argu- ment in [15]. The details are omitted. The operators Cann and C are constructed from a a kernel which is a smooth analog of the truncated kernel p.v. 1 1{|t|≤1}. Nevertheless, our 36 4. L ESTIMATE FOR Hv main theorem follows,1 due to the observation that we can choose a sequence of Schwartz kernels ψ(1+κ)n , for n ∈ Z, which satisfy (4.17) and (4.18), and so that for K(t) := an(1 + κ) nψ(1+κ)n((1 + κ) we have p.v. 1 1{|t|≤1} = K(t)−K(t). Here, for n ≥ 0 we have |an| . 1. And for n < 0, we have |an| . (1+κ)n. The principal sum is thus over n ≥ max(0, ‖v‖Cα), and this corresponds to the operator C. For those n < max(0, ‖v‖Cα), we use the estimate (4.24), and the rapid decay of the coefficient an. Truncation and an Alternate Model Sum There are significant obstacles to proving the boundedness of the model sum Cann on an Lp space, for 1 < p < 2. In this section, we rely upon some naive L2 estimates to define a new model sum which is bounded on Lp, for some 1 < p < 2. Our next Lemma is indicative of the estimates we need. For choices of scl < κann, set AT (ann, scl) := {s ∈ AT (ann) | scl(s) = scl}. Lemma 4.25. For measurable vector fields v and all choices of ann and scl. ∥∥∥ s∈AT (ann,scl) 〈f, ϕs〉φs . ‖f‖2 Proof. The scale and annulus are fixed in this sum, making the Bessel inequality s∈AT (ann,scl) |〈f, ϕs〉|2 . ‖f‖22 evident. For any two tiles s and s′ that contribute to this sum, if ωs 6= ωs′, then φs and φs′ are disjointly supported. And if ωs = ωs′, then Rs and R s are disjoint, but share the same dimensions and orientation in the plane. The rapid decay of the functions φs then gives us the 1In the typical circumstance, one uses a maximal function to pass back and forth between truncated and smooth kernels. This route is forbidden to us; there is no appropriate maximal function to appeal to. TRUNCATION AND AN ALTERNATE MODEL SUM 37 estimate s∈AT (ann,scl) 〈f, ϕs〉φs s∈AT (ann,scl) |〈f, ϕs〉|2 . ‖f‖2 Consider the variant of the operator (4.21) given by (4.26) Φf = s∈AT (ann) scl(s)≥κ−1‖v‖Lip 〈f, ϕs〉φs. As ann is fixed, we shall begin to suppress it in our notations for oper- ators. The difference between Φ and Cann is the absence of the initial . log(1+ ‖v‖Lip) scales in the former. The L2 bound for these missing scales is clearly provided by Lemma 4.25, and so it remains for us to establish (4.27) ‖Φ‖2 . 1, the implied constant being independent of ann, and the Lipschitz norm of v. It is an important fact, the main result of Lacey and Li [15], that (4.28) ‖Φ‖p . 1, 2 < p <∞. This holds without the Lipschitz assumption. We are now at a point where we can be more directly engaged with the construction of our alternate model sum. We only consider tiles with κ−1‖v‖Lip ≤ scl(s) ≤ κann. A parameter is introduced which is used to make a spatial truncation of the functions ϕs; it is (4.29) γ2s := 100 −2 scl(s) ‖v‖Lip Write ϕs = αs + βs where αs = (Tc(Rs)D ζ)ϕs, and ζ is a smooth Schwartz function supported on |x| < 1/2, and equal to 1 on |x| < 1/4. Write for choices of tiles s, (4.30) ψs(y) = ψs−(y) + ψs+(y) where ψs−(y) is a Schwartz function on R, with supp(ψs−) ⊂ γs(scl(s)) −1[−1, 1] , 38 4. L ESTIMATE FOR Hv and equal to ψscl(s)(y) for |y| < 14γs(scl(s)) −1. Then define (4.31) as±(x) = 1ωs2(v(x)) φs(x− yv(x))ψs±(y) dy. Thus, φs = as− + as+. Recalling the notation Sann in Theorem 1.15, define (4.32) A± f := s∈AT (ann) scl(s)≥κ−1‖v‖Lip 〈Sann f, αs〉as± We will write Φ = ΦSann = A+ +A− +B, where B is an operator defined in (4.35) below. The main fact we need concerns A−. Lemma 4.33. There is a choice of 1 < p0 < 2 so that ‖A−‖p . 1, p0 < p <∞. The implied constant is independent of the value of ann, and the Lips- chitz norm of v. The proof of this Lemma is given in the next section, modulo three additional Lemmata stated therein. The following Lemma is important for our approach to the previous Lemma. It is proved below. Lemma 4.34. For each choice of κ−1‖v‖Lip < scl < κann, we have the estimate ∑ s∈AT (ann,scl) |〈Sann f, αs〉|2 . ‖f‖22. Define (4.35) B f := s∈AT (ann) scl(s)≥κ−1‖v‖Lip 〈Sann f, βs〉φs Lemma 4.36. For a Lipschitz vector field v, we have ‖B‖p . 1, 2 ≤ p <∞. Proof. For choices of integers κ−1‖v‖Lip ≤ scl < κann, consider the vector valued operator Tj,k f := {〈Sann f, βs〉√ 1{v(x)∈ωs2}Tc(Rs)D (1 + | · |2)N )(x) | s ∈ AT (ann, scl) where N is a fixed large integer. PROOFS OF LEMMATA 39 Recall that βs is supported off of γsRs. This is bounded linear op- erator from L∞(R2) to ℓ∞(AT (ann, scl)). It has norm. (scl/‖v‖Lip)−10. Routine considerations will verify that Tj,k : L 2(R2) −→ ℓ2(AT (ann, scl)) with a similarly favorable estimate on its norm. By interpolation, we achieve the same estimate for Tj,k from L p(R2) into ℓp(AT (ann, scl)), 2 ≤ p <∞. It is now very easy to conclude the Lemma by summing over scales in a brute force way, and using the methods of Lemma 4.25. � We turn to A+, as defined in (4.32). Lemma 4.37. We have the estimate ‖A+‖p . 1 2 ≤ p <∞. Proof. We redefine the vector valued operator Tj,k to be Tj,k f := {〈Sann f, αs〉√ 1{v(x)∈ωs2}Tc(Rs)D (1 + |x|2)N | s ∈ AT (ann, scl) where N is a fixed large integer. This operator is bounded from Lp(R2) −→ ℓp(AT (ann, scl)) , 2 ≤ p <∞ Its norm is at most . 1. But, for s ∈ AT (ann, scl), we have (4.38) |as+| . (scl/‖v‖Lip)−10|Rs|−1/2(M1Rs)100. Here M denotes the strong maximal function in the plane in the coordi- nates determined by Rs. This permits one to again adapt the estimate of Lemma 4.25 to conclude the Lemma. � Now we conclude that ‖Φ‖2 . 1. And since Φ = A− +A++B, it follows from the Lemmata of this section. Proofs of Lemmata Proof of Lemma 4.33. We have Φ = A−+A+ +B, so from (4.28), Lemma 4.36 and Lemma 4.37, we deduce that ‖A−‖p . 1 for all 2 < p <∞. It remains for us to verify that A− is of restricted weak type p0 for some choice of 1 < p0 < 2. That is, we should verify that for all sets F,G ⊂ R2 of finite measure (4.39) |〈A− 1F , 1G〉| . |F |1/p|G|1−1/p, p0 < p < 2. 40 4. L ESTIMATE FOR Hv Since A− maps L p into itself for 2 < p < ∞, it suffices to consider the case of |F | < |G|. Since we assume only that the vector field is Lipschitz, we can use a dilation to assume that 1 < |G| < 2, and so this set will not explicitly enter into our estimates. We fix the data F ⊂ R2 of finite measure, ann, and vector field v with ‖v‖Lip ≤ κann. Take p0 = 2 − κ2. We need a set of definitions that are inspired by the approach of Lacey and Thiele [20], and are also used in Lacey and Li [15]. For subsets S ⊂ Av := {s ∈ AT (ann) | κ−1‖v‖Lip ≤ scl(s) < κann}, set 〈Sann 1F , αs〉as− Set χ(x) = (1 + |x|)−1000/κ. Define (4.40) χ := χ(p)s = Tc(Rs)D χ, 0 ≤ p ≤ ∞. And set χ̃ s = 1γsRsχ Remark 4.41. It is typical to define a partial order on tiles, following an observation of C. Fefferman [9]. In this case, there doesn’t seem to be an appropriate partial order. Begin with this assumption on the order relation ‘<’ on tiles: (4.42) If ωs ×Rs ∩ ωs′ ×Rs′ 6= ∅, then s and s′ are comparable under ‘<’. It follows from transitivity of a partial order that that one can have tiles s1, . . . , sJ , with sj+1 < sj for 1 ≤ j < J , J ≃ log(‖v‖Lip · ann), and yet the rectangles RsJ and Rs1 can be far apart, namely RsJ ∩ (cJ)Rs1 , for a positive constant c. See Figure 4.3. (We thank the referee for directing us towards this conclusion.) Therefore, one cannot make the order relation transitive, and maintain control of the approximate localization of spatial variables, as one would wish. The partial order is essential to the argument of [9], but while it is used in [20], it is not essential to that argument. We recall a fact about the eccentricity. There is an absolute con- stant K ′ so that for any two tiles s, s′ (4.43) ωs ⊃ ωs′ , Rs ∩ Rs′ 6= ∅ implies Rs ⊂ K ′Rs′ . Figure 3.1 illustrates this in the case where the two rectangles Rs and Rs′ have different widths, which is not the case here. We define an order relation on tiles by s . s′ iff ωs ) ωs′ and Rs ⊂ κ−10Rs′. Thus, (4.42) holds for this order relation, and it is certainly not transitive. PROOFS OF LEMMATA 41 Figure 4.3. The rectangles Rs1 , . . . , RsJ of Remark 4.41. A tree is a collection of tiles T ⊂ Av, for which there is a (non– unique) tile ωT × RT ∈ AT (ann) with Rs ⊂ 100κ−10RT, and ωs ⊃ ωT for all s ∈ T. Here, we deliberately use a somewhat larger constant 100κ−10 than we used in the definition of the order relation ‘..’ For j = 1, 2, call T a i–tree if the tiles for all s, s′ ∈ T, if scl(s) 6= scl(s′), then ωsi∩ωs′i = ∅. 1–trees are especially important. A few tiles in such a tree are depicted in Figure 4.4. Remark 4.44. This remark about the partial order ‘.’ and trees is useful to us below. Suppose that we have two trees T, with top s(T) and T′ with top s(T′). Suppose in addition that s(T′) . s(T). Then, it is the case that T ∪ T′ is a tree with top s(T). Indeed, we must necessarily have ωT ( ωT, since the Rs are from products of a central grid. Also, 100κ−1RT′ ⊂ 100κ−1RT. And so every tile in T′ could also be a tile in T. Our proof is organized around these parameters and functions as- sociated to tiles and sets of tiles. Of particular note here are the first definitions of ‘density,’ which have to be formulated to accommodate the lack of transitivity in the partial order. Note that in the first defi- nition, the supremum is taken over tiles s′ ∈ AT of the same annular parameter as s. We choose the collection AT as it is ‘universal,’ cover- ing all scales in a uniform way, due to different assumptions including 42 4. L ESTIMATE FOR Hv (4.7). dense(S) := sup s′∈AT G∩v−1(ωs′ ) s′ dx | ∃ s , s′′ ∈ S : ωs ⊃ ωs′ ⊃ ωs′′ , Rs ⊂ 100κ−10Rs′ , Rs′ ⊂ 100κ−10Rs′ (4.45) ∆(T)2 := |〈Sann 1F , αs〉|2 1Rs , T is a 1–tree,(4.46) size(S) := sup T is a 1–tree ∆(T) dx.(4.47) Observe that dense(S) only really applies to ‘tree-like’ sets of tiles, and that—and this is important—the tile s′ that appear in (4.45) are not in S, but only assumed to be in AT . Finally, note that dense(s) ≃ G∩v−1(ωs) χ̃(1)s dx with the implied constants only depending upon κ, χ, and other fixed quantities. Observe these points about size. First, it is computed relative to the truncated functions αs, recall (4.29). Second, that for p > 1, (4.48) ‖∆(T)‖p . |F |1/p , because of a standard Lp estimate for a Littlewood-Paley square func- tion. Third, that size(Av(ann)) . 1. And fourth, that one has an estimate of John-Nirenberg type. Lemma 4.49. For a 1-tree T we have the estimate ‖∆(T)‖p . size(T)|RT|1/p , 1 < p <∞ . Proofs of results of this type are well represented in the literature. See [4, 11]. Given a set of tiles, say that count(S) < A iff S is a union of trees T ∈ T for which ∑ |sh(T)| < A. We will also use the notation count(S) . A, implying the existence of an absolute constant K for which count(S) ≤ KA. The principal organizational Lemma is PROOFS OF LEMMATA 43 Figure 4.4. A few possible tiles in a 1–tree. Rectangles ωs are on the left in different shades of gray. Possible locations of Rs are in the same shade of gray. Lemma 4.50. Any finite collection of tiles S ⊂ Av is a union of four subsets Slight, Ssmall, S large, ℓ = 1, 2. They satisfy these properties. size(Ssmall) < size(S),(4.51) dense(Slight) < dense(S),(4.52) and both Sℓlarge are unions of trees T ∈ T ℓ, for which we have the estimates count(S1large) . size(S) −2−κ|F | size(S)−p dense(S)−M |F | + size(S) dense(S)−1 dense(S)−1 (4.53) count(S2large) . size(S) (log 1/ size(S))3|F | size(S)κ/50 dense(S)−1 (4.54) What is most important here is the middle estimate in (4.53). Here, p is as in Conjecture 1.14, and M > 0 is only a function of N in that Conjecture. The estimates that involve size(S)−2|F | are those that follow from orthogonality considerations. The estimates in dense(S)−1 are those that follow from density considerations which are less complicated. However, in the second half of (4.54), the small positive power of size is essential for us. All of these estimates are all variants of those in [20]. The middle estimate of (4.53) is not of this type, and is the key ingredient that permits us to obtain an estimate below L2. Note that 44 4. L ESTIMATE FOR Hv it gives the best bound for collections with moderate density and size. For it we shall appeal to our assumed Conjecture 1.14. Logarithms, such as those that arise in (4.54), arise from our trun- cation arguments, associated with the parameters γs in (4.29). For individual trees, we need two estimates. Lemma 4.55. If T is a 1–tree with − ∆(T) ≥ σ, then we have (4.56) |F ∩ σ−κRT| & σ1+κ|RT|. Lemma 4.57. For trees T we have the estimate (4.58) |〈Sann1F , αs〉〈as−, 1G〉| . Ψ dense(T) size(T) |sh(T)|. Here Ψ(x) = x|log cx|, and inside the logarithm, c is a small fixed constant, to insure that c dense(T) · size(T) < 1 , say. Sum(S) := |〈Sann1F , αs〉〈as−, 1G〉| We want to provide the bound Sum(Av) . |F |1/p for p0 < p < 2. We have the trivial bound (4.59) Sum(S) . Ψ dense(S) size(S) count(S). It is incumbent on us to provide a decomposition of Av into sub- collections for which this last estimate is effective. By inductive application of our principal organizational Lemma 4.50, Av is the union of Sℓδ,σ, ℓ = 1, 2 for δ, σ ∈ 2 := {2n | n ∈ Z , n ≤ 0}, satisfying dense(Sℓδ,σ) . δ,(4.60) size(Sℓδ,σ) . σ,(4.61) count(Sℓδ,σ) . min(σ−2−κ|F |, δ−Mσ−p|F |+ σ1/κδ−1, δ−1) ℓ = 1, min(σ−2(log 1/σ)3|F |, δ−1σκ/50) ℓ = 2 (4.62) Using (4.59), we see that Sum(S1δ,σ) . min(Ψ(δ)σ −1−κ|F |, δ−M+1σ−p+1|F |+ σ1/κ+1, σ) Sum(S2δ,σ) . min(Ψ(δ)σ −1(log 1/σ)4|F |, σ1+κ/50) (4.63) One can check that for ℓ = 1, 2, (4.64) δ,σ∈2 Sum(Sℓδ,σ) . |F |1/p, p0 < p < 2. PROOFS OF LEMMATA 45 This completes the proof of Lemma 4.33, aside from the proof of Lemma 4.50. Proof of (4.64). We can assume that |G| = 1, and that |F | ≤ 1, for otherwise the result follows from the known Lp estimates, for p > 2 and measurable vector fields, see Theorem 1.15. The case of ℓ = 2 in (4.64) is straightforward. Notice that in (4.63), for ℓ = 2, the two terms in the minimum are roughly comparable, ignoring logarithmic terms, for δ|F | ≃ σ2+κ/50 . Therefore, we set T1 = {(δ, σ) ∈ 2× 2 | δ|F | ≤ σ2+κ/50 ≤ |F |} , T2 = {(δ, σ) ∈ 2× 2 | σ2+κ/50 ≤ δ|F |} and T3 = 2× 2− T1 − T2. We can estimate (δ,σ)∈T1 Sum(S2δ,σ) . (δ,σ)∈T1 Ψ(δ)σ−1(log 1/σ)4|F | σ2+κ/50≤|F | σ1+κ/75 . |F |1/p0 , p0 = 2 + κ/50 1 + κ/75 < 2 . Notice that we have absorbed harmless logarithmic terms into a slightly smaller exponent in σ above. The second term is (δ,σ)∈T2 Sum(S2δ,σ) . (δ,σ)∈T2 σ1+κ/50 (δF )1/p1 , p1 = 2 + κ/50 1 + κ/50 < 2 , . |F |1/p1 . 46 4. L ESTIMATE FOR Hv The third term is (δ,σ)∈T3 Sum(S2δ,σ) . (δ,σ)∈T3 Ψ(δ)σ−1(log 1/σ)4|F | σ2+κ/50≥|F | σ−1|F |1−κ/75 . |F |1/p0 . Here, we have again absorbed harmless logarithms into a slightly smaller power of |F |, and p0 < 2 is as in the first term. The novelty in this proof is the proof of (4.64) in the case of ℓ = 1. We comment that if one uses the proof strategy just employed, that is only relying upon the first and last estimates from the minimum in (4.63), in the case of ℓ = 1, one will only show that |F |1/2. In the definitions below, we will have a choice of 0 < τ < 1, where τ = τ(M, p) ≃ M−1·(2−p) will only depend uponM and p in (4.63). (τ enters into the definition of T4 and T5 below.) The choice of 0 < κ < τ will be specified below. T1 = {(δ, σ) ∈ 2× 2 | |F | (2+κ)(1+κ) ≤ σ} , T2 = {(δ, σ) ∈ 2× 2 | σ < |F | 2−κ , δ ≥ σ1/κ} , T3 = {(δ, σ) ∈ 2× 2 | σ < |F | 2−κ , δ > σ1/κ} , T4 = {(δ, σ) ∈ 2× 2 | |F | 2−κ ≤ σ < |F | (2+κ)(1+κ) , δ > στ} , T5 = {(δ, σ) ∈ 2× 2 | |F | 2−κ ≤ σ < |F | (2+κ)(1+κ) , δ ≤ στ} , T (T ) = (δ,σ)∈T Sum(S1δ,σ) . Note that for T1 we can use the first term in the minimum in (4.63). T (T1) . (δ,σ)∈T1 δσ−1−κ|F | σ≥|F | (2+κ)(1+κ) σ−1−κ|F | . |F |1− 2+κ . This last exponent on |F | is strictly larger than 1 as desired. The point of the definition of T1 is that when it comes time to use the middle term PROOFS OF LEMMATA 47 of the minimum for ℓ = 1 in (4.63), we can restrict attention to the δ−M+1σ−p+1|F | . For the collection T2, use the last term in the minimum in (4.63). T (T2) . (δ,σ)∈T2 σ≤|F | σ log 1/σ . |F | 2−κ/2 . Again, for 0 < κ < 1, the exponent on |F | above is strictly greater than 1/2. The term T3 can be controlled with the first term in the minimum in (4.63). T (T3) . (δ,σ)∈T3 δσ−1−κ|F | σ|F | . |F | . The term T4 is the heart of the matter. It is here that we use the middle term in the minimum of (4.63), and that the role of τ becomes clear. We estimate T (T4) . (δ,σ)∈T4 δ−Mσ−p+1|F | δ≥|F |τ δ−M |F |1− . |F |1− −Mτ . Recall that 1 < p < 2, so that 0 < p − 1 < 1. Therefore, for 0 < κ sufficiently small, of the order of 2− p, we will have 1− p− 1 2− κ > + 2−p Therefore, choosing τ ≃ (2 − p)/M will leave us with a power on |F | that is strictly larger than 1 The previous term did not specify κ > 0. Instead it shows that for 0 < κ < 1 sufficiently small, we can make a choice of τ , that is 48 4. L ESTIMATE FOR Hv independent of κ, for which T (T4) admits the required control. The bound in the last term will specify a choice of κ on us. We estimate T (T5) . (δ,σ)∈T5 δσ−1−κ|F | σ≥|F | σ−1−κ|F |1+τ . |F |1+τ+ Choosing κ = τ/6 will result in the estimate which is as required, so our proof is finished. � Remark 4.65. The resolution of Conjecture 1.21 would depend upon refinements of Lemma 4.50, as well as using the restricted weak type approach of [21]. Proof of Lemma 4.34. We only consider tiles s ∈ AT (ann, scl), and sets ω ∈ Ω which are associated to one of these tiles. For an element a = {as} ∈ ℓ2(AT (ann, scl)), s :ωs=ω asSannαs For |ωs| = |ωs′|, note that dist(ωs,ωs′) is measured in units of scl/ann. By a lemma of Cotlar and Stein, it suffices to provide the estimate ′‖2 . ρ−3, ρ = 1 + dist(ω,ω′). Now, the estimate ‖T ‖2 . 1 is obvious. For the case ω 6= ω′, by Schur’s test, it suffices to see that (4.66) sup s′ :ωs′=ω s :ωs=ω |〈Sannαs, Sannαs′〉| . ρ−3. For tiles s′ and s as above, recall that 〈ϕs, ϕs′〉 = 0, note that |Rs′ ∩ Rs| ann dist(ω,ω′) ≃ ρ−1, and in particular, for a fixed s′, let Ss′ be those s for which ρRs∩ρRs′ 6= ∅. Clearly, card(Ss′) . |ρRs| |2ρRs′ ∩ 2ρRs| ρ ≃ ρ2 PROOFS OF LEMMATA 49 If for r > 1, rRs ∩ rRs′ = ∅, then it is routine to show that |〈Sannαs, Sannαs′〉| . r−10 And so we may directly sum over those s 6∈ Ss′ , s 6∈Ss′ |〈Sannαs, Sannαs′〉| . ρ−3. For those s ∈ Ss′, we estimate the inner product in frequency vari- ables. Recalling the definition of αs = (Tc(Rs)D ζ)ϕs, we have α̂s = (Mod−c(Rs) D ζ̂) ∗ ϕ̂s. Recall that ζ is a smooth compactly supported Schwartz function. We estimate the inner product |〈Ŝannαs, Ŝannαs′〉| without appealing to cancellation. Since we choose the function λ̂ to be supported in an annulus 1 < |ξ| < 3 so that λ̂ann = λ̂(·/ann) is supported in the annulus 1 ann < |ξ| < 3 ann. We can restrict our attention to this same range of ξ. In the region |ξ| > ann/4, suppose, without loss of generality, that ξ is closer to ωs than ωs′. Then since ωs and ωs′ are separated by an amount & anndist(ω,ω |α̂s(ξ)α̂s′(ξ)| . χ(2)ωs (ξ)χ dist(ω,ω) . χ(2)ωs (ξ)χ (ξ)ρ−20. Here, χ is the non–negative bump function in (4.40). Hence, we have the estimate ∫ |λ̂ann(ξ)|2|α̂s(ξ)α̂s′(ξ)|dξ . ρ−10. This is summed over the . ρ2 possible choices of s ∈ Ss′ , giving the estimate ∑ s∈Ss′ |〈Sannαs, Sannαs′〉| . ρ−8 . ρ−3. This is the proof of (4.66). And this concludes the proof of Lemma 4.34. Proof of the Principal Organizational Lemma 4.50. Recall that we are to decompose S into four distinct subsets satisfying the favorable estimates of that Lemma. For the remainder of the proof set dense(S) := δ and size(S) := σ. Take Slight to be all those s ∈ S for which there is no tile s′ ∈ AT of density at least δ/2 for which s . s′. It is clear that this set so constructed has density at most δ/2, that this is a set of tiles, and that S1 := S− Slight is also . 50 4. L ESTIMATE FOR Hv The next Lemma and proof comment on the method we use to obtain the middle estimate in (4.53) which depends upon the Lipschitz Kakeya Maximal Function Conjecture 1.14. It will be used to obtain the important inequality (4.82) below. Lemma 4.67. Suppose we have a collection of trees T ∈ T , with these conditions. a: For T ∈ T there is a 1-tree T1 ⊂ T with (4.68) ∆(T1) dx ≥ κσ , b: Each tree has top element s(T) := ωT×RT of density at least c: The collections of tops {s(T) | T ∈ T } are pairwise incompa- rable under the order relation ‘.’. d: For all T ∈ T , γT = γωT×RT ≥ κ−1/2σ−κ/5N . Here, N is the exponent on δ in Conjecture 1.14. Then we have |RT| . δ−Np−1σ−p(1+κ/4)|F |+ σ1/κδ−1.(4.69) |RT| . δ−1.(4.70) Concerning the role of γT, recall from the definition, (4.29), that γs is a quantity that grows as does the ratio scl(s)/‖v‖Lip, hence there are only . log σ−1 scales of tiles that do not satisfy the assumption d above. Proof. Our primary interest is in (4.69), which is a consequence of our assumption about the Lipschitz Kakeya Maximal Functions, Con- jecture 1.12. s(T) := ωT × σ−κ/10NRT . Let us begin by noting that κ−1‖v‖Lip ≤ scl(s(T)) ≤ κann(s(T)), T ∈ T ,(4.71) dense(s(T)) ≥ δσκ/10N , T ∈ T ,(4.72) |F ∩Rs(T)| ≥ σ1+κ/4N |Rs(T)| .(4.73) The conclusion (4.71) is straightforward, as is (4.72). The inequality (4.73) follows from Lemma 4.55. PROOFS OF LEMMATA 51 Note that the length of σ−κ/10NRT satisfies σ−κ/10N L(RT) ≤ γT L(RT) scl(s) ‖v‖Lip ≤ (100‖v‖Lip)−1 . (4.74) This is the condition (1.8) that we impose in the definition of the Lipschitz Kakeya Maximal Functions. Observe that we can regard ann(s(T)) ≃ σκ/10ann as a constant independent of T. The point of these observations is that our assumption about the Lipschitz Kakeya Maximal Function applies to the maximal function formed over the set of tiles {s(T) | T ∈ T }. And it will be applied below. Let Tk be the collection of trees so that T ∈ Tk if k ≥ 0 is the smallest integer such that (4.75) |(2kRT) ∩ v−1(ωT) ∩G| ≥ 220k/κ 2−1δ|RT| . Then since the density of s(T) for every tree T ∈ T is at least δ, we have T = k=0 Tk. We can apply Conjecture 1.12 to these collections, with the value of δ in that Conjecture being 220k/κ 2−1δ. For each Tk, we decompose it by the following algorithm. Initialize T selectedk ← ∅, T stockk ← Tk . While T stockk 6= ∅, select T ∈ T stockk such that scl(s(T)) is minimal. Define Tk(T) by Tk(T) = {T′ ∈ Tk : (2kRT) ∩ (2kRT′) 6= ∅ and ωT ⊂ ωT′} . Update T selectedk ← T selectedk ∪ {T} , T stockk ← T stockk \Tk(T) . Thus we decompose Tk into T∈T selected T′∈Tk(T) {T′} . And ∑ |RT| = T∈T selected T′∈Tk(T) |RT′| . Notice that RT′ ’s are disjoint for all T ′ ∈ Tk(T) and they are contained in 5(2kRT). This is so, since the tops of the trees are assumed to be incomparable with respect to the order relation ‘.’ on tiles. 52 4. L ESTIMATE FOR Hv Thus we have |RT| . T∈T selected 22k|RT| . δ−12−10k/κ T∈T selected |(2kRT) ∩ v−1(ωT) ∩G| . Observe that (2kRT)∩v−1(ωT)’s are disjoint for all T ∈ T selectedk . This and the fact that |G| ≤ 1 proves (4.70). To argue for (4.69), we see |RT| . δ−12−10k/κ T∈T selected (2kRT) ∩ v−1(ωT) ∩G . δ−12−10k/κ (2kRT) ∩G ∣∣∣∣ . At this point, Conjecture 1.12 enters. Observe that we can estimate ∣∣∣ . |{Mδ′,v,(σκ/10ann)−1 1F > σ1+κ/4N}| . (δ′)−Npσ−p(1+κ/4N)|F |. . (δ)−Np2−kσ−p(1+κ/4N)|F |. (4.76) Here, δ′ = 220k/κ 2−1δ, the choice of δ′ permitted to us by (4.75), and we have used (4.73) in the first line, to pass to the Lipschitz Kakeya Maximal Function. Hence, |sh(T)| . . δ−1 2−10k/κ (2kRT) ∩G . δ−1 k : 1≤2k≤σ−κ/10 (2kRT) + δ−1 k : 2k>σ−κ/10 2−10k/κ 2 |G| . On the first sum in the last line, we use (4.76), and on the second, we just sum the geometric series, and recall that |G| = 1. PROOFS OF LEMMATA 53 We can now begin the principal line of reasoning for the proof of Lemma 4.50. The Construction of S1large. We use an orthogonality, or TT ∗ argu- ment that has been used many times before, especially in [20] and [15]. (There is a feature of the current application of the argument that is present due to the fact that we are working on the plane, and it is detailed by Lacey and Li [15].) We may assume that all intervals ωs are contained in the upper half of the unit circle in the plane. Fix S ⊂ Av, and σ = size(S). We construct a collection of trees T 1large for the collection S1, and a corresponding collection of 1–trees T 1,1large, with particular properties. We begin the recursion by initializing T 1large ← ∅, T large ← ∅, S1large ← ∅, Sstock ← S1. In the recursive step, if size(Sstock) < 1 σ1+κ/100, then this recursion stops. Otherwise, we select a tree T ⊂ Sstock such that three conditions are met. a: The top of the tree s(T) (which need not be in the tree) satisfies dense(s(T)) ≥ δ/4. b: T contains a 1–tree T1 with (4.77) − ∆(T1) dx ≥ 1 σ1+κ/100 . c: And that ωT is in the first place minimal and and in the second most clockwise among all possible choices of T. (Since all ωs are in the upper half of the unit circle, this condition can be fulfilled.) We take T to be the maximal tree in Sstock which satisfies these condi- tions. We then update T 1large ← {T} ∪ Tlarge, T large ← {T1} ∪ T large, S1large ← T ∪ S1large Sstock ← Sstock −T. The recursion then repeats. Once the recursion stops, we update S1 ← Sstock It is this collection that we analyze in the next subsection. Note that it is a consequence of the recursion, and Remark 4.44, that the tops of the trees {s(T) | T ∈ T 1large} are pairwise incomparable under .. 54 4. L ESTIMATE FOR Hv The bottom estimate of (4.53) is then immediate from the construc- tion and (4.70). First, we turn to the deduction of the first estimate of (4.53). Let T 1,(1)large be the set T 1,(1)large = T ∈ T 1large : |〈Sann 1F , βs〉|2 < 116σ 2+κ/50|RT| And let T 1,(2)large be the set T 1,(2)large = T ∈ T 1large : |〈Sann 1F , βs〉|2 ≥ 116σ 2+κ/50|RT| In the inner products, we are taking βs, which is supported off of γsRs. Since T ∈ T 1large satisfies (4.78) − ∆(T) dx ≥ 1 σ1+κ/100 , we have ∑ |〈Sann 1F , αs〉|2 ≥ 14σ 2+κ/50|RT| . Thus, if T ∈ T 1,(1)large , we have |〈1F , ϕs〉|2 ≥ 18σ 2+κ/50|RT| . The replacement of αs by ϕs in the inequality above is an important point for us. That we can then drop the Sann is immediate. With this construction and observation, we claim that (4.79) T∈T 1,(1) large |RT| . (log 1/σ)2σ−2−κ/50|F |. Proof of (4.79). This is a variant of the the argument for the ‘Size Lemma’ in [15], and so we will not present all details. Begin by making a further decomposition of the trees T ∈ T 1,(1)large . To each such tree, we have a 1-tree T1 ⊂ T which satisfies (4.77). We decompose T1. Set T1(0) = s ∈ T1 | |〈f, ϕs〉|√ < σ1+κ/100 T1(j) = s ∈ T1 | 4j−1σ1+κ/100 ≤ |〈f, ϕs〉|√ < 4jσ1+κ/100 1 ≤ j ≤ j0 = C log 1/σ . PROOFS OF LEMMATA 55 Now, set T (j) to be those T ∈ T 1,(1)large for which (4.80) s∈T1(j) |〈f, ϕs〉|2 ≥ (2j0)−1σ2+κ/50|RT| , 0 ≤ j ≤ j0 . It is the case that each T ∈ T 1,(1)large is in some T (j), for 0 ≤ j ≤ j0. The central case is that of j = 0. We can apply the ‘Size Lemma’ of [15] to deduce that T∈T (0) |RT| ≤ (2j0)σ−2−κ/50 T∈T (0) s∈T1(0) |〈f, ϕs〉|2 . (log 1/σ)σ−2−κ/50|F | . The point here is that to apply the argument in the ‘Size Lemma’ one needs an average case estimate, namely (4.80), as well as a uniform control, namely the condition defining T1(0). This proves (4.79) in this case. For 1 ≤ j ≤ j0, we can apply the ‘Size Lemma’ argument to the individual tiles in the collection {T1(j) | T ∈ T (j)} . The individual tiles satisfy the definition of a 1-tree. And the defining condition of T1(j) is both the average case estimate, and the uniform control needed to run that argument. In this case we conclude that T∈T (j) s∈T1(j) |〈f, ϕs〉|2 . |F | . Thus, we can estimate T∈T (j) |RT| . (log 1/σ)σ−1−κ/50|F | . This summed over 1 ≤ j ≤ j0 = C log 1/σ proves (4.79). � For T 1,(2)large , we have T∈T 1,(2) large |RT| . σ−2−κ/50 scl≥κ−1‖v‖Lip s:scl(s)=scl |〈Sann 1F , βs〉|2 . σ−2−κ/50|F | scl≥κ−1‖v‖Lip ‖v‖Lip . σ−2−κ/50|F | , 56 4. L ESTIMATE FOR Hv since βs has fast decay. The Bessel inequality in the last display can be obtained by using the same argument in the proof of Lemma 4.34. Hence we get (4.81) T∈T 1,(2) large |RT| . σ−2−κ/50|F |. Combining (4.79) and (4.81), we obtain the first estimate of (4.53). Second, we turn to the deduction of the middle estimate of (4.53), which relies upon the Lipschitz Kakeya Maximal Function. Let T 1,goodlarge be the set T ∈ T 1large : γT ≥ κ−1/2σ−κ/5N And let T 1,badlarge be the set T ∈ T 1large : γT < κ−1/2σ−κ/5N The ‘good’ collection can be controlled by facts which we have already marshaled together. In particular, we have been careful to arrange the construction so that Lemma 4.67 applies. By the main conclusion of that Lemma, (4.69), we have (4.82) T∈T 1,good large |RT| . δ−Mσ−1−3κ/4|F |+ σ1/κδ−1 . Here, M is a large constant that only depends upon N in Conjec- ture 1.14. For T ∈ T 1,badlarge , there are at most K = O(log(σ−κ)) many possible scales for scl(ωT × RT). Let scl(T) = scl(ωT × RT). Thus we have T∈T 1,bad large |RT| . T:scl(T)=2mκ−1‖v‖Lip |RT| . Since T satisfies (4.78), we have |F ∩ γTRT| & σ1+κ/2|RT| . Thus, we get T∈T 1,bad large |RT| . σ−1−κ/2 T:scl(T)=2mκ−1‖v‖Lip 1σ−κRT(x)dx . PROOFS OF LEMMATA 57 For the tiles with a fixed scale, we have the following inequality, which is a consequence of Lemma 4.25. T:scl(T)=2mκ−1‖v‖Lip 1σ−κRT . σ−κ/5δ−1 . Hence we obtain (4.83) T∈T 1,bad large |RT| . δ−1σ−1−3κ/4|F | . Combining (4.82) and (4.83), we obtain the middle estimate of (4.53). Therefore, we complete the proof of (4.53). The Construction of S2large. It is important to keep in mind that we have only removed trees of nearly maximal size, with tops of a given density. In the collection of tiles that remain, there can be trees of large size, but they cannot have a top with nearly maximal density. We repeat the TT∗ construction of the previous step in the proof, with two significant changes. We construct a collection of trees T 2large from the collection S1, and a corresponding collection of 1–trees T 2,1large, with particular properties. We begin the recursion by initializing T 2large ← ∅ , T large ← ∅ , S2large ← ∅ , Sstock ← S1 . In the recursive step, if size(Sstock) < σ/2, then this recursion stops. Otherwise, we select a tree T ⊂ Sstock such that two conditions are a: T satisfies ‖∆(T)‖2 ≥ σ2 |RT| b: ωT is both minimal and most clockwise among all possible choices of T. We take T to be the maximal tree in Sstock which satisfies these condi- tions. We take T1 ⊂ T to be a 1–tree so that (4.84) − ∆(T1) dx ≥ κσ . This last inequality must hold by Lemma 4.49. We then update T 2large ← {T} ∪ Tlarge, T large ← {T1} ∪ T large, Sstock ← Sstock −T. The recursion then repeats. 58 4. L ESTIMATE FOR Hv Once the recursion stops, it is clear that the size of Sstock is at most σ/2, and so we take Ssmall := Sstock. The estimate ∑ T∈T 2 large |RT| . σ−2|F | then is a consequence of the TT ∗ method, as indicated in the previous step of the proof. That is the first estimate claimed in (4.54). What is significant is the second estimate of (4.54), which involves the density. The point to observe is this. Consider any tile s of density at least δ/2. Let Ts be those trees T ∈ T 2large with top ωs(T) ⊃ ωs and Rs(T) ⊂ KRs. By the construction of S1large, we must have ∆(T1) dx ≤ σ1+κ/100 , for the maximal 1–tree T 1 contained in T∈Ts T. But, in addition, the tops of the trees in T 2large are pairwise incomparable with respect to the order relation ‘.,’ hence we conclude that |RT| . σ2+κ/50|Rs|. Moreover, by the construction of Slight, for each T ∈ T 2large we must be able to select some tile s with density at least δ/2 and ωs(T) ⊃ ωs and Rs(T) ⊂ KRs. Thus, we let S∗ be the maximal tiles of density at least δ/2. Then, the inequality (4.70) applies to this collection. And, therefore, T∈T 2 large |RT| ≤ σκ/50 |Rs| . σκ/50δ−1. This completes the proof of second estimate of (4.54). � The Estimates For a Single Tree. The Proof of Lemma 4.55. It is a routine matter to check that for any 1–tree we have ∑ |〈f, ϕs〉|2 . ‖f‖22. Indeed, there is a strengthening of this estimate relevant to our concerns here. Recalling the notation (4.40), we have (4.85) |〈f, ϕs〉|2 ]1/2∥∥∥ . ‖χ(∞)RT f‖p , 1 < p <∞ . PROOFS OF LEMMATA 59 This is variant of the Littlewood-Paley inequalities, with some addi- tional spatial localization in the estimate. Using this inequality for p = 1 + κ/100 and the assumption of the Lemma, we have σ1+κ/100 ≤ ∆T dx ]1+κ/100 1+κ/100 ≤ |RT|−1 |〈f, ϕs〉|2 ]1/2∥∥∥ 1+κ/100 1+κ/100 . |RT|−1 dx.(4.86) This inequality can only hold if |F ∩ σ−κRT| ≥ σ1+κ|RT|. � The Proof of Lemma 4.57. This Lemma is closely related to the Tree Lemma of [15]. Let us recall that result in a form that we need it. We need analogs of the definitions of density and size that do not incorporate truncations of the various functions involved. Define dense(s) := G∩v−1(ωs) (x) dx. (Recall the notation from (4.40).) dense(T) := sup dense(s). Likewise define size(T) := sup ′ is a 1–tree |RT′|−1 |〈1F , ϕs〉|2 Then, the proof of the Tree Lemma of [15] will give us this inequality: For T a tree, (4.87) |〈Sann 1F , ϕs〉〈φs, 1G〉| . dense(T) size(T)|RT|. Now, consider a tree T with dense(T) = δ, and size(T) = σ, where we insist upon using the original definitions of density and size. If in addition, γs ≥ K(σδ)−1 for all s ∈ T, we would then have the inequalities dense(T) . δ, size(T) . σ, 60 4. L ESTIMATE FOR Hv This places (4.87) at our disposal, but this is not quite the estimate we need, as the functions ϕs and φs that occur in (4.87) are not truncated in the appropriate way, and it is this matter that we turn to next. Recall that ϕs = αs + βs , αs(x− yv(x))ψs(y) dy = αs−(x) + αs+(y) . One should recall the displays (4.30), (4.31), and (4.38). As an immediate consequence of the definition of βs, we have∫ |βs(x)| dx . γ−2s |Rs|. Hence, if we replace ϕs by βs, we have |〈Sann 1F , βs〉〈φs, 1G〉| . |Rs||〈φs, 1G〉| γ−1s |Rs| . σδ|RT|. And by a very similar argument, one sees corresponding bounds, in which we replace the φs by different functions. Namely, recalling the definitions of as± in (4.31) and estimate (4.38), we have |Rs||〈as+, 1G〉| . σ (‖v‖Lip scl(s) (x) dx(4.88) (‖v‖Lip scl(s) . σδ|RT| . Similarly, we have |Rs||〈φs − as+ − as−, 1G〉| . σδ|RT|, Putting these estimates together proves our Lemma, in particular (4.58), under the assumption that γs ≥ K(σδ)−1 for all s ∈ T. Assume that T is a tree with scl(s) = scl(s′) for all s, s′ ∈ T. That is, the scale of the tiles in the tree is fixed. Then, T is in particular a 1– tree, so that by an application of the definitions and Cauchy–Schwartz, |〈Sann 1F , αs〉〈as−, 1G〉| ≤ δ |〈Sann 1F , αs〉| ≤ δσ|RT|. PROOFS OF LEMMATA 61 But, γs ≥ 1 increases as does scl(s). Thus, any tree T with γs ≤ K(σδ)−1 for all s ∈ T, is a union of O(|log δσ|) trees for which the last estimate holds. � CHAPTER 5 Almost Orthogonality Between Annuli Application of the Fourier Localization Lemma We are to prove Lemma 4.23, and in doing so rely upon a technical lemma on Fourier localization, Lemma 5.56 below. We can take a choice of 1 < α < 9 , and assume, after a dilation, that ‖v‖Cα = 1. The first inequality we establish is this. Lemma 5.1. Using the notation of of Lemma 4.23, and assuming that ‖v‖Cα . 1, we have the estimate ‖C‖2 . 1, where ann≥1 where the Cann are defined in (4.21). We have already established Lemma 4.22, and so in particular know that ‖Cann‖2 . 1. Due to the imposition of the Fourier restriction in the definition of these operators, it is immediate that CannC∗ann′ ≡ 0 for ann 6= ann′. We establish that Cann′‖2 . max(ann, ann′)−δ , δ = 1 (α− 1) , |log ann(ann′)−1| > 3 . (5.2) Then, it is entirely elementary to see that C is a bounded operator. Let Pann be the Fourier projection of f onto the frequencies ann < |ξ| < 2ann. Observe, ‖Cf‖22 = ann≥1 Cann Pann f ann≥1 ann′>1 〈Cann Pann f, Cann′ Pann′ f〉 ≤ 2‖f‖2 ann≥1 ann′>1 Cann′ Pann′ f‖2 . ‖f‖22 ann≥1 ann′>1 max(ann, ann′)−δ . ‖f‖22. 64 5. ALMOST ORTHOGONALITY There are only O(log ann) possible values of scl that contribute to Cann, and likewise for Cann′. Thus, if we define (5.3) Cann,sclf = s∈AT (ann) scl(s)=scl 〈f, ϕs〉φs , it suffices to prove Lemma 5.4. Using the notation of of Lemma 4.23, and assuming that ‖v‖Cα . 1, we have ann,sclCann′,scl′‖2 . (max(ann, ann′))−δ . Here, we can take δ′ = 1 (α − 1), and the inequality holds for all |log ann(ann′)−1| > 3, 1 < scl ≤ ann and 1 < scl′ ≤ ann′. Proof of Lemma 4.23. In this proof, we assume that Lemma 5.1 and Lemma 5.4 are established. The first Lemma clearly establishes the first (and more important) claim of the Lemma. Let us prove the inequality (4.24). Using the notation of this sec- tion, this inequality is as follows. (5.5) ann=−∞ Cann,scl . (1 + log(1 + scl−1‖v‖Cα)). This inequality holds for all choices of Cα vector fields v. Note that Lemma 5.4 implies immediately ann=3 Cann,scl . 1 , ‖v‖Cα = 1 . We are however in a scale invariant situation, so that this inequality implies this equivalent form, independent of assumption on the norm of the vector field. (5.6) ann≥8‖v‖Cα Cann,scl . 1 . On the other hand, Lemma 4.25, implies that independent of any assumption other than measurability, we have have the inequality ‖Cann,scl‖2 . 1 . To prove (5.5), use the inequality (5.6), and this last inequality together with the simple fact that for a fixed value of scl, there are at most . 1 + log(1 + scl−1‖v‖Cα)) values of ann with scl ≤ ann ≤ 8‖v‖Cα. APPLICATION OF THE FOURIER LOCALIZATION LEMMA 65 We use the notation AT (ann, scl) := {s ∈ AT (ann) : scl(s) = scl} , Observe that as the scale is fixed, we have a Bessel inequality for the functions {ϕs | s ∈ AT (ann, scl)}. Thus, ann,sclCann′,scl′f‖22 = s∈AT (ann,scl) s∈AT (ann′,scl′) 〈φs, φs′〉〈ϕs′, f〉ϕs s∈AT (ann,scl) s∈AT (ann′,scl′) 〈φs, φs′〉〈ϕs′, f〉 At this point, the Schur test suggests itself, and indeed, we need a quantitative version of the test, which we state here. Proposition 5.7. Let A = {ai,j} be a matrix acting on ℓ2(N) by ai,jxj Then, we have the following bound on the operator norm of A. ‖A‖2 . sup |ai,j| · sup |ai,j| We assume that 1 ≤ ann < 1 ′. For a subset S ⊂ AT (ann, scl)× AT (ann′, scl′) Consider the operator and definitions below. AS f = (s,s′)∈S 〈φs, φs′〉〈ϕs′, f〉ϕs , FL(s,S) = s′∈AT (ann′,scl′) |〈φs, φs′〉| , FL(S) = sup FL(s,S) . Here ‘FL’ is for ‘Fourier Localization’ as this term is to be controlled by Lemma 5.56. We will use the notations FL(s′,S), and FL′(S), which are defined similarly, with the roles of s and s′ reversed. By Proposition 5.7, we have the inequality (5.8) ‖AS‖22 . FL(S) · FL′(S) . We shall see that typically FL(S) will be somewhat large, but is bal- anced out by FL′(S). We partition AT (ann, scl) × AT (ann′, scl′) into three disjoint sub- collections Su, u = 1, 2, 3, defined as follows. In this display, (s, s′) ∈ 66 5. ALMOST ORTHOGONALITY AT (ann, scl)×AT (ann′, scl′). (s, s′) | scl ≥ scl ,(5.9) (s, s′) | scl , scl < scl′ ,(5.10) (s, s′) | scl , scl′ < scl .(5.11) A further modification to these collections must be made, but it is not of an essential nature. For an integer j ≥ 1, and (s, s′) ∈ Su, for u = 1, 2, 3, write (s, s′) ∈ Su,j if j is the smallest integer such that 2j+2Rs ∩ 2j+2Rs′ 6= ∅. We apply the inequalities (5.8) to the collections Su,j, to prove the inequalities (5.12) ‖ASu,j‖2 . 2−j(ann′)−δ where δ′ = 1 (α − 1). This proves Lemma 5.4, and so completes the proof of Lemma 5.1. In applying (5.8) it will be very easy to estimate FL(s,S), with a term that decreases like say 2−10j. The difficult part is to estimate either FL(s,S) or FL′(S) by a term with decreases faster than a small power of (ann′)−1. for which we use Lemma 5.56. Considering a term 〈φs, φs′〉, the inner product is trivially zero if ωs ∩ ωs′ = ∅. We assume that this is not the case below. To apply Lemma 5.56, fix e ∈ ω′s ∩ωs. Let α be a Schwartz function on R with α̂ supported on [ann′, 2ann′], and identically one on 3 [ann′, 2ann′]. Set β̂(θ) := α̂(θ − 3 ′). We will convolve φs with β in the direction e, and φs′ with α also in the direction e, thereby obtaining orthogonal functions. Define Ie g(x) = g(x− ye)β(y) dy,(5.13) ∆s = φs − Ie φs ∆s′ = φs′ − Ie φs′ (5.14) By construction, we have 〈φs, φs′〉 = 〈Ie φs +∆s, Ie φs′ +∆s′〉 = 〈Ie φs,∆s′〉+ 〈∆s, Ie φs′〉+ 〈∆s,∆s′〉 . APPLICATION OF THE FOURIER LOCALIZATION LEMMA 67 It falls to us to estimate terms like s∈Sℓ,j |〈∆s, Ie φs′〉|,(5.15) s∈Sℓ,j |〈Ie φs,∆s′〉|,(5.16) s∈Sℓ,j |〈∆s,∆s′〉|.(5.17) as well as the dual expressions, with the roles of s and s′ reversed. The differences ∆s and ∆s′ are frequently controlled by Lemma 5.56. Concerning application of this Lemma to ∆s, observe that Mod−c(ωs)∆s = Mod−c(ωs) φs − [Mod−c(ωs) φs(x− ye)]β̃(y) dy where β̃(y) = e(c(ωs)·e)y β(y). Now the Fourier transform of β is identi- cally one in a neighborhood of the origin of width comparable to ann′, where as |c(ωs) · e| is comparable to ann. Since we can assume that ′ > ann+3, say, the function β̃ meets the hypotheses of Lemma 5.56, namely it is Schwarz function with Fourier transform identically one in a neighborhood of the origin, and the width of that neighborhood is comparable to ann′. And so ∆s is bounded by the bounded by the three terms in (5.57)—(5.59) below. In these estimates, we take 2k ≃ ann′ > 1. By a similar argument, one sees that Lemma 5.56 also applies to ∆s′. We will let ∆s,m, for m = 1, 2, 3, denote the terms that come from (5.57), (5.58), and (5.59) respectively. We use the corresponding no- tation for ∆s,m, for m = 1, 2, 3. A nice feature of these estimates, is that while ∆s and ∆s′ depend upon the choice of e ∈ ωs′ ∩ ωs, the upper bounds in the first two estimates do not depend upon the choice of e. While the third estimate does, the dependence of the set Fs on the choice of e is rather weak. In application of (5.58), the functions ∆s,2 will be very small, due to the term (ann′)−10 which is on the right in (5.58). This term is so much smaller than all other terms involved in this argument that these terms are very easy to control. So we do not explicitly discuss the case of ∆s,2, or ∆s′,2 below. In the analysis of the terms (5.15) and (5.16), we frequently only need to use an inequality such as |Ie φs′| . χ(2)Rs′ . When it comes to the analysis of (5.17), the function ∆s′ obeys the same inequality, so that 68 5. ALMOST ORTHOGONALITY these sums can be controlled by the same analysis that controls (5.15), or (5.16). So we will explicitly discuss these cases below. In order for 〈φs, φs′〉 6= ∅, we must necessarily have ωs ∩ ωs′ 6= ∅. Thus, we update all Sℓ,j as follows. Sℓ,j ← {(s, s′) ∈ Sℓ,j | ωs ∩ ωs′ 6= ∅} . The Proof of (5.12) for S1,j, j ≥ 1. Recall the definition of S1,j from (5.9). In particular, for (s, s′) ∈ S1,j , we must have ωs ⊂ ωs′. We will use the inequality (5.8), and show that for 0 < ǫ < 1, FL(S1,j) . 2−10j(ann′)−eα ′ · ann′ scl · ann(5.18) FL′(S1,j) . 22j(ann′)ǫ · scl · ann ′ · ann′ .(5.19) Notice that in the second estimate, we permit some slow increase in the estimates as a function of 2j and ann′. But, due to the form of the estimate of the Schur test in (5.8), this slow growth is acceptable. The terms inside the square root in these two estimates cancel out. These inequalities conclude the proof of the inequality (5.12) for the collection S1,j , j ≥ 1. We prove (5.18). For this, we use Lemma 5.56. That is, we should bound the several terms s′ : (s,s′)∈S1,j |〈∆s, Ie φs′〉| ,(5.20) s′ : (s,s′)∈S1,j |〈Ie φs,∆s′〉| ,(5.21) s′ : (s,s′)∈S1,j |〈∆s,∆s′〉| .(5.22) Here ∆s and ∆s′ are as in (5.14). And, Ie is defined as in (5.13). We can regard the tile s as fixed, and so fix a choice of e ∈ ωs. In the next two cases, we will need to estimate the same expressions as above. In all three cases, Lemma 5.56 is applied with 2k ≃ ann′, and we can take ǫ in this Lemma to be ǫ = 1 (α− 1). For ease of notation, we set (5.23) α̃ = (α− 1)(1− ǫ)2 − ǫ > 0 As we have already mentioned, we do not explicitly discuss the upper bound on the estimate for (5.22). APPLICATION OF THE FOURIER LOCALIZATION LEMMA 69 The Upper Bound on (5.20). We write ∆s = ∆s,1 + ∆s,2 + ∆s,3, where these three terms are those on the right in (5.57)—(5.59) respec- tively. Note that (5.24) |Ie φs′| . χ(2)Rs′ , since Ie is convolution in the long direction of Rs′ , at the scale of (ann′)−1, which is much smaller than the length of Rs′ in the direc- tion e. Therefore, we can estimate the term in (5.20) by s′ : (s,s′)∈S1,j |〈∆s,1, Ie φs′〉| . (ann′)−eα2−10j s′ : (s,s′)∈S1,j . (ann′)−eα2−10j ′ · ann′ scl · ann .(5.25) This is as required to prove (5.18) for these sums. For the terms associated with ∆s,3, we have s′ : (s,s′)∈S1,j |〈∆s,3, Ie φs′〉| . s′ : (s,s′)∈S1,j |Rs|−1/2 · χ(2)R′s dx . 2−10j|Fs| ann′ · scl′ · ann · scl . 2−10j(ann′)−α+ǫ ′ · ann′ scl · ann . That is, we only rely upon the estimate (5.60). This completes the analysis of (5.20). (As we have commented above, we do not explicitly discuss the case of ∆s,2.) The Upper Bound for (5.21). Since ωs ⊂ ωs′ , the only facts about ∆s′ we need are (ann′)ǫR′s |∆s′| dx . (ann′)−eα+ǫ ′ · ann′ , |∆s′(x)| . (ann′)−eαχ(2)Rs′ (x) , x 6∈ (ann ′)ǫRs′ . (5.26) Indeed, this estimate is a straightforward consequence of the various conclusions of Lemma 5.56. (We will return to this estimate in other cases below.) 70 5. ALMOST ORTHOGONALITY These inequalities, with |Ie φs| . χ(2)Rs , permit us to estimate (5.21) . 2−20j |Rs|−1/2 s′ : (s,s′)∈S1,j (ann′)ǫR′s |∆s′| dx . 2−20j(ann′)−eα scl · ann ′ · ann′ × ♯{s ′ : (s, s′) ∈ S1,j} . 2−20j(ann′)−eα ′ · ann′ scl · ann . which is the required estimate. Here of course we use the estimate ♯{s′ : (s, s′) ∈ S1,j} . 22j ′ · ann′ scl · ann . We now turn to the proof of (5.19), where it is important that we justify the small term √ scl · ann ′ · ann′ on the right in (5.19). We estimate the terms dual to (5.20)—(5.22), namely s : (s,s′)∈S1,j |〈∆s, Ies φs′〉| ,(5.27) s : (s,s′)∈S1,j |〈Ies φs,∆s′〉| ,(5.28) s : (s,s′)∈S1,j |〈∆s,∆s′〉| .(5.29) Here, for each choice of tile s, we make a choice of es ∈ ωs ⊂ ωs′. The Upper Bound on (5.27). We have an inequality analogous to (5.24). (5.30) |Ies φs′| . χ Note that as we can view s′ as fixed, all the tiles {s : (s, s′) ∈ S1,j} have the same approximate spatial location. Let us single out a tile s0 in this collection. Then, for all s, we have Rs ⊂ 2j+2Rs0 . Recalling the specific information about the support of the functions of ∆s from (5.57), (5.59) and (5.61), it follows that s : (s,s′)∈S1,j |∆s| . 22j(ann′)ǫχ(2)2j+2Rs0 . APPLICATION OF THE FOURIER LOCALIZATION LEMMA 71 In particular, we do not claim any decay in ann′ in this estimate. (The small growth of (ann′)ǫ above arises from the overlapping supports of the functions ∆s, as detailed in Lemma 5.56.) Therefore, we can esti- s : (s,s′)∈S1,j |〈∆s, Ies φs′〉| . 22j(ann′)ǫ 2jRs0 . 2−10j(ann′)ǫ scl · ann ′ · ann′ . This is as required in (5.19). Remark 5.31. It is the analysis of the term s : (s,s′)∈S1,j |〈∆s,3, Ies φs′〉| which prevents us from obtaining a decay in ann′, at least in some choices of the parameters scl , ann , scl′, and ann′. The Upper Bound on (5.28). The fact about ∆s′ we need is the simple inequality |∆s′| . χ(2)Rs′ . As in the previous case, we turn to the fact that all the tile {s : (s, s′) ∈ S1,j} have the same approximate spatial location. Single out a tile s0 in this collection, so that Rs ⊂ 2j+2Rs0 for all such s. Our claim is that (5.32) s : (s,s′)∈S1,j |Ies φs| . 22jχ 22jRs (We will have need of related inequalities below.) Suppose that s ∈ {s : (s, s′) ∈ S1,j}. These intervals all have the same length, namely scl/ann. And x 6∈ supp(φs) implies v(x) 6∈ ωs, so that by the Lipschitz assumption on the vector field dist(x, supp(φs)) & dist(v(x),ωs) . This means that (5.33) |Ies φs(x)| . χ 1 + ann′ · dist(v(x),ωs) Here, we recall that the operator Ie is dominated by the operator which averages on spatial scale (ann′)−1 in the direction e. Moreover, we have (5.34) ann′ · dist(ωs,ω) & scl . Here, we partition the unit circle into disjoint intervals ω ∈ Ω of length |ω| ≃ scl/ann, so that for all s ∈ {s : (s, s′) ∈ S1,j}, we have ωs ∈ Ω. 72 5. ALMOST ORTHOGONALITY Figure 5.1. The relative positions of Rs and Rs′ in for pairs (s, s′) ∈ Sℓ, for ℓ = 2 and ℓ = 3 respectively. In fact, the term on the left in (5.34) can be taken to be integer multiples of scl. Combining these observations proves (5.32). Indeed, we can estimate the term in (5.32) as follows. For x, fix ω ∈ Ω with v(x) ∈ ω. Then, s : (s,s′)∈S1,j |Ies φs| . s : (s,s′)∈S1,j 1 + ann′ · dist(ω,ωs) The important point is that the term involving the distance allows us to sum over the possible values of ωs ⊂ ωs′ to conclude (5.32). To finish this case, we can estimate s : (s,s′)∈S1,j |〈Ies φs,∆s′〉| . 2−10j scl · ann ′ · ann′ . This completes the upper bound on (5.28). The Proof (5.12) for S2,j, j ≥ 1. In this case, note that the assumptions imply that we can assume that ωs′ ⊂ ωs, and that di- mensions of the rectangle Rs′ are smaller than those for Rs in both directions. See Figure 5. We should show these two inequalities, in analogy to (5.18) and (5.19). FL(S2,j) . 2−10j(ann′)−eα ′ · ann′ scl · ann(5.35) FL′(S2,j) . 2−10j(ann′)−eα · scl · ann ′ · ann′ .(5.36) Here, α̃ is as in (5.23). APPLICATION OF THE FOURIER LOCALIZATION LEMMA 73 For the proof of (5.35), we should analyze the sums s′ : (s,s′)∈S2,j |〈∆s, Ies′ φs′〉| ,(5.37) s′ : (s,s′)∈S2,j |〈Ies′ φs,∆s′〉| ,(5.38) s′ : (s,s′)∈S2,j |〈∆s,∆s′〉| .(5.39) These inequalities are in analogy to (5.20)—(5.22), and es′ ∈ ωs′ ⊂ ωs. The Upper Bound on (5.37). Fix the tile s. Fix a translate Rs of Rs with 2 jRs ∩ 2jRs = ∅, but 2j+1Rs ∩ 2j+2Rs 6= ∅. Let us consider (5.40) S2,j = {(s, s′) ∈ S2,j | Rs′ ⊂ Rs} and we restrict the the sum in (5.37) to this collection of tiles. Note that with . 22j choices of Rs, we can exhaust the collection S2,j . So we will prove a slightly stronger estimate in the parameter 2j for the restricted collection S2,j. The point of this restriction is that we can appeal to an inequality similar to (5.32). Namely, (5.41) s′ : (s,s′)∈S2,j |Ie′s φs′| . ′ · ann′ scl · ann χ Note that the term in the square root takes care of the differing L2 normalizations of φs′ and χ . Indeed, the proof of (5.32) is easily modified to give this inequality. Next, we observe that the analog of (5.26) holds for ∆s. Just replace s′ in (5.26) with s. It is a consequence that we have s′ : (s,s′)∈S2,j |〈∆s, Ies′ φs′〉| . 2 −12j(ann′)−eα ′ · ann′ scl · ann . This is enough to finish this case. The Upper Bound on (5.38). Let us again appeal to the notations Rs and S2,j as in (5.40). We have the estimates |Ies′ φs| . χ 74 5. ALMOST ORTHOGONALITY As for the sum over ∆s′ , we have an analog of the estimates (5.26). Namely, s′ : (s,s′)∈S2,j (ann′)ǫRs |∆s′,1| dx . (ann′)−eα+ǫ ′ · ann′ scl · ann s′ : (s,s′)∈S2,j |∆s′,1| . ′ · ann′ scl · ann χ , x 6∈ (ann′)ǫRs . Note that we again have to be careful to accommodate the different normalizations here. The proof of (5.26) can be modified to prove this estimate. Putting these two estimates together clearly proves that s′ : (s,s′)∈S2,j |〈Ies′ φs,∆s′〉| . 2 −10j(ann′)−eα ′ · ann′ scl · ann , as is required. We now turn to the proof of the inequality (5.36), which will follow from appropriate upper bounds on the sums below. s : (s,s′)∈S2,j |〈∆s, Ie φs′〉| ,(5.42) s : (s,s′)∈S2,j |〈Ie φs,∆s′〉| ,(5.43) s : (s,s′)∈S2,j |〈∆s,∆s′〉| .(5.44) Here, we can regard s′ as a fixed tile, and e ∈ ωs′ ⊂ ωs. In this case, observe that we have the inequality (5.45) ♯{s : (s, s′) ∈ S1,j} . 22j . This is so since Rs has larger dimensions in both directions than does The Upper Bound on (5.42). We use the decomposition of ∆s = ∆s,1 +∆s,2 +∆s,3. In the first case, we can estimate s : (s,s′)∈S2,j |〈∆s,1, Ie φs′〉| . 22j sup|〈∆s,1, Ie φs′〉| . 2−10j(ann′)−eα scl · ann ′ · ann′ . APPLICATION OF THE FOURIER LOCALIZATION LEMMA 75 For the last case, of ∆s,3, we estimate s : (s,s′)∈S2,j |〈∆s,3, Ie φs′〉| . 22j sup|〈∆s,3, Ie φs′〉| . 22j min |Fs| · scl · ann · scl′ · ann′ , 2−30j scl · ann ′ · ann′ Examining the two terms of the minimum, note that by (5.61), |Fs| · scl · ann · scl′ · ann′ . (ann′)−α+ǫ ′ · ann′ scl · ann . (ann′)−α+1+ǫ ′ · ann′ · scl · ann . (ann′)−α+1+ǫ scl · ann ′ · ann′ . Here it is essential that we have the estimate (5.60) as stated, with |Fs| . (ann′)−α+ǫ|Rs|. This is an estimate of the desired form, but without any decay in the parameter j. The second term in the min- imum does have the decay in j, but does not have the decay in ann′. Taking the geometric mean of these two terms finishes the proof, pro- vided (α−ǫ)/2 > α̃, which we can assume by taking α sufficiently close to one. The Upper Bound on (5.43). Using the inequality |Ie φs| . χ(2)Rs , and the inequalities (5.26) and (5.45), it is easy to see that s : (s,s′)∈S2,j |〈Ie φs,∆s′〉| . 2−12j(ann′)−eα scl · ann ′ · ann′ . This is the required estimate. The Proof of (5.12) for S3,j, j ≥ 1. In this case, we have that the length of the rectangles Rs′ are greater than those of the rectangles Rs, as depicted in Figure 5. We show that FL(S3,j) . (ann′)ǫ ′ · ann′ scl · ann(5.46) FL′(S3,j) . 2−10j(ann′)−eα scl · ann ′ · ann′ .(5.47) In particular, we do not claim any decay in the term FL(S3,j), in fact permitting a small increase in the parameter ann′. Recall that 0 < ǫ < 1 is a small quantity. See (5.23). But due to the form of the estimate in 76 5. ALMOST ORTHOGONALITY Proposition 5.7, with the decay in 2j and ann′ in the estimate (5.47), these two estimates still prove (5.12) for S3,j . For the proof of (5.46), we analyze the sums s′ : (s,s′)∈S3,j |〈∆s, Ies′ φs′〉| ,(5.48) s′ : (s,s′)∈S3,j |〈Ies′ φs,∆s′〉| ,(5.49) s′ : (s,s′)∈S3,j |〈∆s,∆s′〉| .(5.50) Here, es′ ∈ ωs′ ⊂ ωs. The Upper Bound on (5.48). Regard s as fixed. We employ a vari- ant of the notation established in (5.40). Let R̃s be a rectangle with in the same coordinates axes as Rs. In the direction es, let it have length 1/scl′, that is the (longer) length of the rectangles Rs′, and let it have the same width of Rs. Further assume that 2 jRs ∩ R̃s = ∅ but 2j+4Rs ∩ R̃s 6= ∅. (There is an obvious change in these requirements for j = 1.) Then, define S̃3,j = {(s, s′) ∈ S3,j | Rs′ ⊂ R̃s} . With . 22j choices of R̃s, we can exhaust the collection S3,j . Thus, we prove a slightly stronger estimate in the parameter 2j for the collection S̃3,j. The main point here is that we have an analog of the estimate (5.32): s′ : (s,s′)∈ eS3,j |Ie φs′| . The term in the square root takes into account the differing L2 normal- izations between the φs′ and χ . The proof of (5.32) can be modified to prove the estimate above. We also have the analogs of the estimate (5.26). Putting these two together proves that s′ : (s,s′)∈S3,j |〈∆s, Ies′ φs′〉| . 2 −10j(ann′)−eα . 2−10j(ann′)−eα ′ · ann′ scl · ann . APPLICATION OF THE FOURIER LOCALIZATION LEMMA 77 That is, we get the estimate we want with decay in ann′, we do not claim in general. The Upper Bound on (5.49). We use the inequality |Ies′ φs| . χ And we use the decomposition ∆s′ = ∆s′,1 +∆s′,2 +∆s′,3. For the case of ∆s′,1, we have ωs′ ⊂ ωs. And the supports of the functions ∆s′ are well localized with respect to the vector field. See (5.57). Thus, in particular we have s′ : (s,s′)∈S3,j |∆s′| . (ann′)ǫ ′ · ann′ . Hence, we have s′ : (s,s′)∈S3,j |〈χ(2)Rs ,∆s′〉| . (ann ′ · ann′ scl · ann which is the desired estimate. Remark 5.51. It is the analysis of the sum s′ : (s,s′)∈S3,j |〈Ie φs,∆s′,3〉| that prevents us from obtaining decay in the parameter ann′ for cer- tain choices of parameters scl , ann , scl′ and ann′. This is why we have formulated (5.46) the way we have. For the proof of (5.47), we analyze the sums s : (s,s′)∈S3,j |〈∆s, Ie φs′〉| ,(5.52) s : (s,s′)∈S3,j |〈Ie φs,∆s′〉| ,(5.53) s : (s,s′)∈S3,j |〈∆s,∆s′〉| .(5.54) Here es′ ∈ ωs′ ⊂ ωs, and one can regard the interval ωs′ as fixed. It is essential that we obtain the decay in 2j and ann′ in these cases. Indeed, these cases are easier, as the sum is over s. For fixed s′, there is a unique choice of interval ωs ⊃ ωs′. And the rectangles Rs are shorter than Rs′, but wider. Hence, (5.55) ♯{s : (s, s′) ∈ S3,j} . 22j 78 5. ALMOST ORTHOGONALITY The Upper Bound on (5.52). We use the decomposition ∆s = ∆s,1+ ∆s,2 +∆s,3, and the inequality |Ies′ φs′, | . χ For the sum associated with ∆s,1, we have s : (s,s′)∈S3,j |〈∆s,1, Ies′ φs′〉| . (ann ′)−eα s : (s,s′)∈S3,j 〈χ(2)Rs , χ . 2−12j(ann′)−eα ′ · ann scl · ann′ × ♯{s : (s, s′) ∈ S3,j} . 2−10j(ann′)−eα ′ · ann′ scl · ann . This is the required estimate. For the sum associated with ∆s,3, the critical properties are those of the corresponding sets Fs, described in (5.60) and (5.61). Note that the sets ∑ s : (s,s′)∈S3,j 1Fs . (ann ′)2ǫ . On the other hand, s : (s,s′)∈S3,j |Fs| . 22j ′ sup s : (s,s′)∈S3,j . 22j(ann′)−α+ǫ ′ · ann . Here, we have used the estimate (5.55). This permits us to estimate s : (s,s′)∈S3,j |〈∆s,3, Ies′ φs′〉| . 2 −10j(ann′)−α+3ǫ scl · ann′ ′ · ann Note that the parity between the ‘primes’ is broken in this estimate. By inspection, one sees that this last term is at most . 2−10j(ann′)−eα ′ · ann′ scl · ann . Indeed, the claimed inequality amounts to (ann′)−α+3ǫscl . (ann′)−eαscl′ . We have to permit scl′ to be as small as 1, whereas scl can be as big as ann. But α > 1, and ann < ann′, so the inequality above is trivially true. This completes the analysis of (5.52). THE FOURIER LOCALIZATION ESTIMATE 79 The Upper Bound on (5.53). We only need to use the inequality |Ies′ φs| . χ , and the inequalities (5.26). It follows that s : (s,s′)∈S3,j |〈Ies′ φs,∆s′〉| . s : (s,s′)∈S3,j 〈χ(2)Rs , |∆s′|〉 . 2−10j(ann′)−eα ′ · ann′ scl · ann . The Fourier Localization Estimate The precise form of the inequalities quantifying the Fourier local- ization effect follows. Fourier Localization Lemma 5.56. Let 1 < α < 2, ǫ < (α− 1)/20, and v be a vector field with ‖v‖Cα ≤ 1. Let s be a tile with 1 < scl(s) = scl ≤ ann(s) = ann < 1 fs = Mod−c(ωs) φs Let ζ be a smooth function on R, with 1(−2,2) ≤ ζ̂ ≤ 1(−3,3) and set ζ2k(y) = 2 kζ(y2k). We have this inequality valid for all unit vectors e with |e− es| ≤ |ωs|. ∣∣∣fs(x)− fs(x− ye)ζ2k(y) dy .(scl2(α−1)k)−1+ǫχ (x)1fωs(v(x))(5.57) + (2kscl)−10χ (x)(5.58) + |Rs|−1/21Fs(x) ,(5.59) where ω̃s is a sub arc of the unit circle, with ω̃s = λωs, and 1 < λ < 2ǫk. Moreover, the sets Fs ⊂ R2 satisfy |Fs| . 2−(α−ǫ)k(1 + scl−1)α−1|Rs|,(5.60) Fs ⊂ 2ǫkRs ∩ v−1(ω̃s) ∩ ∂(v ·es⊥) ∣∣∣ > 2(1−ǫ)k .(5.61) The appearance of the set Fs is explained in part because the only way for the function φs to oscillate quickly along the direction es is that the vector field moves back and forth across the interval ωs very quickly. This sort of behavior, as it turns out, is the only obstacle to the frequency localization described in this Lemma. Note that the degree of localization improves in k. In (5.57), it is important that we have the localization in terms of the directions 80 5. ALMOST ORTHOGONALITY of the vector field. The terms in (5.58) will be very small in all the instances that we apply this lemma. The third estimate (5.59) is the most complicated, as it depends upon the exceptional set. The form of the exceptional set in (5.61) is not so important, but the size estimate, as a function of α > 1, in (5.60) is. Proof. We collect some elementary estimates. Throughout this argument, ~y := y e ∈ R2. (5.62) |y|>t2−k |y2k||ζ2k(y)| dy . t−N , t > 1. This estimate holds for all N > 1. Likewise, (5.63) |u|>tscl−1 |uscl||sclψ(scl u)| du . t−N , t > 1. More significantly, we have for all x ∈ R2, (5.64) eiξ0y ϕ (x− ~y)ζ2k(y) dy = ϕ (x) − 2k < ξ0 < 2k, where ϕ = Tc(Rs)D ϕ. This is seen by taking the Fourier transform. Likewise, by (4.17), for vectors v0 of unit length,∫ e−2πiuλ0 ϕ (x− uv0)sclψ(scl u) du 6= 0 implies that (5.65) scl ≤ λ0 + ξ · v0 ≤ 98scl, for some ξ ∈ supp(ϕ̂ At this point, it is useful to recall that we have specified the fre- quency support of ϕ to be in a small ball of radius κ in (4.16). This has the implication that (5.66) |ξ · es| ≤ κscl, |ξ · es⊥| ≤ κann ξ ∈ supp(ϕ̂(2)Rs ) We begin the main line of the argument, which comes in two stages. In the first stage, we address the issue of the derivative below exceeding a ‘large’ threshold. e ·Dv(x) · es⊥ = ∂v · es⊥ We shall find that this happens on a relatively small set, the set Fs of the Lemma. Notice that due to the eccentricity of the rectangle Rs, we can only hope to have some control over the derivative in the long direction of the rectangle, and e essentially points in the long direction. We are interested in derivative in the direction es⊥ as that is THE FOURIER LOCALIZATION ESTIMATE 81 the direction that v must move to cross the interval ωs2. A substantial portion of the technicalities below are forced upon us due to the few choices of scales 1 ≤ scl ≤ 2εk, for some small positive ε.1 Let 0 < ε1, ε2 < ǫ to be specified in the argument below. In partic- ular, we take 0 < ε1 ≤ min , κα−1 , 0 < ε2 < (α− 1). We have the estimate |fs(x)|+ fs(x− ~y)ζ2k(y) dy ∣∣∣ . 2−10kχ(2)Rs (x), x 6∈ 2 ε1kRs. This follows from (5.62) and the fact that the direction e differs from es by an no more than the measure of the angle of uncertainty for Rs. This is as claimed in (5.58). We need only consider x ∈ 2ε1kRs. Let us define the sets Fs, as in (5.59). Define λs := 2ε1k scl < 2−2ε1k 8 otherwise Let λωs denote the interval on the unit circle with length λ|ωs|, and the same center as ωs. 2 This is our ω̃ of the Lemma; the set Fs of the Lemma is (5.67) Fs := 2 ε1kRs ∩ v−1(λsωs) ∩ ∂(v ·es⊥) ∣∣∣ > 2(1−ε2)k And so to satisfy (5.61), we should take ε1 < 1/1200. Let us argue that the measure of Fs satisfies (5.60). Fix a line ℓ in the direction of e. We should see that the one dimensional measure (5.68) |ℓ ∩ Fs| . 2−k(α−ǫ)(1 + scl−1)α−1scl−1. For we can then integrate over the choices of ℓ to get the estimate in (5.60). The set ℓ ∩ Fs is viewed as a subset of R. It consists of open intervals An = (an, bn), 1 ≤ n ≤ N . List them so that bn < an+1 for all n. Partition the integers {1, 2, . . . , N} into sets of consecutive integers Iσ = [mσ, nσ]∩N so that for all points x between the left-hand endpoint of Amσ and the right-hand endpoint of Anσ , the derivative ∂(v · es⊥)/∂e has the same sign. Take the intervals of integers Iσ to be maximal with respect to this property. 1The scales of approximate length one are where the smooth character of the vector field helps the least. The argument becomes especially easy in the case that√ ann ≤ scl, as in the case, |ωs| & scl−1. 2We have defined λs this way so that λsωs makes sense. 82 5. ALMOST ORTHOGONALITY For x ∈ Fs, the partial derivative of v, in the direction that is transverse to λsωs, is large with respect to the length of λsωs. Hence, v must pass across λsωs in a small amount of time: |Am| . 2−(1−ε1−ε2)k for all σ. Now consider intervals Anσ and A1+nσ = Amσ+1. By definition, there must be a change of sign of ∂v(x) · es⊥/∂e between these two intervals. And so there is a change in this derivative that is at least as big as 2(1−ε2)k scl . The partial derivative is also Hölder continuous of index α − 1, which implies that Anσ and Amσ+1 cannot be very close, specifically dist(Anσ , Amσ+1) ≥ 2(1−ε2)k As all of the intervals An lie in an interval of length 2 ε1kscl −1, it follows that there can be at most 1 ≤ σ . 2ε1kscl−1 2(1−ε2)k )−α+1 intervals Iσ. Consequently, |ℓ ∩ Fs| . 2−(1−2ε1−ε2+(1−ε2)(α−1))kscl−1 . 2−(α−2ε1−2ε2)kscl−α+1scl−1 We have already required 0 < ε1 < and taking 0 < ε2 < achieve the estimate (5.68). This completes the proof of (5.60). The second stage of the proof begins, in which we make a detailed estimate of the difference in question, seeking to take full advantage of the Fourier properties (5.62)—(5.65), as well as the derivative informa- tion encoded into the set Fs. We consider the difference in (5.57) in the case of x ∈ 2ε1kRs − v−1(λsωs). In particular, x is not in the support of fs, and due to the smoothness of the vector field, the distance of x to the support of fs is at least & 2ε1k so that by (5.62), we can estimate ∣∣∣fs(x)− fs(x− ~y)ζ2k(y) dy ∣∣∣ . (2ε1kscl)−N |Rs|−1/2 which is the estimate (5.58). THE FOURIER LOCALIZATION ESTIMATE 83 We turn to the proof of (5.57). For x ∈ 2ε1kRs ∩ v−1(λsωs), we always have the bound ∣∣∣fs(x)− fs(x− ~y)ζ2k(y) dy ∣∣∣ . 210ε1k/κχ(2)Rs (x) 101λsωs(x). It is essential that we have |e − es| ≤ |ωs| for this to be true, and κ enters in on the right hand side through the definition (4.40). We establish the bound ∣∣∣fs(x)− fs(x− ~y)ζ2k(y) dy ∣∣∣ . (scl2(α−1)k)−1|Rs|−1/2, x ∈ 2ε1kRs ∩ v−1(λsωs) ∩ F cs . (5.69) We take the geometric mean of these two estimates, and specify that 0 < ε1 < κ to conclude (5.57). It remains to consider x ∈ 2ε1kRs ∩ v−1(λsωs) ∩ F cs , and now some detailed calculations are needed. To ease the burden of notation, we exp(x) := e−2πiuc(ωs)·v(x), Φ(x, x′) = ϕ (x− uv(x′)), with the dependency on u being suppressed, and define w(du, d~y) := sclψ(scl u)ζ2k(y) du d~y. In this notation, note that fs = Mod−c(ωs) φs ec(ωs)(x−uv(x)−c(ωs))x ϕ (x− uv(x)) sclψ(sclu) du exp(x)Φ(x, x, )sclψ(sclu) du exp(x)Φ(x, x, )w(du, d~y) , since ζ has integral on R2. In addition, we have fs(x− ~ye)ζ2k(~y) d~y = ec(ωs)(x−uv(x−~y)−c(ωs))x × ϕ(2)Rs (x− uv(x− ~y)) sclψ(sclu) du d~y exp(x− ~y)Φ(x− ~y, x− ~y)w(du, d~y) . 84 5. ALMOST ORTHOGONALITY We are to estimate the difference between these two expressions, which is the difference of Diff1(x) := exp(x)Φ(x, x)− exp(x− ~y)Φ(x− ~y, x)w(du, d~y) Diff2(x) := exp(x− ~y){Φ(x− ~y, x− ~y)− Φ(x− ~y, x)}w(du, d~y) The analysis of both terms is quite similar. We begin with the first term. Note that by (5.64), we have Diff1(x) = {exp(x)− exp(x− ~y)}Φ(x− ~y, x)w(du, d~y). We make a first order approximation to the difference above. Observe exp(x)− exp(x− ~y) = exp(x){1− exp(x− ~y)exp(x)} = exp(x){1− e−2πiu[c(ωs)·Dv(x)·e]~y}(5.70) +O(|u|ann|~y|α). In the Big–Oh term, |u| is typically of the order scl−1, and |~y| is of the order 2−k. Hence, direct integration leads to the estimate of this term |u|ann|y|α|Φ(x− ~y, x)|·|w(du, d~y)| .|Rs|−1/2 .|Rs|−1/2(scl2(α−1)k)−1. This is (5.69). The term left to estimate is Diff ′1(x) := exp(x)(1−e−2πiu[c(ωs)·Dv(x)·e]~y) Φ(x− ~y, x)w(du, d~y) . Observe that by (5.64), the integral in y is zero if |u[c(ωs) ·Dv(x) · e]| ≤ 2k. Here we recall that c(ωs) = ann es⊥. By the definition of Fs, the partial derivative is small, namely |es⊥ ·Dv(x) · e| . 2(1−ε1)k THE FOURIER LOCALIZATION ESTIMATE 85 Hence, the integral in y in Diff ′1(x) can be non–zero only for scl|u| & 2ε1k. By (5.63), it follows that in this case we have the estimate |Diff ′1(x)| . 2−2k|Rs|−1/2 This estimate holds for x ∈ 2ε1kRs∩v−1(λsωs)∩F cs and this completes the proof of the upper bound (5.69) for the first difference. We consider the second difference Diff2. The term v(x− ~y) occurs twice in this term, in exp(x − ~y), and in Φ(x − ~y, x − ~y). We will use the approximation (5.70), and similarly, Φ(x− ~y, x−~y)− Φ(x− ~y, x) (x− ~y − uv(x− ~y))− ϕ(2)Rs (x− ~y − uv(x)) (x− ~y − uv(x)− uDv(x)~y) − ϕ(2)Rs (x− ~y − uv(x)) +O(ann |u||y| = ∆Φ(x, ~y) +O(ann |u||~y|α) The Big–Oh term gives us, upon integration in u and ~y, a term that is no more than . |Rs|−1/2 2−αk . |Rs|−1/2(scl2(α−1)k)−1. This is as required by (5.69). We are left with estimating Diff ′2(x) := e−2πiuc(ωs)·(v(x)−Dv(x)·~y)∆Φ(x, ~y)w(du, d~y). By (5.64), the integral in y is zero if both of these conditions hold. |uc(ωs)Dv(x) · e| < 2k, |[uc(ωs)Dv(x)− ξ − uξDv(x)] · e| < 2k, ξ ∈ supp(ϕ̂(2)Rs ) Both of these conditions are phrased in terms of the derivative which is controlled as x 6∈ Fs. In fact, the first condition already occurred in the first case, and it is satisfied if scl|u| . 2ε1k. Recalling the conditions (5.66), the second condition is also satisfied for the same set of values for u. The application of (5.63) then yields a very small bound after integrating |u| & 2ε1kscl−1. This completes the proof our technical Lemma. � REFERENCES 87 References [1] Angeles Alfonseca, Fernando Soria, and Ana Vargas, A remark on maximal operators along directions in R2, Math. Res. Lett. 10 (2003), no. 1, 41– 49.MR1960122 (2004j:42010) ↑8 [2] Angeles Alfonseca, Strong type inequalities and an almost-orthogonality princi- ple for families of maximal operators along directions in R2, J. London Math. Soc. 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0704.0809

Wide Field Surveys and Astronomical Discovery Space

Wide Field Surveys and Astronomical Discovery Space A.Lawrence Institute for Astronomy, SUPA∗, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ A review for publication in Astronomy and Geophysics Feb 27th 2007 Abstract I review the status of science with wide field surveys. For many decades surveys have been the backbone of astronomy, and the main engine of discovery, as we have mapped the sky at every possible wavelength. Surveys are an efficient use of resources. They are important as a fundamental resource; to map intrinsically large structures; to gain the necessary statistics to address some problems; and to find very rare objects. I summarise major recent wide field surveys - 2MASS, SDSS, 2dfGRS, and UKIDSS - and look at examples of the exciting science they have produced, covering the structure of the Milky Way, the measurement of cosmological parameters, the creation of a new field studying substellar objects, and the ionisation history of the Universe. I then look briefly at upcoming projects in the optical-IR survey arena - VISTA, PanSTARRS,WISE, and LSST. Finally I ask, now we have opened up essentially all wavelength windows, whether the exploration of survey discovery space is ended. I examine other possible axes of discovery space, and find them mostly to be too expensive to explore or otherwise unfruitful, with two exceptions : the first is the time axis, which we have only just begun to explore properly; and the second is the possibility of neutrino astrophysics. 1 Why are wide field surveys important ? Some astronomical experiments are direct, in that measurements are made of some piece of sky, and these measurements are then used for a specific scientific analysis. The essence of a survey however is that extracting science is a two step process. First we summarise the sky, usually by taking an image and then running pattern recognition software to produce a catalogue of objects each with a set of measured parameters. When this summary is made, we can then do the science with the catalogue; the archive becomes the sky. There are many such archives, distributed around the world in online structured databases; querying such databases is a growing mode of scientific analysis. This of course is why survey databases have played such a central role in the worldwide Virtual Observatory initiatives. Why is this two-step process a good thing to do ? Firstly, it is cost effective, because we can performmany experiments using the same data. Secondly, surveys are a resource that can support other experiments. This can mean for example creating samples of objects which are ‘followed up’, i.e. observed in detail, on other facilities (eg getting spectra of galaxy samples). Conversely, interesting objects discovered by other experiments can be matched against objects in the standard survey catalogues, so that one quickly has the optical flux of a new gamma-ray source. (Should this be called follow-down ?). Finally, surveying the sky can produce surprises. First looks in new corners of parameter space have often ∗Scottish Universities Physics Alliance http://arxiv.org/abs/0704.0809v1 Wide Field Surveys : A.Lawrence 2 Table 1: Examples of major astronomical surveys from recent decades Type Survey Examples Radio 3C, PKS, 4C, FIRST IR IRAS-PSC, ELAIS, 2MASS, UKIDSS Optical APM, SuperCOSMOS, SDSS, CFHTLS X-ray 3U, 2A, HEAO-A, 1-XMM z-surveys CfA-z, QDOT, 2dFGRS, SDSS-z discovered completely new populations of objects. Historically, surveys have been the main engine of discovery for astronomy. Why are wide angle surveys important, as opposed to the deepest possible pencil beams ? The key point here is that in Euclidean space, time spent surveying more area increases volume much faster than time spent going deeper. (The argument that wide angles produce large samples faster breaks down when the differential source count slope is flatter than 1, which for example occurs for galaxies fainter than about B∼ 23. Also of course, sometimes, one simply has to go deep, for example to survey at some given large redshift.) Many astronomical problems need large samples of objects to address them. Sometimes this is because one wants accurate function estimation – for example to test theories of structure formation, one wants to estimate the galaxy clustering power spectrum to an accuracy of around 5% in many bins over a wide range of scale. Sometimes large samples are needed to recover a very weak signal from noise – for example the net alignment of many random galaxy ellipticities produced by weak lensing by intervening dark matter. The second reason for maximising volume as quickly as possible is to find rare objects, such as the hoped for Y dwarfs and z = 7 quasars; to a given depth there might be only a handful over the whole sky. Finally, some objects of astronomical study simply have intrinsically large angular scale - for example the Milky Way, the galaxy clustering dipole, or open clusters of stars, which can be tens of degrees across. 2 Major surveys Surveys are the core of astronomy. This has always been true of course, from Ptolemy through the New General Catalogue, to the Carte du Ciel, but it has been certainly been the case in the last few decades. Table 1 lists some of the best known imaging surveys in each wavelength regime. (I have also included a few redshift surveys as a distinct set). This is only a selection, and is biased towards my own favourites, so apologies to those whose own surveys aren’t listed. The point to note is that these names are as immediately recogniseable to every astronomer as are the names of famous telescopes and satellites - Palomar, AAT, Ariel-V, etc. The data in these catalogues are of everyday use and have been the source of many discoveries. Many of the older surveys were classic examples of opening a completely new window on the Universe - 3C, IRAS, and 3U in particular, though I think it is also fair to include the CfA redshift survey in this category, as it gave us the first real feel for the three dimensional structure of the Universe, with bubbles, filaments, and walls. The 1-XMM catalogue is slightly different, in that it wasn’t planned as a coherent single survey, but is the uniformly processed summation of XMM pointings over the sky. Over the last 5-10 years the most important major new surveys have been in the optical-IR - 2MASS, SDSS, 2dFGRS, and now UKIDSS, which started in 2005. I will summarise each of these briefly in turn. Some highlight science results are in the next section. The Two Micron All Sky Survey : 2MASS. Wide Field Surveys : A.Lawrence 3 Figure 1: All sky distribution of 2MASS catalogues. Point sources are shown as white dots. Extended sources are coloured according to estimated redshift, based either on known values, or estimated from K magnitude. Blue are the nearest sources (z < 0.01); green are at moderate distances (0.01 < z < 0.04) and red are the most distant sources that 2MASS resolves (0.04 < z < 0.1). Taken from Jarrett (2004). 2MASS broke new ground, as it was the first real sky survey at near infa-red wavelengths. At near- IR wavelengths we see roughly the same Universe as in the visible light regime, but with some key improvements. Extinction is much less; we can see pretty much clean through the Milky Way, and can find reddened versions of objects such as quasars. Cooler objects such as brown dwarfs can be found, with the most extreme objects essentially invisible in standard optical bands. Cleaner galaxy samples can be constructed, with high redshift objects easier to find. Colour combinations with optical bands have proved especially good at finding rare objects, such as the new T-dwarf class of brown dwarfs. 2MASS used two dedicated 1.3m telescopes, in Mt Hopkins, Arizona, and CTIO, Chile. Each telescope was equipped with a three-channel camera, each channel consisting of a 256×256 HgCdTe array, so that observations could be made simultaneously at J (1.25 microns), H (1.65 microns), and Ks (2.17 microns). One interesting innovation was the use of large pixels, maximising survey speed, requiring micro-stepping to improve sampling. The survey started in June 1997 June and completed in February 2001. The full data release occurred in March 2003, including both an Atlas of images and a catalogue of almost half a billion sources. To a point source limit of 10σ, the catalogue depth is J=16 H=15 Ks=14.7, almost five orders of magnitude deeper than any comparable IR survey. However, for the colours of many astronomical objects, this is still two orders of magnitude shallower than modern optical surveys. The core reference for 2MASS is Skrutskie et al. 2006). Further information can be found at the IPAC (http://www.ipac.caltech.edu/2mass/) and UMASS(http://pegasus.phast.umass.edu/) sites. Data ac- cess is through the IRSA system at http://irsa.ipac.caltech.edu/ . The Sloan Digital Sky Survey : SDSS. The SDSS project has produced a survey of 8,000 square degrees of sky at visible light wavelengths, approximately two magnitudes deeper than the historic Schmidt surveys, and in addition has carried out a spectroscopic survey of objects selected from the imaging survey. The project used a dedicated 2.5m telescope at Apache Point, New Mexico, and a camera covering 1.5 square degrees. Survey operations used a novel drift scan approach; the 30 CCDs on the camera are arranged in five rows each sensitive to a separate filter band (u,g,r,i,z); the telescope is parked in a given position and the sky allowed to drift past. The spectroscopic survey is carried out using a 600-fibre system, on separate nights spliced into the imaging programme. This then required the data processing pipeline to keep up in almost real time. Public data access has been announced in a series of staged releases, culminating in June 2006 with DR5, which contains a catalogue of around 200 million objects, and spectra for around a million http://www.ipac.caltech.edu/2mass/ Wide Field Surveys : A.Lawrence 4 galaxies, quasars, and stars. An extended programme, cunningly called SDSS-II, has now commenced, and is expected to continue through 2008. SDSS has been arguably the most successful survey project of recent times, with many hundreds of scientific papers based directly on its data, and having an impact on a very large range of astronomical topics - large scale structure, the highest redshift quasars, the structure of the Milky Way, and many other things besides. This may seem surprising, as visible light sky surveys covering the whole sky have been available for decades, and available as digitised queryable online databases for some years (eg the Digitised Sky Survey (DSS : see http://archive.stsci.edu/dss/) or the SuperCOSMOS Science Archive (SSA : see http://surveys.roe.ac.uk/ssa). There are several reasons for the success of SDSS. The first reason is of course the spectroscopic database, matched only by 2dF (see below). The second reason is the wider wavelength range, with filters carefully chosen and calibrated to optimise various kinds of search. The third reason is the improvement in quality - not only is SDSS a magnitude or two deeper than the Schmidt surveys, but the seeing is markedly better. The fourth reason, shared by 2MASS, is the quality of the online interface - well calibrated, reliable, and documented data were available promptly, and with the ability to do online analysis rather than just downloading data. This has made it easy for astronomers all over the world to jump in and benefit from SDSS. The core reference for SDSS is York et al. 2000). Further information can be found at http://www.sdss.org, which also contains links to data access via SkyServer. The UKIRT Infrared Deep Sky Survey : UKIDSS. UKIDSS is the near-infrared equivalent of the SDSS, covering only part of the sky, but many times deeper than 2MASS. The project has been designed and implemented by a private consortium, but on behalf of the whole ESO member community, and after a short delay, the world. It uses the Wide Field Camera (WFCAM) on the UK Infrared Telescope (UKIRT) in Hawaii, and is taking roughly half the UKIRT time over 2005-2012. WFCAM has an instantaneous field of view of 0.21 sq.deg, much larger than any previous large facility IR camera. Put together with a 4m telescope, this makes possible an ambitiuous survey. It is estimated that the effective volume of UKIDSS will be 12 times that of 2MASS, and the effective amount of information collected 70 times larger. UKIDSS is not a single survey, but a portfolio of five survey components. Three of these are wide shallow surveys, to K∼ 18−19, and covering a total of ∼ 7000 sq.deg - the Galactic Plane Survey (GPS); the Galactic Clusters Survey (GCS); and the high latitude Large Area Survey (LAS). Then there is a Deep Extragalactic Survey (DXS), covering 35 sq.deg to K ∼ 21, and an Ultra Deep Survey (UDS), covering 0.77 sq.deg. to K ∼ 23. In all cases, there is the maximum possible overlap with other multiwavelength surveys and key areas, such as SDSS, the Lockman Hole, and the Subaru Deep Field. The aim of UKIDSS is to provide a public legacy database, but the design was targeted at some specific goals - for example, to measure the substellar mass function, and its dependence on metallicity; to find quasars at z = 7; to discover Population II brown dwarfs if they exist; to measure galaxy clustering at z = 1 and z = 3 with the same accuracy as at z = 0; and to determine the epoch of spheroid formation. Like SDSS, data are being released in a series of stages. At each stage the data are public to astronomers in all ESO member states, and world-public eighteen months later. Data are made available through a queryable interface at the WFCAM Science Archive (WSA : http://surveys.roe.ac.uk/wsa). Three data releases have occured so far – the “Early Data Release”, and Data Releases One and Two (DR1 and DR2) which contain approximately 10% of the likely full dataset. UKIDSS is summarised in Lawrence et al. (2007) , and technical details of the releases are described in Dye et al. (2006) and Warren et al. (2007). Redshift surveys : 2MRS/6dFGRS; 2dFGRS and SDSS-z. Systematic redshift surveys based on galaxy catalogues from imaging surveys were one of the big success stories of the 1970s–90s, culminating in the all-sky z-survey based on the IRAS galaxies, the PSC-z (Saunders et al. 2000). The most ambitious surveys to date have however been carried out over the last five years. The first example is the construction of a complete all-sky redshift survey based on galaxies in http://archive.stsci.edu/dss/ http://www.sdss.org http://surveys.roe.ac.uk/wsa Wide Field Surveys : A.Lawrence 5 the 2MASS Extended Source Catalog (XCS) to a depth of KS=12.2, containing roughly 100,000 galaxies. In the south, observations are carried out at the UK Schmidt, as part of the 6dfGRS project (Jones et al. 2004, http://www.aao.gov.au/local/www/6df/); in the North observations are being carried out by a CfA team at Mt Hopkins, Arizona (see http://cfa-www.harvard.edu/∼huchra/2mass/). The survey is part way through, but has already been used to measure the dipole anisotropy of the local universe (Erdodgu et al. 2005). Two very successful projects have completed redshift surveys of smaller area, but reaching considerably deeper, containing hundreds of thousands of galaxies. The first, in the Northern sky, is SDSS-z, the spectroscopic component of SDSS, as described above. The second, in the southern sky, is the 2dF Galaxy Redshift Survey (2dFGRS; Colless et al. 2001). This was based on galaxies selected from the APM digitisation of UK Schmidt plates (Maddox et al. 199x), and observed using the Two Degree Field (2dF) facility at the Anglo-Australian Telescope, which has 400 independent fibres. The 2dFGRS obtained spectra for 245591 objects, mainly galaxies, brighter than a nominal extinction-corrected mag- nitude limit of bJ=19.45, covering 1500 square degrees in three regions. The final data release was in June 2003. More information, and data access, is available at http://www.mso.anu.edu.au/2dFGRS/. These two surveys have produced a range of science, but have concentrated on making the best possible measurement of the power spectrum of galaxy clustering, and together with WMAP and supernova programme results, have produced the definitive estimates of the cosmological parameters, leading to the current ‘concordance cosmology’. 3 Recent survey science highlights I have picked out a handful of results from the optical-IR surveys of the last few years, including the first results from UKIDSS, to illustrate the power of the survey approach. Panoramic mapping : the structure of the Milky Way. Two topics which clearly benefit from a map covering 4π sr, and with low extinction, are the structure of the Milky Way, and the structure of the local extragalactic universe. Figure 1, taken from Jarrett (2004), illustrates the impact 2MASS has on both these topics, showing both the Point Source Catalog (mostly stars) and the Extended Source Catalog (mostly galaxies at z<0.1). For the first time, we can see the Milky Way looking like other external galaxies, with disc, bulge, and dust lane. Some of the most important scientific results however have come from looking at subsets of the stellar population. Figure 2, from Majewski et al. (2003), shows the sky distribution of M giants selected from the 2MASS PSC, a selection which traces very large scale structures while removing the dilution of local objects, using a few thousand stars out of the catalogue of half a billion. From APM star counts we already knew of the existence of the Sagittarius dwarf, swallowed by the Milky Way (Ibata, Irwin and Gilmore 1994), but now we can see its complete structure including an extraordinary 150 degree tidal tail. Its orbital plane shows no precession, indicating that the Galactic potential within which it moves is spherical. The Earth is currently close to the debris, which means that some very nearby stars are actually members of the Sagittarius dwarf system. Interestingly, Sagittarius seems to contribute over 75% of of high latitude halo M giants, with no evidence for M giant tidal debris from the Magellanic clouds. SDSS, although not a panoramic survey, has also been very important for Galactic structure and stel- lar populations, with the five widely spread bands making it possible to derive stellar types and so photometric parallaxes. Juric et al. (2005) derive such parallaxes for 48 million stars. They fit a com- bination of oblate halo, thin disk, thick disk, but also find significant ‘localised overdensities’, including the known Monoceros stream, but also a new enhancement towards Virgo that covers 1000 sq.deg. This then maybe another dwarf galaxy swallowed by the Milky Way. Large sample statistics : galaxy clustering and the cosmological parameters. http://www.aao.gov.au/local/www/6df/ http://www.mso.anu.edu.au/2dFGRS/ Wide Field Surveys : A.Lawrence 6 Figure 2: Smoothed maps of the sky in equatorial coordinates showing the 2MASS point source catalogue optimally filtered to show the Sagittarius dwarf; southern arc (top), and the Sagittarius dwarf northern arm (bottom). Two cycles around the sky are plotted to demonstrate the continuity of features. The top panel uses 11 < Ks < 12 and 1.00 < J − Ks < 1.05. The bottom panel uses 12 < Ks < 13 and 1.05 < J −Ks < 1.15. Taken from Majewski et al. (2003). Figure 3: Cone diagram showing projected distribution of galaxies in 2dFGRS. Taken from Peacock (2002). Wide Field Surveys : A.Lawrence 7 Figure 4: (a) Power spectrum from 2dFGRS, compared to various model predictions. Taken from Per- cival et al. (2001). (b) Correlation function of Luminous red Galaxies in the SDSS-z sample, showing the first baryon acoustic oscillation peak. Taken from Eisenstein et al. (2005) The SDSS-z and 2dFGRS surveys illustrate the power of the survey approach in two ways. First, significant volume is needed to map out large scale structures and overcome shot noise on the largest scales. Figure 3 is a cone diagram for all the 2dFGRS galaxies, showing the richness of structure that is only possible to map out with both a large volume and density. Second, large numbers are needed to make a good enough estimate of the power spectrum of galaxy clustering. This is illustrated in Fig 4a, which shows the power spectrum derived from 2dFGRS compared to various model predictions (data from Percival et al. (2001), figure from Peacock (2002)). To distinguish models with differing matter density in the interesting range requires accuracy of a few percent over a very wide range of scales; to have a chance of measuring small scale features predicted by models including a significant baryon fraction requires many samples across this wide range, with of the order 103 galaxies per bin to achieve the required accuracy. These wiggles are due to acoustic oscillations in the baryon component of the universe at early times. In the Percival et al. paper, only a limit could be placed on these oscillations, but they were statistically detected in the fimal 2dFGRS data (Cole et al. 2005). However, in another good example of filtering out a tracer sub-sample from a very large sample, the first baryon peak was much more clearly seen in the correlation function of Luminous Red Galaxies (LRGs) selected from SDSS-z (Eisenstein et al. 2005; Huetsi 2005; see Fig 4). 2dFGRS and SDSS-z were the first redshift surveys to have large enough scale and depth to overlap the fluctuation measurements from the CMB, enabling degeneracies in the estimation of cosmological parameters to be broken, and accuracy to be increased by a factor of several. Several key papers made joint analyses of the galaxy and CMB datasets (Percival et al. 2002; Efstathiou et al. 2002; Tegmark et al. 2003; Pope et al. 2004) arriving at broadly consistent answers. We now know what kind of universe we live in : a geometrically flat universe dominated by vacuum energy (75%), with some kind of cold dark matter at about 21% and ordinary baryons 4%. The equation of state parameter for the dark energy has been limited to w < −0.52 (Percival et al. 2002), and the total mass of the neutrinos to m <1 eV (Tegmark et al. 2003; Elgaroy et al. 2002) The Deep eXtragalactic Survey (DXS) of UKIDSS will produce a galaxy survey over a volume as large as that of 2dFGRS or SDSS, but at z = 1. A redshift survey of this sample is a prime target for future work. Rare objects : Brown Dwarfs. Infrared surveys have transformed the study of the substellar regime, blurring our idea of what it means Wide Field Surveys : A.Lawrence 8 7000 7500 8000 8500 9000 9500 10000 10500 Wavelength Dashed: LBQS composite Dotted: Telfer continuum SDSS J1030+0524 z=6.28 Figure 5: (a) High resolution spectrum of high redshift quasar found in SDSS. Gunn-Peterson troughs due to Lyα and Lyβ are the black sections from 8500Å to 9000Å and from 7000Å to 7500Å. Taken from White et al. (2003); original discovery spectrum in Becker et al. (2001). (b) Spectral energy distributions for a z = 7 quasar and a T-dwarf, compared to filter passbands from SDSS and UKIDSS. Taken from Lawrence et al. (2007). to be a star. For many years, until the first discovery of the very faint IR companion of GL 229 (i.e. GL229B) by Nakajima et al. (1995), the possibility of star-like objects which never ignite nuclear burning was only a speculation. Within a year of the start of 2MASS, Kirkpatrick et al. (1999) had found 20 brown dwarfs in the field, increasing the number of known brown dwarfs by a factor of four, and had defined two new stellar spectral types - L and T. (These strange designations were determined by the fact that various odd stellar types had already used up nearly all the other letters of the alphabet.) The transition from M to L was defined by the change of key atmospheric spectral features from those of metal oxides to metal hydrides and neutral metals; the transition from L to T by the appearance of molecular features such as methane - as seen in solar system planets. The effective temperature for L dwarfs is in the range T∼1500 – 2000 K, and for T-dwarfs T∼1000 –1500 K. As of the time of writing, almost 600 brown dwarfs are known. Most of these are L-dwarfs, but almost 60 T-dwarfs have now been found in a series of 2MASS papers (see Ellis et al. 2005 and references therein). The much deeper UKIDSS search is expected to make significant further advances in two ways. The first is by pushing to ever cooler and fainter objects, hopefully finding examples of a putative new stellar class labelled ‘Y dwarfs’ (the last useable letter left ... see Hewett et al. 2006), finding T-dwarfs further than 10pc, and plausibly finding Population II brown dwarfs if they exist. The second advance expected from UKIDSS is the determination of the substellar mass function, through the Galactic Clusters Survey (GCS), and testing whether it is universal or not. These hopes are already being borne out by early UKIDSS results; Warren et al. (2007b) report the discovery of the coolest known star, classified as T8.5; and in early results from the GCS programme, Lodieu et al. (2007) have found 129 new brown dwarfs in Upper Sco, a significant fraction of all known brown dwarfs, including a dozen below 20 Jupiter masses, finding the mass function in the range 0.3 – 0.01 solar masses to have a slope of index α = 0.6 ± 0.1. Rare objects : the ionisation history of the Universe. An excellent example of the ‘needle in a haystack’ search is looking for very high redshift quasars. Only the most extremely luminous quasars are detectable at these distances, but the space density of such objects is very low; even in a survey with thousands of square degrees there may be only a few present. Luminous and high redshift quasars are interesting for a variety of reasons, but a key target for four decades has been their use as beacons to detect the re-ionisation of the inter-galactic medium. The baryon content of the early universe must have become neutral as it cooled down, but something subsequently re-ionised it, as attempts to find the expected ‘Gunn-Peterson trough’ (Gunn and Peterson 1965) in the spectra of high redshift quasars had failed for many years. This finally changed in 2001 as SDSS broke the z = 6 quasar redshift barrier (Fan et al. 2001) and Becker et al. (2001) made the first Wide Field Surveys : A.Lawrence 9 detection of a Gunn-Peterson trough at z = 6.28. Figure 5 shows the improved spectrum from White et al. (2003). Unfortunately this exciting result seemed to conflict with the CMB measurements by the WMAP year-1 data. The degree of scattering required implied that ionisation had already taken place by z=11 – 30 (Kogut et al. 2003). Rather than being seen as a contradiction, it seems likely that that re-ionisation was not a single sharp-edged event, but an extended and very likely complex affair, perhaps with multiple stages and even spatial inhomogeneity (see White et al. 2003). This opens an entire new field of investigation for understanding the history of the early universe. Rather than a single object locating the transition edge, it is now important to find as many beacons as possible at z>6, and to find some beacons in the range z = 7–8. This is one of the key aims of UKIDSS, in combination with SDSS data, looking for z-dropouts. A problem however is that JHK colours of high-z quasars and T-dwarfs become very similar. For this reason, UKIDSS is using a Y-band filter centred at 1.0 µm. Figure 6 illustrates the point, comparing the spectrum of a quasar redshifted to z = 7 with that of a T-dwarf brown dwarf. 4 Next steps in optical-IR surveys Three key optical-IR survey projects are to begin soon (VISTA, PanSTARRS, and WISE), with the ultimate in wide-field surveys (LSST) now in the planning stage. Here I briefly summarise each of these. VISTA. The Visible and Infrared Survey Telescope for Astronomy (VISTA) is a 4m aperture dedicated survey telescope on Paranal in Chile. It was originally a UK project, aimed ta bothe optical and IR surveys, but became an IR-only ESO telescope during the accession of the UK to ESO. The infrared camera operates at Z, Y , J , H , and KS , and contains 16 arrays each of which has 2048×2048 0.33” pixels, covering 0.6 sq.deg. in each shot. VISTA therefore operates in the same parameter space as UKIDSS, but will survey three times faster, and furthermore, 100% of the time is dedicated to IR surveys. The majority (75%) of the telescope time is reserved for large public surveys. At the time of writing, these surveys are in the process of final approval, but are likely to include a complete hemisphere survey to K=18.5, surveys of the Galactic Bulge and the Milky Way, a thousand sq.deg. survey to K=19.5, a 30 sq.deg. survey to K=21.5, and a 1 sq.deg. survey to K=23. VISTA is expected to begin operations in late 2007. The VISTA web page is at http://www.vista.ac.uk/, and a recent reference is McPherson et al. (2006). PanSTARRS. The power of a survey facility is measured by its étendue, the product of collecting area times field of view. The cost of a telescope, and the difficulty of producing very wide fields, scales steeply with telescope aperture. The idea behind the ‘Panoramic Survey Telescope and Rapid Response System’ (PanSTARRS), a University of Hawaii project, is to produce the maximum étendue per unit cost by building several co-operating wide field telescopes of moderate size. The design has four 1.8m telescopes each with a mosaic array of 64×64 CCD chips covering 7 sq.deg, which will produce an étendue an order of magnitude larger than the SDSS facility. As well as enabling one to produce deep surveys faster, this makes it plausible to cover very large areas of sky repeatedly - thousands of square degrees per night. The prime aim of PanSTARRS is to detect potentially hazardous NEOs, but it will also be used for stellar transits, microlensing studies, and locating distant supernovae to constrain the dark energy problem. The accumulated sky survey will be many times deeper than SDSS, and the expected image quality and stability from Hawaii should allow the best ever mapping of dark matter via weak lensing distortions. A prototype single PanSTARRS system (‘PS1’) has recently been built and is being commissioned at the time of writing. The operation and science analysis for PS1 involves an extended ‘PS1 Science Consortium’ with additional partners, from the US, Uk and Germany. Over three years, it is expected http://www.vista.ac.uk/ Wide Field Surveys : A.Lawrence 10 13 13.5 14 14.5 15 Log(ν) Hz Sky survey comparisons : Extragalactic PS1UHSWISE LRG z=2 quasar z=2 quasar z=7 13 13.5 14 14.5 15 Log(ν) Hz Sky survey comparisons : Brown dwarfs PS1UHSWISE T=1000K D=30pc M=10MJ, T=5Gyr D=1pc Figure 6: Spectral energy distributions of various objects compared to 5σ sensitivities of key sky surveys. Green triangles are JHK sensitivities for a proposed extension to the UKIDSS Large Area Survey, the UKIRT Hemisphere SUrvey (UHS). Blue circles are for the WISE mission, from Mainzer et al 2005. Red circles (PS1) are for the PanSTARRS-1 3π survey, taken from project documentation. The left hand frame compares extragalactic objects - a giant elliptical at z = 2; the mean quasar continuum SED from Elvis et al 1994 redshifted to z = 2; and a high redshift quasar spectrum redshifted to z = 7. The right hand frame compares two model brown dwarf spectra, from Burrows et al (2003, 2006). The red line (lower curve at low frequency) is for an object with effective temperature of 1000K and surface gravity of 4.5, placed at a distance of 50pc. The black line (upper curve at low frequency) is for an object with mass of 10 Jupiter masses and age 5 Gyr, placed at a distance of 1pc. to produce a 3π steradian survey at grizy to z=23 with 12 visits, a Medium Deep Survey visiting 12 7 sq.deg. fields with a 4 day cadence, building up a survey to z=26, and special stellar transit campaigns and microlensing monitoring of M31. The data will become public at the end of this science programme. Information about PanSTARRS can be found at http://pan-starrs.ifa.hawaii.edu/public/ WISE. The Wide Field Infrared Survey Explorer (WISE) is a NASA MIDEX mission scheduled for launch in 2009 that will fill the gap between UKIDSS/VISTA in the near-IR and IRAS and Akari in the far-IR, surveying the sky in four bands simultaneously (3.3, 4.7, 12, and 23µm). The sky survey at 3 and 5µm is completely new territory; as 12 and 23µm WISE covers the same territory as IRAS but will be a thousand times deeper. WISE carries a 40cm cooled telescope with a 47 arcmin field of view. It is designed to have a relatively short lifetime - 7 months - but in this time will make a mid-infrared survey of the entire sky in all four bands. WISE will produce significant advances in a number of areas, but especially for objects expected to have temperatures in the hundreds of degrees - the very coolest brown dwarfs, protoplanetary discs, solar system bodies, and obscured quasars. Information about WISE can be found at http://wise.ssl.berkeley.edu/ and in Mainzer et al. (2006). The depths of PS1, UKIDSS-VISTA, and WISE surveys are well matched and will produce a stunning sky survey dataset over a factor of a hundred in wavelength. This is illustrated in Fig. 6, taken from a recent proposal to extend UKIDSS to a complete hemisphere survey. LSST. The Large Synoptic Survey Telescope (LSST) aims at the maximum possible étendue, aiming at the same kind of science as PanSTARRS - hazardous NEOs, GRBs, supernovae, dark matter mapping via weak lensing - but a factor of several faster; it should be able to produce a survey equivalent to SDSS every few days. The design has an 8.4m telescope with a 10 sq.deg. field of view. The planned standard mode of use is to take 15 second exposures, and keep moving, covering the whole sky visible from LSST http://pan-starrs.ifa.hawaii.edu/public/ http://wise.ssl.berkeley.edu/ Wide Field Surveys : A.Lawrence 11 in bands ugrizy once every three days. This produces 15TB of imaging data every night. The aim is to keep up with this flow in quasi-real time, producing alerts for transient objects within minutes. This requires approximately 60 TFlops of processing power - a huge amount today, but following Moore’s Law, very likely to be equivalent to merely the 500th most powerful computer in the world by 2012.. The LSST data management plan has a hierarchy of archive and data centres, reminiscent of the LHC Grid, with the primary mission facility acting like a ‘beamline’, where a variety of research groups can rent space for their own experiments on the data flowing past. The LSST site has now been chosen (Cerro Pachon in Chile), but the project is not yet fully funded. More information can be found at http://www.lsst.org 5 The end of survey discoveries ? In 1950, the universe seemed to consist of stars, and a sprinkling of dust. Over the last fifty years, the actual diverse and bizarre contents of the universe have been successively revealed as we surveyed the sky at a series of new wavelengths. Radio astronomy has shown us radio galaxies and pulsars; microwave observations have given us molecular clouds and the Big Bang fossil background; IR astronomy has shown us ultraluminous starburst galaxies and brown dwarfs; X-ray astronomy has given us collapsed object binaries and the intra-cluster medium; and submm astronomy has shown us debris disks and the epoch of galaxy formation. As well as revealing strange new objects, these surveys revealed new states of matter (relativistic plasma, degenerate matter, black holes) and new physical processes (bipolar ejection, matter-antimatter annihilation). Having opened up gamma-rays and the submm with GRO and SCUBA, there are no new wavelength windows left. Has this amazing journey of discovery now finished ? Wavelength is not the only possible axis of survey discovery space. Let us step through some other axes and examine their possibilities. In doing this, we will to some extent go over ground already trodden by Harwit (2003), but with a particular emphasis on surveys rather than discovery space in general, and with an eye to what is economically plausible. Photon Flux. Historically, going ever deeper has been as productive as opening new wavelength windows, the classic example of course being the existence of the entire extragalactic universe, which did not become apparent until reaching ten thousand times fainter than naked eye observations, requiring both large telescopes and the ability to integrate. We can now see things ten billion times fainter than the naked eye stars. However, we have reached the era of diminishing returns. The flux reached by a telescope is inversely proportional to diameter D but its cost is proportional to D3. Significant improvements can now only be achieved with world-scale facilities, and orders of magnitude improvements are unthinkable. The easy wins have been covered already - our detectors now achieve close to 100% quantum efficiency; we have gone into space and reduced sky background to a minimum; and multi-night integrations have been used many times. We will keep building bigger telescopes, but it no longer seems the fast track to discovery. Spectral resolution. Detailed spectroscopy of individual objects is of course the key technique of modern astrophysics. Spectroscopic surveys of samples drawn from imaging surveys have been carried out at many wavelengths, and have been particularly important for measuring redshift and so mapping the Universe in 3D; we were not expecting the voids, bubbles and walls that we found in the galaxy distribution in the 1980s. This industry will continue, but there is no obvious new barrier to break. Narrow band imaging surveys centred on specific atomic or molecular features (21cm HI, CO, Hα) have been fruitful, but again its not obvious there is anywhere new to go. Polarization. Polarisation measurements of individual objects are a very important physical diagnostic, but are polarisation surveys plausible ? Surveys of samples of known objects to the 0.1% level have been done, with interesting results but no big surprises. Perhaps blank field imaging surveys in four Stokes parameters would turn up unexpected highly polarised objects ? This has essentially been done http://www.lsst.org Wide Field Surveys : A.Lawrence 12 in radio astronomy but not at other wavelengths. Spatial resolution. This is the dominant big-project target of the next few decades, and of course is the real point of Extremely Large Telescopes. Put together with multi-conjugate Adaptive Optics, we hope to achieve both depth and milli-arcsec resolution at the same time. However, the royal road to high spatial resolution is through interferometry. Surveys with radio interferometers in the twentieth century showed the existence of masers in space, and bulk relativistic outflow. In the twenty first century we will be doing microwave interferometry on the ground (ALMA) and IR interferometry in space (TPF/DARWIN), hoping to directly detect Earth-like planets around nearby stars. So there is excitement for at least some time to come; however, as with photon flux, we are hitting an economic brick wall. Significantly bigger and better experiments will be a very long time coming. Time. The observation of temporal changes has repeatedly brought about revolutionary changes in astronomy, the classic examples being Tycho’s supernova, and the measurement of parallax. The last two decades has seen a renaissance in this area, with an impressive number of important discoveries from relatively cheap monitoring experiments - the discovery of extrasolar planets from velocity wobbles and transits; the discovery of the accelerating universe and dark energy from supernova campaigns; the location of substellar objects from survey proper motions; the existence of Trans-Neptunian Objects, and Near Earth Objects; the final pinning down of gamma-ray burst counterparts; and the limits on dark matter candidates from micro-lensing events. The next decade or two will see more ambitious pho- tometric monitoring experiments, such as PanSTARRS and LSST, and a series of astrometric missions, culminating in GAIA, which will see external galaxies rotating. Overall, the ‘time window’ is well and truly opened up. However, the temporal frequency axis is far from fully explored. My instinct is that this technique will continue to produce surprises for some time. Non-light channels : particles. Cosmic ray studies have been important for many decades, but you can’t really do surveys - indeed the central mystery has alway been where they come from. Dark matter experiments are confronting what is arguably the most important problem in physics, let alone astrophysics, but again no survey is plausible. The big hope is neutrino astrophysics. Neutrinos should emerge from deep in the most fascinating places that we could otherwise never see - supernova cores, the centres of stars, the interior of quasar accretion discs. Measurement of solar neutrinos has solved a long standing problem, and set a challenge for particle physics - but what about the rest of the Universe ? New experiments such as ANTARES (under the sea) and AMANDA (under the ice) seem to be clearly detecting cosmic neutrinos, but no distinct sources have yet emerged. Possibly the next generation (ICECUBE) will get there. This looks like the best bet for genuinely unexpected discoveries in the twenty first century. Non-light channels : gravitational waves. Like neutrinos, we know that gravitational waves have to be there somewhere, and their existence has been indirectly proved by the famous binary pulsar timing experiment. However after many years of exquisite technical development, we still have no direct detection of a gravitational wave. The space interferometer mission LISA should finally detect gravitational waves, unless current predictions are badly wrong. However even LISA will not produce a genuine survey. We will detect many events and understand more astrophysics, but will have essentially no idea where they came from, except that hopefully some will correlate with Gamma-ray bursts. If we see totally unexpected signals, it will be very hard to know what to do next. Hyper-space planes : the Virtual Observatory. As we explore the various possible axes one by one, many if not most of them are running out of steam, or too expensive to pursue. But we are a long way short of exploring the whole space - for example narrow line imaging in all Stokes parameters versus time. This exploration does not necessarily need complex new experiments. More survey-quality datasets come on line every year. As formats, access and query protocols, and analysis tool interaction protocols all get standardised, the virtual universe becomes easier for the e-astronomer to explore, and unexpected results will emerge. This, of course, is the agenda of the worldwide Virtual Observatory initiative. Wide Field Surveys : A.Lawrence 13 6 Conclusions Surveys are perhaps the most cost effective and productive way of doing astronomy. In recent years optical, infra-red, and redshift surveys have produced spectacular results in determining cosmological parameters, finding the smallest stellar objects, decoding the history of the Milky Way, and much else besides. Surveys underway now and over the next few years should also produce impressive science. Having been the main engine of discovery for decades, there is a worry now that we have already explored every axis of discovery space. The best hopes for unexpected discoveries may be in massive time domain surveys, in neutrino astrophysics, and in exploring the full multi-dimensional space through the Virtual Observatory. Wide Field Surveys : A.Lawrence 14 7 REFERENCES Becker, R.H. Fan, X., White, R.L., 2001, AJ, 122, 2850. Burrows, A., Sudarsky, D., Lurine, J.I., 2003 ApJ, 596, 587. Burrows, A., Sudarsky, D., Hubeny, I., 2006 ApJ, 640, 1063. Cole, S., Percival, W.J., Peacock, J.A. et al. 2005, MNRAS, 363 505. Colless M.M., Dalton G.B., Maddox S.J., et al. 2001, MNRAS, 328, 1039. Dye, S., Warren, S.J., Hambly, N.C., et al. 2006, MNRAS, 372, 1227. Eisenstein, D.J., Zehavi, I., Hogg, D.W., et al. 2005, ApJ, 633, 560. Efstathiou G., Moody S., Baugh C., et al. 2002, MNRAS, 330, 29. Elgary, ., Lahav, O., Percival, W.J., et al. 2002, Phys.Rev.Lett., 89, 1301. Ellis, S.C., Tinney, C.G., Burgasser, A.J., Kirkpatrick, J.D., McElwain, M.W., 2005, AJ, 130, 2347. Elvis, M, Wilkes, B.J., McDowell, J.C., Green, R.F., Bechtold, J., Willner, S.P., Oey, M.S., Polomski, E. Cutri, R., 1994, ApJSupp, 95, 1. Erdogdu, P., Huchra, J.P., Lahav, O., et al. 2006, MNRAS, 368, 1515. Fan, X., Narayanan, V.K., Lupton, R.H., et al. 2001, AJ, 122, 2833. Gunn, J.E., Peterson, B.A., 1965, ApJ, 142, 1633. Harwit, M., 2003, Physics Today, November 2003, 38. Hewett, P.C., Warren, S.J., Leggett, S.K., Hodgkin, S.L., 2006 MNRAS, 367, 454. Huetsi, G., 2005, A&A, 449, 891. Ibata, R.A., Gilmore, G., Irwin, M.I. 1994 Nature, 370, 194. Jarrett, T.H. 2004, PASA, 21, 396. Jones, D.H., Saunders, W., Colless, M., et al. 2004, MNRAS, 355, 747. Juric, M., Ivezic, Z., Brooks, A., et al. 2005, ApJ submitted (astro-ph/0510520) Kirkpatrick, J.D., Reid, I.N., Liebert, J., Cutri, R.M., Nelson, B., Beichman, C.A., Dahn, C.C., Monet, D.G., Gizis, J.E., Skrutskie, M.F., 1999, ApJ, 519, 802. Kogut, A., Spergel, D.N., Barnes, C., et al. 2003 ApJSupp, 148, 161. Lawrence, A., Warren, S.J., Almaini, O., et al. 2006 MNRAS submitted (astro-ph/0604426) Lodieu, N., Hambly, N.C., Jameson, R.F., Hodgkin, S.T., Carraro, G., Kendall, T.R., 2007, MNRAS, 374, 372. Maddox, S.J., Sutherland, W.J., Efstathiou, G., Loveday, J., 1990, MNRAS, 243, 692. http://arxiv.org/abs/astro-ph/0510520 http://arxiv.org/abs/astro-ph/0604426 Wide Field Surveys : A.Lawrence 15 Mainzer, A.K., Eisenhardt, P., Wright, E.L., Liu, F-C., Irace, W., Heinrichsen, I., Cutri, R., Duval, V., 2006, Proc SPIE, 6256, 61. 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Warren, S.J., Hambly, N.C., Dye, S., et al. 2007a, MNRAS in press (astro-ph/0610191) Warren, S.J., Mortlock, D.J., Legget, S.K., et al. 2007b, MNRAS submitted White, R.L., Becker, R.H., Fan, X., Strauss, M.A., 2003 AJ, 126, 1. York, D.G., Adelman, J., Anderson, J.E., et al. 2000, AJ, 120, 1579 http://arxiv.org/abs/astro-ph/0610191 Why are wide field surveys important ? Major surveys Recent survey science highlights Next steps in optical-IR surveys The end of survey discoveries ? Conclusions REFERENCES

0704.0810

The Cosmic Foreground Explorer (COFE): A balloon-borne microwave polarimeter to characterize polarized foregrounds

The Cosmic Foreground Explorer (COFE): A balloon-borne microwave polarimeter to characterize polarized foregrounds Rodrigo Leonardi a,b, Brian Williams a, Marco Bersanelli c, Ivan Ferreira b, Philip M. Lubin a, Peter R. Meinhold a, Hugh O’Neill a, Nathan C.Stebor a, Fabrizio Villa d, Thyrso Villela b, Carlos A. Wuensche b aPhysics Department, University of California, Santa Barbara, CA 93106 bInstituto Nacional de Pesquisas Espaciais, Divisão de Astrof́ısica, Caixa Postal 515, 12227-010, São José dos Campos, SP, Brazil cDipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16, 20133, Milan, Italy dINAF - IASF Bologna, Via P. Gobetti, 101, 40129, Bologna, Italy Abstract The COsmic Foreground Explorer (COFE) is a balloon-borne microwave polarime- ter designed to measure the low-frequency and low-ℓ characteristics of dominant diffuse polarized foregrounds. Short duration balloon flights from the Northern and Southern Hemispheres will allow the telescope to cover up to 80% of the sky with an expected sensitivity per pixel better than 100 µK/deg2 from 10 GHz to 20 GHz. This is an important effort toward characterizing the polarized foregrounds for fu- ture CMB experiments, in particular the ones that aim to detect primordial gravity wave signatures in the CMB polarization angular power spectrum. Key words: cosmology: observations, cosmic microwave background, polarization foregrounds 1 Introduction Measurement of polarization anisotropies in the Cosmic Microwave Back- ground (CMB) is one of the great challenges in cosmology today. Very sensitive measurements of these anisotropies, particularly at large angular scales, will Preprint submitted to Elsevier Science 30 October 2018 http://arxiv.org/abs/0704.0810v2 provide unique constraints on the influence of gravitational waves on the pro- duction of structure in the very early Universe and information on the epoch of reionization. Several experiments are running or in the planning stages, and long term devel- opment for a future space mission attacking CMB polarization is underway. To date, nearly all of the effort has been directed towards maximizing the number of detectors in the focal plane to achieve the required sensitivity. Relatively little work is going into sub-orbital efforts to constrain polarization fluctua- tions at the largest angular scales, those most interesting for their impact on understanding the inflationary epoch and ionization history of the universe. This is primarily because of an unproven perception that very low multipoles will not be accessible to any but space-based missions. Indeed, large scale polarization has been searched for with ground based experiments over the last 30 years. The COsmic Foreground Explorer (COFE) is a balloon-borne instrument to measure the low frequency and low-ℓ characteristics of some dominant polarized foregrounds. Good understanding of these foregrounds is critical both for interpreting recent results, e.g. Spergel et al. (2006), and for appropriately planning future CMB missions. The experiment also explores low-ℓ limits to CMB polarization measurements at moderate frequencies from non-space based platforms. We believe that balloon and ground-based mea- surements to characterize in detail the polarized microwave sky are essential to prepare a future space mission dedicated to CMB B-modes. 2 Science The CMB radiation field is an observable that provides direct information from the early Universe. The temperature and polarization characteristics of this field impose constraints on cosmological scenarios relevant to understand the origin and the structure of the Universe. Accurate measurements of the CMB are vital to improve our understanding about geometry, mass-energy composition, and reionization of the Universe. Ultimately, the CMB could also provide indirect detection of a stochastic gravitational background and information from the inflationary epoch itself. Having this big picture in mind, several CMB experiments are now trying to constrain the tensor-to-scalar ratio value and to detect the B-mode signature. Among all practical limitations to primordial tensor amplitude detection, con- tamination due diffuse microwave foreground polarized emission is certainly the fundamental one. This emission presents spatial and frequency variations that are not well known, and the residuals from foreground subtraction are restricting our knowledge of CMB polarization. This is particularly true for fu- ture B-mode experiments that will benefit if accurate determinations of spatial and spectral characteristics of polarized foreground are made. For this reason, multifrequency measurements of the polarized foregrounds in the microwaves is now recognized as a key objective within the CMB community. At low frequencies, foregrounds include synchrotron, free-free, and possible spinning dust emission. Synchrotron dominates the low frequency range of the microwave sky. Its emission is caused by relativistic charged particles interact- ing with the Galactic magnetic field and can be highly polarized. Synchrotron measurements provide better understanding of the Galactic magnetic field structure and the density of relativistic electrons across the Galaxy. Free-free emission becomes more important in the microwave intermediate frequency range, and it is due to electron-ion scattering. Free-free is expected to be un- polarized but this might not be true at the edges of HII clouds. Electrical dipole emission from spinning dust has also been suggested by recent obser- vations at low microwave frequencies, e.g. Finkbeiner et al. (2004). COFE is a balloon-borne microwave polarimeter to measure spatial and low- frequency characteristics of diffuse polarized foregrounds. This is an important effort toward characterizing the polarized foregrounds for future CMB experi- ments, in particular the ones that aim to detect primordial gravitational wave signatures in the CMB polarization angular power spectrum. 3 Instrumentation 3.1 Telescope Amodified BEAST telescope design is the basis for the COFE optics (Childers et al., 2005; Figueiredo et al., 2005; Meinhold, P. R. et al., 2005; Mej́ıa, J. et al., 2005; O’Dwyer, I. J. et al., 2005). It consists of an off-axis Gregorian configuration obeying the DragoneMizuguchi condition (Dragone, 1978; Mizuguchi et al., 1978). The telescope is optimized for minimal cross-polarization contamina- tion and maximum focal plane area. The primary reflector is a 2.2 m off-axis parabolic reflector. The incoming radiation is reflected off of the primary re- flector towards a polarization modulating wave plate then to the secondary reflector. The 0.9 m ellipsoidal secondary reflects the incoming radiation to- ward the array of scalar feed horns that couple the radiation to an array of cryogenic low noise amplifiers. The telescope will be mounted in a gondola that has been simplified from a standard balloon-borne design due to the very light carbon fiber optical elements. A schematic of the optics is shown in Figure 1. 3.2 Polarization modulator COFE will employ a low-loss reflective polarization modulator for measur- ing both Q and U simultaneously. It consists of a linear polarizing wire grid mounted in front of a reflecting plate. The wire grid decomposes the input wave into components, parallel and perpendicular to the wires, reflecting the par- allel component with low loss. The perpendicular component passes through the wire grid and reflects off the back short, passes through the grid again and recombines with the parallel component. The distance between the plate and the grid introduces a phase shift between the two components, effectively rotating the plane of polarization of the input wave. A schematic of the polar- ization modulator is shown in Figure 2. Rotating the grid chops between the two polarization states four times per revolution as shown in Figure 3. Tests of this modulator were performed at 41.5 GHz, using a 70 cm telescope. We measured beam patterns for the rotated polarization states and integrated for extended periods on the sky in Santa Barbara, CA. We were able to deter- mine a 1/f knee lower than 50 mHz and very stable long term offsets. We also demodulated sky data to the two different states and calculated the correct combined sensitivity, as seen in Figure 4. The polarization modulator has a broad bandwidth. We achieved 22 dB isola- tion at 20% bandwidth. The radiometric loss of the elements in the modulator can easily be made very low (of order 0.11%) up to relatively high frequencies. The system works for a very wide range of frequency bands. 3.3 Receiver COFE will use InP MMIC 1 amplifiers integrated into simple total power re- ceivers. All of the RF gain will be integrated into a small compact module inside the vacuum chamber. The module will contain 3 to 4 amplifiers (∼ 75 dB of gain), band pass filter, cryogenic detector diode, and an audio ampli- fier. The module avoids the need for cryo/vacuum waveguide feedthrus on the dewar simplifying the overall design. The audio amplifiers will be within the cryostat vacuum vessel for simplicity and noise reasons, but will be at ambient temperatures. COFE has a modest number of feeds required, and no ortho- mode transducers or hybrid tees, so the passive components are minimal. A schematic of the receiver is shown in Figure 5. 1 Indium Phosphide Monolithic Microwave Integrated Circuit 3.4 Data acquisition/demodulation Data acquisition will use the same technique we have been using in our test system, namely synchronous sampling of analog integrators. We oversample the data by a large factor and perform the demodulation of Q and U Stokes parameters (and other modes for systematic error analysis) in software. This yields the most information and allows a variety of post-flight tests includ- ing null signal analysis and analysis of the DC or total power components (contaminated with 1/f , but still useful for systematic tests). 3.5 Ground-based B-machine prototype A prototype polarimeter for a B-mode project, named B-machine, is being deployed at the WMRS 2 Barcroft facility, CA (118◦14′ W longitude, 37◦35′ N latitude, 3800 m altitude). The WMRS facility is an excellent site for mi- crowave observation because of a cold microwave zenith temperature, low precipitable water vapor, and a high percentage of clear days (Marvil et al., 2006). Many of the components that will be used by the B-machine prototype are useful for COFE as well. For example, the prototype will allow systematic checks of the polarization modulator, and COFE scan strategy. The B-machine prototype will be able to yield some basic higher multipole results on the fore- grounds as well as the polarization signature and establish a data analysis pipeline. The prototype possesses telescope and detector technology identical to COFE. It has 2 Ka-band and 6 Q-band channels centered at 31 and 41.5 GHz with FWHM resolution of 28′ and 20′ respectively. The receiver has been previ- ously used in anisotropy measurements (Childers et al., 2005). The telescope runs at constant elevation while continuously scanning the sky in azimuth. A photograph of B-machine prototype is shown in Figure 6. 4 Performance For any sub-orbital CMB experiment, minimizing atmospheric contamination is important. For the COFE bands, total atmospheric emission at our target altitude of 35 km is less than 1 mK. Common broad band bolometric at- mospheric antenna temperature contributions at balloon altitudes are several hundred mK or more. Since the effective CMB antenna temperature drops 2 White Mountain Research Station with frequency, our effective atmospheric signal is approximately 1000 times less than for a bolometric balloon-borne system. Hence low-ℓ information from a balloon-borne system is very clean by comparison. Figure 7 shows the atmo- sphere and predicted foreground emission over a range of frequencies interest- ing for CMB work (the foreground prediction is calculated from Bennett et al. (2003)). 4.1 Receiver bands and expected receiver sensitivity Receiver sensitivity can be estimated according to the radiometer equation σT = K Tsys + Tsky ∆ν · τ , (1) where σT is the root-mean-square noise, Tsys is the system noise temperature, Tsky is the sky antenna temperature, ∆ν is the bandwidth, τ is the integration time, and K is the sensitivity constant of the receiver. For COFE and B-machine prototype the sensitivity constant of each receiver is K = π . The signal is sine wave modulated, reducing the sensitivity by a factor as compared with a standard Dicke receiver, with an addition factor of from the the standard definition for Q and U in the Rayleigh-Jeans regime of the CMB spectrum. Table 1 shows our estimation of the sensitivity of each receiver. Table 1 – Instrument parameters. COFE B-machine Central frequency (GHz) 10 15 20 31 41.5 FWHM beam (arcmin) 83 55 42 28 20 Tsys (K) 8 10 12 25 27 Tsky (K) at target altitude 3 2.5 2.4 2.3 6.4 13.0 Bandwidth (GHz) 4 4 5 10 7 Number of receivers 3 6 10 2 6 Sensitivity per receiver (µK s) 261 308 318 493 751 Aggregate sensitivity (µK s) 151 126 100 348 307 3 For COFE and B-machine (ground based), we compute expected Tsky antenna temperature at target altitude of 35 km and 3.8 km, respectively. By increasing the number of receivers, future ground-based or balloon-borne experiments can significantly improve aggregate sensitivity. For instance, 30 detectors could reach 61 µK s and 107 µK s at 30 and 40 GHz, respectively. 4.2 Scan strategy, sky coverage and expected map sensitivity COFE uses a simple scan strategy to cover the largest available sky area in each flight. The telescope will be pointed nominally 45◦ from the horizon to minimize ground and balloon pickup, and the gondola will rotate constantly at approximately 1/2 rpm. Data acquisition sample rate will be synchronized with the polarization rotator (at ∼ 30 Hz). For instance, using this strategy, a 24 hour flight from Fort Sumner, NM, allows to cover 59% of the sky area with a median aggregate pixel sensitivity of 92 µK/deg2, 77 µK/deg2, and 61 µK/deg2 at 10 GHz, 15 GHz, and 20 GHz respectively. COFE will acquire data from nearly all of the sky (∼ 93%). This will be achieved in a set of 12 and/or 24 hour flights from the Northern and Southern Hemispheres. Figure 8 provides estimates for sensitivity per square degree pixel over the whole sky for our flight plans. Figure 9 illustrates the expected sky coverage. The B-machine prototype focuses on higher multipoles but uses a similar scan- ning strategy from the ground. For a conservative 60 day observing campaign at WMRS, we expect to cover 56% of the sky with an median aggregate sensi- tivity of 27 µK/deg2, and 23 µK/deg2 at 31 GHz and 41.5 GHz, respectively. 5 Conclusion Over the next few years we will field a balloon-borne telescope to map more than 90% of the sky. Both polarization anisotropy and polarized foregrounds will be measured over several bands. This is an important effort toward char- acterizing the polarized foregrounds for future CMB experiments. In addition to foreground detection, COFE will better characterize the po- larization modulation capability for measuring Q and U simultaneously. As discussed earlier, a large scale ground-based campaign will capitalize on the technology that has been developed by COFE and B-machine prototype. It is clear that our current understanding of the polarization foregrounds limits our ability to make accurate observations of the B-mode signature. COFE will lessen the effect that incomplete models of foregrounds will have on future experiments. 6 Acknowledgments We acknowledge support from the National Aeronautics and Space Admin- istration (NASA), and the California Space Institute (CalSpace). T.V. and C.A.W. acknowledge CNPq Grants 305219/2004-9 and 307433/2004-8, respec- tively. Some of the results have been derived using the HEALPix 4 (Górski et al., 2005) package. References Bennett, C. L. et al. 2003, ApJS, 148, 97 Childers, J. et al. 2005, ApJS, 158, 124 Dragone, C., 1978, The Bell System Technical Journal, 57, 7, 2663 Figueiredo, N. et al. 2005, ApJS, 158, 118 Finkbeiner, D. P. et al. 2004, ApJ, 617, 350 Górski, K. M. et al. 2005, ApJ, 622, 759 Marvil, J. et al. 2006, New Astronomy, 11, 218 Meinhold, P. R. et al. 2005, ApJS, 158, 101 Mej́ıa, J. et al. 2005, ApJS, 158, 109 Mizuguchi, Y., Akagawa, M., and Yokoi, H., 1978, Electronics and Communi- cations In Japan, 61, 58 O’Dwyer, I. J. et al. 2005, ApJS, 158, 93 Spergel, D. N. et al. 2006, astro-ph/0603449 4 http://healpix.jpl.nasa.gov http://arxiv.org/abs/astro-ph/0603449 Fig. 1. Optical schematic for COFE and B-machine prototype telescopes, an off-axis Gregorian configuration optimized for minimal cross-polarization contamination. A 2.2 m parabolic reflector primary, a 0.9 m ellipsoidal secondary, and a 0.3 m rotator grid are shown. Fig. 2. Schematic of the polarization modulator. The input wave is decomposed into its two linear polarization states, parallel and perpendicular to the wires (rep- resented by dots just above the conducting reflector). The perpendicular component is phase shifted from the extra path length. When added back to the parallel com- ponent, the plane of polarization of the input wave is rotated. Fig. 3. Sample signal from a polarized thermal source. A single revolution of the modulator is shown, along with the reference signal to be used for demodulation. Commutating using this signal yields Q, for instance, while demodulating with a reference phase shifted by π/4 gives U . Fig. 4. Sample data from our room temperature radiometer viewing the sky at 41.5 GHz. The undemodulated PSD displays the 1/f knee of the HEMT radiometer of 10 Hz and a white noise of 5.4 mK s. The demodulated data have no visible 1/f and a white noise level consistent with expectation. Fig. 5. Radiometer layout for COFE. Fig. 6. Picture of prototype telescope to be deployed at WMRS. Fig. 7. Atmosphere, CMB, and predicted foreground emission from 5 to 300 GHz. COFE bands run from 10 to 20 GHz. The zenith atmosphere emission is shown at 3.8 and 35 km. The atmospheric emission and lines are mainly due to H2O, O2, and O3. For the target altitude of 35 km, we expect well under 1 mK total emission from the atmosphere. Foreground spectral index β for free-free, synchrotron, and dust were assumed, respectively, as −2.15, −2.7, and 2.2. Fig. 8. Integrated histogram of anticipated aggregate sensitivity per 1 deg2 pixel assuming a 24 hour flight from the Northern Hemisphere (Fort Sumner, NM) and a 24 hour flight from the Southern Hemisphere (Alice Springs, Australia). For each COFE band, we plot the fraction of the entire sky measured with better than a given aggregate sensitivity. The change of the curves slope is due to the fact that 35% of the sky can be observed from both hemispheres using COFE scan strategy. Fig. 9. Sky coverage for COFE assuming a 24 hour flight from the Northern Hemi- sphere (Fort Sumner, NM) and a 24 hour flight from the Southern Hemisphere (Alice Springs, Australia). The region observed contains nearly the entire sky (93%). The darker strip shows the overlap between the two observations. For illustration pur- poses, we show the diffuse Galactic structure obtained adding synchrotron, free-free and dust maps at 23 GHz (Bennett et al., 2003). Introduction Science Instrumentation Telescope Polarization modulator Receiver Data acquisition/demodulation Ground-based B-machine prototype Performance Receiver bands and expected receiver sensitivity Scan strategy, sky coverage and expected map sensitivity Conclusion Acknowledgments

0704.0811

Jet interactions in massive X-ray binaries

To appear in “Massive Stars: Fundamental Parameters and Circumstellar Interactions (2007)”RevMexAA(SC) JET INTERACTIONS IN MASSIVE X-RAY BINARIES Gustavo E. Romero 1,2 RESUMEN Los sistemas binarios masivos de rayos X están formados por un objeto compacto que acreta materia del viento estelar de una estrella donante de tipo temprano. En algunos de estos sistemas, llamados microcuásares, chorros de part́ıculas relativistas son eyectados desde las cercańıas del objeto compacto. Estos chorros interactúan con el campo de fotones de la estrella compañera, con el viento estelar, y, a grandes distancias, con el medio interestelar. En este trabajo se resumirán los principales resultados de tales interacciones con especial énfasis en la producción de fotones de alta enerǵıa y neutrinos. El caso de algún sistema particular, como ser LS I +61 303, será discutido con algún detalle. Además, se presentarán las perspectivas futuras para observaciones a diferentes longitudes de onda para este tipo de objetos. ABSTRACT Massive X-ray binaries are formed by a compact object that accretes matter from the stellar wind of an early-type donor star. In some of these systems, called microquasars, relativistic jets are launched from the surroundings of the compact object. Such jets interact with the photon field of the companion star, the stellar wind, and, at large distances, with the interstellar medium. In this paper I will review the main results of such interactions with particular emphasis on the production of high-energy photons and neutrinos. The case of some specific systems, like LS I +61 303, will be discussed in some detail. Prospects for future observations at different wavelengths of this type of objects will be presented. Key Words: GAMMA RAYS: THEORY — GAMMA RAYS: OBSERVATIONS — JETS AND OUT- FLOWS — STARS: BINARIES — STARS: MICROQUASARS 1. INTRODUCTION Massive stars use to form binary systems. In such systems one of the stars evolves faster than the other. At the end of the lifetime of this star a supernova explosion will occur, and either a neutron star or a black hole will be left behind. If the system is not disrupted by the explosion, the compact object will start to accrete matter from the stellar wind of its early-type companion. Since the matter has angu- lar momentum, it will form an accretion disk around the compact star. The matter will be heated in the disk, losing angular momentum and falling into the potential well. The hot disk will cool through the emission of X-rays. We say then that a massive X- ray binary (HMXRB) is born. There are around 120 HMXRBs detected in the Galaxy so far (Liu et al. 2006). Some of these systems present non-thermal radio emission. This emission is thought to be syn- chrotron radiation produced by relativistic electrons in a jet that is somehow ejected from the surround- 1Facultad de Ciencias Astronómicas y Geof́ısicas, Univer- sidad Nacional de La Plata, Paseo del Bosque, 1900 La Plata, Argentina ([email protected]). 2Instituto Argentino de Radioastronomı́a, Casilla de Correos No. 5, (1894) Villa Elisa, Buenos Aires, Argentina ([email protected]). ings of the compact object. When the jet is resolved at radio wavelengths through interferometric tech- niques or at X-rays, the HMXRB is called a high- mass microquasar (Mirabel et al. 1992). The word ‘microquasar’ (MQ) was coined to em- phasize the similarities between galactic jet sources and extragalactic quasars (Mirabel & Rodŕıguez 1998). These similarities, although important, should not make us to overlook the also important differences between both types of objects. The main difference is, of course, the presence of a donor star in the case of MQs. In high-mass MQs, this star pro- vides a strong photon field, a matter field in the form of a stellar wind, and a gravitational field that can act upon the accretion disk producing a torque and inducing its precession. The photon and matter field constitute targets for the relativistic particles in the jet. The interaction of the jets with these fields can produce a variety of phenomena that are absent in the case of extragalactic jets. The aim of the present article is to review these phenomena. 2. WHAT IS A MICROQUASAR? A microquasar is an accreting X-ray binary sys- tem with non-thermal jets. The basic ingredients of a MQ are shown in Figure 1. They are the compact http://arxiv.org/abs/0704.0811v1 2 GUSTAVO E. ROMERO ACCRETION DISC (hard X-rays) ‘Corona’ CORONA Accretion disc Accreting neutron star or black (radio - ?) (optical - soft X-rays) Mass- donating companion star (IR-optical) Mass-flow > 1radio infrared optical soft-X hard-X gamma-ray COMPANION Fig. 1. Sketch showing the different components of a microquasar and the energy bands at which they emit. Not to scale. From Fender & Maccarone (2004). object, the donor star, the accretion disk, the jets, which usually are relativistic or mildly relativistic, and a region of hot plasma called the ‘corona’ that surrounds the compact object. If the star is an early- type, hot star, the accretion can proceed through capture of the wind material. In the case of low- mass stars and in some close systems, the accretion occurs through the overflow of the Roche lobe. In what follows we will focus only on high-mass MQs. The donor star can produce radiation from the IR up to UV energies. The accretion disk produces soft X-rays, whereas the corona is responsible for hard X-rays that are likely generated by Comptonization of disk photons. The emission of the jets goes from radio wavelengths to, in some cases like LS 5039, gamma-rays. MQs, like blazars, can emit along the entire electromagnetic spectrum. Their spectral en- ergy distribution (SED) is complex, being the result of a number of different radiative processes occurring on different size-scales in the MQs. MQs present different spectral states at X-rays. The two basic state are the ‘soft’ state and the ‘hard’ state. In the former the SED is dominated by a grey- body peak around E ∼ 1 keV, probably due to the contribution of the accretion disk, which extends in this state all the way down to the last stable or- bit around the compact object. In the hard state the peak in the X-rays is shifted toward lower ener- gies and a strong and hard power-law component is present up to energies ∼ 150 keV, in some cases even beyond. This emission is usually interpreted as soft X-ray Comptonization in the corona (e.g. Ichimaru 1977), although some authors have suggested that it could be produced in the jet through external inverse Compton (IC) interactions (Georganopoulos et al. 2002) or through synchrotron mechanism (Markoff et al. 2001, 2003). The sources spend most of the time in the hard state. It is in this state when a steady, self-absorbed radio jet is usually observed. The transition form one state to the other is commonly accompanied by the ejection of superluminal components, that can be de- tected as moving radio blobs (Mirabel & Rodŕıguez 1994, Fender et al. 2004). 3. WHAT ARE JETS MADE OF? One of the most important open issues concern- ing MQs is the nature of the matter content of the jets. We know for sure that relativistic lep- tons with a power-law distribution are present in the jets since we can detect and measure their syn- chrotron radiation. The relativistic outflow can be made of relativistic electron-positron pairs, or al- ternatively it could be a relativistic proton-electron plasma. Another possibility is a plasma formed by a cold electron-proton fluid, where the particles would have a thermal distribution, plus a relativistic con- tent, locally accelerated by shocks (Bosch-Ramon, Romero & Paredes 2006). In this kind of jets, the bulk of the momentum is carried out by the cold plasma, which additionally confines the relativistic component. In any case, the large perturbations observed in the interstellar medium (ISM) around some MQs like Cygnus X-1 (Gallo et al. 2005) and SS 433 (Dubner et al. 1998), strongly suggest that the jets are bari- onic loaded. The direct detection of iron lines in the case of the jets of SS 433 (Kotani et al. 1994, 1996; Migliari et al. 2002) clearly confirms that they con- tain hadrons, at least in this particular object. Since there seems to be a clear correlation between the ac- cretion and ejection of matter in MQs (Mirabel et al. 1998), it is natural to assume that the content of the jets does not basically differ in nature from that of the accreting matter. All these considerations make quite likely the presence of relativistic protons in the jets of MQs. Hence, their radiative signatures can not be neglected in a serious analysis of the radia- tive processes in these sources. 4. JET INTERACTIONS What does happen when a relativistic jet pass through the medium that surrounds a hot, massive star?. The radiation field of the star penetrates freely into the jet and the dominant UV photons will in- teract with relativistic particles in the outflow. The interaction of the stellar wind with the jet will form JET INTERACTIONS IN HMXRBS 3 a boundary layer where shocks will likely be formed, but some level of fluid mixing is expected to occur. The interaction between relativistic particles from the jet and thermal particles of the wind will take place, producing high-energy emission. We can sep- arate the microscopic jet-stellar environment inter- actions in two groups, according to whether they are of leptonic or hadronic nature. Of course, both types of reactions will occur in a specific system, but ac- cording to the given conditions, one type or the other might dominate the high-energy output of the MQ. Let us briefly discuss both cases. 4.1. Leptonic interactions Relativistic electrons and positrons in the jet will IC scatter soft photons up to high energies. The origin of these photons can be diverse: stellar UV photons, X-ray photons from the accretion disk and the hot corona around the compact object, or non- thermal photons produced in the jet by synchrotron mechanism. At high energies, the interaction en- ters in the Klein-Nishina regime, where the cross section decreases dramatically. Opacity effects to gamma-ray propagation due to the presence of the local photon fields can result in the generation of IC cascades within the binary system (Bednarek 2006a, Orellana et al. 2007). Relativistic leptons can in- teract with cold protons and nuclei from the stellar wind producing high-energy emission through rela- tivistic Bremsstrahlung. A number of papers have been devoted to leptonic interactions in MQs in re- cent years, for instance, Atoyan & Aharonian (1999), Markoff et al. (2001, 2003), Georganopolous et al. (2002), Kaufman Bernadó et al. (2002), Romero et al. (2002), Bosch-Ramon & Paredes (2004), Bosch- Ramon et al. (2005a, 2006), Paredes et al. (2006), Dermer & Böttcher (2006), Gupta et al. (2006), Bednarek (2006b), etc. The reader is referred to these papers and references therein for detailed dis- cussions. In Figure 2 we show the broadband SED ex- pected from leptonic interactions in a high-mass MQ. The different contributions are indicated. It can be seen that the synchrotron emission can extend up to MeV energies and that in the GeV-TeV range the dominant process is IC upscattering of stellar pho- tons. Figure 3 shows a detail of the SED at high energies. Notice that absorption by photon-photon annihilation has been taken into account in the fi- nal curve, yielding a soft spectrum around 100 GeV (Bosch-Ramon et al. 2006). −5 −3 −1 1 3 5 7 9 11 13 log(photon energy [eV]) observed SED IC emission seed photons ext. Bremsstr. int. Bremsstr. star IC corona IC disk IC sync. corona Fig. 2. Spectral energy distribution of a high-mass MQ. The different contributions to the total SED are shown. From Bosch-Ramon, Romero & Paredes (2006). 6 7 8 9 10 11 12 13 log(photon energy [eV]) observed SED star IC synchrotron corona IC Fig. 3. High-energy emission from high-mass MQ. Cour- tesy of V. Bosch-Ramon. 4.2. Hadronic interactions The main reaction for proton cooling in a high- mass MQs is pp interaction, through the channels pp → p + p + π0 and pp → p + p + ξ(π+ + π−), where ξ is the π± multiplicity. The neutral pions decay yielding gamma rays, π0 → γ + γ, whereas the charged pions produce neutrinos and e± pairs: π± → µ±νµν̄µ → e ±νeν̄e. The gamma-ray spectrum will mimic at high-energies the spectrum of the par- ent relativistic proton population. In general, since proton losses are not as severe as electron losses in the inner region of the source, we could expect a higher energy cutoff in hadronic-dominated sources. Models for hadronic MQs have been developed 4 GUSTAVO E. ROMERO 7 8 9 10 11 12 13 14 Log E [eV] θ = 0º θ = 30º θ = 60º θ = 90º 0 60 120 180 θ [º] Beaming factor Fig. 4. Spectral energy distributions for the hadronic emission of a high-mass MQ with a smooth stellar wind. Different curves correspond to different viewing angles. From Orellana et al. (2007). by Romero et al. (2003), Romero et al. (2005), Romero & Orellana (2005) and Orellana & Romero (2007). Neutrino production in this kind of models is discussed by Romero & Orellana (2005), Aharo- nian et al. (2006), Benarek (2005) and Christiansen et al (2006). For photo-pion production of neutri- nos, under rather extreme conditions, see Levinson & Waxman (2001). Figure 4 shows different SED obtained from pp interactions for a high-mass MQ with a smooth spherical wind (Orellana et al. 2007). The vari- ous curves correspond to different viewing angles. The total jet power in relativistic protons is Lrel 6 × 1036 erg s−1. The jet is assumed to be perpen- dicular to the orbital plane, but this constraint can be relaxed to allow, for instance, for a precessing jet. Actually, in some systems, the jet could point even in the direction of the star (Butt et al. 2003, Romero & Orellana 2005). In such a case, jet-induced nucle- osynthesis can occur in the stellar atmosphere. The power of the stellar wind might be, for some stars, strong enough as to stop the jet creating a stand- ing shock between the compact object and the star. Protons and electrons might be re-accelerated there up to very high energies, producing a detectable TeV source. All existing models for hadronic MQs assume a smooth wind from the star3. However, it would be quite possible that the wind have some struc- ture, for instance in the form of clumps, a fact that 3See, nonetheless, the paper by Aharonian & Atoyan (1996) that, although not framed in the context of MQs, dis- cusses the interaction of a proton beam with a cloudy medium around a star. would lead to gamma-ray variability on short time scales. If such a variability could be detected by fu- ture Cherenkov telescope arrays, it might be used to infer the structure of the wind. The jet would act as a kind of lantern illuminating the wind in gamma- rays to the observer. Hadronic jets can propagate through the ISM producing hot spots similar to those observed in the case of extragalactic sources (Bosch-Ramon et al. 2005b). Particles re-accelerated at the termination point of the jets, can diffuse in the ambient medium, interacting with diffuse material and producing ex- tended high-energy sources. 5. THE CONTROVERSIAL CASE OF LS I +61 303 LS I +61 303 is a puzzling Be/X-ray binary, which displays gamma-ray variability at high ener- gies. The nature of the compact object and the origin of the high-energy emission is unclear. The detection of jet-like radio features by Massi et al. (2001, 2004) led to the classification of this source as a MQ. This has been recently challenged by Dhawan et al. (2006), who observed the source with the VLBA at different orbital phases concluding that the direction the jet-like feature during the perias- tron passage (opposed to the primary star) supports the scenario of a colliding wind model where the com- pact object is an energetic pulsar (wind power∼ 1036 erg/s). The system has been detected by the MAGIC telescope at E > 200 GeV. The variability is modu- lated with the orbital period. Contrary to the expec- tations the maximum of the gamma-ray emission oc- curred well after the periastron passage. The source was not clearly detected during the periastron (Al- bert et al. 2006). The cause of this could be gamma- ray absorption in the combined photon field of the Be star and its decretion disk (e.g. Orellana & Romero 2007). Figure 5 shows the electromagnetic cascades that might develop close to the periastron passage (which occurs at phase 0.23). According to these simulations the source should be detectable during the periastron, but with longer exposures, and the spectrum will be softer than what was observed at phases 0.6-0.7. The pulsar/Be binary interpretation goes not free of severe problems. The flux at MeV-GeV energies observed by EGRET (Kniffen et al. 1997) accounts for a luminosity of ∼ 1036 erg/s, which would imply an impossible conversion efficiency from wind power to gamma-rays of ∼ 1. In addition, since the pulsar wind would be orders of magnitude stronger than the slow Be equatorial wind, the observed ‘cometary tail’ JET INTERACTIONS IN HMXRBS 5 Fig. 5. Electromagnetic cascades at different phases de- veloped close to the periastron passage of the X-ray bi- nary LS I +61 303 (from Orellana & Romero 2007). radio feature, if interpreted as synchrotron radiation from electrons accelerated at the colliding wind re- gion, should point out toward the primary star, and not opposite to it. It is clear that LS I +61 303 is a interesting and peculiar system that deserves more intensive studies in the near future. 6. CONCLUSIONS MQs are outstanding natural laboratories to study a variety of physical phenomena such as par- ticle acceleration, accretion physics, and particle in- teractions. Observations of gamma-ray emission of high-mass MQs can be used to probe the structure of stellar winds and the nature of the matter content of relativistic jets. How many MQs are there in the Galaxy?. It is difficult to answer this questions, but it seems possi- ble that a significant number of the yet-unidentified variable gamma-ray sources located on the galactic plane (Romero et al. 1999) could be associated with high-mass MQs (Romero et al. 2004, Bosch-Ramon et al. 2005a). In the next few years, new Cherenkov telescope arrays like HESS II, MAGIC II, and VER- ITAS, along with the satellite observatories AGILE and GLAST, will continue detecting these extraor- dinary objects at high energies and helping to pene- trate into their mysteries. Acknowledgments This work has been supported by the Agencies CONICET (PIP 5375) and ANPCyT (PICT 03- 13291 BID 1728/OC-AR). I thank the organizers for a wonderful meeting and a warm hospitality. REFERENCES Albert, J. et al. (MAGIC coll.) 2006, Science, 312, 1771 Aharonian, F. A., & Atoyan, A. M. 1996, Space Sci. Rev., 75, 357 Aharonian, F. A., Anchordoqui, L. A., Khangulyan, D., & Montaruli, T. 2006, Journal of physics: conference series, 39, 408 Atoyan, A. M., & Aharonian, F. A. 1999, MNRAS, 302, Bednarek, W. 2005, ApJ, 631, 466 Bednarek, W. 2006a, MNRAS, 368, 579 Bednarek, W. 2006b, MNRAS, 371, 1737 Bosch-Ramon, V. & Paredes, J. M. 2004, A&A, 417, 1075 Bosch-Ramon, V., Romero, G. E., & Paredes, J. 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0704.0812

The Na I D resonance lines in main sequence late-type stars

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 3 October 2018 (MN LATEX style file v2.2) The Na I D resonance lines in main sequence late-type stars Rodrigo F. Dı́az, Carolina Cincunegui and Pablo J. D. Mauas Instituto de Astronomı́a y Fı́sica del Espacio, CC. 67, suc. 28, 1428. Buenos Aires, Argentina. 3 October 2018 ABSTRACT We study the sodium D lines (D1: 5895.92 Å; D2: 5889.95 Å) in late-type dwarf stars. The stars have spectral types between F6 and M5.5 (B−V between 0.457 and 1.807) and metallicity between [Fe/H] = −0.82 and 0.6. We obtained medium resolution echelle spectra using the 2.15-m telescope at the argentinian observatory CASLEO. The observations have been performed periodically since 1999. The spectra were calibrated in wavelength and in flux. A definition of the pseudo-continuum level is found for all our observations. We also define a continuum level for calibration purposes. The equivalent width of the D lines is computed in detail for all our spectra and related to the colour index (B − V) of the stars. When possible, we perform a careful comparison with previous studies. Finally, we construct a spectral index (R′D) as the ratio between the flux in the D lines, and the bolometric flux. We find that, once corrected for the photospheric contribution, this index can be used as a chromospheric activity indicator in stars with a high level of activity. Additionally, we find that combining some of our results, we obtain a method to calibrate in flux stars of unknown colour. Key words: Stars: late-type - Stars: activity - Stars: chromospheres 1 INTRODUCTION All through the main sequence, the Na I D resonance lines (D1: 5895.92 Å; D2: 5889.95 Å) are ubiquitous absorption features, clearly visible in stars of all spectral types. In particular, for cool stars at the end of the main sequence (late G, K and M) the dou- blet develops strong absorption wings. In the most active flare stars, the D lines show chromospheric emission in their core, a telltale of collision dominated formation processes. Besides the intrinsic interest in understanding how these lines behave for different stellar parameters, the sodium doublet provides a useful diagnostic tool when studying stellar atmospheres. In this regard, Andretta et al. (1997), Short & Doyle (1998) and Mauas (2000) showed that in M dwarfs the sodium D lines provide infor- mation of the conditions in the middle-to-lower chromosphere, and therefore complements the diagnostics of the upper-chromosphere and low transition region provided by Hα. Tripicchio et al. (1997) modelled the equivalent width of the doublet in stars of a wide range of spectral types (between F6 and M5.5). They concluded that the chromosphere is not very important in determining the equivalent width, since it only affects emission in the central core, which provides a small contribution to the doublet strength. How- ever, their model fails to reproduce the tendency shown by their observations for stars with Te f f < 4000 In the present work we study different features of the D lines using medium resolution echelle spectra covering the entire visi- ble spectrum. Our study is focused on stars at the end of the main sequence (from late F to middle M), for which we analyze the con- tinuum and line fluxes and the equivalent width of the doublet. In Sect. 2 we describe the observations and calibration pro- cess. In Sect. 3 we study the continuum flux near the D lines. The equivalent width is treated in Sect. 4, where we also describe a method to obtain an approximate flux calibration for stars of un- known colour index. In Sect. 5 we study the changes observed in the D lines as a consequence of chromospheric activity. Finally, in Sect. 6 we discuss the results and present our conclusions. 2 OBSERVATIONS AND STELLAR SAMPLE 2.1 Observations and calibration of the spectra The observations were obtained at the Complejo Astronómico El Leoncito (CASLEO), in the province of San Juan, Argentina, in 27 observing runs starting in March 1999. At present, the spectra of only 19 of these runs were reduced and calibrated. We used the 2.15-m telescope, equipped with a REOSC echelle spectrograph designed to work between 3500 Å and 7500 Å. As a detector we used a 1024 x 1024 pixel TEK CCD. We obtained spectra covering the whole region between 3860 and 6690 Å, which lies in 24 echelle orders. Due to this complete coverage we can study the effect of chromospheric activity in all the optical spectrum simultaneously. In the present work we concen- trate on the sodium D lines. The study of other lines is done some- where else (see Cincunegui et al. 2006; Buccino & Mauas 2007). We used a 300 µm-width slit which provided a resolving power of R = λ/δλ ≈ 26400. This corresponds to a spectral res- olution of around 0.22 Å at the centre of the D doublet. Addition- ally, we obtained ThAr spectra for the wavelength calibration and medium resolution long slit spectra that were employed for the flux calibration. c© 0000 RAS http://arxiv.org/abs/0704.0812v1 2 Dı́az et al. The observations were reduced and extracted using standard IRAF1 routines. The details of the flux calibration process are pre- sented in Cincunegui & Mauas (2004). 2.2 Stellar sample Since 1999 we have been monitoring around 110 stars of the main sequence on a regular basis. Each star was observed around three times a year, weather permitting. The stellar sample is presented in Table 4. The stars have spectral types between F6 and M5.5 and colour indexes (B − V) between 0.457 and 1.807, from the Hipparcos/Tycho catalogues (Perryman et al. 1997; Hoeg et al. 1997). The metallicity was ob- tained from different sources, as shown in Table 4 and range from [Fe/H] = −0.82 to 0.6. The sample was gathered for different research programmes. All the stars are field stars, and were included either for being sim- ilar to the Sun, for exhibiting high levels of chromospheric activ- ity -as is usually the case for the M stars-, or for having planetary companions. We have also observed a few sub-giant stars for cal- ibration purposes, which were excluded from the present analysis. In previous works we used 18 of the stars included in the sample to calibrate the S index calculated at CASLEO (Cincunegui & Mauas 2002; Cincunegui et al. 2006) to the one obtained at Mount Wilson (Vaughan et al. 1978). These 18 stars also belong to the group of standard stars used by Henry et al. (1996) at Cerro Tololo. Several stars were included to cover the entire range of effective tempera- ture. However, the sample is not uniform in metallicity or effective temperature. Nevertheless, we believe our results are not biassed by this lack of uniformity in our sample. When such a risk is present, we binned the observations (see Sect. 3). Our sample also includes seven stars that fall outside the main sequence when placed in an HR diagram, even though they are clas- sified as dwarf stars in the SIMBAD2 database. For these stars, we calculated the absolute magnitude in the V filter (MV ) using the measurements of parallax and V from the Hipparcos/Tycho cata- logues (Perryman et al. 1997; Hoeg et al. 1997). We list these stars in Table 1. The fourth column shows the luminosity class that cor- responds to the calculated MV (see Allen 1964, §95), obtained as- suming that the spectral type is correct. These stars were excluded from the analysis, together with the rest of the non-main-sequence stars. Also contained in our sample are four stars which are listed in a luminosity class other than V in the SIMBAD database that ac- tually fall well whithin the main sequence. Again, MV values were obtained and compared with the table in Allen (1964) to confirm their placement in the HR diagram. These stars are shown in Ta- ble 2, and were included in the analysis. We also excluded from the analysis the stars with unknown parallax, since for them abso- lute fluxes cannot be determined. Our final sample consists of 652 spectra for 84 stars. 1 IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. 2 http://simbad.u-strasbg.fr Spectral type (B − V) MV Lum Class hd103112 K0 1.05 3.18 IV hd105115 K2/K3 1.41 -0.68 III hd119285 K1p 1.04 3.39 IV hd25069 G9 0.99 2.75 IV hd5869 K4 1.49 1.05 III hd94683 K4 1.78 -2.38 II hd17576 G0 1.78 1.91 III Table 1. Stars located outside the main sequence in an HR diagram despite being classified as dwarf-type stars. The fourth column is the luminosity class we obtained assuming the spectral type is correct Star Spectral type (SIMBAD) (B − V) MV hd52265 G0 III-IV 0.57 4.12 hd57555 GO IV/V 0.66 4.18 hd75289 G0 I 0.58 4.1 hd120136 F6 IV 0.51 3.58 Table 2. Stars located in the main sequence despite not being classified as dwarf-type stars. 3 PSEUDO-CONTINUUM 3.1 Definition In Fig. 1 we show the spectrum in the region of the D lines for four stars of different spectral types. As can be seen in the spectra, in the case of later stars the doublet forms inside a strong molec- ular band, which extends from 5847 Å to 6058 Å and is due to TiO (Mauas & Falchi 1994) Therefore, in order to obtain a reliable estimate of the contin- uum flux near the lines for all spectral types, the windows chosen have to be located far away from the line centre, outside the molec- ular bands. The closest regions are a 10 Å wide window centered at 5805 Å (window V) and a 20 Å wide window centered at 6090 Å (window R). These windows are indicated in Fig. 1 with solid lines. However, even in these regions there are several photospheric lines, and therefore computing the mean flux would lead to a con- siderable underestimation of the continuum flux. To deal with this problem, we considered the mean value of the ten highest points in each window as an indication of the continuum flux in it. Finally, to take into account the shape of the continuum in this region, we computed the flux at the doublet centre (Fcont) as the linear interpo- lation of the values in each window. In Fig. 1, the linear interpolation in each spectrum is shown as a dashed line. We see that this is a good estimation for the contin- uum flux at the centre of the doublet, even for M-type stars. 3.2 Relation with (B-V) We calculated Fcont for all the spectra in our sample in the way explained above. Fcont was normalized to the flux F⊙ , defined in such a way that Fcont/F⊙ = 1 for a star of solar colour (B − V) = 0.62. In Fig. 2 we plot Fcont/F⊙ in logarithmic scale as a function of colour index (B − V). The trend is similar to that shown by the lower end of the main sequence in an HR diagram. c© 0000 RAS, MNRAS 000, 000–000 The Na I D resonance lines in main sequence late-type stars 3 5800 5850 5900 5950 6000 6050 6100 Figure 1. The region of the sodium doublet for stars of different spectral types. The dotted lines mark the windows used for the calculation of the continuum flux. The dashed line is the continuum level as defined in Sect. 3.1. As can be seen in Fig. 2, the stars are not evenly distributed with colour, a fact that might bias our results. To avoid this, we binned the observations every 0.05 units in (B − V). For each bin, we computed the mean value of Fcont , which are plotted as filled squares in Fig. 2. It can be seen in the figure that there exists a break at (B − V) = 1.4, where M stars begin. Therefore, we fitted the binned data with two linear relations: Fcont 1.18 − 1.86 × (B − V) (B − V < 1.4) 7.55 − 6.55 × (B − V) (B − V ≥ 1.4) where F⊙ = 5.7334 × 10 −11 erg cm−2 s−1 Å−1. 3.3 Calibration A relation like that of Eq. 1 can be used to calibrate in flux spectra of stars of known colour index (B − V). However, most spectro- graphs only observe a small wavelength range around the line of interest. For these instruments, the windows at 5805 Å and 6010 Å might fall outside the observed range. For this reason, we studied a relation similar to the one in Eq. 1 for a continuum window closer to the doublet. First, we note that due to the presence of molecular bands, the raw spectra of M-type stars are difficult to normalize, in particular if the observed range is not wide enough. Therefore, we did not try to find a good pseudo-continuum to calibrate M-dwarfs. For F, G and K stars, we considered a window 20 Å wide around 5840 Å, which is indicated in Fig. 1 with dotted lines. In Fig. 3 we show the ratio between the mean flux F̄ in this window and the one obtained earlier, for stars with (B − V) < 1.4. As can be seen, using F̄ underestimates the true continuum, but the percentual difference between both is smaller than 6 per cent. Therefore, the first part of Eq. 1 can be used to fit the mean flux F̄, and to calibrate in flux this region of the spectrum of a star of known (B − V), increasing the error by less than 6 per cent. 4 EQUIVALENT WIDTH 4.1 Definition The equivalent width Wλ of a spectral line is defined as dλ , (2) where Fc is the continuum flux and Fλ is the value of the flux in the line. As we pointed out in the preceeding section, the choice of the continuum level can be subjective, and it can be very difficult for M-dwarfs. On the other hand, the limits of the wings of a line are not always well-defined, and therefore the integration limits λ1 and λ2 also constitute subjective points in the definition of Wλ. In fact, while in earlier F and G stars the photospheric wings are thin and well defined, in later stars the wings are extremely wide. There- fore, the choice of a proper integration window requires a varying width (∆λ = λ2 − λ1), ranging from 14 to 40 Å around the cen- tre of the doublet (5892.94 Å), depending on the spectral type of the star. In Table 3 we list the values of ∆λ we used for differ- ent colours. A similar technique, with varying widths, was used by Tripicchio et al. (1997), although it is not clear from their paper which ∆λ was used for each colour. Note that in stars with spectral type later than M2.5 the ab- sorption wings of the D lines become very large and blended with c© 0000 RAS, MNRAS 000, 000–000 4 Dı́az et al. 0.5 1 1.5 (B-V) Figure 2. log(Fcont/F⊙) vs. colour index (B − V). The small empty squares represent the individual observations, and the filled squares are the mean value in each bin (see text). The solid lines are the best piecewise fit to the data. Note that in several bins there is only one star, and the individual observations are therefore hidden by the filled square. 0.6 0.8 1 1.2 1.4 (B-V) Figure 3. Ratio between Fcont and the mean flux F̄ in the 5830-5850 Å window, as a function of colour index (B − V). The percentage difference remains smaller than 6 per cent for F, G and K stars. the highly-developed absorption bands present (see Fig. 1). There- fore, for these stars a suitable choice of ∆λ is very difficult and the calculation of the equivalent width cannot be considered entirely reliable. Regarding the continuum level, we used both values described in the previous section. The equivalent width computed with F̄, 0.6 0.8 1 1.2 1.4 (B-V) Figure 4. Ratio between Wλ, calculated with a realistic estimation of the continuum flux (Fc = Fcont), and Wapprox, computed using the definition of Fc suitable for the calibration process described in Sect. 4.4 (Fc = F̄). The percentage difference remains smaller than 20 per cent for almost all stars. (B − V) interval ∆λ [Å] ≤ 0.8 14 [0.8 − 1) 16 [1.0 − 1.2) 20 [1.2 − 1.3) 24 [1.3 − 1.5) 28 > 1.5 40 Table 3. Wavelength window ∆λ used in the calculation of the equivalent width for different values of (B − V) (see Equation 2). the aproximate definition of Fc, will be referred to as Wapprox, and the one computed with Fcont will be refered simply as Wλ. The ra- tio between both values is plotted in Fig. 4. Again, using F̄ leads to an underestimation of the equivalent width. The differences are larger than for the continuum flux, but even in this case they remain smaller than 20 per cent. 4.2 Relation with colour index (B − V) In Fig. 5 we illustrate the behaviour of Wλ as a function of the colour index (B − V). In the y-axis we plot log(Wλ/λo), where λo = 5890 Å. The small dots represent the mean value per star and the filled squares show the results of binning the data as explained in the previous section. As empty squares we plot the values of Wapprox, binned in the same way. A quadratic fit to the binned data gives = −4.579 + 1.948 (B − V) − 0.3726 (B − V)2 . (3) c© 0000 RAS, MNRAS 000, 000–000 The Na I D resonance lines in main sequence late-type stars 5 0.5 1 1.5 (B-V) Figure 5. Equivalent width (logarithmic scale) vs (B − V). The small dots are the mean values for each star and the filled squares are the mean values in each bin. The empty squares are the mean values of Wapprox in each bin (see text). The solid line is the best fit to the binned data. The dotted line is the fit to Wapprox and the dashed line is the fit given by Tripicchio et al., transformed to the colour index scale using Eq 5. This fit is shown as a solid line in Fig. 5. It can be seen that, as the temperature decreases, the D lines become wider. It is evident from Fig. 5 that there is a saturation effect for the reddest stars. As shown in Fig. 1, for M stars with (B − V) ≥ 1.4, the lines are so wide that they are almost completely blended. At this point, the doublet behaves as a single line, and the equivalent width grows mainly in the outer wings. Additionally, as cooler stars are also more active, the central line cores are filled in by chromospheric emission, contributing to the saturation effect. 4.3 Comparison Tripicchio et al. (1997) also studied the dependence of the equiv- alent width with effective temperature. They constructed synthetic profiles of the D lines using a modified version of the MULTI code, which takes into account the line blending present in the doublet. They also observed a sample of dwarf and giant stars with spec- tral type between F6 and M5. They used both the modelled atmo- spheres and the observed spectra to obtain a relation between Wλ and effective temperature, given by: = 2.43 · Te f f − 5.73 , (4) which reproduces their observations fairly well for effective tem- peratures higher than 4000 ◦K. In order to compare this relation to our regression we per- formed a change of variables in equation 4 using the relation given by Noyes et al. (1984): log(Te f f ) = 3.908 − 0.234 · (B − V) , (5) valid for stars with 0.4< (B − V) < 1.4. In this way we obtain Wλ as a function of (B − V). The resulting expression is plotted as a dashed line in Fig. 5. It can be seen that the values by Tripicchio et al. (1997) are smaller than ours: in some cases the difference between both ex- pressions is as large as 40 per cent. Unfortunately, they do not men- tion how the continuum level or the integration windows were cho- sen. Although they say they used a wavelength interval of variable width and that they cleared the spectrum of unwanted lines, no de- tails are given, so we could not apply their calculation method to our data. To check whether our larger values of Wλ are due to the weak photospheric lines blended in the profiles, we recomputed the equivalent width in a completely different way, for several of our stars. We considered the raw spectra of 16 stars from our sam- ple. The spectroscopic order in which the D lines are found was extracted, calibrated in wavelegth and normalized. It is worth not- ing that the doublet is located in the middle of this order, where the effects of the blaze function are less severe. However, the presence of molecular bands prevented us from normalizing the spectra of M-type stars. We then computed the equivalent width using the IRAF line- deblending routines included in the SPLOT task, which fit Voigt profiles to the lines, and perform the calculation over the fits. In this way, the presence of photospheric lines has no influence on the obtained value. Due to the severe blending of the D lines, the routine used by IRAF did not produce acceptable results for stars with colour larger than ≈ 1.3. Therefore, this method can only be applied to stars with (B − V) < 1.3. In Fig. 6 we plot as filled squares the value of Wλ calculated from our calibrated spectra Vs. the value obtained using IRAF (WIRAF). The errors in Wλ were calculated assuming a 3 per cent error in Fλ/Fc (see Eq. 2). The error bars in the x-axis were taken as the RMS of the fit done by IRAF times the width of the inte- gration window (see Table 3). The solid line represents the identity relation, and not a fit to the data. It can be seen that the differences beween our calculation and the one done with IRAF are well within the errors. The filled triangles represent the values of Wλ obtained from our fit, rather than from the individual obervations. Note that the points still remain very close to the identity line. As open squares we plot the equivalent width obtained from Trippichio’s relation (Eq. 4) for the value of (B − V) of the stars considered, which in all cases fall below the identity relation. 4.4 Determination of (B − V) The relation shown in Fig. 5 can be used to determine the (B − V) of a star using the equivalent width of the D lines, measured in non- calibrated spectra. Since in many cases the spectral region needed to compute Fcont might fall outside the observed one, we repeat the fit for Wapprox. The best least square fit to the binned values of log(Wapprox/λo) Vs. (B-V) is: Wapprox = −4.528 + 1.734 (B − V) − 0.2442 (B − V)2 , (6) which is valid only for (B−V) < 1.4, and where λo = 5890 Å. This fit is plotted as a dotted line in Fig. 5. To obtain (B − V) from the measured equivalent width, one could invert Eq. 6. However, since the least square estimation of parameters does not treat the dependent and independent variables symmetrically, a more rigorous procedure would be to fit (B−V) as a function of Wapprox. We performed this fit and found that within the errors, both procedures give the same values of (B − V). c© 0000 RAS, MNRAS 000, 000–000 6 Dı́az et al. 2 4 6 8 10 Figure 6. Wλ computed from the calibrated spectra (filled squares) as a funtion of the one given by the IRAF deblend routine (WIRAF). The triangles represent the values obtained from the fit of Eq. 3 and the empty squares represent the values obtained using the fit from Tripicchio et al. (1997). 0.4 0.6 0.8 1 1.2 Figure 7. (B − V) computed from the normalized ([comp]) spectra through the inverse of Eq. 6 as a funtion of (B − V) from the Hipparchos/Tycho catalogue ([cat]). The solid line is the identity function. A good agreement is found between both values. The method was verified using the same 16 stars used in the previous section. The raw spectra were extracted, calibrated in wavelength and normalized using Chebyshev polynomials. All these procedures were carried out using IRAF. Wλ was calculated by direct integration of the flux in the appropiate window (see Ta- ble 3) using SPLOT. Then, the inverse of Eq. 6 was used to obtain (B − V) for these stars. In Fig. 7 we plot the value of (B − V) computed in this way ((B − V)comp) as a funtion of the value obtained from the Hippar- cos/Tycho catalogue ((B−V)cat). To estimate the error in (B−V)comp we assumed a 5 per cent error in Fλ/Fc , in order to consider errors in the normalization procedure. The errors in the x-axis were taken from the Hipparcos/Tycho catalogue. The solid line represents the identity relation. It can be seen that the method gives, in fact, very good estimates of the colour of the stars. Therefore, by measuring the equivalent width of the D doublet an estimate of the colour index (B−V) can be obtained. Afterwards, this value of (B − V) can be used to obtain an approximate flux calibration in the region of the D lines, by means of Eq. 1. In this way, the spectra of stars of unknown colour can be calibrated in flux in this spectral region. 5 ACTIVITY INDEXES AND CHROMOSPHERIC ACTIVITY In this section, we define different activity indexes using the D lines, and study their applicability as chromospheric activity indi- cators. 5.1 N index First, we constructed an index (N) similar to the Mount Wilson S index, described by Vaughan et al. (1978). f1 + f2 fcont where fi is the mean flux in the Di line, integrated in a square win- dow 1 Å wide, and fcont is the mean flux of the continuum calcu- lated as explained in Sect. 3.1. For the earlier stars in our sample, the N index is highly depen- dent on the colour (B−V), as can be seen in Fig. 8. This behaviour changes for stars with (B − V)> 1.1, for which the value of N is roughly constant. Stars with (B − V) < 1.1 were fitted with a linear relation, shown in the figure as a solid line: N(B−V) = −1.584 · (B − V) + 2.027 . (7) In Fig. 9 we show the result of plotting N as a function of the S index, which was obtained from our spectra using the calibration presented in Cincunegui et al. (2006), which relates the S index ob- tained at CASLEO with the one computed at Mount Wilson. Note that since our spectra cover the entire visible range, we measure N and S simultaneously. It can be seen that stars with (B − V) < 1.1 (shown as filled triangles) have smaller values of S, and show a weak anticorrelation between N and S (r = −0.488). We plot the observations for these stars alone in the upper panel of Fig. 10. Despite the considerable scatter, the anticorrelation is evident. However, for these stars the correlation between N and (B−V) is stronger. To correct for this dependence we substracted N(B−V) (Eq. 7) from N. In the lower panel of Fig. 10 we plot N − N(B−V) vs. (B-V). In this case, the correlation dissapears (r = 0.07), imply- ing that the correlation was due only to the dependence of N with colour, and the tendency of cooler stars to be more active. Thre- fore, the N index cannot be used as an activity indicator for stars with (B − V) < 1.1. c© 0000 RAS, MNRAS 000, 000–000 The Na I D resonance lines in main sequence late-type stars 7 0.5 1 1.5 (B - V) Figure 8. N index vs. colour index (B − V). The solid line is the best linear fit for stars with (B − V) < 1.1. Stars with a larger (B − V) do not show any correlation. 0 5 10 15 Figure 9. N vs. S. Values for stars with (B − V) < 1.1 are shown as filled triangles, and those with (B − V) ≥ 1.1 as empty squares. On the other hand, stars with (B − V) ≥ 1.1 present a good correlation between both N and S, as can be seen in Fig. 9. Since for these stars N do not correlate with colour, we do not perform any correction in this case. Since we have simultaneous observations of both indexes, we can explore this apparent correlation for particular stars. In Fig. 11 we plot N vs S for several stars with (B − V) ≥ 1.1, and we show the linear fits to each individual dataset (the slope of each fit is given in the legend). It can be seen that the slope of the relation be- 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 10. Upper panel: N vs. S for stars with (B − V) < 1.1. Lower panel: The dependence on N with colour has been corrected by substracting N(B−V), shown in Eq. 7. The correlation disappears, as is confirmed by the small value of the correlation coefficient (r = 0.07). 0 5 10 GJ551 (0.018) GJ388 (-0.0017) GJ517 (0.084) GJ1 (-0.16) Figure 11. N vs S for selected stars with (B − V) ≥ 1.1. A linear fit to the data of each star is also shown (GJ551: solid line; GJ388: dotted line; GJ517: dashed line; GJ1: dot-dash line). The slope of the fit is given in parentheses in the legend. tween S and N changes from star to star, and the behaviour ranges from correlations with different slopes (GJ551 and GJ517) to cases where N and S are uncorrelated, either with constant S and vari- able N (GJ1) or viceversa (GJ388). A similar effect was observed comparing the flux of the Hα line to the flux in the H and K lines (Cincunegui et al. 2006). In that case, we found correlations with c© 0000 RAS, MNRAS 000, 000–000 8 Dı́az et al. different slopes, anti-correlations, and cases where no correlation is present. Therefore, the N index cannot be used to compare activity lev- els of different stars. However, it can be useful to compare activity levels on the same star at different times, for those stars which show a correlation between S and N. For example, N can be used to trace activity cycles in late-type stars, as we did with Proxima Centauri using the flux of the Hα emission (see Cincunegui et al. 2007). 5.2 R′D Noyes et al. (1984, hereafter N84) noted that the S index is sensitive to the flux in the continuum windows and to the photospheric radi- ation present in the H and K bandpasses, both of which depend on spectral type. Despite the apparent independence of N with (B−V) seen in Fig. 8 for stars with (B − V) ≥ 1.1, the results of the pre- vious section seem to indicate that N also shows a dependence on photospheric flux, even for stars with (B − V) ≥ 1.1. This depen- dence must be corrected more thoroughly in order to compare stars of different spectral type. Following N84, we define a new index R′D = RD − R D , (8) where RD = (F1+F2)/σT e f f , F1 and F2 are the fluxes in the D1 and D2 windows at the stellar surface, and σ is the Stefan-Boltzmann constant. Note that RD can be expressed as: f1 + f2 fcont · N , (9) where fbol is the relative bolometric flux. Both expressions in Eq. 9 have the virtue of depending on fluxes on Earth instead of fluxes at the stellar surface. fbol was computed interpolating the values of the bolometric correction provided by Johnson (1966). Defined in this way, RD is independent of the photospheric flux in the continuum window. RphotD is the photospheric contribution to the flux in the win- dows of the D lines. For the H and K lines, N84 integrated the flux outside the H1 and K1 minima, and used these values to calculate Rphot, which is substracted from RHK . However, the D lines are usu- ally in absorption. For this reason, we had to correct for the photo- spheric contribution in a different way. Since even for very inactive stars the D lines are not completely dark, the minimum line flux in a very inactive star must be photospheric in origin. Therefore, we assumed that the basal flux of the D lines is a reasonable estimation of the photospheric contribution. In Fig. 12 we plot the flux in the D1 line as a function of the colour index (B−V), in logarithmic scale. The solid line represents the basal flux, computed using a third order polynomial. The spec- tra with the D lines in emission were excluded from the fit. The resulting expression is: log(Fmin1 ) = −6.527 − 11.86· (B − V) + 10.87 · (B − V) −4.218 · (B − V)3 . For the D2 line, a similar expression is obtained: log(Fmin2 ) = −6.374 − 12.48· (B − V) + 11.37 · (B − V) −4.344 · (B − V)3 . Contrary to what happens for the H and K lines, this term is appreciable for all the range of our observations. Even for stars with Balmer lines in emission, the photospheric correction can be up to 50 per cent of the value of RD. 0.5 1 1.5 (B-V) Figure 12. Flux in the D1 line as a function of colour index (B − V). The solid line represents our estimation of the minimum flux. Those spectra with D lines in emission were not considered for the fit. Following N84, we substracted the same value of RphotD to all stars of the same spectral type. In this way, we have constructed an index eliminating the dependence on spectral type which arises from the photospheric flux in the integration bandpasses. R′D should therefore depend exclusively on chromospheric flux and be inde- pendent of spectral type. 5.3 Chromospheric activity According to N84, R′HK = RHK−Rphot is proportional to the fraction of nonradiative energy flux in the convective zone. Since this flux then heats the chromosphere by means of the magnetic field, R′HK should be a good activity indicator. We computed R′HK for all our spectra using a relation analogue to Eq. 9. As Rphot we used the expression given in N84, valid in the range 0.44 < (B − V) < 0.82, log(Rphot) = −4.02 − 1.4 (B − V) . (12) As noted by N84 the photospheric correction is unimportant for active stars. Indeed, we found that the correction is less than 10 per cent virtually for all stars with B − V > 1.1. Therefore, we used Eq. 12 for all the range of our observations. To study its applicability as an activity indicator, R′D was plot- ted against R′HK . The result is presented in Fig. 13, where the filled triangles represent stars which exhibit the Balmer lines in emission. No correlation seems to be present when we consider the complete stellar sample. Indeed the correlation coefficient is r = 0.094. How- ever, if we consider only the most active stars –those which exhibit Balmer lines in emission – an excelent correlation is found between both indexes. This is shown in Fig. 14, where we plot R′D vs R only for these stars. The solid line is the best linear fit to the data: R′D = 2.931 × 10 + 0.1388 · R′HK , (13) with correlation coefficient r = 0.978. This result demonstrates that c© 0000 RAS, MNRAS 000, 000–000 The Na I D resonance lines in main sequence late-type stars 9 0 0.0001 0.00015 0.0002 0.0001 Figure 13. R′D vs R HK . Filled triangles represent stars with Balmer lines in emission. No clear correlation is present, and the correlation coefficient for the complete sample is r = 0.094. R′D can be used as an activity indicator for active stars. Note that de- spite presenting the Balmer lines in emission, not all stars in Fig. 14 present the D lines in emission, and this does not seem to be a con- dition for the use of R′D as an activity indicator. R D is particularly useful to study the cooler stars, which tend to be more active and where the emission in the region of the H and K Ca II lines is sub- stantially smaller (even an order of magnitude) than the flux in the region of the D doublet. 6 SUMMARY A program was started in 1999 to study chromospheric activity and atmospheres of main sequence stars in the Southern Hemisphere. In this paper we analyze the sodium D lines. We use medium reso- lution echelle spectra, obtained in CASLEO, which were calibrated in wavelength and in flux. We constructed an HR diagram for all the stars in our sample taking the colour index (B−V) and the visual magnitude V from the Hipparcos/Tycho catalogues. We found that several stars, in spite of being classified as dwarfs in the Simbad database, fall outside the main sequence. We present these stars in Table 1 and defer a more detailed analysis for future work. Conversely, in Table 2 we show four stars that were not classified as dwarfs even though they fall within the main sequence. In Sect. 3 we define a pseudo-continuum level useful for all spectral types. Since in M stars the D lines are formed inside a TiO molecular band, we consider two windows located quite far from the doublet. In each window, the mean value of the ten high- est points was calculated and the pseudo-continuum was defined as the linear interpolation between both windows. The continuum flux obtained in this way shows a very tight correlation with colour in- dex (B − V). Since most observations do no include regions too far from the lines, for stars with (B− V) < 1.4 we also defined another pseudo-continuum, which uses a window located closer to the D 0 0.0001 0.00015 0.0002 Figure 14. R′D vs R HK for stars with Balmer lines in emission. The solid line is the best linear fit to the data, with correlation coefficient r = 0.977. The excelent correlation indicates that R′D is a good chromospheric activity indicator. lines, and also correlates strongly with (B − V). We showed that this correlation can be used to obtain an approximate calibration in flux in this spectral region. In Sect. 4 we describe in detail the computation of the equiv- alent width. We use a wavelength interval which range from 14 Å for F6 stars to 40 Å for the coolest stars in our sample. In M-type stars, the large photospheric wings, blended with deep molecular bands, difficult an accurate calculation. We find a good correlation between equivalent width and colour index (B−V) for all the range of observations. The relation obtained was then used to determine the colour of dwarf-type stars from the equivalent width of the D lines. We find values of (B − V) in good agreement with those from the Hippar- cos/Tycho catalogues. Since equivalent width is a characteristic of line profiles that do not require high resolution spectra to be mea- sured, this fact could become a useful tool for subsequent studies. Finally, in Sect. 5 we study how the flux in the D lines changes with changing levels of chromospheric activity. We define an index (N) analogue to Mount Wilson S index and find it is strongly cor- related with colour for stars with (B − V) < 1.1. This fact produces an apparent correlation between N and S, which disappears when the colour dependence is taken into account. On the other hand, in stars with (B−V) ≥ 1.1, N is independent of colour and is correlated with the S index. However, the slope between N and S varies when different stars are studied individually, varying from tight correla- tions to cases where no correlation is present. This fact restricts the use of the N index as an activity indicator when different stars are compared. However, the N index may be useful when comparing different activity levels on individual stars, specially in later stars with little emission in the region of the CaII H & K lines. In order to compare activity levels on stars of different spectral types we define an improved index taking into account the photo- spheric contribution to the flux both in the lines and in the contin- uum windows. First we construct RD as the ratio between absolute c© 0000 RAS, MNRAS 000, 000–000 10 Dı́az et al. line fluxes and stellar luminosity. Then, the photospheric contribu- tion in the lines bandpasses, which was computed using the basal flux in the D lines, was substracted from RD to define R D. As was expected, the earlier stars in our sample do not show any sign of correlation between both indexes. Indeed, it has been known for a long time that the cores of the Na I D lines in the solar atmosphere remain dark from center to limb while the Ca II H&K lines show increasing line reversal. Only in stars with higher atomic densities, as the M-dwarfs, the D lines can be expected to have collision rates high enough to respond to the temperature changes in the chromo- sphere (e.g. Mauas 2000). Because of this, earlier theoretical works (see Andretta et al. 1997; Short & Doyle 1998) have restricted their attention to this type of stars. However, in this work the new R′D in- dex was found to correlate well with R′HK for the most active stars in our sample (those which exhibit the Balmer lines in emission), even though some of these stars do not present a line reversal at the core of the D lines. Therefore, R′D is also a good activity indicator for these stars. For the rest of the spectra, no correlation was found. The CCD and data acquisition system at CASLEO has been partly financed by R. M. Rich through U.S. NSF grant AST-90- 15827. This research has made use of the SIMBAD database, op- erated at CDS, Strasbourg, France. We thankfully acknowledge the comments and suggestions of the referee (Ian Short), which helped us to improve our original manuscript. REFERENCES Allen C. W., 1964, Astrophysical Quantities. Astrophysical Quan- tities, London: Athlone Press (2nd edition), 1964 Andretta V., Doyle J. G., Byrne P. B., 1997, A&A, 322, 266 Buccino A., Mauas P. J. D., 2007, in preparation Cayrel de Strobel G., Soubiran C., Friel E. D., Ralite N., Francois P., 1997, A&AS, 124, 299 Cayrel de Strobel G., Soubiran C., Ralite N., 2001, A&A, 373, Cincunegui C., Dı́az R. F., Mauas P. J. D., 2007, A&A, in press Cincunegui C., Dı́az R. F., Mauas P. J. D., 2007, A&A, 461, 1107 Cincunegui C., Mauas P. J. D., 2002, in ESA SP-477: Solspa 2001, Proceedings of the Second Solar Cycle and Space Weather Eu- roconference Cycles in other stars. pp 91–94 Cincunegui C., Mauas P. J. D., 2004, A&A, 414, 699 Henry T. J., Soderblom D. R., Donahue R. A., Baliunas S. L., 1996, AJ, 111, 439 Hoeg E., Bässgen G., Bastian U., Egret D., Fabricius C., Groß- mann V., Halbwachs J. L., Makarov V. V., Perryman M. A. C., Schwekendiek P., Wagner K., Wicenec A., 1997, A&A, 323, L57 Johnson H. L., 1966, ARA&A, 4, 193 Mauas P. J. D., 2000, ApJ, 539, 858 Mauas P. J. D., Falchi A., 1994, A&A, 281, 129 Nordström B., Mayor M., Andersen J., Holmberg J., Pont F., Jørgensen B. R., Olsen E. H., Udry S., Mowlavi N., 2004, A&A, 418, 989 Noyes R. W., Hartmann L. W., Baliunas S. L., Duncan D. K., Vaughan A. H., 1984, ApJ, 279, 763 Perryman M. A. C., Lindegren L., Kovalevsky J., Hoeg E., Bas- tian U., Bernacca P. L., Crézé M., Donati F., Grenon M., van Leeuwen F., van der Marel H., Mignard F., Murray C. A., Le Poole R. S., Schrijver H., Turon C., Arenou F., Froeschlé M., Petersen C. S., 1997, A&A, 323, L49 Short C. I., Doyle J. G., 1998, A&A, 336, 613 Tripicchio A., Severino G., Covino E., Terranegra L., Garcia Lopez R. J., 1997, A&A, 327, 681 Vaughan A. H., Preston G. W., Wilson O. C., 1978, PASP, 90, 267 c© 0000 RAS, MNRAS 000, 000–000 Table 4. List of stars used in the analysis and average measurements. Nobs is the number of times the star has been observed. V is the visual magnitude in the UBV system. [Fe/H] is the iron abundance relative to the Sun, obtained from references (a): Cayrel de Strobel et al. (2001); (b): Nordström et al. (2004); (c): Cayrel de Strobel et al. (1997). < Fcont > is the continuum flux (see Sect. 3.1). < F̄ > is the mean flux in a window 20 Å wide around 5840 Å. < Wλ > is the equivalent width of the sodium doublet and < Wapprox > is the equivalent width computed using F̄ as pseudo-continuum (see Sect. 4.1), in angstroms. Cols. 10 and 11 give the average absolute flux in a 1 Å window centered at the D1 and D2 lines, respectively. All fluxes are in units of [erg cm−2 s−1]. < N > is the value of the N index, calculated as explained in Sect. 5.1. Columns 13 and 14 give average value for the RD index and the corresponding photospheric correction R D (see Sect. 5.2). Finally, in the last column we present the mean values of R HK . Colour index (B − V), magnitude V and spectral type have been obtained from the Hipparcos/Tycho catalogues (Perryman et al. 1997; Hoeg et al. 1997). Star Nobs Sp. type (B − V) V [Fe/H] < Fcont > < F̄ > < Wλ > < Wapprox > < FD1 > < FD2 > < N > < RD > R D < R hd28246 5 F6V 0.457 6.380 0.02b 9.651e-12 9.591e-12 1.189 1.108 6.330e-12 5.7 3 9 e - 1 2 1.251 1.825e-04 1.761e-04 5.737e-05 hd38393 12 F7V 0.481 3.590 -0.12a 1.293e-10 1.286e-10 1.222 1.153 8.595e-11 7 . 8 6 2e-11 1.272 1.897e-04 1.780e-04 3.084e-05 hd120136 3 F6IV 0.508 4.500 0.32a 5.665e-11 5.618e-11 1.602 1.500 3.622e-11 3 .270e-11 1.217 1.829e-04 1.037e-04 4.078e-05 hd16673 9 F6V 0.524 5.790 -0.01a 1.711e-11 1.704e-11 1.243 1.192 1.098e-11 9. 567e-12 1.201 1.784e-04 1.529e-04 4.929e-05 hd114762 3 F9V 0.525 7.300 -0.82a 4.346e-12 4.325e-12 1.069 1.008 2.913e-12 2 .641e-12 1.278 1.938e-04 1.716e-04 2.693e-05 hd35850 5 F7V 0.553 6.300 0.00a 1.063e-11 1.057e-11 1.374 1.296 7.466e-12 6.6 73e-12 1.330 1.940e-04 1.198e-04 1.376e-04 hd17051 5 G0V 0.561 5.400 -0.04a 2.536e-11 2.505e-11 1.679 1.526 1.423e-11 1. 233e-11 1.047 1.582e-04 1.176e-04 4.635e-05 hd45067 11 F8V 0.564 5.880 -0.16a 1.572e-11 1.562e-11 1.340 1.255 9.635e-12 8 .603e-12 1.160 1.686e-04 4.831e-05 2.549e-05 hd52265 2 G0III-IV 0.572 6.290 0.21a 1.108e-11 1.091e-11 1.541 1.347 5.911e-1 2 5.201e-12 1.003 1.490e-04 9.101e-05 2.567e-05 hd19994 9 F8V 0.575 5.070 0.15a 3.438e-11 3.385e-11 1.736 1.543 1.932e-11 1.7 17e-11 1.062 1.587e-04 4.529e-05 3.132e-05 hd130948 6 G1V 0.576 5.860 -0.20a 1.640e-11 1.622e-11 1.885 1.744 9.382e-12 8 .210e-12 1.072 1.582e-04 1.445e-04 5.599e-05 hd75289 8 G0Ia 0.578 6.350 0.28a 1.054e-11 1.039e-11 1.651 1.472 6.067e-12 5. 247e-12 1.074 1.596e-04 8.565e-05 3.365e-05 hd215768 5 G0V 0.589 7.490 -0.20b 3.678e-12 3.642e-12 1.466 1.342 2.083e-12 1 . 7 9 5 e - 1 2 1.054 1.550e-04 1.193e-04 6.666e-05 hd213240 2 G0/1V 0.603 6.810 0.13b 6.975e-12 6.907e-12 1.975 1.856 3.737e-12 3 . 2 1 1 e - 12 0.996 1.484e-04 5.347e-05 2.894e-05 hd43587 7 F9V 0.610 5.700 -0.08a 1.932e-11 1.908e-11 1.812 1.656 1.103e-11 9. 575e-12 1.067 1.583e-04 8.101e-05 2.246e-05 hd197076 6 G5V 0.611 6.430 -0.20b 9.906e-12 9.812e-12 1.560 1.440 5.488e-12 4 . 6 9 6 e - 1 2 1.028 1.533e-04 1.334e-04 2.160e-05 hd202996 2 G0V 0.614 7.460 0.00b 3.735e-12 3.690e-12 1.778 1.629 2.355e-12 2. 0 4 7 e - 1 2 1.179 1.711e-04 5.219e-05 6.028e-05 hd45270 6 G1V 0.614 6.530 -0.14b 9.022e-12 8.913e-12 1.770 1.620 5.030e-12 4. 4 6 4 e - 1 2 1.052 1.567e-04 1.141e-04 8.372e-05 hd165185 10 G5V 0.615 5.940 -0.06a 1.540e-11 1.523e-11 1.727 1.589 8.778e-12 7.63 2e-12 1.066 1.573e-04 1.203e-04 6.079e-05 hd48189 6 G1.5V 0.624 6.150 -0.18b 1.259e-11 1.248e-11 1.878 1.770 7.036e-12 6 . 3 7 0 e - 1 2 1.065 1.559e-04 8.756e-05 8.837e-05 hd147513 11 G5V 0.625 5.370 0.03a 2.656e-11 2.618e-11 1.754 1.577 1.508e-11 1 .271e-11 1.046 1.576e-04 1.201e-04 6.066e-05 hd150433 7 G0 0.631 7.210 -0.47b 4.702e-12 4.649e-12 1.753 1.613 2.666e-12 2. 294 e - 1 2 1.055 1.530e-04 1.185e-04 2.568e-05 hd219709 7 G2/3V 0.632 7.500 -0.02a 3.607e-12 3.572e-12 1.925 1.808 1.922e-12 1.6 48e-12 0.990 1.438e-04 8.602e-05 4.212e-05 hd30495 11 G3V 0.632 5.490 -0.13a 2.326e-11 2.304e-11 1.696 1.576 1.228e-11 1 .047e-11 0.978 1.439e-04 1.187e-04 6.058e-05 hd202628 9 G2Va 0.637 6.750 -0.14a 7.346e-12 7.261e-12 1.954 1.812 3.955e-12 3.401e-12 1.001 1.480e-04 1.141e-04 4.981e-05 hd38858 5 G4V 0.639 5.970 -0.25b 1.511e-11 1.492e-11 1.872 1.710 8.241e-12 7. 0 4 6 e - 1 2 1.011 1.498e-04 1.279e-04 2.778e-05 hd20766 12 G2.5V 0.641 5.530 -0.22a 2.242e-11 2.225e-11 1.833 1.742 1.199e-11 9.930e-12 0.978 1.430e-04 1.384e-04 4.897e-05 hd59967 5 G4V 0.641 6.660 -0.19b 8.012e-12 7.884e-12 1.826 1.629 4.420e-12 3. 7 6 3 e - 1 2 1.021 1.512e-04 1.214e-04 7.739e-05 hd187923 10 G0V 0.642 6.160 -0.20a 1.368e-11 1.350e-11 1.809 1.646 7.548e-12 6.660e-12 1.038 1.655e-04 4.708e-05 2.116e-05 hd19467 10 G3V 0.645 6.970 -0.13b 5.923e-12 5.861e-12 1.736 1.607 3.277e-12 2 . 8 7 4 e - 1 2 1.039 1.508e-04 7.483e-05 2.271e-05 hd189567 8 G3V 0.648 6.070 -0.30a 1.395e-11 1.377e-11 1.752 1.595 7.994e-12 6 .906e-12 1.068 1.592e-04 1.008e-04 3.051e-05 hd210918 11 G5V 0.648 6.230 -0.18a 1.154e-11 1.139e-11 1.788 1.623 6.353e-12 5.686e-12 1.043 1.491e-04 7.484e-05 2.604e-05 hd173560 9 G3/5V 0.649 8.690 -0.21b 1.272e-12 1.248e-12 1.667 1.427 7.344e-13 6 . 3 8 5 e - 13 1.079 1.637e-04 4.757e-05 3.793e-05 hr6060 15 G2Va 0.652 5.490 0.05a 2.369e-11 2.331e-11 1.962 1.765 1.221e-11 1. 033e-11 0.951 1.408e-04 9.134e-05 2.526e-05 hd11131 9 G0 0.654 6.720 -0.06a 7.558e-12 7.474e-12 1.726 1.587 4.098e-12 3.4 39e-12 0.997 1.460e-04 1.038e-04 7.184e-05 hd20619 11 G1.5V 0.655 7.050 -0.20a 5.339e-12 5.293e-12 1.738 1.632 2.916e-12 2.486e-12 1.012 1.417e-04 1.212e-04 3.218e-05 hd217343 2 G3V 0.655 7.470 -0.18b 3.768e-12 3.711e-12 1.995 1.810 2.075e-12 1 . 8 0 5 e - 1 2 1.030 1.499e-04 1.060e-04 1.057e-04 hd4308 12 G5V 0.655 6.550 -0.47a 8.907e-12 8.808e-12 1.652 1.513 4.651e-12 3. 991e-12 0.970 1.431e-04 9.757e-05 2.719e-05 hd1835 13 G3V 0.659 6.390 -0.01a 1.067e-11 1.049e-11 2.116 1.921 5.225e-12 4. 551e-12 0.917 1.393e-04 9.379e-05 6.410e-05 hd213941 9 G5V 0.670 7.580 -0.42a 3.335e-12 3.313e-12 2.057 1.977 1.759e-12 1 .508e-12 0.979 1.386e-04 1.031e-04 3.985e-05 Table 4 – continued Star Nobs Sp. type (B − V) V [Fe/H] < Fcont > < F̄ > < Wλ > < Wapprox > < FD1 > < FD2 > < N > < RD > R D < R hd197214 10 G5V 0.671 6.950 -0.5b 6.138e-12 6.048e-12 1.983 1.804 3.224e-12 2 . 7 1 9 e - 1 2 0.968 1.411e-04 1.186e-04 3.608e-05 hd172051 13 G5V 0.673 5.850 -0.29b 1.699e-11 1.677e-11 2.078 1.923 8.668e-12 7 . 3 6 7 e - 1 2 0.944 1.380e-04 1.266e-04 2.725e-05 hd19034 8 G5 0.677 8.080 -0.39b 2.167e-12 2.148e-12 1.870 1.767 1.128e-12 9.4 4 3 e - 1 3 0 .957 1.389e-04 1.297e-04 2.861e-05 hd203019 3 G5V 0.687 7.840 0.07b 2.710e-12 2.652e-12 2.330 2.078 1.256e-12 1. 1 2 3 e - 1 2 0.878 1.272e-04 9.105e-05 9.492e-05 hd202917 5 G5V 0.690 8.650 -0.16b 1.269e-12 1.250e-12 2.377 2.198 6.377e-13 5 . 5 0 5 e - 1 3 0.936 1.338e-04 1.183e-04 1.416e-04 hd12759 4 G3V 0.694 7.300 -0.13b 4.585e-12 4.501e-12 2.245 2.026 2.427e-12 2. 0 9 3 e - 1 2 0.986 1.465e-04 2.882e-05 7.779e-05 hd128620 7 G2V 0.710 -0.010 0.22a 3.844e-09 3.777e-09 2.123 1.912 1.943e-09 1 . 5 9 1e-09 0.919 1.355e-04 4.082e-05 2.221e-05 hd41824 11 G6V 0.712 6.600 -0.09b 9.098e-12 8.924e-12 2.456 2.231 4.401e-12 3 .864 e - 1 2 0.909 1.394e-04 3.642e-05 1.074e-04 hd117176 3 G4V 0.714 4.970 -0.11a 3.835e-11 3.783e-11 1.947 1.780 2.074e-11 1 .897e-11 1.036 1.492e-04 2.157e-05 1.581e-05 hd3443 9 K1V 0.715 5.570 -0.16c 2.212e-11 2.186e-11 2.440 2.305 1.051e-11 8.7 90e-12 0.872 1.259e-04 5.059e-05 2.837e-05 hd3795 9 G3/5V 0.718 6.140 -0.70a 1.299e-11 1.288e-11 1.692 1.584 7.208e-12 6 .159e-12 1.029 1.472e-04 2.478e-05 2.282e-05 hd203244 1 G5V 0.723 6.980 -0.21a 6.150e-12 6.069e-12 2.284 2.128 3.265e-12 2 .754e-12 0.979 1.434e-04 1.010e-04 7.445e-05 hd10700 11 G8V 0.727 3.490 -0.59a 1.490e-10 1.475e-10 2.344 2.227 6.720e-11 5 .630e-11 0.829 1.180e-04 1.243e-04 3.117e-05 hd152391 11 G8V 0.749 6.650 -0.06b 8.579e-12 8.407e-12 2.613 2.379 3.882e-12 3.26 3 e - 1 2 0.833 1.238e-04 9.103e-05 6.268e-05 hd36435 5 G6/8V 0.755 6.990 -0.02a 6.071e-12 5.984e-12 2.425 2.255 2.798e-12 2.433e-12 0.862 1.235e-04 8.948e-05 6.689e-05 hd13445 5 K1V 0.812 6.120 -0.21a 1.365e-11 1.338e-11 3.400 3.147 4.822e-12 3. 947e-12 0.643 8.931e-05 8.795e-05 3.455e-05 hd26965 10 K1V 0.820 4.430 -0.25a 6.876e-11 6.776e-11 3.413 3.228 2.547e-11 2 .081e-11 0.673 9.885e-05 8.235e-05 2.275e-05 hd177996 9 K1V 0.862 7.890 -0.28b 2.876e-12 2.798e-12 4.034 3.699 1.131e-12 9 .836 e - 1 3 0.735 1.048e-04 3.767e-05 7.872e-05 hd17925 7 K1V 0.862 6.050 0.10a 1.648e-11 1.608e-11 3.898 3.596 5.892e-12 4.9 04e-12 0.655 9.825e-05 6.478e-05 8.305e-05 hd22049 13 K2V 0.881 3.720 -0.14a 1.318e-10 1.292e-10 3.905 3.653 4.355e-11 3 .628e-11 0.606 8.335e-05 6.921e-05 5.299e-05 hd128621 8 K1V 0.900 1.350 0.24a 1.220e-09 1.183e-09 4.576 4.216 3.709e-10 3. 049e-10 0.554 7.802e-05 3.906e-05 2.154e-05 gl349 12 K3V 1.002 7.200 -0.15b 5.535e-12 5.373e-12 6.086 5.667 1.353e-12 1.1 42e- 1 2 0 .451 5.558e-05 4.579e-05 5.335e-05 gl542 9 K3V 1.017 6.660 0.26a 9.770e-12 9.361e-12 7.693 7.154 1.959e-12 1.678 e-12 0.372 4.834e-05 2.846e-05 1.566e-05 hd188088 6 K3/4V 1.017 6.220 – 1.499e-11 1.433e-11 7.920 7.369 3.456e-12 2.903e-12 0.425 5.636e-05 1.315e-05 4.730e-05 hd131977 9 K4V 1.024 5.720 0.03 2.381e-11 2.322e-11 8.199 7.903 4.477e-12 3.802e-12 0.348 4.588e-05 4.552e-05 2.660e-05 gl610 9 K3/4V 1.043 7.390 -0.03a 4.892e-12 4.719e-12 7.510 7.054 1.128e-12 9. 638e-13 0.428 5.266e-05 1.849e-05 1.505e-05 hd32147 10 K3V 1.049 6.220 0.34a 1.468e-11 1.415e-11 7.691 7.232 2.773e-12 2. 353e-12 0.349 4.360e-05 2.653e-05 1.978e-05 gl845 8 K4.5V 1.056 4.690 -0.23a 6.041e-11 5.768e-11 7.782 7.203 1.179e-11 1. 008e-11 0.362 4.505e-05 3.618e-05 2.748e-05 gl435 8 K4/5V 1.064 7.770 – 3.576e-12 3.467e-12 7.456 7.062 7.230e-13 6.108e-13 0.373 4.640e-05 4.852e-05 2.499e-05 hd216803 2 K4V 1.094 6.480 – 1.181e-11 1.159e-11 6.814 6.562 2.636e-12 2.258e-12 0.414 4.993e-05 3.080e-05 6.597e-05 hd156026 7 K5V 1.144 6.330 -0.21a 1.398e-11 1.357e-11 8.456 8.105 2.623e-12 2 .313e-12 0.353 4.114e-05 2.798e-05 3.156e-05 gl416 7 K4V 1.203 9.060 – 1.178e-12 1.121e-12 10.178 9.482 2.018e-13 1.851e-13 0.329 3.623e-05 1.301e-05 2.745e-05 gl517 7 K5Ve 1.210 9.240 -0.15c 9.272e-13 8.849e-13 9.795 9.116 2.115e-13 1.9 72e-13 0.441 4.464e-05 1.877e-05 1.643e-04 hd157881 9 K5 1.359 7.540 -0.20a 5.122e-12 4.868e-12 13.782 13.040 8.167e-13 7.218e-13 0.300 2.634e-05 3.709e-06 3.227e-05 gl526 5 M1.5 1.435 8.460 – 2.297e-12 – 15.084 – 3.925e-13 3.433e-13 0.320 2.379e-05 4.811e-06 1.880e-05 gl536 5 M1 1.461 9.710 – 7.087e-13 – 14.472 – 1.436e-13 1.307e-13 0.387 2.547e-05 2.607e-06 1.920e-05 gl1 9 M1.5 1.462 8.560 – 2.062e-12 – 15.579 – 2.574e-13 2.163e-13 0.230 1.520e-05 4.825e-06 6.587e-06 hd180617 8 M2.5 1.464 9.120 – 1.220e-12 – 16.324 – 1.837e-13 1.661e-13 0.287 1.866e-05 4.279e-06 2.185e-05 gl479 6 M2 1.470 10.650 – 3.116e-13 – 16.527 – 5.484e-14 5.032e-14 0.338 2.246e-05 5.738e-06 3.258e-05 hd36395 8 M1.5 1.474 7.970 0.60c 3.719e-12 – 15.876 – 6.313e-13 5.648e-13 0 .32 2 2.133e-05 1.291e-06 4.688e-05 hd42581 8 M1/2V 1.487 8.150 – 3.000e-12 – 14.864 – 5.104e-13 4.495e-13 0.320 1.895e-05 1.113e-06 2.533e-05 gl388 6 M3.5Ve 1.540 9.430 – 9.761e-13 – 22.636 – 2.003e-13 2.289e-13 0.440 1.987e-05 1.501e-06 1.122e-04 gl699 11 M4Ve 1.570 9.540 – 9.174e-13 – 26.649 – 7.240e-14 6.531e-14 0.150 4.653e-06 4.012e-06 1.006e-05 gl551 13 M5.5Ve 1.807 11.010 – 2.303e-13 – 28.131 – 4.582e-14 5.835e-14 0.452 5.566e-06 2.657e-08 1.883e-05 Introduction Observations and stellar sample Observations and calibration of the spectra Stellar sample Pseudo-continuum Definition Relation with (B-V) Calibration Equivalent width Definition Relation with colour index (B-V) Comparison Determination of (B-V) Activity indexes and chromospheric activity N index Chromospheric activity Summary

0704.0813

Dynamics of Bose-Einstein Condensates

Dynamics of Bose-Einstein Condensates Benjamin Schlein Department of Mathematics, University of California at Davis, CA 95616, USA October 15, 2018 Abstract We report on some recent results concerning the dynamics of Bose-Einstein condensates, ob- tained in a series of joint papers [5, 6] with L. Erdős and H.-T. Yau. Starting from many body quantum dynamics, we present a rigorous derivation of a cubic nonlinear Schrödinger equation known as the Gross-Pitaevskii equation for the time evolution of the condensate wave function. 1 Introduction Bosonic systems at very low temperature are characterized by the fact that a macroscopic fraction of the particles collapses into a single one-particle state. Although this phenomenon, known as Bose-Einstein condensation, was already predicted in the early days of quantum mechanics, the first empirical evidence for its existence was only obtained in 1995, in experiments performed by groups led by Cornell and Wieman at the University of Colorado at Boulder and by Ketterle at MIT (see [2, 4]). In these important experiments, atomic gases were initially trapped by magnetic fields and cooled down at very low temperatures. Then the magnetic traps were switched off and the consequent time evolution of the gas was observed; for sufficiently small temperatures, the particles remained close together and the gas moved as a single particle, a clear sign for the existence of condensation. In the last years important progress has also been achieved in the theoretical understanding of Bose-Einstein condensation. In [10], Lieb, Yngvason, and Seiringer considered a trapped Bose gas consisting of N three-dimensional particles described by the Hamiltonian (−∆j + Vext(xj)) + Va(xi − xj), (1.1) where Vext is an external confining potential and Va(x) is a repulsive interaction potential with scattering length a (here and in the rest of the paper we use the notation ∇j = ∇xj and ∆j = ∆xj). Letting N → ∞ and a→ 0 with Na = a0 fixed, they showed that the ground state energy E(N) of (1.1) divided by the number of particle N converges to N→∞, Na=a0 = min ϕ∈L2(R3): ‖ϕ‖=1 EGP(ϕ) where EGP is the Gross-Pitaevskii energy functional EGP(ϕ) = |∇ϕ(x)|2 + Vext(x)|ϕ(x)| 2 + 4πa0|ϕ(x)| . (1.2) http://arxiv.org/abs/0704.0813v1 Later, in [9], Lieb and Seiringer also proved that trapped Bose gases characterized by the Gross- Pitaevskii scaling Na = a0 = const exhibit Bose-Einstein condensation in the ground state. More precisely, they showed that, if ψN is the ground state wave function of the Hamiltonian (1.1) and N denotes the corresponding one-particle marginal (defined as the partial trace of the density matrix γN = |ψN 〉〈ψN | over the last N − 1 particles, with the convention that Tr γ N = 1 for all N), N → |φGP〉〈φGP| as N → ∞ . (1.3) Here φGP ∈ L 2(R3) is the minimizer of the Gross-Pitaevskii energy functional (1.2). The interpre- tation of this result is straightforward; in the limit of large N , all particles, apart from a fraction vanishing asN → ∞, are in the same one-particle state described by the wave-function φGP ∈ L 2(R3). In this sense the ground state of (1.1) exhibits complete Bose-Einstein condensation into φGP. In joint works with L. Erdős and H.-T. Yau (see [5, 6, 7]), we prove that the Gross-Pitaevskii theory can also be used to describe the dynamics of Bose-Einstein condensates. In the Gross- Pitaevskii scaling (characterized by the fact that the scattering length of the interaction potential is of the order 1/N) we show, under some conditions on the interaction potential and on the initial N - particle wave function, that complete Bose-Einstein condensation is preserved by the time evolution. Moreover we prove that the dynamics of the condensate wave function is governed by the time- dependent Gross-Pitaevskii equation associated with the energy functional (1.2). As an example, consider the experimental set-up described above, where the dynamics of an initially confined gas is observed after removing the traps. Mathematically, the trapped gas can be described by the Hamiltonian (1.1), where the confining potential Vext models the magnetic traps. When cooled down at very low temperatures, the system essentially relaxes to the ground state ψN of (1.1); from [9] it follows that at time t = 0, immediately before switching off the traps, the system exhibits complete Bose-Einstein condensation into φGP in the sense (1.3). At time t = 0 the traps are turned off, and one observes the evolution of the system generated by the translation invariant Hamiltonian HN = − Va(xi − xj) . Our results (stated in more details in Section 3 below) imply that, if ψN,t = e −iHN tψN is the time evolution of the initial wave function ψN and if γ N,t denotes the one-particle marginal associated with ψN,t, then, for any fixed time t ∈ R, N,t → |ϕt〉〈ϕt| as N → ∞ where ϕt is the solution of the nonlinear time-dependent Gross-Pitaevskii equation i∂tϕt = −∆ϕt + 8πa0|ϕt| 2ϕt (1.4) with the initial data ϕt=0 = φGP. In other words, we prove that at arbitrary time t ∈ R, the system still exhibits complete condensation, and the time-evolution of the condensate wave function is determined by the Gross-Pitaevskii equation (1.4). The goal of this manuscript is to illustrate the main ideas of the proof of the results obtained in [5, 6, 7]. The paper is organized as follows. In Section 2 we define the model more precisely, and we give a heuristic argument to explain the emergence of the Gross-Pitaevskii equation (1.4). In Section 3 we present our main results. In Section 4 we illustrate the general strategy used to prove the main results and, finally, in Sections 5 and 6 we discuss the two most important parts of the proof in some more details. 2 Heuristic Derivation of the Gross-Pitaevskii Equation To describe the interaction among the particles we choose a positive, spherical symmetric, compactly supported, smooth function V (x). We denote the scattering length of V by a0. Recall that the scattering length of V is defined by the spherical symmetric solution to the zero energy equation ( V (x) f(x) = 0 f(x) → 1 as |x| → ∞ . (2.1) The scattering length of V is defined then by a0 = lim |x|→∞ |x| − |x|f(x) . This limit can be proven to exist if V decays sufficiently fast at infinity. Note that, since we assumed V to have compact support, we have f(x) = 1− (2.2) for |x| sufficiently large. Another equivalent characterization of the scattering length is given by 8πa0 = dxV (x)f(x) . (2.3) To recover the Gross-Pitaevskii scaling, we define VN (x) = N 2V (Nx). By scaling it is clear that the scattering length of VN equals a = a0/N . In fact if f(x) is the solution to (2.1), it is clear that fN (x) = f(Nx) solves ( VN (x) fN (x) = 0 (2.4) with the boundary condition fN (x) → 1 as |x| → ∞. From (2.2), we obtain fN (x) = 1− N |x| for |x| large enough. In particular the scattering length a of VN is given by a = a0/N . We consider the dynamics generated by the translation invariant Hamiltonian −∆j + VN (xi − xj) (2.5) acting on the Hilbert space L2s(R 3N ,dx1 . . . dxN), the bosonic subspace of L 2(R3N ,dx1 . . . dxN) con- sisting of all permutation symmetric functions (although it is possible to extend our analysis to include an external potential, to keep the discussion as simple as possible we only consider the translation invariant case (2.5)). We consider solutions ψN,t of the N -body Schrödinger equation i∂tψN,t = HNψN,t . (2.6) Let γN,t = |ψN,t〉〈ψN,t| denote the density matrix associated with ψN,t, defined as the orthogonal projection onto ψN,t. In order to study the limit N → ∞, we introduce the marginal densities of γN,t. For k = 1, . . . , N , we define the k-particle density matrix γ N,t associated with ψN,t by taking the partial trace of γN,t over the last N − k particles. In other words, γ N,t is defined as the positive trace class operator on L2s(R 3k) with kernel given by N,t(xk;x dxN−k ψN,t(xk,xN−k)ψN,t(x k,xN−k) . (2.7) Here and in the rest of the paper we use the notation x = (x1, x2, . . . , xN ), xk = (x1, x2, . . . , xk), x′k = (x 2, . . . , x k), and xN−k = (xk+1, xk+2, . . . , xN ). We consider initial wave functions ψN,0 exhibiting complete condensation in a one-particle state ϕ. Thus at time t = 0, we assume that N,0 → |ϕ〉〈ϕ| as N → ∞ . (2.8) It turns out that the last equation immediately implies that N,0 → |ϕ〉〈ϕ| ⊗k as N → ∞ (2.9) for every fixed k ∈ N (the argument, due to Lieb and Seiringer, can be found in [9], after Theorem 1). It is also interesting to notice that the convergence (2.8) (and (2.9)) in the trace class norm is equivalent to the convergence in the weak* topology defined on the space of trace class operators on 3 (or R3k, for (2.9)); we thank A. Michelangeli for pointing out this fact to us (the proof is based on general arguments, such as Grümm’s Convergence Theorem). Starting from the Schrödinger equation (2.6) for the wave function ψN,t, we can derive evolution equations for the marginal densities γ N,t. The dynamics of the marginals is governed by a hierarchy of N coupled equations usually known as the BBGKY hierarchy. N,t = −∆j, γ VN (xi − xj), γ + (N − k) Trk+1 VN (xj − xk+1), γ (k+1) (2.10) Here Trk+1 denotes the partial trace over the (k + 1)-th particle. Next we study the limit N → ∞ of the density γ N,t for fixed k ∈ N. For simplicity we fix k = 1. From (2.10), the evolution equation for the one-particle density matrix, written in terms of its kernel N,t(x1;x 1) is given by N,t(x1, x −∆1 +∆ N,t(x1;x + (N − 1) VN (x1 − x2)− VN (x 1 − x2) N,t(x1, x2;x 1, x2) . (2.11) Suppose now that γ ∞,t and γ ∞,t are limit points (with respect to the weak* topology) of γ N,t and, respectively, γ N,t as N → ∞. Since, formally, (N − 1)VN (x) = (N − 1)N 2V (Nx) ≃ N3V (Nx) → b0δ(x) with b0 = dxV (x) as N → ∞, we could naively expect the limit points γ ∞,t and γ ∞,t to satisfy the limiting equation ∞,t(x1;x −∆1 +∆ ∞,t(x1;x 1) + b0 δ(x1 − x2)− δ(x 1 − x2) ∞,t(x1, x2;x 1, x2) . (2.12) From (2.9) we have, at time t = 0, ∞,0(x1;x 1) = ϕ(x1)ϕ(x ∞,0(x1, x2;x 2) = ϕ(x1)ϕ(x2)ϕ(x 1)ϕ(x (2.13) If condensation is really preserved by the time evolution, also at time t 6= 0 we have ∞,t(x1;x 1) = ϕt(x1)ϕt(x ∞,t(x1, x2;x 2) = ϕt(x1)ϕt(x2)ϕt(x 1)ϕt(x (2.14) Inserting (2.14) in (2.12), we obtain the self-consistent equation i∂tϕt = −∆ϕt + b0|ϕt| 2ϕt (2.15) for the condensate wave function ϕt. This equation has the same form as the time-dependent Gross- Pitaevskii equation (1.4), but a different coefficient in front of the nonlinearity (b0 instead of 8πa0). The reason why we obtain the wrong coupling constant in (2.15) is that going from (2.11) to (2.12), we took the two limits (N − 1)VN (x) → b0δ(x) and γ N,t → γ ∞,t (2.16) independently from each other. However, since the scattering length of the interaction is of the order 1/N , the two-particle density γ N,t develops a short scale correlation structure on the length scale 1/N , which is exactly the same length scale on which the potential VN varies. For this reason the two limits in (2.16) cannot be taken independently. In order to obtain the correct Gross-Pitaevskii equation (1.4) we need to take into account the correlations among the particles, and the short scale structure they create in the marginal density γ To describe the correlations among the particles we make use of the solution fN (x) to the zero energy scattering equation (2.4). Assuming that the function fN (xi−xj) gives a good approximation for the correlations between particles i and j, we may expect that the one- and two-particle densities associated with the evolution of a condensate are given, for large but finite N , by N,t(x1;x 1) ≃ ϕt(x1)ϕt(x N,t(x1, x2;x 2) ≃ fN (x1 − x2)fN (x 1 − x 2)ϕt(x1)ϕt(x2)ϕt(x 1)ϕt(x (2.17) Inserting this ansatz into (2.11), we obtain a new self-consistent equation i∂tϕt = −∆ϕt + (N − 1) dxfN (x)VN (x) = −∆ϕt + dxf(Nx)V (Nx) = −∆ϕt + 8πa0|ϕt| (2.18) because of (2.3). This is exactly the Gross-Pitaevskii equation (1.4), with the correct coupling constant in front of the nonlinearity. Note that the presence of the correlation functions fN (x1−x2) and fN (x 2) in (2.17) does not contradict complete condensation of the system at time t. On the contrary, in the weak limit N → ∞, the function fN converges to one, and therefore γ N,t and γ N,t converge to |ϕt〉〈ϕt| and |ϕt〉〈ϕt| respectively. The correlations described by the function fN can only produce nontrivial effects on the macroscopic dynamics of the system because of the singularity of the interaction potential VN . From this heuristic argument it is clear that, in order to obtain a rigorous derivation of the Gross- Pitaevskii equation (2.18), we need to identify the short scale structure of the marginal densities and prove that, in a very good approximation, it can be described by the function fN as in (2.17). In other words, we need to show a very strong separation of scales in the marginal density γ N,t (and, more generally, in the k-particle density γ N,t) associated with the solution of the N -body Schrödinger equation; the Gross-Pitaevskii theory can only be correct if γ N,t has a regular part, which factorizes for large N into the product of k copies of the orthogonal projection |ϕt〉〈ϕt|, and a time independent singular part, due to the correlations among the particles, and described by products of the functions fN (xi − xj), 1 ≤ i, j ≤ k. 3 Main Results To prove our main results we need to assume the interaction potential to be sufficiently weak. To measure the strength of the potential, we introduce the dimensionless quantity α = sup |x|2V (x) + V (x) . (3.1) Apart from the smallness assumption on the potential, we also need to assume that the correlations characterizing the initial N -particle wave function are sufficiently weak. We define therefore the notion of asymptotically factorized wave functions. We say that a family of permutation symmetric wave functions ψN is asymptotically factorized if there exists ϕ ∈ L 2(R3) and, for any fixed k ≥ 1, there exists a family ξ (N−k) N ∈ L 3(N−k)) such that ∥∥∥ψN − ϕ⊗k ⊗ ξ (N−k) ∥∥∥→ 0 as N → ∞ . (3.2) It is simple to check that, if ψN is asymptotically factorized, then it exhibits complete Bose-Einstein condensation in the one-particle state ϕ (in the sense that the one-particle density associated with ψN satisfy γ N → |ϕ〉〈ϕ| as N → ∞). Asymptotic factorization is therefore a stronger condition than complete condensation, and it provides more control on the correlations of ψN . Theorem 3.1. Assume that V (x) is a positive, smooth, spherical symmetric, and compactly sup- ported potential such that α (defined in (3.1)) is sufficiently small. Consider an asymptotically factorized family of wave functions ψN ∈ L 3N ), exhibiting complete Bose-Einstein condensation in a one-particle state ϕ ∈ H1(R3), in the sense that N → |ϕ〉〈ϕ| as N → ∞ (3.3) where γ N denotes the one-particle density associated with ψN . Then, for any fixed t ∈ R, the one- particle density γ N,t associated with the solution ψN,t of the N -particle Schrödinger equation (2.6) satisfies N,t → |ϕt〉〈ϕt| as N → ∞ (3.4) where ϕt is the solution to the time-dependent Gross-Pitaevskii equation i∂tϕt = −∆ϕt + 8πa0|ϕt| 2ϕt (3.5) with initial data ϕt=0 = ϕ. The convergence in (3.3) and (3.4) is in the trace norm topology (which in this case is equivalent to the weak* topology defined on the space of trace class operators on R3). Moreover, from (3.4) we also get convergence of higher marginal. For every k ≥ 1, we have N,t → |ϕt〉〈ϕt| ⊗k as N → ∞. Theorem 3.1 can be used to describe the dynamics of condensates satisfying the condition of asymptotic factorization. The following two corollaries provide examples of such initial data. The simplest example of N -particle wave function satisfying the assumption of asymptotic fac- torization is given by a product state. Corollary 3.2. Under the assumptions on V (x) stated in Theorem 3.1, let ψN (x) = j=1 ϕ(xj) for an arbitrary ϕ ∈ H1(R3). Then, for any t ∈ R, N,t → |ϕt〉〈ϕt| as N → ∞ where ϕt is a solution of the Gross-Pitaevskii equation (3.5) with initial data ϕt=0 = ϕ. The second application of Theorem 3.1 gives a mathematical description of the results of the experiments depicted in the introduction. (−∆j + Vext(xj)) + VN (xi − xj) (3.6) with a confining potential Vext. Let ψN be the ground state of H N . By [9], ψN exhibits complete Bose Einstein condensation into the minimizer φGP of the Gross-Pitaevskii energy functional EGP defined in (1.2). In other words N → |φGP〉〈φGP| as N → ∞ . In [5], we demonstrate that ψN also satisfies the condition (3.2) of asymptotic factorization. From this observation, we obtain the following corollary. Corollary 3.3. Under the assumptions on V (x) stated in Theorem 3.1, let ψN be the ground state of (3.6), and denote by γ N,t the one-particle density associated with the solution ψN,t = e −iHN tψN of the Schrödinger equation (2.6). Then, for any fixed t ∈ R, N,t → |ϕt〉〈ϕt| as N → ∞ where ϕt is the solution of the Gross-Pitaevskii equation (3.5) with initial data ϕt=0 = φGP. Although the second corollary describes physically more realistic situations, also the first corollary has interesting consequences. In Section 2, we observed that the emergence of the scattering length in the Gross-Pitaevskii equation is an effect due to the correlations. The fact that the Gross-Pitaevskii equation describes the dynamics of the condensate also if the initial wave function is completely uncorrelated, as in Corollary 3.2, implies that the N -body Schrödinger dynamics generates the singular correlation structure in very short times. Of course, when the wave function develops correlations on the length scale 1/N , the energy associated with this length scale decreases; since the total energy is conserved by the Schrödinger evolution, we must conclude that together with the short scale structure at scales of order 1/N , the N -body dynamics also produces oscillations on intermediate length scales 1/N ≪ ℓ ≪ 1, which carry the excess energy (the difference between the energy of the factorized wave function and the energy of the wave function with correlations on the length scale 1/N) and which have no effect on the macroscopic dynamics (because only variations of the wave function on length scales of order one and order 1/N affect the macroscopic dynamics described by the Gross-Pitaevskii equation). 4 General Strategy of the Proof and Previous Results In this section we illustrate the strategy used to prove Theorem 3.1. The proof is divided into three main steps. Step 1. Compactness of γ N,t. Recall, from (2.7), the definition of the marginal densities γ associated with the solution ψN,t = exp(−iHN t)ψN of the N -body Schrödinger equation. By defini- tion, for any N ∈ N and t ∈ R, γ N,t is a positive operator in L k = L 1(L2(R3k)) (the space of trace class operators on L2(R3k)) with trace equal to one. For fixed t ∈ R and k ≥ 1, it follows by standard general argument (Banach-Alaouglu Theorem) that the sequence {γ N,t}N≥k is compact with respect to the weak* topology of L1k. Note here that L k has a weak* topology because L k = K k, where Kk = K(L 2(R3k)) is the space of compact operators on L2(R3k). To make sure that we can find subsequences of γ N,t which converge for all times in a certain interval, we fix T > 0 and consider the space C([0, T ],L1k) of all functions of t ∈ [0, T ] with values in L k which are continuous with respect to the weak* topology on L1k. Since Kk is separable, it follows that the weak* topology on the unit ball of L1k is metrizable; this allows us to prove the equicontinuity of the densities γ N,t, and to obtain compactness of the sequences {γ N,t}N≥k in C([0, T ],L Step 2. Convergence to an infinite hierarchy. By Step 1 we know that, as N → ∞, the family of marginal densities ΓN,t = {γ k=1 has at least one limit point Γ∞,t = {γ ∞,t}k≥1 in⊕ k≥1C([0, T ],L k) with respect to the product topology. Next, we derive evolution equations for the limiting densities γ ∞,t. Starting from the BBGKY hierarchy (2.10) for the family ΓN,t, we prove that any limit point Γ∞,t satisfies the infinite hierarchy of equations ∞,t = −∆j, γ + 8πa0 Trk+1 δ(xj − xk+1), γ (k+1) (4.1) for k ≥ 1. It is at this point, in the derivation of this infinite hierarchy, that we need to identify the singular part of the densities γ (k+1) N,t . The emergence of the scattering length in the second term on the right hand side of (4.1) is due to short scale structure of γ (k+1) N,t . It is worth noticing that the infinite hierarchy (4.1) has a factorized solution. In fact, it is simple to see that the infinite family t = |ϕt〉〈ϕt| ⊗k for k ≥ 1 (4.2) solves (4.1) if and only if ϕt is a solution to the Gross-Pitaevskii equation (3.5). Step 3. Uniqueness of the solution to the infinite hierarchy. To conclude the proof of Theorem 3.1, we show that the infinite hierarchy (4.1) has a unique solution. This implies immediately that the densities γ N,t converge; in fact, a compact sequence with at most one limit point is always convergent. Moreover, since we know that the factorized densities (4.2) are a solution, it also follows that, for any k ≥ 1, N,t → |ϕt〉〈ϕt| ⊗k as N → ∞ with respect to the weak* topology of L1k. Similar strategies have been used to obtain rigorous derivations of the nonlinear Hartree equation i∂tϕt = −∆ϕt + (v ∗ |ϕt| 2)ϕt (4.3) for the dynamics of initially factorized wave functions in bosonic many particle mean field models, characterized by the Hamiltonian HmfN = −∆j + v(xi − xj) . (4.4) In this context, the approach outlined above was introduced by Spohn in [11], who applied it to derive (4.3) in the case of a bounded potential v. In [8], Erdős and Yau extended Spohn’s result to the case of a Coulomb interaction v(x) = ±1/|x| (partial results for the Coulomb case, in particular the convergence to the infinite hierarchy, were also obtained by Bardos, Golse, and Mauser, see [3]). More recently, Adami, Golse, and Teta used the same approach in [1] for one-dimensional systems with dynamics generated by a Hamiltonian of the form (4.4) with an N -dependent pair potential vN (x) = N βV (Nβx), β < 1. In the limit N → ∞, they obtain the nonlinear Schrödinger equation i∂tϕt = −∆ϕt + b0|ϕt| 2ϕt with b0 = V (x)dx . Notice that the Hamiltonian (2.5) has the same form as the mean field Hamiltonian (4.4), with an N -dependent pair potential vN (x) = N 3V (Nx). Of course, one may also ask what happens if we consider the mean field Hamiltonian (4.4) with the N -dependent potential vN (x) = N 3βV (Nβx), for β 6= 1. If β < 1, the short scale structure developed by the solution of the Schrödinger equation is still characterized by the length scale 1/N (because the scattering length of N3β−1V (Nβx) is still of order 1/N); but this time the potential varies on much larger scales, of the order N−β ≫ N−1. For this reason, if β < 1, the scattering length does not appear in the effective macroscopic equation (8πa0 is replaced by b0 = dxV (x)). In [6] (and previously in [5] for 0 < β < 1/2) we prove in fact that Corollary 3.2 can be extended to include the case 0 < β < 1 as follows. Theorem 4.1. Suppose ψN (x) = j=1 ϕ(xj), for some ϕ ∈ H 1(R3). Let ψN,t = e −iHβ,N tψN with the mean-field Hamiltonian Hβ,N = −∆j + N3βV (Nβ(xi − xj)) for a positive, spherical symmetric, compactly supported, and smooth potential V such that α (defined in (3.1)) is sufficiently small. Let γ N,t be the one-particle density associated with ψN,t. Then, if 0 < β ≤ 1 we have, for any fixed t ∈ R, γ N,t → |ϕt〉〈ϕt| as N → ∞. Here ϕt is the solution to the nonlinear Schrödinger equation i∂tϕt = −∆ϕt + σ|ϕt| with initial data ϕt=0 = ϕ and with 8πa0 if β = 1 b0 if 0 < β < 1 5 Convergence to the Infinite Hierarchy In this section we give some more details concerning Step 2 in the strategy outlined above. We consider a limit point Γ∞,t = {γ ∞,t}k≥1 of the sequence ΓN,t = {γ k=1 and we prove that Γ∞,t satisfies the infinite hierarchy (4.1). To this end we use that, for finite N , the family ΓN,t satisfies the BBGKY hierarchy (2.10), and we show the convergence of each term in (2.10) to the corresponding term in the infinite hierarchy (4.1) (the second term on the r.h.s. of (2.10) is of smaller order and can be proven to vanish in the limit N → ∞). The main difficulty consists in proving the convergence of the last term on the right hand side of (2.10) to the last term on the right hand side of (4.1). In particular, we need to show that in the limit N → ∞ we can replace the potential (N − k)N2V (N(xj −xk+1)) ≃ N 3V (Nx) in the last term on the r.h.s. of (2.10) by 8πa0δ(xj − xk+1) . In terms of kernels we have to prove that dxk+1 N3V (N(xj − xk+1))− 8πa0δ(xj − xk+1) (k+1) N,t (xk, xk+1,x k, xk+1) → 0 (5.1) as N → ∞. It is enough to prove the convergence (5.1) in a weak sense, after testing the expression against a smooth k-particle kernel J (k)(xk;x k). Note, however, that the observable J (k) does not help to perform the integration over the variable xk+1. The problem here is that, formally, the N -dependent potential N3V (N(xj − xk+1)) does not converge towards 8πa0δ(xj − xk+1) as N → ∞ (it converges towards b0δ(xj − xk+1), with b0 =∫ dxV (x)). Eq. (5.1) is only correct because of the correlations between xj and xk+1 hidden in the density γ (k+1) N,t . Therefore, to prove (5.1), we start by factoring out the correlations explicitly, and by proving that, as N → ∞, dxk+1 N3V (N(xj − xk+1))fN (xj − xk+1)− 8πa0δ(xj − xk+1) ) γ(k+1)N,t (xk, xk+1,x k, xk+1) fN (xj − xk+1) (5.2) where fN (x) is the solution to the zero energy scattering equation (2.4). Then, in a second step, we use the fact that fN → 1 in the weak limitN → ∞, to prove that the ratio γ (k+1) N,t /fN (xj−xk+1) converges to the same limiting density γ (k+1) ∞,t as γ (k+1) N,t . Eq. (5.2) looks now much better than (5.1) because, formally, N3V (N(xj − xk+1))fN (xj − xk+1) does converge to 8πa0δ(xj − xk+1). To prove that (5.2) is indeed correct, we only need some regularity of the ratio γ (k+1) N,t (xk, xk+1;x k, xk+1)/fN (xj − xk+1) in the variables xj and xk+1. In terms of the N -particle wave function ψN,t we need regularity of ψN,t(x)/fN (xi−xj) in the variables xi, xj, for any i 6= j. To establish the required regularity we use the following energy estimate. Proposition 5.1. Consider the Hamiltonian HN defined in (2.5), with a positive, spherical sym- metric, smooth and compactly supported potential V . Suppose that α (defined in (3.1)) is sufficiently small. Then there exists C = C(α) > 0 such that 〈ψ,H2Nψ〉 ≥ CN ∣∣∣∣∇i∇j fN(xi − xj) . (5.3) for all i 6= j and for all ψ ∈ L2s(R 3N ,dx). Making use of this energy estimate it is possible to deduce strong a-priori bounds on the solution ψN,t of the Schrödinger equation (2.6). These bounds have the form ∣∣∣∣∇i∇j ψN,t(x) fN(xi − xj) ≤ C (5.4) uniformly in N ∈ N and t ∈ R. To prove (5.4) we use that, by (5.3), and because of the conservation of the energy along the time evolution, ∣∣∣∣∇i∇j ψN,t(x) fN (xi − xj) ≤ CN−2〈ψN,t,H NψN,t〉 = CN −2〈ψN,0,H NψN,0〉 . (5.5) From (5.5) and using an approximation argument on the initial wave function to make sure that the expectation of H2N at time t = 0 is of the order N 2, we obtain (5.4). The bounds (5.4) are then sufficient to prove the convergence (5.1) (using a non-standard Poincaré inequality; see Lemma 7.2 in [6]). Remark that the a-priori bounds (5.4) do not hold true if we do not divide the solution ψN,t of the Schrödinger equation by fN(xi − xj) (replacing ψN,t(x)/fN (xi − xj) by ψN (x) the integral in (5.4) would be of order N). It is only after removing the singular factor fN (xi − xj) from ψN,t(x) that we can prove useful bounds on the regular part of the wave function. It is through the a-priori bounds (5.4) that we identify the correlation structure of the wave function ψN,t and that we show that, when xi and xj are close to each other, ψN,t(x) can be approximated by the time independent singular factor fN(xi − xj), which varies on the length scale 1/N , multiplied with a regular part (regular in the sense that it satisfy the bounds (5.4)). It is therefore through (5.4) that we establish the strong separation of scales in the wave function ψN,t and in the marginal densities γ N,t which is of fundamental importance for the Gross-Pitaevskii theory. Since it is quite short and it shows why the solution fN (xi − xj) to the zero energy scattering equation (2.1) can be used to describe the two-particle correlations, we reproduce in the following the proof Proposition 5.1. Note that this is the only step in the proof of our main theorem where the smallness of constant α, measuring the strength of the interaction potential, is used. The positivity of the interaction potential, on the other hand, also plays an important role in many other parts of the proof. Proof of Proposition 5.1. We decompose the Hamiltonian (2.5) as hj with hj = −∆j + i 6=j VN (xi − xj) . For an arbitrary permutation symmetric wave function ψ and for any fixed i 6= j, we have 〈ψ,H2Nψ〉 = N〈ψ, h iψ〉+N(N − 1)〈ψ, hihjψ〉 ≥ N(N − 1)〈ψ, hihjψ〉 . Using the positivity of the potential, we find 〈ψ,H2Nψ〉 ≥ N(N − 1) −∆i + VN (xi − xj) −∆j + VN (xi − xj) . (5.6) Next, we define φ(x) by ψ(x) = fN (xi−xj)φ(x) (φ is well defined because fN (x) > 0 for all x ∈ R note that the definition of the function φ depends on the choice of i, j. Then fN (xi − xj) ∆i (fN (xi − xj)φ(x)) = ∆iφ(x) + (∆fN )(xi − xj) fN (xi − xj) φ(x) + ∇fN (xi − xj) fN (xi − xj) ∇iφ(x) . From (2.1) it follows that fN (xi − xj) −∆i + VN (xi − xj) fN (xi − xj)φ(x) = Liφ(x) and analogously fN (xi − xj) −∆j + VN (xi − xj) fN (xi − xj)φ(x) = Ljφ(x) where we defined Lℓ = −∆ℓ + 2 ∇ℓ fN (xi − xj) fN (xi − xj) ∇ℓ, for ℓ = i, j . Remark that, for ℓ = i, j, the operator Lℓ satisfies dx f2N (xi−xj) Lℓ φ(x) ψ(x) = dx f2N (xi−xj) φ(x) Lℓ ψ(x) = dx f2N (xi−xj) ∇ℓ φ(x) ∇ℓ ψ(x) . Therefore, from (5.6), we obtain 〈ψ,H2Nψ〉 ≥ N(N − 1) dx f2N (xi − xj) Li φ(x)Lj φ(x) = N(N − 1) dx f2N (xi − xj) ∇iφ(x)∇iLj φ(x) = N(N − 1) dx f2N (xi − xj) ∇iφ(x)Lj ∇iφ(x) +N(N − 1) dx f2N(xi − xj) ∇iφ(x) [∇i, Lj ]φ(x) = N(N − 1) dx f2N (xi − xj) |∇j∇iφ(x)| +N(N − 1) dx f2N(xi − xj) ∇fN(xi − xj) fN (xi − xj) ∇iφ(x)∇jφ(x) . (5.7) To control the second term on the right hand side of the last equation we use bounds on the function fN , which can be derived from the zero energy scattering equation (2.1): 1− Cα ≤ fN (x) ≤ 1, |∇fN (x)| ≤ C , |∇2fN(x)| ≤ C (5.8) for constants C independent of N and of the potential V (recall the definition of the dimensionless constant α from (3.1)). Therefore, for α < 1, dx f2N (xi − xj) ∇fN(xi − xj) fN (xi − xj) ∇iφ(x)∇jφ(x) |xi − xj | |∇iφ(x)| |∇jφ(x)| |xi − xj |2 |∇iφ(x)| 2 + |∇jφ(x)| dx |∇i∇jφ(x)| (5.9) where we used Hardy inequality. Thus, from (5.7), and using again the first bound in (5.8), we obtain 〈ψ,H2Nψ〉 ≥ N(N − 1)(1 −Cα) dx |∇i∇jφ(x)| which implies (5.3). 6 Uniqueness of the Solution to the Infinite Hierarchy In this section we discuss the main ideas used to prove the uniqueness of the solution to the infinite hierarchy (Step 3 in the strategy outlined in Section 4). First of all, we need to specify in which class of family of densities Γt = {γ t }k≥1 we want to prove the uniqueness of the solution to the infinite hierarchy (4.1). Clearly, the proof of the uniqueness is simpler if we can restrict our attention to smaller classes. But of course, in order to apply the uniqueness result to prove Theorem 3.1, we need to make sure that any limit point of the sequence ΓN,t = {γ k=1 is in the class for which we can prove uniqueness. We are going to prove uniqueness for all families Γt = {γ t }k≥1 ∈ C([0, T ],L1k) with t ‖Hk := Tr ∣∣∣(1−∆1)1/2 . . . (1−∆k)1/2 γ t (1−∆k) 1/2 . . . (1−∆1) ∣∣∣ ≤ Ck (6.1) for all t ∈ [0, T ] and for all k ≥ 1 (with a constant C independent of k). The following proposition guarantees that any limit point of the sequence ΓN,t satisfies (6.1). Proposition 6.1. Assume the same conditions as in Proposition 5.1. Suppose that Γ∞,t = {γ ∞,t}k≥1 is a limit point of ΓN,t = {γ k=1 with respect to the product topology on k≥1C([0, T ],L k). Then ∞,t ≥ 0 and there exists a constant C such that Tr (1−∆1) . . . (1−∆k)γ ∞,t ≤ C k (6.2) for all k ≥ 1 and t ∈ [0, T ]. Because of Proposition 6.1, it is enough to prove the uniqueness of the infinite hierarchy (4.1) in the following sense. Theorem 6.2. Suppose that Γ = {γ(k)}k≥1 is such that ‖γ(k)‖Hk ≤ C k (6.3) for all k ≥ 1 (the norm ‖.‖Hk is defined in (6.1)). Then there exists at most one solution Γt = t }k≥1 ∈ C([0, T ],Lk) of (4.1) such that Γt=0 = Γ and t ‖Hk ≤ C k (6.4) for all k ≥ 1 and all t ∈ [0, T ] (with the same constant C as in (6.3)). In the next two subsections we explain the main ideas of the proofs of Proposition 6.1 and Theorem 6.2. 6.1 Higher Order Energy Estimates The main difficulty in proving Proposition 6.1 is the fact that the estimate (6.2) does not hold true if we replace γ ∞,t by the marginal density γ N,t. More precisely, Tr (1−∆1) . . . (1−∆k)γ N,t ≤ C k (6.5) cannot hold true with a constant C independent of N . In fact, for finite N and k > 1, the k-particle density γ N,t still contains the short scale structure due to the correlations among the particles. Therefore, when we take derivatives of γ N,t as in (6.5), the singular structure (which varies on a length scale of order 1/N) generates contributions which diverge in the limit N → ∞. To overcome this problem, we cutoff the wave function ψN,t when two or more particles come at distances smaller than some intermediate length scale ℓ, with N−1 ≪ ℓ ≪ 1 (more precisely, the cutoff will be effective only when one or more particles come close to one of the variable xj over which we want to take derivatives). For fixed j = 1, . . . , N , we define θj ∈ C ∞(R3N ) such that θj(x) ≃ 1 if |xi − xj| ≫ ℓ for all i 6= j 0 if there exists i 6= j with |xi − xj | . ℓ It is important, for our analysis, that θj controls its derivatives (in the sense that, for example, |∇iθj| ≤ Cℓ j ); for this reason we cannot use standard compactly supported cutoffs, but instead we have to construct appropriate functions which decay exponentially when particles come close together. Making use of the functions θj(x), we prove the following higher order energy estimates. Proposition 6.3. Choose ℓ ≪ 1 such that Nℓ2 ≫ 1. Suppose that α is small enough. Then there exist constants C1 and C2 such that, for any ψ ∈ L 3N ), 〈ψ, (HN + C1N) kψ〉 ≥ C2N dx θ1(x) . . . θk−1(x) |∇1 . . .∇kψ(x)| 2 . (6.6) The meaning of the bounds (6.6) is clear. We can control the L2-norm of the k-th derivative ∇1 . . .∇kψ by the expectation of the k-th power of the energy per particle, if we only integrate over configurations where the first k − 1 particles are “isolated” (in the sense that there is no particle at distances smaller than ℓ from x1, x2, . . . , xk−1). In this sense the energy estimate in Proposition 5.1 (which, compared with Proposition 6.3, is restricted to k = 2) is more precise than (6.6), because it identifies and controls the singularity of the wave function exactly in the region cutoff from the integral on the right side of (6.6). The point is that, while Proposition 5.1 is used to identify the two-particle correlations in the marginal densities γ N,t (which are essential for the emergence of the scattering length a0 in the infinite hierarchy (4.1)), we only need Proposition 6.3 to establish properties of the limiting densities; this is why we can introduce cutoffs in (6.6), provided we can show their effect to vanish in the limit N → ∞. Note that we can allow one “free derivative”; in (6.6) we take the derivative over xk although there is no cutoff θk(x). The reason is that the correlation structure becomes singular, in the L sense, only when we derive it twice (if one uses the zero energy solution fN introduced in (2.1) to describe the correlations, this can be seen by observing that ∇fN (x) ≃ 1/|x|, which is locally square integrable). Remark that the condition Nℓ2 ≫ 1 is a consequence of the fact that, if ℓ is too small, the error due to the localization of the kinetic energy on distances of order ℓ cannot be controlled. The proof of Proposition 6.3 is based on induction over k; for details see Section 9 in [6]. From the estimates (6.6), using the preservation of the expectation ofHkN along the time evolution and a regularization of the initial N -particle wave function ψN , we obtain the following bounds for the solution ψN,t = e −iHN tψN of the Schrödinger equation (2.6). dx θ1(x) . . . θk−1(x) |∇1 . . .∇kψN,t(x)| ≤ Ck (6.7) uniformly in N and t, and for all k ≥ 1. Translating these bounds in the language of the density matrix γN,t, we obtain Tr θ1 . . . θk−1∇1 . . .∇kγN,t∇ 1 . . .∇ k ≤ C k . (6.8) The idea now is to use the freedom in the choice of the cutoff length ℓ. If we fix the position of all particles but xj, it is clear that the cutoff θj is effective in a volume at most of the order Nℓ 3. If we choose now ℓ such that Nℓ3 → 0 as N → ∞ (which is of course compatible with the condition that Nℓ2 ≫ 1), then we can expect that, in the limit of large N , the cutoff becomes negligible. This approach yields in fact the desired results; starting from (6.8), and choosing ℓ such that Nℓ3 ≪ 1, we can complete the proof of Proposition 6.1 (see Proposition 6.3 in [6] for more details). 6.2 Expansion in Feynman Graphs To prove Theorem 6.2, we start by rewriting the infinite hierarchy (4.1) in the integral form γt = U (k)(t)γ0 + 8iπa0 ds U (k)(t− s)Trk+1 δ(xj − xk+1), γ (k+1) = U (k)(t)γ0 + ds U (k)(t− s)B(k)γ(k+1)s , (6.9) where U (k)(t) denotes the free evolution of k particles, U (k)(t)γ(k) = eit ∆jγ(k)e−it and the collision operator B(k) maps (k+1)-particle operators into k-particle operators according to B(k)γ(k+1) = 8iπa0 Trk+1 δ(xj − xk+1), γ (k+1) (6.10) (recall that Trk+1 denotes the partial trace over the (k + 1)-th particle). Iterating (6.9) n times we obtain the Duhamel type series t = U (k)(t)γ m,t + η n,t (6.11) m,t = ds1 . . . ∫ sm−1 dsm U (k)(t− s1)B (k)U (k+1)(s1 − s2)B (k+1) . . . B(k+m−1)U (k+m)(sm)γ (k+m) · · · ds1 . . . ∫ sm−1 dsm U (k)(t− s1)Trk+1 δ(xj1 − xk+1), U (k+1)(s1 − s2)Trk+2 δ(xj2 − xk+2), . . .Trk+m δ(xjm − xk+m),U (k+m)(sm)γ (k+m) . . . (6.12) and the error term n,t = ds2 . . . ∫ sn−1 dsn U (k)(t− s1)B (k)U (k+1)(s1 − s2)B (k+1) . . . B(k+n−1)γ(k+m)sn . (6.13) Note that the error term (6.13) has exactly the same form as the terms in (6.12), with the only difference that the last free evolution is replaced by the full evolution γ (k+m) 2k+2m leaves2k roots Vertices: Figure 1: A Feynman graph in Fm,k and its two types of vertices To prove the uniqueness of the infinite hierarchy, it is enough to prove that the error term (6.13) converges to zero as n → ∞ (in some norm, or even only after testing it against a sufficiently large class of smooth observables). The main problem here is that the delta function in the collision operator B(k) cannot be controlled by the kinetic energy (in the sense that, in three dimensions, the operator inequality δ(x) ≤ C(1 − ∆) does not hold true). For this reason, the a-priori estimates t ‖Hk ≤ C k are not sufficient to show that (6.13) converges to zero, as n→ ∞. Instead, we have to make use of the smoothing effects of the free evolutions U (k+j)(sj − sj+1) in (6.13) (in a similar way, Stricharzt estimates are used to prove the well-posedness of nonlinear Schrödinger equations). To this end, we rewrite each term in the series (6.11) as a sum of contributions associated with certain Feynman graphs, and then we prove the convergence of the Duhamel expansion by controlling each contribution separately. The details of the diagrammatic expansion can be found in Section 9 of [5]. Here we only present the main ideas. We start by considering the term ξ m,t in (6.12). After multiplying it with a compact k-particle observable J (k) and taking the trace, we expand the result as Tr J (k)ξ m,t = Λ∈Fm,k KΛ,t (6.14) where KΛ,t is the contribution associated with the Feynman graph Λ. Here Fm,k denotes the set of all graphs consisting of 2k disjoint, paired, oriented, and rooted trees with m vertices. An example of a graph in Fm,k is drawn in Figure 1. Each vertex has one of the two forms drawn in Figure 1, with one “father”-edge on the left (closer to the root of the tree) and three “son”-edges on the right. One of the son edge is marked (the one drawn on the same level as the father edge; the other two son edges are drawn below). Graphs in Fm,k have 2k + 3m edges, 2k roots (the edges on the very left), and 2k + 2m leaves (the edges on the very right). It is possible to show that the number of different graphs in Fm,k is bounded by 2 4m+k. The particular form of the graphs in Fm,k is due to the quantum mechanical nature of the expansion; the presence of a commutator in the collision operator (6.10) implies that, for every B(k+j) in (6.12), we can choose whether to write the interaction on the left or on the right of the density. When we draw the corresponding vertex in a graph in Fm,k, we have to choose whether to attach it on the incoming or on the outgoing edge. Graphs in Fm,k are characterized by a natural partial ordering among the vertices (v ≺ v the vertex v is on the path from v′ to the roots); there is, however, no total ordering. The absence of total ordering among the vertices is the consequence of a rearrangement of the summands on the r.h.s. of (6.12); by removing the order between times associated with non-ordered vertices we significantly reduce the number of terms in the expansion. In fact, while (6.12) contains (m+ k)!/k! summands, in (6.14) we are only summing over 24m+k contributions. The price we have to pay is that the apparent gain of a factor 1/m! because of the ordering of the time integrals in (6.12) is lost in the new expansion (6.14). However, since the time integrations are already needed to smooth out singularities, and since they cannot be used simultaneously for smoothing and for gaining a factor 1/m!, it seems very difficult to make use of the apparent factor 1/m! in (6.12). In fact, we find that the expansion (6.14) is better suited for analyzing the cumulative space-time smoothing effects of the multiple free evolutions than (6.12). Because of the pairing of the 2k trees, there is a natural pairing between the 2k roots of the graph. Moreover, it is also possible to define a natural pairing of the leaves of the graph (this is evident in Figure 1); two leaves ℓ1 and ℓ2 are paired if there exists an edge e1 on the path from ℓ1 back to the roots, and an edge e2 on the path from ℓ2 to the roots, such that e1 and e2 are the two unmarked son-edges of the same vertex (or, if there is no unmarked sons in the path from ℓ1 and ℓ2 to the roots, if the two roots connected to ℓ1 and ℓ2 are paired). For Λ ∈ Fm,k, we denote by E(Λ), V (Λ), R(Λ) and L(Λ) the set of all edges, vertices, roots and, respectively, leaves in the graph Λ. For every edge e ∈ E(Λ), we introduce a three-dimensional momentum variable pe and a one-dimensional frequency variable αe. Then, denoting by γ̂ (k+m) 0 and by Ĵ (k) the kernels of the density γ (k+m) 0 and of the observable J (k) in Fourier space, the contribution KΛ,t in (6.14) is given by KΛ,t = e∈E(Λ) dpedαe αe − p2e + iτeµe v∈V (Λ) × exp e∈R(Λ) τe(αe + iτeµe)  Ĵ (k) {pe}e∈R(Λ) (k+m) {pe}e∈L(Λ) (6.15) Here τe = ±1, according to the orientation of the edge e. We observe from (6.15) that the momenta of the roots of Λ are the variables of the kernel of J (k), while the momenta of the leaves of Λ are the variables of the kernel of γ (k+m) 0 (this also explain why roots and leaves of Λ need to be paired). The denominators (αe−p e+iτeµe) −1 are called propagators; they correspond to the free evolutions in the expansion (6.12) and they enter the expression (6.15) through the formula eit(α+iµ) α− p2 + iµ (here and in (6.15) the measure dα is defined by dα = d′α/(2πi) where d′α is the Lebesgue measure on R). The regularization factors µe in (6.15) have to be chosen such that µfather = e= son µe at every vertex. The delta-functions in (6.15) express momentum and frequency conservation (the sum over e ∈ v denotes the sum over all edges adjacent to the vertex v; here ±αe = αe if the edge points towards the vertex, while ±αe = −αe if the edge points out of the vertex, and analogously for ±pe). An analogous expansion can be obtained for the error term η n,t in (6.13). The problem now is to analyze the integral (6.15) (and the corresponding integral for the error term). Through an appropriate choice of the regularization factors µe one can extract the time dependence of KΛ,t and show that |KΛ,t| ≤ C k+m tm/4 e∈E(Γ) dαedpe 〈αe − p2e〉 v∈V (Γ) ∣∣∣Ĵ (k) {pe}e∈R(Γ) ) ∣∣∣ ∣∣∣γ̂(k+m)0 {pe}e∈L(Γ) ) ∣∣∣ (6.16) where we introduced the notation 〈x〉 = (1 + x2)1/2. Because of the singularity of the interaction at zero, we may be faced here with an ultraviolet problem; we have to show that all integrations in (6.16) are finite in the regime of large momenta and large frequency. Because of (6.3), we know that the kernel γ̂ (k+m) 0 ({pe}e∈L(Λ)) in (6.16) provides decay in the momenta of the leaves. From (6.3) we have, in momentum space, dp1 . . . dpn (p 1 + 1) . . . (p n + 1) γ̂ 0 (p1, . . . , pn; p1, . . . , pn) ≤ C for all n ≥ 1. Power counting implies that (k+m) 0 ({pe}e∈L(Λ))| . e∈L(Λ) −5/2 . (6.17) Using this decay in the momenta of the leaves and the decay of the propagators 〈αe−p −1, e ∈ E(Λ), we can prove the finiteness of all the momentum and frequency integrals in (6.15). Heuristically, this can be seen using a simple power counting argument. Fix κ≫ 1, and cutoff all momenta |pe| ≥ κ and all frequencies |αe| ≥ κ 2. Each pe-integral scales then as κ 3, and each αe-integral scales as κ 2. Since we have 2k + 3m edges in Λ, we have 2k + 3m momentum- and frequency integrations. However, because of the m delta functions (due to momentum and frequency conservation), we effectively only have to perform 2k + 2m momentum- and frequency-integrations. Therefore the whole integral in (6.15) carries a volume factor of the order κ5(2k+2m) = κ10k+10m. Now, since there are 2k + 2m leaves in the graph Λ, the estimate (6.17) guarantees a decay of the order κ−5/2(2k+2m) = κ−5k−5m. The 2k + 3m propagators, on the other hand, provide a decay of the order κ−2(2k+3m) = κ−4k−6m. Choosing the observable J (k) so that Ĵ (k) decays sufficiently fast at infinity, we can also gain an additional decay κ−6k. Since κ10k+10m · κ−5k−5m−4k−6m−6k = κ−m−5k ≪ 1 for κ ≫ 1, we can expect (6.15) to converge in the large momentum and large frequency regime. Remark the importance of the decay provided by the free evolution (through the propagators); without making use of it, we would not be able to prove the uniqueness of the infinite hierarchy. This heuristic argument is clearly far from rigorous. To obtain a rigorous proof, we use an integration scheme dictated by the structure of the graph Λ; we start by integrating the momenta and the frequency of the leaves (for which (6.17) provides sufficient decay). The point here is that when we perform the integrations over the momenta of the leaves we have to propagate the decay to the next edges on the left. We move iteratively from the right to the left of the graph, until we reach the roots; at every step we integrate the frequencies and momenta of the son edges of a fixed vertex and as a result we obtain decay in the momentum of the father edge. When we reach the roots, we use the decay of the kernel Ĵ (k) to complete the integration scheme. In a typical step, we α upuα rpr Figure 2: Integration scheme: a typical vertex consider a vertex as the one drawn in Figure 2 and we assume to have decay in the momenta of the three son-edges, in the form |pe| −λ, e = u, d,w (for some 2 < λ < 5/2). Then we integrate over the frequencies αu, αd, αw and the momenta pu, pd, pw of the son-edges and as a result we obtain a decaying factor |pr| −λ in the momentum of the father edge. In other words, we prove bounds of the dαudαddαwdpudpddpw |pu|λ|pd| λ|pw|λ δ(αr = αu + αd − αw)δ(pr = pu + pd − pw) 〈αu − p u〉〈αd − p d〉〈αw − p const |pr|λ . (6.18) Power counting implies that (6.18) can only be correct if λ > 2. On the other hand, to start the integration scheme we need λ < 5/2 (from (6.17) this is the decay in the momenta of the leaves, obtained from the a-priori estimates). It turns out that, choosing λ = 2 + ε for a sufficiently small ε > 0, (6.18) can be made precise, and the integration scheme can be completed. After integrating all the frequency and momentum variables, from (6.16) we obtain that |KΛ,t| ≤ C k+m tm/4 for every Λ ∈ Fm,k. Since the number of diagrams in Fm,k is bounded by C k+m, it follows immediately that ∣∣∣Tr J (k) ξ(k)m,t ∣∣∣ ≤ Ck+mtm/4 . Note that, from (6.12), one may expect ξ m,t to be proportional to t m. The reason why we only get a bound proportional to tm/4 is that we effectively use part of the time integration to control the singularity of the potentials. Note that the only property of γ (k+m) 0 used in the analysis of (6.15) is the estimate (6.3), which provides the necessary decay in the momenta of the leaves. However, since the a-priori bound (6.4) hold uniformly in time, we can use a similar argument to bound the contribution arising from the error term η n,t in (6.13) (as explained above, also η n,t can be expanded analogously to (6.14), with contributions associated to Feynman graphs similar to (6.15); the difference, of course, is that these contributions will depend on γ (k+n) s for all s ∈ [0, t], while (6.15) only depends on the initial data). Thus, we also obtain ∣∣∣Tr J (k) η(k)n,t ∣∣∣ ≤ Ck+n tn/4 . (6.19) This bound immediately implies the uniqueness. In fact, given two solutions Γ1,t = {γ 1,t }k≥1 and Γ2,t = {γ 2,t }k≥1 of the infinite hierarchy (4.1), both satisfying the a-priori bounds (6.4) and with the same initial data, we can expand both in a Duhamel series of order n as in (6.11). If we fix k ≥ 1, and consider the difference between γ 1,t and γ 2,t , all terms (6.12) cancel out because they only depend on the initial data. Therefore, from (6.19), we immediately obtain that, for arbitrary (sufficiently smooth) compact k-particle operators J (k), ∣∣∣TrJ (k) 1,t − γ )∣∣∣ ≤ 2Ck+n tn/4 Since it is independent of n, the left side has to vanish for all t < 1/C4. This proves uniqueness for short times. But then, since the a-priori bounds hold uniformly in time, the argument can be repeated to prove uniqueness for all times. References [1] Adami, R.; Golse, F.; Teta, A.: Rigorous derivation of the cubic NLS in dimension one. Preprint: Univ. Texas Math. Physics Archive, www.ma.utexas.edu, No. 05-211. [2] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Science 269 (1995), 198. [3] Bardos, C.; Golse, F.; Mauser, N.: Weak coupling limit of the N -particle Schrödinger equation. Methods Appl. Anal. 7 (2000), 275–293. [4] K. B. Davis, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Phys. Rev. Lett. 75 (1995), 3969. [5] Erdős, L.; Schlein, B.; Yau, H.-T.: Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math. 167 (2007), no. 3, 515-614. [6] Erdős, L.; Schlein, B.; Yau, H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. To appear in Ann. of Math. Preprint arXiv:math-ph/0606017. [7] Erdős, L.; Schlein, B.; Yau, H.-T.: Rigorous derivation of the Gross-Pitaevskii equation. Phys. Rev. Lett. 98 (2007), no. 4, 040404. [8] Erdős, L.; Yau, H.-T.: Derivation of the nonlinear Schrödinger equation from a many body Coulomb system. Adv. Theor. Math. Phys. 5 (2001), no. 6, 1169–1205. [9] Lieb, E.H.; Seiringer, R.: Proof of Bose-Einstein condensation for dilute trapped gases. Phys. Rev. Lett. 88 (2002), no. 17, 170409. [10] Lieb, E.H.; Seiringer, R.; Yngvason, J.: Bosons in a trap: a rigorous derivation of the Gross- Pitaevskii energy functional. Phys. Rev A 61 (2000), no. 4, 043602. [11] Spohn, H.: Kinetic Equations from Hamiltonian Dynamics. Rev. Mod. Phys. 52 (1980), no. 3, 569–615. http://arxiv.org/abs/math-ph/0606017 Introduction Heuristic Derivation of the Gross-Pitaevskii Equation Main Results General Strategy of the Proof and Previous Results Convergence to the Infinite Hierarchy Uniqueness of the Solution to the Infinite Hierarchy Higher Order Energy Estimates Expansion in Feynman Graphs

0704.0814

Photons as quasi-charged particles

Photons as quasi-charged particles K.-P. Marzlin, Jürgen Appel, A. I. Lvovsky Institute for Quantum Information Science, University of Calgary, Calgary, Alberta T2N 1N4, Canada (Dated: November 3, 2018) The Schrödinger motion of a charged quantum particle in an electromagnetic potential can be simulated by the paraxial dynamics of photons propagating through a spatially inhomogeneous medium. The inhomogeneity induces geometric effects that generate an artificial vector potential to which signal photons are coupled. This phenomenon can be implemented with slow light propagating through an a gas of double-Λ atoms in an electromagnetically-induced transparency setting with spatially varied control fields. It can lead to a reduced dispersion of signal photons and a topological phase shift of Aharonov-Bohm type. Introduction.— It is known since the ground-breaking work of Berry on geometric phases [1] that artificial gauge potentials can be induced if the spatial dynamics of a sys- tem that obeys a wave equation is confined in a certain way. For instance, if the internal Hamiltonian of neu- tral atoms contains an energy barrier but the spin eigen- states are spatially varying, gauge field dynamics can be induced [2]. In the limit of ray optics, moving atomic en- sembles could simulate the propagation of light around a black hole or generate topological phase factors of the Aharonov-Bohm type [3], and inhomogeneous dielectric media could generally exhibit geometric effects such as an optical spin-Hall effect and the optical Magnus force In this paper, we propose to use electromagnetically in- duced transparency (EIT) to generate an artificial vector potential for the paraxial dynamics of signal photons that simulates quantum dynamics of charged particles in a static electromagnetic field. Not only the ray of light but also its mode structure is affected, resulting in a paraxial wave equation that is equivalent to the Schrödinger equa- tion for charged particles. Furthermore, the form of the artificial vector potential can be easily controlled through spatial variations in the control fields. We suggest con- figurations that generate homogeneous quasi-electric and magnetic fields as well as a vector potential of Aharonov- Bohm type. Although the treatment in this paper is based on EIT, the effect presented here is more general: it will occur in any medium that supports a set of discrete eigenmodes for a propagating signal fields with different indices of refraction. If the parameters governing these eigenmodes vary in space, the signal modes will adiabatically follow, acquiring geometric phases that affect their paraxial dy- namics. Review of EIT with multi-Λ atoms.— The effect takes place in an atomic multi-Λ system, in which two ground states are coupled to Q excited states by Q pairs of con- trol (Ωq) and signal (âq) fields (Fig. 1). An experimen- tally relevant example of such system is the fundamental D1 transition in atomic rubidium, where both the ground and excited levels are split into two hyperfine sublevels [5]. We assume that the detunings are small so each sig- nal field âq interacts only with the respective transition |B〉 ↔ |Aq〉 with the associated atomic operator σ̂B,Aq and vacuum Rabi frequency gq. In this case, the parax- ial wave equation for each signal mode can be cast into the form âq = iNgqσ̂B,Aq , (1) where the wave propagates along the z axis, ∆⊥ = ∂ ∂2y , N is the number of atoms and k is the wavevector which we assume approximately independent of q. In Ref. [6] we have constructed a unitary transformation âq = Wqs b̂s (2) that maps the original field modes aq to a new set of modes b̂q, such that one and only one of the new modes, b̂Q = R∗q âq, (3) (where Rq ≡ Ωq/(gqΩ⊥) and Ω⊥ ≡ q=1 |Ωq/gq|2 de- pend on the control fields) couples only to an atomic dark state and experiences EIT [6, 7, 8]. All other superposi- tions of field modes are absorbed. This transformation is given explicitly by Wqq′ = γwqw q′−δqq′ , with γ = RQ+1 and wq = γ −1(δQq +Rq). The EIT mode b̂Q interacts with the multi-Λ atoms in the same fashion as does the signal field in a regu- lar 3-level system. While propagating through the EIT medium, it gives rise to a dark-state polariton associ- ated with zero interaction energy [9]. All other modes couple to atomic states whose energy levels are Stark- shifted by the interaction with either the pump field or the other signal modes b̂q (q 6= Q). The resulting energy gap guarantees that, if the amplitudes and phases of the control fields are slowly changed, the composition of the dark-state polariton, and hence the EIT mode b̂Q, will adiabatically follow. It has been proposed [6] and ex- perimentally demonstrated [5] that a variation in time of the control fields can therefore be used to adiabatically http://arxiv.org/abs/0704.0814v2 FIG. 1: Multi Λ-system: Q excited states |Aq〉 are each cou- pled by a classical control field Ωq to the ground state |C〉 and by a quantized field âq with detuning δ to state |B〉. transfer optical states between signal modes. In this pa- per, we focus on spatial propagation of the EIT mode under control fields that are constant in time, but varied in space. Derivation of the gauge potential.— We proceed by ex- pressing Eq. (1) in terms of the new signal modes b̂q. Employing the vector notation ~a = {â1, · · · , âQ} and ~σB,A = {g1σ̂B,A1 , · · · , gQσ̂B,AQ} we get b = iN~σB,A. (4) Throughout the paper, the double arrow denotes a Q × Q matrix. Because W depends on space and time, the differential operators have to be applied to both W and b. As a result, transformation (2) brings about additional terms into the equation of motion, that can be written in form of a minimal coupling scheme by introducing the Hermitian gauge field i ≡ i W †∂i W, (5) where i = t, x, y, z. We multiply both sides of Eq. (4) by W † and exploit the unitarity of W to show that ∂i W † = W †(∂i W † from which it follows that − W †∂2i i + i∂iA i. The dynamic equation for the b̂ modes can then be written as i∂t +A b = − ic∂z + cA b (6) (−i∇⊥ −A 2~̂b− W †N~σBA with ∇⊥ = (∂x, ∂y). This equation has the structure of a 2+2 dimensional field theory with minimal coupling. Under the assumption that the control fields do not depend on t and z we can make a temporal Fourier trans- formation of the slowly varying amplitudes, which results in the paraxial wave equation b(δ) = (−i∇⊥ −A ~σBA(δ). (7) The gauge potential is given explicitly by ⊥ = i R∗q(∇⊥Rq)~w~w† − iγ(∇⊥ ~w)~w† + iγ∗ ~w∇⊥ ~w† . The full matrix A ⊥ is a pure gauge: it has emerged solely as a consequence of the unitary transformation (2), which reflects our choice to describe the system in terms of the new modes b̂q rather than the original modes âq. How- ever, this choice is motivated by the fact that the EIT mode b̂Q is the only mode that is not absorbed. Ab- sorption of other modes b̂q (with q 6= Q) means that the index of refraction for these modes has a significant imaginary part. This separates the EIT mode b̂Q from other b-modes and ensures that it will adiabatically follow variations of the control fields. Therefore, when analyz- ing the evolution of b̂Q, we can neglect the off-diagonal terms in the matrix (−i∇⊥ −A 2 in Eq. (7) and write i∂z b̂Q(δ) = −( ~σBA)Q(δ)− b̂Q(δ) (9) (−i∇⊥ −A⊥)Qq(−i∇⊥ −A⊥)qQb̂Q(δ). This equation does not include the whole matrix A Consequently, this potential no longer acts like a pure gauge but attains physical significance in determining the spatial dynamics of the EIT mode. The first term on the right-hand side of Eq. (9), re- sponsible for the interaction of the light field with the EIT medium, takes the same form as the susceptibility of EIT in a single Λ-system. Neglecting decoherence, we can write it as [6] ( W † N ~σBA)Q(δ) = b̂Q, with the EIT group velocity vEIT = cΩ 2/(Ng2). Note that vEIT depends on the spatial position because Ω does. This transforms Eq. (9) to i∂z b̂Q = (−i∇⊥ −AQQ)2 − b̂Q (10) AQQ = i R∗q∇⊥Rq = − |Rq|2∇⊥Arg(Rq), |(A⊥)Qq|2 = −A2QQ + |∇⊥Rq|2 (11) being, respectively, the “quasi-vector” and “quasi-scalar” potentials. We see that the paraxial spatial evolution of the EIT signal mode is governed by the equation that is identi- cal (up to coefficients) to the Schrödinger equation of a charged particle in an electromagnetic field. This is the main result of this work. By arranging the control field in a certain configuration, one can control the spatial prop- agation of the signal mode through the EIT medium. Some steering of the EIT mode is possible even in a single-Λ system by affecting the term δ/vEIT in Eq. (10), which results in nonuniform refraction for this mode [10, 11]. The action of quasi-gauge fields (11) is fun- damentally different: deflection of the signal field occurs not due to refraction (the refraction index on resonance is 1), but due to adiabatic following. The case of two control fields: homogeneous electric and magnetic quasi-fields.— Of particular practical im- portance is the simplest non-trivial case with Q = 2. We parametrize the control fields by writing R1,2 = 1/2±Rei(φ±θ). The corresponding Rabi frequencies are then Ωi = h(x, y) giRi, with h(x, y) being an arbi- trary common prefactor. This parametrization yields the gauge potentials AQQ = −∇⊥φ− 2R∇⊥θ; (12) (∇⊥R)2 1− 4R2 + (∇⊥θ) 2(1 − 4R2). Similarly to usual electrodynamics we can use a gauge transformation [12], A′QQ = AQQ + ∇⊥f , to eliminate the term ∇⊥φ from Eq. (12). The common phase φ of the control fields therefore does not contribute and can be set to zero. A simple way to generate a term that corresponds to a one-dimensional scalar potential V (x) for a Schrödinger particle is to choose R = 0 and θ = 2kV (x′) This choice of control fields leads to AQQ = 0 and Φ = 2kV (x). For the special case of a constant electric quasi-field along the x axis, V (x) = −Fx and subsequently θ = − 4kF |x|3/3, (13) where x < 0 is assumed for the region of interest. A res- onant (δ = 0) Gaussian solution to Eq. (10) is displayed in Fig. 2(a). The center of the Gaussian beam is shifted by an amount xctr = Fz 2/2k, which is equivalent to the motion of a charged particle in a constant electric field. The control field phase profile (13) can be implemented using, for example, a phase plate. The assumption that the control fields do not depend on z implies that the Fresnel number for these fields must be above 1, i.e. that the characteristic transverse distance over which these fields significantly change must be larger than ∼ where L is the EIT cell length. This imposes a limi- tation on the magnitude of the electric quasi-field: from Eq. (13) we find F <∼ λ−1/2L−3/2 and thus xctr <∼ Assuming that the signal field also has a Fresnel number of at least 1, and thus satisfies 2zR >∼ L (with Rayleigh length zR = kw 2/2, w being the signal beam width at the cell entrance), we find that in a realistic experiment, the maximum possible signal beam displacement due to the quasi-electric field is on the order of the signal beam width w. To generate a homogeneous magnetic quasi-field along the z-axis the quantity B = ∇ × AQQ = 2∇⊥θ × ∇⊥R should be constant. However, it seems difficult to si- multaneously achieve a vanishing electric quasi-field E = −∇⊥Φ. A choice that minimizes the electric quasi- field around the origin is given by θ = B/2x and B/2 y. The quasi-potentials then become AQQ = −B y ex, which corresponds to the Landau gauge in stan- dard electrodynamics, and Φ = B + 2B3y4 + O(y6). If Φ is neglected, a Gaussian solution to the paraxial wave equation is given by bQ = N cscu(z) exp cotu(z)∆x2 +∆x · pc ∆x∆y − 1 pc,xpc,y + ycpc,y where we have set ∆x ≡ (x − xc, y − yc), u(z) ≡ Bz/(2k) − i tanh−1(2η) and η ≡ Bw2/4. Here xc = x0 + (k/B)(x 0 sin(Bz/k) + x̃ 0(1 − cos(Bz/k)) denotes the classical spiral trajectory of a charged particle in a magnetic field, with x′c = dxc/dz, initial position x0 and initial velocity x′0. For convenience we also have defined x̃′0 = (y 0,−x′0) and the classical canonical momentum pc. We remark that pc,x is a constant of motion. The evolution of the signal mode is displayed in Fig. 2(b). A surprising feature of solution (14) is that the diffrac- tive divergence of the signal beam is reduced: the width squared of the Gaussian, Re(iB cotu) 1 + 4η2 − (1 − 4η2) cos(Bz varies periodically with z instead of monotonically in- creasing. This effect is known for electron wavepackets [13] and can be understood as a consequence of the cir- cular motion of particles in a magnetic field: instead of dispersing, two-dimensional particles in a magnetic field will simply move on circles of different size (depending on their velocity), but with the same angular velocity. The particle cloud will therefore not spread but “breathe”. It remains to show that non-adiabatic coupling to other modes can be suppressed for realistic experimental pa- rameters. This is the case if the strength of the gauge field terms coupling bQ to other modes, which for the quasi-magnetic field are of the order B/(2k), are much smaller than the difference in the respective linear sus- ceptibilities χ1. For the EIT mode bQ, χ1 = δ/vEIT with vEIT defined above Eq. (10); for the other modes it can be approximated by the susceptibility of a two- level medium, χ1 = −4Ng2(δ − iγ/2)/(cγ2). Evalu- ating this relation at resonance leads to the condition η ≪ (kw)2n3π/2, with n ≡ N/(V k3) being the number of atoms in the volume k−3, which can easily be fulfilled in an experiment. Aharonov-Bohm potential for photons.— One of the most intriguing phenomena of charged quantum particles FIG. 2: Paraxial propagation of a signal beam over twice the Rayleigh length in the presence (solid) and absence (grey) of a constant (a) electric field along the x axis and (b) magnetic field along z. The dashed line represents the center of the grey beam. The effect of the fields is somewhat exaggerated. in electromagnetic fields is the Aharonov-Bohm (AB) ef- fect [14]. Its two astonishing features are (i) a phase shift induced by the vector potential in a region in which elec- tric and magnetic fields are absent, and (ii) its topological nature: the phase shift does not depend on the particle trajectory as long as it encloses a magnetic flux. Because (unlike genuine electromagnetism) the potential (5) is a differential function of the control fields, it is impossible to simulate feature (i) with quasi-charged photons. How- ever, we will show here that a mathematically equivalent topological phase shift does exist for the optical case. To generate an AB potential for photons we propose to use two counter-rotating Laguerre-Gaussian control fields, i.e., fields that possess an orbital angular momen- tum. If these control fields are spatially wider than the signal fields, the corresponding Rabi frequencies can be approximated in cylindrical coordinates (r, ϕ) by Ω1 = g1s1re iϕ and Ω2 = g2s2re −iϕ. The gauge potentials (12) then become AQQ = −2R/r ~eϕ and Φ = (1 − 4R2)/r2, with R = 1 (|s1|2 − |s2|2)/(|s1|2 + |s2|2). The potential AQQ corresponds exactly to an Aharonov-Bohm poten- tial for charged particles as it is created by a solenoid. Solutions of the paraxial wave equation (10) can be found in cylindrical coordinates by expanding the field mode as bQ = r m∈ZZ Bm(z, r) exp(imϕ). Because of Ω ∼ r, the EIT group velocity can be written as vEIT = ṽ r2 with ṽ ≡ c |s1|2 + |s2|2/N . Exact solutions are given by Bessel functions, Bm = e −iκ2z/(2k) κrJν(κr) + βm κrYν(κr) with ν = 1 +m2 + 4Rm− 2kṽδ. For monochromatic signal fields this corresponds to a rotation of the trans- verse mode structure. For R = ±1/2 the potential trans- fers a unit amount of angular momentum to the signal light, but generally the amount can vary continuously be- tween −h̄ and h̄. Signal photons in the EIT mode there- fore form a two-dimensional bosonic quantum system in an Aharonov-Bohm potential. Conclusion.— We showed that EIT in a multi-Λ sys- tem can be used to generate a variety of geometric ef- fects on propagating signal pulses that mimic the be- havior of a charged particle in an electromagnetic field. We found specific arrangements of two spatially inhomo- geneous pump fields in a double-Λ system which gener- ate quasi-gauge potentials which correspond to constant electric and magnetic fields. Furthermore topological ef- fects like the Aharonov-Bohm phase shift can be induced. The latter is significantly different from the proposal of Ref. [3] in that it is based on spatially inhomogeneous pump fields rather than the Doppler effect in moving me- This paper investigated EIT in systems with two ground levels. In such a system, there is only one EIT mode, which results in an Abelian U(1) gauge theory, making the physics analogous to electromagnetism. By extending to multiple ground levels, it may be possible to obtain multiple EIT modes and model non-Abelian gauge potentials. This will be explored in a future publication. We thank David Feder and Alexis Morris for fruit- ful discussions. This work was supported by iCORE, NSERC, CIAR, QuantumWorks and CFI. [1] M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984). [2] R. Dum and M. Olshanii, Phys. Rev. Lett. 76, 1788 (1996); J. Ruseckas et al., Phys. Rev. Lett. 95, 010404 (2005); K. Osterloh et al., Phys. Rev. Lett. 95, 010403 (2005). [3] U. Leonhardt and P. Piwnicki, Phys. Rev. A 60, 4301 (1999). [4] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003); M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. E 74, 066610 (2006); K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. 96, 073903 (2006); K. Bliokh, Phys. Rev. Lett. 97, 043901 (2006); C. Duval, Z. Hor- vath, and P. Horvathy, J.Geom.Phys. 57, 925 (2007); C. Duval, Z. Horvathy, and P. A. Horvathy, Phys. Rev. D 74, 021701 (2006); S. Raghu and F. D. M. Haldane, cond-mat/0602501. [5] F. Vewinger et al., quant-ph/0611181. [6] J. Appel, K.-P. Marzlin, and A. I. Lvovsky, Phys. Rev. A 73, 013804 (2006). [7] X.-J. Liu, H. Jing, and M.-L. Ge, Eur. Phys. J. D 40, 297 (2006); see also quant-ph/0403171. [8] S. A. Moiseev and B. S. Ham, Phys. Rev. A 73, 033812 (2006). [9] M. Fleischauer and M. D. Lukin, Phys. Rev. A 65, 022314 (2002). [10] A. G. Truscott et al., Phys. Rev. Lett. 82, 1438 (1999); R. Kapoor and G. S. Agarwal, Phys. Rev. A 61, 053818 (2000). [11] L. Karpa and M. Weitz, Nature Phys. 2, 332 (2006). [12] Note that this gauge transformation acts on the EIT mode b̂Q only and is therefore different from the gauge transformation discussed above. [13] H. Takagi, M. Ishida, and N. Sawaki, Jpn. J. Appl. Phys 40, 1973 (2001). [14] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). http://arxiv.org/abs/cond-mat/0602501 http://arxiv.org/abs/quant-ph/0611181 http://arxiv.org/abs/quant-ph/0403171

0704.0815

Exchange of quantum states between coupled oscillators

Exchange of quantum states between coupled oscillators D. Portes Jr. and H. Rodrigues Centro Federal de Educação Tecnológica do Rio de Janeiro Departamento de Educação Superior-DEPES. S. B. Duarte∗ Centro Brasileiro de Pesquisas F́ısicas /CNPq, Rua Dr. Xavier Sigaud 150, CEP 22.290-180, Rio de Janeiro (RJ), Brazil. B. Baseia Instituto de F́ısica, Universidade Federal de Goiás PO-Box-131, 74.001-970, Goiania(GO), Brazil. December 9, 2018 Abstract Exchange of quantum states between two interacting harmonic oscilla- tor along their evolution time is discussed. It is analyzed the conditions for such exchange starting from a generic initial state and demonstrating that the effect occurs exactly only for the particular states C0 |0 > +CN |N >, which includes the interesting qubits components |0〉, |1〉. It is also deter- mined the relation between the coupling constant and characteristic fre- quencies of the oscillators to have the complete exchange. 1 Introduction The engineering of quantum states of light fields and oscillators became an in- teresting topic in the last years, due to its applications in : (i) fundamentals of quantum mechanics (preparation of Schrodinger-cat states [1], their super- position [2] and measurement of their decoherence [3], etc.); (ii) determination of certain properties of a system (phase distribution P(θ) [4], Wigner [5] and Husimi [6] functions, etc.); (iii) proposals for practical applications (quantum lithography [7], quantum communication [8] - e.g., via hole-burning in Fock ∗corresponding author, e-mail : [email protected] http://arxiv.org/abs/0704.0815v2 space [9] - quantum teleportation [10], etc). However, a difficult situation ap- pears when one wants to prepare a state of a system offering hard access [11]. In this case the difficulty may be circumvented by coupling the system having hard access to a second system offering easy access, in which a desired state is prepared with subsequent transfer to the first one. The success of this operation depends on the model-Hamiltonian and on the initial state describing the whole system. Although the problem of two interacting harmonic oscillators has been ex- haustively studied in the literature, the discussion about exchange of nonclas- sical states between them is scarce. The coupled quantum oscillation problem was considered earlier in [12, 13, 14], where the authors of those papers were interested only in the energy of the system. Later on, in Ref [15] a full exchange between quantum two-mode harmonic oscillators was presented, however the issue was only concerned with the particular transfer of coherent states. In Ref. [16] we have studied the transfer of certain properties (statistics and squeezing) and in Ref. [17] we have studied the transfer of the most relevant part of the state of a sub-system to another, through the simultaneous transfer of the number and phase distributions, Pn and P (θ) 1 [17]; the solutions were found numerically since the models were not exactly soluble. In the present work we employ a distinct model-Hamiltonian, allowing us to treat the problem analytically permitting us to analyze the transfer of generic states. We show in which way one can get exact exchange of the states between two interacting sub-systems. Exchange of states means simultaneous transfer of states in two opposite directions ; so, it is more significant than the transfer of states in one direction as studied in [17]. In the present case the transfer of a state from the “easy-oscillator” to the “hard-oscillator” is observed by simply monitoring the state of the easy-oscillator during the time evolution of the whole system. For brevity, hereafter the easy- and the hard-oscillator will be referred to as O1 and O2, respectively. The Sect. II introduces the model-Hamiltonian allowing us to obtain the evolution operator for this coupled system. In the Sect. III we consider differ- ent types of initial states describing the entire system to study the mentioned effect between the O1 and the O2 ( Sub-Sects. (A), (B),and (C) ), includ- ing superpositions of states representing the qubits |0〉 and |1〉. The Sect. IV contains the comments and conclusion. 2 Model-Hamiltonian: evolution operator We start from the Hamiltonian H/h̄ = ω1a 1 a1 + ω2a 2 a2 + λ a+1 a2 + a1a , (1) 1Since the number and phase are canonically conjugate operators they are complementary, in the sense that simultaneous transfer of number and phase distributions, Pn and P (θ), concerns the transfer of the major part of the state describing a system. where a+i (ai) stands for the raising (lowering) operator of the i− th oscillator, i = 1, 2; ωi and λ are real parameters standing for the i-th oscillator frequency and coupling constant, respectively. The equations of motion for the operators a1(t) and a2(t) can be solved analytically, a1(t) = c2e−iω t + s2e−iω a1(0) + cs t − e−iω a2(0), (2) a2(t) = c2e−iω t + s2e−iω a2(0) + cs t − e−iω a1(0), where, ω′1 = ω1 + λ , (3) ω′2 = ω2 − λ x2 + 1 x2 + 1 ω1 − ω2 . (5) The parameter s and c satisfy the condition c2+s2 = 1, they define the auxiliary operators a′1 = c a1 + s a2 , (6) a′2 = −s a1 + c a2 , which decouple the above Hamiltonian. The following relations also hold: ω′1 + ω 2 = ω1 + ω2 , (7) ω′1 − ω′2 = It is convenient for our purposes to find the time dependent state vector or density operator in the Schrodinger picture. One formal prescription is to work with Wigner representation of the state and obtain the time-dependent density operator from the Wigner function[19], for which the time evolution is easily obtained. However, it is a hard task to restore analytical or numerical values for the density matrix ρ(t) in the Fock basis from the time dependent Wigner function. To overcome this difficult we will show that for the Hamiltonian given by Eq.(1) there is an analytical expression for the evolution operator U(t), which defines the solution of the Schrodinger equation, allowing us to get directly the matrix ρ(t) in the Fock basis. This kind of approach was already used in Ref [18], but only treating the system in the resonant case (ω1 = ω2). In [18] the author studied the transfer of state starting from the particular one photon state. Our results permit one to obtain an analytical expression for the matrix element U(t), for the Hamiltonian (1) not restricted to the resonant case and permitting easy application to a generic initial state. Consequently, the problem of transfer of states can be more comfortably discussed using the present results. To obtain the operator U(t), we define the (auxiliary) unitary operator Us(t) which is associated to a rotation and decouples the Hamiltonian, U−1s ai Us = a i . (8) We have, U−1s = U−s , (9) in view of the reverse transformation a1 = c a 1 − s a′2 , (10) a2 = s a 1 + c a We denote {|n1, n2〉0} as representing the Fock′s basis, eigenvectors of the (old) number operator Ni = a i ai , whereas {|n1, n2〉s} is the same for the (new) number operator Ni(s) = a i. We have, Us|n1, n2〉s = |n1, n2〉0, (11) |n1, n2〉s = U−s|n1, n2〉0. If we represent Us in the Fock ′s basis {|n1, n2〉0}, we obtain n1, n2 m1, m2 = 0〈n1, n2|Us|m1,m2〉0 (12) = s〈n1, n2|m1,m2〉0. Next, to reconstruct the operator Us in the Fock’s basis, we start from s〈n1, n2| a′1|m1,m2〉0 = s〈n1, n2| (c a1 + s a2) |m1,m2〉0, (13) Since the operators a′i act on the basis {|n1, n2〉s} whereas the ai act on the basis {|n1, n2〉0}, we get n1 + 1s〈n1 + 1, n2|m1,m2〉0 = c m1 s〈n1, n2|m1 − 1,m2〉0 (14) m2 s〈n1, n2|m1,m2 − 1〉0, which, after using the Eq.(12), leads to n1, n2 m1, m2 n1−1,n2 m1−1,m2 n1−1,n2 m1,m2−1 , (15) and similarly, repeating the procedure for the operator a′2, we find n1, n2 m1, m2 n1,n2−1 m1−1,m2 n1,n2−1 m1,m2−1 . (16) Using the Eqs. (15), (16) plus the unitary condition U †sUs = UsU s = 1 we obtain, after a lengthy calculation, the expression n1, n2 m1, m2 = δn1+n2, m1+m2 n1!n2! m1!m2! (−1)n2 cm1−n2 sm2+n2 (17) min(n2,m2) k=max(0,m2−n1) (−1)−k n2 − k (U−s) n1, n2 m1, m2 = (−1)m2−n2 (Us)n1, n2m1, m2 . (18) The time evolution operator U(t) may be written in the basis {|n1, n2〉s} as U(t) = k1,k2 |k1, k2〉s e−i(k1ω + k2ω s〈k1, k2| , (19) for H is diagonal in this basis. Finally from the Eqs.(12) and (19) we obtain the expression n1, n2 m1, m2 k1,k2 e−i(k1ω + k2ω )t (U−s) n1, n2 k1, k2 (U−s) m1, m2 k1, k2 , (20) restricted to n1 + n2 = k1 + k2 = m1 +m2 , whereas U n1, n2 m1, m2 = 0 otherwise. The evolution operator obtained in Eq.(20) allows us to study the time evo- lution of the whole state describing our bipartite system composed by coupled oscillators, represented by the Hamiltonian in the Eq.(1). In the next section we will study the exchange of states between these oscillators and, as a natural assumption, we will suppose the O2 initially in its ground state |0〉. The O1 is assumed to be previously prepared in various initial states, firstly starting from an arbitrary state |φ〉. 3 Exchange of generic state Let us consider that the whole (bipartite) system is initially in the state |Ψ(0)〉 = |φ〉 ⊗ |0〉 , (21) whose components in the Fock’s basis are given by, |Ψ(0)〉 = Cn, 0(0)|n, 0〉 , (22) since Cn1, n2(0) = 0 for n2 6= 0. In the Schrodinger representation, the coeffi- cients Cn1,n2(t) are obtained from Cn1,n2(t) = 〈n1, n2|U(t)|Ψ(0)〉, which, using Eq. (22) and the constraint n1 + n2 = n, results in the form Cn1,n2(t) = Cn1+n2,0(0)U(t) n1,n2 n1+n2,0 . (23) In particular, we have that Cn,0(t) = Cn,0(0)U(t) n,0 , (24) C0,n(t) = Cn,0(0)U(t) n,0 . (25) The exchange of states between the oscillators will occur after an instant τ ,when C0,n(τ ) = Cn,0(0) and |Ψ(τ)〉 = C0, n(τ )|0, n〉 , (26) or, |Ψ(τ )〉 = |0〉 ⊗ |φ〉. This shows that exchange of states allows us to verify the transfer of states to the O2 by monitoring the time evolution of the O1. From the Eqs. (17) and (18) we have, n−l,l = (n− l)!l! cn−l sl , (27) n−l,l = (n− l)!l! (−1) cl sn−l . (28) The substitution of the Eqs. (27) and (28) in the Eq. (20) results n,0 = (−1) (n− l)!l! (−1)n−l cnsne−i (n−l) ω t e−i l ω t . (29) where we recognize the Newton’s binomial expression, n,0 = (−1) e−i ω t − e−i ω or, replacing the auxiliary parameters ω′1, ω 2 by ω1, ω2 and λ (cf. Eq. (7)), n,0 = e ω1+ω2 −2 i s c sin( λ , (31) and, consequently, C0,n(t) = Cn,0(0) e−i ω1+ω2 −2 i s c sin( λ . (32) In a similar way we get, Cn,0(t) = Cn,0(0) e−i ω1+ω2 c2e−i t + s2ei . (33) From Eq. (32) we see that a partial exchange of states will occur when λt/sc = (2k + 1)π, i.e., in the time intervals τk = (sc/λ) (2k + 1)π. The effect attains the highest efficiency when the product sc is maximum, i.e., when s = c = 1/ 2 and τk = (k + 1/2)π/λ. According to the Eq. (4) this implies x = 0 and the resonance condition ω1 = ω2 = ω (cf. Eq. (5)), C0,n(τk) = (−i)n Cn,0(0) e−i ω n τk . (34) However, we note that even at resonance we obtain no exchange of states, due to the presence of the phase factor exp ωτk + affecting the coefficients of the state describing both oscillators in the Fock’s representation. In this gen- eral case we obtain ∣C0,n(τk) ∣Cn,0(0) ∣, which means exchange of statistics between the two oscillators. This can also be seen comparing both reduced density matrix, ρ m1, m2(τk) and ρ m1, m2(0), in the Fock’s representation, ρ(2)m1, m2(τk) = e −i (ωτk+π2 ) (m1−m2) ρ(1)m1, m2(0) , (35) which exhibits the distinction between their off-diagonal elements. As well known, while the state of a system offers its complete description, the same is not true for the statistics, which contains only partial informations of the system. 3.1 The complete exchange of state It is shown in the last section that it is not possible to have a complete exchange of states for a generic initial state because the phases are not transferred (see Eq.35). Here we show that when the state of oscillator O1 is given by the super- position C0|0〉+ CN |N〉 whereas O2 is in the vacuum state, complete exchange of states occurs. Note that this state includes in particular the important case C0|0〉+ C1|1〉 using the qubits |0〉, |1〉 having potential applications in quantum communication [20] and in quantum computation [21]. It was shown that this state exhibits squeezed fluctuations [22]. Next, let us consider the whole system initially in the superposed state |Ψ(0)〉 = C0,0(0)|0, 0〉+ CN,0(0)|N, 0〉 . (36) In this case we verify perfect exchange of states between the oscillators for a convenient choice of the parameters involved. Assuming the resonance condition in the Eq.(32) we have, for C0,0(t) = C0,0(0), C0,N (t) = CN,0(0) e−i (ω t+π/2)N sinN (λ t) . (37) Partial exchange of states will occur when t = τ0 = π/(2λ),which results in C0,N (τk) = C N,0(0) e−i π/2(ω/λ+1)N , (38) whose meaning is the exchange of statistics. The exchange of states becomes complete (exact) when C0,N (τk) = C N,0(0), namely, when , (39) with m integers. Taking m = 1 and ω in the microwave domain (ω ∼ 109Hz) the time spent to transfer the state C0|0 > + C1|1 > from the O1 to the O2 results τ0 = π/(2λ) ∼ 10−9s, since λ = ω/3 (cf. Eq.(39)), which is smaller than the typical decoherence time for such systems (τd ∼ 10−3s), as it should. Note that the previous initial state C0|0〉+CN |N〉 describing the O1 includes the Fock states |N〉, obtained from C0 = 0 and CN = 1. In this case exact exchange of states no longer requires the Eq. (39). The reason comes from the phase factor appearing in the Eq. (39), now becoming a global phase with no physical relevance. In this case the exchange of states is exact for any instant tk = τ0 + 2πk/λ. 4 Comments and Conclusion An analytical procedure applied to a convenient model-Hamiltonian describing two coupled oscillators allows us to get the exact evolution operator for the entire system (Sect. II). This approach, through the use of distinct initial states and parameters (Sub-Sects. (A), (B) of Sect. III), makes easy the study of exchange of states between such sub-systems. In all cases we have shown that the fidelity of the process is maximum when the resonance condition, ω1 = ω2, is attained. Assuming the O2 always in the vacuum state we find, sub- Section by sub-Section, that: (A) partial exchange of states is achieved when the initial state of the O1 is arbitrary, for the time intervals t = τk = (k + 1/2)π/λ; the efficiency of partial exchange is maximum when the product sc is maximum (sc = 1/2); however, while the occurrence of exchange of states is partial, exchange of statistics is obtained exactly, as shown in the Eqs. (34), (35); (B) exact exchange of states occurs when the O1 starts from the initial superposed state C0|0〉 + CN |N〉, in the time intervals tk = τ0 + 2πk/λ, with the requirement in Eq. (39). If the Eq.(39) is not obeyed, exchange of states will occur at the same time intervals, but now the effect is only partial; Exact exchange of states is also found in the particular case of (B), setting C0 = 0 and CN = 1, which means the O1 starting from a Fock state |N〉. In this case the exchange of states occurs exactly at the same time intervals found in (B), no matter the Eq. (39) is obeyed or not. As final remarks we mention that exchange of states and its efficiency could be investigated for other model-Hamiltonians and, as explained before, the ef- fect goes beyond those studied in [16] and [17]. To our knowledge, exchange of states in coupled systems and even exchange of certain properties, are subjects receiving little attention in the literature [23] - with the remarkable exception of quantum teleportation [21], an effect having a very distinct nature (requiring the presence of quantum channels and entangled states), which occurs in the absence of coupling between the two sub-systems. In the context of teleporta- tion, exchange of states appears with the name ”identity interchange” [24] and ”two-way teleportation” [25]. 4.1 Acknowledgements The authors thank the CNPq (SBD, BB) and FAPERJ (DPJ) for the partial supports. 4.2 References References [1] B.Yurke, D. Stoler, Phys. Rev. Lett. 57 (1986) 13. [2] L. Davidovich et al., Phys. Rev. Lett. 71 (1993) 2360. [3] M. Brune et al., Phys. Rev. Lett. 77 (1996) 4887; D.M. Meekhof et al., Phys. Rev. Lett. 76 (1996) 1796. [4] D.T. Pegg, S. M. Barnett, Phys. Rev. Lett. 76 (1996) 4148. [5] L.G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78 (1997) 2547. [6] M. H.Y. Moussa, B. Baseia, Phys. Lett. A 238 (1998) 223. [7] G. Bjork, L.L. Sanchez-Soto, J. D. Soderholm, Phys. Rev. Lett. 86 (2001) 4516. [8] See, e.g., S.L. Braustein, P. van Loock, Rev. Mod. Phys. 77 (2005) 513. [9] B. Baseia, J.M.C. Malbouisson, Chinese Phys. Lett. 18 ,1467 (2001); Phys. Lett. A 290 (2001) 234; A.T. Avelar, B. Baseia, Opt. Commun. 239 (2004) 281; Phys. Rev. A 72 (2005) 67508; B. Escher et al., Phys. Rev. A 70 (2004) 025801. [10] B. Julsgaard et al., Nature, 413 (2001) 400, and references therein. [11] F. Dietrich et al., Phys. Rev. Lett. 62 (1989) 403; D.J. Heizein et al., Phys. Rev. Lett. 66 (1991) 2080. [12] J. Tucker and D. F. Walls, Ann. Phys. (N.Y.) 52, 1 (1969). [13] E.Y.C. Lu, Phys. Rev. A 8, 1053 (1973). [14] M.S. Abdalla, J. Phys. A: Math. Gen. 29, 1997 (1996). [15] Marcos C de Oliveira et al, Journal Optics B 1 (1999) 610. [16] H. Rodrigues et al., Physica A 311 (2002)188. [17] D. Portes Jr., et al., Physica A 329 (2003) 391. [18] Lee E. Estes, Thomas H. Keil, and Lorenzo M. Narducci, Physical Review 175,1 (1968) 286. [19] B. R. Mollow, Physical Review 162,5 (1967) 1256. [20] S. J. van Enk, J. I. Cirac, P. Zoller, Phys. Rev. Lett. 78 (1997) 4293. [21] P. W. Shor, Phys. Rev. A 52 (1995) R2493. [22] K. Wodkiewicz et al, Phys. Rev. A 35, (1987) 2567. [23] A. S. M. de Castro, V.V. Dodonov , J. Opt. B: Quantum Semiclass. Opt. 4 (2002) 191. [24] M. H. Y. Moussa, Phys. Rev. A 55, (1997) R3287. [25] L. Vaidman, N. Yoran, Phys. Rev. A 59 (1999) 116. Introduction Model-Hamiltonian: evolution operator Exchange of generic state The complete exchange of state Comments and Conclusion Acknowledgements References

0704.0816

3-D Simulations of Ergospheric Disk Driven Poynting Jets

3-D Simulations of Ergospheric Disk Driven Poynting Jets Brian Punsly 4014 Emerald Street No.116, Torrance CA, USA 90503 and International Center for Relativistic Astrophysics, I.C.R.A.,University of Rome La Sapienza, I-00185 Roma, Italy [email protected] or [email protected] ABSTRACT This Letter reports on 3-dimensional simulations of Kerr black hole magne- tospheres that obey the general relativistic equations of perfect magnetohydro- dynamics (MHD). In particular, we study powerful Poynting flux dominated jets that are driven from dense gas in the equatorial plane in the ergosphere. The physics of which has been previously studied in the simplified limit of an ergop- sheric disk. For high spin black holes, a/M > 0.95, the ergospheric disk is promi- nent in the 3-D simulations and is responsible for greatly enhanced Poynting flux emission. Any large scale poloidal magnetic flux that is trapped in the equatorial region leads to an enormous release of electromagnetic energy that dwarfs the jet energy produced by magnetic flux threading the event horizon. The implication is that magnetic flux threading the equatorial plane of the ergosphere is a likely prerequisite for the central engine of powerful FRII quasars. Subject headings: Black hole physics - magnetohydrodynamics -galaxies: jets— galaxies: active — accretion disks 1. Introduction Recent studies of luminous radio quasars indicate that the power of the radio jet can exceed the bolometric luminosity associated with the accretion flow thermal emission (Punsly 2006b, 2007). This has proven to be quite challenging for current 3-D numerical simulations of MHD black hole magnetospheres. Based on table 4 of Hawley and Krolik (2006) and the related discussion of Punsly (2006b, 2007), the most promising 3-D simulations for achieving this level of efficiency are those of the highest spin, a/M ≈ 1 (where the black hole mass, M , and the angular momentum per unit mass, a, are in geometrized units). More generally, such high spins have been inferred in some black hole systems based on observational constraints (McClintock et al 2006). Thus, there is tremendous astronomical relevance to these highest http://arxiv.org/abs/0704.0816v1 – 2 – spin configurations, in particular the physical origin of the relativistic Poynting jet. The first generation of long term 3-D simulations produced one Poynting flux powerhouse, the a/M = 0.995 simulation, KDE (De Villiers et al 2003, 2005a; Hirose et al 2004; Krolik et al 2005). The source of most of the Poynting flux was clearly shown to be outside the event horizon in KDE (Punsly 2006a). However, without access to the original data, the details of the physical mechanism could not be ascertained. A second generation of 3-D simulations were developed in Hawley and Krolik (2006), the highest spin case was KDJ, a/M = 0.99, with by far the most powerful Poynting jet within the new family of simulations; three times the Poynting flux (in units of the accretion rate of mass energy) of the next closest simulation KDH, a/M = 0.95. The last three data dumps, at simulation times, t = 9840 M, t = 9920 M and t = 10000 M, were generously made available to this author. The late time behavior of the simulations is established after t = 2000 M (when the large transients due to the funnel formation have died off) making these data dumps of particular interest for studying the Poynting jet (Hawley and Krolik 2006). This paper studies the origin of the Poynting jet at these late times. The analysis of the data from the KDJ simulation clearly indicates that the Poynting flux in the outgoing jet is dominated by large flares. Typically, one expects the turbulence in the field variables to mask the dynamics of Poynting flux creation in an individual time slice of one of the 3-D simulations (Punsly 2006a). Surprisingly, the flares are of such a large magnitude that they clearly standout above the background field fluctuations as evidenced by figure 1. The flares are created in the equatorial accretion flow deep in the egosphere between the inner calculational boundary at r=1.203 M and r= 1.6 M (the event horizon is at r= 1.141 M). Powerful beams of Poynting flux emerge perpendicular to the equatorial plane in the ergospheric flares and much of the energy flux is diverted outward along approximately radial trajectories that are closely aligned with the poloidal magnetic field direction in the jet (see figure 1). The situation is unsteady, whenever some vertical magnetic flux is captured in the accretion flow it tends to be asymetrically distributed and concentrated in either the northern or southern hemisphere. This hemisphere then receives a huge injection of electromagnetic energy on time scales ∼ 60M . The source of Poynting flux in KDJ resembles a nonstationary version of the ergospheric disk (see Punsly and Coroniti (1990) and chapter 8 of Punsly (2001) for a review). The ergospheric disk is modeled in the limit of negligible accretion and it is the most direct manifestation of gravitohydromagnetics (GHM) Punsly (2001). A GHM dynamo arises when the magnetic field impedes the inflow of gas in the ergosphere, i.e., vertical flux in an equatorial accretion flow. The strong gravitational force will impart stress to the magnetic field in an effort to move the plasma through the obstructing flux. In particular, the metric induced frame dragging force will twist up the field azimuthally. These stresses are coupled – 3 – into the accretion vortex around a black hole by large scale magnetic flux, and propagate outward as a relativistic Poynting jet. The more obstinate the obstruction, the more powerful the jet. There are two defining characteristics that distinguish the GHM dynamo from a Blandford-Znajek (B-Z) process, Blandford and Znajek (1977), on field lines that thread the ergopshere: 1. The B-Z process is electrodynamic so there is no source within the ergosphere, it appears as if the energy flux is emerging from the horizon. In the GHM mechanism, the source of Poynting flux is in the ergospheric equatorial accretion flow. 2. In a B-Z process in a magnetosphere shaped by the accretion vortex, the field line angular velocity is, ΩF ≈ ΩH/2 (where ΩH is the angular velocity of the horizon) near the pole and decreases with latitude to ≈ ΩH/5 near the equatorial plane of the inner ergosphere (Phinney 1983). In GHM, since the magnetic flux is anchored by the inertia of the accretion flow in the inner ergosphere, frame dragging enforces dφ/dt ≈ ΩH . One therefore has the condition, ΩF ≈ ΩH . In order to understand the physical origin of the Poynting flux, these two issues are studied below. 2. The KDJ Simulation The simulation is performed in the Kerr metric (that of a rotating, uncharged black hole), gµν . Calculations are carried out in Boyer-Lindquist (B-L) coordinates (r, θ, φ, t). The reader should refer to Hawley and Krolik (2006) for details of the simulation. We only give a brief overview. The initial state is a torus of gas in equilibrium that is threaded by concentric loops of weak magnetic flux that foliate the surfaces of constant pressure. The magnetic loops are twisted azimuthally by the differentially rotating gas. This creates significant magnetic stress that removes angular momentum from the gas, initiating a strong inflow that is permeated by magneto-rotational instabilities (MRI). The end result is that after t = a few hundred M, accreted poloidal magnetic flux gets trapped in the accretion vortex or funnel (with an opening angle of ∼ 60◦ at the horizon tapering to ∼ 35◦ at r > 20M). This region is the black hole magnetosphere and it supports a Poynting jet. The surrounding accretion flow is very turbulent. In order to understand the source of the strong flares of radial Poynting flux, one needs to merely consider the conservation of global, redshifted, or equivalently the B-L coordinate evaluated energy flux (Thorne et al 1986). In general, the divergence of the – 4 – Fig. 1.— The source of Poynting flux. The left hand column is Sθ and the right hand column is Sr in KDJ, both averaged over azimuth, at (from top to bottom) t= 9840 M, t = 9920 M and t= 10000 M. The relative units (based on code variables) are in a color bar to right of each plot for comparison of magnitudes between the six plots. The contours on the Sθ plots are of the density, scaled from the peak value within the frame at relative levels 0.5 and 0.1. The contours on the Sr plots are of Sθ scaled from the peak within the frame at relative levels 0.67 and 0.33. The inside of the inner calculational boundary (r=1.203 M) is black. The calculational boundary near the poles is at 8.1◦ and 171.9◦. Notice that any contribution from an electrodynamic effect associated with the horizon appears minimal. The white contour is the stationary limit surface. There is no data clipping, so plot values that exceed the limits of the color bar appear white. – 5 – Fig. 2.— The central engine. The left hand column is Bθ and the right hand column is ΩF in KDJ, both averaged over azimuth, at (from top to bottom) t= 9840 M, t = 9920 M and t= 10000 M. The relative units (based on code variables) are in a color bar to right of each plot for comparison of magnitudes between the plots. The calculational boundaries are the same as figure 1. The contours on the Bθ plots are of the density, scaled from the peak value within the frame at relative levels 0.5 and 0.1. There is no data clipping, so plot values that exceed the limits of the color bar appear white. – 6 – time component of the stress-energy tensor in a coordinate system can be expanded as, T νt ;ν = (1/ −g)[∂( −g T νt )/∂(x ν)] + Γ µ , where Γ t β is the connection coefficient and g = −(r2 + a2 cos2 θ)2 sin2 θ is the determinant of the metric. However, the Kerr metric has a Killing vector (the metric is time stationary) dual to the B-L time coordinate. Thus, there is a conservation law associated with the time component of the divergence of the stress- energy tensor. Consequently, if one expands out the inhomogeneous connection coefficient term in the expression above, it will equate to zero. The conservation of energy evaluated in B-L coordinates reduces to, ∂( −g T νt )/∂(x ν) = 0, where the four-momentum −T νt has two components: one from the fluid, −(T νt )fluid, and one from the electromagnetic field, −(T νt )EM. The reduction to a homogeneous equation with only partial derivatives is the reason why the global conservation of energy can be expressed in integral form in (3.70) of Thorne et al (1986). It follows that the poloidal components of the redshifted Poynting flux are Sθ = − −g (T θt )EM and S r = − −g (T rt )EM. We can use these simple expressions to understand the primary source of the Poynting jet in KDJ. Figure 1 is a plot of Sθ (on the left) and Sr (on the right) in KDJ at the last three time steps of data collection. Each frame is the average over azimuth of each time step. This greatly reduces the fluctuations as the accretion vortex is a cauldron of strong MHD waves. The individual φ = constant slices show the same dominant behavior, however it is embedded in large MHD fluctuations. On the left hand column of figure 1, density contours have been superimposed on the images to indicate the location of the equatorial accretion flow. The density is evaluated in B-L coordinates with contours at 0.5 and 0.1 of the peak value within r < 2.5M . Notice that in all three left hand frames, Sθ is created primarily in regions of very high accretion flow density. In all three of the right hand frames of figure 1, there is an enhanced Sr that emanates from the ergosphere (defined by the interior of the stationary limit, rs = M + M2 − a2 cos2 θ, note that there are 40 grid points between r = 1.203M and rs at θ = π/2). This radial energy beam diminishes precipitously just outside the horizon, near the equatorial plane in all three time steps. The region in which Sr diminishes is adjacent to a region of strong Sθ that orig- inates in the inertially dominated accretion flow in the inner ergosphere, 1.2M < r < 1.6M (this region is resolved by 28 radial grid zones). In fact, if one looks at the conservation of energy equation, the term ∂(Sθ)/∂θ is sufficiently large to be the source of ∂(Sr)/∂r at the base of the radial beam in all three frames. This does not preclude the transfer of energy to and from the plasma. It merely states that the magnitude is sufficient to source Sr. In general, the hydrodynamic energy flux is negligible in the funnel. In order to illustrate this, contours of Sθ are superimposed on the color plots of Sr. The contour levels are chosen to be 2/3 and 1/3 of the maximum value of Sθ emerging from the dense equatorial accretion flow. One clearly sees Sθ switching off where Sr switches on. We conclude that a vertical Poynting flux created in the equatorial accretion flow is the source of the strong beams of Sr. This establishes condition 1 of the Introduction. – 7 – The left column of figure 2 contains plots of the magnetic field component, Bθ ≡ Frφ, at the three time steps. At every location in which Sθ is strong in figure 1, there is a pronounced enhancement in Bθ in figure 2. Recall that the sign of Sθ is not determined by the sign of Bθ. These intense flux patches penetrate the inertially dominated equatorial accretion flow in all three frames. The density contours indicate that the regions of enhanced vertical field greatly disrupt the equatorial inflow. As noted in the introduction, a GHM interaction is likely to occur when the magnetic field impedes the inflow in the ergosphere. The regions of large Bθ are compact compared to the global field configuration of the jet, only ∼ 1.0M − 2.0M long. Considering the turbulent, differentially rotating plasma in which they are embedded, these are most likely highly enhanced regions of twisted magnetic loops created by the MRI. The strength of Bθ at the base of the flares is comparable to, or exceeds the radial magnetic field strength. The situation is clearly very unsteady and vertical flux is constantly shifting from hemisphere to hemisphere. The time slice t = 10000 M, although primarily a southern hemisphere event, also has a significant contribution in the northern hemisphere (see the blue fan-like plume of vertical Poynting flux in figure 1). The GHM interaction is provided by the vertical flux that links the equatorial plasma to the relatively slowly rotating plasma of the magnetosphere within the accretion vortex. The vertical flux transmits huge torsional stresses from the accretion flow to the magnetosphere. Further corroboration of this interpretation can be found by looking at the values of ΩF in the vicinity of the Sr flares. In a non-axisymmetric, non-time stationary flow, there is still a well defined notion of ΩF : the rate at which a frame of reference at fixed r and θ would have to rotate so that the poloidal component of the electric field, E⊥, that is orthogonal to the poloidal magnetic field, BP , vanishes. This was first derived in Punsly (1991) (see the extended discussion in Punsly (2001) for the various physical interpretations), and has recently been written out in B-L coordinates in Hawley and Krolik (2006) in terms of the plasma three-velocity, vi and the Faraday tensor as ΩF = v φ − Fθr rFφθ + gθθv (Fφθ)2grr + (Frφ)2gθθ . (2-1) This expression was studied in the context of the simulation KDH, a/M = 0.95, in Hawley and Krolik (2006). They found that a long term time and azimuth average yielded ΩF ≈ 1/3ΩH and there was no enhancement at high latitudes as was anticipated by Phinney (1983). The t = 10000 M time slice of KDH was generously provided to this author. At t = 10000 M, there are no strong flares emerging from the equatorial accretion flow. Inside the funnel at r < 10M , at t=10000 M, 0 < ΩF < 0.5ΩH . The right hand column of figure 2 is ΩF plotted at three different time steps for KDJ. By comparison to figure 1, notice that each flare in Sr is enveloped by a region of enhanced – 8 – ΩF , typically 0.7ΩH < ΩF < 1.2ΩH . The regions of the funnel outside the ergosphere are devoid of large flares in Sr and typically have 0 < ΩF < 0.5ΩH , similar to what is seen in KDH.. Unlike KDH, there are huge enhancements in ΩF at lower latitudes in the funnel. It seems reasonable to associate this large difference in the peak values of ΩF in KDJ and KDH (at t= 10000 M) with the spatially and temporally coincident flares in Sr that occur in KDJ. Furthermore, this greatly enhanced value of ΩF indicates a different physical origin for ΩF in the flares than for the remainder of the funnel or in KDH at t = 10000 M. The most straightforward interpretation is that it is a direct consequence of the fact that the flares originate on magnetic flux that is locked into approximate corotation with the dense accreting equatorial plasma (i.e., the inertially dominated equatorial plasma anchors the magnetic flux). In the inner ergosphere, frame dragging enforces 0.7ΩH < dφ/dt < 1.0ΩH on the accretion flow. This establishes condition 2 of the Introduction. 3. Discussion In this Letter we showed that in the last three data dumps of the 3-D MHD numerical simulation, KDJ, the dominant source of Poynting flux originated near the equatorial plane deep in the ergopshere. The phenomenon is unsteady and is triggered by large scale vertical flux that is anchored in the inertially dominated equatorial accretion flow. The situation typifies the ergospheric disk in virtually every aspect, even though there is an intense accre- tion flow. There is one exception, unlike the ergospheric disk, the anchoring plasma rarely achieves the global negative energy condition that is defined by the four-velocity, −Ut < 0, because of the flood of incoming positive energy plasma from the accretion flow. The plasma attains −Ut < 0 only near the base of the strongest flares seen in the φ = constant slices. The switch-on of a powerful beam of Sr outside the horizon at r ≈ 1.3M in the a/M = 0.995 simulation, KDE, of Krolik et al (2005) was demonstrated in Punsly (2006a). It seems likely the the source of Sr in KDE is Sθ from an ergopsheric disk. The ergospheric disk appears to switch on at a/M > 0.95 as evidenced by the factor of 3 weaker Poynting flux in KDH. Furthermore, if the funnel opening angle at the horizon in KDH at t= 10000 M is typical within ±5◦ then figure 5 and table 4 of Hawley and Krolik (2006) indicate that only 35% to 40% of the funnel Poynting flux at large distances is created outside the horizon during the course of the simulation. A plausible reason is given by the plots of Bθ in figure 2. The vertical magnetic flux at the equatorial plane is located at r < 1.55M . The power in the ergospheric disk jet ∼ [Bθ(SA)(ΩH)]2, where SA is the proper surface area of the equatorial plane threaded by vertical magnetic flux (Semenov et al 2004; Punsly 2001). The proper surface area in the ergospheric equatorial plane increases dramatically – 9 – at high spin, diverging at a = M . For example, between the inner calculational boundary and 1.55 M the surface area is only significant for a/M > 0.95 and grows quickly with a/M , exceeding twice the surface area of the horizon for a/M = 0.99. Thus, if Bθ in the inner ergosphere were independent of spin to first order, then a strong ergospheric disk jet would switch-on in the 3-D simulations at a/M > 0.95. Note that if the inner boundary were truly the event horizon instead of the inner calculational boundary then this argument would indicate that the ergospheric disk would likely be very powerful even at a/M = 0.95 and the switch-on would occur at a/M ≈ 0.9. The implication is that a significant amount of large scale magnetic flux threading the equatorial plane of the ergopshere (which implies a large black hole spin based on geometrical considerations) catalyzes the formation of the most powerful Poynting jets around black holes. Thus, we are now considering initial conditions in simulations that are conducive to producing significant vertical flux in the equatorial plane of the ergosphere. It should be noted that 2-D simulations from a similar initial state of torii threaded by magnetic loops have been studied in McKinney and Gammie (2004). However, the magnetic flux evolution can be much different in this setting as discussed in Punsly (2006a) and poloidal flux configurations conducive to GHM could be highly suppressed. In summary, there are no interchange instabilities, so flux tubes cannot pass by each other or move around each other in the extra degree of freedom provided by the azimuth. Thus, there is a tendency for flux tubes to get pushed into the hole by the accretion flow. This is in contrast to the formation of the ergospheric disk in Punsly and Coroniti (1990) in which buoyant flux tubes are created by reconnection at the inner edge of the ergospheric disk and recycle back out into the outer ergosphere by interchange instabilities. Ideally, a full 3-D simulation with a detailed treatment of resistive MHD reconnection is preferred for studying the relevant GHM physics. I would like to thank Jean-Pierre DeVilliers for sharing his deep understanding of the numerical code and these simulations. I was also very fortunate that Julian Krolik and John Hawley were willing to share their data in the best spirit of science. REFERENCES Blandford, R. and Znajek, R. 1977, MNRAS. 179, 433 De Villiers, J-P., Hawley, J., Krolik, 2003, ApJ 599 1238 De Villiers, J-P., Hawley, J., Krolik, J.,Hirose, S. 2005, ApJ 620 878 – 10 – De Villiers, J-P., Staff, J., Ouyed, R.. 2005, astro-ph 0502225 Hawley, J., Krolik, K. 2006, ApJ 641 103 Hirose, S., Krolik, K., De Villiers, J., Hawley, J. 2004, ApJ 606, 1083 Krolik, K., Hawley, J., Hirose, S. 2005, ApJ 622, 1008 McKinney, J. and Gammie, C. 2004, ApJ 611 977 McClintock, J.E. et al 2006, ApJ 652, 518 Phinney, E.S. 1983, PhD Dissertation University of Cambridge. Punsly, B., Coroniti, F.V. 1990, ApJ 354 583 Punsly, B. 1991, ApJ 372 424 Punsly, B. 2001, Black Hole Gravitohydromagnetics (Springer-Verlag, New York) Punsly, B. 2006, MNRAS 366 29 Punsly, B. 2006, ApJL 651 L17 Punsly, B. 2007, MNRAS 374 10 Semenov, V., Dyadechkin, S. and Punsly, B. 2004, Science 305978 Thorne, K., Price, R. and Macdonald, D. 1986, Black Holes: The Membrane Paradigm (Yale University Press, New Haven) This preprint was prepared with the AAS LATEX macros v5.2. http://arxiv.org/abs/astro-ph/0502225 Introduction The KDJ Simulation Discussion

0704.0817

An S_3-symmetric Littlewood-Richardson rule

AN S3-SYMMETRIC LITTLEWOOD-RICHARDSON RULE HUGH THOMAS AND ALEXANDER YONG 1. INTRODUCTION Fix Young shapes λ, µ, ν ⊆ Λ := ℓ × k. Viewing the Littlewood-Richardson coeffi- cients Cλ,µ,ν as intersection numbers of three Schubert varieties in general position in the Grassmannian of ℓ-dimensional planes in Cℓ+k, one has the obvious S3-symmetries: (1) Cλ,µ,ν = Cµ,ν,λ = Cν,λ,µ = Cµ,λ,ν = Cν,µ,λ = Cλ,ν,µ. There is interest in the combinatorics of these symmetries; previously studied Littlewood- Richardson rules for Cλ,µ,ν manifest at most three of the six, see, e.g., [BeZe91, KnTaWo01, VaPa05, HeKa06] and the references therein. We construct a carton rule for Cλ,µ,ν that transparently and uniformly explains all symmetries (1). Figure 1 depicts a carton, i.e., a three-dimensional box with a grid drawn rectilinearly on the six faces of its surface. Along the “∅, Tλ, Tµ” face, the grid has (|µ| + 1) × (|λ| + 1) vertices (including ones on the bounding edges of the face); the remainder of the grid is similarly determined. Define a carton filling to be an assignment of a Young diagram to each vertex of the grid so the shapes increase one box at a time while moving away from ∅, and so that, for any subgrid α − β γ − δ the Fomin growth assumptions hold: (F1) if α is the unique shape containing γ and contained in β, then δ = α; (F2) otherwise there is a unique such shape other than α, and this shape is δ. Notice that these conditions are symmetric in α and δ. Initially, assign the shapes ∅ and Λ to opposite corners, as in Figure 1. A standard Young tableau T ∈ SYT(σ/π) of shape σ/π is equivalent to a shape chain in Young’s lattice, e.g., T = 1 5 ↔ − − − − − by starting with the unfilled inner shape, and adding one box at a time to indicate the box containing 1, then 2, etc. Fix a choice of standard tableaux Tλ, Tµ and Tν of respective shapes λ, µ and ν. Initialize the indicated edges with the shape chains for these tableaux. Let CARTONSλ,µ,ν be all carton fillings with the above initial data. Main Theorem. The Littlewood-Richardson coefficient Cλ,µ,ν equals #CARTONSλ,µ,ν. This rule manifests bijections between CARTONSλ,µ,ν and CARTONSδ,ǫ,ζ for permutations (δ, ǫ, ζ) of (λ, µ, ν). We remark that it can be stated as counting lattice points of a 0, 1- polytope, and readily extends to the setting of [ThYo06, ThYo07]. Date: April 4, 2007. http://arxiv.org/abs/0704.0817v1 FIGURE 1. The carton rule counts Cλ,µ,ν by assigning Young diagrams to the vertices of the six faces The Theorem is proved in Section 2, starting from the jeu de taquin formulation of the Littlewood-Richardson rule, and using Fomin’s growth diagram ideas. In Figure 2 we give an example of the Main Theorem. An extended example is given in Section 3. ν = 1 FIGURE 2. The “front” three faces of a carton filling for C(2),(2,1),(1) = 1. The choices of tableaux Tλ, Tµ and Tν are as shown. 2. TABLEAU FACTS AND PROOF OF THE MAIN THEOREM 2.1. Tableau sliding. There is a partial order ≺ on the rectangle Λ where x ≺ y if x is weakly northwest of y. Given T ∈ SYT(ν/λ) consider x ∈ λ, maximal in ≺ subject to being less than some box of ν/λ. Associate another standard tableau jdtx(T), called the jeu de taquin slide of T into x: Let y be the box of ν/λ with the smallest label, among those covering x. Move the label of y to x, leaving y vacant. Look for boxes of ν/λ covering y and repeat, moving into y the smallest label among those boxes covering it. Then jdtx(T) results when no further slides are possible. The rectification of T is the iteration of jeu de taquin slides until terminating at a straight shape standard tableau rectification(T). We can impose a specific “inner” U ∈ SYT(λ) that encodes the order in which the jeu de taquin slides are done. For example, if U = 1 2 3 T = 1 5 → 1 5 → 1 5 → 1 5 → 1 3 5 = rectification(T). Placing these chains one atop another (starting with T ’s and ending with rectification(T)’s) gives a Fomin growth diagram, which in this case is given in Table 1. Growth diagrams satisfy (F1) and (F2); see Fomin’s [St99, Appendix 2] for more. (3, 1) (4, 1) (4, 2) (4, 3) (4, 3, 1) (5, 3, 1) (5, 3, 2) (3) (4) (4, 1) (4, 2) (4, 2, 1) (5, 2, 1) (5, 2, 2) (2) (3) (3, 1) (3, 2) (3, 2, 1) (4, 2, 1) (4, 2, 2) (1) (2) (2, 1) (2, 2) (2, 2, 1) (3, 2, 1) (3, 2, 2) ∅ (1) (1, 1) (2, 1) (2, 1, 1) (3, 1, 1) (3, 2, 1) TABLE 1. A growth diagram Given a top row U and left column T , define infusion1(U, T) to be the bottom row of the growth diagram, and infusion2(U, T) to be the right column. We write infusion(U, T) for the ordered pair (infusion1(U, T), infusion2(U, T)). Growth diagrams are transpose symmetric, because (F1) and (F2) are. So one has the infusion involution: infusion(infusion(U, T)) = (U, T). If U is a straight shape, infusion1(U, T) = rectification(T), while infusion2(U, T) en- codes the order in which squares were vacated in the jeu de taquin process. Also, given T ∈ SYT(ν/λ), consider x ∈ Λ\ν minimal subject to being larger than some element of ν/λ. The reverse jeu de taquin slide revjdtx(T) of T into x is defined similarly to a jeu de taquin slide, except we move into x the largest of the labels among boxes in ν/λ covered by x. We define reverse rectification revrectification(T) similarly. The first fundamental theorem of jeu de taquin asserts (rev)rectification is well-defined. Fix Tµ ∈ SYT(µ). The number of T ∈ SYT(ν/λ) such that rectification(T) = Tµ equals Cλ,µ,ν∨ = C λ,µ where ν ∨ is the straight shape obtained as the 180-degree rotation of Λ\ν. This is Schützenberger’s jeu de taquin formulation of the Littlewood-Richardson rule, see [St99, Appendix 2]. By a slide we mean either kind of jeu de taquin slide. Consider two equivalence rela- tions on a pair of tableaux T and U. Tableaux are jeu de taquin equivalent if one can be obtained from the other by a sequence of slides. They are dual equivalent if any such se- quence results in tableaux of the same shape [Ha92]. Facts: Tableaux of the same straight shape are dual equivalent. A common application of slides to dual equivalent tableaux produces dual equivalent tableaux. A pair of tableaux that are both jeu de taquin and dual equivalent must be equal. Recall Schützenberger’s evacuation map. For T ∈ SYT(λ), let T̂ be obtained by erasing the entry 1 (in the northwest corner c) of T and subtracting 1 from the remaining entries. Let ∆(T) = jdtc(T̂). The evacuation evac(T) ∈ SYT(λ) is defined by the shape chain ∅ = shape(∆|λ|(T)) − shape(∆|λ|−1(T)) − . . .− shape(∆1(T)) − T. This map is an involution: evac(evac(T)) = T . 2.2. Proof of the rule. Given T ∈SYT(α) for a straight shape α, define T̃ ∈SYT(rotate(α)) where rotate(α) = Λ \ α∨ by computing evac(T) ∈ SYT(α), replacing entry i with |α|− i+ 1 throughout and rotating the resulting tableau 180-degrees and placing it at the bottom right corner of Λ. We need the following well-known fact. The proof we give extends straightforwardly to the setup of [ThYo06, ThYo07], allowing a “cominuscule” version of the Main Theorem. Lemma 2.1. Suppose α, β and γ are shapes where Tβ ∈ SYT(β), Tγ∨/α ∈ SYT(γ ∨/α) and rectification(Tγ∨/α) = Tβ. Then revrectification(Tγ∨/α) = revrectification(Tβ) = T̃β. Proof. It suffices to show that revrectification(Tβ) = T̃β. We induct on |β| = n. Let U be the tableau obtained by removing the box labeled 1 from Tβ. So U is of skew shape β/(1). Let V = rectification(U), a tableau of shape κ ⊂ β. By induction, revrectification(V) = Ṽ . Now revrectification(Tβ) and revrectification(V) agree except that the former has a label 1 in a box that does not appear in the latter. In [ThYo06, Proposition 4.6], we showed that the reverse rectification of a tableau of shape β is nec- essarily of shape rotate(β). Thus, the location of 1 in revrectification(Tβ) must be the box κ∨/β∨. This is the 180-degree rotation of the box β/κ in Λ, the latter being the position of n in evac(Tβ). Thus, 1 is located in the desired position in revrectification(Tβ). The rest of the statement follows from the fact that the rest of revrectification(Tβ) agrees with Ṽ , and the entries of evac(Tβ) other than n agree with evac(V). � Corollary 2.2. Fix a carton filling. The uninitialized corners of the “∅, Tλ, Tµ” and “∅, Tν, Tλ” faces are ν∨ and µ∨ respectively. The remaining “sixth” corner (the unique one not visible in Figure 1) is λ∨. Thus, we can speak of the edges λ∨−Λ, µ∨−Λ and ν∨−Λ. These represent the shape chains of T̃λ, T̃µ and T̃ν respectively. Proof. Think of the union of the faces “∅, Tµ, Tλ” with the adjacent face involving Λ and the “sixth corner” as a single growth diagram. This diagram computes revrectification(Tλ) and records the result along the edge involving Λ and diagonally opposite to the Tλ edge. But the lemma asserts this result is T̃λ and hence the “sixth corner vertex” is assigned λ The other conclusions are proved similarly. � Corollary 2.2 affords us the convenience of referring to a face by its corner vertices. Note any carton filling gives a growth diagram on the face ∅ − µ − ν∨ − λ for which the edge λ− ν∨ is a standard tableau of shape ν∨/λ rectifying to Tµ. By the jeu de taquin Littlewood-Richardson rule, fillings of this face count Cλ,µ,ν. Hence it suffices to show that any such growth diagram for this face extends uniquely to a filling of the entire carton. If such an extension exists, it is unique: λ− ν∨ and Λ− ν∨ determine, by (F1), (F2) and the Corollary, the face λ−µ∨−Λ−ν∨. Similarly, these two faces determine the remaining faces (in order): ∅− ν− µ∨− λ =⇒ ν− λ∨−Λ− µ∨ =⇒ µ− λ∨−Λ− ν∨ =⇒ ∅− ν− λ∨ − µ. We now show that with a filling of ∅−µ−ν∨−λ (together with the extra edges given in the Corollary) one can extend it to fill the carton, using the conclusions of Corollary 2.2. ∅− µ− ν∨− λ: This is given by labeling λ − ν∨ with Tν∨/λ ∈ SYT(ν ∨/λ) that witnesses Cλ,µ,ν = C λ,µ, i.e., it rectifies to Tµ. By the infusion involution, the edge µ−ν ∨ representing Tν∨/µ := infusion2(Tλ, Tν∨/λ) ∈ SYT(ν ∨/µ) witnesses Cν λ− µ∨−Λ− ν∨: Build the growth diagram using Tν∨/λ and T̃ν from the edges λ−ν ∨ and ν∨ − Λ respectively. The only boundary condition we need to check is that infusion2(Tν∨/λ, T̃ν) = T̃µ, which is true by the Lemma. The newly determined edge λ − µ∨ is infusion1(Tν∨/λ, T̃ν). This is a tableau Tµ∨/λ ∈ SYT(µ ∨/λ) which rectifies to Tν, i.e., one that witnesses C ∅− ν− µ∨− λ: Using the determined edges λ−µ∨ and ∅−λ we obtain a growth diagram with edge ∅ − ν representing infusion1(Tλ, Tµ∨/λ), which equals Tν, as desired. By the infusion involution, ν− µ∨ represents a tableau Tµ∨/ν ∈ SYT(µ ∨/ν) that witnesses C ν− λ∨ −Λ− µ∨: This time, we grow the face using the edges ν−µ∨ and µ∨−Λ. The edge λ∨ − Λ is thus infusion2(Tµ∨/ν, T̃µ) which indeed equals T̃λ, by the Lemma. The newly determined edge ν−λ∨ is Tλ∨/ν := infusion1(Tµ∨/ν, T̃µ) ∈ SYT(λ ∨/ν) that witnesses Cλ µ− λ∨−Λ− ν∨: Growing the face using µ−ν∨ and ν∨−Λ we find that the edge λ∨−Λ equals infusion2(Tν∨/µ, T̃ν), which is the already determined T̃λ. The newly determined edge µ− λ∨ is Tλ∨/µ := infusion1(Tν∨/µ, T̃ν) ∈ SYT(λ ∨/µ) that witnesses Cλ ∅− ν− λ∨− µ: We grow this final face using ∅ − ν and ν − λ∨, but need to make two consistency checks. First the edge ∅− µ is infusion1(Tν, Tλ∨/ν) which clearly is Tµ. It remains to check that (2) infusion2(Tν, Tλ∨/ν) = Tλ∨/µ. (Notice that infusion2(Tν, Tλ∨/ν) is a filling of λ ∨/µ that rectifies to Tν, just as Tλ∨/µ does. However it is not clear a priori that they are the same.) To prove (2), we need a definition. For tableaux A and B of respective (skew) shapes α and γ/α let A ⋆ B be their concatenation as a (nonstandard) tableau. If C is a tableau of shape Λ/γ and α is a straight shape, A ⋆ B ⋆ C is a layered tableau of shape Λ. Example 2.3. Let α = (2, 1) and γ = (4, 2, 1). We have A = 1 2 , B = 2 3 , C = 1 4 5 , and A ⋆ B ⋆ C = 1 2 2 3 3 1 2 3 4 1 4 5 We have used boldface and underlining to distinguish the entries from A,B and C. Let I1 and I2 be operators on layered tableaux defined by I1 : A ⋆ B ⋆ C 7→ infusion1(A,B) ⋆ infusion2(A,B) ⋆ C I2 : A ⋆ B ⋆ C 7→ A ⋆ infusion1(B,C) ⋆ infusion2(B,C). The infusion involution says I21 and I 2 are the identity operator. In fact, the following crucial “braid identity” holds, showing I1 and I2 generate a representation of S3: Proposition 2.4. I1 ◦ I2 ◦ I1 = I2 ◦ I1 ◦ I2. In view of the Proposition, (2) follows by setting α = λ, γ = ν∨, A = Tλ, B = Tλ∨/ν and C = T̃ν, since the assertion merely says the “middle” tableau in (3) I1 ◦ I2 ◦ I1 ◦ I2(Tλ ⋆ Tν∨/λ ⋆ T̃ν) and I2 ◦ I1(Tλ ⋆ Tν∨/λ ⋆ T̃ν) are the same, whereas we even have equality of the two layered tableaux. Proof of Proposition 2.4: By the Lemma it follows that (4) C̃ ⋆ B ⋆ à := I1 ◦ I2 ◦ I1(A ⋆ B ⋆ C), and (5) C̃ ⋆ B̂ ⋆ à := I2 ◦ I1 ◦ I2(A ⋆ B ⋆ C), where the shapes of C̃, B and à are respectively the 180 degree rotations of the shapes of A, B and C, and we know the fillings of C̃ and à in terms of evacuation. We thus also know the shapes of B and B̂ are the same. It remains to show B equals B̂. We first study (4). Let B ′ = infusion1(A,B), and A ′ = infusion2(A,B). Now (6) I1 ◦ I2(B ⋆A ′ ⋆ C) = C̃ ⋆ B ⋆ Ã, and the skew tableau B ⋆ à is obtained by treating B ′ ⋆ A ′ as a single standard tableau by valuing an entry i of A ′ as |B ′| + i, evacuating, rotating the result 180 degrees and finally sending the entry i from A ′ (respectively B ′) to |A ′| − i+ 1 (respectively |B ′| − i+ 1). Example 2.5. Continuing the previous example, B ′ ⋆A ′ = 1 2 3 2 and evac(B ′ ⋆A ′) = A ′′ ⋆ B ′′ = 1 3 3 4 7→ B ⋆ à = 3 1 2 1 3 Let us focus on evac(B ′⋆A ′) = A ′′⋆B ′′ and momentarily ignore the subsequent rotation and complementation of entries. Fomin has shown in [St99, Appendix 2] that a fruitful way to think of evac(X) is with triangular growth diagrams obtained by placing the shape chain of X, then ∆(X), ∆2(X), . . . on top of one another (slanted left to right); see Figure 3. We begin with the data B ′ ⋆ A ′ along the left side of the diagram. This is given as a concatenation of two shape chains, one from ∅−µ and then one from µ−γ where µ is the shape of B ′. Applying (F1) and (F2) from Section 1, we obtain A ′′ ⋆ B ′′ = evac(B ′ ⋆ A ′), ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ B ′ evac(B ′) A ′′A FIGURE 3. A triangular growth diagram to compute evac(B ′ ⋆A ′) given along the righthand side as a pair of chains connected at α. It follows from the construction of growth diagrams that the thick lines represent evac(B ′) and A = evac(A ′′) (the latter being a consequence of the Lemma). These thick lines together with the vertex γ define a rectangular growth diagram, so: (7) infusion(A,B ′′) = (evac(B ′), A ′). In particular, the shape of rectification(B ′′) = µ. On the other hand, by definition (8) infusion(A,B) = (B ′, A ′), and B rectifies to a tableau of shape µ also. By (7) and (8) combined with the aforementioned results of [Ha92] we see B and B ′′ are dual equivalent, as they both are obtained by an application of the same sequence of slides (encoded by A ′) to a pair of tableaux of the same straight shape. Moreover, by (7) B ′′ is jeu de taquin equivalent to evac(B ′) = evac(rectification(B)). Carrying out the analogous “reverse” analysis on I2 ◦ I1 ◦ I2(A ⋆ B ⋆ C) we let C ◦ = infusion1(B,C) and B◦ = infusion2(B,C) and study I2 ◦ I1(A ⋆ C ⋆ B◦) = C̃ ⋆ B̂ ⋆ Ã. Parallel to (6) we consider the “reverse evacuation” B◦◦ ⋆ C◦◦ of C◦ ⋆ B◦ using reverse jeu de taquin slides. Then we similarly conclude B◦◦ is dual equivalent to B (and thus B ′′). Also B◦◦ is jeu de taquin equivalent to the reverse evacuation of revrectification(B). Thus by the lemma, B ′′ and B◦◦ are jeu de taquin equivalent. Hence B ′′ = B◦◦ and so B = B̂, as required. Example 2.6. Revisiting Example 2.3, C◦⋆B◦ = 2 3 4 5 3 1 1 2 4 with reverse evacuation 3 4 1 1 2 2 3 4 5 and B◦◦ (boldface in the latter tableau) is also B ′′ from Example 2.5. Rotation and comple- mentation gives B̂, which is B from Example 2.5, as desired. � 3. AN EXTENDED EXAMPLE OF THE MAIN THEOREM Let Λ = 3 × 4 = , λ = (2, 1) = , µ = (3, 1) = , ν = (3, 2) = . Therefore λ∨ = (4, 3, 2), µ∨ = (4, 3, 1) and ν∨ = (4, 2, 1). Also Tλ = 1 2 , Tµ = 1 2 3 , Tν = 1 2 3 , T̃λ = , T̃µ = 1 2 4 , and T̃ν = 1 4 5 Here Cλ,µ,ν = 1, and we now give the unique carton filling. We begin with Tν∨/λ = . Then we have the following sides of the carton, described as growth diagrams with the obvious identifications of boundaries (the partitions correspond to shapes placed on the vertices of the carton and the diagrams have been oriented to be consistent with Figure 1): (2, 1) = λ (2, 1, 1) (3, 1, 1) (4, 1, 1) (4, 2, 1) = ν∨ (2) (2, 1) (3, 1) (4, 1) (4, 2) (1) (1, 1) (2, 1) (3, 1) (3, 2) ∅ (1) (2) (3) (3, 1) = µ TABLE 2. ∅− µ− ν∨− λ (4, 3, 1) = µ∨ (4, 3, 2) (4, 3, 3) (4, 4, 3) (4, 4, 4) = Λ (4, 2, 1) (4, 2, 2) (4, 3, 2) (4, 4, 2) (4, 4, 3) (4, 2) (4, 2, 1) (4, 3, 1) (4, 4, 1) (4, 4, 2) (3, 2) (3, 2, 1) (3, 3, 1) (4, 3, 1) (4, 3, 2) (2, 2) (2, 2, 1) (3, 2, 1) (4, 2, 1) (4, 2, 2) (2, 1) = λ (2, 1, 1) (3, 1, 1) (4, 1, 1) (4, 2, 1) = ν∨ TABLE 3. λ− µ∨−Λ− ν∨ (4, 3, 1) = µ∨ (4, 2, 1) (4, 2) (3, 2) (2, 2) (2, 1) = λ (4, 2, 1) (4, 1, 1) (4, 1) (3, 1) (2, 1) (2) (3, 2, 1) (3, 1, 1) (3, 1) (2, 1) (1, 1) (1) (3, 2) = ν (3, 1) (3) (2) (1) ∅ TABLE 4. ∅− ν− µ∨− λ (4, 3, 1) = µ∨ (4, 3, 2) (4, 3, 3) (4, 4, 3) (4, 4, 4) = Λ (4, 2, 1) (4, 2, 2) (4, 3, 2) (4, 4, 2) (4, 4, 3) (3, 2, 1) (3, 2, 2) (3, 3, 2) (4, 3, 2) (4, 4, 2) (3, 2) = ν (3, 2, 1) (3, 3, 1) (4, 3, 1) (4, 3, 2) = λ∨ TABLE 5. ν− λ∨−Λ− µ∨ ACKNOWLEDGMENTS HT was partially supported by an NSERC Discovery Grant and AY by NSF grant 0601010. This work was partially completed while HT was a visitor at the Centre de Recherches Mathématiques and the University of Minnesota, and while AY was at the Fields Institute and the University of Michigan, Ann Arbor. We thank the Banff Interna- tional Research Station for a common visit during the “Schubert calculus and Schubert (4, 4, 4) = Λ (4, 4, 3) (4, 4, 2) (4, 3, 2) (4, 2, 2) (4, 2, 1) = ν∨ (4, 4, 3) (4, 4, 2) (4, 4, 1) (4, 3, 1) (4, 2, 1) (4, 2) (4, 3, 3) (4, 3, 2) (4, 3, 1) (3, 3, 1) (3, 2, 1) (3, 2) (4, 3, 2) = λ∨ (4, 2, 2) (4, 2, 1) (3, 2, 1) (3, 1, 1) (3, 1) = µ TABLE 6. λ∨−Λ− ν∨− µ (3, 2) = ν (3, 2, 1) (3, 3, 1) (4, 3, 1) (4, 3, 2) = λ∨ (3, 1) (3, 1, 1) (3, 2, 1) (4, 2, 1) (4, 2, 2) (3) (3, 1) (3, 2) (4, 2) (4, 2, 1) (2) (2, 1) (2, 2) (3, 2) (3, 2, 1) (1) (1, 1) (2, 1) (3, 1) (3, 1, 1) ∅ (1) (2) (3) (3, 1) = µ TABLE 7. ∅− ν− λ∨− µ geometry” workshop organized by Jim Carrell and Frank Sottile. We also thank Sergey Fomin, Allen Knutson, Ezra Miller, Vic Reiner, Luis Serrano, David Speyer and Dennis Stanton for helpful discussions. REFERENCES [BeZe91] A. D. Berenstein and A. V. Zelevinsky, Triple multiplicities of sℓ(r + 1) and the spectrum of the exterior algebra of the adjoint representation, Journal of Algebraic Combinatorics, 1 (1992), 7–22. [Ha92] M. Haiman, Dual equivalence with applications, including a conjecture of Proctor, Discrete Math. 99 (1992), 79–113. [HeKa06] A. Henriques and J. Kamnitzer, The octahedron recurrence and gln crystals, Adv. Math., 206 (2006), no. 1, 211–249. [KnTaWo01] A. Knutson, T. Tao and C. Woodward, The honeycomb model of GL(n) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone, J. Amer. Math. Soc., 17 (2004), 19–48. [St99] R. P. Stanley, Enumerative Combinatorics, Volume 2 (with an appendix by S. Fomin), Cambridge University Press, 1999. [VaPa05] E. Vallejo and I. Pak, Combinatorics and geometry of Littlewood-Richardson cones, Europ. J. Com- binat., 26(2005), 995–1008. [ThYo06] H. Thomas and A. Yong, A combinatorial rule for (co)minuscule Schubert calculus, preprint math.AG/0608273. [ThYo07] , Cominuscule tableau combinatorics, preprint math.CO/0701215. DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF NEW BRUNSWICK, FREDERICTON, NEW BRUNSWICK, E3B 5A3, CANADA E-mail address: [email protected] DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MINNESOTA, MINNEAPOLIS, MN 55455, USA E-mail address: [email protected] 1. Introduction 2. Tableau facts and proof of the Main Theorem 2.1. Tableau sliding 2.2. Proof of the rule 3. An extended example of the main theorem Acknowledgments References

0704.0818

Two-scale structure of the electron dissipation region during collisionless magnetic reconnection

Submitted to Physical Review Letters Two-scale structure of the electron dissipation region during collisionless magnetic reconnection M. A. Shay∗ Department of Physics & Astronomy, 217 Sharp Lab, University of Delaware, Newark, DE 19716 J. F. Drake, M. Swisdak University of Maryland, College Park, MD, 20742 (Dated: November 1, 2018) Particle in cell (PIC) simulations of collisionless magnetic reconnection are presented that demon- strate that the electron dissipation region develops a distinct two-scale structure along the outflow direction. The length of the electron current layer is found to decrease with decreasing electron mass, approaching the ion inertial length for a proton-electron plasma. A surprise, however, is that the electrons form a high-velocity outflow jet that remains decoupled from the magnetic field and extends large distances downstream from the x-line. The rate of reconnection remains fast in very large systems, independent of boundary conditions and the mass of electrons. PACS numbers: Valid PACS appear here Magnetic reconnection drives the release of magnetic energy in explosive events such as disruptions in labo- ratory experiments, magnetic substorms in the Earth’s magnetosphere and flares in the solar corona. Recon- nection in these events is typically collisionless because reconnection electric fields exceed the Dreicer runaway field. Since magnetic field lines reconnect in a boundary layer, the “dissipation region”, whose structure may limit the rate of release of energy, understanding the structure of this boundary layer and its impact on reconnection is critical to understanding the observations. Because of their ability to carry large currents the dynamics of electrons continues to be a topic of interest. Early sim- ulations of reconnection suggested that the rate of re- connection was not sensitive to electron dynamics [1, 2] and this insensitivity was attributed to the coupling to whistler dynamics at the small spatial scales of the dis- sipation region [3, 4]. The results of more recent kinetic PIC simulations have called into question these results by suggesting that the electron current layer stretches along the outflow direction and the rate of reconnection drops[5, 6]. The fast rates of reconnection obtained from earlier simulations[1, 3, 7] were attributed to the influ- ence of periodicity[5]. We present particle-in-cell (PIC) simulations with var- ious electron masses and computational domain sizes and an analytic model that demonstrate that collisionless re- connection remains fast even in very large collisionless systems. The reconnection rate stabilizes before the pe- riodicity of the boundary conditions can impact the dy- namics. The electron current layer develops a distinct two-scale structure along the outflow direction that had not been identified in earlier simulations. The out-of- plane electron current driven by the reconnection elec- ∗Electronic address: [email protected]; URL: http://www.physics.udel.edu/~shay tric field has a length that decreases with the electron mass, scaling as (me/mi) 3/8, which extrapolates to about an ion inertial length di = c/ωpi for the electron-proton mass ratio. The surprise is that a jet of outflowing elec- trons with velocity close to the electron Alfven speed cAe extends up to several 10’s of di from the x-line. Remark- ably, the electrons are able to jet across the magnetic field over such enormous distances because momentum trans- port transverse to the jet effectively “blocks” the flow of the out-of-plane current in this region. The momentum transport causing this “current blocking” effect has the same source (the off diagonal pressure tensor[1]), but is much stronger than that which balances the reconnection electric field at the x-line. Our simulations are performed with the particle-in-cell code p3d [8, 9]. The results are presented in normal- ized units: the magnetic field to the asymptotic value of the reversed field, the density to the value at the cen- ter of the current sheet minus the uniform background density, velocities to the Alfvén speed vA, lengths to the ion inertial length di, times to the inverse ion cy- clotron frequency Ω−1ci , and temperatures to miv A. We consider a system periodic in the x− y plane where flow into and away from the x-line are parallel to ŷ and x̂, respectively. The reconnection electric field is parallel to ẑ. The initial equilibrium consists of two Harris cur- rent sheets superimposed on a ambient population of uni- form density. The reconnection magnetic field is given by Bx = tanh[(y − Ly/4)/w0] − tanh[(y − 3Ly/4)/w0] − 1, where w0 and Ly are the half-width of the initial current sheets and the box size in the ŷ direction. The electron and ion temperatures, Te = 1/12 and Ti = 5/12, are ini- tially uniform. The initial density profile is the usual Har- ris form plus a uniform background of 0.2. The simula- tions presented here are two-dimensional,i.e., ∂/∂z = 0. Reconnection is initiated with a small initial magnetic perturbation that produces a single magnetic island on each current layer. We have explored the dependence of the rate of recon- http://arxiv.org/abs/0704.0818v1 mailto:[email protected] http://www.physics.udel.edu/~shay FIG. 1: (color online). Reconnection electric field versus time: (a) 204.8 × 102.4, (b) 102.4 × 51.2, (c) 51.2 × 25.6. w0 is the initial current sheet width. nection on the system size in a series of simulations with three different system sizes and three different mass ra- tios. For mi/me = 25, the grid scale ∆ = 0.05 and the speed of light c = 15. For mi/me = 100, ∆ = 0.025 and c = 20. For mi/me = 400, ∆ = 0.0125 and c = 40. The reconnection rate versus time is plotted for our simula- tions in Fig. 1. The reconnection rate is determined by taking the time derivative of the total magnetic flux be- tween the x-line and the center of the magnetic island. The rate increases with time, undergoes a modest over- shoot that is more pronounced in the smaller domains, and approaches a quasi-steady rate of around 0.14, in- dependent of the domain size. Earlier suggestions [5] that reconnection rates would plunge until elongated cur- rent layers spawned secondary magnetic islands are not borne out in these simulations. The rates of reconnec- tion approach constant values even in the absence of sec- ondary islands, which for anti-parallel reconnection typ- ically only occur transiently due to initial conditions[10]. Even these transient islands can be largely eliminated by a suitable choice of the initial current layer width w0 (a larger value of w0 is required for the larger domains). A critical issue is whether the periodicity in the x direc- tion can influence the rate of reconnection [5]. In each of the simulations we have identified the time at which the ion outflows from the x-line meet at the center of the mag- netic island. This occurs at t ≈ 155 for the largest simu- lation shown in Fig. 1a. The plasma at the x-line can not be affected by the downstream conditions until t ≈ 255, when a pressure perturbation can propagate back up- stream to the x-line at the magnetosonic speed. This is well after the end of the simulation. The electrons are ejected from the x-line at a velocity of around cAe ≫ cA and therefore might be able to follow field lines back to the x-line. During the traversal time δt = Lx/cAe, the amount of reconnected flux is vinB0L/cAe, where vin is the inflow velocity into the x-line. Using the conservation of the canonical momentum in the z-direction, the condi- tion that an electron with a velocity cAe can not cross this flux to access the x-line reduces to L > di(cA/vin) ∼ 7di, which is easily satisfied for the simulations in Fig. 1. The fact that the reconnection rates for all of the simulation domains in Fig. 1 are essentially identical further sup- ports this conclusion. Also shown in Fig. 1 in the dashed lines are the rates of reconnection for mi/me = 100 in (b) and mi/me = 400 in (c). Consistent with simulations in smaller domains [1, 2], the rate of reconnection is insensitive to the electron mass. We now proceed to explore the structure of the electron current layer. Shown in Fig. 2 is a blow-up around the x- line of the out-of-plane electron velocity for mi/me = 25 and two simulation domains, 204.8 × 102.4 in (a) and 51.2 × 25.6 in (b), and for mi/me = 400 in a simula- tion domain of 51.2 × 25.6 in (c). All of the data is taken in the phase where the reconnection rate and the lengths of the region of intense out-of-plane current are stationary. Reconnection forms intense current layers that have a well-defined length (half widths of around 7di and independent of the size of computational domain formi/me = 25) and then open up forming the open out- flow jet that characterizes Hall reconnection [3, 7]. The current layer in the case of mi/me = 400 in Fig. 2c is dis- tinctly shorter than the smaller mass ratio current layers in Fig. 2a,b, suggesting that the length of the electron current layer depends on the electron mass and would be shorter for realistic proton-electron mass ratios. Shown in Fig. 3a is a blow-up around the x-line of the electron outflow velocity vex for the mi/me = 25, 204.8 × 102.4 run corresponding to Fig. 2a. In contrast with the out-of-plane current the electrons form an out- flow jet that extends a very large distance downstream from the x-line. This outflow jet continued to grow in length until the end of the simulation. This simulation, along with others at differing mass ratios, reveals that the peak outflow velocity is very close to the electron Alfven speed [8, 11]. One might expect that because of the colli- mation of the outflow jet and its length, the reconnection rate would drop. However, this is not the case. While there is an intense jet in the core of the reconnection exhaust, the exhaust as a whole quickly begins to open up downstream of the current layer (Jz). The jet itself therefore does not act as a nozzle to limit the rate of FIG. 2: (color online). Blowups around the x-line of the out- of-plane electron velocity for: (a)mi/me = 25, simulation size 204.8× 102.4, (b) mi/me = 25, 51.2× 25.6, and (c) mi/me = 400, 51.2 × 25.6. FIG. 3: (color online). Blowups around the x-line for system size 204.8× 102.4 with mi/me = 25. (a) The electron outflow velocity vex. (b) Momentum flux vectors, Γ = pexzx̂ + peyzŷ (vectors in box surrounding x-line are multiplied by 20), with a background color plot of | (Ez + (ve ×B/c)z)/Ez | . reconnection: the rate of reconnection remains constant even as the length of the outflow jet varies in time. To understand how the electrons can form such an ex- tended outflow jet while the out-of-plane current layer remains localized, we examine the out-of-plane compo- nent of the fluid electron momentum equation along the symmetry line of the outflow direction. In steady state Ez = − mevex vexBy − ∇ · Γ, (1) where ve is the electron bulk velocity, Γ = pexzx̂+peyzŷ is the flux of z-directed electron momentum in the recon- nection plane (not including convection of momentum) with pe the electron pressure tensor. In Fig. 4a we plot all of the terms in this equation along a cut though the x-line along the outflow direction from a simulation with mi/me = 100 and Lx × Ly = 102.4 × 51.2. The data has been averaged between t = 116.2 and t = 117.0. The electric field (black) is balanced by the sum (red) of the electron inertia (dashed blue), the Lorentz force (solid blue) and the divergence of the momentum flux (green). The major contributions to momentum balance come from the Lorentz force and the divergence of the momentum flux. At the x-line the electric field drive is balanced by the momentum transport [1, 12]. The surprise is that the Lorentz force, rather than simply in- creasing downstream from the x-line to balance the re- connection electric field, instead strongly overshoots the reconnection electric field far downstream of x-line. This tendency was seen in earlier simulations [12] but there was no clear separation of scales because of the small size of these earlier simulations. Downsteam from the x-line the electrons are streaming much faster than the mag- netic field lines. Thus, in a reference frame of the moving electrons the z-directed electric field has reversed direc- tion compared with the x-line. This electric field tries to drive a current opposite to that at the x-line. Evidence for this reversed current appears downstream of the x-line in Fig. 2c. In spite of the strength of the effective elec- tric field, the reversed current carried by the electrons is small. As at the x-line, the momentum transfer to electrons in this extended outflow region is balanced by momentum transport. The momentum flux around the x-line is shown as a 2-D vector plot in Fig. 3 for the same run as in (a). The momentum flux has been multiplied by 20 in the box surrounding the x-line. The data for this figure has been averaged between t = 172.5 and 174.5. The background color plot is of | (Ez+(ve×B/c)z)/Ez |, which is & 1 where the electrons are not frozen-in. Evi- dent is the outward flow of momentum around the x-line and the much stronger outward flow of negative momen- tum in an extended downstream region. The momentum transport is so large that the out-of-plane current down- stream is effectively “blocked”. The force associated with this “blocking effect” drives the flow of the large-scale jet of electrons downstream of the x-line. We define the length ∆x of the inner dissipation re- gion as the distance from the x-line to the point where the Lorentz force vexBy/c crosses the reconnection elec- tric field Ez. At this location the effective out-of-plane electric field seen by the electrons reverses sign, causing the electron current jez to be driven in reverse, which allows the separatrices to open up. Thus, the inner dis- sipation region defines the spatial extent of the magnetic nozzle that develops during reconnection. Since the sim- ulations presented in this paper use artificial values of me, it is essential to understand the me scaling of ∆x so that this important length can be calculated for a proton- electron plasma. The momentum equation of electrons in the outflow direction yields a steady state equation for ex) = vezBy, (2) where vez ∼ cAe. Thus, the profile of By along the out- flow direction and its dependence on me must be deter- mined. This profile is shown formi/me = 25 (system size 102.4× 51.2), 100 (102.4× 51.2) and 400 (51.2× 25.6) in Fig. 4b. Surprisingly, the profile of By is apparently in- dependent of me. Our original expectation was because of the continuity of the flow of magnetic flux into and out of the x-line that By ∼ B0vin/cAe ∝ m e , where the outflow velocity eventually rises to cAe. However, since the electrons are not frozen into the magnetic field until far downstream, the expected scaling fails. To calculate vex we approximate By by a linear ramp and integrate Eq. (2). Setting the Lorentz force equation to the re- connection electric field, we then obtain an equation for )3/8 ( )1/2 ( diBy′ di. (3) For the three simulations shown in Fig. 4b the simula- tions yield 2.9di, 1.8di and 1.0di for mi/me = 25, 100 and 400, respectively, which is in reasonable accord with the scaling. Extrapolating to a mass-ratio of 1836, we predict ∆x ∼ 0.6di. In contrast the outer dissipation region can extend to 10’s of di. We have shown that the electron current layer that forms during reconnection stabilizes at a finite length, independent of the periodicity of the simulation domain, and aside from transients from initial conditions remains largely stable to secondary island formation. Reconnec- tion remains fast with normalized reconnection rates of around 0.14. The length of the electron current layer ∆x scales as m e . Since the width δ of the current layer scales with the electron skin depth c/ωpe, the aspect- ratio δ/∆x ∝ (me/mi) 1/8. Extrapolating from our mi/me = 400 simulations to mi/me = 1836 should not significantly change the aspect-ratio and we therefore ex- pect the current layer to remain stable for real mass ra- tios. The structure of the current layer is important to the design of NASA’s magnetospheric multiscale mission (MMS), which will be the first mission with the time reso- lution to measure the electron current layers that develop during reconnection. The length of the out-of-plane elec- tron current layer projects to around c/ωpi for a proton- electron plasma while the the outflow jet, which supports a strong Hall (out-of-plane) magnetic field, extends 10’s of c/ωpi from the x-line. FIG. 4: (color online). Results for simulation size 102.4×51.2 with mi/me = 25 and 100; and 51.2×25.6 with mi/me = 400. (a) Cuts through the x-line of the contributions to Ohm’s law for mi/me = 100. 1 → −me/eve · ∇vez, 2 → −ẑ · ve × B/e, 3 → −ẑ · (∇ ·Pe)/(nee), 4 → sum of 1,2,3. (b) Cuts through x-line of By for the three different mi/me. This work was supported in part by NSF, NASA and Acknowledgments This work was supported in part by NASA and the NSF. Computations were carried out at the National Energy Research Scientific Computing Center. [1] M. Hesse et al., Phys. Plasmas 6, 1781 (1999). [2] M. A. Shay and J. F. Drake, Geophys. Res. Lett. 25, 3759 (1998). [3] J. Birn et al., J. Geophys. Res. 106, 3715 (2001). [4] B. N. Rogers et al., Phys. Rev. Lett. 87, 195004 (2001). [5] W. Daughton et al., Phys. Plasmas 13, 072101 (2006). [6] K. Fujimoto, Phys. Plasmas 13, 072904 (2006). [7] M. A. Shay et al., Geophys. Res. Lett. 26, 2163 (1999). [8] M. A. Shay et al., J. Geophys. Res. 106, 3751 (2001). [9] A. Zeiler et al., J. Geophys. Res. 107, 1230 (2002), doi:10.1029/2001JA000287. [10] J. F. Drake et al., Geophys. Res. Lett. 33, L13105 (2006), doi:10.1029/2006GL025957. [11] M. Hoshino et al., J. Geophys. Res. 106, 25979 (2001). [12] P. L. Pritchett, J. Geophys. Res. 106, 3783 (2001).

0704.0819

Position-Velocity Diagrams for the Maser Emission coming from a Keplerian Ring

Accepted by The Astrophysical Journal Preprint typeset using LATEX style emulateapj v. 03/07/07 POSITION–VELOCITY DIAGRAMS FOR THE MASER EMISSION COMING FROM A KEPLERIAN RING Lucero Uscanga, Centro de Radioastronomı́a y Astrof́ısica, Universidad Nacional Autónoma de México and Apartado Postal 3-72, 58089 Morelia, Michoacán, Mexico Jorge Cantó, Instituto de Astronomı́a, Universidad Nacional Autónoma de México and Apartado Postal 70-264, 04510 México, DF, Mexico Alejandro C. Raga Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México and Apartado Postal 70-543, 04510 México, DF, Mexico Accepted by The Astrophysical Journal ABSTRACT We have studied the maser emission from a thin, planar, gaseous ring in Keplerian rotation around a central mass observed edge-on. The absorption coefficient within the ring is assumed to follow a power law dependence with the distance from the central mass as, κ = κ0r −q. We have calculated position-velocity diagrams for the most intense maser features, for different values of the exponent q. We have found that, depending on the value of q, these diagrams can be qualitatively different. The most intense maser emission at a given velocity can either come mainly from regions close to the inner or outer edges of the amplifying ring or from the line perpendicular to the line of sight and passing through the central mass (as is commonly assumed). Particularly, when q > 1 the position-velocity diagram is qualitatively similar to the one observed for the water maser emission in the nucleus of the galaxy NGC 4258. In the context of this simple model, we conclude that in this object the absorption coefficient depends on the radius of the amplifying ring as a decreasing function, in order to have significant emission coming from the inner edge of the ring. Subject headings: galaxies: individual (NGC 4258) — galaxies: nuclei — masers 1. INTRODUCTION The a priori probability of seeing a thin disk nearly edge-on is very small. It is given by p ≃ 0.125 (h/R)2, where h is the thickness of the disk and R is its radius. Typically h/R ≃ 0.01 and thus p ≃ 1.25 × 10−5. Sur- prisingly however, the maser emission observed in sev- eral cosmic sources has been successfully modeled as coming from a ring or truncated disk in Keplerian ro- tation (around a massive object) seen edge-on. For in- stance: circumstellar disks in star-forming regions as in S255 (Cesaroni 1990) and MWC 349 (Ponomarev et al. 1994), and also circumnuclear disks around black holes of galactic nuclei as in NGC 4258 (Watson & Wallin 1994; Miyoshi et al. 1995). In general, the maser emis- sion from a Keplerian disk observed edge-on produces a triple-peaked spectrum (Elmegreen & Morris 1979); but Ponomarev et al. (1994) showed that, there is a transi- tion from triple- to double-peaked spectra as the width of the amplifying ring decreases. NGC 4258 is a Seyfert 2/LINER located at a dis- tance of 7.2 ± 0.3 Mpc (Herrnstein et al. 1999). The water maser emission (22 GHz) toward this galaxy was first detected by Claussen et al. (1984). Shortly after- wards it was shown that the water masers are confined in a very small region (∼1.3 pc) at the center of NGC 4258 (Claussen & Lo 1986). Subsequently, Nakai et al. (1993) discovered water maser emission with velocity off- Electronic address: [email protected] Electronic address: [email protected] sets ±1000 km s−1 from the already known emission at the galactic systemic velocity of ≃472 km s−1. They sug- gested that the high-velocity emission could arise from masers orbiting a massive central black hole, or ejected in a bipolar outflow. Using the Very Long Baseline Array (VLBA), Miyoshi et al. (1995) simultaneously observed the systemic and high-velocity water maser emission in NGC 4258, finding that the spatial distribution and line- of-sight velocities of the water masers trace a thin molec- ular ring in Keplerian rotation around a massive black hole of 3.6×107 M⊙ seen nearly edge-on. The position- velocity (PV) diagram for the maser emission shows dis- tinct Keplerian orbits (with deviations < 1%) defined by the high-velocity maser emission that arises on the ring diameter perpendicular to the line of sight, as well as a line traced by the systemic maser emission that arises from material on the inner edge of the amplifying ring, this linear dependence is a consequence of the change in the line-of-sight projection of the rotation velocity. By monitoring both systematic and high-velocity wa- ter maser emission of NGC 4258 over periods of sev- eral years with different radio telescopes, a signif- icant centripetal acceleration was observed only for the maser features near the galactic systemic velocity. The systemic maser features drift at a mean rate of ∼9 km s−1yr−1 (Haschick et al. 1994; Greenhill et al. 1995; Nakai et al. 1995; Bragg et al. 2000) while the high-velocity maser features drift by .1 km s−1yr−1 (Greenhill et al. 1995). In a recent spectroscopic study, Bragg et al. (2000) detected accelerations for http://arxiv.org/abs/0704.0819v1 mailto:[email protected] mailto:[email protected] 2 USCANGA, CANTÓ, & RAGA the high-velocity features in the range of −0.77 to 0.38 km s−1 yr−1. These measurements indicate that the systemic water masers lie within a relatively nar- row range of radii, on the near side of the ring at the proximity of its inner edge, while the high-velocity wa- ter masers are located near the ring diameter (between −13.o6 and 9.o3 of the mid-line, Bragg et al. 2000). In addition, the deviation of the high-velocity masers from a straight line passing through the systemic masers in the plane of the sky suggests that the rotating disk is slightly warped (Herrnstein et al. 1996, 1999, 2005). Previously, Watson & Wallin (1994) demonstrated that the maser emission from a rapidly rotating, thin Keplerian ring viewed edge-on can reproduce the general features of the observed 22 GHz radiation from the nu- cleus of NGC 4258, including the high-velocity satellites. However, it is important to point out that their assump- tion of a uniform absorption coefficient within the ampli- fying ring results in a PV diagram for the most intense masers that is qualitatively different from the observed one. While their model predicts that the maser emission at velocities around the systemic velocity of the galaxy comes mainly from the outer edge of the ring, the ob- servations indicate that this emission is actually coming from the inner parts of the truncated disk. In this paper we show that this discrepancy can be resolved if the absorption coefficient decreases with dis- tance from the central mass. The model is presented in §2. The main results are described in §3. Finally, the conclusions are discussed in §4. 2. MODEL We study the maser emission that arises from a thin, planar, gaseous ring in Keplerian rotation around a mas- sive central object when it is observed edge-on. The mas- ing gas is located between R0 and R, the inner and outer radii of the ring, respectively. For simplicity, we assume that the disk is transparent to the maser radiation at radii smaller than R0 and greater than R, although the inner region is probably thermalized due to the higher gas density, and actually it would absorb a significant fraction of the maser radiation produced in the far side of the ring (see §4). The absorption coefficient is assumed to follow a power law function of the distance from the central mass within the amplifying ring as, κ = κ0r The distances are measured in units of R, and the veloc- ities are measured in units of vout, the rotation velocity at the outer edge of the ring (see Figure 1). For the case of an unsaturated maser and neglecting the spontaneous emission, the intensity of the maser ra- diation from a line of sight with impact parameter y at a velocity vr is I(vr, y) = I0e τ(vr,y) , (1) where the optical depth or gain along the line of sight is given by τ(vr , y) = 2 κ0 ∫ xmax (x2+y2)−q/2 exp −(v − vr)2 xmin = r20 − y2 for 0 ≤ |y| ≤ r0 , 0 for r0 < |y| ≤ 1 , xmax = 1− y2 . The line-of-sight velocity component of the gas at the position (x, y) can be expressed as v = y/(x2 + y2)3/4. Here I0 and ∆vD are the background intensity and the Doppler width, respectively, which are supposed to be uniform inside the amplifying ring. The Doppler width ∆vD is related with the FWHM of the velocity distribu- tion of the emitting particles as ∆vD = FWHM/ 4 ln 2. We have numerically solved equations (1) and (2), and we have also calculated the y-positions (impact parame- ters) of maximum maser intensity for each specific value of the velocity vr. When we have found two local max- ima, we have kept both. With this information, we have constructed the PV diagrams using the positions of the observer’s line of sight with maximum emission at each velocity. This way to construct the PV diagrams was previously used by Uscanga et al. (2005). We show the results using the following values for the model parameters which seem to be appropriate for mod- eling maser emission in the galaxy NGC 4258. The back- ground intensity is I0 = 1.3 × 10−5 Jy beam−1, corre- sponding to a radio continuum source with a temper- ature of 106 K (Watson & Wallin 1994). The dimen- sionless inner radius r0 = 0.51, using the estimated val- ues for the inner and outer radii of 4.1 and 8.0 mas re- spectively, given by Miyoshi et al. (1995). The Doppler width ∆vD = 0.007vout which combined with an outer rotation velocity of 770 km s−1 (Miyoshi et al. 1995), gives a Doppler width ≃ 5 km s−1, similar to the value used by Watson & Wallin (1994). We have used some representative values of the exponent q, specifically q = 0, 1/2, 15/8, 5 for Models I, II, III and IV, respec- tively. In Model I, we study the simplest situation of a uniform absorption coefficient. In Model III, we choose q = 15/8, that corresponds to the density dependence with the radius of an accretion disk, i.e., Frank et al. (1992). Finally, in Models II and IV, we explore two other different values of the exponent q in order to study how it changes the results. In all the models, the value for the absorption coefficient κ0 is mainly determined by the requirement that the intensity at the peak of the central component (13 Jy beam−1) is compatible with the observational data when the background intensity is 1.3×10−5 Jy beam−1. Other values of I0, r0, ∆vD, and κ0 give qualitatively similar results. We present the results in the next section; but let us first discuss briefly some important concepts in order to understand these results. In general, the observed emis- sion at a given velocity coming from a specific position in a nebula has contributions of the whole material along the line of sight. However, when the velocity gradient along the line of sight is greater than the dispersion ve- locity (thermal or turbulent) of the emitting material, the main contribution to the emission is actually coming from a narrow region around the point with a line-of- sight velocity equal to the observation velocity. The esti- mated width of the region is 2l, where l is the correlation distance defined as l ≡ ∆vD |dv/dx| , (3) here dv/dx is the line-of-sight velocity gradient. In this approximation, known as Sobolev’s approximation or the approximation of high velocity gradient, the observed in- tensity is given by the following expression I(vr) = I0(vr)e −τ(vr) + S(vr)(1 − e−τ(vr)) , (4) where I0 is the background intensity, S is the source func- tion and τ is the optical depth given by τ(vr) = κ(2l) , (5) where κ is the absorption coefficient. For the case of maser emission, the value of κ is in- trinsically negative and τ is also negative, therefore the factor e−τ(vr) becomes an amplification factor. Because of this reason, the relative contribution at a given veloc- ity of the correlation region is even more important with respect to the remainder of the emitting material than in the case of non-maser emission. Consequently, the ap- proximation given by equations (4) and (5) is suitable for maser emission. As shown in the next section, for a gaseous ring of in- ner radius R0 and outer radius R in Keplerian rotation and seen edge-on, the emission either comes preferen- tially from the inner or outer edges of the ring or from the line perpendicular to the line of sight and passing through the ring center. In the first two cases, it is easy to show that the expected PV diagram will be a straight line. When the emission comes from the outer edge, the slope of the straight line is equal to one (measuring the distances in units of the outer radius of the ring and the velocities in units of the rotation velocity at that point), whereas if the emission comes from the inner edge, the slope of the straight line is equal to 1/r 0 . On the other hand, when the emission arises from the line perpendic- ular to the line of sight, the PV diagram will be a curve with the form 1/y1/2, where y is the impact parameter of the observation (see Figure 2). 3. RESULTS The PV diagrams for the maser emission peak are point-symmetric, consequently we only discuss positive velocities from now on (see Figures 3 and 4). • Model I (q = 0) – With this value of the expo- nent q, we are considering the simplest situation, a uniform or constant absorption coefficient. The strongest maser emission either comes mostly from the outer edge of the ring at velocities lower than 1, or from the mid-line of the ring perpendicular to the line of sight and passing through the cen- tral mass at greater velocities. The filled squares, circles, and triangles mark the regions of strongest maser emission at each velocity. • Model II (q = 1/2) – The results are qualitatively similar to those of Model I. • Model III (q = 15/8) – This value of exponent q corresponds to the density dependence with radius of an accretion disk (ρ ∝ r−15/8). The strongest maser emission either comes mainly from the inner edge of the ring at low velocities (velocities near the systemic velocity), or from the outer edge at velocities close to 1. On the other hand, at ve- locities greater than 1, the most intense emission comes predominantly from the mid-line of the ring perpendicular to the line of sight. • Model IV (q = 5) – The strongest maser emission comes mainly from the inner edge of the ring at velocities lower than 1. At greater velocities, the most intense emission can either come mainly from the inner edge or from the mid-line of the ring per- pendicular to the line of sight. In summary, from the results of Models I–IV (see Fig- ure 3), we found that the most intense maser emission can be around the inner or outer edges of the ring, or the mid-line of the ring perpendicular to the line of sight depending on the velocity and also on the value of q. In fact, the PV diagrams are qualitatively different when q < 1 or q > 1. In the first case, for q < 1 (including the simplest situation with a uniform absorption coeffi- cient, q = 0) and vr < 1, the PV diagram corresponds to a straight line with slope 1; for vr > 1, the diagram corresponds to a Keplerian curve. In the second case, for q > 1 and vr < 1, the PV diagram corresponds to a straight line with a slope that depends on the inner ra- dius of the ring. At velocities close to 1 the slope changes to 1; for vr > 1, the diagram corresponds to a Keplerian curve and also a straight line with a slope that depends on the inner radius under circumstances such as in Model It is also important to realize that when q > 1, the optical depth or gain presents two local maxima within a certain velocity range. Either local maxima may be a global maximum. For vr < 1, the local maximum can be either at the inner and/or outer edges of the ring, while for vr > 1, they are located at the inner edge and/or mid-line of the ring (see Figure 5). As shown in the top panels of Figure 5 (Model III, q = 15/8), the relative difference between the two lo- cal maxima is not very significant. However, when the value of q is higher (like in Model IV, q = 5 shown in the bottom panels) the relative difference becomes more important. In order to estimate the velocity vc at which the global maximum of the optical depth changes its locus, we have calculated analytical approximations for the largest value of the optical depth or gain that corresponds to the max- imum intensity at the inner and outer edges of the ring, and also at the mid-line of the ring perpendicular to the line of sight. The detailed calculations are presented in the Appendix. The following equations give the local maximum depth as a function of the velocity in each neighborhood τ(vr) ≃ πκ0∆vD 1− r0v2r inner edge , (6) τ(vr) ≃ πκ0∆vD 1− v2r outer edge , (7) τ(vr) ≃ r mid-line . (8) The velocity vc is estimated by combining equations (6) and (7), or equations (6) and (8) according to the value of vc (when vc < 1 or vc > 1, respectively). The results are 2(1−q) 0 − 1 2(1−q) 0 − r0 for vc < 1, (9) 4 USCANGA, CANTÓ, & RAGA 2(1−q) 0 − v c + r0v c = 0 for vc > 1, (10) which are presented in Figure 6 for some representative values of the exponent q. The bottom plot of Figure 6 shows vc as function of the inner radius r0 for different values of q, from equation (9a). For q < 1, there is no solution to equation (9a). When q = 1, vc = 0 for any value of r0. That is, the optical depth has a maximum and its locus is around the outer edge of the ring, and vc is meaningless as we have defined it. When q > 1, the optical depth presents two local maxima, and vc is different from zero and its value depends on r0. This velocity corresponds to the value at which the locus of the global maximum changes from the inner edge to the outer edge of the ring. As a consequence, there is a slope change in the PV diagrams at velocities lower than the rotation velocity at the outer edge of the ring. For instance, when q = 15/8 and r0 = 0.51, the slope change occurs at vc = 0.906. In other words, the locus of the global maximum of the optical depth changes from the inner to the outer edge of the ring at this velocity vc. The top plot of Figure 6 also shows vc as function of the inner radius r0 using specific values of q and ∆vD in equation (9b); in this case, 5 and 0.007vout, respectively. As an example, when ∆vD = 0.007vout, q = 5 and r0 = 0.51, then vc = 1.077. Stated differently, at that velocity vc, the largest value of both the optical depth and the intensity changes its locus from the inner edge to the mid-line of the ring perpendicular to the line of sight. The remarkable water maser emission in the nucleus of the galaxy NGC 4258 traces a PV diagram where the detected emission around the systemic velocity of the galaxy comes from the inner edge of the amplifying ring; this emission delineates a straight line just as the straight line that connects points C and D in Figure 2 (see Figure 3 of Miyoshi et al. 1995). According to our model re- sults, this implies that the absorption coefficient within the molecular ring of NGC 4258 is not uniform, instead it must be a decreasing function of the distance from the central mass, i.e., κ = κ0r −q with q > 1. Moreover, the observed red/blue-shifted emission at high velocities that arises from the mid-line of the ring perpendicular to the line of sight traces a Keplerian curve such as is indicated by the model results (see Figure 7). Simply stated, when q > 1 the PV diagram is qualitatively similar to the one observed for the water maser emission detected in the nucleus of NGC 4258. As an example, in Figure 7 we show a comparison be- tween the results of Model III (q = 15/8) and the water maser emission in NGC 4258. The detected emission arises from the inner edge of the amplifying ring and the mid-line perpendicular to the line of sight. The locus of the observed maser emission coincides with the locus of the most intense maser emission as indicated by the sizes of the circles in Figure 7. The model results indicate that there is emission coming from the outer edge of the ring at velocities close to 1, nevertheless the sizes of the circles indicate that this emission is very weak. Maybe maser emission is not detected from this locus for this reason. Additionally, our model results also indicate that the intense maser emission at the inner edge of the ring ex- tends neither to velocities very different from the sys- temic velocity nor to impact parameters very different from zero, as is indicated by the size of the circles in the PV diagram shown in Figure 7. Furthermore, accord- ing to the size of the circles, the other locus of intense maser emission is the mid-line of the ring perpendicular to the line of sight, precisely the locus of the red/blue- shifted maser emission at high velocities that describes Keplerian curves in the PV diagram. 4. DISCUSSION AND CONCLUSIONS In our model, we have assumed that the gas in the re- gion inside the masing ring is transparent to the maser radiation. This implies that the most intense maser emis- sion at low velocities (velocities near the systemic veloc- ity) comes mainly from the outer edge of ring (for q < 1) or from the inner edge (for q > 1), either the near or far side of the ring, as is indicated in Figure 4. Mea- surements of positive acceleration of the maser emission around the systemic velocity show that this emission cer- tainly comes from the near side of the ring at the prox- imity of its inner edge (e.g., Greenhill et al. 1995). If we suppose that the gas inside the masing ring is thermal- ized probably due to its higher density then an important fraction of the maser emission from the backside of the ring would be absorbed and the detected emission would come from the front side of the ring at the outer or inner edge depending on the value of q. For instance, consider- ing absorption and emission from the gas located inside the masing ring, the difference in the intensity for a line of sight that passes through both the inner absorbing region and the front side of the masing ring from the in- tensity for a line of sight that passes through both the backside of the masing ring and the inner absorbing re- gion is S(1−e−τ2)(eτ1−1) where S is the source function of the gas inside the masing ring, τ2 is the optical depth in this region, and τ1 is the optical depth for the front side of the ring. If τ2 >> τ1, then the detected emission would be the radiation amplified by the front side of the masing ring. Also, we have made a simplifying assumption about the geometry of the masing ring in NGC 4258, consider- ing that the amplifying ring is strictly flat. Despite the observations indicate an apparent warp in the maser dis- tribution of this galaxy, Kartje et al. (1999) presented a model in which the disk does not require to be physically warped in order to the masing gas become exposed to the central continuum radiation. In this scenario, dusty clouds provide the shielding of the high-energy contin- uum, which is required for the gas to remain molecu- lar. They found that a flat-disk model of the irradiated ring could be applied to a source like NGC 4258 only if the water abundance is higher than the value implied by equilibrium photoionization-driven chemistry. A very important result from their study (based on radiative and kinematic considerations) was that, even if the disk in NGC 4258 is warped, the maser-emitting gas must be clumpy, instead of homogeneous as in the scenario pre- viously proposed by Neufeld & Maloney (1995). An important result of our model shows that the as- sumption, commonly used, that considers a uniform or constant absorption coefficient within the masing ring in Keplerian rotation around the nucleus of NGC 4258 is not appropriate. For example, Wallin et al. (1998) supposed that κ was constant considering that the locus of the maser emission from NGC 4258 was determined mainly by the velocity gradients in a Keplerian velocity field indicating some uniformity of κ, at least on length scales comparable to the coherence or correlation length resulting from the Keplerian velocity gradients. On the contrary, from our analysis, we conclude that a constant absorption coefficient would result in a PV diagram qual- itatively different from the observed one, since the most intense maser emission would come predominantly from a narrow region close to the outer edge of the ring instead of a narrow region close to the inner edge of the ring, as indicated by the observations. Necessarily, the absorp- tion coefficient must be a decreasing function of distance from the central mass (i.e., κ = κ0r −q with q > 1) to have significant emission coming from the inner edge of the amplifying ring and hence explain the form of the PV diagram delineated by the water masers in NGC 4258. When comparing our edge-on disk model with the observations of NGC 4258, it is clear that we need a κ ∝ r−q radial dependence for the absorption coefficient with q > 1 (so as to favour the emission from the in- ner edge of the disk, see above) in order to reproduce the observations. In reality, the fact that the disk of NGC 4258 is warped introduces geometrical effects which might favour the inner disk edge emission (over the one of the outer edge). One will need to compute more complex, 3D transfer models to see whether or not these geomet- rical effects are sufficient to explain the PV diagrams of the NGC 4258 masers without introducing the radially dependent absorption coefficient which is required by the edge-on disk models described in the present paper. J. C. and A. C. R. acknowledge support from CONA- CyT grants 41320 and 43103, and DGAPA-UNAM. L. U. acknowledges support from DGAPA-UNAM. We sin- cerely thank J. M. Torrelles and Y. Gómez for useful comments, which contributed to improve an earlier ver- sion of this manuscript. L. U. gives special thanks to M. R. Pestalozzi and M. Elitzur for valuable comments on this work. We also thank an anonymous referee for helpful comments on the manuscript. APPENDIX ANALYTICAL APPROXIMATIONS FOR THE OPTICAL DEPTH In this appendix, we describe how to obtain the analytical approximations for the optical depth given by equations (6)–(8). First, we define w = (v − vr)/∆vD then we can change the variables in equation (2) and rewrite it as τ(vr , y) = 2κ0 ∫ wmax (x2 + y2)−q/2 exp(−w2)dw , (A1) where wmin = 0 − vr wmax = y − vr Note that wmin > wmax since r0 ≤ 1. Using the expressions for the line-of-sight velocity v = y/(x2 + y2)3/4 and the previously defined variable w = (v− vr)/∆vD, we can write x = (y/(vr + w∆vD)) 4/3 − y2 . At zero order around w = 0 we obtain (x2 + y2)−q/2 ≃ − 2∆vD(y/vr) 2(2−q)/3 (y/vr)4/3 − y2 . (A2) Additionally ∫ wmax exp(−w2)dw = erf(wmax)− erf(wmin) , (A3) where erf(w) is the error function, defined as erf(w) ≡ (2/ exp(−t2)dt. Finally, substituting equations (A2) and (A3) into (A1), we obtain the approximation for the optical depth τ(vr , y) ≃ πκ0∆vD(y/vr) 2(2−q)/3 (y/vr)4/3 − y2 erf(wmin)− erf(wmax) . (A4) Around the inner edge of the ring, vr ≃ y/r3/20 and the maximum value of [erf(wmin)− erf(wmax)] = 2. Substituting these approximations into equation (A4), we obtain equation (6). Similarly, around the outer edge of the ring, vr ≃ y, and the maximum value of [erf(wmin)− erf(wmax)] also equals 2. Then, equation (A4) reduces to equation (7) for the optical depth at the outer edge of the ring. In order to find an approximation for the local maximum of optical depth at the mid-line of the ring perpendicular to the line of sight, we expand the expression for the velocity along the line of sight around x = 0 to obtain v ≃ 1 , (A5) 6 USCANGA, CANTÓ, & RAGA therefore = −∆vD , (A6) and thus 5/4 . 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The masing gas exists between radii R0 and R. At radii smaller than R0 and greater than R the disk is transparent to the maser radiation. The observer is on the plane of the disk. All the distances are measured in units of R, the outer radius of the amplifying ring; therefore the variables x, y, and r0 are dimensionless. 8 USCANGA, CANTÓ, & RAGA v = y 1/2v = 1/y v = y/r0 Fig. 2.— PV diagram for the maser emission of a gaseous ring with inner radius R0 and outer radius R in Keplerian rotation observed edge-on. The straight line that connects points A and B has a slope equal to 1, while the straight line that connects points C and D has a slope equal to 1/r Fig. 3.— From top to bottom results of Models I, II, III, and IV. The left panels show PV diagrams for the maser emission peak. The filled squares, circles, and triangles represent the strongest maser emission which is coming from regions a, b, and c, respectively, indicated in Figure 4. The straight lines or curves represent the velocity dependences of the regions where this emission arises. The central panels show PV diagrams for the maser emission peak. Because of the point-symmetric shape of these diagrams, only positive velocities are shown. The radii of the open circles are proportional to the maximum maser intensity at each position and velocity. The right panels show the logarithm of the ratio between the maximum intensity and the background intensity as a function of the velocity. 10 USCANGA, CANTÓ, & RAGA :a v = y b : v = 1 / y x x x b b b Observer Observer Observer Model IIIModels I and II Model IV :c v = y / r0 Fig. 4.— Schematic representation of the results of Models I, II, III, and IV. The filled squares, circles, and triangles represent the most intense maser emission that is coming from regions a, b, and c, respectively. These regions are very narrow because the correlation distance is very small; that is, the width ∆vD is much smaller than the line-of-sight velocity gradient. Besides, the exponential amplification of the intensity emphasizes small changes in the optical depth. Fig. 5.— Left : PV diagrams for the maser emission in grey scale with intensity contours overlaid. The darker regions show the locus of the strongest emission in these diagrams, which potentially could be detected, depending on the sensitivity cutoff of the observations. Right : Close-up to the PV diagrams showing the maser emission at low velocities. 12 USCANGA, CANTÓ, & RAGA Fig. 6.— vc as function of the inner radius of the ring, r0. Bottom: For vc < 1, it is computed from equation (9a) for some representative values of q. For q < 1, there is no solution to equation (9a). Top: For vc > 1, it is computed from equation (9b). This is the solution to equation (9b), using ∆vD = 0.007vout and q = 5. Fig. 7.— Comparison between the calculated PV diagram for the maser emission peak in Model III (q = 15/8) and the PV diagram delineated by the water masers observed in NGC 4258 (Miyoshi et al. 1995). The radii of the open circles are proportional to the maser intensity at each position and velocity. The dots represent the observed maser spots in NGC 4258. We have subtracted the ring systemic velocity of 476 km s−1 from the observed local standard of rest velocity of the maser spots in order to compare the observed PV diagram with the modeled one. The positions and velocities are in units of the outer radius of the ring (8 mas) and the rotation velocity at the outer edge of the ring (770 km s−1), respectively.

0704.0820

Coupling of whispering-gallery modes in size-mismatched microdisk photonic molecules

Microsoft Word - Boriskina_OL2007.DOC Coupling of whispering-gallery modes in size-mismatched microdisk photonic molecules Svetlana V. Boriskina School of Radiophysics, V. Karazin Kharkov National University, Kharkov 61077, Ukraine Mechanisms of whispering-gallery (WG) modes coupling in microdisk photonic molecules (PMs) with slight and significant size mismatch are numerically investigated. The results reveal two different scenarios of modes interaction depending on the degree of this mismatch and offer new insight into how PM parameters can be tuned to control and modify WG-modes wavelengths and Q-factors. From a practical point of view, these findings offer a way to fabricate PM microlaser structures that exhibit low thresholds and directional emission, and at the same time are more tolerant to fabrication errors than previously explored coupled-cavity structures composed of identical microresonators. © 2007 Optical Society of America. OCIS codes: 130.3120, 140.3410, 140.4780, 140.5960, 230.5750, 260.3160 During the last decade, photonic molecules1 (clusters of electromagnetically-coupled optical microcavities) have gone a long way from a useful illustration of parallels between behavior of photons and electrons and now hold promise of new insights into physics of light-matter interactions and also of a variety of practical applications, including microlasers, tunable filters and switches, coupled-cavity waveguides, sensors, etc2-10. The simplest PM composed of two identical optical microcavities has been widely used as a test-bed to demonstrate shift and splitting of cavity modes and formation of a spectrum of bonding and anti-bonding PM supermodes1- 4. I have recently shown how arranging identical WG-mode microdisks into pre-designed high-symmetry configurations yields quasi-single-mode PMs with dramatically increased Q- factors6, enhanced sensitivity to the environmental changes7, and/or directional emission patterns8. In all these structures, size uniformity of microcavities is an important issue in successful realization of PM-based optical components. The motivation for studying interactions of optical modes in a photonic molecule with size mismatch9,10 stems from two sources. First, precise and repeatable fabrication of identical microcavities, which in many cases are tiny structures having just several microns in diameter, is highly challenging. Second, a systematic study of double-cavity PMs with various degrees of cavities size mismatch can reveal new mechanisms of manipulating their optical properties thus paving way to improving or adding new functionalities to PM-based photonic devices. Such study has never been performed before, and is a focus of this letter. Despite its simplicity, the double-cavity structure provides useful insight into the general mechanisms of WG-modes coupling and offers new design ideas for more complex structures. The Muller boundary integral equations framework previously developed by the author to model PMs composed of identical cavities7 has been modified to study size-mismatched PMs. In the following, the term “microcavity mode” encompasses a complex value of the mode eigenfrequency and the corresponding eigenvector (i.e., modal spatial field distribution). The PM under study is composed of a pair of side- coupled microdisks of radii RA, RB and refractive indices nA, nB separated by an airgap of width w (Fig. 1a). The microdisks of thicknesses much smaller than their diameters are considered. Thus, instead of the 3-D boundary value problem for the Maxwell equations, we solve its 2-D equivalent. In the following analysis, we search for the TE (transverse-electric) WG-modes, which are dominant in thin disks. At wavelength λ = 1.521 μm, a 2-D cavity with radius 1.1 μm and effective refractive index 63.2TE =effn (2-D equivalent of a 200 nm-thick GaInAsP disk)2 supports WG8,1-mode with one radial field variation and eight azimuthal field variations (Q = 5243). This mode (like all other WG-modes in circular cavities) is double- degenerate due to the symmetry of the structure. WG-mode degeneracy is removed if two (or more) cavities are brought close together1-9, and four non- degenerate WG-supermodes of different symmetry appear in the double-disk PM spectrum instead of every WG-mode of an isolated cavity. Fig. 2 (b and c) shows the wavelength migration and Q-factors change of these modes with the change of the radius of one of the cavities. The modes are labeled according to the symmetry of their field patterns along the y- and x- axes, respectively (e.g., OE supermode has odd symmetry with respect to y-axis and even symmetry with respect to x-axis). OE and OO modes are termed “anti-bonding” modes, while EO and EE modes are termed “bonding” ones. Bonding and anti-bonding supermodes group into nearly-degenerate doublets as seen in Fig. 1b. The values of real parts of eigenfrequencies of two modes in a doublet are so close to each other that they cannot be distinguished (Fig. 1b), while their imaginary parts differ, resulting in different Q-factors of these supermodes (Fig. 1c). Thus, in practice only two peaks are observed in a symmetrical double-cavity PM lasing spectrum (see Fig. 2 in Ref. 9), where the narrow high- intensity peak corresponds to the high-Q anti-bonding mode doublet, and the wider low-intensity peak corresponds to the bonding one. 0.8 1.0 1.2 1.4 (single disk) Radius of microdisk B, R (μm) w (a) RA=1.1 μm RB Fig 1. (a) A geometry of a PM composed of two microdisks of different radii; (b) wavelengths migration and (c) Q-factors change of PM supermodes as a function of the radius of disk B (RA = 1.1 µm, w = 400 nm). The insets show the magnetic field distribution of the bonding (EE) and the anti- bonding (OE) WG8,1 supermodes in the symmetrical (RA = RB = 1.1 µm) PM. Here and thereafter, corresponding characteristics of the WG8,1 mode of an isolated cavity with radius 1.1 µm are plotted for comparison (dash-dot line). 1.05 1.10 1.15 1.05 1.10 1.15 Superm ode Q -factor Radius of microdisk B, R (μm) Fig. 2. Supermodes wavelengths (left) and Q-factors (right) in the vicinity of anti-crossing point AC1 (RB = RA). Mode switching (see the modal near-field distributions at points A and B shown in the insets) at the anti-crossing point and Q-factor enhancement of one of the supermodes can be observed. Careful study of Figs. 1 b,c reveals a number of so-called exceptional points (corresponding to certain values of the varied parameter), where PM supermodes couple following either crossing (C) or avoided crossing (AC) scenarios. The behavior of wavelengths and Q-factors of the four supermodes in the vicinity of these exceptional points is shown in more detail in Figs. 2-4 (for the points AC1, AC2, and C1, respectively). The phenomenon of coupling of complex eigenvalues of matrices dependent on parameters under the change of these parameters is of a general nature and is observed in many physical systems11. Usually, frequency anti-crossing (crossing) is accompanied by crossing (anti-crossing) of the corresponding widths of the resonance states. Furthermore, at the points of avoided frequency crossing (points AC1-AC3 in Fig. 1), eigenmodes interchange their identities, i.e., Q-factors and field distributions. 0.89 0.90 0.91 0.92 0.89 0.90 0.91 0.92 Superm ode Q -factor Radius of microdisk B, R (μm) Fig. 3. Supermodes wavelengths (left) and Q-factors (right) in the vicinity of anti-crossing point AC2. Wavelengths repulsion accompanied by the linewidths crossing is observed. The insets demonstrate mode switching at the anti-crossing point (the modal near-field distributions shown in the upper(lower) insets correspond to the complex frequency values at the points labeled as A and D (B and C), respectively. In the context of coupling of WG-modes in microcavities, this interchange offers exciting new prospects for manipulating the PM optical characteristics, e.g. for realization of optical flip-flops9. For example, the coupling of modes with avoided frequency crossing scenario observed in Figs. 2 and 3 makes possible switching of field intensity between two microdisks. To realize such switching in practice, carrier-induced refractive index change of one of the disks induced by nonuniform pumping can be used. This effect was observed experimentally11 in a PM composed of nearly-identical microdisks (similar to the case shown in Fig. 2). If the microcavities are severely size- mismatched, their WG-modes couple with the frequency crossing scenario. This situation is demonstrated in Fig. 4, and the numerical data indicate that such coupling may spoil significantly the Q-factors of the high-Q modes in the larger microdisk. However, optical coupling between cavities with strongly detuned WG-modes makes possible broad spectral transmission effects in coupled resonator optical waveguides (CROWs)10, coupled-resonator induced transparency12, and significant reducing of CROW bend radiation loses13. 0.72 0.75 0.78 0.72 0.75 0.78 Superm ode Q -factor Radius of microdisk B, R (μm) Fig. 4. Supermodes wavelengths (left) and Q-factors (right) in the vicinity of the crossing point C1. Wavelength crossing accompanied by damping of the high-Q supermodes is observed. The insets show supermodes near-field portraits at the crossing point. 0.4 0.6 0.8 1.473 1.474 0.4 0.6 0.8 5x103 6x103 7x103 8x103 9x103 Superm ode Q -factor Disk-to-disk separation (μm) Fig. 5. Resonance wavelength (left) and Q-factor (right) of an anti-bonding WG-supermode in a three-disk PM. The central disk of radius 1.065 μm is separated from the side disks of radii 1.1 μm by airgaps of 400 nm width. Supermode near-field portrait and directional far-field emission pattern at the point labeled as A are shown in the inset. Finally, enhancement of the Q-factor of a single supermode in a double-disk PM in comparison to its single- cavity value can be observed in Fig. 2 in a relatively wide range of cavities radii detuning: 14 nm < ΔR < 53 nm (ΔR = |RA - RB|). Note that all the other PM supermodes have significantly lower Q-factors in this range of the parameter change. This effect offers a way for selective enhancement of the Q-factor of a single supermode that (unlike symmetry- enhanced Q-factor boost in polygonal PMs)6,7 does not rely on exact cavity size-matching. A possible realization of a PM-based structure designed by making use of this mechanism of selective mode enhancement is presented in Fig. 5. It consists of three coupled microcavities, with the central cavity radius detuned by ΔR = 35 nm from the side cavities radii. By adjusting the width of the airgaps between microcavities, noticeable Q-factor enhancement of one anti- bonding supermode is achieved without shifting the supermode wavelength (Fig. 5). Furthermore, such PM demonstrates directional light emission, which cannot be achieved in isolated WG-mode microdisks (see inset to Fig. 5). Our studies also indicate that this directional emission pattern is preserved if the disk-to-disk distance is varied. It should also be noted that other system parameters can be tuned to manipulate resonance wavelengths and Q- factors of microcavities through mode coupling at exceptional points. Among these are: the refractive index of the cavity substrate, and the size and/or position of a hole pierced in the cavity, which can be adjusted to enhance a WG-mode Q- factor14,15 or to achieve directional emission on a high-Q WG-mode16. In summary, a comprehensive numerical study was performed to elucidate the mechanisms of modes coupling in PMs with various degrees of cavities size mismatch. The study offers an alternative approach to design novel PM- based components with improved functionalities. In contrast to PM structures composed of identical cavities that may require fabrication accuracy beyond the capabilities of modern technology, the proposed approach does not rely on precise cavities size-matching to achieve the desired device performance. This approach paves the way for new designs of more complex PM structures and arrays, which may eventually lead to new capabilities and applications in micro- and nano-photonics. I wish to thank Vasily Astratov for discussions and Jan Wiersig for bringing his recent paper16 to my attention. This work has been partially supported by the NATO Collaborative Linkage Grant CBP.NUKR.CLG 982430. Svetlana Boriskina’s e-mail address is [email protected]. References 1. M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, and P. A. Knipp, Phys. Rev. Lett. 81, 2582-2586 (1998). 2. A. Nakagawa, S. Ishii, and T. Baba, Appl. Phys. Lett. 86, 041112 (2005). 3. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, IEEE J. Select. Topics Quantum. Electron. 12(1), 78-85 (2006). 4. Y. P. Rakovich, J. F. Donegan, M. Gerlach, A. L. Bradley, T. M. Connolly, J. J. Boland, N. Gaponik, and A. Rogach, Phys. Rev. A 70, 051801(R) (2004). 5. B. Moller, U. Woggon, and M. V. Artemyev, J. Opt. A: Pure Appl. Opt. 8, S113–S121 (2006). 6. S. V. Boriskina, Opt. Lett. 31(3), 338-340 (2006). 7. S. V. Boriskina, J. Opt. Soc. Am. B 23(8), 1565-1573 (2006). 8. S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, IEEE J. Select. Topics Quantum Electron., 12(6), (2006). 9. S. Ishii, A. Nakagawa, and T. Baba, IEEE J. Select. Topics Quantum. Electron. 12(1), 71-77 (2006). 10. A. V. Kanaev, V. N. Astratov, and W. Cai, Appl. Phys. Lett. 88, 111111 (2006). 11. W. D. Heiss, Phys. Rev. E 61, 929–932 (2000). 12. D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, Phys. Rev. A 69, 063804 (2004). 13. S. V. Pishko, P. Sewell, T. M. Benson, and S. V. Boriskina, submitted to J. Lightwave Technol. (2007). 14. S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, J. Lightwave Technol. 20(8) 1563-1572 (2002). 15. X.-S. Luo, Y.-Z. Huang, and Q. Chen, Opt. Lett. 31(8), 1073-1075 (2006). 16. J. Wiersig and M. Hentschel, Phys. Rev. A 73, 031802 (2006).

0704.0821

Spin solid phases of spin 1 and spin 3/2 antiferromagnets on a cubic lattice

Spin solid phases of spin 1 and spin 3/2 antiferromagnets on a cubic lattice. Karol Gregor and Olexei I. Motrunich Department of Physics, California Institute of Technology, Pasadena, CA 91125 (Dated: October 30, 2018) We study spin S = 1 and S = 3/2 Heisenberg antiferromagnets on a cubic lattice focusing on spin solid states. Using Schwinger boson formulation for spins, we start in a U(1) spin liquid phase proximate to Neel phase and explore possible confining paramagnetic phases as we transition away from the spin liquid by the process of monopole condensation. Electromagnetic duality is used to rewrite the theory in terms of monopoles. For spin 1 we find several candidate phases of which the most natural one is a phase with spins organized into parallel Haldane chains. For spin 3/2 we find that the most natural phase has spins organized into parallel ladders. As a by-product, we also write a Landau theory of the ordering in two special classical frustrated XY models on the cubic lattice, one of which is the fully frustrated XY model. In a particular limit our approach maps to a dimer model with 2S dimers coming out of every site, and we find the same spin solid phases in this regime as well. PACS numbers: I. INTRODUCTION A simple, nontrivial, and physically common example of a regular system of quantum objects is a collection of spins on a lattice. This is easiest to analyze if the in- teractions do not compete and all prefer the same spin state; the resulting phases have been known for a long time and include ferromagnetic and Neel states. A much richer situation of current interest is when interactions compete. The frustration together with quantum fluctu- ations can destroy the magnetic order and produce spin solid or spin liquid phases. In a spin solid, spins combine into larger singlet objects such as valence bonds which form an ordered pattern on a lattice. Such phases have been found in nature,1,2,3 and also in numerical studies of model Hamiltonians.4,5,6 A spin liquid, on the other hand, is a featureless paramagnet, which can be crudely viewed as a quantum superposition of many valence bond configurations, thus the name “resonating valence bonds” (RVB) state. So far there are only few experimental can- didates, but on the theoretical side the existence of spin liquids in many varieties and our understanding of them is well established (see Ref. 7 for a recent collection of references and also a very recent example of the so-called Coulomb phase in 3d, which is the spin liquid relevant to the present work). In this paper we look for natural spin solid phases of spin 1 and spin 3/2 on a cubic lattice. A direct study of spin Hamiltonians that can stabilize such phases is diffi- cult but can be done in some cases with Quantum Monte Carlo. Which phases are realized will of course depend on the specific model: For example, Refs. 4,5 found valence bond solids in spin 1/2 systems with ring exchanges on the square and cubic lattices. Refs. 6,8 found spin solid phases for a spin 1 model with biquadratic interaction on the anisotropic square lattice, but only magnetically ordered phases on the isotropic square and cubic lattices. Here we follow instead a more phenomenological approach.9,10,11,12 A systematic and commonly used route to achieve this, and the one we start with, is to generalize the spins to a representation of higher symme- try group, here taken to be SU(N).9,10 The problem can be solved exactly in the N → ∞ limit and one can con- sider fluctuations around this limit to get long distance properties of the system. This approach, while difficult to connect with the actual microscopic SU(2) spin system, nevertheless gives us some guidance about what phases to expect and gives us a form of the effective field theory. Here it results in a gauge theory which naturally exhibits deconfined (liquid) and confined (solid) phases, and we expect that if a microscopic spin system has such phases, they should be described by this theory. FIG. 1: The most natural spin solid phase for S = 1 on the cubic lattice. The thick lines denote links with large spin-spin correlations suggesting that the spins organize into Haldane chains along one lattice direction. One of the spin liquid phases expected in 3d is the so-called Coulomb phase. It is a compact U(1) gauge theory coupled to matter in the deconfined phase, where the matter fields (spinons) are gapped, gauge field (emer- gent photon) is gapless, and monopoles (which arise due http://arxiv.org/abs/0704.0821v1 FIG. 2: The most natural spin solid phase for S = 3/2 on the cubic lattice. The drawn bold lines denote links with large spin-spin correlations suggesting that the spins organize into ladders. to compactness) are gapped. In addition, importantly, there are spin Berry phases that lead to the presence of a background charge in the gauge theory formulation. This makes the confined phases nontrivial in that they break lattice symmetries and therefore correspond to various spin solids. The transition occurs because the monopoles condense, and the theory can be equivalently analyzed in terms of them by employing standard electro-magnetic duality. The background charge causes monopoles to acquire a phase when they hop around a plaquette.12 This leads to a nontrivial monopole condensation pat- tern, which then corresponds to a spin solid phase. In 2d the physics is similar, except that the monopoles are instantons and they always proliferate, so there is no Coulomb spin liquid. This approach was first used by Read and Sachdev10 on the square lattice. The spin solids for spin 1/2 on the cubic lattice were analyzed in Ref. 12 and near several different Coulomb spin liquids in Ref. 13. Ref. 14 was led (in a different context) to a gauge theory with background charges on a diamond lattice which was attacked using analogous techniques. For the spins on the cubic lattice, the analysis depends only on the spin magnitude. Any case can be mapped onto S = 0, 1/2, and 1 in 2d and S = 0, 1/2, 1, and 3/2 in 3d. Only the spin 1/2 case was considered so far, but these results cannot be transferred to the other spins since each requires a separate analysis. This is the task of the present work. We find that the most nat- ural phases for spin 1 and 3/2 are the ones shown on Figures 1 and 2. In the S = 1 case the spins organize into Haldane chains. This is easiest to understand in the standard picture where we break spin 1 into two spin 1/2’s and form singlets with spin 1/2’s of spins on either side. Similarly, in the S = 3/2 case we break spin 3/2 into three spin 1/2’s and form singlets on the bonds of the ladders. Several approaches that we have taken and used in different parameter regimes suggest the same spin solid states, which gives us confidence that these phases are very natural in the two cases. II. SCHWINGER BOSONS, DUAL REFORMULATION, AND A BASIC PHASE DIAGRAM A. Schwinger bosons We begin by briefly reviewing the standard technique of large N for spins.9,10 This maps (approximately) our spin system into a theory of spinons coupled to a U(1) gauge field in the presence of static background charges. Our main work is the analysis of this theory, while the purpose of the review here is to establish the connection with the properties of the original spin system. The basic steps in the derivation are as follows. We generalize the SU(2) spin to SU(N) spin and denote it by Sβα(i). We write the spins in terms of Schwinger bosons: Sβα(i) = b α(i)b β(i) sublattice A , Sβα(j) = −b̄β†(j)b̄α(j) sublattice B , (1) where the b, b̄’s are bosonic operators that transform un- der the fundamental representation of SU(N) if the index is on the top and under its conjugate if the index is on the bottom. To get the Hilbert space of the spins we need to restrict the boson occupations as b†α(i)b α(i) = nc , b̄α†(j)b̄α(j) = nc , (2) where nc corresponds to the spin length. The SU(N) spin Hamiltonian is 〈i,j〉 Sβα(i)S β (j) , (3) which reduces to the SU(2) Heisenberg spin model for spin S when N = 2 and nc = 2S. Next we write the system in the path integral pic- ture, imposing the constraints (2) by Lagrange multi- pliers. The spin interaction contains quartic terms; to get action that is quadratic in the boson fields, we use Hubbard-Stratonovich transformation and obtain b†α(i) + iλ(i) bα(i)− iλ(i)nc b̄α†(j) + iλ(j) b̄α(j)− iλ(j)nc 〈i,j〉 |Qij |2 −Q∗ijbα(i)b̄α(j) + h.c. (4) The path integral goes over b, b̄, Q, λ. We can now integrate out the b’s. The resulting ex- pression will have coefficient N in front of it. At large N it can be approximated by its saddle point value. Our departing point is such a “mean field” with uni- form Qr,r+m̂(τ) = Q̄ and λ(r, τ) = λ̄ and assuming gapped b spectrum; this represents a Coulomb spin liq- uid, which is a stable phase in three dimensions. The effective theory is obtained by considering the fluctua- tions of the fields, Qr,r+m̂(τ) = [Q̄ + qm(r, τ)]e iαm(r,τ) and λ(r, τ) = λ̄ + iα0(r, τ). Here r runs over all sites of the cubic lattice and m̂ = x̂, ŷ, ẑ denotes one of the direc- tions in 3d. The amplitude fields qm are massive, and so are the fields αm and α0 near the wavevector (0, 0, 0). On the other hand, the fields αm and α0 near the wavevector (π, π, π) are massless and describe the gauge field (pho- ton) of the Coulomb phase, am ≡ α(π,π,π)m , aτ ≡ α(π,π,π)0 . For details of the derivation, see the original Ref. 10 (our notation is slightly different compared to these papers, which use a two-site unit cell labeling instead). As emphasized in Refs. 10,11, we also have to con- sider effect of Berry phases, which is crucial for the un- derstanding of the spin solid states. A very convenient encapsulation of the low-energy degrees of freedom and the Berry phases is provided by the following re-latticized Euclidean action:15,16 Daiµe −Sa−SB , (5) Sa = −β i,µ<ν cos(∇µaν −∇νaµ) , SB = i ηiaiτ . Here we have a compact U(1) gauge field a residing on the links of a (3+1)d space-time lattice and described by the action term Sa. The SB term comes from detailed consideration of the Berry phases, and ηi is 2S on one sublattice of the spatial lattice and −2S on the other one. In the Hamiltonian language this has a simple in- terpretation as a background charge of value 2S on one sublattice and −2S on the other one: H = u E2rm − κ r,m<n cos(∇man −∇nam) , (6) (∇ ·E)r = ηr = ±2S , (7) where Em are electric fields residing on the links of the 3d cubic lattice and conjugate to am. Thus we obtained a compact U(1) gauge theory in the presence of background charge.17,18,19 Throughout, we will assume the spinons are gapped and are integrated out. Note that even though we start in the Coulomb phase where the gauge field is decon- fined, the above action also provides access to confining paramagnetic phases, and this will be our main focus. To sum up, we will be describing spin solid phases that are proximate to the simple Coulomb phase; the latter with the specified Berry phases encoded in the staggered background charge is in turn appropriate in the vicinity of the conventional Neel phase. Since we will continue with (5) we need a way to con- nect the variables there to the original spin variables. This is done as follows. The nearest neighbor spin-spin correlation 〈Sr ·Sr′〉 is proportional to the bond variable |Qrr′|2. To get the connection between the fluctuation of the magnitude of Q and the gauge fields we have to also keep the massive amplitude fields qm in the above derivation when integrating out the b’s. One finds that the (π, π, π) component couples to the gauge fields in the action as follows: δS = iγ (π,π,π) m (∂maτ − ∂τam), with some coupling parameter γ. On the other hand, in the derivation of the path integral from the Hamiltonian formulation of the gauge theory, the electric field is cou- pled to the gauge field in the same way, i.e., via a term Em(∂maτ − ∂τam) in the action. Thus the electric field gives the fluctuation of the staggered nearest neigh- bor spin-spin correlation function. B. Electro-magnetic duality We now proceed to the analysis of the model (5). We are interested in the confining phases, which will neces- sarily break lattice symmetries for spins S = 1/2, 1, 3/2 studied here. The confinement occurs due to condensa- tion of monopoles. Therefore we would like to express the theory in terms of them. This can be done by the standard electro-magnetic duality. The duality maps the theory of a compact U(1) gauge field without charges into a theory of a noncompact gauge field coupled to charges – the monopoles of the original theory. The noncompact- ness comes from the fact that we have dropped the elec- tric charges in the original theory; had we retained them, we would have obtained a compact dual gauge field whose monopoles would correspond to the original charges. The new variables reside on the lattice dual to the original lat- tice. The background charge of the original theory gives rise to a static dual magnetic flux emanating out of the center of each cube as drawn in Figure 3. This flux al- ternates in sign from one cube to the next and frustrates the monopole hopping. Therefore we obtain a theory of monopoles with frustrated hopping that are coupled to the dual noncompact gauge field.12 The duality can be done explicitly with various approximations clearly dis- played as is written in Appendix A. Explicitly, the partition function is DL e−Sdual , (8) Sdual = ∑ (∂L)2 λ cos(L + L0 −∇θ) , (9) where L is the dual gauge field, (∂L)µν = ∇µLν −∇νLµ is the four dimensional curl, L0 is the frustration that re- sults from the original background charge and ultimately z even z odd /32πS FIG. 3: a) Original background electric charges 2S and −2S on the two sublattices give rise to the static dual magnetic fluxes as seen by the monopoles. b) Gauge choice for L0 that realizes these fluxes modulo 2π. from Berry phases, and θ is the monopole field. A con- venient choice of L0 that produces the appropriate static fluxes is shown in Figure 3; all subsequent work is done in this gauge. The advantage of the dual formulation is that it has no sign problem and can be in principle studied by Monte Carlo. A sketch of the phase diagram is on Figure 4. In the bottom left side of the diagram, the monopoles are gapped and the system is in the deconfined phase, which correspond to the Coulomb spin liquid in the spin model. At large enough λ and 1/β, the monopoles condense. They can condense in various patterns which translate to various spin solid phases of the original model. Dual- ity relates the original field theory (5) to the large λ part of the dual theory. It is hard to analyze the transition in the large λ limit. Instead we look at three different places in this phase diagram. At 1/β = ∞ the system becomes frustrated XY model. First we analyze the phase tran- sition looking for ordering of the XY spins as we cross the phase boundary to the ordered phase. This gives us the most likely monopole condensation patterns near the transition. Next we look at the classical ground state of the XY model in the upper right corner of the phase diagram as approached from 1/β = ∞. Finally we look near the same point but in the limit λ ≫ 1/β. FIG. 4: Sketch of the expected phase diagram for the dual action Eq. (9). III. ANALYSIS 1,2: FRUSTRATED XY MODEL AT 1/β = ∞ A. Outline of the Analysis In this section we describe in general terms the anal- ysis in the 1/β = ∞ limit where the dual action Eq. (9) reduces to a frustrated XY model. We look at the phase transition and the classical ground state. In the Analysis 1, we consider the transition in the spirit of the Landau theory. We identify the relevant low- energy fields, write the most general quartic potential consistent with the symmetries, and study it in mean field. The approach is the same for each spin S, but the details are unique in each case and are contained in Subsections III B and III C for spin 1 and spin 3/2 respectively (spin 1/2 was considered using this approach in Ref. 12). More explicitly, the mean field derivation is done as follows. The mean field theory of XY spins is described by the continuous soft spin action |∂τΦR|2 − 〈RR′〉 (tRR′Φ RΦR′ + c.c.) + V (|φR|2) with some potential V (|ΦR|2) = r0|ΦR|2 + u0|ΦR|4 + · · ·. After crossing the transition from the disordered side, the system enters a phase with non-zero ΦR that minimize this action. The initial step is to minimize the kinetic energy. This will turn out to have a three-dimensional manifold of minima for spin 1 and four-dimensional one for spin 3/2. One then expands around these minima and writes all terms to a given order that are allowed by symmetry. In both cases the degeneracy is lifted at the fourth order. We will find that for spin 1 there are three independent quartic terms and we manage to draw a gen- eral phase diagram of the Landau theory. For spin 3/2, there are five such terms and the parameter space is too rich for us to describe the phase diagram completely. In this case we confine ourselves to examining the potential obtained from the most natural microscopic fourth order term and determining its mean field phase. In the Analysis 2, we find the classical ground state of the frustrated XY model – the state in the upper right point of the phase diagram Figure 4 – by a direct mini- mization of the hard-spin action (9). We use the following method: For some system size, we start with a random configuration of spins. We pick a random spin and min- imize its (local) energy and repeat this process until the total energy converges. Different starting configurations will lead to different final energies, because sometimes the system gets stuck in some local minima. We repeat this procedure for many starting configurations and also for different system sizes. We then select the configurations with the same lowest energy, which gives the absolute minimum of the potential. The case of spin 3/2, which corresponds to a fully frustrated XY model, was already considered some time ago in Ref. 20, and our method produces results in agreement with that work. For both spin 1 and spin 3/2, we find that the clas- sical ground state coincides with the most natural state identified in the mean field theory near the transition. This suggests that there is only one XY-ordered phase along the 1/β = ∞ line in Fig. 4, which could in princi- ple be tested in Monte Carlo studies of the corresponding frustrated XY models. We have described how to find the phases of the dual action in the 1/β = ∞ limit. However we are interested in the phases of the original spin model. To make the connection we calculate the energies and staggered curls of the monopole currents in the dual model and relate them to variables in the original spin problem. These variables are the plaquette energy and the bond expecta- tion value respectively. This allows us to determine the spin solid patterns. The mapping of the first variable, the energy, is sim- ple. Energy simply maps to energy. In the dual model we can calculate the energy for each bond, which is ǫ = 2Re(tRR′Φ RΦR′). The center of a bond of the dual lattice coincides with the center of a plaquette of the orig- inal lattice, and so the calculated energy is the plaquette energy of the original model. The connection of the staggered curls of the monopole current to the original bond variables is established as follows. The monopole current is given by JM = 2Im(tRR′Φ RΦR′). In terms of the original gauge the- ory, Eq. (5), just as the electric current produces mag- netic field, the magnetic current produces electric field. The resulting electric field is given by the analog of Biot- Savart law. However, approximately, if we have a loop of the magnetic current, the electric field it produces in the center is proportional to the circulation of the current, which is what we call the curl of the monopole current. As we described in the preceding section, the electric field is proportional to the staggered fluctuation of the near- est neighbor spin-spin correlation function, therefore the claimed connection. We will use this extensively in the detailed treatment of spin 1 and spin 3/2 below. B. Results: Spin 1 1. Analysis 1: Phase transition of the XY model Now we turn to finding the phases for spin 1. We choose the gauge shown on Figure 3. In this case the hopping amplitudes in (10) are given by tR,R+x̂ = (−1)z 3 + i(−1)x+y , (11) tR,R+ŷ = (−1)z 3− i(−1)x+y , (12) tR,R+ẑ = 1 . (13) The band structure has three minima and hence the space of ground states of the kinetic energy is three dimen- sional. Convenient choice of the basis is the following: (−1)x+y+z − (−1)x+y 3 + i[(−1)z + Ψ2 = i (−1)x+y+z − (−1)x+y 3− i[(−1)z + Ψ3 = − (−1)y + i(−1)x√ A general kinetic energy ground state can be written as Φ(R) = φ1Ψ1(R) + φ2Ψ2(R) + φ3Ψ3(R) (14) with complex fields φ1,2,3. This degeneracy will be lifted by nonlinear terms. To find out how, we would like to write the Landau theory for the φ’s, including all terms that are allowed by symmetry. Thus we need to find how the φ’s transform under the lattice symmetries. The generators of the symmetries are the translations by one lattice spacing in the x,y,z directions, 90 degree rotations around the x,y,z axes (it suffices to consider two out of three rotations), and mirror reflections. Note that the fluxes seen by the monopoles (and encoded in the complex phases of the hopping amplitudes tRR′) change sign under unit translations. The original spin problem is translationally invariant, and this is represented in the dual action (10) as follows. The fluxes remain unchanged if the t’s are also conjugated after the translation, and there is a gauge transformation that brings such mod- ified t’s to the original themselves. The action of the symmetry on the field Φ is then a combined application of the translation of the coordinates, conjugation, and gauge transformation. Similar considerations apply for the 90 degree rotations performed here about the dual lattice axes. After carrying through this analysis, the transformation properties of φ’s are remarkably simple: Tx Ty Tz Rx Ry Rz φ1 → – + + + φ∗3 φ∗2 φ2 → + – + φ∗3 + φ∗1 φ3 → + + – φ∗2 φ∗1 + In this table, “+” or “−” stands for φi → φ∗i or φi → −φ∗i respectively. We see that φ1 can be loosely associated with the x direction, φ2 with y, and φ3 with z. We should also point out that under mirror symmetries in the dual lattice planes the fields transform simply φi → φi. There is only one invariant term at the quadratic level: V (2) = m(|φ1|2 + |φ2|2 + |φ3|2) , (15) where m is a constant. There are three independent al- lowed terms at the quartic level, and the most general quartic potential can be written in the form V (4) = u(|φ1|2 + |φ2|2 + |φ3|2)2 + v(|φ1|4 + |φ2|4 + |φ3|4) + w(φ∗21 φ 2 + φ 3 + φ 1 + c.c.) , (16) where u, v, w are constants. To find the phases of the Landau theory, we simply need to minimize this potential. Before we start describ- ing the phases, however, it is useful to introduce bilinears of the fields. The reason is that these are gauge indepen- dent objects whereas the form of Ψ1,2,3 and hence the transformation properties of φ1,2,3 are gauge dependent. We consider the following bilinears: B0 = |φ1|2 + |φ2|2 + |φ3|2 , (|φ1|2 + |φ2|2 − 2|φ3|2) , F2 = |φ1|2 − |φ2|2 , Dx = φ 3φ2 + φ 2φ3 , Dy = φ 1φ3 + φ 3φ1 , Dz = φ 2φ1 + φ 1φ2 , Nx = i(φ 3φ2 − φ∗2φ3) , Ny = i(φ 1φ3 − φ∗3φ1) , Nz = i(φ 2φ1 − φ∗1φ2) . The B0 and the groups of F ’s, D’s and N ’s form ir- reducible representations of dimensions 1, 2, 3, and 3 respectively. The transformation properties of these bi- linears are displayed in the following table Tx Ty Tz Rx Ry Rz B0 + + + + + + F1 + + + − 12F1 + F2 − 12F1 − F2 + + + Dx + − − + Dz Dy Dy − + − Dz + Dx Dz − − + Dy Dx + Nx − + + + Nz Ny Ny + − + Nz + Nx Nz + + − Ny Nx + We should also add that all bilinears transform trivially under mirror symmetries in the dual lattice planes. We next calculate the energies and the staggered curls of the monopole currents, which, as described in Subsec. III A, are related to the plaquette ener- gies and bond variables of the original spin prob- lem. To repeat, the energy is given by ǫµ(R) = 2Re(tR,R+µ̂Φ RΦR+µ̂), and the monopole current is given by Jµ(R) = 2Im(tR,R+µ̂Φ RΦR+µ̂). The staggered curl of the monopole current is what the name suggests, for example, fz ≡ (−1)x+y+z[Jx(R) + Jy(R + x̂) − Jy(R) − Jx(R+ ŷ)]. The energies and staggered curls of the monopole cur- rents are bilinears in φ and thus can be expressed in terms of the B0, . . . , Nz. They are − 2(−1)y+zDx 3 [(−1)yNy + (−1)zNz] , (17) fx = 3 − 4(−1)xNx . (18) The components in the other directions are obtained from these by the appropriate rotations using the table, which for all bilinears except for F ’s gives the same result as the obvious permutation of indices. More generally, while the numerical coefficients in these expressions are obtained from the bare monopole hopping problem, the overall structure of the contributing terms is dictated by the symmetries – one only needs to remember that ǫx and fx are associated with scalars residing on respectively pla- quettes and bonds of the original spin lattice and also that the rotations and mirrors quoted here are about the axes and planes passing through the dual lattice sites. With the above results in hand, we now turn to analyz- ing phases of the Landau theory. The phase diagram is obtained simply by minimizing the potential (15)+(16) and is shown in Figure 5. The different phases are de- scribed in the following. In each case the ground state has finite degeneracy; we display few such states and the others are obtained from them by obvious permutations; we display nonzero bilinears, the energies, and the stag- gered curls of the monopole currents for the first listed state. FIG. 5: Phase diagram of the Landau theory for spin 1 ob- tained by minimizing the potential (15)+(16) for m < 0 and u > 0 (the latter choice is made for concreteness). In the ”Quartic unstable” region on the left the potential to quartic order is asymptotically negative and we would have to include sixth order terms to stabilize it. The cross denotes the pa- rameter point obtained by simply expanding the microscopic potential |Φ|4 in terms of the slowly varying fields φ1,2,3. Phase 1. There are three degenerate states. The values in one of them are φ1 = 1, φ2 = φ3 = 0; (19) B0 = 1; F1 = , F2 = 1; (20) ǫx = 2, ǫy = ǫz = 1; (21) fx = 4 3; fy = fz = −2 3. (22) The bond variables are drawn on the original spin lattice in Figure 1; they suggest that the spins are organized into Haldane chains along the x direction. The values of plaquette energies are consistent with this: the plaque- ttes in the xy and xz planes are the same and differ from the plaquettes in the yz plane, ǫz = ǫy 6= ǫx. Phase 2. There are six degenerate states. The values in one of them are φ1 = 0, φ2 = 1, φ3 = ±i; (23) B0 = 2; F1 = − 1√3 , F2 = −1; Nx = 2; (24) ǫx = 2, ǫy = ǫz = 3+ 2 3(−1)x; (25) fx = −4[ 3 + 2(−1)x], fy = fz = 2 3. (26) The corresponding drawing of the bond variables on the original spin lattice is in Figure 6, suggesting that in this phase the spins combine into singlets and form a colum- nar dimer state along one direction. Permuting the val- ues of φ1,2,3 gives six degenerate states that correspond to six possible ways of placing such columnar solid onto the cubic lattice. Phase 3. There are eight degenerate states specified as follows: φ1 = 1, φ2 = e iα2 , φ3 = e iα3 , (27) {α2, α3} = ±{2π/3,−2π/3}, ± {2π/3, π/3}, ± {π/3, 2π/3}, ± {π/3,−π/3}; (28) FIG. 6: Phase 2 of spin 1. The thick lines denote the positions where the bond variables are strongest and dashed lines where they are weakest. This suggests that the spins organize into singlets (dimers) and form a columnar order. B0 = 3; Dx = Dy = Dz = −1; Nx = Ny = Nz = 3 (29) ǫx = 4 + 2(−1)y+z + 3[(−1)y + (−1)z], etc., (30) fx = −4 3(−1)x, etc. (31) The nearest neighbor spin-spin correlation has higher ex- pectation value on the sides of the cubes shown in Figure 7, which suggests that this phase corresponds to a box state. There are eight possible ways of placing such box state onto the cubic lattice. FIG. 7: Phase 3 of spin 1. The bond variables have higher expectation values on the cubes shown. Phase 4. There are four degenerate states: φ1 = 1, φ2 = e iα2 , φ3 = e iα3 ; (32) α2 = 0, π; α3 = 0, π; (33) B0 = 3; Dx = Dy = Dz = 2; (34) ǫx = 4− 4(−1)y+z, ǫy = 4− 4(−1)z+x, ǫz = 4− 4(−1)x+y; (35) fx = fy = fz = 0. (36) This state breaks lattice symmetries as can be seen from the plaquette energies. However, because the bond vari- ables fx,y,z are zero, we do not know a simple interpre- tation of this phase in terms of the original spins; some finer characterization than what we use here is needed to establish this state. This concludes the discussion of the general phase dia- gram of the Landau theory including quadratic and quar- tic terms. Higher-order interactions may stabilize some other phases, but the presented states are the most nat- ural ones. The actual lowest-energy state depends on the parameters u, v, w, unknown apriori. If we are to guess which of the four phases is the most likely candidate in the specific frustrated XY model, we can consider the simplest microscopic quartic potential |Φ|4. When ex- panded in terms of the continuum fields, we find u = 2, v = −1, w = −1/2; this point is denoted by the cross in Figure 5 and lies in the Phase 1, i.e., the Haldane chains phase. 2. Analysis 2: The ground state of the XY model Minimizing the classical energy of the hard spin XY model as described in Sec. III A, we find that the ground state configurations coincide with the condensate wave- functions of the phase 1 and hence the state is that of the phase 1. In particular note that each wavefunction Ψ1,2,3 has the same length |Φ| on all sites. The XY angles of spins in this gauge in the three ground states are (0,−30,−30, 0, 60,−90,−90, 60) , (37) (0, 30, 30, 0,−60, 90, 90,−60) , (38) (0,−90, 90, 180, 0,−90, 90, 180) , (39) where the convention is that we vary position on the cube in the x direction first, then in the y direction, and then in the z. 3. Discussion and extension to anisotropic system Some remarks are in order. First, it is useful to note that the doublet F1,2 can be interpreted as an order parameter of the Haldane chains phase. Indeed, one can readily see that the transformation properties of F1 and F2 coincide with those of (Qx +Qy − 2Qz)/ 3 and Qx − Qy respectively, where Qm is the bond variable in the direction m̂. On the other hand, Nx transforms as (−1)xQx and similarly for Ny and Nz, so ~N can be viewed as an order parameter of the valence bond solids such as the columnar Phase 2 or the box Phase 3. In the columnar phase, it is suggestive to view each strong bond in Fig. 6 as representing a singlet formed by two spin-1’s, which can be also drawn as two spin-1/2 valence bonds connecting the two sites. However, we should be cautious with such interpretation, since we can only tell that the deviations of the bond variables from their mean value will have the displayed pattern. The actual state needs to be studied by constructing the corresponding spin wavefunction. For example, the Haldane phase of a spin 1 chain is stable to weak dimerization and should be viewed as a solid formed by single-strength bonds along the chains, so such distinct possibilities should be kept in mind. Let us now assume that the system is in the Phase 1. It is also interesting to ask what happens when we stretch the lattice in one of the axis directions, say the z-direction. In this case the Rx and Ry rotations are no longer symmetries but the other transformations are. At the quadratic level, the translation symmetries already prohibit all terms except B0 and F ’s. Then from Rz we see that only F1 is allowed. Thus at the quadratic level one new term is allowed. In principle we should look at the new allowed terms at the quartic level, however we will assume that this quadratic term is leading but small compared to the terms that were there before we broke the symmetry. We find that if the F1 comes with a positive pre-factor, out of the three ground states it selects the state with chains running along the z direction whereas if it comes with a negative pre-factor it selects the two states with chains running along the x and y directions. This has a simple interpretation in terms of spins. If the coupling in the z direction is stronger than in the other directions the state with maximum number of bonds in this directions is selected which is the state with chains running in the z-direction. In the opposite case, the states with fewest bonds in the z direction are se- lected which are the states with chains running in the x or y directions. C. Results: Spin 3/2 1. Analysis 1: Phase transition of the XY model We choose the gauge as shown on Figure 3 with S = 3/2. The hopping amplitudes are (−1)z 1 + i(−1)x+y , (40) (−1)z 1− i(−1)x+y , (41) tz = 1 . (42) The band structure has four minima and hence the space of the ground states of kinetic energy is four-dimensional. Unlike the spin 1 case where this space was three- dimensional and simple basis vectors were found corre- sponding to the three directions of the physical space, there is no such form in the spin 3/2 case. The four wavefunctions that give us relatively simple subsequent analysis are the following Ψ1 = (−1)x cosβ − i(−1)x+y+z sinβ Ψ2 = i(−1)y cosβ + i(−1)x+y+z sinβ 1 + i(−1)x+y√ cosβ − i(−1)x+y+z sinβ 1− i(−1)x+y√ cosβ + i(−1)x+y+z sinβ where cosβ = 3 + 1 , sinβ = . (43) We again write Φ(R) = i=1 φiΨi(R). The transfor- mation properties of the slow fields φ1,2,3,4 are derived in the same manner as in the spin 1 case. The symmetries Tx : ~φ → τ3σ0 ~φ∗ , (44) Ty : ~φ → τ0σ0 ~φ∗ , (45) Tz : ~φ → τ1σ0 ~φ∗ , (46) Ry : ~φ → τ1e−i ~φ∗ , (47) Rz : ~φ → e−i σ1 ~φ∗ . (48) Here ~φ, ~φ∗ are column vectors, and we introduced two sets of Pauli matrices: τ matrices that act on the blocs {1, 2} and {3, 4}, and σ matrices that act within each bloc (τ0 and σ0 are the corresponding identity matrices). At the quadratic order there is one invariant term V (2) = m |φi|2 . (49) At the quartic order there are five invariant terms. The expressions in terms of φ are rather complicated and not very illuminating, particularly since φ’s depend on the choice of gauge and the basis. Instead, we will use gauge invariant bilinears of φ to which we now turn. There are 16 bilinears and they can be conveniently organized using tensor product of the introduced two sets of Pauli matrices, namely φ†τµσνφ with µ, ν = 0, 1, 2, 3. These break up into irreducible representa- tions of the cubic lattice symmetry group. There are two one-dimensional, one two-dimensional, and four three- dimensional representations. The convenient combina- tions that we use are B0 = φ †τ0σ0φ , C = φ†τ0σ2φ , F1 = φ †τ0σ1φ , F2 = φ †τ0σ3φ , ~D = (Dx, Dy, Dz) = φ †~τσ2φ , ~N = (Nx, Ny, Nz) = φ †~τσ0φ , Mx = φ †τ1(−1 σ3)φ , My = φ †τ2(−1 σ3)φ , Mz = φ †τ3σ1φ , Kx = φ σ1 − 1 σ3)φ , Ky = φ †τ2(− σ3)φ , Kz = φ †τ3σ3φ . The transformation properties of these bilinears are in the following table Tx Ty Tz Rx Ry Rz B0 + + + + + + C − − − + + + F1 + + + − 12F1 + F2 − 12F1 − F2 + + + Dx + − − + Dz Dy Dy − + − Dz + Dx Dz − − + Dy Dx + Nx − + + + Nz Ny Ny + − + Nz + Nx Nz + + − Ny Nx + Mx − + + + Mz My My + − + Mz + Mx Mz + + − My Mx + Kx − + + − −Kz −Ky Ky + − + −Kz − −Kx Kz + + − −Ky −Kx − The energies and staggered curls of monopole cur- rents in term of these bilinears are B0 − 2(−1)y+zDx 2 [(−1)yMy + (−1)zMz] [(−1)yKy − (−1)zKz] , (50) fx = 2 2(F1 + 3F2) + 8(−1)x√ Nx . (51) The components in the other directions are obtained from these by simple rotations of the coordinates. Our general discussion following similar expressions (17) and (18) in the spin 1 case apply here as well (for ease of compari- son, we are using similar labels for objects with identical transformation properties in the two cases). However, a word of warning is in order here, which will be explained in Sec. III C 3 below. Observe, for example, that ~N and ~M have identical transformation properties and therefore should enter similarly in any expression. The absence of M ’s in the expression for ǫx and the absence of N ’s in the expression for fx is due to their different eigenval- ues under an additional artificial symmetry present in the frustrated XY model, namely a charge conjugation symmetry defined later, which is also present in our bare kinetic term and thus in the above expressions. This symmetry is not physical in the original spin model and will not be used here; it is therefore important to note that the degeneracy of the four slow modes obtained from the bare kinetic term is protected at the quadratic level by the physical lattice symmetries. There are five independent fourth order terms in φ al- lowed by translation and rotation symmetries: I1 = B 0 , (52) I2 = C 2 , (53) I3 = N z , (54) I4 = M z , (55) I5 = NxMx +NyMy +NzMz . (56) As we have said earlier, because the number of invariant terms is large, we will not attempt to draw the phase diagram of the Landau’s theory. Instead we look at the natural microscopic term V (4) = |Φ|4 = 4 I4 , (57) where the second equality is obtained after some calcu- lation keeping only non-oscillatory terms. This potential does not have any continuous symme- try left other than the global U(1) transformation of all fields. In fact the dimensions of the subgroups of SU(4) that keep the terms I1, . . . , I5 invariant are 15, 7, 6, 0, 0 respectively. The potential (57) achieves global mini- mum at twelve discrete points. As an illustration, we consider the following four minima that are associated with the z direction in the sense to become clear below: (φ1, φ2, φ3, φ4) = (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1). (58) The four states can be related to each other by transla- tions in the z direction and rotations about the z axis. Besides B0 = 1, the only nonzero bilinears in these states are (F2, Nz,Kz) = (1, 1, 1), (−1, 1,−1), (1,−1,−1), and (−1,−1, 1) respectively. The energies are ∓ (−1)z , (59) ± (−1)z , (60) , (61) where the upper sign corresponds to the first and fourth minima and the lower sign to the other two. The staggered curls of monopole currents are respec- tively fx, fy, fz = 8(−1)z√ , (62) 8(−1)z√ , (63) 6,− 8(−1) , (64) 6,− 8(−1) . (65) The staggered curls are interpreted as the strength (above some mean) of the expectation value of nearest neighbor spin-spin correlation function. The above val- ues imply that the spins organize themselves into ladders as shown in Figure 2, obtained by drawing say the pos- itive bonds for the first of the above minima. The four listed states correspond to the four different positions of ladders with rungs oriented along the z-axis. The other eight minima are obtained by 90 degree rotations around the x and y axes and we will not write the specific values of the variables. The ladder state is natural for S = 3/2 system, in the picture where spin 3/2 breaks up into three spin 1/2’s and each of them forms a bond with some other neighboring spin 1/2. 2. Analysis 2: The ground state of the XY model We can use the same procedure as in the case of spin 1 to find the classical ground state of the appropriate XY model. In fact, this was already done in Ref. 20 because this problem is the fully frustrated XY model (FFXY), which is of interest by itself, and we can use the available results. We find that the ground state con- figurations coincide with the condensate wavefunctions obtained above. Thus, in each of the four displayed states (58), the microscopic boson field Φ is given precisely by one of the four wavefunctions Ψ1,...,4. One can see that |Φ| = 1 on all lattice sites, and the complex phases of Φ can be interpreted as angles of the hard-spin XY model. For example, for Φ = Ψ1 the angles are (−β, π + β, β, π − β, β, π − β,−β, π + β) , (66) listed in the same order as in Eq. (39). All other ground states can be obtained by appropriate symmetry trans- formations. The agreement of the two analyses suggests that there is only one ordered phases in the FFXY model, which is also supported by the available Monte Carlo studies.20,21 3. Remark on charge conjugation symmetry in the FFXY It is worth to point out that the fully frustrated XY model has an additional charge conjugation symmetry. Indeed, since π and −π fluxes are indistinguishable, tRR′ and t∗RR′ are related by a gauge transformation, tRR′ = eiγRt∗RR′e −iγR′ , and so the action remains invariant under the following unitary transformation: C : ΦR → eiγRΦ∗R . (67) In terms of the continuum fields, this becomes C : ~φ → τ2σ2 ~φ∗ . (68) In particular, the bilinears Nx,y,z are odd under C while Mx,y,z are even, so if this symmetry is included, the I5 quartic term is not allowed (this is why this term did not appear in Eq. 57 since both the microscopic |Φ|4 and the bare quadratic terms in Eq. 10 have this additional sym- metry). Thus, the complete field theory for the FFXY model is a φ4 theory with four complex fields and inde- pendent quartic terms I1,...,4. One consequence of the charge conjugation symmetry is that, for example, if we draw the Ψ1 state using neg- ative values of the staggered curls fx,y,z as opposed to using positive values which was done in Fig. 2, we would obtain another set of ladders that go perpendicularly to the ones displayed and are shifted up by one lattice spac- ing. To put this in other words, the Ψ1 and Ψ4 states that can be related by a translation in the z direction followed by a rotation around the z axis are also related by C. In this sense, each of the states Eq. (58) does not define a direction in the x-y plane since the correlations in the x and y directions are related by the charge conjugation symmetry. Tracing back to the original gauge theory formulation, this symmetry is present in the simplest model Eq. (5) for S = 3/2 that we wrote down and the corresponding sim- plest “dimer model” Hamiltonian Eq. (6). Specifically, the transformation E → 1−E on the links oriented from one sublattice to the other, or equivalently 1 ↔ 0 in the dimer language, takes the model corresponding to spin S to the one corresponding to spin 3−S, while the S = 3/2 case maps back onto itself. This symmetry is useful in the specific models, but there is no corresponding sym- metry in the microscopic derivation from the spin model, and therefore it was not used in the preceding analysis. Let us look what happens to the ground states when we add small term that breaks the charge conjugation sym- metry, the I5, to the potential. Using general arguments it is easy to check that the twelve minima will shift but not split, and the twelve-fold degeneracy remains since all are related to each other by lattice symmetries. Fur- thermore, each ground state stays translationally invari- ant along the ladders and perpendicular to the plane of ladders (otherwise, if this were not true, there would be more than twelve states). In other words, the states still have the structure of ladders. However since the charge conjugation is broken, it is no longer true that the nega- tive bonds are of the same magnitude as the correspond- ing positive ones. This makes sense when interpreted in terms of spins. In the picture where spin 3/2 breaks up into three spin 1/2 and ladders of valence bonds are formed, the links that belong to these ladders are differ- ent from the links without bonds (which also form lad- ders). For example, the system is entangled along the former but not along the latter. Thus these two should not be related by any symmetry. Explicitly, the four states in Eq. (58) become (1, δ, 0, 0), (δ, 1, 0, 0), (0, 0, 1, δ), (0, 0, δ, 1), (69) with appropriate δ obtained from minimization. There is now an additional non-zero bilinear Mz, and also both F1 and F2 are non-zero. The expressions for the ener- gies and staggered curls in the x and y directions are no longer related, and we can then associate a unique x or y direction with each of the four states. These are ladders with rungs oriented in the z direction and are related to each other by the z translations and rotations. 4. Extension to anisotropic system As in the spin 1 case, we ask what happens when we stretch the system along one axis, say the z-direction. Again, the Rx and Ry rotations are no longer symmetries but the translations and Rz are. At the quadratic level, the translation symmetries already prohibit all terms ex- cept B0 and F ’s. Then from Rz we see that only F1 is allowed. Thus at the quadratic level one new term is allowed. We find that if the F1 comes with a positive pre-factor, out of the twelve ground states it selects four with the ladders that lie entirely in the x-y plane, whereas if it comes with a negative pre-factor it selects four states with the ladders running along the z-direction. Note that this breaking up into groups of four is a consequence of the remaining symmetries in the system. These results have a simple physical interpretation for the spin system. If the coupling in the z direction is weaker than in the other directions, the states with fewest bonds in the z direction are selected which are the states with the ladders lying in the x-y plane. On the other hand, if the coupling in the z direction is stronger, the states with the largest number of bonds in the z direction are selected, which are the states with ladders oriented in the z direction. IV. ANALYSIS 3: MAPPING TO DIMERS AT λ ≫ 1/β ≫ 1 Here we look at the right hand corner of the phase diagram Fig. 4 in the regime with λ ≫ 1/β ≫ 1, where as we will see the system can be mapped to dimers.17,18,19 The analysis proceeds as follows. First we gauge away the ∇θ in Eq. (9) to obtain Sdual = ∑ (∂L)2 λ cos(L+ L0) , (70) Because we assume λ ≫ 1/β ≫ 1 the configurations that contribute significantly to the partition function can be written in the form L = −L0 + 2πn + δL where n is an integer and δL is small. Note that the λ term does not depend on n and the 1/β term has a gauge invariance n → n+∇m where m’s are integers on sites. The partition function can be written as a sum over the gauge equivalent classes. These classes are in one-to- one correspondence with the fluxes j = ∂n which are integers on plaquettes, where ∂n is the four dimensional curl (∂n)µν = ∂µnν − ∂νmµ. Consider first configurations with δL = 0. Some con- figurations of j minimize the action and we denote them by jgs. As we show below, there is an extensive num- ber of them in all our cases. The configurations with j that are not jgs are at energy of at least ∼ 1/β higher. Now turning on δL, if we show that the typical energy of excitation in δL around a given j is much smaller then 1/β then we can neglect all configurations which are not around jgs. We will assume that this is true and show this self-consistently below. We define Jgs = −∂L0/(2π) + jgs. We expand the action to the second order and drop the terms that do not depend on Jgs, δL to obtain ∑ 4πJgs · (∂δL) + (∂δL)2 (δL)2 . (71) This is just a gaussian integral. There are two quadratic terms and the first one has 1/β in front and contains two derivatives while the second has λ in front and contains no derivatives. Since we are on a lattice the derivatives are of order one. Since λ ≫ β, the first term can be neglected. Next we sum by parts and integrate out the δL. Before we do this however, we notice that the cou- pling is ferromagnetic in time direction and L0 has zero time components and its spatial components do not de- pend on time. This implies that the jgs and Jgs have zero time components and their spatial components do not depend on time. Thus we drop time components and time derivatives from the action and treat the Jgs and L0 as three-dimensional. Now we integrate out the δL and obtain Seff [J gs] = − 1 8π2β2λ (∇× Jgs)2 (72) Thus, to obtain a ground state, we need to maximize the sum of the squares of curls of Jgs. Let us check the consistency of our approach. From (71), δL ∼ ∇× Jgs/(λβ) and so energy∼ 1/(λβ2). This needs to be much smaller then 1/β which implies λ ≫ 1/β which is what we assumed. FIG. 8: a) L0/(2π) where S = 1/2, 1, 3/2 is the spin. The link variables switch orientations under elementary translation in the x or y direction. b) The fluxes (∇ × L0)/(2π). This figure is similar to Fig. 3 with 2π’s removed to simplify the discussion of the dimer ground states. Now let us turn to the specific cases of spins. Since the spin 1/2 case has not been considered using this approach before, we will add it here for completeness. The gauge choice for L0 and the fluxes∇×L0/(2π) through the faces of the spatial cubes are shown on Figure 8 with S = 1/2. It is easy to see that the set of ground states consists of all configurations with precisely one −5/6 and five 1/6 fluxes Jgs coming out of every site of one sublattice of the original spin lattice (and coming into every site on the other sublattice). Jgs = −∇ × L0/(2π) is one such configuration in the spin 1/2 case, but there are many more. Associating the −5/6 plaquettes with dimers on the links of the original spin lattice, the set of the ground states is thus the set of dimer configurations with one dimer coming out of every site. Now turn to the case of spin 1. The fluxes (∇ × L0)/(2π) are shown on Figure 8 with S = 1. If we try Jgs = −∇× L0/(2π), each cube contributes 1/β energy term proportional to 5(1/3)2 + (5/3)2 = 10/3. However we can do better. Using L = −L0+2πn, if we pick n = 1 on the upper link on the front face and zero elsewhere on the cube in Fig. 8, we lower the magnitude of the flux on the upper face, at the expense of increasing the flux through the front face. The energy of this cube is then 4(1/3)2 + 2(2/3)2 = 4/3, which is lower. It is easy to show that this is the lowest we can achieve and that the ground state configurations have two fluxes of value −2/3 and four fluxes of value 1/3 coming out of every site of one sublattice of the original spin lattice. Associating the 2/3 links with dimers, the set of the ground states is thus the set of dimer configurations with two dimers coming out of every spin site. Finally, in the S = 3/2 case, it is easy to see that the ground state configurations have precisely three −1/2 and three 1/2 fluxes coming out of every site of one sub- lattice of the original cubic lattice. Associating the −1/2 links with dimers, the set of the ground states is thus the set of dimer configurations with three dimers coming out of every spin site. Thus, as claimed, in each case there is an extensive number of Jgs’s. To find the true ground state, we need to minimize (72) among these dimer configurations. It is not hard to show that for the spin 1/2 we get columnar state, for spin 1 the Haldane chains state of Fig. 1 and for spin 3/2 the ladder state of Fig. 2. Finally we note that defining Egs = S/3− Jgs, the set of Egs is the set of electric fields on links, cf. Eq. (6), with the property that the magnitude of each is either zero or one (which can be imposed by minimizing the energy term E2); the mapping between such electric fields and dimers above is the standard one on the cubic lattice17,18,19. The final ground state selection is obtain by maximizing (∇× E)2. V. CONCLUSIONS In this paper we looked for spin solid phases in the sys- tem of spin 1 and 3/2 on the cubic lattice. We wrote the spins in terms of Schwinger bosons, assumed the uniform Coulomb spin liquid phase and by process of monopole condensation transitioned into spin solid phases. Using the duality we rewrote the system in terms of monopoles coupled to a noncompact U(1) gauge field, Eq. (9), and analyzed this theory in three different limits shown in Figure 4. In the first two limits the theory becomes a frustrated XY model. For spin 1 the frustrating flux through every plaquette is 2π/3, while for spin 3/2 it is π. In the first approach, using symmetries we wrote the Landau’s the- ory near the ordering transition. It is a φ4 theory with φ a complex vector with three components for S = 1 and four components for S = 3/2. At the quadratic level only the rotationally invariant mass term is allowed. At the quartic level there are three allowed terms for spin 1 and five for spin 3/2. For spin 1 we draw a mean field phase diagram Figure 5. For spin 3/2 we didn’t attempt it due to a large number of parameters. In both cases we also considered the most natural microscopic potential and found that it selects a state with parallel Haldane chains of Figure 1 for S = 1 and a state with parallel ladders of Figure 2 for S = 3/2. These are natural states for the spin systems to be in, in the picture where spin 1 breaks up into two and spin 3/2 into three spin 1/2’s and each such spin 1/2 forms a singlet bond with another spin 1/2 of some neighbor. In the second approach we looked at the classical ground states of the frustrated XY models and found that these actually describe the same phases as the most natural ones identified near the transition. In the third approach the theory becomes a dimer model with 2S dimers coming out of every site. Dimer configurations with parallel lines for spin 1 and parallel ladders for spin 3/2 are selected, which is the same re- sult as in the other two limits suggesting that these are indeed the most natural valence bond solids in the corre- sponding spin systems. It would be interesting to look for such spin solid phases in Quantum Monte Carlo studies of models on the cubic lattice.5,8 It is also worth noting14 that if we consider our quan- tum 3d systems at a finite temperature, we obtain simply the corresponding classical 3d dimer models, e.g., with the classical energy given by the first term in Eq. (6). Our results then provide appropriate long-wavelength (dual) description of the dimer ordering patterns transitioning out of the so-called Coulomb phase of the classical dimer models,22,23,24 stressing in particular a composite char- acter of the naive order parameters for the valence bond solid phases. It would interesting to explore such 3d clas- sical dimer models and their transitions further. APPENDIX A: CLASSICAL U(1) DUALITY WITH BACKGROUND CHARGE In this section we derive duality for classical com- pact U(1) gauge theory.12,25 However we will use a gen- eral notation of antisymmetric tensors, or differential forms which are fields of antisymmetric tensors. Thus the derivation will work not only for the gauge theory, whose objects are one dimensional, but for general n- dimensional objects. For n = 0 this is the vortex duality of the XY model and for n = 1 the duality of the gauge theory. The further advantage of this derivation is that the formulas are simpler and more transparent. First we give the basic notations and properties of antisymmetric tensors. An n-dimensional antisymmet- ric tensor ω in d dimensions is a collection of numbers ωµ1,µ2,...,µn , where µv = 1, . . . , d, which is completely an- tisymmetric. A differential form ω(~r) is a field of these tensors. We define two operations. First is the exterior deriva- tive ∂. The derivative of ω, denoted ∂ω is the (n+1)-form (∂ω)µ1,µ2,...,µn+1 = (−1)p∂µp1ωµp2 ,...,µpn (A1) where the sum is over all permutations of the n+1 indices and (−1)p is −1 if the permutation is odd and 1 if it is even. Thus for example for n = 1, a vector field, (∂ω)12 = ∂1ω2 − ∂2ω1 and hence this is the curl of a vector field. The second operation that we define is the star opera- tor that takes n-form to (d− n)-form (∗ω)ν1,...,νd−n = ǫν1,...,νd−n,µd−n+1,...,µdωµd−n+1,...,µd where ǫ is the fully antisymmetric tensor in d dimensions and repeated indices are summed over. For example in three dimensions for n = 2, (∗ω)1 = 12 (ω23 − ω32). Note that ∗∗ = (−1)n(d−n). A common operator is divergence which in this nota- tion is proportional to ∗∂∗. As easily checked, (∇ · ω)µ1,...,µn−1 ≡ ∂νων,µ1,...,µn−1 (A3) = (−1)(n−1)(d−n)(∗∂ ∗ ω)µ1,...,µn−1(A4) For a vector field this is the standard divergence. We will work on the lattice. The variables are defined on discrete points. We will define the coordinates of a given variable to be those of the center of the object the variable belongs to. For example the x component of a one form ω in d = 3 lies on a link pointing in x direction and it is denoted by ωx(x+1/2, y, z). The ∂ now denotes the difference operator. For example the curl of the ω is (∂ω)xy(x + 1/2, y + 1/2, z) = ωy(x + 1, y + 1/2, z) − ωy(x, y+1/2, z)−ωx(x+1/2, y+1, z)+ωx(x+1/2, y, z). Finally we will write the integration (summation) by parts ω · ∂φ = − (∇ · ω) · φ+ surface term (A5) where the dot is the sum over the component by com- ponent product of two forms of the same n. Note that ∗ω1 · ∗ω2 = ω1 · ω2. Because we use periodic boundary conditions below, the surface term will be zero. Now we are ready to turn to the duality. Let a be an n-form in d dimensions where its variables are defined on the unit circle. The action is S = −β cos(∂a)− i η · a (A6) In the first term one takes every component at every point, takes cosine of it and sums. In the second term the n-form η denotes the background charge. For the action considered in this paper, the first term is the Sa and the second term the SB in (5), while the η is the four dimensional vector with the time component being ±2S and the other components being zero. The duality proceeds by the following steps. cos(∂a)+i (∂a−2πp)2+i (∂a−2π∂−1q′)2+i J·(∂a−2π∂−1q′)+i J·∂−1q′ ×∆(∇ · J − η) (A7) All numerical factors are dropped throughout, while the sign “≈” is used when an approximation is being made that does not change the qualitative aspects. In the second line we use the Villain form of the cosine. In the third line we have written the field p = ∂α+∂−1q′ as a curl of α plus a field of a particular monopole current configuration q′, ∂−1q′. The ∂−1 denotes a particular configuration of p that gives the monopole currents - that satisfies q′ = ∂p. Then we shifted a → a − α. The summation over α extends the integration of a over the whole real line. The prime on q′ denotes that fact that we are summing over fields for which ∂q′ = 0. The third line can be obtained from the fourth one by completing the square, shifting J and integrating it out. In the fifth line, the ∆ denotes that the operator inside of it is zero. This line is obtained from the fourth one by integrating (summing) by parts and integrating out the Next, as shown explicitly below, in our case there are fields J0 and L0 such that η = ∇ · J0 (A8) J0 = (∗P∂L0)/2π (A9) ∂J0 = 0 (A10) The P shifts a real number by a multiple of 2π so that the result is in the interval (−π, π]. Using (A8) in (A7) we see ∂ ∗ (J − J0) = 0 and hence we can write J = J0 + (∗∂L)/2π (A11) for some field L. To substitute this into (A7) we notice the following J2 = J20 + (∗∂L/2π)2 + 2J0 · ∗∂L/2π ≃ J20 + (∗∂L/2π)2. The ≃ denotes that these expressions are equal under integration, which follows from Eq. (A10). Also ∂−1q′ · ∗∂L ≃ − ∗ ∂∂−1q′ · L = − ∗ q′ · L ≡ −Q′ · L, ′·∗P∂L0 = ei∂ ′·∗∂L0 ≃ e−iQ ′·L0, where Q′ ≡ ∗q′, and we have dropped inconsequential ± signs; in the last line, the P can be removed because the resulting expression, which is in the exponent, differs from the original one by a multiple of 2π. With this we can proceed to complete the duality (∂L)2 ′·(L+L0) (∂L)2 Q·(L+L0−∂θ) (∂L)2 (L+L0−∂θ−2πp)2 (∂L)2 −λ cos(L+L0−∂θ) (A12) In the first line the summation over Q′ is over integer fields Q′ with zero divergence ∇ · Q′ = 0 - currents. In the second line we introduced θ that imposes this con- straint as a Lagrange multiplier and summed by parts. In the third line we added a small term Q2/2λ and as- sumed that it is not going to change the basic behavior of the system. Then we summed out Q, which introduced integer p because Q is an integer (this is the Poisson sum- mation formula). The second term is the Villain form of cosine. In the last line we approximated it by cosine. To complete it remains to find J0 and L0. The η has values ητ (x, y, z, τ + 1/2) = (−1)x+y+z2S and zero for other components. As easily checked (J0)τx(x+ 1/2, y, z, τ + 1/2) = (−1)x+y+z (A13) and similarly for y and z with other components (other then the ones obtained by permutation of indices) being zero. This gives the right η and satisfies ∂J0 = 0. The L0 can be chosen as on the Fig. 3. In the final expression (A12) the L is 1-form and hence a gauge field. The θ is 0-form - a number on a circle - a matter field. Thus we obtained a noncompact U(1) gauge theory coupled to scalar fields of monopoles with frustrated hopping. 1 S. Taniguchi et al., J. Phys. Soc. Jpn. 64, 2758 (1995); 2 D. S. Chow, P. Wzietek, D. Fogliatti, B. Alavi, D. J. Tan- tillo, C. A. Merlic, and S. E. Brown, Phys. Rev. Lett. 81, 3984 (1998). 3 H. Kageyama, K. Yoshimura, R. Stern, N.V. Mushnikov, K. Onizuka, M. Kato, K. Kosuge, C.P. Slichter, T. Goto, and Y. Ueda, Phys. Rev. Lett. 82, 3168 (1999); H. Kageyama, M. Nishi, N. Aso, K. Onizuka, T. Yosihama, K. Nukui, K. Kodama, K. Kakurai, and Y. Ueda, Phys. Rev. Lett. 84, 5876 (2000); 4 A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett. 89, 247201 (2002). 5 K. S. D. Beach and A. W. Sandvik, cond-mat/0612126. 6 K. Harada, N. Kawashima, and M. 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0704.0822

Braided quantum field theories and their symmetries

YITP-07-14 Braided quantum field theories and their symmetries Yuya Sasai ∗ and Naoki Sasakura † Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Abstract Braided quantum field theories proposed by Oeckl can provide a framework for defining quantum field theories having Hopf algebra symmetries. In quantum field the- ories, symmetries lead to non-perturbative relations among correlation functions. We discuss Hopf algebra symmetries and such relations in braided quantum field theories. We give the four algebraic conditions between Hopf algebra symmetries and braided quantum field theories, which are required for the relations to hold. As concrete exam- ples, we apply our discussions to the Poincaré symmetries of two examples of noncom- mutative field theories. One is the effective quantum field theory of three-dimensional quantum gravity coupled with spinless particles given by Freidel and Livine, and the other is noncommutative field theory on Moyal plane. We also comment on quantum field theory on κ-Minkowski spacetime. ∗ e-mail: [email protected] † e-mail: [email protected] http://arxiv.org/abs/0704.0822v6 1 Introduction Symmetry is one of the most important notions in quantum field theory. In many examples, it is useful in investigating properties of quantum field theories non-perturbatively, is a guiding principle in constructing field theories for various purposes such as grand unification, or gives powerful methods in finding exact solutions. It also plays important roles in actual renormalization procedures. Therefore it should be interesting to study symmetries also in noncommutative field theories [1, 2, 3, 4, 5], which may result from some quantum gravity effects [6]. A difficulty in the study in this direction is the apparent violation of basic symme- tries such as Poincaré symmetry in the noncommutativity of spacetime. For example, the Moyal plane [xµ, xν ] = iθµν is translational invariant, but is not Lorentz or rotational invariant. Another example is the three-dimensional spacetime with noncommutativity [xi, xj] = iκǫijkxk (i, j, k = 1, 2, 3) [7, 8, 9, 10] with a noncommutativity parameter κ. This noncommutative spacetime is Lorentz-invariant, but is not invariant under the translational transformation xi → xi + ai with c-number ai. In fact, a naive construction of noncom- mutative quantum field theory on this spacetime leads to rather disastrous violations of energy-momentum conservation [10]: the violations coming from the non-planar diagrams do not vanish in the commutative limit κ→ 0 as in the UV/IR mixing phenomena [11]. In recent years, however, there has been interesting conceptual progress in understanding symmetries in noncommutative field theories: the symmetry transformations in noncommu- tative spacetime are not the usual Lie-algebraic type, but should be generalized to have Hopf algebraic structures. The Moyal plane was pointed out to be invariant under the twisted Poincaré transformation in [12, 13, 14] and under the twisted diffeomorphism in [15, 16, 17, 18]. There have been various proposals to implement the twisted Poincaré in- variance in quantum field theories [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. As for the noncommutative spacetime with [xi, xj ] = iκǫijkxk, a noncommutative quantum field theory was derived as the effective field theory of three-dimensional quantum gravity with matters [31]. Its essential difference from the naive construction mentioned above is the nontrivial braiding for each crossing in non-planar Feynman diagrams. With this braiding, there ex- ists a kind of conserved energy-momentum in the amplitudes, and the energy-momentum operators have Hopf algebraic structures. Our aim of this paper is to systematically understand these Hopf algebraic symmetries and their consequences in noncommutative field theories in the framework of braided quan- tum field theories proposed by Oeckl [34]. In the usual quantum field theories, symmetries give non-perturbative relations among correlation functions. We will see that such relations have natural extensions to the Hopf algebraic symmetries in braided quantum field theories, and will obtain the four conditions for the relations to hold. These conditions should be interpreted as the criteria of the symmetries in braided quantum field theories. This paper is organized as follows. In the following section, we review braided quantum field theory. This review part follows faithfully the original paper [34], but figures are more extensively used in the proofs and the explanations to make this paper self-contained and intuitively understandable. We start with braided category and braided Hopf algebra. Then correlation functions of braided quantum field theory are represented in terms of them. Finally braided Feynman rules are given. In Section 3, we first review the axioms of action1 of an algebra on vector spaces. Then we consider the relations among correlation functions in braided quantum field theory. We find that four algebraic conditions are required for the relations to hold. Then, as concrete examples, we discuss whether the noncommutative field theories mentioned above have the Poincaré symmetry by checking the four conditions. In the former case, we find that the twisted Poincaré symmetry is implemented only after the introduction of a non-trivial braid- ing factor, which agrees with the previous proposal in [21, 35]. In the latter case, we find that the theory has a kind of translational symmetry, which is different from the usual one by multi-field contributions. We also give some examples of such relations among correlation functions and the implications. The final section is devoted to summary and comments. We comment on quantum field theory on κ-Minkowski spacetime whose noncommutativity of coordinates is [x0, xj] = xj (j = 1, 2, 3) [36]. 2 Review of braided quantum field theory 2.1 Braided categories and braided Hopf algebras First of all, we review braided categories and braided Hopf algebras [34, 37]. Braided cate- gories are composed of an object X , which is a vector space, a dual object X∗, which is a dual vector space, and morphisms ev : X∗ ⊗X → k (evaluation), (1) coev : k → X ⊗X∗ (coevaluation), (2) where k is a c-number. The composition of the two morphisms in an obvious way makes the identity. Then the braided categories have also an invertible morphism ψV,W : V ⊗W → W ⊗ V (braiding), (3) where V,W are any pair of vector spaces. Generally the inverse of braiding is not equal to the braiding itself. The braiding is required to be compatible with the tensor product such that ψU,V⊗W = (id⊗ ψU,W ) ◦ (ψU,V ⊗ id), ψU⊗V,W = (ψU,W ⊗ id) ◦ (id⊗ ψV,W ). (4) 1We use the italic symbol to distinguish it from the action S. Figure 1: The evaluation, coevaluation, braiding and its inverse. Then the braiding is also required to be intersectional under any morphisms in a Hopf algebra. For example, ψZ,W (Q⊗ id) = (id⊗Q)ψV,W for any Q : V → Z, ψV,Z(id⊗Q) = (Q⊗ id)ψV,W for any Q : W → Z, (5) where Z is a vector space. We can represent these axioms in pictorial ways [38]. We write the morphisms, ev, coev, ψ, downwards as in Figure 1. Thus the axioms (4) are represented as in Figure 2, and the axioms (5) are represented as in Figure 3. Next we consider the polynomials of X , X̂ := Xn, with X0 := 1 and Xn := X ⊗ · · · ⊗X︸ ︷︷ ︸ n times , (6) where 1 is the trivial one-dimensional space. X̂ naturally has the structure of a braided Hopf algebra via · (product) : X̂⊗̂X̂ → X̂, (7) η (unit) : k → X̂ ; η(1) = 1, (8) ∆ (coproduct) : X̂ → X̂⊗̂X̂ ; ∆φ = φ⊗̂1+ 1⊗̂φ, and ∆(1) = 1⊗̂1, (9) ǫ (counit) : X̂ → k ; ǫ(φ) = 0, and ǫ(1) = 1, (10) S (antipode) : X̂ → X̂ ; Sφ = −φ, and S(1) = 1, (11) where φ ∈ X . The tensor product ⊗ is the same as the usual product of Xs, while the new tensor product ⊗̂ is the tensor product of X̂s. The coproduct ∆, counit ǫ, antipode S of the Figure 2: The axioms of braiding (4). Figure 3: The axioms of braiding (5). Figure 4: The axioms of coproduct, counit, antipode for products. products of Xs are defined inductively by ∆ ◦ · = (·⊗̂·) ◦ (id⊗̂ψ⊗̂id) ◦ (∆⊗̂∆), (12) ǫ ◦ · = · ◦ (ǫ⊗̂ǫ), (13) S ◦ · = · ◦ ψ ◦ (S⊗̂S). (14) These axioms are diagrammatically represented in Figure 4. 2.2 Braided quantum field theory Next we represent braided quantum field theory [34] in terms of the braided category and the braided Hopf algebra. We take the vector space X as the space of a field φ(x), where x denotes a general index for independent modes of the field. Thus X̂ is the space of polynomials of the fields such as φ(x1)φ(x2) · · ·φ(xn), and 1 correspond to the constant field of unit. We also take the dual vector space X∗ as the space of differentials δ/δφ(x). We take the evaluation and the coevaluation as follows, δφ(x) ⊗ φ(x′) → δ(x− x′), (15) coev : 1 → φ(x)⊗ δ δφ(x) , (16) Figure 5: The differentials on X̂. where the distribution and the integration should symbolically be understood, and their detailed forms, which may contain non-trivial measures, depend on each case. The differential on X̂ is defined by diff := (êv ⊗ id) ◦ (id⊗∆); X∗ ⊗ X̂ → X̂, (17) where êv|X∗⊗Xn = ev for n = 1, 0 for n 6= 1. Diagrammatically this is given by Figure 5. To see whether the map diff gives really the differential of products, let us compute the differential of φ(x)φ(y) as a simple example using the definition (17). This becomes δφ(x′) ⊗ φ(x)φ(y) = (êv ⊗ id) ◦ (id⊗∆) δφ(x′) ⊗ φ(x)φ(y) = (êv ⊗ id) ◦ δφ(x′) ⊗∆(φ(x)φ(y)) = (êv ⊗ id) ◦ δφ(x′) ⊗ (φ(x)φ(y)⊗̂1 + φ(x)⊗̂φ(y) + ψ(φ(x)⊗̂φ(y)) + 1⊗̂φ(x)φ(y)) = δ(x′ − x)⊗ φ(y) + (êv ⊗ id) ◦ δφ(x′) ⊗ ψ(φ(x)⊗̂φ(y)) where we have used the axiom (12) in deriving the third line. If the braiding is trivial, we find that the differential (17) satisfies the usual Leibniz rule. Figure 6: Diagram of ψn,m. Generally we find a braided Leibniz rule ∂(αβ) = ∂(α)β + ψ−1(∂ ⊗ α)(β) (20) ∂(α) = (ev ⊗ idn−1)(∂ ⊗ [n]ψα), (21) where ∂ ∈ X∗, α, β ∈ X̂ , and we have used a simplified notation ∂(α) := diff(∂ ⊗ α). (22) Here n is the degree of α, and [n]ψ is called a braided integer defined by [n]ψ := id n + ψ ⊗ idn−2 + · · ·+ ψn−2,1 ⊗ id + ψn−1,1, (23) where ψn,m is a braiding morphism given in Figure 6. The proofs of the formula (20), (21) are in Appendix A. Now we define a Gaussian integration, which defines the path integral. The definition is given by ∫ ∂(αw) := 0 for ∂ ∈ X∗, α ∈ X̂, (24) where w ∈ X̂ is a Gaussian weight. In field theory, w is the exponential of the free part of the action, e−S0 . In order to obtain a formula for correlation functions, we define a morphism γ : X∗ → X such that ∂(w) := −γ(∂)w. (25) This morphism is assumed to be commutative with the braiding as in (5). If w = e−S0 , γ(∂) = ∂(S0). In field theory, this is the kinetic part of the action, or the inverse of the propagator. Starting from (24), we can represent correlation functions of a free field theory in terms of the braided category and the braided Hopf algebra. This is the analog of the Wick theorem in braided quantum field theory. The definition of the free n point correlation function is given by Z(0)n (α) := , (26) where the degree of α is n. Algebraically, this is given by 2 = ev ◦ (γ−1 ⊗ id) ◦ ψ, (27) 2n = (Z n ◦ [2n− 1]′ψ!!, (28) 2n−1 = 0, (29) where [2n− 1]′ψ!! := ([1]′ψ ⊗ id2n−1) ◦ ([3]′ψ ⊗ id2n−3) ◦ · · · ◦ ([2n− 1]′ψ ⊗ id), (30) ψ := id n + idn−2 ⊗ ψ−1 + · · ·+ ψ−11,n−1 = ψ−11,n−1 ◦ [n]ψ. (31) The proofs of (27), (28), (29) are in Appendix B. Next we consider correlation functions with the existence of an interaction. For S = S0 + λSint, a correlation function is perturbatively given by Zn(α) = αe−S∫ α(1− λSint + · · · )e−S0∫ (1− λSint + · · · )e−S0 , (32) where α ∈ Xn. Introducing a morphism Sint : k → Xk, where k is the degree of Sint, the correlation function is algebraically given by n − λZ(0)n+k ◦ (id n ⊗ Sint) + 12λ n+2k ◦ (id n ⊗ Sint ⊗ Sint) + · · · 1− λZ(0) ◦ Sint + 12λ2Z 2k ◦ (Sint ⊗ Sint) + · · · . (33) Acting Zn on α ∈ Xn, we obtain the correlation function (32). One can obviously extend Sint to include various interaction terms. 2.3 Braided Feynman rules From the results in the preceding subsection, a correlation function can be represented by summation of diagrams obeying the following rules below. Figure 7: Propagator (left) and vertex (right). Figure 8: The braiding ψ (left) and its inverse ψ−1 (right). • An n-point function Zn is a morphism Xn → k. Thus a Feynman diagram starts with n strands at the top and must be closed at the bottom. • The propagator Z(0)2 : X ⊗X → k is represented by the left of Figure 7, which is the abbreviation of Figure 9. • The interaction vertex Sint : k → Xk is represented by the right of Figure 7. Generally the order of the strands is noncommutative. • The two kinds of crossings, which are represented in Figure 8, correspond to the braid- ing and its inverse. • Any Feynman diagram is built out of propagators, vertices, and crossings, and is closed at the bottom. 3 Symmetries in braided quantum field theory In this section, we discuss symmetries in braided quantum field theory. In order to represent symmetry transformations on fields, we review general description of an action in Section Figure 9: The propagator, which is abbreviated in the left figure of Figure 7. 3.1. In Section 3.2, we study relations among correlation functions. We find four conditions for such relations to follow from a symmetry algebra. In Section 3.3 and 3.4, we treat two examples of (braided) noncommutative field theories and discuss their Poincaré symmetries. 3.1 General description of an action We review an action of a general Hopf algebra on vector spaces in a mathematical language [37, 39]. An action αV is a map αV : A⊗ V → V , where A is an arbitrary Hopf algebra and V is a vector space (in our case, A is a symmetry algebra, and V = X or X∗). We will denote the coproduct and the counit of the Hopf algebra2 by ∆′ and ǫ′ to distinguish them from those of the braided Hopf algebra of fields in Section 2. We do not write all the axioms of an action, but our important axioms are the following. • αV satisfies the following condition. αV ◦ (· ⊗ id) = αV ◦ (id⊗ αV ), (34) where the equality acts on A ⊗ A ⊗ V . This means that αV ((a · b) ⊗ V ) = αV (a ⊗ (αV (b⊗ V ))), where a, b ∈ A. In short we can write this as (a · b) ⊲ V = a ⊲ (b ⊲ V ). (35) • An action on 1, which is in a vector space, is defined by αV (a⊗ 1) = ǫ′(a)1, (36) where ǫ′(a) is the counit of an algebra a ∈ A. • An action on a tensor product of vector spaces V,W is defined by αV⊗W (a) := ((αV ⊗ αW ) ◦∆′)(a) = αV (a (1))⊗ αW (ai(2)), a ∈ A, (37) where ∆′(a) = ai(1) ⊗ ai(2) is the coproduct of the Hopf algebra A. In the case of a usual Lie-algebraic transformation, its coproduct is given by ∆′(a) = a ⊗ 1 + 1 ⊗ a, where 1 is in A. This gives the usual Leibnitz rule. • Since a Hopf algebra has the coassociativity that ((∆′ ⊗ id) ◦∆′)(a) = ((id⊗∆′) ◦∆′)(a), (38) 2We omit the antipode. the action on a tensor product of vector spaces, which is obtained by the multiple operations of ∆′ on a, is actually unique. An important consequence is that one can divide the action on a tensor product of vector spaces as a⊲(V1 ⊗ · · · ⊗ Vk−1 ⊗ Vk ⊗ · · · ⊗ Vn) =∑ ai(1) ⊲ (V1 ⊗ · · · ⊗ Vk−1)⊗ ai(2) ⊲ (Vk ⊗ · · · ⊗ Vn) (39) for any k. 3.2 Symmetry relations among correlation functions and their al- gebraic descriptions The expression of the correlation functions (33) is perturbative in interactions, but is a full order algebraic description. Therefore we can discuss the symmetry of the theory and the implied relations among correlation functions by using this expression. We may even expect that the relations will hold non-perturbatively. In usual quantum field theory, if a field theory has a certain symmetry, there is a relation among the correlation functions in the form, 〈φ(x1) · · · δaφ(xi) · · ·φ(xn)〉 = 0, (40) where δaφ(x) is a variation of a field under a transformation a, on the assumption that the path integral measure and the action are invariant under the transformation. If the coproduct of a symmetry algebra is not the usual Lie-algebraic type and thus the Leibniz rule is deformed, the relation will generally have the form, c(bi)a 〈φ(x1) · · · δbφ(xi) · · ·φ(xn)〉 +c(bi)(cj)a 〈φ(x1) · · · δbφ(xi) · · · δcφ(xj) · · ·φ(xn)〉 +c(bi)(cj)(dk)a 〈φ(x1) · · · δbφ(xi) · · · δcφ(xj) · · · δdφ(xk) · · ·φ(xn)〉 + · · · = 0, (41) where c···a are some coefficients. Its essential difference from (40) is the multi-field contribu- tions. In our algebraic language, the relation can be written as Zn(a ⊲ χ) = ǫ ′(a)Zn(χ), for a ∈ A, χ ∈ Xn. (42) This is equivalent to Figure 10 in our diagrammatic representation. Then we consider what an algebraic structure is required for (42) to hold for any a and χ, i.e. the theory is invariant under the Hopf algebra transformation A. Figure 10: A relation among correlation functions in the diagrammatic representation. n is the number of external legs, k is the order of the interaction, and p is the order of the perturbation. n+ kp is even. Let us write the coproduct of an element a ∈ A as ∆′(a) = f s ⊗ gs, (43) where f s, gs ∈ A. Since the coproduct must satisfy the Hopf algebra axiom [37], (ǫ′ ⊗ id)∆′(a) = (id⊗ ǫ′)∆′(a) = a, (44) f s, gs must satisfy ǫ′(f s)⊗ gs = f s ⊗ ǫ′(gs) = a. (45) For all the relations among correlation functions to hold, we find the following four conditions for any action a ∈ A. • (Condition 1) Sint must satisfy a ⊲ Sint = ǫ ′(a)Sint. (46) • (Condition 2) The braiding ψ is an intertwining operator. That is ψ(a ⊲ (V ⊗W )) = a ⊲ ψ(V ⊗W ). (47) • (Condition 3) γ−1 and a are commutative, a ⊲ (γ−1(V )) = γ−1(a ⊲ V ). (48) • (Condition 4) Under an action a, the evaluation map follows ev(a ⊲ (X∗ ⊗X)) = ǫ′(a)ev(X∗ ⊗X). (49) Condition 1 to 4 are diagrammatically represented in Figure 11. It is clear that, when the algebra A is generated from a finite number of its independent elements, it is enough for these generators to satisfy these conditions. Condition 1 is the requirement of the symmetry at the classical level for the interaction. We can extend this condition to (a ⊲ Xn)⊗ Spint = a ⊲ (Xn ⊗ S int). (50) The proof is the following. From a coproduct (43) and its coassociativity (39), the right hand side of (50) is equal to (f s ⊲ (Xn ⊗ Sp−1int ))⊗ gs ⊲ Sint (51) Since Condition 1 implies gs ⊲ Sint = ǫ ′(gs)Sint, (52) (51) becomes (f s ⊲ (Xn ⊗ Sp−1int ))⊗ ǫ′(gs)Sint = a ⊲ (Xn ⊗ Sp−1int )⊗ Sint, (53) where we have used (45). Iterating this procedure, we obtain the left-hand side of (50). Condition 2,3,4 can also be extended to [n + kp− 1]ψ!! ◦ (a ⊲ Xn+kp) = a ⊲ [n+ kp− 1]ψ!! Xn+kp, (54) (γ−1 ⊗ id) 2 ◦ (a ⊲ Xn+kp) = a ⊲ (γ−1 ⊗ id) 2 Xn+kp, (55) 2 (a ⊲ (X∗ ⊗X) 2 ) = ǫ′(a) ev 2 (X∗ ⊗X) 2 . (56) We can find that these extended conditions (50), (54), (55), (56) can be represented as in Figure 12. In the diagrammatic language, the relation among correlation functions holds if an action can pass downwards through a Feynman diagram and satisfies (36). 3.3 Symmetries of the effective noncommutative field theory of three-dimensional quantum gravity coupled with scalar parti- In this subsection, we discuss the Poincaré symmetry of the effective noncommutative field theory of three-dimensional quantum gravity coupled with scalar particles, which was ob- tained in [31] by studying the Ponzano-Regge model [40] coupled with spinless particles. The Figure 11: Conditions 1,2,3, and 4. Figure 12: A relation among correlation functions is satisfied if the four conditions (46), (47), (48), (49) are satisfied. symmetries of this theory is also known as DSU(2), which was discussed in [32, 33]. We first review the field theory [10, 31]. Let φ(x) be a scalar field on a three-dimensional space x = (x1, x2, x3). Its Fourier transformation is given by φ(x) = dgφ̃(g)e tr(Xg), (57) where κ is a constant, X = ixiσi, and g = P 0−iκP iσi ∈ SO(3)3 with Pauli matrices σi. Here∫ dg is the Haar measure of SO(3) and P 0 = ± 1− κ2P iPi by definition. In the following discussions, we will only deal with the Euclidean case, but the Lorentzian case can also be treated in a similar manner by replacing SO(3) with SL(2, R). The definition of the star product is given by tr(Xg1) ⋆ e tr(Xg2) := e tr(Xg1g2). (58) Differentiating both hands sides of (58) with respect to P i1 := P i(g1) and P 2 := P j(g2) and then taking the limit P i1, P 2 → 0, one finds the SO(3) Lie-algebraic space-time noncommu- tativity [7, 8, 9], [xi, xj]⋆ = 2iκǫ ijkxk. (59) For example, the action4 of a φ3 theory is (∂iφ ⋆ ∂iφ)(x)− M2(φ ⋆ φ)(x) + (φ ⋆ φ ⋆ φ)(x) , (60) where M2 = sin . Its momentum representation is P 2(g)−M2 φ̃(g)φ̃(g−1) dg1dg2dg3δ(g1g2g3)φ̃(g1)φ̃(g2)φ̃(g3), (61) from which it is straightforward to read the Feynman rules. Some quantum properties of this scalar field theory were analyzed in [10]. As can be seen from (59), the naive translational symmetry is violated. In fact, the violation is rather disastrous. There exists a kind of conserved energy-momentum in the amplitudes of the tree and the planar loop diagrams, but this energy-momentum is not conserved in the non- planar loop diagrams. Moreover, the violation of the energy-momentum conservation does not vanish in the commutative limit κ → 0 due to a mechanism similar to the UV/IR phenomena [11]. 3The identification g ∼ −g is implicitly assumed. 4Since in the Ponzano-Regge model the definition of the weight of partition function is eiS despite of Euclidean theory, the sign of the mass term is not the usual one. In the effective field theory of quantum gravity coupled with spinless particles, however, the Feynman rules contain also a non-trivial braiding rule for each crossing, which comes from a flatness condition in a graph of intersecting particles [31]. This can be incorporated as a braiding between the scalar fields, ψ(φ̃(g1)φ̃(g2)) = φ̃(g2)φ̃(g 2 g1g2), (62) in the braided quantum field theory. From the direct analysis of the Feynman graphs with this braiding rule, one can easily find that the energy-momentum mentioned above is conserved also in the non-planar diagrams. This suggests the existence of a translational symmetry in the quantum field theory. In the sequel, we will discuss the embedding of this field theory into the framework of braided quantum field theory, and will check the four conditions for its translational and rotational symmetries. We use the momentum representation, and take X as the space of φ̃(g) and X∗ as that δφ̃(g) . We take the braided Hopf algebra of the fields as follows, ∆ : φ̃(g) → φ̃(g)⊗̂1 + 1⊗̂φ̃(g), (63) ǫ : φ̃(g) → 0, (64) S : φ̃(g) → −φ̃(g), (65) ψ : φ̃(g1)⊗ φ̃(g2) → φ̃(g2)⊗ φ̃(g−12 g1g2). (66) The evaluation and coevaluation maps are given by δφ̃(g) ⊗ φ̃(g′) → δ(g−1g′), (67) coev : 1 → dgφ̃(g)⊗ δ δφ̃(g) . (68) From γ(∂) = ∂S0 = (P 2(g)−m2)φ̃(g−1), γ−1(φ̃(g)) = P 2(g−1)−m2 δφ̃(g−1) . (69) From the algebraic consistencies in Figure 13, the braidings between X and X∗ and the braiding between X∗s are determined to be δφ̃(g1) ⊗ φ̃(g2) = φ̃(g2)⊗ δφ̃(g−12 g1g2) , (70) φ̃(g1)⊗ δφ̃(g2) δφ̃(g2) ⊗ φ̃(g2g1g−12 ), (71) δφ̃(g1) δφ̃(g2) δφ̃(g2) δφ̃(g2g1g . (72) Figure 13: The algebraic consistency conditions of coevaluation map and X , X∗. In this derivation, we have used the invariance of the Haar measure d(g−1g′g) = dg′. Now we consider a translational transformation of the field. If we shift xi to xi + ǫi, a field φ(x) becomes φ(x) → φ(x+ ǫ) dgφ̃(g)ei(x+ǫ) iPi(g) dg(1 + iǫiPi(g))φ̃(g)e ixiPi(g). (73) Thus in the momentum representation, the translational transformation corresponds to an action P i ⊲ φ̃(g) = P i(g)φ̃(g), P 0 ⊲ φ̃(g) = P 0(g)φ̃(g). (74) From the requirement that the star product (58) conserve a kind of momentum, the action on a product of fields should be P i ⊲ (φ̃(g1)φ̃(g2)) = P i(g1g2)φ̃(g1)φ̃(g2) = (P 01P 2 + P 1 + κǫ 2 )φ̃(g1)φ̃(g2), (75) P 0 ⊲ (φ̃(g1)φ̃(g2)) = (P 2 − κ2P i1P2i)φ̃(g1)φ̃(g2). (76) This determines the coproduct of P i, P 0 as ∆′(P i) = P 0 ⊗ P i + P i ⊗ P 0 + κǫijkP j ⊗ P k, (77) ∆′(P 0) = P 0 ⊗ P 0 − κ2P i ⊗ Pi. (78) This coproduct satisfies the coassociativity, which essentially comes from the associativity of the group multiplication. From the axiom (44), the counit of P i, P 0 is given by ǫ′(P i) = ǫ′(P 0) = 0. (79) Since the conservation of momentum under the coevaluation map (68) requires that the action of P i on dg(φ̃(g)⊗ δ δφ̃(g) ) vanish from (36), the action of P i on δ δφ̃(g) must be P i ⊲ δφ̃(g) = P i(g−1) δφ̃(g) . (80) In the following, we see that the momentum algebra satisfies the four conditions (46), (47), (48), (49). Condition 1 is satisfied since P i ⊲ Sint dg1dg2dg3δ(g1g2g3)P i ⊲ (φ̃(g1)φ̃(g2)φ̃(g3)) dg1dg2dg3δ(g1g2g3)P i(g1g2g3)(φ̃(g1)φ̃(g2)φ̃(g3)) = 0. (81) Condition 2 is satisfied since ψ(P i ⊲ (φ̃(g1)φ̃(g2))) = P i(g1g2)(φ̃(g2)φ̃(g 2 g1g2)), P i ⊲ ψ(φ̃(g1)φ̃(g2)) = P i(g2g 2 g1g2)(φ̃(g2)φ̃(g 2 g1g2)). Condition 3 is satisfied since P i ⊲ γ−1(φ̃(g)) = P 2(g−1)−m2 P i(g) δφ̃(g−1) γ−1(P i ⊲ φ̃(g)) = P 2(g−1)−m2 P i(g) δφ̃(g−1) Condition 4 is satisfied since P i ⊲ δφ̃(g1) ⊗ φ̃(g2) = P i(g−11 g2) ev δφ̃(g1) ⊗ φ̃(g2) = 0. (82) Thus we find that the effective braided noncommutative field theory of three-dimensional quantum gravity coupled with spinless particles has the translational symmetry. Next we consider a rotational symmetry. The rotational symmetry corresponds to an action Λ ⊲ φ̃(g) = φ̃(h−1gh), (83) which is the usual Lie-group one. The action on the tensor product is Λ ⊲ (φ̃(g1)⊗ φ̃(g2)) = φ̃(h−1g1h)⊗ φ̃(h−1g2h). (84) Thus the coproduct of the rotational symmetry is given by ∆′(Λ) = Λ⊗ Λ. (85) From the axiom (44), the counit of Λ is given by ǫ′(Λ) = 1. (86) Condition 1 is satisfied since Λ ⊲ Sint dg1dg2dg3δ(g1g2g3)Λ ⊲ (φ̃(g1)φ̃(g2)φ̃(g3)) dg1dg2dg3δ(g1g2g3)(φ̃(h −1g1h)φ̃(h −1g2h)φ̃(h −1g3h)) =ǫ′(Λ)Sint. (87) Condition 2 is satisfied since ψ(Λ ⊲ (φ̃(g1)⊗ φ̃(g2))) = φ̃(h−1g2h)⊗ φ̃(h−1g−12 g1g2h) Λ ⊲ ψ(φ̃(g1)⊗ φ̃(g2)) = φ̃(h−1g2h)⊗ φ̃(h−1g−12 g1g2h). (88) Condition 3 is satisfied since Λ ⊲ γ−1(φ̃(g)) = P 2(g−1)−m2 δφ̃(h−1g−1h) γ−1(Λ ⊲ φ̃(g)) = P 2(h−1g−1h)−m2 δφ̃(h−1g−1h) P 2(g−1)−m2 δφ̃(h−1g−1h) . (89) Condition 4 is satisfied since δφ̃(g1) ⊗ φ̃(g2) δφ̃(h−1g1h) ⊗ φ̃(h−1g2h) = δ(g−11 g2) = ǫ′(Λ)ev δφ̃(g1) ⊗ φ̃(g2) . (90) Thus we find that this braided noncommutative field theory has also the rotational sym- metry. 3.4 Twisted Poincaré symmetry of noncommutative field theory on Moyal plane In this subsection, we discuss the twisted Poincaré symmetry of noncommutative field theory on Moyal plane [xµ, xν ] = iθµν . For example, the action of a φ3 theory is given by (∂µφ ∗ ∂µφ)(x)− m2(φ ∗ φ)(x) + λ (φ ∗ φ ∗ φ)(x) , (91) where the star product is given by φ(x) ∗ φ(x) = e θµν∂xµ∂ νφ(x)φ(y) . (92) In the momentum representation, the action is (p2 −m2)φ̃(p)φ̃(−p) dDp1d µνp2νδ(p1 + p2 + p3)φ̃(p1)φ̃(p2)φ̃(p3) . (93) We take X as the space of φ̃(p) and X∗ as that of δ δφ̃(p) . Then we take the braided Hopf algebra as follows: ∆ : φ̃(p) → φ̃(p)⊗̂1+ 1⊗̂φ̃(p), (94) ǫ : φ̃(p) → 0, (95) S : φ̃(p) → −φ̃(p). (96) From γ(∂) = ∂S0 = (p 2 −m2)φ̃(−p), γ−1(φ̃(p)) = p2 −m2 δφ̃(−p) . (97) Let us consider the twisted Poincaré symmetry [12, 13, 14]. The coproduct and the counit of the twisted Poincaré algebra is given by ∆′(P µ) = P µ ⊗ 1+ 1⊗ P µ, ǫ′(P µ) = 0, ∆′(Mµν) =Mµν ⊗ 1+ 1⊗Mµν θαβ [(δµαP ν − δναP µ)⊗ Pβ + Pα ⊗ (δ ν − δνβP µ)], ǫ′(Mµν) = 0. (98) Thus the action of the twisted Lorentz algebra on the tensor product is Mµν ⊲ (φ̃(p1)⊗ φ̃(p2)) =Mµν ⊲ φ̃(p1)⊗ φ̃(p2) + φ̃(p1)⊗Mµν ⊲ φ̃(p2) θαβ [(δµαP ν − δναP µ) ⊲ φ̃(p1)⊗ Pβ ⊲ φ̃(p2) + Pα ⊲ φ̃(p1)⊗ (δµβP ν − δνβP µ) ⊲ φ̃(p2)], (99) where Mµν ⊲ φ̃(p) = i(pµ∂/∂pν − pν∂/∂pµ)φ̃(p) and P µ ⊲ φ̃(p) = pµφ̃(p). The actions of Mµν and P µ on δ δφ̃(p) Mµν ⊲ δφ̃(p) = i(pµ∂/∂pν − pν∂/∂pµ) δφ̃(p) , (100) P µ ⊲ δφ̃(p) = −pµ δ δφ̃(p) . (101) One easily finds that three conditions (46), (48), (49) are satisfied, but (47) is not if the braiding is trivial. In order to keep the invariance, the braiding must be taken as ψ(φ̃(p1)⊗ φ̃(p2)) = eiθ αβp2α⊗p1β(φ̃(p2)⊗ φ̃(p1)). (102) This agrees with the previous proposal [21, 35]. We can easily check that the translational symmetry holds since the coproduct ∆′(P µ) follows the usual Leibniz rule. 3.5 Relations among correlation functions : Examples Now we have checked, in all orders of perturbation, that the two theories in the preceding sections have symmetry relations among correlation functions implied by the Hopf algebra symmetries. In Section 3.3 we gave how the translational generator acts on a product of fields in (75), (76) in the momentum representation. Since the physical meaning of this Hopf algebra transformation is not so clear, it would be interesting to see explicitly the symmetry relations among correlation functions. The same thing is also true in the case of the twisted Lorentz symmetry in Section 3.4. In this subsection, we work out explicitly some relations among correlation functions in the two theories. In the effective quantum field theory of quantum gravity, the action of the translational generators on a correlation function is given by 〈φ̃(g1) · · · φ̃(gn)〉 → iǫiPi(g1 · · · gn)〈φ̃(g1) · · · φ̃(gn)〉 (103) in the momentum representation, where ǫi is an infinitesimal parameter. Thus we obtain a relation, Pi(g1 · · · gn)〈φ̃(g1) · · · φ̃(gn)〉 = 0. (104) This is a (modified) momentum conservation law; the correlation function has support only on the vanishing momentum subspace, Pi(g1 · · · gn) = 0. This all-order relation in the quantum field theory would be a simple but an important implication of the Hopf algebraic translational symmetry. This provides a good example of the physical importance of a Hopf algebraic symmetry: a Hopf algebra symmetry leads to a (modified) conservation law. It would also be interesting to see the relations in the coordinate representations, where the fields are defined by φ(x) = eip·xφ̃(p). As explicitly noted in the preceding subsections, we stress that the basis of the spaces X of the field variables in the path integrals are parameterized in terms of momenta, and that φ(x) are defined by some c-number linear combinations of them. Therefore, an action a ∈ A of a symmetry transformation acts as a ⊲ φ(x) = eip·x(a ⊲ φ̃(p)), (105) and the symmetry relations of correlation functions can be obtained by some inverse Fourier transformations (with possible non-trivial measures) of those in momentum representations. For example, in the case of the two point function, after the inverse Fourier transforma- tion, the relation among correlation functions is given by 〈∂iφ(x1)φ(x2) + φ(x1)∂iφ(x2)〉 = 0, (106) where we have used the relation (104). Interestingly, this is the usual relation in a transla- tionally invariant quantum field theory. In the case of the three point function, however, the relation is given by 〈∂iφ(x1) 1 + κ2∂2φ(x2) 1 + κ2∂2φ(x3) + 1 + κ2∂2φ(x1)∂ iφ(x2) 1 + κ2∂2φ(x3) 1 + κ2∂2φ(x1) 1 + κ2∂2φ(x2)∂ iφ(x3) + iκǫ 1 + κ2∂2φ(x1)∂jφ(x2)∂kφ(x3) + iκǫijk∂jφ(x1) 1 + κ2∂2φ(x2)∂kφ(x3)− iκǫijk∂jφ(x1)∂kφ(x2) 1 + κ2∂2φ(x3) + κ2∂jφ(x1)∂ jφ(x2)∂ iφ(x3)− κ2∂iφ(x1)∂kφ(x2)∂kφ(x3) + κ2∂kφ(x1)∂ iφ(x2)∂ kφ(x3)〉 = 0. (107) This is quite a non-trivial relation among correlation functions, and would be hard to find, if the Hopf algebra symmetry in the quantum field theory was not noticed. This would be another interesting example implying the physical importance of a Hopf algebra symmetry. In general, the relation has the form, ∂xli − i κǫijk∂xlj∂xmk +O(κ 2))〈φ(x1) · · ·φ(xn)〉 = 0. (108) In the κ → 0 limit, the relation approaches the usual relation. Thus the Hopf algebra sym- metry is a kind of translational symmetry modified by adding κ dependent higher derivative multi-field contributions. We can proceed in a similar manner for the twisted Lorentz symmetry. We have a general form of such a symmetry relation as Mµν ⊲ 〈φ̃(p1) · · · φ̃(pn)〉 = 0. (109) In the case of the two point function, the relation is given by 〈(xµ1∂ν − xν1∂µ)φ(x1)φ(x2) + φ(x1)(x ν − xν2∂µ)φ(x2)〉 = 0, (110) where we have used the momentum conservation. This is the same relation as that in a Lorentz invariant quantum field theory. In the case of the three point function, the relation is given by 〈(xµ1∂ν − xν1∂µ)φ(x1)φ(x2)φ(x3) + φ(x1)(x ν − xν2∂µ)φ(x2)φ(x3) + φ(x1)φ(x2)(x ν − xν3∂µ)φ(x3) iθαµ(∂αφ(x1)∂ νφ(x2)φ(x3) + ∂αφ(x1)φ(x2)∂ νφ(x3) + φ(x1)∂αφ(x2)∂ νφ(x3) − ∂νφ(x1)∂αφ(x2)φ(x3)− ∂νφ(x1)φ(x2)∂αφ(x3)− φ(x1)∂νφ(x2)∂αφ(x3)) iθαν(∂αφ(x1)∂ µφ(x2)φ(x3) + ∂αφ(x1)φ(x2)∂ µφ(x3) + φ(x1)∂αφ(x2)∂ µφ(x3) − ∂µφ(x1)∂αφ(x2)φ(x3)− ∂µφ(x1)φ(x2)∂αφ(x3)− φ(x1)∂µφ(x2)∂αφ(x3))〉 = 0. (111) In general, the relation among correlation functions has the from, ((x1µ∂x1ν − x1ν∂x1µ) + · · ·+ (xnµ∂xnν − xnν∂xmν) +O(θ))〈φ(x1) · · ·φ(xn)〉 = 0 (112) in the coordinate representation. The leading terms corresponds to the usual Lorentz trans- formation xµ → xµ + ǫµνxν . The above symmetry relations on Moyal plane can be represented in similar manners as the usual commutative cases, if we use star products. In the papers [24, 25, 26, 27, 28, 29, 30], they have pointed out that in coordinate representation, correlation functions on Moyal plane should be defined with star products extended to non-coincident points (see also [43]) instead of usual products since the usual commutative commutation relation [x i , x j ] = 0 (i, j = 1, · · · , n) is not invariant under the twisted Poincaré transformation. Carrying out Fourier transformation of the symmetry relation (109) in momentum representation to such a noncommutative coordinate representation, we obtain the symmetry relations in star tensor products. Namely (110) becomes 〈((xµ1∂ν − xν1∂µ)φ(x1)) ∗ φ(x2) + φ(x1) ∗ ((x ν − xν2∂µ)φ(x2))〉 = 0, (113) and (111) becomes 〈((xµ1∂ν − xν1∂µ)φ(x1)) ∗ φ(x2) ∗ φ(x3) + φ(x1) ∗ ((xµ2∂ν − xν2∂µ)φ(x2)) ∗ φ(x3) + φ(x1) ∗ φ(x2) ∗ ((x ν − xν3∂µ)φ(x3)) iθαµ(∂αφ(x1) ∗ ∂νφ(x2) ∗ φ(x3) + ∂αφ(x1) ∗ φ(x2) ∗ ∂νφ(x3) + φ(x1) ∗ ∂αφ(x2) ∗ ∂νφ(x3) − ∂νφ(x1) ∗ ∂αφ(x2) ∗ φ(x3)− ∂νφ(x1) ∗ φ(x2)∂α ∗ φ(x3)− φ(x1) ∗ ∂νφ(x2) ∗ ∂αφ(x3)) iθαν(∂αφ(x1) ∗ ∂µφ(x2) ∗ φ(x3) + ∂αφ(x1) ∗ φ(x2)∂µ ∗ φ(x3) + φ(x1) ∗ ∂αφ(x2) ∗ ∂µφ(x3) − ∂µφ(x1) ∗ ∂αφ(x2) ∗ φ(x3)− ∂µφ(x1) ∗ φ(x2) ∗ ∂αφ(x3)− φ(x1) ∗ ∂µφ(x2) ∗ ∂αφ(x3))〉 = 0. (114) More generally we can derive the symmetry relations of correlation functions for tensor fields φα1···αn(x) ≡ ∂α1 · · ·∂αnφ(x). For example in the case of the three point function of the tensor fields, the symmetry relation becomes 〈((M1µν)α1···αl δ1···δlφδ1···δl(x1)) ∗ φβ1···βm(x2) ∗ φγ1···γn(x3) + φα1···αl(x1) ∗ ((M 2µν)β1···βm δ1···δmφδ1···δm(x2)) ∗ φγ1···γn(x3) + φα1···αl(x1) ∗ φβ1···βm(x2) ∗ ((M 3µν)γ1···γn δ1···δnφδ1···δn(x3)) θαµ[∂αφα1···αl(x1) ∗ ∂ νφβ1···βm(x2) ∗ φγ1···γn(x3) + ∂αφα1···αl(x1) ∗ φβ1···βm(x2) ∗ ∂ νφγ1···γn(x3) + φα1···αl(x1) ∗ ∂αφβ1···βm(x2) ∗ ∂ νφγ1···γn(x3) − ∂νφα1···αl(x1) ∗ ∂αφβ1···βm(x2) ∗ φγ1···γn(x3) − ∂νφα1···αl(x1) ∗ φβ1···βm(x2)∂α ∗ φγ1···γn(x3) − φα1···αl(x1) ∗ ∂ νφβ1···βm(x2) ∗ ∂αφγ1···γn(x3)] θαν [∂αφα1···αl(x1) ∗ ∂ µφβ1···βm(x2) ∗ φγ1···γn(x3) + ∂αφα1···αl(x1) ∗ φβ1···βm(x2)∂ µ ∗ φγ1···γn(x3) + φα1···αl(x1) ∗ ∂αφβ1···βm(x2) ∗ ∂ µφγ1···γn(x3) − ∂µφα1···αl(x1) ∗ ∂αφβ1···βm(x2) ∗ φγ1···γn(x3) − ∂µφα1···αl(x1) ∗ φβ1···βm(x2) ∗ ∂αφγ1···γn(x3) − φα1···αl(x1) ∗ ∂ µφβ1···βm(x2) ∗ ∂αφγ1···γn(x3)]〉 = 0, (115) where (Mµν)α1···αn β1···βn = (Lµν)α1···αn β1···βn + (Sµν)α1···αn β1···βn (Lµν)α1···αn β1···βn = i(xµ∂ν − xν∂µ)δα1β1 · · · δαnβn (Sµν)α1···αn β1···βn = i(ηνβ1δ{α1 β2 · · · δαn}βn − ηµβ1δ{α1νδα2β2 · · · δαn}βn) (116) If we bring the operators (M iµν)α1···αn β1···βn (i = 1, 2, 3) out of the star products, θµν depen- dent terms are canceled. The final expressions are just the usual Lorentz rotations on the coordinates and the tensorial indices in the correlation functions. This is fully consistent with the discussions in [29]. 3.6 Origin of Hopf algebra symmetries To study more the meaning of these additional terms, let us see closer the transformation properties of the star products. In the latter case, it is known that the θµν dependence of the twisted Lorentz transformation (99) comes from the Lorentz transformation of θµν itself [41]. To see this, let us consider an infinitesimal Lorentz transformation, Λµν = δ ν + ǫ The transformation of θµν is given by θµν → θµν + ǫµρθρν + ǫνρθµρ := θµν + δθµν . (117) If one considers not only the transformation of the coordinates, x ′µ = xµ + ǫµνxν , but also (117), and assumes that φ(x) ∗θ φ(x) and φ′(x′) ∗θ+δθ φ′(x′) be equal, one obtains, after the Fourier transformation, φ̃′(p1)⊗ φ̃′(p2) (ǫµνMµν ⊗ 1+ 1⊗ ǫµνMµν + δθµνPµ ⊗ Pν) φ̃(p1)⊗ φ̃(p2) ǫµν∆′Mµν φ̃(p1)⊗ φ̃(p2), (118) which agrees with (99). This shows that the additional part of the coproduct of Mµν takes into account the transformation of the non-dynamical background parameter θµν . The former case can be discussed in a similar manner. The definition of the star product is given by iPi(g1) ⋆x e ixiPi(g2) = eix iPi(g1g2), (119) where we have explicitly indicated the coordinate where the star product is taken. Then we recognize that ei(x+ǫ) iPi(g1) ⋆x+ǫ e i(x+ǫ)iPi(g2) and ei(x+ǫ) iPi(g1) ⋆x e i(x+ǫ)iPi(g2) give distinct values. Namely, if the coordinate of the star product is also shifted, ei(x+ǫ) iPi(g1) ⋆x+ǫ e i(x+ǫ)iPi(g2) = ei(x+ǫ) iPi(g1g2), (120) but, if not, ei(x+ǫ) iPi(g1) ⋆x e i(x+ǫ)iPi(g2) = eiǫ iPi(g1)eiǫ iPi(g2)eix iPi(g1g2). (121) Therefore, if we take the translational transformation as (120), and carry out the same procedure in deriving (59), we always obtain a translational invariant commutation relation5, [(x+ ǫ)i, (x+ ǫ)j ]⋆x+ǫ = 2iκǫ ijk(x+ ǫ)k. (122) 5There is a similar discussion in [42]. Now, assuming that φ(x) ⋆x φ(x) and φ ′(x′) ⋆x′ φ ′(x′) be equal under the translation xi → ′i = xi + ǫi, we obtain, after the Fourier transformation, φ̃′(g1)φ̃ ′(g2) = (1− iǫiPi(g1g2))φ̃(g1)φ̃(g2), (123) which is the same as (75). From these two examples, we anticipate that the multi-field contributions in (41) comes from the transformation properties of the star products. 4 Summary and comments We have discussed symmetries in noncommutative field theories in the framework of braided quantum field theory. We have obtained the algebraic conditions for a Hopf algebra to be a symmetry of a braided quantum field theory, by discussing the conditions for the relations among correlation functions generated from the transformation algebra to hold. Then we have applied our discussions to the Poincaré symmetries in the effective noncommutative field theory of three-dimensional quantum gravity coupled with spinless particles and in the noncommutative field theory on Moyal plane. In the former case we can understand the braiding between fields, which was derived from the three-dimensional quantum gravity computation, from the viewpoint of the translational symmetry of the noncommutative field theory on a Lie-algebraic noncommutative spacetime. In the latter case we have found that the twisted Lorentz symmetry on Moyal plane is a symmetry of the quantum field theory only after the inclusion of the nontrivial braiding factor, which is in agreement with the previous proposal [28, 35]. Then we have discussed the meaning of the Hopf algebra symmetries from the viewpoint of coordinate representation. In the recent research a noncommutative field theory on κ-Minkowski spacetime is dis- cussed [36]. Since this noncommutativity of the coordinates is given by [x0, xj] = i xj , this noncommutative field theory will not have the naive translational symmetry. We may intro- duce a non-trivial braiding between fields as in the effective field theory discussed in Section 3.3 to keep the momentum conservation. However, while the effective field theory has the braided category structure because of the invariance of the Haar measure d(g−1g′g) = dg′, the measure of the momentum space of the field theory on κ-Minkowski spacetime is only left-invariant [36]. Therefore it is not clear to us whether we can embed this field theory on κ-Minkowski spacetime into the framework of braided quantum field theory. Acknowledgments We would like to thank S. Terashima and S. Sasaki for useful discussions and comments, and would also like to thank L. Freidel for stimulating discussions and explaining their recent results during his stay in Yukawa Institute for Theoretical Physics after the 21st Nishinomiya- Yukawa Memorial Symposium. Y.S. was supported in part by JSPS Research Fellowships for Young Scientists. N.S. was supported in part by the Grant-in-Aid for Scientific Research No.13135213, No.16540244 and No.18340061 from the Ministry of Education, Science, Sports and Culture of Japan. A The proofs of the formula (20), (21) We give the proofs of the formula (20), (21) using diagrams. At first we use the formula êv(∂ ⊗ αβ) = êv(∂ ⊗ α)ǫ(β) + êv(∂ ⊗ β)ǫ(α), (124) where α, β ∈ X̂. This is clear from the definition of êv. Figure 14 gives the proof of (20). In the first line, we use the axiom (12), and in the second line we use the lemma (124). We find the last line from the property of counit. Next we prove (21). By using the braided Leibniz rule (20) as α ∈ X ⊗ X̂, the left-hand side of (21) becomes Figure 15. The first term of Figure 15 becomes (ev ⊗ idn−1)(∂ ⊗ idnα) by using the definition of coproduct (9). In the second term of Figure 15, we divide X̂ into X ⊗ X̂ and iterate the same as we did above. For example, if the degree of X̂ is 3, the second term of Figure 15 can be reduced as in Figure 16. We have used ∆X = X⊗̂1+ 1⊗̂X in the second line of Figure 16. The result agrees with (21). In the same way, we can obtain the formula (21) in general. B The proofs of (27), (28), (29) From the definition of γ (25), we find that αaw = −αdiff(γ−1(a)⊗ w), (125) for a ∈ X and α ∈ X̂ . On the other hand, adding γ−1 and ψ to the braided Leibniz rule (20) as in Figure 17, we find that αdiff(γ−1(a)⊗ w) = diff(ψ(α⊗ γ−1(a))w)− (diff ◦ ψ(α⊗ γ−1(a)))w. (126) Combining (125), (126), we obtain αaw = −diff(ψ(α⊗ γ−1(a))w) + (diff ◦ ψ(α⊗ γ−1(a)))w. (127) Integrating the both hand sides of (127) and using (24), we find that Z(0)(αa) = Z(0)(diff ◦ ψ(α⊗ γ−1(a))). (128) If α is b ∈ X , Z(0)(ba) = Z(0)(diff ◦ ψ(b⊗ γ−1(a))) = ev ◦ ψ(b⊗ γ−1(a)) = ev ◦ (γ−1 ⊗ id) ◦ ψ(b⊗ a). (129) Figure 14: The proof of (20). Figure 15: The left-hand side of (21). Figure 16: The second term of Figure 15. 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Schmudgen, “Quantum groups and their representations,” Berlin, Germany: Springer (1997) 552 p [40] G. Ponzano and T. Regge, in “Spectroscopic and Group Theoretical Methods in Physics” ed. F. Bloch, North-Holland, Amsterdam, (1968). [41] L. Alvarez-Gaume, F. Meyer and M. A. Vazquez-Mozo, “Comments on noncommutative gravity,” Nucl. Phys. B 753, 92 (2006) [arXiv:hep-th/0605113]. [42] A. Agostini, G. Amelino-Camelia, M. Arzano, A. Marciano and R. A. Tac- chi, “Generalizing the Noether theorem for Hopf-algebra spacetime symmetries,” arXiv:hep-th/0607221. [43] R. J. Szabo, “Quantum field theory on noncommutative spaces,” Phys. Rept. 378, 207 (2003) [arXiv:hep-th/0109162]. http://arxiv.org/abs/hep-th/0703245 http://arxiv.org/abs/hep-th/0502106 http://arxiv.org/abs/gr-qc/0612145 http://arxiv.org/abs/gr-qc/0612144 http://arxiv.org/abs/hep-th/9906225 http://arxiv.org/abs/hep-th/0003018 http://arxiv.org/abs/hep-th/0612170 http://arxiv.org/abs/hep-th/9212151 http://arxiv.org/abs/hep-th/0605113 http://arxiv.org/abs/hep-th/0607221 http://arxiv.org/abs/hep-th/0109162 Introduction Review of braided quantum field theory Braided categories and braided Hopf algebras Braided quantum field theory Braided Feynman rules Symmetries in braided quantum field theory General description of an action Symmetry relations among correlation functions and their algebraic descriptions Symmetries of the effective noncommutative field theory of three-dimensional quantum gravity coupled with scalar particles Twisted Poincaré symmetry of noncommutative field theory on Moyal plane Relations among correlation functions : Examples Origin of Hopf algebra symmetries Summary and comments The proofs of the formula (??), (??) The proofs of (??), (??), (??)

0704.0823

Chromospheric Flares

Coimbra Solar Physics Meeting on The Physics of Chromospheric Plasmas ASP Conference Series, Vol. xxx, 2007 Petr Heinzel, Ivan Dorotovič and Robert J. Rutten, eds. Chromospheric Flares Hugh S. Hudson Space Sciences Laboratory, University of California, Berkeley Abstract. In this topical review I revisit the “chromospheric flare”. This should currently be an outdated concept, because modern data seem to rule out the possiblity of a major flare happening independently in the chromosphere alone, but the chromosphere still plays a major observational role in many ways. It is the source of the bulk of a flare’s radiant energy – in particular the visi- ble/UV continuum radiation. It also provides tracers that guide us to the coronal source of the energy, even though we do not yet understand the propagation of the energy from its storage in the corona to its release in the chromosphere. The formation of chromospheric radiations during a flare presents several difficult and interesting physical problems. 1. Introduction Solar flares first revealed themselves as visual perturbations of the solar atmo- sphere (“white light flares”) and hence immediately were construed as a pho- tospheric process. With the invention of spectroscopic techniques, though, it became clear that chromospheric emission lines such as Hα revealed flare pres- ence much more readily. This led to the concept of the “chromospheric flare” and to a great deal of observational material on Hα flares and eruptions, as reviewed by Smith & Smith (1963), Zirin (1966), or Švestka (1976), for example. At some point, prior to the discovery of coronal flare effects, the misinterpretations of the Hα line profile even led to the incorrect idea that a flare was a sudden cooling of the solar atmosphere. In any case, a perturbation of the lower solar atmosphere violent enough to affect the solar luminosity itself (“white light”) implies a large energy content. Our view of flares now emphasizes the high temperatures and non-thermal effects seen in the corona, and we generally believe the chromospheric effects themselves to be secondary in nature. This may be true, but nonetheless the modern observations confirm the fact that the lower solar atmosphere dominates the radiant energy budget of a flare via the UV and white-light continua. Some- how, therefore, the energy stored in the solar corona rapidly focuses down into regions visible in chromospheric signatures; this accounts for the high contrast of flare effects there. Thus the “chromospheric flare” remains essential to our understanding of the overall processes involved. The chromosphere nowhere exists as a well-defined layer with a reproducible height structure. In this paper I use the term interchangeably with “lower solar atmosphere,” embracing the phenomena of the visible photosphere through the transition region. During flares the structure of these “layers” and the physical conditions within them may change drastically. The changes generally http://arxiv.org/abs/0704.0823v1 2 Hudson happen so fast and on such small spatial scales that we cannot observe them comprehensively. Understanding the impulsive phase in the chromosphere may therefore seem like something of a lost cause from the the point of view of theory, especially in view of our inability to understand the quiet chromosphere any better than we do. The data repeatedly reveal that we simply have not yet resolved the spatial or temporal structures involved in the impulsive phase, and that without knowing the geometry of the physical structure, we cannot really comprehend its physics. The TRACE (Handy et al. 1999) and RHESSI (Lin et al. 2002) observations have provided more than one recent breakthrough, however, and it may be that we are beginning to understand the gradual phase of a flare at least. This review is organized around several topics involving the behavior of the chromosphere during a flare. These include the process of “chromospheric evaporation” (Section 4), flare energetics (Section 5), the mechanisms of flare continuum emission (Section 6), and the inference of flare structure from the morphology of the chromospheric flare (Section 7 and Section 9). In Section 2 and Section 3 we give an overview of the history of chromospheric flares and show a cartoon to establish a working model of a solar flare. Sections 8 and 11 discuss large-scale magnetic reconnection and theoretical ideas, and Section 10 presents a γ-ray mystery. 2. Historical Development Although it was the white-light continuum that initially revealed the existence of solar flares, the advent of spectroscopy (e.g., Hale 1930) allowed their regular observation via the Hα line (see Švestka 1966 for a discussion of the historical de- velopment of these observations). This strong absorption line actually becomes an emission line during bright flares, and Hα limb observations frequently show prominences and eruptions. Hα observers came to recognize a particular flare morphology, the so-called two-ribbon flare. Bruzek (1964) described the pat- terns followed by these events, which provided strong evidence that the solar corona had to play a major role in flare development. Figure 1 reproduces one of Bruzek’s sketches, and then illustrates in a cartoon (due to Anzer & Pneuman 1982) how this morphology led to our standard magnetic-reconnection scenario that tries to embrace the X-ray observations and the coronal mass ejections (CMEs) as well as the chromospheric ribbon structures. In this standard picture a solar flare develops in a complicated manner that involves restructuring of the coronal magnetic field in such a way as to release energy. The immediate effects of this energy release are to produce broad-band “impulsive phase” emissions and to drive chromospheric gas up into coronal magnetic loops, the process we term “chromospheric evaporation.” A part of the field magnetic structure may actually erupt and open out into the solar wind, in the sense that the field lines stretch out past the Alfvé n critical point of the flow. This opening may consist of rising loops which then take the form of a coronal mass ejection (CME), or it may involve interactions with previously open field (a process often termed “interchange reconnection” nowadays; see Heyvaerts et al. 1977). If a CME does accompany the flare, as it almost invariably does for flares of GOES class X or greater, the energy involved in mass motions may Chromospheric Flares 3 Figure 1. Left: one of Bruzek’s (1964) sketches, showing a flare with ribbons on the disk and its equivalent Hα “loop prominence system” over the limb. This key observational pattern led directly to the formation of our standard flare model (right), in the form presented by Anzer & Pneuman (1982). be comparable to the luminous energy (e.g., Emslie et al. 2005). Generally the observations are limited in resolution, both temporal and spatial, and especially in spectral coverage. Thus we often resort to a cartoon that serves to identify how the essential parts of a flare relate to one another. Soft X-ray observations show hot loops in the gradual phase of a flare. These result from the material “evaporated” from the chromosphere and have anomalously high gas pressure (but still low plasma β; however see Gary 2001). Whereas the pressure at the base of the corona normally is of order 0.1 dyne cm−2, a bright flare loop can achieve 103 dyne cm−2. This over-dense and over-hot coro- nal loop gradually cools, and in its final stages the remaining plasma returns to a more chromospheric state and suddenly becomes visible in Hα (Goldsmith 1971). The loops that have reached this state then form Bruzek’s Hα loop prominence system (Figure 1). During the ribbon expansion another important phenomenon occurs: hard X-ray emission appears at the footpoints of the coronal loops that are in the pro- cess of being filled by chromospheric evaporation (Hoyng et al. 1981). The hard X-rays show that a substantial part of flare energy appears in the form of non- thermal electrons (Kane & Donnelly 1971; Lin & Hudson 1976; Holman et al. 2003). The hard X-ray signature (and hence the energetic dominance of these electrons) is present whether or not the flare develops the two-ribbon morphol- ogy or has a CME association. The hard X-ray emission occurs in the impulsive phase of the flare, contem- poraneously with the period of chromospheric evaporation that fills the coro- nal loops and with the acceleration phase of the associated CME (Zhang et al. 2001). In Section 5 we describe this phase of the flare with the thick-target model (Kane & Donnelly 1971) which Hudson (1972) identified with the energy source of white-light flare continuum. 3. The Flare Spectrum A (major) flare can be observed at almost any wavelength in a fast-rise/slow- decay time profile, with some (e.g., the white-light continuum) having a more impulsive variability, and others (e.g., the Balmer lines) having a more gradual pattern (Figure 2, right). We generally describe a flare as consisting of a foot- 4 Hudson Figure 2. Left: Line widths of the Balmer-series lines, from the classic paper by Suemoto & Hiei (1959). The inferred densities added to the curves are logne = 13.5 and 13.3; the inferred filling factor is small, suggesting either filamentary structure or thin layering. Right: Typical time series of flare radiations, distinguishing the impulsive phase from the gradual phase (see Kane & Donnelly 1971). point and ribbon structures in the lower atmosphere, coronal loops, and various kinds of ejecta. The impulsive phase is typically associated with the footpoint structures, and the gradual phase with the flare ribbons. Nowadays imaging spectroscopy in principle allows us to study these regions independently. Flare spectroscopy began with the observation of the Balmer series, which shows broad lines tending towards emission profiles as the flare gets more en- ergetic. Early observations of the higher members of the sequence allowed the inference of a relatively high density and of a small filling factor (Suemoto & Hiei 1959; see the left panel of Figure 2). Such observations refer to what we would now call the gradual phase of the flare (see the right panel of Figure 2 for a sketch of the temporal development of a flare). In the impulsive phase the con- tinuum appears in emission, as noted originally by Carrington and by Hodgson independently. The weak photospheric metallic lines may also go weakly into emission (or are filled in by continuum) and the recent observations of Xu et al. (2004) show that flare effects can appear even at the “opacity minimum” region of the spectrum, where one would expect much higher densities. In fact a single density could never properly describe such a heterogeneous structure, but each spectral band provides its own clues. At the time of writing no proper analy- sis of spectroscopic “response functions” (e.g., Uitenbroek 2005) for any of the signatures has yet been attempted, so our inference of flare structure from the spectroscopy alone is weak. The continuum radiation seen in white light and the UV constitutes the bulk of flare radiated energy (Kane & Donnelly 1971; Woods et al. 2006). TRACE imaging of this emission component shows it to consist of unresolved, intensely Chromospheric Flares 5 bright fine structures (Hudson et al. 2006). The thick-target model invokes fast electrons (energies above about 10 keV) to transport coronal energy into the chromosphere. Here collisional losses provide the heating and footpoint emis- sions that accompany the hard X-ray bremsstrahlung. The thick-target model does not explain the particle acceleration, nor show how the footpoint sources can be so intermittent. We return to this question in Section 7. The spectra emitted at the footpoints of the flaring coronal loops have contributions over an exceptionally broad wavelength range, as sketched in the right panel of Figure 2. The prototypical observable is the hard X-ray flux, which imaging observations show to be concentrated at the footpoints (Hoyng et al. 1981), but impulsive footpoint emissions also occur in many spectral windows ranging from the microwaves (limited presumably by opacity) to the γ-rays (limited presumably by detection sensitivity). There is a large body of work on the Hα line alone, both observation and theory. Berlicki (2007) reviews the Hα spectroscopic material in detail in these proceedings. A strong absorption line forms across a wide range of continuum optical depths, and in principle this single line might provide sufficient information to infer the physical structure of the flare everywhere. In practice the complexities of the radiative transfer and of the flare motions, especially in the impulsive phase, make this information ambiguous (see Berlicki 2007). 4. Chromospheric Evaporation The motions most directly relevant to the chromosphere are often called “chro- mospheric evaporation,” even though the direct Doppler signatures of this mo- tion are normally found in lines formed at higher temperatures (but see Berlicki et al. 2005). That this process occurs (even if it is not “evaporation” strictly speak- ing) was suggested by the early observations of loop prominence systems (e.g., Bruzek 1964) with their “coronal rain,” and Neupert (1968) established its as- sociation with non-thermal processes such as bursts of microwave synchrotron radiation. The thermal microwave spectrum (e.g., Hudson & Ohki 1972) made it particularly clear that the gradual phase of a solar flare involves the temporary levitation of chromospheric material into the corona, as opposed to the process that might be imagined from the earlier term “sporadic coronal condensation” (e.g., Waldmeier 1963). The flows involved in chromospheric evaporation are along the field direction and serve to create systems of coronal loops with rel- atively high gas pressure and therefore higher (but still probably low) plasma beta. The early observational indications of chromospheric evaporation actually came from blueshifts in EUV and soft X-ray lines (e.g., Antonucci et al. 1982; Acton et al. 1982) such as those from FeXXV or CaXIX. Figure 3 shows an image-resolved view of Doppler shifts in an evaporative flow (Czaykowska et al. 1999). The chromospheric effects are more subtle and in fact the impulsive-phase evaporation is difficult to disentangle from other effects (Schmieder et al. 1987). The high-temperature blueshifts correspond to upward velocities of some hun- dreds km/s and seldom appear in the absence of a stationary emission line; in other words, hot plasma has already accumulated in coronal loops as the process continues. Based on theory and simulations (Fisher et al. 1985) one can distin- 6 Hudson Figure 3. Imaging spectroscopy from SOHO/CDS of EUV emission lines in the gradual phase of a two-ribbon flare, showing the clear signature of blueshifted upflows in the expected locations along the flare ribbons. This is “gentle” evaporation not associated with strong hard X-ray emission (from Czaykowska et al. 1999). Note that CDS produces images by scanning in one spatial dimension, so that each image (while monochromatic) is not instanta- neous. guish “explosive” and “gentle” evaporation, depending upon the physics of en- ergy deposition (e.g., Abbett & Hawley 1999). In explosive evaporation, driven hypothetically by an electron beam, one has the additional complication of a “chromospheric condensation” that produces a redshift as well. Schmieder et al. (1987) and Berlicki et al. (2005) survey our overall understanding. It would be fair to comment that the explosive evaporation stage remains ill-understood, even though it principle it describes the key physics of sudden mass injections into flare loops. Chromospheric Flares 7 -500 0 500 1000 1500 2000 2500 3000 VAL C Height, km -500 0 500 1000 1500 2000 2500 3000 VAL C Height, km 10000 Figure 4. Characteristic radiative cooling time (upper) as a function of height in the VAL-C model, crudely estimated as described in the text. The lower panel shows the temperature in this model. 5. Energetics and Magnetic Field We can use the standard VAL-C model (Vernazza et al. 1981), as discussed further in the Appendix, to discuss the energetics. First we establish that the chromosphere and the rest of the lower solar atmosphere (i.e., that for which τ5000 < 1) have negligible heat capacity and limited time scales. Figure 4 shows an estimate of the radiative time scale in the VAL-C model (Vernazza et al. 1981). This shows 3σ(z)kT/L⊙, where σ is the surface density as a function of height about the τ5000 = 1 layer, and L⊙ the solar luminosity. The time scale decreases below 1 sec only above z ∼ 515 km, near the temperature minimum in the VAL-C model. Above this height any energy injected into the system will tend to radiate rapidly, resulting in a direct energy balance between input and output energy, rather than a local storage and release. At lower altitudes we would not expect to see rapid variability. The model also allows us to ask whether the chromosphere itself can store energy comparable to that released in a major flare or CME. Table 1 gives some order-of-magnitude properties for a chromospheric area of 1019 cm2, showing both possible sources (bold) and sinks (italics) of energy. For the magnetic field we simply assume 10 or 1000 G as representative cases. Using the total magnetic energy in this manner is an upper limit, since the actual free energy would depend on its degree of non-potentiality. We find that magnetic energy storage limited to the volume of the chromosphere will not suffice, unless unobservably small-scale fields there somehow dominate. The gravitational potential energy also will not suffice. Estimates of this sort confirm the idea that the flare energy must reside in the corona prior to the event. The estimate of gravitational potential energy is somewhat more ambiguous. The Table shows the value needed to displace the entire atmosphere by its total thickness, the equivalent of roughly 3′′ in the VAL-C model. There does not seem to be any evidence for such a displacement, although I am not aware of any searches. It is likely that the stresses that store energy in the coronal field 8 Hudson have their origin deeper in the convection zone, rather than in the atmosphere (McClymont & Fisher 1989). Actually the observable changes of gravitational energy are even of the wrong sign, given that we normally observe only outward motions, (against gravity) during a flare. Table 1. Properties of a chromospheric volume of area 1019 cm2 Parameter VAL-C VAL-C above Tmin Mass 4×1019 gram 5×1017 gram Magnetic energy 1×1028erg 8×1027 erg (10 G) Magnetic energy 1×1032 erg 8×1031 erg (103 G) Gravitational energya 3×1032 erg 3×1030 erg Thermal energy 2×1031 erg 3×1029 erg Kinetic energy 3×1029 erg 3×1027 erg Ionization energy 4×1032 erg 6×1030 erg aPotential energy for a vertical displacement of 2.5 × 108 cm From Table 1 one concludes that the chromosphere probably does play a dominant role in the energetics of a solar flare, at least as described by a semi- empirical model such as VAL-C. This just restates the conventional wisdom, namely that the flare energy needs to be stored magnetically in the corona, rather than in the chromosphere where the radiation forms. Note that this is backwards from the relationship for steady emissions: the requirement for chromospheric heating is larger than that for coronal heating, so it is possible to argue that the steady-state corona actually forms as a result of energy leakage from the process of chromospheric heating (e.g., Scudder 1994). We can make a similar estimate for energy flowing up from the photosphere. The Alfvén speed at τ5000 = 1 ranges from 3 to 30 km s −1, depending on the field strength, in the VAL-C model (see Appendix). Below the surface of the Sun vA drops rapidly because of the increase of hydrogen ionization. Thus chromo- spheric flare energy cannot have been stored just below the photosphere, since it could not propagate upwards rapidly enough (McClymont & Fisher 1989). This again supports the conventional wisdom that flare energy resides in the corona prior to the event. To drive chromospheric radiations from coronal energy sources requires ef- ficient energy transport, which is normally thought to be in the form of non- thermal particles (the “thick target model”, Brown 1971; Hudson 1972) or in the form of thermal conduction as in the formation of the classical transition region. Both of these mechanisms provide interesting physical problems, but the impulsive phase of the flare (where the thick-target model usually is thought to apply) certainly remains less understood. Section 11 comments on models. The magnetic field in the chromosphere is decisively important but ill- understood. The plasma beta is generally low (see Appendix), so just as in the corona the dynamics depends more on the behavior of the field itself than to the other forces at work. Generally we believe that the subphotospheric field exists in fibrils, implying the existence of sheath currents that isolate the flux tubes from their unmagnetized environment. On the other hand, the dominance of plasma pressure in the chromosphere as well as the corona implies that the Chromospheric Flares 9 field must rapidly expand to become space-filling. Longcope & Welsch (2000) discuss the physics involved in this process as flux emerges from the interior. The effect of the flux emergence must be to create current systems linking the sources of magnetic stress below the photosphere, with the non-potential fields containing the coronal free energy. A full theory of how this works does not exist, and we must add to the uncertainty the possibilty of unresolved fields (e.g., Trujillo Bueno et al. 2004). Their suggested factor of 100 in B2 would clearly affect the estimate of magnetic energy given in Table 1 and perhaps change everything. We note in this context that the “impulse response” flares (White et al. 1992) have scales so small that one could argue for an entirely chromospheric origin. 6. Energetics and the Formation of the Continuum The formation of the optical/UV emission spectrum of a solar flare has from the outset presented a special challenge, since (a) it represents so much energy, and (b) it appears in what should be the stablest layer of the solar atmosphere. The recent observations of rapid variability and spatial intermittency make this all the more interesting, and these observations – now from space – also help to intercompare events; previous catalogs of white-light flares (e.g., Neidig 1989 and references therein) had to be based on spotty observations made with a wide variety of instruments. Observationally, the continuum appears to have two classes, with most events (“Type I” spectra) showing evidence for recombination radiation via the presence of the Balmer edge and sometimes the Paschen edge as well. A few events (e.g., Machado & Rust 1974) show spectra with weak or unobservable Balmer jumps, implicating H− continuum as observed in normal photospheric radiation. The spectra in the latter class (“Type II”) suggests a relationship to Ellerman bombs (Chen et al. 2001). However, the physics of Ellerman bombs appears to be quite different from that of solar flares (e.g., Pariat et al. 2004), though. The strong suggestion from correlations is that non-thermal electrons phys- ically transport flare energy from the corona, where it had been stored in the current systems of non-potential field structures, into the radiating layers. The hard X-ray bremsstrahlung results from the collisional energy losses of these particles, and other signatures (such as the optical/UV continuum) depends on secondary effects. Proposed mechanisms include direct heating, heating in the presence of non-thermal ionization, and radiative backwarming. In some manner these effects (or others not imagined) must provide the emissivity ǫν , to support the observed spectrum. Note that the emissivity is often expressed in terms of the source function Sν = ǫν/κν via the opacity κν . In a steady state one would have energy balance between the input (e.g., electrons) and the contin- uum. Fletcher et al. (2007) have now shown that this implies energy transport by low-energy electrons, below 25 keV, as opposed to the 50 keV or higher sug- gested by some earlier authors. Such low-energy electrons have little penetrating power and could not directly heat the photosphere itself from a coronal accel- eration site. Thus either the continuum arises from altered conditions in the chromosphere, or some mechanism must be devised to link the chromosphere and photosphere not involving the thick-target electrons. 10 Hudson “Radiative backwarming” – for example Balmer and Paschen continuum excited in the chromosphere and then penetrating down to and heating a deeper layer – could in principle provide a vertical step between energy source and sink. One problem with this is that the weaker backwarming energy fluxes might not cause appreciable heating in the denser atmosphere, and thus not be able to contribute to the observed continuum excess, because of the short radiative time scale. This idea is a variant of the mechanism of non-thermal ionization originally proposed by Hudson (1972) in the “specific ionization approximation,” which involves no radiative-transfer theory and simply assumes ion-electron pairs to be created locally at a mean energy (∼30 eV per ion pair). Finally, the rapid variability observed in the continuuum, even at 1.56 µm (Xu et al. 2006) provides a clear argument that the continuum forms at the temperature minimum or above (see Section 5, especially Figure 4). Early proponents of particle heating as an explanation for white-light flares also considered protons as an energy source (Najita & Orrall 1970; Švestka 1970). This made sense, because protons at energies even below those char- acteristic of γ-ray emission-line excitation can penetrate more deeply than the electrons that produce hard X rays. It makes even more sense now that we have the suggestion that ion acceleration in solar flares may rival electron accelera- tion energetically (Ramaty et al. 1995). Simnett & Haines (1990) suggest that particle acceleration in solar flares involves a neutral beam, implying that the major energy content (and hence the optical/UV continuum) would originate in the ion component. This idea does not appear to explain the apparent simul- taneity of the footpoint sources (Sakao et al. 1996), and at present we do not understand the plasma physics of the particle acceleration and propagation well enough even to identify the location of the acceleration region. 7. Flare Structures Inferred from Chromospheric Signatures The continuum kernels may move systematically for perhaps tens of seconds and generally have short lifetimes. We illustrate this in Figure 5 (from Fletcher et al. 2004). This shows the motions of individual UV bright points within the flare ribbon structure. Such motions are only apparent motions, as in a deflagration wave, because they exceed the estimated photospheric Alfvén speed (see Sec- tion 4 and the Appendix). Figure 6 (from Hudson et al. 2006) makes the same point for a different flare, using TRACE white-light observations. The basic picture one gets from such observations is that the white light/UV continuum of a flare appears in compact structures that are essentially unresolved in space and in time within the present observational limits. These bright points con- tain enormous energy and thus must map directly to the energy source. We do not know if the fragmentation (intermittency) results from this mapping or is intrinsic to the basic energy-release mechanism. How do the small chromospheric sources map into the corona, where the flare energy must reside on a large scale before its release? A strong literature has grown up regarding this point, interpreting the ribbon motions as measures of flux transfer in the standard magnetic-reconnection model (Poletto & Kopp 1986; see also literature cited, for example, by Isobe et al. 2005). The flux transfer in the photosphere is taken to measure the coronal inflow into the re- Chromospheric Flares 11 Figure 5. Flare footpoint apparent motions deduced from TRACE UV ob- servations. Each squiggle represents the track of a bright point visible for several consecutive images at a few-second cadence, with the black dot show- ing the beginning of each track (Fletcher et al. 2004). Figure 6. Intermittent structure seen in TRACE white-light images of an M-class flare on July 24, 2004. The individual frames have dimensions 32′′× 64′′. Note the presence of bright features consistent with the TRACE angular resolution, and which change from frame to frame over the 30-second interval. These observations do not appear to resolve the fluctuations either in space or in time (Hudson et al. 2006). connecting current sheet, which appears to correlate with the radiated energy as seen in hard X-rays, UV, or Hα. Figure 1 (right) shows the assumed geometry linking the chromosphere and corona. The analysis extends to the multiple si- multaneous UV footpoints apparently moving within the ribbons as they evolve, as noted in Figure 5 above. The analyses suggest a strong relationship between energy release and the inferred coronal Alfvén speed. 12 Hudson Figure 7. Left: UV ribbons (TRACE observations) from a flare of Novem- ber 23, 2000. The gray scale shows the time sequence of brightening. Right: Correlation between pixel brightness in Ribbon A and the inferred reconnec- tion rate (from Saba et al. 2006). 8. Dynamics and Magnetic Reconnection To release energy from coronal magnetic field in a largely “frozen-field” plasma, a flare must involve mass motions. We often do observe apparent motions, both parallel and perpendicular to the field as indicated by the image striations (“loops”). Most of the observable motions are outward, leading to the idea of a “magnetic explosion” (e.g., Moore et al. 2001). Motions apparently perpendicu- lar to the magnetic field may become coronal mass ejections (CMEs) and contain a great deal of energy (e.g., Emslie et al. 2005). These perpendicular motions also are involved in flare energy release; for example the large-scale magnetic reconnection involved in many flare models (Figure 1, right panel) necessarily involve “shrinkage” (e.g., Švestka et al. 1987; Forbes & Acton 1996). Note that this process is more of a magnetic implosion than a magnetic explosion (Hudson 2000). The motion of flare footpoints and ribbons is (we believe) only apparent, because of the low Alfvén speed vA in the photosphere, where the field is tem- porarily anchored (“line-tied”). For B =1000 G and n = 1017 cm−3 we find vA ∼ 6 km s −1; observations often suggest motions an order of magnitude faster (e.g., Schrijver et al. 2006). The motions therefore represent a wave-light confla- gration moving through a relatively fixed magnetic-field structure. It is natural to imagine that this sequence of field lines links to the coronal energy-release site, which the standard model identifies with a current sheet that mediates large-scale magnetic reconnection. Figure 7 shows one example of the result of an analysis of the apparent motion of a flare ribbon (Saba et al. 2006). This and other similar analyses reveal a tendency for the “reconnection rate” to correlate with the pixel brightness. The reconnection rate is the rate at which flux is swept out in the ribbonmotion, often expressed as an electric field from E = v×B (the so-called “reconnection electric field”). In this picture the flare ribbons are identified with “quasi-separatrix structures” where magnetic reconnection can take place most directly. Chromospheric Flares 13 9. Surges, Sprays, and Jets Chromospheric material also appears in the corona in the form of surges and sprays, which may have a close relationship to the flare process (e.g., Engvold 1980). In addition, of course, we observe filaments and prominences in chromo- spheric lines, and these also have a flare/CME association, but too tangential for discussion in this review. Surges and sprays are Hα ejecta, rising into the corona as a result of chro- mospheric magnetic activity. The literature traditionally distinguishes them by apparent velocity, with the faster-moving sprays taken to have stronger flare associations. Surges often appear to return to the Sun, while sprays acceler- ate beyond the escape velocity and do not return. Both appear to move along the magnetic field lines, but unlike the evaporation flow the surges and sprays incorporate material at chromospheric temperatures. Modern soft X-ray and EUV data (Yohkoh, SOHO, and TRACE) have had sufficient time resolution to reveal the phenomenon of X-ray jets (Shibata et al. 1992); see also the UV observations of Dere et al. 1983. These tightly-collimated structures at X-ray temperatures have a strong correlation with surges and sprays, and indeed presumably lead to the jet-like CMEs seen at much greater altitudes (Wang & Sheeley 2002). These events have a strong association with emerging flux, and indeed the X-ray jets invariably have an association with mi- croflares and originate in the chromosphere near the microflare loop(s) (Shibata et al. 1992). As Zirin famously remarked, most emerging flux emerges within active regions, and that is where the jets preferentially occur. The site is frequently in the leading part of the sunspot group. Figure 8 (Canfield et al. 1996) shows the sequence of events in an explanation of these phenomena invoking magnetic reconnection to allow chromospheric material access to open fields. Note that this scenario imposes two requirements on the chromosphere: there must be open and closed fields juxtaposed, and a large-scale reconnection process must be able to proceed under chromospheric conditions. The Canfield et al. (1996) observations strongly imply that this process requires the presence of vertical electric currents supporting the observed twisting motions. The surges, sprays, and jets, not to mention flares and CMEs, underscore the time dependence and three-dimensionality characterizing what is often char- acterized as a thin time-independent layer for convenience. The subject of spicules is outside the scope of this review, but we note that they represent a form of activity that occurs ubiquitously outside the magnetic active regions. 10. A Chromospheric γ-ray Mystery The γ-ray observations of solar flares have begun, as did the radio and X-ray observations before them, to open new windows on flare physics. Share et al. (2004) have made a discovery that is difficult to understand and which in- volves chromospheric material. They report observations of the line width of the 0.511 MeV γ-ray emission line formed by positron annihilation (Figure 9). This emission requires a complicated chain of events: the acceleration of high-energy ions, their collisional braking and nuclear interactions in the solar atmosphere, the emission of secondary positrons by the excited nuclei, the collisional braking 14 Hudson Figure 8. A mechanism for jet/surge formation involving emerging flux (upper left), with magnetic reconnection against already-open fields (upper right), which may lead to a high-temperature ejection (the jet) entraining chromospheric material (the surge). The cartoon at lower right describes the observations of (Canfield et al. 1996), who find a spinning motion suggesting that the process must occur in a 3D configuration rather than that of the cartoons left and above. of these energetic positrons in turn, and finally their recombination with ambient electrons to produce the 0.511 MeV γ-rays. Because the γ-ray observations are so insensitive, this process requires an energetically significant level of particle acceleration that is possibly distinct from the well-known electron acceleration in the impulsive phase. The mystery comes in the line width of the emission line. Surprisingly the pioneering RHESSI observations of Share et al. (2004) showed it to be broad enough to resolve. The likeliest source of this line broadening is Doppler mo- tions in the positron-annihilation region. This requires the existence of a large column density (of order gram cm−2) at transition-region temperatures; the transition region under hydrostatic conditions would be many orders of mag- nitude thinner (see also Figure 11). According to Schrijver et al. (2006), the excitation of the footpoint regions during the the time of intense particle accel- eration only continues for some tens of seconds at most. This would represent the time scale for the apparent motion of a foopoint source across its diameter. The γ-ray observations, on the other hand, require minutes of integration for a statistically significant line-profile measurement. We therefore are confronted with a major problem. What is the structure of the flaring atmosphere that permits the formation of the broad 0.511 MeV γ-ray line? Recent spectroscopic observations of the impulsive phase in the UV, as viewed off the limb (Raymond et al. 2007) make a conventional explanation difficult. Chromospheric Flares 15 500 505 510 515 520 Energy (keV) 0.000 0.005 0.010 0.015 Figure 9. RHESSI γ-ray observations of the 0.511 MeV line of positron annihilation (Share et al. 2004). The two line profiles are from different inte- grations in the late phase of the X17 flare of 28 October 2003; for the broader line the authors suggest thermal broadening, which would require a large column depth of transition-region temperatures during the flare. 11. Theory and Modeling To understand the chromospheric spectrum of a solar flare we must understand the formation of the radiation and its transfer in the context of the motions produced by (or producing) the flare. The representation of the spectrum by a “semi-empirical model” represents one shortcut; in such an approach (e.g., the standard VAL model that we use in the Appendix) one attempts to construct a model atmosphere capable of describing the spectrum even if it may not be physically self-consistent. Such descriptions may however be sufficient in the gradual phase of a flare when the flare loops no longer have energy input and simply evolve by cooling and draining. Even here, however, we do not have a good understanding of the “moss” regions that form at the footpoints of these high-pressure loops (but see Berger et al. 1999). So far as I am aware there is no literature specifically on “spreading moss,” the similar structure that appears in association with flare ribbons. A more complete approach to the physics comes from “radiation hydrody- namics” physical models, most recently those of Allred et al. (2005); see Berlicki (2007) for a fuller description. Such models solve the equations of hydrodynam- ics and radiative transfer simultaneously and can thus deal with chromospheric evaporation and the formation of the high-pressure flare loops. This frame- work is necessary if we are to be able to understand the flare impulsive phase (e.g., Heinzel 2003). Even these models do not have sufficient realism, though, since they work currently in one dimension and thus cannot follow the time development of the excitation properly; the high-resolution observations of UV and white light by TRACE clearly show that the energy release has unresolved scales. Further, as pointed out by Hudson (1972), the ionization of the chromo- sphere (and hence the formation of the continua) cannot be described by a fluid 16 Hudson Figure 10. Continuum emission in the near infrared (1.56 µm, the “opacity minimum” region) during an X10 flare (Xu et al. 2004). Red shows the IR emissions, contours show the RHESSI 50-100 keV X-ray sources. The IR contrast relative to the preflare photosphere reached ∼20% in this event. approximation, or even by non-LTE radiative transfer that assumes a unique temperature. At present there has been little effort to create an electrodynamic theory of chromospheric flare processes, even though non-thermal particles are widely thought to provide the dominant energy in at least the impulsive phase. In the gradual phase there is interesting physics associated with heat conduction because the transition region would have to become so steep that classical con- ductivity estimates have difficulty (Shoub 1983). A more complete theory would have to take plasma effects into account and would probably contain elements of theories of the terrestrial aurora that are now largely missing from the solar lexicon. This lack of self-consistency in the modeling probably means that we have major gaps in our understanding of, for example, the evaporation process as it affects the fractionation of the elements and of the ionization states of the flare plasma. The Appendix gives estimates of the ranges of some the key plasma parameters in the chromosphere. 12. Conclusions This article has reviewed chromospheric flare observations from the point of view of the newest available information – Yohkoh, SOHO, TRACE and RHESSI, for example, but not Hinode or STEREO (already launched), nor much less FASR or ATST (not launched yet at the time of writing). spite of the high quality of the data prior to these missions, we still find major unsolved problems: • How does the chromosphere obtain all of the energy that it radiates? • How can flare effects appear at great depths in the photosphere? • How is the anomalous 0.511 MeV line width produced? Chromospheric Flares 17 • What are the elements of an electrodynamic theory of chromospheric flares? In my view the solution of these problems cannot be found in chromospheric observations alone, because the physical processes involve much broader regions of the solar atmosphere. Even providing answers to these specific questions may not reveal the plasma physics responsible for flare occurrence, which may involve spatial scales too fine ever to resolve. But we can hope that new observations from space and from the ground, in wavelengths ranging from the radio to the γ-rays, will enable us to continue our current rapid progress, and can speculate that eventually numerical tools will supplement the theory well enough for us to achieve full comprehension of the important properties of flares. To get to this point we will need to deal with the chromosphere, as messy as it is. One important task that is probably within our grasp now is the compu- tation of response functions for physical models of flares. At present these are restricted to very limited numerical explorations of the radiative transfer within the framework of one-dimensional radiation hydrodynamics (e.g., Allred et al. 2005). The energy transport in these models has been restricted to simplistic representations of particle beams for energy transport, and do not take account of complicated flare geometries, waves, or various elements of plasma physics. Future developments of chromospheric flare theory will need to complete the picture in a more self-consistent manner. Acknowledgments This work has been supported by NASA under grant NAG5- 12878 and contract NAS5-38099. I thank W. Abbett for a critical reading. I am also grateful to Rob Rutten for LaTeX instruction, and to Bart de Pontieu for meticulous keyboard entry. Appendix: plasma parameters The lower solar atmosphere marks the transition layer between regions of strik- ing physical differences, and as one goes further up in height the tools of plasma physics should become more important. This Appendix evaluates for conve- nience several basic plasma-physics parameters for the conditions of the staple VAL-C atmospheric model (Vernazza et al. 1981)1. This is a “semi-empirical model” in which interprets a set of observations in terms of the theory of ra- diative transfer, but without any effort to have self-consistent physics. Such a model can accurately represent the spectrum but may or may not provide a good starting point for physical analysis. Because the optical depth of a spec- tral feature is the key parameter determining its structure, one often sees the model parameters plotted against continuum optical depth τ5000 evaluated at 5000Å. Just for illustration, Figure 11 shows the VAL-C temperature separately as a function of height, column mass, and optical depth. Note that features prominent in one display may appear to be negligible in another The VAL-C model is an “average quiet Sun” model, and like all static 1D models, it cannot describe the variability of the physical parameters that theory The VAL-C parameters are available within SolarSoft as the procedure VAL C MODEL.PRO. 18 Hudson 10000 1000 100 10 Height, km 10-10 10-8 10-6 10-4 10-2 100 Continuum opacity 10-8 10-6 10-4 10-2 100 102 Column mass, g/cm3 Figure 11. Illustration of the structure of a semi-empirical model, using three different independent variables: the VAL-C temperature plotted against height, optical depth, and column mass. and observation require (see other papers in these proceedings, e.g., Carlsson’s review). Thus we should regard the plasma parameters estimated here as order- of-magnitude estimates only and note especially that the vertical scales, which depend in the model on the inferred optical-depth scale, may be systematically displaced. The VAL-C model explicitly does not represent a chromosphere perturbed by a flare. Vernazza et al. (1981) and many other authors give more appropriate models derived by similar techniques for flares as well as other structures. As the discussion of the γ-ray signatures in Section 10 suggests, though, a pow- erful flare may be able distort the lower solar atmosphere essentially beyond all recognition (especially in the impulsive phase). To estimate representative plasma parameters I have therefore chosen just to start with the basic VAL-C model, and we simply assume constant values of B at 10 G and 1000 G. The actual magnetic field may vary through this region (the “canopy”) but the de- tails are little-understood. The γ-ray literature usually uses a parametrization of the magnetic field strength B ∝ Pαg (Zweibel & Haber 1983), where Pg is the gas pressure. The most complicated behavior of the plasma parameters happens prefer- entially near the top of the VAL-C model range (for example, Figure 12 shows that the collision frequency decreases below the plasma and Larmor frequencies) above the helium ionization level (or even below this level for strong magnetic fields). Because VAL-C ignores time dependences and 3D structure, and as- sumes Te = Ti, we can expect that it has diminished fidelity as one approaches the unstable transition region; thus one should be especially careful not to take these approximations too literally. The following notes correspond to each panel of the figure. Most of the plasma-physics formulae used in this Appendix are from Chen (1984). Chromospheric Flares 19 Figure 12. Various plasma parameters in the VAL-C model. We have as- sumed representative B values of 10 G and 1000 G. The different panels show the following, left to right and top to bottom: (a) Temperature. (b) Den- sities: solid, total hydrogen density; dotted, electron density; dashed, He I density; dash-dot, He II density. (c) Plasma beta: solid, for 1000 G; dotted, for 10 G; light solid, electron density as a fraction of total hydrogen den- sity; dash-dot, the plasma parameter. (d) Frequencies. Solid, electron and ion plasma frequencies; dotted, electron gyrofrequencies for 10 and 1000 G; dashed, electron and ion collision frequencies; dash-dot, electron/neutral col- lision frequency. (e) Velocities: Solid, electron and ion thermal velocities; dashed, Alfvén speeds for 10 and 1000 G. (f) Scale lengths: solid, electron Larmor radii for 10 G and 100 G; dotted, Debye length; dashed, ion and electron inertial lengths. Temperature: The VAL-C model, like all of the semi-empirical models, sets Te = Ti. It therefore cannot support plasma processes dependent upon dif- ferent ion and electron temperatures, or more complicated particle distribution functions (e.g. Scudder 1994). Densities: Total hydrogen density, electron density, and densities of He I and He II. Dimensionless parameters: We approximate the plasma beta as 2(nH + 2ne)kT B2/8π with nH the hydrogen density, ne the electron density, Figure 12(c) gives the number of electrons in a Debye sphere as the “plasma parameter” Λ. 20 Hudson Frequencies: The plasma frequency, the electron and proton Larmor frequen- cies, and the electron and ion and collision frequencies νei = 2.4 × 10 −6nelnΛ/T eV ; νii = 0.05 × 4νei; νeH = (nH/ne)νe with ne in cm −3, TeV the temperature in eV, using Z = 1.2 and the Coulomb logarithm lnΛ = 23 - ln(n0.5e T eV ) (Chen 1984; De Pontieu et al. 2001). Note that the collision frequencies are small compared with the plasma and Larmor frequencies above about 1000 km in this model. This means generally that plasma processes must have strong effects on the physical parameters of the atmosphere in this region. 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0704.0824

On the (3,N) Maurer-Cartan equation

On the (3, N) Maurer-Cartan equation Mauricio Angel, Jaime Camacaro and Rafael Dı́az Abstract Deformations of the 3-differential of 3-differential graded algebras are controlled by the (3, N) Maurer-Cartan equation. We find explicit formulae for the coefficients appearing in that equation, introduce new geometric examples of N -differential graded algebras, and use these results to study N Lie algebroids. AMS Subject Classification: 53B99, 18G99, 18G99. Keywords: Lie algebroids, N -complexes, Higher differentials. 1 Introduction In this work we study deformations of the N -differential of a N -differential graded algebra. According to Kapranov [18] and Mayer [24, 25] a N -complex over a field k is a Z-graded k- vector space V = n∈Z Vn together with a degree one linear map d : V −→ V such that dN = 0. Remarkably, there are at least two generalizations of the notion of differential graded algebras to the context of N -complexes. A choice, introduced first by Kerner in [20, 21] and further studied by Dubois-Violette [13, 14] and Kapranov [18], is to fix a primitive N -th root of unity q and define a q-differential graded algebra A to be a Z-graded associative algebra together with a linear operator d : A −→ A of degree one such that d(ab) = d(a)b + qāad(b) and dN = 0. There are several interesting examples and constructions of q-differential graded algebras [1, 2, 6, 8, 9, 15, 16, 19, 21]. We work within the framework of N -differential graded algebras (N -dga) introduced in [4]. This notion does not depend on the choice of a N -th primitive root of unity, and thus it is better adapted for differential geometric applications. A N -differential graded algebra A consist of a Z-graded associative algebra A = n∈Z An together with a degree one linear map d : A −→ A such that dN = 0 and d(ab) = d(a)b + (−1)āad(b) for a, b ∈ A. The main question regarding this definition is whether there are interesting examples of N -differential graded algebras. Much work still needs to be done, but already a variety of examples has been constructed in [4, 5]. These examples may be classified as follows: • Deformations of 2-dga into N -dga. This is the simplest and most direct way to construct N -differential graded algebras. Take a differential graded algebra A with differential d and consider the deformed derivation d+e where e : A −→ A is a degree one derivation. It http://arxiv.org/abs/0704.0824v3 is possible to write down explicitly the equations that determine under which conditions d+ e is a N -differential, and thus turns A into a N -differential graded algebra. In other words one can explicitly write down the condition (d+ e)N = 0. • N flat connections. Let E be a vector bundle over a manifold M provided with a flat connection ∂E . Differential forms on M with values in End(E) form a differential graded algebra. An End(E)-valued one form T determines a deformation of this algebra into a N -differential graded algebra with differential of the form ∂E + [T, ] if and only if T is a N -flat connection, i.e., the curvature of T is N -nilpotent. • Differential forms of depth N ≥ 3. Attached to each affine manifold M there is a (dim(M)(N − 1) + 1)-differential graded algebra ΩN (M), called de algebra of differential forms of depth N on M , constructed as the usual differential forms allowing higher order differentials, i.e., for affine coordinates xi on M , there are higher order differentials d for 1 ≤ j ≤ N − 1. • Deformations of N -differential graded algebras into M -differential graded algebras. If we are given a N -differential graded algebra A with differential d, one can study under which condition a deformed derivation d + e, where e is a degree one derivation of A, turns A into a M -differential graded algebra, i.e., one can determine conditions ensuring that (d + e)M = 0. In [4] we showed that e must satisfy a system of non-linear equations, which we called the (N,M) Maurer-Cartan equation. • Algebras AN∞. This is not so much an example of N -differential graded algebras but rather a homotopy generalization of such notion. AN∞ algebras are studied in [7]. This paper has three main goals. One is to introduce geometric examples of N - differential graded algebras. We first review the constructions of N -differential graded algebras outlined above and then proceed to consider the new examples: • Differential forms on finitely generated simplicial sets. We construct a contravariant func- tor ΩN : set ∆op −→ N ildga from the category of simplicial sets generated in finite dimen- sions to N ildga, the category of nilpotent differential graded algebras, i.e., N -differential graded algebras for some N ≥ 1. For a simplicial set s we let ΩN (s) be the algebra of algebraic differential forms of depth N on the algebro-geometric realization of s. For each integer K we define functor Sing≤K : Top −→ set ∆op , thus we obtain contravariant func- tors ΩN ◦Sing≤K : Top −→ N ildga assigning to each topological space X a nil-differential graded algebra. • Difference forms on finitely generated simplicial sets. We construct a contravariant functor DN defined on set ∆op with values in a category whose objects are graded algebras which are also N -complexes for some N , with the N -differential satisfying a twisted Leibnitz rule. For a simplicial set s we let DN (s) be the algebra of difference forms of depth N on the integral lattice in the algebro-geometric realization of s. Again, for each integer K ≥ 0 we obtain a functor DN ◦ Sing≤K defined on Top assigning to each topological space X a twisted nil-differential graded algebra. Our second goal is to study the construction of N -differential graded algebras as defor- mations of 3-differential graded algebras. Although in [4] a general theory solving this sort of problem was proposed, our aim here is to provided a solution as explicit as possible. We consider exact and infinitesimal deformations of 3-differentials in Section 3. Our final goal in this work is to find applications of N -differential graded algebras to Lie algebroids. In Section 4 we review the concept of Lie algebroids introduced by Pradines [27], which generalizes both Lie algebras and tangent bundles of manifolds. A Lie algebroid E may be defined as a vector bundle together with a degree one differential d on Γ( E∗). We generalize this notion to the world of N -complexes, that is we introduce the concept of N Lie algebroids and construct several examples of such objects. 2 Examples of N-differential graded algebras In this section we give a brief summary of the known examples of N -dgas and introduce new examples of N -dgas of geometric nature. Definition 1. Let N ≥ 1 be an integer. A N -complex is a pair (A, d), where A is a Z-graded vector space and d : A −→ A is a degree one linear map such that dN = 0. Clearly a N -complex is also a M -complex for M ≥ N . N -complexes are also referred to as N -differential graded vector spaces. A N -complex (A, d) such that dN−1 6= 0 is said to be a proper N -complex. Let (A, d) be a N -complex and (B, d) be a M -complex, a morphism f : (A, d) −→ (B, d) is a linear map f : A −→ B such that df = fd. One of the most interesting features of N -complexes is that they carry cohomological information. Let (A, d) be a N -complex, a ∈ Ai is p-closed if dp(a) = 0, and is p-exact if there exists b ∈ Ai−N+p such that dN−p(b) = a, for 1 ≤ p < N . The cohomology groups of (A, d) are the spaces i(A) = Ker{dp : Ai −→ Ai+p}/Im{dN−p : Ai−N+p −→ Ai}, for i ∈ Z and p = 1, 2, ..., N−1. Definition 2. A N -differential graded algebra (N -dga) over a field k, is a triple (A,m, d) where m : A⊗A −→ A and d : A −→ A are linear maps such that: 1. dN = 0, i.e., (A, d) is a N -complex. 2. (A,m) is a graded associative algebra. 3. d satisfies the graded Leibnitz rule d(ab) = d(a)b+ (−1)āad(b). The simplest way to obtain N -differential graded algebras is deforming differential graded algebras. Let Der(A) be the Lie algebra of derivations on a graded algebra A. Recall that a degree one derivation d on A, induces a degree one derivation, also denoted by d, on End(A). Let A be a 2-dga and e ∈ Der(A). It is shown in [4] that e defines a deformation of A into a N -differential graded algebra if and only if (d + e)N = 0, or equivalently, if and only if the curvature Fe = d(e) + e 2 of e satisfies (Fe) 2 = 0 if N is even, or (Fe) 2 (d + e) = 0 if N is odd. For example, consider the trivial bundle M × Rn over M . A connection on M × Rn is a gl(n)-valued one form a on M , and its curvature is Fa = da + [a, a]. Let Ω(M,gl(n)) be the graded algebra of gl(n)-valued forms on M . Thus the pair (Ω(M,gl(n)), d + [a, ]) defines a N -dga if and only if (Fa) 2 = 0 for N even, or (Fa) 2 (d+ a) = 0 for N odd. Differential forms of depth N on simplicial sets Fix an integer N ≥ 3. We are going to construct the (n(N − 1) + 1)-differential graded algebra ΩN (R n) of algebraic differential forms of depth N on Rn. Let x1, ..., xn be coordinates on R and for 0 ≤ i ≤ n and 0 ≤ j < N, let djxi be a variable of degree j. We identify d 0xi with xi. Definition 3. The (n(N − 1) + 1)-differential graded algebra ΩN (R n) is given by • ΩN (R n) = R[djxi]/ djxid kxi | j, k ≥ 1 as a graded algebras. • The (n(N − 1) + 1)-differential d : ΩN (R n) −→ ΩN (R n) is given by d(djxi) = d j+1xi, for 0 ≤ j ≤ N − 2, and d(dN−1xi) = 0. One can show that d is (n(N − 1) + 1)-differential as follows: 1. It is easy to check that ΩN (R) is a N -dga. 2. If A is a N -dga and B is a P -dga, then A⊗B is a (N + P − 1)-dga. 3. ΩN (R n) = ΩN (R) We often write ΩN (x1, ..., xn) instead of ΩN (R n) to indicate that a choice of affine coordi- nates (x1, ..., xn) on R n has been made. Let ∆ be the category such that its objects are non-negative integers; morphisms in ∆(n,m) are order preserving maps f : {0, ..., n} −→ {0, ...,m}. The category of simplicial sets Set∆ the category of contravariant functors ∆ −→ Set. Explicitly, a simplicial set s : ∆op −→ Set is a functorial correspondence assigning: • A set sn for each integer n ≥ 0. Elements of sn are called simplices of dimension n. • A map s(f) : sm −→ sn for each f ∈ ∆(n,m). Let Aff be the category of affine varieties, and let A : ∆ −→ Aff be the functor sending n ≥ 0, into the affine variety A(x0, ..., xn) = ∆n = {(x0, ..., xn) ∈ R n | x0 + ... + xn = 1}. A sends f ∈ ∆(n,m) into A(f) : A(x0, ..., xn) → A(x0, ..., xm) given by A(f) ∗(xj) = f(i)=j xi, for 0 ≤ j ≤ m. Forms of depth N on the cosimplicial affine variety A are defined by the functor ΩN : ∆ op −→ N ildga sending n ≥ 0 into ΩN (n) = ΩN (x0, ..., xn)/ 〈x0 + ...+ xn − 1, dx0 + ...+ dxn〉 . A map f ∈ ∆(n,m) induces a morphisms ΩN (f) : ΩN (m) −→ ΩN (n) given for 0 ≤ j ≤ m by ΩN (f)(xj) = f(i)=j xi and ΩN (f)(dxj) = f(i)=j Let set∆ be the full subcategory of Set∆ whose objects are simplicial sets generated in finite dimensions, i.e., simplicial sets s for which there is an integer K such that for each p ∈ si, i ≥ K, there exists q ∈ sj, j ≤ K, with p = s(f)(q) for some f ∈ ∆(p, q). We are ready to define the contravariant functor ΩN : set ∆op −→ N ildga announced in the introduction. The nil-differential graded algebra ΩN (s) = i=0 Ω N (s) associated with s is given by ΩiN (s) = {a ∈ ΩiN (n) | as(f)(p) = ΩN (f)(ap) for p ∈ sm and f ∈ ∆(n,m)}. A natural transformation l : s −→ t induces a map ΩN(l) : ΩN (t) −→ ΩN (s) given by the rule [ΩN (l)(a)]p = al(p) for a ∈ ΩN (t) and p ∈ sn. For each integer K ≥ 0 there is functor ( )≤K : Set ∆op −→ set∆ sending a simplicial set s, into the simplicial set s≤K generated by simplices in s of dimension lesser or equal to K. The singular functor Sing : Top −→ Set∆ sends a topological space X into the simplicial set Sing(X) such that Singn(X) = {f : ∆n −→ X | f is continous }. Thus, for each pair of integers N and K we have constructed a functor ΩN ◦ ( )≤K ◦ Sing : Top −→ N ildga sending a topological space X into the nil-differential graded algebra ΩN (Sing≤K(X)). Difference forms of depth N on simplicial sets Next we construct difference forms of higher depth on finitely generated simplicial sets. Dif- ference forms on discrete affine space were introduced by Zeilberger in [28]. We proceed to construct a discrete analogue of the functors from topological spaces to nil-differential graded algebras introduced above. First, we construct DN (Z n) the algebra of difference forms of depth N on Zn. Let F (Zn,R) be the algebra concentrated in degree zero of R-valued functions on the lattice Zn. Introduce variables δjmi of degree j for 1 ≤ i ≤ n and 1 ≤ j < N . The graded algebra of difference forms of depth N on Zn is given by DN (Z n) = F (Zn,R)⊗ R[δjmi]/ δjmiδ kmi | j, k ≥ 1 A form ω ∈ DN (Z n) can be written as ω = I ωIdmI where I : {1, .., n} −→ N is any map and dmI = i=1 d I(i)mi. The degree of dmI is |I| = i=1 I(i). The finite difference ∆i(g) of g ∈ F (Zn,R) along the i-direction is given by ∆i(g)(m) = g(m+ ei)− g(m), where the vectors ei are the canonical generators of Z n and m ∈ Zn. The difference operator δ is defined for 1 ≤ j ≤ N − 2 by the rules δ(g) = ∆i(g)δmi, δ(δ jmi) = δ j+1mi and δ(δ N−1mi) = 0. It is not difficult to check that if ω = I ωIdmI , then δω = J(δω)JdmJ where (δω)J = J(i)=1 (−1)|J<i|∆iωJ−ei + J(i)≥2 (−1)|J<i|ωJ−ei . From the later formula we see that (δω)J is a linear combination of (differences of) functions ωK with |K| < |J |. This fact implies that δ is nilpotent, indeed, one can check that δn(N−1)+1 = 0. All together we have proved the following result. Theorem 4. DN (Z n) is a graded algebra and the difference operator δ gives DN (Z n) the structure of a (n(N − 1) + 1)-complex. One can check that δ satisfies a twisted Leibnitz rule, so DN (Z n) is actually pretty close of being a N -dga. Let Zn,1 ⊆ Zn+1 consists of tuples (m0, ...,mn) such that m0 + ... + mn = 1. Consider the functor DN defined on ∆ op sending n ≥ 0 into DN (n) = F (Z n,1,R)⊗ δjmiδ kmi, δm0 + ...+ δmn A map f ∈ ∆(n, k) induces a morphisms DN (f) : DN (k) −→ DN (n) given for g ∈ F (Z k,1,R) and 0 ≤ j ≤ k by DN (f)(g)(m0, ...,mn) = g f(i)=0 mi, ..., f(i)=k  and DN (f)(δmj) = f(i)=j We extend DN to the functor defined on set ∆op sending a finitely generated simplicial set s into DN (s) = i=0 D N (s) where DiN (s) = DiN (n) | as(f)(p) = DN (f)(ap) for p ∈ sk and f ∈ ∆(n, k) A natural transformation l : s −→ t induces a map ΩN (l) : ΩN (t) −→ ΩN (s) by the rule [DN (l)(a)]p = al(p) for a ∈ DN (t) and p ∈ sn. Thus for given integers N and K we have constructed a functor DN ◦ ( )≤K ◦ Sing on Top sending a topological space X into a sort of nil-differential graded algebra satisfying a twisted Leibnitz rule DN (Sing≤K(X)) . It would be interesting to compute the cohomology groups of the algebra of difference forms of higher depth on known simplicial sets. Even in the case of forms of depth 2 these groups have seldom been studied. 3 On the (3,N) curvature Recall that a discrete quantum mechanical system is given by the following data: 1. A directed graph with set of vertices V and set of directed edges E. The Hilbert space of the system is H = CV . 2. A map ω : E −→ R assigning a weight to each edge. 3. Operators Un : H −→ H for n ∈ N given by (Unf)(y) = ωn(y, x)f(x) where the discretized kernel ωn(y, x) is given by ωn(y, x) = γ∈Pn(x,y) ω(e). Pn(x, y) denotes the set of paths in Γ from x to y of length n, i.e., sequences (e1, · · · , en) of edges such that s(e1) = x, t(ei) = s(ei+1), for i = 1, ..., n − 1 and t(en) = y. Let us introduce some notation. For s = (s1, ..., sn) ∈ N n we set l(s) = n and |s| = i si. For 1 < i ≤ n we set s<i = (s1, ..., si−1); also we set s>n = s<1 = ∅. N (∞) is equal to where by convention N(0) = {∅}. Let A be a 3-dga and e be a degree one derivation on A. For s ∈ Nn we let e(s) = e(s1) · · · e(sn), where e(l) = dl(e) if l ≥ 1, e(0) = e and e∅ = 1. For N ∈ N, we set EN = s ∈ N(∞) | s 6= ∅ and |s|+ l(s) ≤ N and for s ∈ EN we let N(s) ∈ Z be given by N(s) = N − |s| − l(s). The following data defines a discrete quantum mechanical system: 1. The set of vertices is N(∞). 2. There is a unique directed edge from s to t if and only if t ∈ {(0, s), s, (s + ei)}, where ei ∈ N l(s) are the canonical vectors. 3. Edges are weighted according to the table: source target weight s (0, s) 1 s s (−1)|s|+l(s) s (s+ ei) (−1) |s<i|+i−1 PN (∅, s) consists of paths γ = (e1, ..., eN ), such that s(e1) = ∅, t(eN ) = s and s(el+1) = t(el). The weight ω(γ) of a path γ ∈ PN (∅, s) is given by ω(γ) = i=1 ω(ei). The following result, proved in [4], tell us when d+ e defines a deformation of a 3-dga into a N -dga. Theorem 5. d+ e defines a deformation of the 3-dga A into a N -dga if and only if the (3, N) Maurer-Cartan equation holds co + c1d+ c2d 2 = 0, where for 0 ≤ k ≤ 2 we set N(s)=k c(s,N)e(s) and c(s,N) = γ∈PN (∅,s) ω(γ). Exact deformations Let us first consider the deformation of a 3-dga into a 3-dga. According to Theorem 5 the derivation d+ e defines a 3-dga if and only if co + c1d+ c2d 2 = 0 where N(s)=k c(s, 3)e(s) and c(s, 3) = γ∈P3(∅,s) ω(γ). Let us compute the coefficients ck. We have that E3 = {(0), (1), (2), (0, 0), (1, 0), (0, 1), (0, 0, 0)} . Let us first compute c0. There are four vectors s in E3 such that N(s) = 0, these are (2), (1, 0), (0, 1) and (0, 0, 0). The only path from ∅ to (2) of length 3 is ∅ −→ (0) −→ (1) −→ (2) of weight 1. Since e(2) = d2(e), then we have that c((2), 3) = d2(e). The unique path from ∅ to (1, 0) of length 3 is ∅ −→ (0) −→ (0, 0) −→ (1, 0) of weight 1. Since e(1,0) = d(e)e we have that c((1, 0), 3) = d(e)e. There are two paths from ∅ to (0, 1) of length 3, namely ∅ −→ (0) −→ (0, 0) −→ (0, 1) ∅ −→ (0) −→ (1) −→ (0, 1) of weight −1 and 1, respectively. Thus c((1, 0), 3) = 0 since the sum of the weights vanishes. The unique path from ∅ to (0, 0, 0) of length 3 is ∅ −→ (0) −→ (0, 0) −→ (0, 0, 0) of weight 1. Since e(0,0,0) = e3, then c((0, 0, 0), 3) = e3. Thus we have shown that c0 = d 2(e) + d(e)e + e3. We proceed to compute c1. The vectors in E3 such that N(s) = 1 are (1) and (0, 0). Paths from ∅ to (1) of length 3 are ∅ −→ ∅ −→ (0) −→ (1) ∅ −→ (0) −→ (0) −→ (1) ∅ −→ (0) −→ (1) −→ (1) of weight 1, −1 and 1, respectively. Since e(1) = d(e), then c((1), 3) = d(e). Paths from ∅ to (0, 0) of length 3 are ∅ −→ (0) −→ (0) −→ (0, 0) ∅ −→ ∅ −→ (0) −→ (0, 0) ∅ −→ (0) −→ (0, 0) −→ (0, 0). The corresponding weights are, respectively, −1, 1 and 1. We have that e(0,0) = e2, thus c((0, 0), 3) = e2 and c1 = d(e) + e Finally we compute c2. (0) is the only vector in E3 such that N(s) = 2. The paths from ∅ to (0) of length 3 are ∅ −→ ∅ −→ ∅ −→ (0) ∅ −→ ∅ −→ (0) −→ (0) ∅ −→ (0) −→ (0) −→ (0). The corresponding weights are, respectively, 1, −1 and 1. Since e(0) = e then c2 = c((0), 3) = e. Altogether we have proven the following result. Theorem 6. d+ e defines a deformation of the 3-dga A into a 3-dga if and only if (d2(e) + d(e)e + e3) + (d(e) + e2)d+ ed2 = 0. Consider now deformations of a 3-dga into a 4-dga. Again by Theorem 5 we must have c0 + c1d+ c2d 2 = 0. We proceed to compute the coefficients ck. We have that E4 = {(0), (1), (2), (0, 0), (1, 0), (0, 1), (2, 0), (0, 2), (1, 1), (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 0, 0, 0)}. (0) is the only vector in E4 such that N(s) = 3. Paths of length 4 from ∅ to (0) are of the form ∅ → · · · → ∅︸ ︷︷ ︸ → (0) → · · · → (0)︸ ︷︷ ︸ with weight (−1)j , where i + j = 3, thus we have that c3 = (1− 1 + 1− 1)e = 0. We compute c2. Vectors in E4 with N(s) = 2 are (0, 0) and (1). Paths from ∅ to (0, 0) of length 2 are of the form ∅ → · · · → ∅︸ ︷︷ ︸ → (0) → · · · → (0)︸ ︷︷ ︸ → (0, 0) → · · · → (0, 0)︸ ︷︷ ︸ of weight i+j+k=2(−1) j = 2, thus c((0, 0), 4)e(0,0) = 2e2. Paths from ∅ to (1) of length 2 are of the form ∅ → · · · → ∅︸ ︷︷ ︸ → (0) → · · · → (0)︸ ︷︷ ︸ → (1) → · · · → (1)︸ ︷︷ ︸ with weight i+j+k=2(−1) j = 2, thus c((1), 4)e(1) = 2d(e) and c2 = 2(e 2 + d(e)). Let us now compute c1. Vectors in E4 with N(s) = 1 are (0, 0, 0), (1, 0), (0, 1) and (2). Paths from ∅ to (0, 0, 0) are of 5 types. Paths of the form ∅ → · · · → ∅︸ ︷︷ ︸ → (0) → · · · → (0)︸ ︷︷ ︸ → (0, 0) → · · · → (0, 0)︸ ︷︷ ︸ → (0, 0, 0) → · · · → (0, 0, 0)︸ ︷︷ ︸ with weight i+j+k+l=1(−1) j(−1)l, so that c((0, 0, 0), 4)e(0,0,0) = (1− 1 + 1− 1)e3 = 0. Paths of the form ∅ → · · · → ∅︸ ︷︷ ︸ → (0) → · · · → (0)︸ ︷︷ ︸ → (1) → · · · → (1)︸ ︷︷ ︸ → (0, 1) → · · · → (0, 1)︸ ︷︷ ︸ with weight ∑ i+j+k+l=1 (−1)j(−1)l = 1− 1 + 1− 1 = 0 . Path of the form ∅ → · · · → ∅︸ ︷︷ ︸ → (0) → · · · → (0)︸ ︷︷ ︸ → (0, 0) → · · · → (0, 0)︸ ︷︷ ︸ → (0, 1) → · · · → (0, 1)︸ ︷︷ ︸ of weight i+j+k+l=1(−1) j(−1)(−1)l so that c((0, 1), 4)e(0,1) = ((1 − 1 + 1− 1) + (1− 1 + 1− 1))ed(e) = 0. Paths of the form ∅ → · · · → ∅︸ ︷︷ ︸ → (0) → · · · → (0)︸ ︷︷ ︸ → (0, 0) → · · · → (0, 0)︸ ︷︷ ︸ (1, 0) → · · · → (1, 0)︸ ︷︷ ︸ of weight i+j+k+l=1(−1) j(−1)l, thus c((1, 0), 4)e(1,0) = (1 − 1 + 1 − 1)d(e)e = 0. There are also paths of the form ∅ → · · · → ∅︸ ︷︷ ︸ → (0) → · · · → (0)︸ ︷︷ ︸ → (1) → · · · → (1)︸ ︷︷ ︸ → (2) → · · · → (2)︸ ︷︷ ︸ of weight i+j+k+l=1(−1) j(−1)l, so we have c((2), 4)e(2) = (1 − 1 + 1− 1)d2(e) = 0. We have shown that c1 = c((0, 0, 0), 4)e (0,0,0) + c((0, 1), 4)e(0,1) + c((1, 0), 4)e(1,0) + c((2), 4)e(2) = 0. Let us compute c0. There are several types of paths in this case. Path ∅ −→ (0) −→ (0, 0) −→ (0, 0, 0) −→ (0, 0, 0, 0) of weight 1, thus cq((0, 0, 0, 0), 4)a (0,0,0,0) = a4. Paths ∅ −→ (0) −→ (0, 0) −→ (0, 0, 0) −→ (0, 0, 1) ∅ −→ (0) −→ (1) −→ (0, 1) −→ (0, 0, 1) ∅ −→ (0) −→ (0, 0) −→ (0, 1) −→ (0, 0, 1) of weight 1, thus we have that c((0, 0, 1), 4)e(0,0,1) = e2d(e). Paths ∅ −→ (0) −→ (0, 0) −→ (1, 0) −→ (0, 1, 0) ∅ −→ (0) −→ (0, 0) −→ (0, 0, 0) −→ (0, 1, 0) of weight 0, thus c((0, 1, 0), 4)e(0,1,0) = 0ad(a)a = 0. Path ∅ −→ (0) −→ (0, 0) −→ (0, 0, 0) −→ (1, 0, 0) of weight 1, thus c((1, 0, 0), 4)e(1,0,0) = d(e)e2. Path ∅ −→ (0) −→ (0, 0) −→ (1, 0) −→ (2, 0) of weight 1, so c((2, 0), 4)e(2,0) = d2(e)e. Paths ∅ −→ (0) −→ (0, 0) −→ (0, 1) −→ (0, 2) ∅ −→ (0) −→ (1) −→ (0, 1) −→ (0, 2) ∅ −→ (0) −→ (1) −→ (2) −→ (0, 2) of weight 1, so that c((0, 2), 4)e(0,2) = ed2(e). There are also paths ∅ −→ (0) −→ (0, 0) −→ (1, 0) −→ (1, 1) ∅ −→ (0) −→ (1) −→ (0, 1) −→ (1, 1) of weight 2, so that c((1, 1), 4)e(1,1) = (d(e))2. We see that c0 = e 4 + e2d(e) + d(e)e2 + d2(e)e + ed2(e) + (d(e))2. All together we have shown the following result. Theorem 7. d+ e defines a deformation of the 3-dga A into a 4-dga if and only if (e4 + e2d(e) + d(e)e2 + d2(e)e + ed2(e) + (d(e))2) + 2(e2 + d(e))d2 = 0. Infinitesimal deformations Let t be a formal parameter such that t2 = 0. Theorem 8. Let (A, d) be a N -dga and e a degree one derivation on A, then we have (d+ te)N = t p∈Par(k,N−k+1) (−1)w(p)  dN−k−1(e)dN−k−1, where Par(k,N − k + 1) = {p = (p1, · · · , pN−k+1) | N−k+1∑ pi = k} and w(p) = N−k+1∑ (i− 1)pi. Proof. From Theorem 5 we know that DN = k=0 ckd k. Since t2 = 0, then (te)(s) = (te)(s1) · · · (te)(sl(s)) = tl(s)e(s) = 0 unless l(s) ≤ 1. On the other hand we have that EN = {(0), (1), · · · , (N − 1)}. Suppose that l(s) = 1 and N(s) = N − |s| − l(s) = k, thus |s| = N − k − 1. The unique vector s in EN of length 1 such that |s| = N − k − 1 is s = (N − k − 1). Therefore N(s)=k c(s,N)e(s) = c((N − k − 1), N)e(s) = c((N − k − 1), N)dN−k−1(e). A path from ∅ to (N − k − 1) of length N must be of the form ∅→ · · · →︸ ︷︷ ︸ ∅ → (0)→ · · · →︸ ︷︷ ︸ (0) → (1)→ · · · →︸ ︷︷ ︸ (1) → · · · → (N − k − 1)→ · · · →︸ ︷︷ ︸ pN−k+1 (N − k − 1) with (p1 + 1) + (p1 + 1) + · · ·+ (pN−k + 1) + pN−k+1 = N , i.e., ∑N−k+1 i=1 pi = k. The weight of such path is (−1)0p1(−1)(2−1)p2(−1)(3−1)p2 ...(−1)(N−k)pN−k+1 = (−1)w(p), thus we get that c((N − k − 1), N) = γ∈PN (∅,s) ω(γ) = p∈Par(k,N−k+1) (−1)w(p). Corollary 9. e defines an infinitesimal deformation of theN -dga (A, d) into theN -dga (A, d+e) if and only if p∈Par(k,N−k+1) (−1)w(p)  dN−k−1(e)dN−k−1 = 0. (1) 4 N Lie algebroids In this section we introduce the notion of N Lie algebroids and construct examples of such structures. We first review the notion of Lie algebroids, provide some examples, and write the definition of Lie algebroids in a convenient way for our purposes. Lie algebroids We review basic ideas around the notion of Lie algebroids; the interested reader will find much more information in [12, 23, 27]. The notion of Lie algebroids has gained much attention in the last few years because of its interplay with various branches of mathematics and theoretical physics, see [10, 11, 17]. We center our attention on the basic definitions and constructions of Lie algebroids and its relation with graded manifolds and differential graded algebras. Definition 10. A Lie algebroid is a vector bundle π : E −→ M together with: • A Lie bracket [ , ] on the space Γ(E) of sections of E. • A vector bundle map ρ : E −→ TM over the identity, called the anchor, such that the induced map ρ : Γ(E) −→ Γ(TM) is a Lie algebra morphism. • The identity [v, fw] = f [v,w]+(ρ(v)f)w must hold for sections v,w of E and f a smooth function on M . Let (x1, ..., xn) be coordinates on a local chart U ⊂ M , and let {eα | α = 1, . . . , r} be a basis of local sections of π : E|U −→ U . Local coordinates on E|U are given by (x i, yα). Locally the Lie bracket and the anchor are given by [eα, eβ ]E = C eγ and ρ(eα) = ρ , respectively. The smooth functions C , ρiα are the structural functions of the Lie algebroid. The condition for ρ to be a Lie algebra homomorphism is written in local coordinates as = ρiγ C The other compatibility condition between ρ and [ , ] is given by cycl(α,β,γ) + Cµαν C where the sum is over indices α, β, γ such that the map 1, 2, 3 −→ α, β, γ is a cyclic permuta- tion. The simplest examples of Lie algebroids are described below; the reader will find further examples in the references listed at the beginning of this section. Example 11. A finite dimensional Lie algebra g may be regarded as a vector bundle over a single point. Sections are elements of g, the Lie bracket is that of g, and the anchor map is identically zero. The structural functions C are the structural constants c of g and ρiα = 0. Example 12. The tangent bundle π : TM −→ M with anchor the identity map ITB on TB and with the usual bracket on vector fields. Exterior differential algebra of Lie algebroids Sections Γ( E) of a Lie algebroid E play the rôle of vector fields on a manifold and are called E vector fields. Sections of the dual bundle π : E∗ −→ M are called E 1-forms. Similarly sections Γ( E∗) of E∗ are called E forms. The degree of a E form in Γ( E∗) is k. Let us state and sketch the proof of a result of fundamental importance for the rest of this work. Theorem 13. Let E be a vector bundle. E is a Lie algebroid if and only if Γ( E∗) is a differential graded algebra. A differential on E∗ is the same as a degree one vector field v on E[−1] such that v2 = 0. Above E[−1] denotes the graded manifold whose underlying space is E with fibers placed in degree one. If E is a Lie algebroid one defines a differential d : Γ(∧kE∗) −→ Γ(∧k+1E∗) as follows: dθ(v1, . . . , vk+1) = (−1)i+1ρ(vi)θ(v1, . . . , v̂i, . . . , vk+1) (−1)i+jθ([vi, vj ], v1, . . . , v̂i, . . . , v̂j , . . . vk+1), for v1, . . . , vk+1 ∈ Γ(E). The axioms for a Lie algebroid imply that: 1. d2 = 0; 2. If f ∈ C∞(M) and v ∈ Γ(E), then 〈df, v〉 = ρ(v)f ; 3. d is a derivation of degree 1, i.e., d(θ ∧ ζ) = dθ ∧ ζ + (−1)θθ ∧ ζ. Conversely, assume that d is a degree one derivation on Γ( E∗) satisfying d2 = 0. Then E is a Lie algebroid with the structural maps ρ and [ , ] given by ρ(v)f = df(v) , θ([v,w]) = ρ(v)θ(w) − ρ(w)θ(v)− dθ(v,w), for v,w ∈ Γ(E), f ∈ C∞(M) and θ ∈ Γ( E). In local coordinates d is determined by dxi = ρiα e α and deγ = C eα ∧ eβ, where {eα | α = 1, . . . , r} is the dual basis of {eα | α = 1, . . . , r}. It is not hard to see that the conditions d2xi = 0 and d2eα = 0 are equivalent to the structural equations defining a Lie algebroid. Let us compute the exterior algebra of a few Lie algebroids. Example 14. To the trivial Lie algebroid structure on a vector bundle E corresponds to the exterior algebra E∗ with vanishing differential. Example 15. Chevalley-Eilenberg differential on ∗ arises from the Lie algebroid g −→ {•} of Example 11. The Chevalley-Eilenberg differential d takes the form dθ(v1, . . . , vk+1) = (−1)i+jθ([vi, vj ], v1, . . . , v̂i, . . . , v̂j , . . . vk+1), for vi ∈ g and θ ∈ Example 16. The differential associated with the tangent bundle TM −→ M Lie algebroid is de Rham differential. N Lie algebroids We are ready to introduce the main concept of this section. In the light of Theorem 13 it is rather natural to define a N Lie algebroid as a vector bundle E together with a degree one N -nilpotent vector field v on the graded manifold E[−1]. That definition, useful as it might be, rules out some significant examples that we would not like to exclude, thus, we prefer the more inclusive definition given below. Though not strictly necessary for our definition of N Lie algebroids, the study of nilpotent vector fields on graded manifolds is of independent interest, and we shall say a few words about them. Indeed our next result gives an explicit formula for the N -th power of a graded vector field. Let x1, ..., xm be local coordinates on a graded manifold and ∂1, ..., ∂m be the corresponding vector fields. We recall that if xi is a variable of degree xi, then ∂i is of degree −xi, and dxi is of degree xi + 1. Let a 1, ..., am be functions of homogeneous degree depending on x1, ..., xm. For L a linearly ordered set and f : L −→ [m] a map we define f(i) and ∂f = ∂f(i). Also we define the sign s(f) by the rule ∂f = s(f)∂ |f−1(1)| 1 ... ∂ |f−1(m)| Let p : N −→ Z2 be the map such that p(n) is 1 if n is even and −1 otherwise. Using induction on N one can show that: Theorem 17. (ai∂i) s(f, α)( α−1(i) af(i))∂f | α−1(N+1)⊔N where the sum runs over f : [N ] −→ [m] and α : [N − 1] −→ [2, N + 1] such that α(i) > i. The sign s(f, α) is given by s(f, α) = p( s<j<α(s) f(j) + xsf |α−1(j)∩[s+1,N−1] ). Corollary 18. (ai∂i) cI∂I , where I : [m] −→ N is such that 1 ≤ |I| := I(1) + ...+ I(N) ≤ N , ∂I = i=1 ∂ i , and S(f, α) (∂f(α−1(i))a f(i)) where the sum runs over maps α : [N − 1] −→ [2, N + 1] with α(i) > i for i ∈ [N − 1], and f : [N ] −→ [m] such that |{j ∈ α−1(N + 1) ⊔ {N} | f(j) = i}| = I(i), for i ∈ [m]. The sign S(f, α) is given by S(f, α) = s(f, α)s(f |α−1(N+1)⊔{N}) . Corollary 19. (ai∂i) N = 0 if and only if cI = 0 for I as above. For example for N = 2 one gets (ai∂i) p(xiaj)aiaj∂i∂j + ai∂i(aj)∂j . For N = 3 we get that (ai∂i) i,j,k ai∂i(aj)∂j(ak)∂k + p(xiaj)aiaj∂i∂j(ak)∂k + p(xiak)aiaj∂j(ak)∂i∂k + p(xjak)ai∂i(aj)ak∂j∂k + xiaj)aiaj∂i(ak)∂j∂k + p(xjak + xiaj + xiak)aiajak∂i∂j∂k. For N = 4 the corresponding expression have 24 terms and we won’t spell it out. We return to the problem of defining N Lie algebroids. We need some general remarks on differential operators on associative algebras. Given an associative algebra A we let DO(A) be the algebra of differential operators on A, i.e., the subalgebra of End(A) generated by A ⊂ End(A) and Der(A) ⊂ End(A), the space of derivations of A. Thus DO(A) is generated as a vector space by operators of the form x1 ◦x2 ◦ · · · ◦xn ∈ End(A) where xi is in A⊔Der(A). Notice that DO(A) admits a natural filtration ∅ = DO≤−1(A) ⊆ DO≤0(A) ⊆ DO≤1(A) ⊆ · · · ⊆ DO≤k(A) ⊆ · · · ⊆ DO(A), whereDO≤k(A) ⊆ DO(A) is the subspace generated by operators x1◦x2◦· · ·◦xn, where at most k operators among the xi belong to Der(A). Thus DO(A) admits the following decomposition as graded vector space DO(A) = DOk(A) := DO≤k(A)/DO≤k−1(A). Clearly DO0(A) = A and if A is either commutative or graded commutative, then DO1(A) = Der(A). The projection map π1 : DO(A) −→ DO1(A) induces a non-associative product ⋄ : DO1(A)⊗DO1(A) −→ DO1(A) given by s ⋄ t = π1(s ◦ t) for s, t ∈ DO1(A). In particular if A is commutative or graded commutative we obtain a non-associative product ⋄ : Der(A)⊗Der(A) −→ Der(A). To avoid unnecessary use of parenthesis we assume that in the iterated applications of ⋄ we associate in the minimal form from right to left. Definition 20. A N Lie algebroid is a vector bundle E together with a degree one derivation d : Γ( E∗) −→ Γ( E∗), such that the result of N ⋄-compositions of d with itself vanishes, i.e., d ⋄ d ⋄ · · · ⋄ d = 0. The notions of Lie algebroids and 2 Lie algebroids agree; indeed it is easy to check that d ◦ d = d ⋄ d for any degree one derivation d : Γ( E∗) −→ Γ( E∗). Let us now illustrate with an example the difference between the condition d◦d◦· · ·◦d = 0 and the much weaker condition d ⋄ d ⋄ · · · ⋄ d = 0. Let C[x1, ..., xn] be the free graded algebra generated by graded variables xi for 1 ≤ i ≤ n. A derivation on C[x1, ..., xn] is a vector field ∂ = ai∂i where ai ∈ C[x1, ..., xn]. The condition ∂N = 0 is rather strong and restrictive, it might be tackled with the methods provided above. In contrast, the condition ∂ ⋄ ∂ ⋄ · · · ⋄ ∂ = 0 is much simpler and indeed it is equivalent to the condition ∂N (xi) = 0 for 1 ≤ i ≤ n. Definition 21. A N Lie algebra is a vector space g together with a degree one derivation d on ∗ such that the N -th ⋄-composition of d with itself vanishes. Our next result characterizes 3 Lie algebras in more familiar terms. For integers k1, k2, ..., kl such that k1 + k2 + · · ·+ kl = n, we let Sh(k1, k2, · · · , kl) be the set of permutations σ : {1, · · · , n} −→ {1, · · · , n} such that σ is increasing on the intervals [ki + 1, ki+1] for 0 ≤ i ≤ l, k0 = 1 and kl+1 = n. Assume we are given a map [ , ] : g −→ g. Theorem 22. The pair (g, [ , ]) is a 3 Lie algebra if and only if for v1, v2, v3, v4 ∈ g we have σ∈Sh(2,1,1) sgn(σ)[[[vσ(1), vσ(2)], vσ(3)]vσ(4)] = σ∈Sh(2,2) sgn(σ)[[vσ(1) , vσ(2)], [vσ(3), vσ(4)]], Proof. One can show that a degree one differential on ∗ is necessarily the Chevalley-Eilenberg operator dθ(v1, . . . , vn+1) = (−1)i+jθ([vi, vj ], v1, . . . , v̂i, . . . , v̂j , . . . vn+1) , where [ , ] : g −→ g is an antisymmetric operator. We remark that we are not assuming, at this point, that the bracket [ , ] satisfies any further identity. Jacobi identity arises when the square of d is set to be equal to zero, but we do not do that since we want to investigate the weaker condition that the third ⋄-power of d be equal to zero. For θ ∈ ∗ = g∗ the Chevalley-Eilenberg operator takes the simple form dθ(v1, v2) = −θ([v1, v2]). Moreover a further application of d to dθ yields d2θ(v1, v2, v3) = σ∈Sh(2,1) sgn(σ)θ([[vσ(1), vσ(2)], vσ(3)]). From the last equation it is evident that Jacobi identity is equivalent to the condition d2 = 0. We do not assume assume that Jacobi identity holds and proceed to compute the third ⋄-power of d. We obtain that d ⋄ d ⋄ dθ(v1, v2, v3, v4) = σ∈Sh(2,1,1) sgn(σ)θ([[[vσ(1), vσ(2)], vσ(3)]vσ(4)]) σ∈Sh(2,2) sgn(σ)θ([[vσ(1), vσ(2)], [vσ(3), vσ(4)]]). Thus d ⋄ d ⋄ d = 0 if and only if the condition from the statement of the Theorem holds. Using local coordinates θ1, ..., θm on the graded manifold g[−1], it is not hard to show that a vector field of degree one on g[−1] can be written as where the constants C may be identified with the structural constants of [·, ·]. The square of the vector field ∂ is given by Cσδ εθ Cσγ εθ αθβθε Cσδ γθ αθβθδ θαθβCσδ εθ Using the antisymmetry properties of C and the commutation rules for θα one can write together the first to terms. We find that ∂ ⋄ ∂ = Cσγ εθ αθβθε The condition ∂ ⋄ ∂ = 0 is equivalent to Jacobi identity. We assume that ∂ ⋄ ∂ 6= 0 and proceed to compute consider the condition ∂ ⋄ ∂ ⋄ ∂ = 0. We have that ∂ ◦ (∂ ⋄ ∂) = Cνλµθ Cσγ εθ αθβθε Using carefully the properties of C and θα we find that ∂ ◦ (∂ ⋄ ∂) = CνλµC Cσγ εθ λθµθβθε CνλµC Cσγ νθ λθµθαθβ CνλµC Cσγ εθ λθµθαθβθε Therefore we have shown that ∂ ⋄ (∂ ⋄ ∂) = CνλµC Cσγ ǫ − CσγαC θλθµθβθε Thus the condition ∂ ⋄ (∂ ⋄ ∂) = 0 is equivalent to the following equations for fixed σ: λ,µ,β,ε CνλµC Cσγ ǫ − CσγαC θλθµθβθε = 0 . Let us now go back to the case of Lie algebroids as opposed to Lie algebras. There is a natural degree one vector field on the graded manifold T[−1]R n, namely, de Rham differential. We now investigate whether it is possible to deform, infinitesimally, de Rham differential into a 3-differential. In local coordinates (x1, . . . , xn, θ1, . . . , θn) on T[−1]R n, with xi of degree zero and θi of degree 1, de Rham operator takes the form ∂ = δiαθ Let t be a formal infinitesimal parameter such that t2 = 0. We are going to show that any set of functions aiα of degree zero on T[−1]R n determine a deformation of de Rham operator into a 3-⋄ nilpotent operator given by δiα + ta Theorem 23. ∂a ⋄ ∂a = t θα θβ and ∂a ⋄ (∂a ⋄ ∂a) = 0. Proof. ∂2a = δiα + ta θα θβ θα θβ θα θβ Since t2 = 0 the third term on the right hand side of the expression above vanishes. The second term also vanishes because it is a contraction of even and odd indices. So we get that ∂a ⋄ ∂a = t θα θβ The third power of ∂a is given by ∂a ⋄ (∂a ⋄ ∂a) = t ∂xγ∂xα θγ θα θβ It also vanishes because it includes a contraction of even and odd indices. The nilpotency condition for the operator ∂a ⋄∂a is θα θβ = 0 for j = 1, . . . , n. It is not hard to find examples of matrices a such that ∂a ⋄ ∂a = 0, for example  (x4)2 x1 x1 x2 x2 x3 x2 x3 x3 x2 x4 x4 x4 x1 x4 x3  More importantly there are also matrices a such that ∂a ⋄ ∂a 6= 0, for example  x1 x4 x1 x1 x1 x2 x2 x4 x2 x2 x3 x3 x3 x4 x3 x4 x4 x4 x1 x4  We now consider full deformations as opposed to infinitesimal ones. Let δiα + a be a vector field. We think of ∂a as a deformation of de Rham differential with deformation parameters aiα. Theorem 24. ∂a ⋄ (∂a ⋄ ∂a) = δlγ + a ){∂aiα + aiα ∂xl∂xi θγθαθβ Proof. Since ∂2a = δiα + a ∂a ⋄ ∂a = δiα + a ) ∂ajβ we get ∂a ⋄ (∂a ⋄ ∂a) = δlγ + a δiα + a ) ∂ajβ δlγ + a ){∂aiα + aiα ∂xl∂xi θγθαθβ Corollary 25. ∂a ⋄ (∂a ⋄ ∂a) = 0 if for fixed indices α, β, λ, j the following identity holds δlγ + a ){∂aiα + aiα ∂xl∂xi θγθαθβ = 0. Corollary 26. Each matrix A = (A ) ∈ Mn(R) such that A 2 = 0 determines a 3 Lie algebroid structure on TRn with differential given by (δiα +A αxα)dx Our final result describes explicitly the conditions defining a 3 Lie algebroid. Let E be a vector bundle over M . A vector field on E[−1] of degree one is given in local coordinates by ∂ = ρiαθ where ρiα and C are functions of the bosonic variables only. Theorem 27. ∂ ⋄ (∂ ⋄ ∂) = 0 if and only if for fixed γ and i the following identity holds: Cασ µ) Cλµσ − CαλµC CαβµC θνθσθµθβ = 0 , Cǫσν− Cǫσν + ρ θσθνθγ = 0 . Proof. We sketch the rather long proof. For ∂ = ρiαθ θαθβ ∂ , we have ∂ ⋄ ∂ = θλθµθβ As in the previous theorem one finds that the condition ∂ ⋄ (∂ ⋄ ∂) = 0 is equivalent to the following identities Cασ µ) Cλµσ − Cλνσ − θνθσθµθβ = 0 , Cǫσν − ρ θσθνθγ Needless to say further research is necessary in order to have a better grasp of the meaning and applications of the notion of N Lie algebroids. We expect that this approach will lead towards new forms of infinitesimal symmetries, and for that reason alone it should find appli- cations in various problems in mathematical physics. In our forthcoming work [3] we are going to discuss some applications of N Lie algebroids in the context of Batalin-Vilkovisky algebras and the master equation. Acknowledgment Thanks to Takashi Kimura, Juan Carlos Moreno and Jim Stasheff. References [1] V. Abramov, R. Kerner, Exterior differentials of higher order and ther covariant generalization, J. Math. Phys. 41 (8) (2000) 5598-5614. [2] V. Abramov, R. Kerner, On certain realizations of the q-deformed exterior differential calculus, Rep. Math. Phys. 43 (1999) 179-194. [3] M. Angel, J. Camacaro, R. Dı́az, Batalin-Vilkovisky algebras andN -complexes, in preparation. [4] M. Angel, R. Dı́az, N-differential graded algebras, J. Pure App. Alg. 210 (3) (2007) 673-683. [5] M. Angel, R. Dı́az, N -flat connections, in S. Paycha, B. Uribe (Eds.), Geometric and Topological Methods for Quantum Field Theory, Contemp. Math. 432, Amer. Math. Soc., Providence, pp. 163- 172, 2007. [6] M. Angel, R. Dı́az. On the q-analoque of the Maurer-Cartan equation, Adv. Stud. Contemp. Math. 12 (2) (2006) 315-322. [7] M. Angel, R. Dı́az, AN -algebras, preprint, arXiv.math.QA/0612661. [8] N. Bazunova, Construction of graded differential algebra with ternary differential, in J. Fuchs, J. Mickelsson, Grigori Rozenblioum and Alexander Stolin (Eds.), Noncommutative geometry and rep- resentation theory in mathematical physics, Contemp. Math. 391, Amer. Math. Soc., Providence, pp. 1-9, 2005. [9] N. Bazunova, Non-coordinate case of graded differential algebra with ternary differential, J. Nonlinear Math. Phys. 13 (2006) 21-26. [10] J. R. Camacaro, Lie algebroid exterior algebra in gauge field theories, in Groups, Geometry and Physics, Monogr. Acad. Ci. Zaragoza 29, Zaragoza, pp. 57-64, 2006. [11] J.F. Cariñena, Lie groupoids and algebroids in classical and quantum mechanics, in Symmetries in Quantum Mechanics and Quantum Optics, Universidad de Burgos, Burgos, pp. 67-81, 1999. [12] A.C. da Silva, A. Weinstein, Lectures on geometrical models for noncommutative algebra, Berkeley Mathematical Lecture Notes 10, Amer. Math. Soc., Providence, 1999. [13] M. Dubois-Violette, Generalized differential spaces with dN = 0 and the q-differential calculus, Czech J. of Phys. 46 (1996) 1227- 1233. [14] M. Dubois-Violette, Generalized homologies for dN = 0 and graded q-differential algebras, in M. Henneaux, J. Krasil’shchik, A. Vinogradov (Eds.), Secondary Calculus and Cohomological Physics, Contemp. Maths. 219, Amer. Math. Soc., Providence, pp. 69-79, 1998. [15] M. Dubois-Violette, Lectures on differentials, generalized differentials and some examples re- lated to theoretical physics, in R. Coquereaux, A. Garcia, R. Trinchero (Eds.),Quantum Symmetries in Theoretical Physics and Mathematics, Contemp. Maths. 294, Amer. Math. Soc., Providence, pp. 59-94, 2002. [16] M. Dubois-Violette, R. Kerner, Universal q-differential calculus and q-analog of homological algebra, Acta Math. Univ. Comenianae LXV (2) (1996) 175-188. [17] N. P. Landsman, Lie groups and Lie algebroids in physics and noncommutative geometry, J. Geom. Phys. 56 (2006) 24-54. [18] M.M. Kapranov, On the q-analog of homological algebra, preprint, arXiv.q-alg/9611005. [19] C. Kassel, M. Wambst, Algèbre homologique des N-complexes et homologie de Hochschild aux racines de l’unité, Publ. Res. Inst. Math. Sci. Kyoto University 34 (2) (1998) 91-114. [20] R. Kerner, The cubic chessboard, Class. Quantum Grav. 14 (1997) A203-A225. [21] R. Kerner, Z3-graded exterior differential calculus and gauge theories of higher order, Lett. Math. Phys. 36 (1996) 441-454. [22] R. Kerner, B. Niemeyer, Covariant q-differential calculus and its deformations at qN = 1, Lett. Math. Phys. 45 (1998) 161-176. http://arxiv.org/abs/math/0612661 http://arxiv.org/abs/q-alg/9611005 [23] K. C. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Math. Soc. Lecture Note Series 213, Cambridge Univ. Press, Cambridge, 2005. [24] W. Mayer, A new homology theory I, Ann. of Math. 43 (1942) 370-380. [25] W. Mayer, A new homology theory II, Ann. of Math. 43 (1942) 594-605. [26] A. Sitarz, On the tensor product construction for q-differential algebras, Lett. Math. Phys. 44 (1998). [27] J. Pradines, Théorie de Lie pour les groupöıdes différentiables. Relations entre propiétés locales et globales, C. R. Acad. Sci. Paris Sér. I Math. 236 (1966) 907-910. [28] D. Zeilberger; Closed forms (pun inteded!), in A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math. 143, Amer. Math. Soc., Providence, pp. 579-607, 1993. [email protected], [email protected], [email protected] Introduction Examples of N-differential graded algebras On the (3,N) curvature N Lie algebroids

0704.0825

Electronic structure of kinetic energy driven superconductors in the presence of bilayer splitting

Electronic structure of kinetic energy driven superconductors in the presence of bilayer splitting Yu Lan,1 Jihong Qin,2 and Shiping Feng1 Department of Physics, Beijing Normal University, Beijing 100875, China Department of Physics, Beijing University of Science and Technology, Beijing 100083, China (Dated: November 17, 2018) Within the framework of the kinetic energy driven superconductivity, the electronic structure of bilayer cuprate superconductors in the superconducting state is studied. It is shown that the electron spectrum of bilayer cuprate superconductors is split into the bonding and antibonding components by the bilayer splitting, then the observed peak-dip-hump structure around the [π, 0] point is mainly caused by this bilayer splitting, with the superconducting peak being related to the antibonding component, and the hump being formed by the bonding component. The spectral weight increases with increasing the doping concentration. In analogy to the normal state case, both electron antibonding peak and bonding hump have the weak dispersions around the [π, 0] point. PACS numbers: 74.20.Mn, 74.20.-z, 74.25.Jb I. INTRODUCTION The parent compounds of cuprate superconductors are the Mott insulators with an antiferromagnetic (AF) long- range order (AFLRO), then via the charge carrier doping, one can drive these materials through a metal-insulating transition and enter the superconducting (SC) dome1,2,3. It has become clear in the past twenty years that cuprate superconductors are among the most complex systems studied in condensed matter physics1,2,3. The compli- cations arise mainly from (1) a layered crystal structure with one or more CuO2 planes per unit cell separated by insulating layers which leads to a quasi-two-dimensional electronic structure, and (2) extreme sensitivity of the physical properties to the compositions (stoichiometry) which control the carrier density in the CuO2 plane 1,2,3. As a consequence, both experimental investigation and theoretical understanding are extremely difficult. By virtue of systematic studies using the angle-resolved photoemission spectroscopy (ARPES), the low-energy electronic structure of cuprate superconductors in the SC state is well-established by now2,3, where an agree- ment has emerged that the electronic quasiparticle-like excitations are well defined, and are the entities par- ticipating in the SC pairing. In particular, the lowest energy states are located at the [π, 0] point of the Bril- louin zone, where the d-wave SC gap function is max- imal, then the most contributions of the electron spec- tral function come from the [π, 0] point2,3. Moreover, some ARPES experimental results unambiguously estab- lished the Bogoliubov-quasiparticle nature of the sharp SC quasiparticle peak near the [π, 0] point4,5, then the SC coherence of the quasiparticle peak is described by the simple Bardeen-Cooper-Schrieffer (BCS) formalism6. However, there are numerous anomalies for different fam- ilies of cuprate superconductors, which complicate the physical properties of the electronic structure2,3. Among these anomalies is the dramatic change in the spectral lineshape around the [π, 0] point first observed on the bi- layer cuprate superconductor Bi2Sr2CaCu2O8+δ, where a sharp quasiparticle peak develops at the lowest bind- ing energy, followed by a dip and a hump, giving rise to the so-called peak-dip-hump (PDH) structure in the elec- tron spectrum7,8,9. Later, this PDH structure was also found in YBa2Cu3O7−δ 10 and in Bi2Sr2Ca2Cu3O10+δ Furthermore, although the sharp quasiparticle peaks are identified in the SC state along the entire Fermi surface, the PDH structure is most strongly developed around the [π, 0] point2,7,8,9,10,11. The appearance of the PDH structure in bilayer cuprate superconductors in the SC state is the mostly remarkable effect, however, its full understanding is still a challenging issue. The earlier works2,12 gave the main impetus for a phenomenological description of the single- particle excitations in terms of an interaction between quasiparticles and collective modes, which is of fun- damental relevance to the nature of superconductivity and the pairing mechanism in cuprate superconductors. However, the different interpretive scenario has been proposed2,13. This followed from the observation of the bilayer splitting (BS) for both normal and SC states in a wide doping range14,15,16. This BS of the CuO2 plane derives the electronic structure in the bonding and antibonding bands due to the present of CuO2 bilayer blocks in the unit cell of bilayer cuprate superconduc- tors, then the main features of the PDH structure is caused by the BS13,14,15,16, with the peak and hump corresponding to the antibonding and bonding bands, respectively. Furthermore, some ARPES experimental data measured above and below the SC transition tem- perature show that this PDH structure is totally unre- lated to superconductivity14. The recent ARPES exper- imental results reported by several groups support this scenario, and most convincingly suggested that the PDH structure originates from the BS at any doping levels17. To the best of our knowledge, this PDH structure in bi- layer cuprate superconductors has not been treated start- ing from a microscopic SC theory. Within the single layer t-t′-J model, the electronic structure of the single layer cuprate superconductors in the SC state has been discussed18 based on the frame- work of the kinetic energy driven superconductivity19, and the main features of the ARPES experiments on the single layer cuprate superconductors have been repro- duced, including the doping and temperature dependence of the electron spectrum and quasiparticle dispersion. In http://arxiv.org/abs/0704.0825v2 this paper, we study the electronic structure of bilayer cuprate superconductors in the SC state along with this line. Within the kinetic energy driven SC mechanism19, we employed the t-t′-J model by considering the bilayer interaction, and then show explicitly that the BS occurs due to this bilayer interaction. In this case, the elec- tron spectrum is split into the bonding and antibond- ing components by this BS, then the SC peak is closely related to the antibonding component, while the hump is mainly formed by the bonding component. In other words, the well pronounced PDH structure in the electron spectrum of bilayer cuprate superconductors is mainly caused by the BS. Furthermore, the spectral weight in the [π, 0] point increases with increasing the doping con- centration. In analogy to the normal-state case14,20,21,22, both electron antibonding peak and bonding hump have the weak dispersions around the [π, 0] point, in qualita- tive agreement with the experimental observation on bi- layer cuprate superconductors in the SC state2,7,8,9,10,11. The paper is organized as follows. The basic formal- ism is presented in Sec. II, where we generalize the kinetic energy driven superconductivity from the previ- ous single layer case18,19 to the bilayer case, and then evaluate explicitly the longitudinal and transverse com- ponents of the electron normal and anomalous Green’s functions (hence the bonding and antibonding electron spectral functions). Within this theoretical framework, we discuss the electronic structure of bilayer cuprate su- perconductors in the SC state in Sec. III. It is shown that the striking PDH structure in bilayer cuprate supercon- ductors is closely related to the BS. Finally, we give a summary and discussions in Sec. IV. II. FORMALISM It has been shown from the ARPES experiments2,23 that the two-dimensional t-t′-J model is of particular relevance to the low energy features of cuprate super- conductors. For discussions of the physical properties of bilayer cuprate superconductors, the t-t′-J model can be expressed by including the bilayer interactions as, H = −t iη̂aσ iaσCi+η̂aσ + t iτ̂aσ iaσCi+τ̂aσ t⊥(i)(C i1σCi2σ +H.c.) + µ iaσCiaσ Sia · Si+η̂a + J⊥ Si1 · Si2, (1) supplemented by an important on-site local constraint iaσCiaσ ≤ 1 to avoid the double occupancy, where η̂ = ±x̂,±ŷ representing the nearest neighbors of a given site i, τ̂ = ±x̂± ŷ representing the next nearest neighbors of a given site i, a = 1, 2 is plane index, C iaσ (Ciaσ) is the electron creation (annihilation) operator, Sia = iaσCia/2 is the spin operator with the Pauli matrices σ = (σx, σy, σz), µ is the chemical potential, and the interlayer coherent hopping has the form, t⊥(k) = (cos kx − cos ky) 2, (2) which is strongly anisotropic and follows the theoret- ical predictions24. In particular, this momentum de- pendent form (2) has been experimentally verified14,15. For this t-t′-J model (1), it has been argued that cru- cial requirement is to impose the electron single occu- pancy local constraint for a proper understanding of the physical properties of cuprate superconductors. To in- corporate the electron single occupancy local constraint, the charge-spin separation (CSS) fermion-spin theory has been proposed25, where the constrained electron opera- tors are decoupled as, Cia↑ = h ia and Cia↓ = h with the spinful fermion operator hiaσ = e −iΦiaσhia rep- resents the charge degree of freedom together with some effects of the spin configuration rearrangements due to the presence of the doped hole itself (dressed holon), while the spin operator Sia represents the spin degree of freedom, then the bilayer t-t′-J Hamiltonian (1) can be expressed in this CSS fermion-spin representation as, H = t i+η̂a↑hia↑S i+η̂a + h i+η̂a↓hia↓S i+η̂a) i+τ̂a↑hia↑S i+τ̂a + h i+τ̂a↓hia↓S i+τ̂a) t⊥(i)(h i2↑hi1↑S i2 + h i1↑hi2↑S i2↓hi1↓S i2 + h i1↓hi2↓S i1)− µ iaσhiaσ + Jeff Sia · Si+η̂a + Jeff⊥ Si1 · Si2, (3) where Jeff = J(1 − δ) 2, Jeff⊥ = J⊥(1 − δ) 2, and δ = iaσhiaσ〉 = 〈h iahia〉 is the doping concentration. It has been shown that the electron single occupancy local con- straint is satisfied in analytical calculations within this CSS fermion-spin theory, and the double spinful fermion occupancy are ruled out automatically25. Although in common sense hiaσ is not a real spinful fermion, it be- haves like a spinful fermion25. As in the single layer case18, the kinetic energy terms in the bilayer t-t′-J model have been transferred as the dressed holon-spin interactions, which can induce the dressed holon pair- ing state (hence the electron Cooper pairing state) by exchanging spin excitations in the higher power of the doping concentration. Before calculation of the electron normal and anomalous Green’s functions of the bilayer system in the SC state, we firstly introduce the SC order parameter. As we have mentioned above, there are two coupled CuO2 planes in the unit cell, and in this case, the SC order parameters for the electron Cooper pair is a matrix ∆ = ∆L + σx∆T , with the longitudinal and transverse SC order parameters in the CSS fermion-spin theory can be expressed as, ∆L = 〈C i+η̂a↓ − C i+η̂a↑〉 = 〈hia↑hi+η̂a↓S i+η̂a − hia↓hi+η̂a↑S i+η̂a〉 = −χ1∆hL, (4a) ∆T = 〈C i2↓ − C = 〈hi1↑hi2↓S i2 − hi1↓hi2↑S = −χ⊥∆hT , (4b) respectively, where the spin correlation functions χ1 = 〈S+iaS i+η̂a〉 and χ⊥ = 〈S i2〉, and the longitudinal and transverse dressed holon pairing order parameters ∆hL = 〈hi+η̂a↓hia↑ − hi+η̂a↑hia↓〉 and ∆hT = 〈hi2↓hi1↑ − hi2↑hi1↓〉. Within the t-J type model, robust indications of superconductivity with the d-wave symmetry in doped cuprates have been found by using numerical techniques26. On the other hand, it has been argued that the SC transition in doped cuprates is determined by the need to reduce the frustrated kinetic energy27. Although it is not necessary for the strong coupling of the electron quasiparticles and a pairing boson in their arguments27, a series of the inelastic neutron scattering experimental results provide a clear link between the electron quasi- particles and magnetic excitations28,29. In particular, an impurity-substitution effect on the low energy dynamics has been studied by virtue of the ARPES measurement30, this impurity-substitution effect is a magnetic analogue of the isotope effect used for the conventional super- conductors. These experimental results30 reveal that the impurity-induced changes in the electron self-energy show a good correspondence to those of the magnetic excitations, indicating the importance of the magnetic fluctuation to the electron pairing in cuprate supercon- ductors. Recently, we19 have developed the kinetic en- ergy driven SC mechanism based on the CSS fermion- spin theory25, where the dressed holons interact occur- ring directly through the kinetic energy by exchanging spin excitations, leading to a net attractive force between dressed holons, then the electron Cooper pairs originat- ing from the dressed holon pairing state are due to the charge-spin recombination, and their condensation re- veals the SC ground-state. Within this SC mechanism19, the doping and temperature dependence of the electron spectral function of the single layer cuprate supercon- ductors in the SC state has been discussed18. In this section, our main goal is to generalize these analytical calculations from the single layer case to the bilayer sys- tem. As in the case for the SC order parameter, the full dressed holon normal and anomalous Green’s functions can also be expressed as g(k, ω) = gL(k, ω) + σxgT (k, ω) and ℑ†(k, ω) = ℑ L(k, ω) + σxℑ L(k, ω), respectively. We now can follow the previous discussions for the single layer case18,19, and evaluate explicitly these correspond- ing longitudinal and transverse parts of the full dressed holon normal and anomalous Green’s functions as [see the Appendix], gL(k, ω) = ν=1,2 U2hνk ω − Ehνk V 2hνk ω + Ehνk , (5a) gT (k, ω) = ν=1,2 (−1)ν+1Z U2hνk ω − Ehνk V 2hνk ω + Ehνk , (5b) L(k, ω) = − ν=1,2 2Ehνk ω − Ehνk ω + Ehνk , (5c) T (k, ω) = − ν=1,2 (−1)ν+1Z 2Ehνk ω − Ehνk ω + Ehνk , (5d) where the dressed holon quasiparticle coherence fac- tors U2hνk = [1 + ξ̄νk/Ehνk]/2 and V hνk = [1 − ξ̄νk/Ehνk]/2, the dressed holon quasiparticle disper- sion Ehνk = [ξ̄νk]2+ | ∆̄ (k) |2, the renormalized dressed holon excitation spectrum ξ̄νk = Z ξνk, with the mean-field (MF) dressed holon excitation spectrum ξνk = Ztχ1γk − Zt k − µ+ (−1) ν+1χ⊥t⊥(k), where the spin correlation function χ2 = 〈S i+τ̂a〉, γk = (1/Z) eik·η̂, γ′k = (1/Z) eik·τ̂ , Z is the num- ber of the nearest neighbor or next nearest neighbor sites, the renormalized dressed holon pair gap func- tion ∆̄ (k) = Z [∆̄hL(k) + (−1) ν+1∆̄hT (k)], with ν = 1 ( ν = 2) for the bonding (antibonding) case, where ∆̄hL(k) = Σ 2L (k, ω) |ω=0= ∆̄hLγ , with γ (coskx − cosky)/2, ∆̄hT (k) = Σ 2T (k, ω) |ω=0= ∆̄hT , the dressed holon quasiparticle coherent weights Z (1)−1 hF1 − Z hF2, Z (2)−1 = Z−1 hF1 + Z hF2, with Z hF1 = 1 − Σ 1L (k0, ω) |ω=0, and Z hF2 = Σ 1T (k0, ω) |ω=0 , where k0 = [π, 0], Σ 1L (k, ω) and Σ 1T (k, ω) are the cor- responding antisymmetric parts of the longitudinal and transverse dressed holon self-energy functions Σ 1L (k, ω) and Σ 1T (k, ω), while the longitudinal and transverse parts of the dressed holon self-energy functions Σ 1 (k, ω) and Σ 2 (k, ω) have been evaluated as, 1L (k, iωn) = p+q+k gL(p+ k, ipm + iωn)ΠLL(p,q, ipm) p+q+k gT (p+ k, ipm + iωn)ΠTL(p,q, ipm)], (6a) 1T (k, iωn) = p+q+k gT (p+ k, ipm + iωn)ΠTT (p,q, ipm) p+q+k gL(p+ k, ipm + iωn)ΠLT (p,q, ipm)], (6b) 2L (k, iωn) = p+q+k L(−p− k,−ipm − iωn)ΠLL(p,q, ipm) p+q+k T (−p− k,−ipm − iωn)ΠTL(p,q, ipm)], (6c) 2T (k, iωn) = p+q+k (−p− k,−ipm − iωn)ΠTT (p,q, ipm) p+q+k L(−p− k,−ipm − iωn)ΠLT (p,q, ipm)], (6d) where R = [Z(tγk − t ′γ′k)] 2 + t2⊥(k), R = 2Z(tγk − )t⊥(k), and the spin bubbles Πη,η′(p,q, ipm) = (1/β) η (q, iqm)D η′ (q + p, iqm+ ipm), with η = L, T and η′ = L, T , and the MF spin Green’s function D(0)(k, ω) = D (k, ω) + σxD (k, ω), with the cor- responding longitudinal and transverse parts have been given by22, L (k, ω) = ν=1,2 ω2 − ω2 , (7a) (k, ω) = ν=1,2 (−1)ν+1 ω2 − ω2 , (7b) where Bνk = λ(A1γk−A2)−λ ′(2χz2γ k−χ2)−Jeff⊥[χ⊥+ 2χz⊥(−1) ν ][ǫ⊥(k)+(−1) ν ], A1 = 2ǫ‖χ 1+χ1, A2 = ǫ‖χ1+ 2χz1, λ = 2ZJeff , λ ′ = 4Zφ2t ′, ǫ‖ = 1+2tφ1/Jeff , ǫ⊥(k) = 1 + 4φ⊥t⊥(k)/Jeff⊥, the spin correlation functions χ 〈SziaS i+η̂a〉, χ 2 = 〈S i+τ̂a〉, χ ⊥ = 〈S i2〉, the dressed holon particle-hole order parameters φ1 = 〈h iaσhi+η̂aσ〉, φ2 = 〈h iaσhi+τ̂aσ〉, φ⊥ = 〈h i1σhi2σ〉, and the MF spin excitation spectrum, ω2νk = λ A4 − αǫ‖χ 1γk − αǫ‖χ1 (1 − ǫ‖γk) + αχz1 − αχ1γk (ǫ‖ − γk) Z − 1 γ′k + + λλ′α χz1(1 − ǫ‖γk)γ k − C2)(ǫ‖ − γk) + γ′k(C 2 − ǫ‖χ 2γk)− ǫ‖(C2 − χ2γk) + λJeff⊥α ǫ⊥(k)(ǫ‖ − γk)[C⊥ + χ1(−1) + (1− ǫ‖γk)[C ⊥ + χ 1ǫ⊥(k)(−1) ν ] + [ǫ⊥(k) + (−1) ǫ‖(C⊥ − χ⊥γk) + (C ⊥ − ǫ‖χ ⊥γk)(−1) + λ′Jeff⊥α γ′k[C ⊥ + χ 2ǫ⊥(k)(−1) ǫ⊥(k)[C ⊥ + χ2(−1) k − C ⊥) + χ k(−1) [ǫ⊥(k) + (−1) J2eff⊥[ǫ⊥(k) + (−1) ν ]2, (8) where A3 = αC1 + (1− α)/2Z, A4 = αC 1 + (1− α)/4Z, A5 = αC3 + (1 − α)/2Z, and the spin correla- tion functions C1 = (1/Z η̂η̂′ i+η̂aS i+η̂′a C2 = (1/Z i+η̂aS i+τ̂a〉, C3 = (1/Z2) τ̂ τ̂ ′ i+τ̂aS i+τ̂ ′a 〉, Cz1 = (1/Z2) η̂η̂′ 〈Szi+η̂aS i+η̂′a 〉, Cz2 = (1/Z2) 〈Szi+η̂aS i+τ̂a〉, C⊥ = (1/Z) 〈S+i1S i+η̂2〉, C′⊥ = (1/Z) 〈S+i1S i+τ̂2〉, C ⊥ = (1/Z) 〈Szi1S i+η̂2〉, and C′ ⊥ = (1/Z) 〈Szi1S i+τ̂2〉. In order to satisfy the sum rule of the spin correlation function 〈S+iaS ia〉 = 1/2 in the case without AFLRO, the important decoupling parameter α has been introduced in the above calcu- lation as in the single layer case18,19,22, which can be regarded as the vertex correction. With the help of the longitudinal and transverse parts of the full dressed holon normal and anomalous Green’s functions in Eq. (5) and MF spin Green’s function in Eq. (7), we now can calculate the electron nor- mal and anomalous Green’s functions G(i − j, t − t′) = 〈〈Ciσ(t);C ′)〉〉 = GL(i− j, t− t ′) + σxGT (i− j, t− t and Γ†(i−j, t−t′) = 〈〈C i↑(t);C ′)〉〉 = Γ L(i−j, t−t T (i−j, t−t ′), where these longitudinal and transverse parts are the convolutions of the corresponding longitudi- nal and transverse parts of the full dressed holon normal and anomalous Green’s functions and MF spin Green’s function in the CSS fermion-spin theory, and can be eval- uated explicitly as, GL(k, ω) = L(1)µν (k,p) U2hµp−k ω + Ehµp−k − ωνp V 2hµp−k ω − Ehµp−k + ωνp + L(2)µν (k,p) U2hµp−k ω + Ehµp−k + ωνp V 2hµp−k ω − Ehµp−k − ωνp , (9a) GT (k, ω) = (−1)µ+νZ L(1)µν (k,p) U2hµp−k ω + Ehµp−k − ωνp V 2hµp−k ω − Ehµp−k + ωνp + L(2)µν (k,p) U2hµp−k ω + Ehµp−k + ωνp V 2hµp−k ω − Ehµp−k − ωνp , (9b) L(k, ω) = (p− k) 2Ehµp−k L(1)µν (k,p) ω − Ehµp−k + ωνp ω + Ehµp−k − ωνp + L(2)µν (k,p) ω − Ehµp−k − ωνp ω + Ehµp−k + ωνp , (9c) (k, ω) = (−1)µ+νZ (p− k) 2Ehµp−k L(1)µν (k,p) ω − Ehµp−k + ωνp ω + Ehµp−k − ωνp + L(2)µν (k,p) ω − Ehµp−k − ωνp ω + Ehµp−k + ωνp , (9d) where L µν (k,p) = [coth(βωνp/2) − th(βEhµp−k/2)]/2 and L µν (k,p) = [coth(βωνp/2) + th(βEhµp−k/2)]/2, then the longitudinal and transverse parts of the electron spectral function AL(k, ω) = −2ImGL(k, ω) and AT (k, ω) = −2ImGT (k, ω) and SC gap func- tion ∆L(k) = (1/β) L(k, iωn) and ∆T (k) = (1/β) (k, iωn) are obtained as, AL(k, ω) = π {L(1)µν (k,p)[U hµp−kδ(ω + Ehµp−k − ωνp) + V hµp−kδ(ω − Ehµp−k + ωνp)] + L(2)µν (k,p)[U hµp−kδ(ω + Ehµp−k + ωνp) + V hµp−kδ(ω − Ehµp−k − ωνp)]}, (10a) AT (k, ω) = π (−1)µ+νZ {L(1)µν (k,p)[U hµp−kδ(ω + Ehµp−k − ωνp) + V hµp−kδ(ω − Ehµp−k + ωνp)] + L(2)µν (k,p)[U hµp−kδ(ω + Ehµp−k + ωνp) + V hµp−kδ(ω − Ehµp−k − ωνp)]}, (10b) ∆L(k) = − p,µ,ν (p− k) Ehµp−k βEhµp−k]coth[ βωνp], (10c) ∆T (k) = − p,µ,ν (−1)µ+νZ (p− k) Ehµp−k βEhµp−k]coth[ βωνp]. (10d) With the above longitudinal and transverse parts of the SC gap functions in Eqs. (10c) and (10d), the corre- sponding longitudinal and transverse SC gap parameters are obtained as ∆L = −χ1∆hL and ∆T = −χ⊥∆hT , respectively. In the bilayer coupling case, the more ap- propriate classification is in terms of the spectral func- tion and SC gap function within the basis of the an- tibonding and bonding components13,14,15,16,17. In this case, the electron spectral function and SC gap parame- ter can be transformed from the plane representation to the antibonding-bonding representation as, A(a)(k, ω) = [AL(k, ω)−AT (k, ω)], (11a) A(b)(k, ω) = [AL(k, ω) +AT (k, ω)], (11b) ∆(a) = ∆L −∆T , (11c) ∆(b) = ∆L +∆T . (11d) respectively, then the antibonding and bonding parts have odd and even symmetries, respectively. III. ELECTRON STRUCTURE OF BILAYER CUPRATE SUPERCONDUCTORS We now begin to discuss the effect of the bilayer in- teraction on the electronic structure in the SC state. We first plot, in Fig. 1, the antibonding (solid line) and bonding (dashed line) electron spectral functions in the [π, 0] point for parameters t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35 with temperature T = 0.002J at the doping concentration δ = 0.15. In comparison with the single layer case18, the electron spectrum of the bilayer system has been split into the bonding and antibonging compo- nents, with the bonding and antibonding SC quasipar- ticle peaks in the [π, 0] point are located at the differ- ent positions. In this sense, the differentiation between the bonding and antibonding components of the electron spectral function is essential. The antibonding spectrum consists of a low energy antibonding peak, corresponding to the SC peak, and the bonding spectrum has a higher energy bonding peak, corresponding to the hump, while the spectral dip is in between them, then the total con- tributions for the electron spectrum from both antibond- ing and bonding components give rise to the PDH struc- ture. Although the simple bilayer t-t′-J model (1) can- not be regarded as a comprehensive model for a quanti- tative comparison with bilayer cuprate superconductors, our present results for the SC state are in qualitative agreement with the major experimental observations on bilayer cuprate superconductors2,7,8,9,10,11,16. We now turn to discuss the doping evolution of the electron spectrum of bilayer cuprate superconductors in ( )/J -1.0 -.5 0.0 .5 Bonding Antibonding FIG. 1: The antibonding (solid line) and bonding (dashed line) electron spectral functions in the [π, 0] point for t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35 with T = 0.002J at δ = 0.15. ( )/J -1.0 -.5 0.0 .5 FIG. 2: The electron spectral functions at [π, 0] point for t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35 with T = 0.002J at δ = 0.09 (solid line), δ = 0.12 (dashed line), and δ = 0.15 (dotted line). the SC state. We have calculated the electron spec- trum at different doping concentrations, and the result of the electron spectral functions in the [π, 0] point for t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35 with T = 0.002J at δ = 0.09 (solid line), δ = 0.12 (dashed line), and δ = 0.15 (dotted line) are plotted in Fig. 2. In compari- son with the corresponding ARPES experimental results of the bilayer cuprate superconductor Bi2Sr2CaCu2O8+δ in the SC state in Ref.12, it is obviously that the doping evolution of the spectral weight of the bilayer supercon- Bonding Band Antibonding Band (-0.2 , ) (0, ) (0.2 , ) FIG. 3: The positions of the antibonding peaks and bonding humps in the electron spectrum as a function of momentum along the direction [−0.2π, π] → [0, π] → [0.2π, π] with T = 0.002J at δ = 0.15 for t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35. ductor Bi2Sr2CaCu2O8+δ is reproduced. With increas- ing the doping concentration, both SC peak and hump become sharper, and then the spectral weights increase in intensity. Furthermore, we have also calculated the electron spectrum with different temperatures, and the results show that the spectral weights of both SC peak and hump are suppressed with increasing temperatures. Our these results are also qualitatively consistent with the ARPES experimental results on bilayer cuprate su- perconductors in the SC state2,9,12. To better perceive the anomalous form of the antibond- ing and bonding electron spectral functions as a function of energy ω for k in the vicinity of the [π, 0] point, we have made a series of calculations for the electron spec- tral function at different momenta, and the results show that the sharp SC peak from the electron antibonding spectral function and hump from the bonding spectral function persist in a very large momentum space region around the [π, 0] point. To show this point clearly, we plot the positions of the antibonding peak and bonding hump in the electron spectrum as a function of momen- tum along the direction [−0.2π, π] → [0, π] → [0.2π, π] with T = 0.002J at δ = 0.15 for t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35 in Fig. 3. Our result shows that there are two branches in the quasiparticle dispersion, with upper branch corresponding to the antibonding quasi- particle dispersion, and lower branch corresponding to the bonding quasiparticle dispersion. Furthermore, the BS reaches its maximum at the [π, 0] point. Our present result also shows that in analogy to the two flat bands ap- peared in the normal state22, both electron antibonding peak and bonding hump have a weak dispersion around the [π, 0] point, in qualitative agreement with the ARPES experimental measurements on bilayer cuprate supercon- ductors in the SC state2,7,8,9,10,11,14. In the above calculations, we find that although the antibonding SC peak and bonding hump have different dispersions, the transverse part of the SC gap param- eter ∆T ≈ 0. To show this point clearly, we plot the antibonding and bonding gap parameters in Eqs. (11c) 00.00 0.05 0.10 0.15 0.20 0.25 0.30 00.00 FIG. 4: The antibonding (solid line) and bonding (dashed line) gap parameters as a function of the doping concentration with T = 0.002J for t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35. and (11d) as a function of the doping concentration with T = 0.002J for t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35 in Fig. 4. As seen from Fig. 4, both antibonding and bond- ing gap parameters have the same d-wave SC gap mag- nitude in a given doping concentration, i.e., ∆a ≈ ∆b. This result shows that although there is a single elec- tron interlayer coherent hopping (2) in bilayer cuprate superconductors in the SC state, the electron interlayer pairing interaction vanishes. This reflects that in the present kinetic energy driven SC mechanism, the weak dressed holon-spin interaction due to the interlayer co- herent hopping (2) from the kinetic energy terms in Eq. (3) does not induce the dressed holon interlayer pair- ing state by exchanging spin excitations in the higher power of the doping concentration. This is different from the dressed holon-spin interaction due to the intralayer hopping from the kinetic energy terms in Eq. (3), it can induce superconductivity by exchanging spin excita- tions in the higher power of the doping concentration19. Our this result is also consistent with the ARPES ex- perimental results of the bilayer cuprate superconductor Bi(Pb)2Sr2CaCu2O8+δ 14,16, where the SC gap separately for the bonding and antibonding bands has been mea- sured, and it is found that both d-wave SC gaps from the antibonding and bonding components are identical within the experimental uncertainties. To our present understanding, two main reasons why the electronic structure of bilayer cuprate superconduc- tors in the SC state can be described qualitatively in the framework of the kinetic energy driven supercon- ductivity by considering the bilayer interaction are as follows. Firstly, the bilayer interaction causes the BS, this leads to that the full electron normal (anomalous) Green’s function is divided into the longitudinal and transverse parts, respectively, then the bonding and an- tibonding electron spectral functions (SC gap functions) are obtained from these longitudinal and transverse parts of the electron normal (anomalous) Green’s function, respectively. Although the transverse part of the SC gap parameter ∆T ≈ 0, the antibonding peak around the [π, 0] point is always at lower binding energy than the bonding peak (hump) due to the BS. In this sense, the PDH structure in the bilayer cuprate superconduc- tors in the SC state is mainly caused by the BS. Sec- ondly, the SC state in the kinetic energy driven SC mechanism is the conventional BCS like as in the sin- gle layer case18,19. This can be understood from the electron normal and anomalous Green’s functions in Eq. (9). Since the spins center around the [π, π] point in the MF level18,19,22, then the main contributions for the spins comes from the [π, π] point. In this case, the longitudinal and transverse parts of the electron nor- mal and anomalous Green’s functions in Eq. (9) can be approximately reduced in terms of ωνp=[π,π] ∼ 0 and one of the self-consistent equations22 1/2 = 〈S+iaS ia〉 = 1/(4N) (Bνk/ωνk)coth[(1/2)βωνk] as, GL(k, ω) ≈ ν=1,2 ω − Eνk V 2νk ω + Eνk (12a) GT (k, ω) ≈ ν=1,2 (−1)ν+1Z ω − Eνk V 2νk ω + Eνk (12b) (k, ω) = ν=1,2 z (k) ω − Eνk ω + Eνk (12c) T (k, ω) = ν=1,2 (−1)ν+1Z z (k) ω − Eνk ω + Eνk , (12d) where the electron coherent weights Z FA = Z /2, the electron quasiparticle coherence factors U2νk ≈ V hνk−kA and V 2νk ≈ U hνk−kA , the SC gap function ∆̄ z (k) ≈ (k − kA) and the electron quasiparticle spectrum Eνk ≈ Ehνk−kA , with kA = [π, π]. As in the sin- gle layer case18,19, this reflects that the hole-like dressed holon quasiparticle coherence factors Vhνk and Uhνk and hole-like dressed holon quasiparticle spectrum Ehνk have been transferred into the electron quasiparticle coher- ence factors Uνk and Vνk and electron quasiparticle spec- trum Eνk, respectively, by the convolutions of the corre- sponding longitudinal and transverse parts of the MF spin Green’s function and full dressed holon normal and anomalous Green’s functions due to the charge-spin recombination27. As a result, these electron normal and anomalous Green’s functions in Eq. (12) are typical bi- layer BCS like6. This also reflects that as in the single layer case18,19, the dressed holon pairs condense with the d-wave symmetry in a wide range of the doping concen- tration, then the electron Cooper pairs originating from the dressed holon pairing state are due to the charge- spin recombination, and their condensation automati- cally gives the electron quasiparticle character. These are why the basic bilayer BCS formalism6 is still valid in discussions of SC coherence of the quasiparticle peak and hump, although the pairing mechanism is driven by the intralayer kinetic energy by exchanging spin excitations, and other exotic magnetic scattering28,29 is beyond the BCS formalism. IV. SUMMARY AND DISCUSSIONS We have studied the electronic structure of bilayer cuprate superconductors in the SC state based on the kinetic energy driven SC mechanism19. Our results show that the electron spectrum of bilayer cuprate supercon- ductors is split into the bonding and antibonding com- ponents by the BS, then the observed PDH structure around the [π, 0] point is mainly caused by this BS, with the SC peak being related to the antibonding compo- nent, and the hump being formed by the bonding com- ponent. The spectral weight increases with increasing the doping concentration. In analogy to the two flat bands appeared in the normal state, the antibonding and bonding quasiparticles around the [π, 0] point dis- perse weakly with momentum, in qualitative agreement with the experimental observation on the bilayer cuprate superconductors2,7,8,9,10,11. Our these results also show that the bilayer interaction has significant contributions to the electronic structure of bilayer cuprate supercon- ductors in the SC state. It has been shown from the ARPES experiments2,14 that the BS has been detected in both normal and SC states, and then the electron spectral functions display the double-peak structure in the normal state and PDH structure in the SC state. Recently, we22 have studied the electron spectrum of bilayer cuprate superconductors in the normal state, and shown that the double-peak struc- ture in the electron spectrum in the normal state is dom- inated by the BS. On the other hand, although the anti- bonding and bonding SC peaks have different dispersions, the antibonding and bonding parts have the same d-wave SC gap amplitude as mentioned above. Incorporating our previous discussions for the normal state case22 and the present studies for the SC state case, we therefore find that the one of the important roles of the interlayer co- herent hopping (2) is to split the electron spectrum of the bilayer system into the bonding and antibonding compo- nents in both normal and SC states. As a consequence, the well pronounced PDH structure of bilayer cuprate su- perconductors in the SC state and double-peak structure in the normal state are mainly caused by the BS. Acknowledgments The authors would like to thank Dr. H. Guo and Dr. L. Cheng for the helpful discussions. This work was sup- ported by the National Natural Science Foundation of China under Grant No. 90403005, and the funds from the Ministry of Science and Technology of China under Grant Nos. 2006CB601002 and 2006CB921300. APPENDIX A: DRESSED HOLON BCS TYPE NORMAL AND ANOMALOUS GREEN’S FUNCTIONS IN BILAYER CUPRATE SUPERCONDUCTORS In the single layer case, it has been shown19 that the dressed holon-spin interactions from the kinetic en- ergy terms of the t-t′-J model are quite strong, and in the case without AFLRO, these interactions can in- duce the dressed holon pairing state (then the electron Cooper pairing state) by exchanging spin excitations in the higher power of the doping concentration. Following their discussions18,19, we obtain in terms of Eliashberg’s strong coupling theory31 that the self-consistent equa- tions that satisfied by the full dressed holon normal and anomalous Green’s functions in the bilayer system in the SC state as, g(k, ω) = g(0)(k, ω) + g(0)(k, ω) 1 (k, ω)g(k, ω) 2 (−k,−ω)ℑ †(k, ω) , (A1a) ℑ†(k, ω) = g(0)(−k,−ω) 1 (−k,−ω)ℑ †(−k,−ω) 2 (−k,−ω)g(k, ω) , (A1b) respectively, where the MF dressed holon normal Green’s function22 g(0)(k, ω) = g (k, ω) + σxg (k, ω), with the longitudinal and transverse parts are evaluated as L (k, ω) = (1/2) ν=1,2(ω − ξνk) −1 and g T (k, ω) = (1/2) ν=1,2(−1) ν+1(ω − ξνk) −1, respectively, while the dressed holon self-energy functions Σ 1 (k, ω) = 1L (k, ω) + σxΣ 1T (k, ω) and Σ 2 (k, ω) = Σ 2L (k, ω) + 2T (k, ω), with the corresponding longitudinal and transverse parts have been given in Eq. (6). In the previous discussions of the electronic struc- ture for the single layer cuprate superconductors in the SC state18, it has been shown the self-energy function 2 (k, ω) describes the effective dressed holon pair gap function, while the self-energy function Σ 1 (k, ω) de- scribes the quasiparticle coherence. Since Σ 2 (k, ω) is an even function of ω, while Σ 1 (k, ω) is not, therefore for the convenience, the self-energy function Σ 1 (k, ω) can be broken up into its symmetric and antisymmetric parts as, Σ 1 (k, ω) = Σ 1e (k, ω)+ωΣ 1o (k, ω), then both 1e (k, ω) and Σ 1o (k, ω) are even functions of ω. Now we can define the dressed holon quasiparticle coherent weights in the present bilayer system as Z−1 hF1(k, ω) = 1 − Σ 1L (k, ω) and Z hF2(k, ω) = Σ 1T (k, ω). As in the single layer case18, we only discuss the low-energy behav- ior of the electronic structure of bilayer cuprate super- conductors, which means that the effective dressed holon pair gap functions and quasiparticle coherent weights can be discussed in the static limit, i.e., ∆̄h(k) = 2 (k, ω) |ω=0= ∆̄hL(k) + σx∆̄hT (k), Z hF1(k) = 1 − 1L (k, ω) |ω=0 and Z hF2(k) = Σ 1T (k, ω) |ω=0. As in the single layer case18, although ZhF1(k) and ZhF2(k) still are a function of k, the wave vector dependence may be unimportant. This followed from the ARPES experiments2 that in the SC-state of bilayer cuprate su- perconductors, the lowest energy states are located at the [π, 0] point, which indicates that the majority con- tribution for the electron spectrum comes from the [π, 0] point. In this case, the wave vector k in ZhF1(k) and ZhF2(k) can be chosen as Z hF1 = 1−Σ 1L (k) |k=[π,0] and hF2 = Σ 1T (k) |k=[π,0]. With the help of the above dis- cussions, the corresponding longitudinal and transverse parts of the dressed holon normal and anomalous Green’s functions in Eqs. (A1a) and (A1b) now can be obtained explicitly as, gL(k, ω) = ν=1,2 U2hνk ω − Ehνk V 2hνk ω + Ehνk , (A2a) gT (k, ω) = ν=1,2 (−1)ν+1Z U2hνk ω − Ehνk V 2hνk ω + Ehνk , (A2b) L(k, ω) = − ν=1,2 2Ehνk ω − Ehνk ω + Ehνk , (A2c) T (k, ω) = − ν=1,2 (−1)ν+1Z 2Ehνk ω − Ehνk ω + Ehνk , (A2d) with the dressed holon effective gap parameters and quasiparticle coherent weights satisfy the following four equations, ∆̄hL = − k,q,p ν,ν′,ν′′ k−p+qCνν′′ (k+ q) (ν′′) Bν′pBνq ων′pωνq (ν′′) νν′ν′′(q,p) + F νν′ν′′(k,q,p) [ων′p − ωνq]2 − E hν′′k νν′ν′′(q,p) + F νν′ν′′(k,q,p) [ων′p + ωνq]2 − E hν′′k , (A3a) ∆̄hT = − k,q,p ν,ν′,ν′′ (−1)ν+ν ′+ν′′+1Cνν′′(k+ q) (ν′′) Bν′pBνq ων′pωνq (ν′′) νν′ν′′(q,p) + F νν′ν′′(k,q,p) [ων′p − ωνq]2 − E hν′′k νν′ν′′(q,p) + F νν′ν′′(k,q,p) [ων′p + ωνq]2 − E hν′′k , (A3b) = 1 + ν,ν′,ν′′ [1 + (−1)ν+ν ′+ν′′+1]Cνν′′ (p+ k0) (ν′′) Bν′pBνq ων′pωνq νν′ν′′(q,p) [ων′p − ωνq + Ehν′′p−q+k0 ] νν′ν′′(q,p) [ων′p − ωνq − Ehν′′p−q+k0 ] νν′ν′′(q,p) [ων′p + ωνq + Ehν′′p−q+k0] νν′ν′′ (q,p) [ων′p + ωνq − Ehν′′p−q+k0 ] , (A3c) = 1 + ν,ν′,ν′′ [1− (−1)ν+ν ′+ν′′+1]Cνν′′ (p+ k0) (ν′′) Bν′pBνq ων′pωνq νν′ν′′(q,p) [ων′p − ωνq + Ehν′′p−q+k0 ] νν′ν′′(q,p) [ων′p − ωνq − Ehν′′p−q+k0 ] νν′ν′′(q,p) [ων′p + ωνq + Ehν′′p−q+k0] νν′ν′′ (q,p) [ων′p + ωνq − Ehν′′p−q+k0 ] , (A3d) where Cνν′′ (k) = [Z(tγk − t ′γ′k) + (−1) ν+ν′′t⊥(k)] νν′ν′′ (q,p) = nB(ωνq)+nB(ων′p)+2nB(ωνq)nB(ων′p), νν′ν′′(k,q,p) = [2nF (Ehν′′k) − 1][ων′p − ωνq][nB(ωνq) − nB(ων′p)]/Ehν′′k, F νν′ν′′(q,p) = 1 + nB(ωνq) + nB(ων′p) + 2nB(ωνq)nB(ων′p), νν′ν′′ (k,q,p) = [2nF (Ehν′′k) − 1][ων′p + ωνq][1 + nB(ωνq) + nB(ων′p)]/Ehν′′k, H νν′ν′′(q,p) = nF (Ehν′′p−q+k0)[nB(ων′p) − nB(ωνq)] + nB(ωνq)[1 + nB(ων′p)], H νν′ν′′(q,p) = nF (Ehν′′p−q+k0)[nB(ωνq) − nB(ων′p)] + nB(ων′p)[1 + nB(ωνq)], H νν′ν′′(q,p) = [1 − nF (Ehν′′p−q+k0)][1 + nB(ωνq) + nB(ων′p)] + nB(ωνq)nB(ων′p), H νν′ν′′ (q,p) = nF (Ehν′′p−q+k0)[1 + nB(ωνq)+nB(ων′p)]+nB(ωνq)nB(ων′p), and k0 = [π, 0]. 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0704.0826

9.7 micrometer Silicate Absorption in a Damped Lyman-alpha Absorber at z=0.52

Submitted to ApJ Letters (Revised Version including Referee’s Comments) Preprint typeset using LATEX style emulateapj v. 08/22/09 9.7 µ M SILICATE ABSORPTION IN A DAMPED LYMAN-α ABSORBER AT Z = 0.52 Varsha P. Kulkarni1, Donald G. York2,3, Giovanni Vladilo4, Daniel E. Welty2 Submitted to ApJ Letters (Revised Version including Referee’s Comments) ABSTRACT We report a detection of the 9.7 µm silicate absorption feature in a damped Lyman-α (DLA) system at zabs = 0.524 toward AO0235+164, using the Infrared Spectrograph (IRS) onboard the Spitzer Space Telescope. The feature shows a broad shallow profile over ≈ 8-12 µm in the absorber rest frame and appears to be > 15 σ significant in equivalent width. The feature is fit reasonably well by the silicate absorption profiles for laboratory amorphous olivine or diffuse Galactic interstellar clouds. To our knowledge, this is the first indication of 9.7 µm silicate absorption in a DLA. We discuss potential implications of this finding for the nature of the dust in quasar absorbers. Although the feature is relatively shallow (τ9.7 ≈ 0.08− 0.09), it is ≈ 2 times deeper than expected from extrapolation of the τ9.7 vs. E(B − V ) relation known for diffuse Galactic interstellar clouds. Further studies of the 9.7 µm silicate feature in quasar absorbers will open a new window on the dust in distant galaxies. Subject headings: Quasars: absorption lines–ISM:dust 1. INTRODUCTION Damped Lyman-alpha (DLA) absorption systems in quasar spectra dominate the neutral gas content in galax- ies and offer venues for studying the evolution of metals and dust in galaxies. Recent observations, however, sug- gest that the majority of DLAs have low metallcities at all redshifts studied (0 . z . 4), with the mean metal- licity reaching at most ≈ 10 − 20% solar at the lowest redshifts (see, e.g., Prochaska et al. 2003; Kulkarni et al. 2005, 2007; Péroux et al. 2006; and references therein). These results appear to contradict the predictions of a near-solar global mean interstellar metallicity of galaxies at z ∼ 0 in most chemical evolution models based on the cosmic star formation history inferred from galaxy imag- ing surveys such as the Hubble Deep Field (HDF) (e.g., Madau et al. 1996). Furthermore, for a large fraction of the DLAs, the SFRs inferred from emission-line imag- ing searches fall far below the global predictions (e.g., Kulkarni et al. 2006, and references therein). A possible explanation of these puzzles is that the cur- rent DLA samples are biased due to dust selection ef- fects, i.e. that the more dusty and more metal-rich ab- sorbers obscure the background quasars more, making them harder to observe (e.g., Fall & Pei 1993; Boissé et al. 1998; Vladilo & Péroux 2005). DLAs are known to have some dust, based on both the (generally mild) deple- tions of refractory elements and the (typically slight) red- dening of the background quasars (e.g., Pei et al. 1991; Pettini et al. 1997; Kulkarni et al. 1997). Combining ∼ 800 quasar spectra from the Sloan Digital Sky Survey (SDSS), York et al. (2006b) found a small but significant amount of dust in absorbers at 1 < z < 2, with E(B−V ) of 0.02-0.09 for 9 of their 27 sub-samples (see also Khare et al. 2007). York et al. (2006b) also showed that the extinction in the composite spectra is best fitted by a 1 Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208; E-mail: [email protected] 2 Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637 3 Also, Enrico Fermi Institute 4 INAF, Osservatorio Astronomico di Trieste, Trieste, Italy Small Magellanic Cloud (SMC) curve (with no 2175 Å bump). Some recent studies suggest that dusty DLAs could hide as much as 17% of the total metal content at z ∼ 2, and more at lower z (Bouché et al. 2005). To un- derstand whether this is the case, and to understand the role of dust in quasar absorbers in general, it is essential to directly probe the basic properties of the dust. Recently, a small number of very dusty quasar ab- sorbers have been discovered, via various signatures of the dust in optical and UV observations: substantial red- dening of the background quasars, large element deple- tions (e.g., for Cr, Fe), and/or a detectable 2175 Å bump (e.g., Junkkarinen et al. 2004; Wang et al. 2004). It is not yet clear, however, whether the dust in these systems is similar to that in the Milky Way or SMC or LMC. The 2175 Å bump is generally, though not conclusively, attributed to carbonaceous grains. The silicate compo- nent of the dust, believed to comprise ≈ 70% of the core mass of interstellar dust grains in the Milky Way (see, e.g., Draine 2003) has not yet been probed in quasar ab- sorbers. A unique opportunity to study this important dust component is provided by the Spitzer IRS (Werner et al. 2004; Houck et al. 2004), which provides the spec- tral coverage, sensitivity, and resolution needed for the detection of the strongest of the silicate spectral features near 9.7 µm. The 9.7 µm feature, thought to arise in Si-O stretching vibrations, is seen in a wide range of Galac- tic and extragalactic environments (e.g., Whittet 1987 and references therein; Spoon et al. 2006; Imanishi et al. 2007). We have been carrying out an exploratory study of the silicate dust in quasar absorbers by searching for the 9.7 µm absorption feature with the Spitzer IRS. Here we report on the detection of the 9.7 µm feature in one of the systems studied, while the remaining three systems observed recently will be reported in a separate paper (Kulkarni et al. 2007b, in preparation). 2. OBSERVATIONS AND DATA ANALYSIS The DLA at zabs = 0.524 (Junkkarinen et al. 2004) toward the blazar AO 0235+164 (zem = 0.94) offers an excellent venue for comparing dust in a distant galaxy http://arxiv.org/abs/0704.0826v2 2 Kulkarni et al. with that in near-by galaxies. It has one of the largest H I column densities seen in DLAs (log NHI = 21.70) and shows 21-cm absorption (Roberts et al. 1976). It also shows X-ray absorption, consistent with a metallic- ity of 0.7 solar (Junkkarinen et al. 2004). Candidate absorber galaxies (much fainter than the blazar) within a few arcseconds from the blazar sightline have been de- tected (e.g., Smith et al. 1977; Yanny et al. 1989; Chun et al. 2006). This absorber is one of a very few DLAs producing appreciable reddening [E(B−V ) = 0.23 in the absorber rest frame] and detection of a strong broad 2175 Å extinction bump (Junkkarinen et al. 2004). Finally, this absorber is the only DLA with detections of several diffuse interstellar bands (Junkkarinen et al. 2004; York et al. 2006a). All of these data suggest that this absorber is very dusty and may contain molecular gas. The observations were obtained with the Spitzer IRS on January 30, 2006 (UT) as GO program 20757 (PI V. P. Kulkarni). IRS modules Short-Low 1 (SL1) and Long-Low 2 (LL2) were used to cover 7.5-21.4 µm in the observed frame (4.9-14.1 µm in the DLA rest frame). The target was acquired with high-accuracy peakup using a near-by bright star. The IRS standard staring mode was used, with 2-pixel slit widths of 3.6” for SL1 and 10.5” for LL2. Integration times were 60 s ×8 cycles for SL1 and 120 s ×11 cycles for LL2. For each cycle, observations were performed at both nod positions A and B (offset by 1/3 the slit length), so the total integration times were 960 s and 2640 s, respectively, for SL1 and LL2. The data were processed using the IRS S15.0 calibra- tion pipeline (the latest version available at present), Im- age Reduction and Analysis Facility (IRAF5), and In- teractive Data Language (IDL). As detailed below, the S15.0 pipeline yielded significant improvements for the reliable detection and measurement of weak, broad fea- tures in our spectra. The pipeline performs a number of standard processing steps to produce the basic calibrated data (BCD) files (see, e.g., the IRS Data Handbook at http://ssc.spitzer.caltech.edu/irs/dh). Subtraction of the sky (mostly zodiacal light) was performed by sub- tracting the coadded frames at nod position B from those at nod position A, and vice versa. The 1-dimentional spectra were extracted from the 2-dimensional images us- ing the Spitzer IRS Custom Extraction (SPICE) software using the default extraction windows, and flux calibrated using the standard S15.0 flux calibration files. The spec- tra from the two nod positions were averaged together, and the corresponding flux uncertainties calculated us- ing both measurement uncertainties and “sampling un- certainties” between the two nod positions. The absolute flux levels in the different IRS mod- ules were scaled to match the continuum levels in the overlapping regions, using the bonus segment available in the LL2 images. There was no mismatch between the SL1 and LL2 flux levels; we used the SL1 data for λ < 14.23µm and LL2 data for λ > 14.23µm. The data at λ > 20µm bonus segment level had to be scaled up by 5.5% to match with the LL2 data at λ < 20µm. Fig. 1(a) shows the final merged spectrum of AO0235+164. The 5 IRAF is distributed by the National Optical Astronomy Ob- servatories, which are operated by the Association of Universities for Research in Astronomy, Inc. (AURA), under cooperative agree- ment with the National Science Foundation S/N achieved per unbinned pixel in the final spectrum, determined from rms fluctuations in the continuum re- gions, is ≈ 100. The error bars denote 1 σ uncertainties. The dashed line in Fig. 1(a) shows an estimate of the power-law continuum of the quasar. This line joins the observed continuum fluxes at 5.6 and 7.1 µm in the ab- sorber rest frame and is extrapolated to the remaining wavelength region. These wavelengths are chosen to be in regions free of any other potential emission or absorp- tion features (e.g., Imanishi et al. 2007). In principle, significant 9.7 µm emission at the quasar redshift could affect continuum determination redward of the suspected silicate absorption feature from the DLA. However, (a) our spectrum does not extend that far to the red, (b) the power law provides a good fit to the continuum in our data, and (c) the 9.7 µm emission is not particularly strong in most quasars (e.g., Hao et al. 2007). 3. RESULTS The spectrum shown in Fig. 1(a) exhibits a broad absorption feature between about 12.4 and 18.3 µm rel- ative to the power law continuum. The flux decline from the continuum begins near the long wavelength end of SL1 and continues smoothly into the LL2 data. The broad feature is centered at 15.41 µm (10.11 µm in the DLA rest frame). The observed frame equivalent width is 0.31µm, with a 1 σ uncertainty of 0.014-0.020 µm, in- cluding contributions from photon noise and continuum fitting uncertainties (Sembach & Savage 1992). We have performed several checks of our data analy- sis to see whether the observed broad feature could be an artifact. Since the possible silicate feature is broad and shallow, extending from the long wavelength end of SL1 through most of LL2, flux calibration and contin- uum fitting are critical issues. In the S14 pipeline ver- sion of these data, the possible silicate feature was some- what stronger than in the S15 version. These differences are due to a low-level non-linearity problem in the S14 pipeline, which produces a 4% tilt in LL2 spectra and a 5% mismatch at the SL1/LL2 boundary. This problem has been eliminated in the S15 pipeline, and we find no mismatch at the SL1/LL2 boundary in the S15 data. The possible silicate feature does not show any visi- ble signature of the “teardrop” feature known to exist near 14.1 µm in some SL1 data (see, e.g., the IRS data handbook). The beginning of decline in flux at the long- wavelength end of SL1 matches smoothly with the flux at the short-wavelength end of LL2 (which does not suffer from the teardrop problem). Our results do not change much even if the SL1 data are truncated at 14 µm to avoid the region potentially affected by the teardrop (the region 14-14.23 µm is a small fraction of the whole fea- ture stretching out to 18.3 µm in the observed frame). Inaccuracies in pointing (which can affect SL1 and LL2 fluxes at the ±1% level) also do not appear to be sig- nificant for our data. Based on an examination of the spectral images and the pointing difference keywords in the data file headers, the telescope pointing was accu- rate to within 0.09-0.11” for LL2 and within 0.22-0.29” for SL1. Integrating a Gaussian intensity distribution from a point source with the Spitzer point spread func- tion over the known slit dimensions (57′′ × 3.6′′ for SL1, 168′′×10.5′′ for LL2), we estimate that the effect of such an offset would be about 0.26% for SL1 and 0.05% for http://ssc.spitzer.caltech.edu/irs/dh Silicate Feature in A DLA 3 LL2, far too small to account for the observed feature. We also compared our results with IRS spectra from the literature for quasars without strong absorption sys- tems (e.g., Sturm et al. 2006; Hao et al. 2007), and did not find the broad absorption feature from our data in those other quasars. In fact, quasar spectra in general show no silicate absorption, but rather (generally rela- tively weak) silicate emission at the quasar emission red- shift. We also compared our IRS data for AO0235+164 with those for other targets in our study. The feature seen in AO0235+164 is not seen at the same observed wavelength in the other objects, suggesting that it is not an instrumental artifact. [In fact, in Kulkarni et al. 2007b (in prep.), we will report the possible detection of redshifted broad 9.7 µm silicate absorption in other parts of the Spitzer spectral coverage toward other quasars.] Given the results of the above tests and the fact that the DLA toward AO0235+164 is already known to be dusty (from detection of 2175 Å bump and diffuse inter- stellar bands and reddening of the background quasar), it seems very likely that the feature detected is the broad 9.7 µm silicate feature arising in the absorber galaxy. 4. DISCUSSION The suggested silicate feature in the DLA toward AO0235+164 is relatively shallow/weak compared to the silicate features typically observed in Galactic interstel- lar material (ISM) because of the modest reddening and lower amounts of dust in quasar absorbers than in the Milky Way. Indeed, the dust-to-gas ratio in the DLA toward AO0235+164 is estimated to be 0.19 times the Galactic value (Junkkarinen et al. 2004). On the other hand, the observed feature is stronger than expected from E(B−V ) = 0.23±0.01 for this absorber (Junkkari- nen et al. 2004). In Galactic diffuse interstellar clouds, the peak optical depth in the 9.7 µm silicate feature (τ9.7) is observed to correlate with the reddening along the line of sight, with τ9.7 = AV /18.5 (e.g., Whittet 1987). Ex- trapolating this relation, and assuming RV = 3.1, one would expect τ9.7 ≈ 0.039 for the DLA in AO0235+164. Our observations, however, indicate τ9.7 ≈ 0.08 for this DLA, ∼ 2 times higher than expected from the relation for Galactic diffuse ISM. The dust in this absorber may thus be somewhat richer in silicates than typical Galac- tic dust. We note, however, that the silicate feature is also known to be stronger in the Galactic Center region, perhaps due to fewer carbon stars (and thus less carbona- ceous dust) there (e.g., Roche & Aitken 1985). If future observations of other DLAs also reveal material richer in silicates, it might indicate that those DLAs probe denser regions near the centers of the respective galaxies. The Galactic interstellar 9.7 µm feature is generally broad and relatively featureless, which is taken as an in- dication that interstellar silicates are largely amorphous. (Crystalline silicates would produce structure within the broad feature.) In principle, silicate grains may be com- posed of a mixture of pyroxene-like [(MgxFe1−x)SiO3] and olivine-like [(MgxFe1−x)2SiO4] silicates, with the shape and central wavelength of the 9.7 µm absorp- tion somewhat dependent on the exact composition (e.g., Kemper et al. 2004; Chiar & Tielens 2006). Fig. 1(b) shows a closer view of the data, normalized by the power law continuum shown in Fig. 1(a), and binned by a fac- tor of 3. The dotted and short-dashed curves are fits based on silicate emissivities derived from observations of the M supergiant µ Cep and of the Orion Trapezium region (e.g., Roche & Aitken 1984; Hanner et al. 1995), which are taken to be representative of diffuse Galactic ISM and denser molecular material, respectively . The long-dashed and dot-dashed curves are fits based on the silicate absorption profile observed toward the Galactic Center Source GCS3, and on laboratory measurements for amorphous olivine (Spoon et al. 2006). The shape of the silicate profile observed toward AO 0235+164 is most similar to that of laboratory amorphous olivine, but the µ-Cep and GCS3 templates also yield reasonable fits. The DLA silicate profile does not exhibit the red- ward extension seen for the Trapezium profile, suggest- ing that the DLA dust resembles dust in diffuse Galactic clouds more than that in molecular clouds. Using χ2 minimization for 8.0-13.3 µm in the DLA rest frame, the peak optical depth values τ9.7 for the laboratory olivine, GCS3, µ Cep, and Trapezium templates are 0.081±0.018, 0.088 ± 0.020, 0.083 ± 0.018, and 0.071 ± 0.016, respec- tively for the binned data (0.081± 0.020, 0.091± 0.023, 0.084± 0.021, and 0.069± 0.017, respectively, for the un- binned data). The error bars on τ9.7 correspond to op- tical depths that give reduced χ2 larger by 1.0 than the minimum values. The respective reduced χ2 values are 1.22, 1.32, 1.51, and 1.92 for the binned data (1.82, 2.08, 2.10, and 2.65 for the unbinned data). It is interesting to note that the best-fit astronomical template is GCS3, consistent with the enhanced τ9.7/E(B−V ) ratio seen in the DLA as toward the Galactic center. While the min- imum reduced χ2 values are greater than 1.0, they are similar to those found in other studies of the silicate ab- sorption toward both Galactic and extragalactic sources (e.g., Hanner et al. 1995; Bowey et al. 1998; Roche et al. 2006, 2007). Indeed, we do not expect a perfect fit, since possible differences in dust grain size and chemical composition can alter the shape of the silicate feature, in- cluding the peak wavelength and the FWHM (Bowey et al. 1998 and references therein). Higher S/N and higher resolution data would be needed to shed further light on the specific types of silicates present in DLAs. With a larger absorber sample, it would be possible to explore correlations between the strengths of the 9.7 µm silicate feature and the 2175 Å extinction bump (which is thought to be produced by a carbonaceous component of the dust). For example, it would be interesting to understand whether the relative amounts of silicate and carbonaceous dust vary with redshift or with the gas- phase abundances of C or Si. High-S/N observations of other possible features (e.g., the 18.5 µm silicate feature or the 3.0 µm H2O ice feature) would provide additional constraints on dust composition. (While those features are generally weaker than the 9.7 µm feature in the Milky Way, the 3.0 µm feature can be stronger than the 9.7 µm feature in highly reddened molecular sightlines.) Our exploratory study has demonstrated the potential of the Spitzer IRS to study dust in quasar absorbers. It would be very interesting to obtain similar spectra for other dusty quasar absorbers. The E(B − V ) val- ues for dusty absorbers such as that reported here (0.23) are much larger than those for typical Mg II absorbers [E(B − V ) of 0.002; York et al. 2006b]. These relatively large reddening values are comparable to some of those 4 Kulkarni et al. for Ly-break galaxies (LBGs), which show E(B − V ) up to 0.4 and a median E(B − V ) of ≈ 0.15 at z ∼ 2 and z ∼ 3 (Shapley et al. 2001, 2005; Papovich et al. 2001). Such dusty absorbers appear to be chemically more evolved (Wild et al. 2006) than typical DLAs, and may possibly provide a link in terms of SFRs, masses, metallicities, and dust content between the primarily metal-poor and dust-poor general DLA population with low SFRs and the actively star-forming, metal-rich, and dust-rich LBGs. Further Spitzer IRS observations of more dusty quasar absorbers thus will help to open a new window on this interesting class of distant galaxies. This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technol- ogy under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL/Caltech. VPK acknowledges support from NSF grant AST-0607739 to University of South Carolina. DEW acknowledges support from NASA LTSA grant NAG5-11413 to the University of Chicago. We are grate- ful to the Spitzer Science Center staff for helpful advice on data analysis and to an anonymous referee for helpful comments. Facilities: SST (IRS). REFERENCES Boissé, P., Le Brun, V., Bergeron, J., & Deharveng, J.-M. 1998, A&A, 333, 841 Bouché, N., Lehnert, M. D., & Péroux, C. 2005, MNRAS, 364, Bowey, J. E., Adamson, A. J., & Whittet, D. C. B. 1998, MNRAS, 298, 131 Chiar, J. E., & Tielens, A. G. G. M. 2006, ApJ, 637, 774 Chun, M. R. et al. 2006, AJ, 131, 686 Draine, B. T. 2003, ARAA, 41, 241 Fall, S. M., & Pei, Y. C. 1993, ApJ, 402, 479 Hao, L., Weedman, D. W., Spoon, H. W. W., Marshall, J. A., Levenson, N., Elitzur, M., & Houck, J. R. 2007, ApJ, 655, L77 Hanner, M. S., Brooke, T. Y., & Tokunaga, A. 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G. et al. 2006b, MNRAS, 367, 945 http://arxiv.org/abs/astro-ph/0608126 Silicate Feature in A DLA 5 0.85 0.95 1.05 1.15 1.25 1.35 −1.65 −1.55 −1.45 −1.35 −1.25 −1.15 −1.05 SL order 1 LL order 2 LL bonus order log λobserved 5.0 10.06.0 8.0 12.0 Rest Wavelength( Q0235+164 z abs = 0.524 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 Rest Wavelength tau_9.7=0.083, mu Cep tau_9.7=0.071, Trapezium tau_9.7=0.088, GCS3 tau_9.7=0.081, Lab Olivine ( m)µ Fig. 1.— (a) Left: Spitzer IRS spectrum of AO0235+164. The lower scale for the abscissa denotes the logarithm of the observed wavelength in µm; rest frame wavelengths at the absorber redshift are shown at the top. The errorbars denote 1 σ flux uncertainties. The dashed line shows a power law estimate of the continuum. (b) Right: A closer look at the suggested silicate feature. The abscissa denotes the rest frame wavelength at the DLA redshift. The data points show the spectrum, normalized by the power law continuum and binned by a factor of 3. The errorbars denote 1 σ uncertainties. The smooth curves show profiles for four templates of silicate optical depth, based on observations for three Galactic sightlines and laboratory measurements for amorphous olivine.

0704.0827

Photometry of the Globular Cluster NGC 5466: Red Giants and Blue Stragglers

Photometry of the Globular Cluster NGC 5466: Red Giants and Blue Stragglers Nassissie Fekadu and Eric L. Sandquist1,2 Department of Astronomy, San Diego state University, 5500 Campanile Drive, San Diego, CA 92182 [email protected], [email protected] Michael Bolte1 University of California Observatories, University of California, Santa Cruz, CA 95064 [email protected] ABSTRACT We present wide-field BV I photometry for about 11,500 stars in the low- metallicity cluster NGC 5466. We have detected the red giant branch bump for the first time, although it is at least 0.2 mag fainter than expected relative to the turnoff. The number of red giants (relative to main sequence turnoff stars) is in excellent agreement with stellar models from the Yonsei-Yale and Teramo groups, and slightly high compared to Victoria-Regina models. This adds to evidence that an abnormally large ratio of red giant to main-sequence stars is not correlated with cluster metallicity. We discuss theoretical predictions from different research groups and find that the inclusion or exclusion of helium diffusion and strong limit Coulomb interactions may be partly responsible. We also examine indicators of dynamical history: the mass function expo- nent and the blue straggler frequency. NGC 5466 has a very shallow mass func- tion, consistent with large mass loss and recently-discovered tidal tails. The blue straggler sample is significantly more centrally concentrated than the HB or RGB stars. We see no evidence of an upturn in the blue straggler frequency at 1Visiting Astronomer, Kitt Peak National Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation. 2Guest User, Canadian Astronomy Data Centre, which is operated by the Dominion Astrophysical Ob- servatory for the National Research Council of Canada’s Herzberg Institute of Astrophysics. http://arxiv.org/abs/0704.0827v1 – 2 – large distances from the center. Dynamical friction timescales indicate that the stragglers should be more concentrated if the cluster’s present density structure has existed for most of its history. NGC 5466 also has an unusually low central density compared to clusters of similar luminosity. In spite of this, the specific frequency of blue stragglers that puts it right on the frequency – cluster MV relation observed for other clusters. Subject headings: equation of state — globular clusters: individual (NGC 5466) — stars: blue stragglers — stars: evolution — stars: luminosity function 1. Introduction The luminosity function (LF) is an observational tool used for analyzing the post- main sequence evolutionary phases of low-mass (≈ 0.5-0.8M⊙) metal-poor stars in Galactic globular clusters (GGC). Because of their age and richness, GGC typically contain hun- dreds of stars that have evolved off the main sequence. The numbers of stars in evolved phases are directly related to the evolutionary timescales and fuel consumed in each phase (Renzini & Fusi Pecci 1988), so that they present us with an opportunity to test this aspect of stellar evolution models. The results of the most stringent tests have been mixed. Repeated studies of the metal- poor cluster M30 (Bolte 1994; Bergbusch 1996; Guhathakurta et al. 1998; Sandquist et al. 1999) have found an excess number of red giant branch (RGB) stars relative to main sequence (MS) stars. Stetson (1991) also uncovered an apparent excess of stars in a combined LF of the metal-poor clusters M68, NGC 6397, and M92. However, the LFs of more metal-rich clusters show no discrepancy (M5: Sandquist et al. 1996; M3: Rood et al. 1999; M12: Hargis, Sandquist, & Bolte 2004). In a survey of 18 clusters, Zoccali & Piotto (2000) found good agreement with model predictions with the possible exception of clusters at the high metallicity end. In this paper we present BV I photometry of NGC 5466, a high galactic latitude globular cluster (l = 42.2◦ and b = 73.6◦), located in the constellation of Boötes (α = 14h05m27.s4, δ = +28◦32′04′′ at a distance of R=15.9 kpc; Harris 1996). NGC 5466 is a loose cluster (rc = 1. ′64) with extremely low metallicity ([Fe/H]= −2.22) and subject to little or no reddening, (E(B − V ) ≃ 0) (Harris 1996). In §2, we describe the process leading to the calibrated photometry, and compare with previous studies of the cluster. In §3, we compare the observed color-magnitude diagram and observed luminosity function with theoretical models, focusing on the relative number – 3 – of stars on the lower RGB and around the MS turnoff. Finally, in §4, we present a new examination of the blue straggler population of NGC 5466. 2. Observations and Data Reduction The data used in this study were obtained with the Kitt Peak National Observatory (KPNO) 0.9 m telescope (0.′′68 pix−1) on the nights of UT dates 1995 May 4, May 5, and May 9. A complete list of the image frames, exposure times, and observing conditions is given in Table 1. The images obtained on the three nights were processed using IRAF1 tasks and pack- ages. The reduction involved subtraction and trimming of the overscan region of all images, subtraction of a master bias frame from flats and object frames, and flat fielding of the object frames using images taken at twilight. Profile-fitting photometry was done using the DAOPHOTII/ALLSTAR programs (Stetson 1987). We also reduced archival ground-based photometry of the cluster core taken with the High-Resolution Camera (HRCam) on the 3.6 m Canada-France-Hawaii Telescope (CFHT). The V and I images were taken 30 and 31 May 1992 (observers J. Heasley and C. Christian), and have not previously been described in the literature. The CCD images had 1024 × 1024 pixels, 0.′′13 per pixel, and excellent seeing (0.4 - 0.5 arcsec). The images were reduced using the archived bias and twilight flat frames, and following a procedure similar to that for the KPNO data. This allowed us to get excellent photometry to 2 magnitudes below the turnoff in the cluster core. These images were used entirely for blue straggler identification (see §4). 2.1. Calibration against Primary Standard Stars The conditions at KPNO on 1995 May 9 were photometric, and Landolt standard star fields were observed at a range of air masses to determine photometric transformation coeffi- cients. The standard values used for the calibration were chosen from the large compilation of Stetson (2000), which is set to be on the same photometric scale as the earlier Landolt (1992) values. We conducted photometry on the standard stars and isolated cluster stars using multiple 1IRAF(Image Reduction and Analysis Facility) is distributed by the National Optical Astronomy Ob- servatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation. – 4 – synthetic apertures. We then used the DAOGROW (Stetson 1990) program to construct growth curves to extrapolate measurements to a common aperture size. Using the CCDSTD program, the standard star transformation equations were found to be: b = B + ao + (−0.069± 0.005)(B − V ) + (0.255± 0.014)(X − 1.0) v = V + bo + (0.027± 0.004)(B − V ) + (0.172± 0.010)(X − 1.0) i = I + co + (−0.013± 0.005)(V − I) + (0.145± 0.018)(X − 1.0) where X is the airmass, v, b and i are instrumental magnitudes, and V , B, and I are standard magnitudes. These calibration equations are different than those used for our analysis of M10 Pollard et al. (2005) and M12 Hargis et al. (2004), which were observed on the same night, because our I−band exposures did not go as deep as the B and V exposures. As a result, the (B − V ) color was a better choice for calibrating the V photometry down to our faintest observed stars. The calibrated measurements for the standard stars are compared with catalogue values in Fig. 1. We note that there is slight evidence of trend in the residuals for the I band versus magnitude, which might indicate nonlinearity. This impression is caused by one observation of the PG1323-086 field. We did, however, have an additional observation of the same field on the same night having the same exposure time that does not show the same (small) trend. Because we do not have any reason to eliminate the frame and because its elimination has a minimal effect on the transformation coefficients, we have decided to retain the measurements from the image. 2.2. Calibration against Secondary Standard Stars Aperture photometry for 165 cluster stars was used to calibrate the point-spread function (PSF) photometry for the cluster. These secondary standard stars were chosen based on relatively low measurement errors and location in relatively uncrowded regions of the cluster. They were chosen from the asymptotic giant branch (AGB), upper RGB, and horizontal branch (HB) of the cluster in order to cover the entire range of colors covered by cluster stars. The PSF-fitting photometry for the three nights of data was combined and averaged after zero-point differences among the frames had been determined and corrected. The zero- point corrections to the standard system were determined after fixing the color-dependent terms at the values measured in the primary standard star calibration. (This was also done in our studies of M10 and M12.) In Fig. 2, it can be seen that this procedure does not introduce systematic color- or magnitude-dependent errors. – 5 – 2.3. Comparison with Previous Studies We compared our photometric data set to those of Jeon et al. (2004), Rosenberg et al. (2000), and Stetson (2000). The magnitude and color comparisons (BV I in this study versus V I in Stetson and Rosenberg et al., and BV in Jeon et al.) as function of magnitude and color are shown in Figs. 3 – 5. Though our calibrated magnitudes are slightly brighter than those of Stetson (2000), the differences are small, and there is no color trend. The offsets compared to the Rosenberg et al. (2000) are larger, but again there are no clear color trends. The offsets compared to the Jeon et al. (2004) data are also significant, but more notable are slight trends with color. 2.4. Calculation of the Luminosity Function Artificial star tests were performed to empirically measure the precision of our photom- etry and to correct for incompleteness in the detection of stars. We followed the procedure described by Hargis et al. (2004) for the calculation of incompleteness corrections as function of position and magnitude. The inputs used for producing the artificial star tests were the reduced B and V CCD frames, PSFs for each object frame, fiducial lines, and an estimate for the initial LF (Sandquist et al. 1996). Artificial stars were randomly placed in cells on a spatial grid and the entire grid was then shifted randomly from run to run in order to ensure the whole imaged field was tested (Piotto & Zoccali 1999). Each star was placed in a consistent po- sition relative to the cluster center on each image. If a detected star was found to coincide with the input position of an artificial star, it was added to the archive. The new images were reduced using the same procedure applied to the original data set. In this study, a total of 100,000 artificial stars from 50 separate runs were added. The number of artificial stars per trial was chosen so that the effects of crowding on the photometry was qualitatively unchanged. The recovered artificial stars were used to calculate 1) median magnitude and color biases (δV and δB−V , where δ = median[output − input]), 2) median external error esti- mates (σext(V ) = median[δV −median(δV )]/0.6745 and σext(B − V )), and 3) total recovery probabilities (F (V ), which is the fraction of the stars that were recovered with any output magnitude) in bins according to projected radius and magnitude. The values for the above quantities are plotted in Figs.6 – 8. Finally, an initial estimate of the “true” LF and the error distribution, magnitude biases and the total recovery probability (F ) were used to compute the completeness fraction f – 6 – (the ratio of the predicted number of stars to the actual number of observed stars). The completeness fraction results are shown in Figure 9. We then interpolated to compute f for the radial distance and magnitude of each detected star. For each observed star f−1 was added to the appropriate magnitude bin to determined the observed LF. (Note that the completeness fraction was set to 1.0 for star brighter than the turnoff.) The observed LF along with the upper and lower 1 σ error bars on log N are listed in Table 2. 3. Discussion 3.1. Reddening, Metallicity, Distance Modulus, and Age Because NGC 5466 resides at high galactic latitude, it suffers little if any reddening. Though Schlegel et al. (1998) found a reddening of E(B − V ) = 0.02 from the maps of dust IR emission, we adopted E(B − V ) = 0.0 (Rosenberg et al. 2000). For our interests in this paper, the small difference is of small importance. Most of the comparisons below between observations and theory are relative, in which reference points (like the turnoff) are used to determine magnitude and/or color shifts. This has the benefit of minimizing the influence of uncertainties in reddening and distance modulus (see below). As for abundances, there is only one high-resolution measurement for a cluster star, and it is for the anomalous Cepheid V19. McCarthy & Nemec (1997) find [Fe/H]= −1.92±0.05, while Pritzl et al. (2005) find [Fe/H]= −2.05 using the same data. Typically quoted metal- licity values include [Fe/H] = −2.17 (Zinn 1980) from photoelectric photometry of integrated light in selected filter bands, and [Fe/H] = −2.22, which was derived by Zinn & West (1984) (ultimately from low-resolution spectral scans by Searle & Zinn 1978). When converted to the widely-used metallicity scale of Carretta & Gratton (1997), this becomes [Fe/H] = −2.14. More work could certainly be done on the composition of NGC 5466 stars, but the evidence so far points to an abundance [Fe/H] . −2.0. Though the range in the above quoted metal- licity values is relatively large for a globular cluster, the exact value is not critical for our purposes since we will primarily be concerned with relative comparisons. Our photometry does not extend faint enough to derive a new distance modulus from subdwarf fitting to the main sequence. Harris (1996) obtained (m − M)V = 16.0 by cali- brating the observed luminosity level of the horizontal branch with the relation MV (HB) = 0.15[Fe/H]+0.80 and adopting a reddening, E(B−V )=0.0 and a metallicity, [Fe/H]=−2.22. Ferraro et al. (1999) determined distance moduli (m−M)V = 16.16 from their zero-age HB estimate, assuming no reddening and metallicity on the Carretta & Gratton (1997) scale. We will consider distance moduli in this range. – 7 – Most previous age estimates of NGC 5466’s age have it older than the recent deter- mination of the age of the universe (13.7+0.13 −0.17 Gyr) obtained by the Wilkinson Microwave Anisotropy Probe (WMAP) team (Spergel et al. 2006). Recent homogeneous studies of GGC indicate that NGC 5466 is coeval with clusters of similar metallicity (Salaris & Weiss 2002; Rosenberg et al. 2000). As a result, we will primarily consider ages in the range of 12 to 13 3.2. The Color-Magnitude Diagram The color-magnitude diagrams (CMDs) for NGC 5466 show well-defined RGB, AGB, and HB sequences (see Fig. 10), and stars extending from the tip of the RGB down to V ≈ 22.5. Fiducial sequences for the MS and lower RGB were determined from the mode of the color distribution of stars in magnitude bins. The SGB position was determined using the magnitude distribution of the stars in color bins. The fiducial line for the rest of the RGB was obtained from the mean color of stars in magnitude bins. The fiducial points are listed in Table 3. A comparison of the fiducial points derived for NGC 5466 with theoretical isochrones for a range of ages from the Teramo (Cassisi et al. 2004), Victoria-Regina (VandenBerg et al. 2006), and Yonsei-Yale (Demarque et al. 2004) groups is displayed in Figures 11 and 12. The isochrones have been shifted in color and magnitude (aligning the turnoff colors and the magnitudes of the main sequence point 0.05 mag redder than the turnoff) according to the technique of Vandenberg et al. (1990). This has the advantage of removing some of systematic uncertainties associated with the color-Teff transformations. [In our comparisons, we found that the Yonsei-Yale models could not match the fiducial line for any reasonable set of input parameters when the transformations of Lejeune et al. (1998) were used. Therefore, we only utilize models using the Green, Demarque, & King (1987) transformations. Even then, we could not find a match with the slope of the upper giant branch for reasonable metallicities ([Fe/H] . −1.9).] On the whole, the shape of the fiducial matches the models well on the main sequence, subgiant branch, and lower giant branch. Neither the Teramo nor the Victoria-Regina models include element diffusion processes, while the Yonsei-Yale isochrones only include He diffusion. However, differences in Teff -color transformations are likely to be the cause of some of the differences seen. – 8 – 3.3. The Luminosity Function The number of stars at a given luminosity in post-main-sequence phases is directly pro- portional to the lifetime spent at that luminosity. It is well known that the LF of the RGB probes the chemical stratification inside a star because the hydrogen abundance being sam- pled by the thin hydrogen-fusion shell affects the rate of evolution, and hence star counts on the RGB (Renzini & Fusi Pecci 1988). Setting aside the short pause at the RGB bump, RGB evolution accelerates in a very regular way that is ultimately related to the structure of degenerate core. The relationship between core mass and radius forces the fusion shell to function at strictly controlled density and temperature conditions, which leads to a relation- ship between core mass and luminosity. This causes the LF to be particularly sensitive to certain physical details, which we discuss in §3.3.1 below. Fig. 13 shows the observed luminosity function compared to theoretical LFs for the la- beled values of metallicity, exponent of the initial mass function, and a range of age estimates for NGC 5466, assuming our preferred distance modulus (m −M)V = 16.00. The theoret- ical models were normalized to the observed LF at V ≈ 21.3 (sufficiently faint that stellar evolution effects are minimized). The models agree well with the observed LF, implying an age of approximately 12 - 13 Gyr. 3.3.1. Relative RGB and MS Numbers Gallart et al. (2005) recently discussed theoretical luminosity functions calculated by different groups. One of the primary differences they noted was in the number of giant stars relative to main sequence stars. In order to show these differences in a parameter-independent way, we follow the method of Vandenberg et al. (1990). In Fig. 14, the theoretical LFs were shifted so that a point on the main sequence 0.05 mag redder than the turnoff color point matched the corresponding point on the cluster fiducial line. The reason for using the point (VTO+0.05) rather than the MSTO itself is that the MS has a significant slope and curvature at this point, making it possible to accurately measure the point in both observational data and isochrones (Vandenberg et al. 1990). The theoretical models were normalized to the two bins in the observed LF on either side of the turnoff (V = 19.83 and 20.13). As can be seen, age-related differences between the theoretical LFs nearly disappear when this procedure is applied (see also Stetson 1991; Vandenberg et al. 1998). However, when different sets of models are compared, there are small differences in the number of RGB stars relative to MS stars, with the Victoria-Regina models predicting the smallest number of giants and the Yonsei-Yale models predicting the largest. – 9 – To quantify the differences, we computed the ratio of the number of stars on the lower giant branch to the number of stars near the main sequence. Pollard et al. (2005) introduced this ratio and showed that it is insensitive to age and heavy-element abundance. For the main sequence population, we used star counts in the two bins on either side of the turnoff (19.682 < V < 20.282), and for the red giant branch we used the counts in the range 16.982 < V < 18.482. We derived model values from the same magnitude ranges relative to the VTO+0.05 point on the main sequence. Values are compared in Table 4. The error in the observed value is dominated by Poisson statistical scatter. The Yonsei-Yale models are in best agreement with the observations, the Victoria-Regina models are out of agreement by more than 2 σ, and the Teramo models are in between. It is worth examining the possible causes of this difference both because it may help improve the physics inputs for the models and because red giant stars are some of the largest contributors to the integrated light of old stellar populations. Gallart et al. (2005) tabulated most of the main physics inputs for the most widely-used model sets2. Earlier studies (Stetson 1991; Vandenberg et al. 1998) have shown that the LF-shifting method used above eliminates nearly all sensitivity to model input parameters like mass function, age, convective mixing length, and composition inputs with the exception of helium abundance, which we examine first. Older models (e.g. Fig. 9 of Ratcliff 1987, Fig. 7 of Stetson 1991, Fig. 3 of Vandenberg et al. 1998) seem to agree that an increase in initial helium abundance Y in non-diffusive models by 0.1 results in an increase in the relative number of stars on the RGB (more precisely, a reduction in the relative number of main sequence stars) by about 0.07-0.08 in logN . The Teramo models (Y = 0.245) predict about 12% more giant stars relative to main sequence stars compared to the Victoria-Regina models (Y = 0.235)3. This difference corresponds to a shift of 0.05 in logN , which is about an order of magnitude too large for the helium abundance difference. The Yonsei-Yale models have the lowest assumed helium abundance (Y = 0.23), but are the only set of the three that include helium diffusion. The inclusion of helium dif- fusion reduces the age derived from the turnoff of a globular cluster by about 10-15% 2Since the Gallart et al. (2005) review was published, the Teramo group found an error in the evolution scheme they used on the giant branch, which brings their models into better agreement with other groups. We use their updated models in the comparisons here 3The Teramo models also assume a larger α-element enhancement ([α/Fe] = 0.4) than the Victoria-Regina models ([α/Fe] = 0.3), which would tend to reduce the number of giants relative to main sequence stars. However, because the relative number of RGB and MS stars is not sensitive to small differences in heavy element abundance, this difference is probably unimportant. – 10 – (Proffitt & Vandenberg 1991; Straniero et al. 1997; VandenBerg et al. 2002), thanks to the inward motion of helium. According to theoretical models (e.g. Fig. 8 of Proffitt & Vandenberg 1991), diffusion has a small effect on the LFs (∼ a few times 10−2 in logN , increasing for increasing age), but it does increase the number of giants relative to MS stars. He diffusion reduces the total core hydrogen fuel supply available to an MS star, but in itself this does not strongly modify the LF, just changes the brightness of the turnoff. This magnitude change is eliminated in our LF shifting procedure. The chemical composition profile left in the star after it leaves the main sequence has a greater impact. Diffusion reduces the H abundance in the fusion regions, thereby decreasing the evolutionary timescale. According to the models of Proffitt & Michaud (1991), the changes to the He abundance profile are most considerable immediately below the surface convection zone, and just outside the nuclear fusion regions (where the composition gradient slows the inward settling of helium). However, over most of the star, the changes in Y are limited to 0.01 - 0.02. Because the core portion of the composition profile is consumed on the subgiant branch, the evolution timescales for giant stars are only affected in a minor way, and the appearance of a deep convection zone on the lower giant branch wipes out most of the effects of diffusion for the upper giant branch evolution. In spite of this, the Yonsei-Yale models have almost 23% more giants than the Victoria- Regina models, and more than 9% more giants than the Teramo models (relative to MS stars). The lower helium abundance in the Yonsei-Yale models compared to the other models should partially counteract what effects helium diffusion might have had on the RGB/MS ratio as well. Thus, it appears that neither He abundance nor He diffusion can completely explain the differences between the Yonsei-Yale models and the other groups. We can ask whether the LFs show similar disagreements at other metallicities. Hargis et al. (2004) made comparisons between the Victoria-Regina and Yonsei-Yale theoretical LFs and observational LFs for the clusters M3 (Rood et al. 1999), M5 (Sandquist et al. 1996), M12, and M30. The overall impression from those comparisons was again that the Yonsei-Yale models (having He diffusion) predict more giant stars relative to main-sequence stars than do the Victoria-Regina models. In Fig. 15, we compare the LFs for these clusters with the Teramo models. The degree of agreement or disagreement can be quantified with num- ber ratios of lower RGB and MSTO stars, similar to the ones we computed earlier for comparisons with NGC 5466. Our calculations are shown in Table 4. As can be seen, uncer- tainties in the metallicity scale have some effect on the comparisons with the observations. The Carretta & Gratton (1997) scale has higher [Fe/H] values than the Zinn & West (1984) scale, and thus results in lower number ratios. Fig. 16 shows the results of comparing the observed ratios with the models for different – 11 – [Fe/H] scales. On the Zinn & West 1984 scale (right panels), the observed values seem to be in agreement to within about 1 − 1.5σ for most of the models, with the exception of the the lowest metallicity clusters (M30 and NGC 5466) and the lowest helium abundance (Victoria-Regina) models. On the Carretta & Gratton 1997 scale, the Yonsei-Yale models have the best overall agreement, although the Teramo models only deviate noticeably for the lowest metallicity clusters. The Victoria-Regina models predict too few giants for all of the clusters. The differences from model to model (as opposed to models versus observation) point toward deficiencies elsewhere in the physics or computational algorithms used in the stel- lar evolution codes. 4 The RGB LF is a robust prediction of the models because there is a strong core mass — luminosity relationship: the conditions in the hydrogen fusion shell of the giant are strongly dependent on the structure of the degenerate core and are al- most independent of the details of the mass or structure of the envelope. As a result of this, we can focus on factors affecting core structure. (As a non-standard physics exam- ple, Vandenberg et al. 1998 describe the way in which core rotation relieves a giant star of some of the need to support itself by gas pressure, which reduces the core temperature and lengthens the evolutionary timescale.) Because model-to-model differences appear even on the faint end of the giant branch, we can set aside factors that only become important to the structure of the star near the tip of the giant branch [such as neutrino losses and conductive opacities; see Bjork & Chaboyer 2006, for example], even though there are significant theo- retical uncertainties in these quantities. Nuclear reaction rates in the fusion shell can also be neglected, partly because the uncertainties in the reactions appear to be relatively small Adelberger et al. (1998), but also because small changes in the reaction rates require only tiny changes in the shell temperature to get the same energy production. This leads us to examine the equation of state (EoS) in the core. Although the behavior of degenerate electrons is thought to be very well understood, their interactions with nuclei can have a measureable effect on the pressure. Particles of like charge tend to cluster together, which modifies the free energy of the gas and reduces the gas pressure for given density and temperature. Harpaz & Kovetz (1988) looked at the effects of the inclusion of Coulomb interactions on giant stars, and their results are corroborated by those of Cassisi et al. (2003). They found that for a given core mass the fusion shell temperature was higher when the Coulomb interactions were included, which leads to faster processing of hydrogen. Thus this is another example (like core rotation) where modification of the pressure support of the core affects the evolutionary timescales, which results in 4For a comparison of these physical inputs for the different theoretical groups, see Table 1 of Gallart et al. (2005). – 12 – changes to the luminosity function. The Coulomb corrections to the pressure become more important with increasing density for the core, but are small compared to the contribution of the degenerate electrons. All of the model sets we have considered here incorporate Coulomb interactions in some form. The Teramo group used the most sophisticated “EOS1” version of the FreeEOS5, which incorporates Coulomb corrections in a form that matches limits in both the weak (Debye-Hückel) and strong (once-component plasma) Coulomb interaction limits as well as (less importantly) electron exchange interactions. The strong interaction limit is most relevant for giant star cores since the strong interaction parameter where ζ is the rms nuclear charge and r0 is the average internuclear distance. The Yonsei-Yale group used the OPAL EoS tables (Rogers et al. 1996), but falls back on the group’s older EoS [e.g. (Guenther et al. 1992)] for conditions for high densities and temperatures outside the OPAL tables. While the most recent OPAL tables probably contain the most complete physical description of the Coulomb effect, the OPAL tables they used (Y.-C. Kim, private communication) were computed prior to recent improvements to account for relativistic electrons (Rogers & Nayfonov 2002), and as a result cut off at log ρ > 5.0. The Yale EoS at higher densities only includes the Coulomb effect in the weak Debye-Hückel limit, which for the highest densities in the core. The Victoria-Regina models also use a modified version of the EFF EoS (Eggleton et al. 1973), with a correction for Coulomb corrections in the weak Debye-Hückel limit (VandenBerg et al. 2000). The differences in the implementation of the Coulomb effect may explain the fact that the Teramo models generally predict more giants (relative to the main sequence) than the Victoria-Regina models do. However, the smaller Coulomb corrections in the Yonsei-Yale models would tend to result in fewer giants than the Teramo models (although the effects of helium diffusion work in the opposite direction). So, we are unable to completely reconcile the differences in the luminosity functions from the three groups. Obviously more detailed study is needed by all of the modelling groups to identify the causes of the differences, but such a study is beyond the scope of this paper. Still, we believe that helium diffusion and strong interaction Coulomb corrections are physical effects that should be considered first. There is, for example, good evidence from helioseismology 5FreeEOS is available at http://freeeos.sourceforge.net/, and the discussion of the implementation of the Coulomb effect can be found at http://freeeos.sourceforge.net/coulomb.pdf. http://freeeos.sourceforge.net/ http://freeeos.sourceforge.net/coulomb.pdf – 13 – for helium diffusion in the Sun Bahcall et al. (1995), despite the surface convection and meridional flow (e.g., Hathaway 1996). It is expected that helium diffusion should also act in globular cluster stars. A detailed study of the effect of equation of state uncertainties has yet to be done [see, for example, Bjork & Chaboyer (2006) for a study of uncertainties in other physical inputs]. Use of FreeEoS would make a study of equation of state effects most stringent since it appears to be capable of modelling the most sophisticated tabular EoS (OPAL), while also having the flexibility to allow individual bits of the physics to be “turned off”. As a final warning about the observations, we should remember the LF of the clus- ter M10. Pollard et al. (2005) found that unusual variations in numbers of RGB stars at different brightness levels in M10 (a virtual twin to M12). In particular there seemed to a significant excess in the number of stars near the RGB bump in brightness, while the lower RGB appeared normal (compared to Victoria-Regina and Yonsei-Yale models). A similar excess may be present in the RGB LF of M13 (Cho et al. 2005). These kinds of variations cannot be explained by the “global” physics that should apply to all globular cluster stars. These anomalies point toward fluctuations in the stellar initial mass function or composition-dependent effects. 3.3.2. The RGB Bump A second feature of the LF presented here is a noticeable RGB bump. Typically, the RGB bump appears as a peak in the differential LF and as a change of slope in the cumulative LF (CLF). The bump provides a measure of the maximum depth reached by the outer convection zone during first dredge-up since it is the result of a pause in the star’s evolution when the shell fusion source begins consuming material of constant, lower helium content (Fusi Pecci et al. 1990). Unfortunately, the number of stars occupying the bump gets smaller and the luminosity of the bump increases as the metallicity of the cluster decreases, making the bump harder to detect in metal-poor clusters. A small peak appears in our differential LF at V ≈ 16.2, and a significant (2.5 − 3.5σ) change in slope occurs at the same position as the peak in the differential LF, as shown in Fig. 17. The relative brightness of the bump can be measured by comparing to the V -magnitude of the HB at the level of the RR Lyrae instability strip ∆V HB = Vbump−VHB (Ferraro et al. 1999). This indicator is a function of the total metallicity and the age of the cluster: an increase in metallicity and/or a decrease in age are accompanied by a decrease in luminosity of the bump (Ferraro & Montegriffo 2000). We find Vbump = 16.20 ± 0.05 mag, and VHB = 16.52 ± 0.11 (from interpolation between the average magnitudes of non-variable HB stars – 14 – at the blue and red ends of the RR Lyrae instability strip), giving ∆V HB = −0.32 ± 0.12. In the compilation of Ferraro et al. (1999), a zero-age HB reference point was calculated using the relation VZAHB = VHB +0.106[Fe/H] +0.236[Fe/H]+ 0.193. Ferraro et al. found VZAHB = 16.62 ± 0.10, which is consistent with the value obtained here (VZAHB = 16.65 ± 0.11). Our value of ∆V ZAHB = −0.45 ± 0.12 is considerably lower than tabulated values for other clusters with similar metallicities (M68: −0.60 ± 0.07; M92: −0.65 ± 0.12; M15: −0.65 ± 0.09). In a separate compilation, Zoccali et al. (1999) measured a smaller value ZAHB = −0.45 ± 0.11 for M15, in better agreement with the value for NGC 5466. (The difference is primarily because Zoccali et al. measured the bump position to be 0.16 mag fainter than Ferraro et al. .) Clearly there is still some need for more precise comparisons of bumps in metal-poors clusters with theory. We believe, however, that the brightness of the bump should ultimately be judged using hydrogen-fusing stars as references because it avoids any effects of the poorly-understood physical processes (such as the helium flash and/or mass loss) associated with the creation of an HB star. Using the cluster LF (as seen in Figure 14), we again find that the observed RGB bump is fainter than model values by at least 0.3 mag when the models are shifted to match the cluster’s main sequence. Hargis et al. (2004) did similar comparisons between theoretical models and luminosity functions for M3, M5, M12, and M30 taken from the literature. With the exception of M30 (because the bump could not be identified), the position of the bump relative to the turnoff region agreed well with theory. NGC 5466 is thus the most metal-poor cluster this comparison has been done for. So at present we are left with the question of whether this might result from the cluster’s low metallicity, or whether we have been the unfortunate victims of a fluctuation in the number of giant stars in this low-mass cluster. We therefore encourage the examination of the luminosity function of more massive metal-poor clusters to settle the question. 3.3.3. Mass Function Exponent Two recent papers (Belokurov et al. 2006; Grillmair & Johnson 2006) reported the dis- covery of tidal streams covering many degrees around NGC 5466 in Sloan Digital Sky Sur- vey images. Gnedin & Ostriker (1997) examined the Milky Way globular clusters and found NGC 5466 to be a cluster that has probably been strongly affected by disk shocking in the recent past. In our examination of the LF, we found a rather low value for the global main se- quence mass function slope. The mass function for a cluster is typically expressed as a power law (N(M) ∝ M−(1+x)), where the slope x = 1.35 is the standard Salpeter value. Generally, the present-day power-law index x varies from cluster to cluster. The mass function slopes – 15 – that best fit the upper LF of NGC 5466 around and above the MSTO have −1 . x . 0. (Note that the best fit slope does depend on the models being used: the Yonsei-Yale models require a flatter slope than the Victoria-Regina and Teramo models.) Such a shallow mass function slope is unusual for a metal-poor cluster. For example, the cluster NGC 5053 has similar metallicity, position relative to the galactic center and plane, and density structure, but still has a steep x ∼ 2 mass function (Fahlman et al. 1991). Djorgovski et al. (1993) found that mass function slopes in the range of 0.5 ≤ M ≤ 0.8 are influenced primarily by the cluster’s position in the galaxy, and to some extent by cluster metallicity. Based on both of those factors, NGC 5466 should have a larger mass function slope (x ∼ 3 according to the multivariate formula in Djorgovski et al. 1993). Like other halo clusters, NGC 5466’s orbit is quite eccentric and will take the cluster more than 30 kpc away from the Galactic center (Dinescu et al. 1999), but it is currently on its way back into the halo after two relatively recent passes through the Galactic disk. Recent losses of low-mass stars may explain the recent identification of strong tidal tails near the cluster by Belokurov et al. (2006). 4. Blue Stragglers Blue stragglers (BSSs) were first identified by Sandage (1953) in the globular cluster M3. These stars are more massive than the turnoff mass and occupy the space in the CMD just bluer and brighter than the MSTO. Blue stragglers are found in clusters, and relatively more frequently in lower-luminosity clusters (Ferraro et al. 1993; Preston & Sneden 2000; Piotto et al. 2004; Sandquist 2005). From the various models proposed for the formation mechanism of BSSs, the “collision” theory (involving strong gravitational interactions be- tween previously unassociated single or binary stars) and the “mass-transfer” theory (in which the more massive star in a binary evolves and during its expansion transfers mass to its companion) are the strongest possibilities. There is a continuing interest in the study of BSSs because they may provide insight into the recent dynamical history of a cluster. In order to identify BSSs over the entire observed area of the cluster out to a radius of 11.′6, we used photometry from three datasets. In the core of the cluster we used the CFHT data presented here for the first time. Outside of the CFHT field, we used the BV photometry of Jeon et al. 2004, which covered a field 11.′6 on a side centered on the cluster. Finally, we used our KPNO data for the least crowded outskirts of the cluster. Even though NGC 5466 is a very low density cluster, the spatial resolution of the KPNO data was such that blends of stars would have resulted in the spurious identification of 10 objects as BSS in the intermediate portion of the field. 48 BSS candidates were identified in NGC 5466 by Nemec & Harris (1987), all located – 16 – between 0.′1 and 5.′6 from the cluster center. In spite of the low cluster density, we find new BSS candidates at all radii and luminosity levels, and find several of their candidates are spurious. According to the CFHT photometry, the object with ID 45 from Nemec & Harris is a blend of several fainter stars, none of which is a BSS. In addition, IDs 6 and 24 were identified as blends of stars using the Jeon et al. (2004) dataset. Our BSS list is presented in Table 5. The list includes the nine known SX Phoenicis stars (ID 27, 29, 35, 38, 39, and 49, Nemec & Mateo 1990; ID 3 (SX Phe 3), 36 (SX Phe 2), and 50 (SX Phe 1), Jeon et al. 2004) and the three eclipsing binaries (ID 19, 30, and 31; Mateo et al. 1990). New straggler candidates were given ID numbers that build upon the Nemec & Harris (1987) list. Figure 18 shows the CMDs used to select the 94 identified BSSs in each of the three datasets. In order to use the BSSs to constrain cluster dynamics, we compared the normalized cumulative radial distribution of the BSSs to the population of the giant branch, as shown in Fig. 19. Nemec & Harris (1987) found a 97.8% probability that their BSS sample was more centrally concentrated than red giants in the same magnitude range. Because their photometry was taken in conditions of poorer seeing (compared to our CFHT photometry and that of Jeon et al.), their samples are likely to be somewhat incomplete near the cluster center. Our RGB sample contains 350 stars with magnitude V < 18.5. Kolmogorov-Smirnov (K−S) probability tests were used to test the hypothesis that both populations were drawn from the same parent population. The K−S probability that the BSSs are drawn from the same radial distribution as the RGB population is 8.1 × 10−7, and 2.4 × 10−4 for the comparison with the HB population. By contrast there is a probability of 0.27 that the RGB and HB samples are drawn from the same population. The concentration of BSSs toward the cluster center as compared to the RGB and HB samples is consistent with the idea that they are more massive than individual RGB stars, and as a result have been segregated by mass deeper within the cluster potential well. Piotto et al. (2004) recently used samples of stragglers from the cores of 56 globular clusters to show that there was a strong correlation between FHBBSS = NBSS/NHB and inte- grated cluster V magnitude, and a weaker anti-correlation with central density. Sandquist (2005) examined an additional 13 low-luminosity globular clusters using similar selection criteria. NGC 5466 is an interesting cluster in relation to these samples because it has an integrated luminosity that puts it at the faint end of the Piotto et al. sample (MVt = −6.96; Harris 1996), but with a central density that is nearly an order of magnitude lower than any of their clusters [log(ρ0/(LV,⊙pc −3)) = 0.88] but comparable to clusters in the Sandquist sample. To put NGC 5466 in the context of the Piotto et al. and Sandquist samples, we selected a subset of our BSSs that satisfied the selection criteria in those studies (brighter – 17 – than the MSTO, and bluer than the MSTO by 0.05 in B − V color). From mode fitting to the turnoff region in the Jeon et al. (2004) and CFHT data, we find (B−V )TO = 0.367 and (V − I)TO = 0.511. A color offset of 0.05 in B − V corresponds to an offset of about 0.075 in V − I (VandenBerg & Clem 2003) for NGC 5466’s metallicity. We find that 75 BSSs are brighter than the cluster turnoff (VTO = 19.99±0.05) and 0.05 bluer than the cluster turnoff in B − V . We have identified 97 HB stars in our datasets for NGC 5466, which gives a specific frequency FHBBSS = 0.77± 0.12 (with the error estimate from Poisson statistics). When compared to the Piotto et al. values (see Fig. 20), NGC 5466 falls within the general trend versus MVt despite the cluster’s low central density. On the other hand, NGC 5466 has a lower FHBBSS value than other clusters of similar central density (but lower total luminosity). As discussed by Sandquist (2005), this provides additional evidence that the plateau in FHBBSS seen for clusters with log ρ0 . 2.5 is a result of the correlation between cluster integrated magnitude and central density. The lowest luminosity clusters in the Sandquist sample (E3 and Palomar 13) have central densities comparable to that of NGC 5466, but BSS frequencies that are several times higher. Another moderate-luminosity, low-density cluster (NGC 5053; Hiner et al., in preparation) similar to NGC 5466 shows a comparably low straggler frequency. BSSs produced via purely collisional means are not likely to show this kind of behavior. More likely is the scenario proposed by Davies et al. (2004) in which binary stars that would normally produce BSSs are destroyed earlier in the cluster’s history. More direct observational support for that hypothesis is needed though — for example, a detailed study of the variation of the binary star fractions as a function of integrated cluster magnitude. Fig. 21 shows that the number and frequency RBSS = NBSS/N Lsample/L sample of BSSs relative to the integrated V -band flux (derived from a King model profile) as a func- tion of radius. Both frequencies increase toward the cluster center, and neither shows signs of rising toward larger radii. As recent studies of denser clusters show (M3: Ferraro et al. 1993; M55: Zaggia et al. 1997; 47 Tuc: Ferraro et al. 2004; NGC 6752: Sabbi et al. 2004), the BSS frequency generally decreases at intermediate radii and rises again at larger dis- tances. However, the cluster Palomar 13 (Clark et al. 2004), which has a central density similar to NGC 5466, shows no sign of an increase in straggler frequency at large distance. In more massive clusters, the minimum in the BSS frequency is reached approximately where the timescale for dynamical friction equals the age of the cluster (Warren et al. 2006). A similar calculation for the current structure of NGC 5466 indicates that this occurs at about 270′′(about 2.8rc). This appears to be in the outer reaches of the core straggler distribution. – 18 – Because it is likely that NGC 5466 has lost a significant fraction of its mass, we expect that the current density structure of the cluster has not existed throughout its history and that NGC 5466 might have been able to dynamically relax stragglers to its core from larger distances earlier in its history. This may be showing that the global BSS population differs significantly between low-density/low-mass clusters and high-density/high-mass clusters. A lack of stragglers at large distance may be signature of large-scale tidal stripping of the cluster, which would remove both stragglers that would normally have formed in primordial binaries in the outer reaches of the cluster and ones that formed in the core but were given velocity kicks into orbits that would take them into the outer reaches. The case against a rise at large radius in Palomar 13 is stronger because the cluster has been surveyed out to 19 core radii, while in NGC 5466, we have only surveyed out to about 10 core radii (or 7.5 half-mass radii). Still, NGC 5466 probably should have an even more concentrated distribution of stragglers if its current density structure has existed for most of its history. In more massive clusters, the secondary rise in straggler frequency is observed between 8 and 10 rc. Unfortunately, further study of the stragglers in NGC 5466 will probably be complicated by the strong tidal tails observed in the cluster. 5. Conclusions Examinations of the luminosity functions of globular clusters continue to produce inter- esting tests of astrophysics. In this study, we found that NGC 5466 has a luminosity function that is in better overall agreement with theoretical models than the anomalous cluster M30, which has a similar low metallicity. In addition, we found that the relative numbers of red giant and main sequence stars may produce a fairly sensitive test of the physics near the core of red giants — specifically, helium diffusion and Coulomb interactions. However, we are not yet able to fully explain the differences between sets of theoretical models. Recent discoveries of large tidal tails associated with NGC 5466 suggest that this cluster has been strongly disrupted by interactions with the Galaxy. Our measured flat (−1 . x . 0) main-sequence luminosity function is unusual for a low-metallicity halo cluster. It is, however, consistent with the emerging picture of mass-segregation followed by tidal stripping. We have thoroughly re-examined the blue straggler population in the cluster, and de- tected a total of 94. The radial distribution of stragglers is clearly more centrally concen- trated than the RGB and HB populations. 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L. 2003, AJ, 126, 778 Vandenberg, D. A., Larson, A. M., & de Propris, R. 1998, PASP, 110, 98 VandenBerg, D. A., Richard, O., Michaud, G., & Richer, J. 2002, ApJ, 571, 487 VandenBerg, D. A., Swenson, F. J., Rogers, F. J., Iglesias, C. A., & Alexander, D. R. 2000, ApJ, 532, 430 – 23 – Warren, S. R., Sandquist, E. L., & Bolte, M. 2006, ApJ, 648, 1026 Zaggia, S. R., Piotto, G., & Capaccioli, M. 1997, A&A, 327, 1004 Zoccali, M., Cassisi, S., Piotto, G., Bono, G., & Salaris, M. 1999, ApJ, 518, L49 Zoccali, M., & Piotto, G. 2000, A&A, 358, 943 Zinn, R. & West, M. J. 1984, ApJS, 55, 45 Zinn, R. 1980, ApJ, 241, 602 This preprint was prepared with the AAS LATEX macros v5.2. – 24 – Fig. 1.— Photometric residuals (in the sense of this study minus those of Landolt 1992 and Stetson 2000) of primary standard stars. The median residuals are listed in the panels with the semi-interquartile range (half the magnitude difference between the 25% and 75% points in the ordered list of residuals) given in parentheses. – 25 – Fig. 2.— Photometric residuals (in the sense of the final PSF photometry minus standard aperture photometry values) of secondary standard stars. – 26 – Fig. 3.— Residuals (in the sense of this study minus Stetson 2000) from the star-by-star comparison. The median residuals are listed in the panels with the semi-interquartile range (see Fig.1) given in parentheses. – 27 – Fig. 4.— Residuals (in the sense of this study minus Rosenberg et al. 2000) from the star- by-star comparison. The median residuals and the plots versus color have been restricted to brighter stars (V < 20 and I < 19) to make the comparisons clearer. The numbers in parentheses are the semi-interquartile ranges (see Fig.1). – 28 – Fig. 5.— Residuals (in the sense of this study minus Jeon et al. 2004) from the star-by-star comparison. The median residuals and the plots versus color have been restricted to brighter stars (B < 20 and V < 19.5) to make the comparisons clearer. The numbers in parentheses are the semi-interquartile ranges (see Fig.1). – 29 – Fig. 6.— External V magnitude errors σext(V ) as a function of radius and magnitude determined from artificial star tests, with exponential fits shown by the solid lines. – 30 – Fig. 7.— Magnitude biases δ(V ) determined from artificial star tests as a function of radius and magnitude. – 31 – Fig. 8.— Total recovery probability F (V ) determined from artificial star tests as a function of radius and magnitude. – 32 – Fig. 9.— Completeness fraction f(V ) determined from artificial star tests as a function of radius and magnitude. – 33 – Fig. 10.— Color-magnitude diagrams for all stars measured in the KPNO and CFHT images. The BV fiducial (Table 3) is also plotted in the left panel. – 34 – Fig. 11.— Comparison of the observed fiducial sequence of NGC 5466 with the isochrones of the Teramo, Victoria-Regina, and Yonsei-Yale groups. The isochrones have been shifted horizontally so that the turnoff colors align, and shifted vertically to align the main sequence point 0.05 mag redder than the turnoff. On the giant branch, the ages increase from the reddest to the bluest isochrone. – 35 – Fig. 12.— Same as Fig. 11, except for more metal-rich models. – 36 – Fig. 13.— Comparison of the observed V−band LF of NGC 5466 with theoretical models of the Yonsei-Yale, Teramo, and Victoria-Regina groups assuming (m−M)V = 16. – 37 – Fig. 14.— Comparison of the observed V−band LF of NGC 5466 with theoretical models of the Victoria-Regina, Yonsei-Yale, and Teramo groups using magnitude shifts that bring the main sequence point 0.05 mag redder than the turnoff into alignment. The models have been normalized to the two bins on either side of the turnoff (V = 19.94). – 38 – Fig. 15.— Comparison of the observed LFs of M3, M5, M12, and M30 with theoretical models of the Teramo group using magnitude shifts that bring the main sequence point 0.05 mag redder than the turnoff into alignment. The models have been normalized to bins on either side of the turnoff. – 39 – Fig. 16.— Fractional difference between the observed RGB-MS number ratios (for M5, M12, M3, M30, and NGC 5466, from left to right) and the theoretical predictions from the Yonsei- Yale, Teramo, and Victoria-Regina models. The left panels use the Carretta & Gratton (1997) metallicity scale, and the right panels use the Zinn & West (1984) scale. The sense is (observed − theoretical) / observed. – 40 – Fig. 17.— The cumulative luminosity function for bright RGB stars derived from the pho- tometry of Jeon et al. (2004; 286 stars) and from the KPNO dataset (338 stars) presented here. Dotted lines show fits to the data for stars above and below the position of the apparent bump. – 41 – Fig. 18.— Blue straggler selection for NGC 5466. The stars plotted in each panel show the entire sample used for the selection: stars from Jeon et al. (2004) in the middle panel are only those stars outside the CFHT field, and KPNO stars in the right panel are only those outside the Jeon et al. field. Open squares show stragglers identified by Nemec & Harris (1987), and open circles are new candidates. – 42 – Fig. 19.— Normalized cumulative radial distributions for RGB stars (dashed line), HB stars (dotted line), and BSSs (solid line). – 43 – Fig. 20.— Relative frequencies of blue stragglers as a function of cluster absolute magnitude and central density. The solid square is NGC 5466, the open squares are globular clusters from Sandquist (2005), and all other points are from Piotto et al. (2004). Open circles are post-core-collapse clusters. In the left panel, symbols represent clusters in different ranges of central density from the Piotto et al. sample: log ρ0 < 2.8: open triangles; 2.8 < log ρ0 < 3.6: filled triangles; 3.6 < log ρ0 < 4.4: stars; log ρ0 > 4.4: filled circles. – 44 – Fig. 21.— Frequency of BSS relative to the integrated V -band flux of detected cluster stars (top panel) and specific frequency of blue stragglers relative as a function of radius (bottom panel). – 45 – Table 1. Photometric Observation Log for NGC 5466 UT Date Filters N Exposure Time (s) Airmass 1995 May 4 B,V 1,1 60 1.01,1.12 1995 May 4 B 2 300 1.03, 1.03 1995 May 4 B,V 2,1 600 1.0,1.11,1.0 1995 May 5 B,I 2,2 300 1.03,1.01,1.02,1.01 1995 May 9 B,V ,I 1,1,1 60 1.12,1.13,1.14 – 46 – Table 2. V -Band Luminosity Function V logN σhigh σlow 13.532 0.5005 0.1761 0.3010 13.832 −0.1015 0.3010 1.0000 14.132 −0.1015 0.3010 1.0000 14.432 0.5975 0.1605 0.2575 14.732 0.8985 0.1193 0.1651 15.032 0.6767 0.1487 0.2279 15.332 0.6767 0.1487 0.2279 15.632 0.5976 0.1606 0.2575 15.932 0.8986 0.1193 0.1651 16.232 1.2410 0.0839 0.1041 16.532 1.0127 0.1063 0.1411 16.832 1.0451 0.1029 0.1351 17.132 1.3306 0.0764 0.0928 17.432 1.4432 0.0678 0.0804 17.732 1.4777 0.0728 0.0876 18.032 1.6136 0.0630 0.0738 18.332 1.7575 0.0540 0.0617 18.632 1.7577 0.0540 0.0617 18.932 1.9624 0.0433 0.0481 19.232 2.2928 0.0301 0.0324 19.532 2.5143 0.0236 0.0249 19.832 2.6996 0.0192 0.0201 20.132 2.7738 0.0178 0.0185 20.432 2.8830 0.0159 0.0165 20.732 2.9327 0.0151 0.0157 21.032 3.0036 0.0142 0.0146 21.332 3.0496 0.0137 0.0142 21.632 3.1179 0.0132 0.0136 – 47 – – 48 – Table 3. Fiducial sequence for NGC 5466 V B − V Na 22.198 0.578 518 21.999 0.564 598 21.798 0.549 724 21.596 0.532 710 21.398 0.498 673 21.200 0.470 670 21.004 0.459 653 20.802 0.450 614 20.600 0.431 568 20.400 0.418 546 20.199 0.404 470 20.001 0.395 453 19.805 0.396 382 19.599 0.403 273 19.400 0.436 235 19.206 0.495 155 19.005 0.543 84 18.823 0.586 70 18.588 0.607 43 18.401 0.616 52 18.204 0.621 40 17.989 0.636 34 17.820 0.647 31 17.586 0.654 23 17.403 0.672 20 17.194 0.677 19 16.988 0.693 11 16.784 0.722 10 16.613 0.725 6 16.400 0.752 12 16.199 0.771 16 – 49 – Table 3—Continued V B − V Na 16.066 0.779 5 15.825 0.809 7 15.602 0.827 4 15.407 0.857 6 15.111 0.917 1 15.006 0.930 5 14.786 0.946 2 14.590 0.986 11 14.438 1.044 2 aNumber of stars used to determine fiducial point – 50 – Table 4. RGB-MSTO Number Ratios Sourcea TO Sample RGB Sample Y NRGB/NMSTO [Fe/H] NGC 5466 19.682 < V < 20.282 16.982 < V < 18.482 0.162± 0.013 VR 0.235 0.132 −2.22 VR 0.235 0.130 −2.14 T 0.245 0.148 −2.22 T 0.245 0.146 −2.14 YY 0.230 0.162 −2.22 YY 0.230 0.160 −2.14 M3 18.80 < V < 19.40 16.40 < V < 18.00 0.168± 0.008 VR 0.235 0.170 −1.66 VR 0.235 0.158 −1.34 T 0.246 0.181 −1.66 T 0.248 0.170 −1.34 YY 0.230 0.185 −1.66 YY 0.230 0.162 −1.34 M5 19.13 < B < 19.73 16.33 < B < 17.93 0.110± 0.006 VR 0.235 0.106 −1.40 VR 0.235 0.097 −1.11 T 0.248 0.114 −1.40 T 0.251 0.106 −1.11 YY 0.230 0.119 −1.40 YY 0.230 0.105 −1.11 M12 18.14 < V < 18.74 15.59 < V < 17.24 0.158± 0.011 VR 0.235 0.144 −1.40 VR 0.235 0.118 −1.14 T 0.248 0.155 −1.40 T 0.251 0.144 −1.14 YY 0.230 0.150 −1.40 YY 0.230 0.137 −1.14 M30 18.33 < V < 18.93 15.78 < V < 17.43 0.214± 0.017 VR 0.235 0.174 −2.13 VR 0.235 0.153 −1.91 – 51 – Table 4—Continued Sourcea TO Sample RGB Sample Y NRGB/NMSTO [Fe/H] T 0.245 0.173 −2.13 T 0.246 0.165 −1.91 YY 0.230 0.194 −2.13 YY 0.230 0.180 −1.91 aVR: Victoria-Regina models (no diffusion); YY: Yonsei-Yale models (He diffusion); T: Teramo models (no diffusion). All models are for an age of 12 Gyr. – 52 – Table 5. Selected Star Populations in NGC 5466 ID ∆α(′′) ∆δ(′′) B σB V σV I σI Alternate ID Ref. a Notes Blue Stragglers 1 137.74 32.86 19.1477 19.0038 605 J 1 137.74 32.86 19.1446 0.0122 19.0182 0.0280 18.7606 0.0400 809 K 2 −12.18 −5.60 18.7591 0.0116 18.5972 0.0061 10313 C 2 −12.18 −5.60 18.8461 0.0147 18.7209 0.0258 18.6245 0.0504 2176 K 3 −8.68 14.99 19.1710 0.0100 18.9595 0.0059 9339 C SX Phe (3) 3 −8.68 14.99 19.4373 0.0155 19.2441 0.0296 19.1073 0.0563 2129 K 4 −77.31 83.56 19.2057 19.0546 646 J 4 −77.31 83.56 19.2266 0.0123 19.0548 0.0256 18.8465 0.0460 2880 K 5 −81.70 65.06 18.5511 18.2697 389 J 5 −81.70 65.06 18.5632 0.0121 18.2740 0.0243 17.8309 0.0255 2895 K 7 −132.96 −3.18 18.8010 18.7046 504 J 7 −132.96 −3.18 18.8482 0.0113 18.7379 0.0217 18.5538 0.0386 3289 K 8 −144.85 3.07 18.8436 18.6640 498 J 8 −144.85 3.07 18.8726 0.0120 18.6345 0.0234 18.3367 0.0298 3358 K 9 −90.90 83.38 19.1757 19.0256 626 J 9 −90.90 83.38 19.1385 0.0144 18.9711 0.0257 18.7508 0.0387 2973 K 10 −44.94 96.04 19.3627 19.1882 751 J 10 −44.94 96.04 19.4145 0.0139 19.2190 0.0261 18.9077 0.0494 2520 K aPhotometry Sources: K: KPNO data from this paper, C: CFHT data from this paper, J: Jeon et al. (2004) Note. — The complete version of this table is in the electronic edition of the Journal. The printed edition contains only a sample. Introduction Observations and Data Reduction Calibration against Primary Standard Stars Calibration against Secondary Standard Stars Comparison with Previous Studies Calculation of the Luminosity Function Discussion Reddening, Metallicity, Distance Modulus, and Age The Color-Magnitude Diagram The Luminosity Function Relative RGB and MS Numbers The RGB Bump Mass Function Exponent Blue Stragglers Conclusions

0704.0829

Quark-Antiquark and Diquark Condensates in Vacuum in a 3D Two-Flavor Gross-Neveu Model

Quark-Antiquark and Diquark Condensates in Vacuum in a 3D Two-Flavor Gross-Neveu Model∗ Zhou Bang-Rong College of Physical Sciences, Graduate School of the Chinese Academy of Sciences, Beijing 100049, China and CCAST (World Laboratory), P.O.Box 8730, Beijing 100080, China (Dated:) The effective potential analysis indicates that, in a 3D two-flavor Gross-Neveu model in vacuum, depending on less or bigger than the critical value 2/3 of GS/HP , where GS and HP are respectively the coupling constants of scalar quark-antiquark channel and pseudoscalar diquark channel, the system will have the ground state with pure diquark condensates or with pure quark-antiquark condensates, but no the one with coexistence of the two forms of condensates. The similarities and differences in the interplay between the quark-antiquark and the diquark condensates in vacuum in the 2D, 3D and 4D two-flavor four-fermion interaction models are summarized. PACS numbers: 12.38Aw; 12.38.Lg; 12.10.Dm; 11.15.Pg Keywords: 3D Gross-Neveu model, quark-antiquark and diquark condensates, effective potential I. INTRODUCTION It has been shown by effective potential approach that in a two-flavor 4D Nambu-Jona-Lasinio (NJL) model [1], even when temperature T = 0 and quark chemical poten- tial µ = 0, i.e. in vacuum, there could exist mutual com- petition between the quark-antiquark condensates and the diquark condensates [2]. Similar situation has also emerged from a 2D two-flavor Gross-Neveu (GN) model [3] except some difference in the details of the results [4]. An interesting question is that if such mutual competi- tion between the two forms of condensates is a general characteristic of this kind of two-flavor four-fermion in- teraction models? For answer to this question, on the basis of research on the 4D NJL model and the 2D GN model, we will continue to examine a 3D two-flavor GN model in similar way. The results will certainly deepen our understanding of the feature of the four-fermion in- teraction models. We will use the effective potential in the mean field approximation which is equivalent to the leading order of 1/N expansion. It is indicated that a 3D GN model is renormalizable in 1/N expansion [5]. II. MODEL AND ITS SYMMETRIES The Lagrangian of the model will be expressed by L = q̄iγµ∂µq +GS [(q̄q)2 + (q̄~τq)2] A=2,5,7 (q̄τ2λAq C)(q̄Cτ2λAq). (1) All the denotations used in Eq.(1) are the same as the ones in the 2D GN model given in Ref.[4], except that ∗The project supported by the National Natural Science Founda- tion of China under Grant No.10475113. the dimension of space-time is changed from 2 to 3 and the coupling constant HS of scalar diquark interaction channel is replaced by the coupling constant HP of pseu- doscalar diquark interaction channel. Now the matrices γµ(µ = 0, 1, 2) and the charge conjugate matrix C are taken to be 2× 2 ones and have the explicit forms , γ1 = , γ2 = It is emphasized that, in 3D case, no ”γ5” matrix can be defined, hence the third term in the right-handed side of Eq.(1) will be the only possible color-anti-triplet di- quark interaction channel which could lead to Lorentz- invariant diquark condensates, where we note that the matrix Cτ2λA is antisymmetric. Without ”γ5”, the La- grangian (1) will have no chiral symmetry. Except this, it is not difficult to verify that the symmetries of L include: 1. continuous flavor and color symmetries SUf (2) ⊗ SUc(3)⊗ Uf (1); 2. discrete symmetry R: q → −q; 3. parity P : q(t, ~x) → γ0q(t,−~x) and qC(t, ~x) → −γ0qC(t,−~x); 4. time reversal T : q(t, ~x) → γ2q(−t, ~x) and qC(t, ~x) → −γ2qC(−t, ~x); 5. charge conjugate C: q ↔ qC ; 6. special parity P1: q(t, x1, x2) → γ1q(t,−x1, x2) and qC(t, x1, x2) → −γ1q(t,−x1, x2); 7. special parity P2: q(t, x1, x2) → γ2q(t, x1,−x2) and qC(t, x1, x2) → −γ2qC(t, x1,−x2). If the quark-antiquark condensates 〈q̄q〉 could be formed, then the time reversal T , the special parities P1 and P2 will be spontaneously broken [6]. If the diquark conden- sates 〈q̄Cτ2λ2q〉 could be formed, then the color symme- try SUc(3) will be spontaneously broken down to SUc(2) http://arxiv.org/abs/0704.0829v2 and the flavor number Uf (1) will be spontaneously bro- ken but a ”rotated” electric charge U (1) and a ”rotated” quark number U ′q(1) leave unbroken [7]. In addition, the parity P will be spontaneously broken, though all the other discrete symmetries survive. This implies that the diquark condensates 〈q̄Cτ2λ2q〉 will be a pseudoscalar. In this paper we will neglect discussions of the Gold- stone bosons induced by breakdown of the continuous symmetries and pay our main attention to the problem of interplay between the above two forms of condensates. III. EFFECTIVE POTENTIAL IN MEAN FIELD APPROXIMATION Define the order parameters in the 3D GN model by σ = −2GS〈q̄q〉 and ∆ = −2HP 〈q̄Cτ2λ2q〉, (3) then in the mean field approximation, the Lagrangian (1) can be rewritten by L = Ψ̄(x)S−1(x)Ψ(x) − , (4) where Ψ(x) = qC(x) and Ψ̄(x) = q̄(x) q̄C(x) are the expressions of the quark fields in the Nambu- Gorkov basis [8]. In the momentum space, the inverse propagator S−1(x) for the quark fields may be expressed S−1(p) = 6p− σ −τ2λ2∆ −τ2λ2∆∗ 6p− σ , 6p = γµpµ. (5) The effective potential corresponding to L given by Eq.(4) becomes V (σ, |∆|) = σ (2π)3 Tr lnS−1(p)S0(p). Similar to the case of the 2D NG model [4], the calcu- lations of Tr for (red, green) and blue color degrees of freedom can be made separately thus Eq.(6) will be re- duced to V (σ, |∆|) = σ (2π)3 p2 − (σ − |∆|)2 + iε p2 + iε p2 − (σ + |∆|)2 + iε p2 + iε p2 − σ2 + iε p2 + iε After the Wick rotation, we may define and calculate in 3D Euclidean momentum space I(a2) = (2π)3 p̄2 + a2 a3 arctan a2Λ − π , if Λ ≫ |a|, (8) where Λ is the 3D Euclidean momentum cut-off. Assume that Λ ≫ |σ − |∆||, Λ ≫ σ + |∆| and Λ ≫ σ, then by means of Eq.(8) we will obtain the final expression of the effective potential in the 3D GN model V (σ, |∆|) = σ (3σ2 + 2|∆|2)Λ 6σ2|∆|+ 2|∆|3 + σ3 +2θ(σ − |∆|)(σ − |∆|)3 . (9) IV. GROUND STATES Equation (9) provide the possibility to discuss the ground states of the model analytically. The extreme value conditions ∂V (σ, |∆|)/∂σ = 0 and ∂V (σ, |∆|)/∂|∆| = 0 will lead to the equations θ(σ − |∆|)(σ − |∆|)2 = 0, (10) [σ2 − θ(σ − |∆|)(σ − |∆|)2] = 0. (11) Define the expressions = AC −B2, where A, B and C represent the second order derivatives of V (σ, |∆|) with the explicit expressions θ(σ − |∆|)(σ − |∆|), ∂σ∂|∆| ∂|∆|∂σ [σ − θ(σ − |∆|)(σ − |∆|)], ∂|∆|2 θ(σ − |∆|)(σ − |∆|). (12) Equations (10) and (11) have the four different solutions which will be discussed in proper order as follows. (i) (σ, |∆|)=(0,0). It is a maximum point of V (σ, |∆|), since in this case we have < 0 and K = A assuming Eqs. (10) and (11) have solutions of non-zero σ and |∆|. (ii) (σ, |∆|)=(σ1 ,0), where the non-zero σ1 satisfies the equation = 0. (13) When Eq. (13) is used, we obtain K = = A > 0, if Hence (σ1,0) will be a minimum point of V (σ, |∆|) when GS/HP > 2/3. (iii) (σ, |∆|)= (0, ∆1), where non-zero ∆1 obeys the equation = 0. (14) By using Eq.(14) we may get , K = A Obviously, (0,∆1) will be a minimum point of V (σ, |∆|) when GS/HP < 2/3. (iv) (σ, |∆|)=(σ2,∆2). In view of existence of the func- tion θ(σ− |∆|) in Eqs.(10) and (11), we have to consider the case of σ2 > ∆2 and σ2 < ∆2 respectively. (a) σ2 > ∆2. In this case, Eqs.(10) and (11) will be- + 2∆2 ) = 0, From them we can get − 2∆2 > 0, K = − Thus it is turned out that (σ2, ∆2) will be neither a maximum nor a minimum point of V (σ, |∆|) if σ2 > ∆2. (b) σ2 < ∆2. Now Eqs. (10) and (11) are changed into 4∆2 + σ2 = 0, (15) = 0. (16) Hence we will have the results that , K = − 8σ2∆2), from which it may be deduced that only if σ2 < ( 17− 4)∆2, (17) (σ2,∆2) is just a minimum point of V (σ, |∆|). On the other hand, from Eqs. (15) and (16) obeyed by σ2 and ∆2 we may get 13− 1 ∆2 + σ2 13 + 1 ∆2 − σ2 . (18) Equation (18) indicates that for the minimum point (σ2,∆2) satisfying Eq.(17) one will certainly have GS/HP > 2/3. Taking this and the result obtained in case (ii) into account we see that if GS/HP > 2/3 the effective potential V (σ, |∆|) will have two possible min- imum points (σ1, 0) and (σ2,∆2). To determine which one of the two minimum points is the least value point of V , we must make a comparison between V (σ1, 0) and V (σ2,∆2) with the constraint given by Eq.(17). In fact, it is easy to find out that when Eq.(13) is used, V (σ1, 0) = − , (19) and that when Eqs. (15) and (16) are used, V (σ2,∆2) = − + 3σ2 . (20) By comparing Eq.(13) with Eq.(15) we may obtain the relation 3σ1 = σ2 + 4∆2. (21) By means of Eqs.(19)-(21) it is easy to verify that V (σ1, 0)− V (σ2,∆2) = − (23∆3 − 4σ3 (8∆2 − 7σ2) when Eq.(17) is satisfied. This result indicates that when GS/HP > 2/3, the least value point of V (σ, |∆|) will be (σ1, 0) but not (σ2,∆2). In summary, if the necessary conditions GSΛ > π and HPΛ > π 2/8 for non-zero σ and ∆ are satisfied, then the least value points of the effective potential V (σ, |∆|) will be at (σ, |∆|) = (0,∆1) (σ1, 0) 0 ≤ GS/HP < 2/3 GS/HP > 2/3 . (22) As a result, in the ground state of the 3D two-flavor GN model, depending on that the ratio GS/HP is either big- ger or less than 2/3, one will have either pure quark- antiquark condensates or pure diquark condensates, but no coexistence of the two forms of condensates could hap- V. CONCLUDING REMARKS The result (22) in the 3D GN model can be compared with the ones in the 4D NJL model and in the 2D GN model. The minimal points of the effective potential V (σ, |∆|) for the latter models have been obtained and are located respectively at (σ, |∆|) = (0, ∆1) (σ2, ∆2) (σ1, 0) 0 ≤ GS/HS < 2/[3(1 + C)] 2/[3(1 + C)] < GS/HS < 2/3 GS/HS > 2/3 with C = (2HSΛ /π2 − 1)/3 and Λ4 denoting the 4D Euclidean momentum cutoff in the 4D two-flavor NJL model, if the necessary conditions GSΛ > π2/3 and > π2/2 for non-zero σ and ∆ are satisfied [2], and (σ, |∆|) = (0, ∆1) (σ2, ∆2) (σ1, 0) GS/HS = 0 0 < GS/HS < 2/3 GS/HS > 2/3 in the 2D two-flavor GN model [4]. In Eqs.(23) and (24), GS and HS always represent the coupling constants in scalar quark-antiquark channel and scalar diquark chan- nel separately. By a comparison among Eqs.(22)-(24) it may be found that the three models lead to very similar results. In all the three models, the interplay between the quark- antiquark and the diquark condensates in vacuum de- pends on the ratio GS/HD (D = S for the 4D and 2D model and D = P for the 3D model). In particular, the diquark condensates could emerge (in separate or coex- istent pattern) only if GS/HD < 2/3. This is probably a general characteristic of the considered two-flavor four- fermion models, since in these models the color number of the quarks participating in the diquark condensates and in the quark-antiquark condensates is just 2 and 3 respectively. However, there are also some differences in the pattern realizing the diquark condensates among the three models, though the pure quark-antiquark con- densates arise only if GS/HD > 2/3 in all of them. In the 2D GN model, the pure diquark condensates emerge only if GS/HS = 0 and this is different from the 4D NJL model where the pure diquark condensates may arise if GS/HS is in a finite region below 2/3. Another difference is that in the 3D GN model, there is no coexistence of the quark-antuquark condensates and the diquark conden- sates but such coexistence is clearly displayed in the 4D and 2D model. This implies that in the 3D GN model, GS/HP = 2/3 becomes the critical value which distin- guishes between the ground states with the pure diquark condensates and with the pure quark-antiquark conden- sates. It is also indicated that if the two-flavor four-fermion interaction models are assumed to be simulations of QCD (of course, only the 4D NJL model is just the true one) and the four-fermion interactions are supposed to come from the heavy color gluon exchange interactions −g(q̄γµλaq)2 (a = 1, · · · , 8;µ = 0, · · · , D − 1) via the Fierz transformation [7], then one will find that in all the three models, for the case of two flavors and three colors the ratio GS/HD are always equal to 4/3 which is larger than the above critical value 2/3. From this we can con- clude that there will be only the pure quark-antiquark condensates and no diquark condensates in the ground states of all these models in vacuum. [1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345; 124 (1961) 246. [2] Zhou Bang-Rong, Commun. Theor. Phys. 47 (2007) 95. [3] D.J. Gross and A. Neveu, Phys. Rev. D 10 (1974) 3235. [4] Zhou Bang-Rong, Commun. Theor. Phys., 47 (2007) 520. [5] B. Rosenstein, B. J. Warr, and S. H. Park, Phys. Rep. 205 (1991) 59. [6] Bang-Rong Zhou, Phys. Lett. B444 (1998) 455. [7] M. Buballa, Phys. Rep. 407 (2005) 205. [8] Y. Nambu, Phys. ReV. 117 (1960) 648; L. P. Gorkov, JETP 7 (1958) 993.

0704.0831

On packet lengths and overhead for random linear coding over the erasure channel

On packet lengths and overhead for random linear coding over the erasure channel Brooke Shrader and Anthony Ephremides Electrical and Computer Engineering Dept and Institute for Systems Research University of Maryland College Park, MD 20742 bshrader, [email protected] Abstract— We assess the practicality of random network cod- ing by illuminating the issue of overhead and considering it in conjunction with increasingly long packets sent over the erasure channel. We show that the transmission of increasingly long packets, consisting of either of an increasing number of symbols per packet or an increasing symbol alphabet size, results in a data rate approaching zero over the erasure channel. This result is due to an erasure probability that increases with packet length. Numerical results for a particular modulation scheme demonstrate a data rate of approximately zero for a large, but finite-length packet. Our results suggest a reduction in the performance gains offered by random network coding. I. INTRODUCTION It has been shown that network coding allows a source node to multicast information at a rate approaching the maximum flow of a network as the symbol alphabet size approaches infinity [1], [2]. An infinitely large symbol alphabet would correspond to transmission of infinitely long blocks of data, which is neither possible nor practical. Subsequent works have shown that a finite alphabet is sufficient to achieve the maximum flow and a number of works have provided bounds on the necessary alphabet size. For example, in [3] the alphabet size is upper bounded by the product of the number of sources with the number of receivers, while in [4] a lower bound on the alphabet size is given by the number of receivers. Random linear coding was proposed to allow for distributed implementation [5]. In random network coding, a randomly generated encoding vector is typically communicated to the receiver by appending it to the header of the transmitted packet. The overhead inherent in communicating the encoding vector becomes negligible as the number of symbols in the packet grows large. Regarding the alphabet size for random network coding, a lower bound of 2 times the number of receivers is given in [4]. The “length” of a transmitted packet (i.e., the number of bits conveyed by the packet) depends on both the symbol alphabet and the number of symbols per packet. From the previous works listed above, it is clear that transmitting sufficiently long packets is crucial to ensuring the existence of network codes and to allowing random linear coding to operate with low overhead. However, long packets are more susceptible to noise, inter- ference, congestion, and other adverse channel effects. This point has been ignored in previous works on network coding. In particular, a number of previous works [6]–[8] consider transmission over an erasure channel, in which a packet is either dropped with a probability ǫ or received without error. In a wireline network, the erasure channel is used to model dropped packets due to buffer overflows. Clearly, longer packets take up more space in memory, so the erasure probability will increase as the packet length grows. In the case of a wireless network, there are a number of reasons that the erasure probability will increase with the length of the packet, as listed below. • If we try to fit more bits into the channel using mod- ulation, then for fixed transmission power, points in the signal constellation will move closer together and errors are more likely. • If we try to fit more bits into the channel by decreasing symbol duration, then we are constrained by bandwidth, a carefully-controlled resource in wireless systems. • Longer packets are more susceptible to the effects of fading. In this work, we model the erasure probability as a func- tion of the packet length and investigate the implications on network coding performance. This is somewhat reminiscent of [9], in which the erasure probability is a function of the link distance in a wireless network, and the effect on capacity is investigated. Our emphasis will be on a wireless channel with a fixed bandwidth, for which we will associate erasure probability with the probability of symbol error for a given modulation scheme. We note that a careful examination of packet lengths in data networks is not a new idea; most notably, many researchers participated in a dispute over the packet size for the Asyn- chronous Transfer Mode (ATM) standard in the 1980s. And still many tradeoffs are being studied today at the network level regarding the packet length. The interplay between packet lengths and coding arises because of the way that network coding unifies different layers of the protocol stack. Another work which examines a similar problem is [10], in which packet headers and packet lengths are analyzed for Reed- Solomon coding in video applications. http://arxiv.org/abs/0704.0831v1 II. THROUGHPUT OF RANDOM LINEAR CODING We consider the following setting. A source node has K units of information {s1, s2, . . . , sK} that it wants to transmit. We will refer to each of these information units as packets and let each packet be given by a vector of n q-ary symbols, where q is the symbol alphabet (the size of a finite field) and n is the number of symbols per packet. We consider values of q which are powers of 2, i.e., q = 2u for some u ≥ 1. The length of a packet is given by n log2 q bits. In Section III we will consider the case where {s1, s2, . . . , sK} are packets of uncoded information, whereas in Section IV we will assume that a (deterministic) error-correcting code has been applied in order to form {s1, s2, . . . , sK}. The source generates random linear combinations by form- ing the modulo-q sum i=1 αisi, where αi are chosen ran- domly and uniformly from the set {0, 1, 2, . . . , q − 1}. Note that the resulting random linear combination is a packet with length n log2 q bits. With each random linear combination transmitted, the source appends a packet header identifying αi, which requires an additional K log2 q bits of overhead with every n log2 q bits transmitted. A receiver will collect these random linear combinations until it has received enough to decode. For transmission over an erasure channel, decoding can be performed by Gaussian elimination. Let N denote the number of random linear combinations needed for decoding. Each of the N random linear combinations represents a linear equation in {s1, s2, . . . , sK} and the distribution of N is given Pr(N≤j)=Pr{a random K×j matrix has rank K}. Note that Pr(N ≤ j) is equal to zero for j < K . For j ≥ K , the distribution can be found following the procedure in [11] and is given by Pr(N≤j) = (q j−1)(qj−q)(qj−q2) . . . (qj−qK−1) (1− q−j+i) The expected value of N can be found from the above distribution and is shown to be given by E[N ] = K + (1− q−j+i) (1− q−i)−1 (1) Clearly, as q → ∞, E[N ] → K . We let ǫ(n log2 q) denote the erasure probability on the channel, which is an increasing function of the packet length n log2 q. The expected number of transmissions needed for the receiver to decode the original K packets is given by E[N ] 1− ǫ(n log2 q) Over the course of these transmissions, the average number of packets received for each transmission is K(1− ǫ(n log2 q)) E[N ] We account for the overhead by scaling the number of packets received per transmission by n/(n+K), which is the ratio of number of information symbols to the total number of symbols (information plus overhead) sent with each transmission. We define S as the effective portion of each transmission which contains message information; S is a measure of throughput in packets per transmission and takes values between 0 and 1. E[N ] (1− ǫ(n log2 q)) (2) If the erasure probability is constant, ǫ(n log2 q) = e, then for n, q → ∞, corresponding to infinitely long packets, the value of S approaches 1− e, which is the Shannon capacity of the erasure channel. From the expression in (2), we can identify a tradeoff between the packet length and the throughput. As the number of information symbols per packet n grows large, the effect of overhead becomes negligible, but the erasure probability grows and the transmissions are more likely to fail. Alternatively, if n approaches zero, transmissions are more likely to succeed, but the overwhelming amount of overhead means that no information can be transmitted. In a similar manner, as the alphabet size q grows, the random linear coding becomes more efficient in the sense that E[N ] → K , but again, the erasure probability increases and transmissions are likely to be unsuccessful. This tradeoff demonstrates that the alphabet size, packet length, and overhead must be carefully weighed in determining the performance of random linear coding over the erasure channel. III. PERFORMANCE WITHOUT PRE-CODING In this section we consider the case where {s1, s2, . . . , sK} are uncoded information symbols. We define the data rate in bits per transmission as R, where R = Sn log2 q. (3) The data rate R accounts for the fact that each received packet contains n log2 q bits of information. Since the packets {s1, s2, . . . , sK} consist of uncoded information, for every random linear combination sent, all n symbols must be re- ceived without error. Then Pr(erasure) = 1− (1− Pr(symbol error))n and Pr(symbol error) will correspond to the symbol error probability for q-ary modulation over the channel. We denote the probability of symbol error by Pq and note that it is independent of n but depends on q as well as features of the channel such as pathloss, signal-to-noise ratio (SNR), and fading effects. We will consider a wireless channel of limited bandwidth, for which modulation techniques such as pulse amplitude modulation (PAM), phase shift keying (PSK), and 0 500 1000 1500 2000 Symbols per packet, n 0 500 1000 1500 2000 Fig. 1. Throughput S (solid line) and data rate R (dotted line) versus symbols per packet n without pre-coding for QAM modulation, K = 80, q = 8, and SNR per bit γb=3.5dB. quadrature amplitude modulation (QAM) are appropriate. For these modulation techniques, Pq → 1 as q → ∞ [12]. In this setting, the erasure probability is given by 1− ǫ(n log2 q) = (1− Pq)n. (4) From the above expression we note that, as suggested by our discussion on the tradeoff between throughput and packet length, as n → ∞, both S and R approach zero exponentially fast. Additionally, for the modulation schemes mentioned above, S and R approach zero as q → ∞, albeit at a slower rate. As a numerical example, we have plotted the throughput S and data rate R for QAM modulation over an additive white Gaussian noise (AWGN) channel. In this case the symbol error probability for the optimum detector is approximated by [12] Pq ≈ 1− 3γb log2 q q − 1 where γb is the SNR per bit and Q is the complementary cumulative distribution function for the Gaussian distribution. The above expression for Pq holds with equality for log2 q even. The results are shown in Figure 1, where q is fixed and n is varied, and in Figure 2, where n is fixed and q is varied. In all cases, the throughput and data rate are concave functions, admitting optimum values for n and q. Furthermore, if n grows large as q is fixed or if q grows large as n is fixed, the throughput approaches zero. To improve the performance described above, a number of techniques might be employed, including enhanced receivers, diversity techniques, and channel coding. IV. PERFORMANCE WITH PRE-CODING In this section we examine the effect of using a deter- ministic pre-code prior to performing random linear coding. We assume that an uncoded packet σi, consisting of k q- ary information symbols, is passed through an encoder to 1 1.5 2 2.5 3 3.5 4 4.5 5 Symbol alphabet size, log 1 1.5 2 2.5 3 3.5 4 4.5 5 Fig. 2. Throughput S (solid line) and data rate R (dotted line) versus log of the symbol alphabet log q for QAM modulation, K = 80, n = 200, SNR per bit γb=3.5dB, and no pre-coding. add redundancy, producing the packet si consisting of n q- ary symbols. Thus a q-ary code of rate Rpc = k/n is used to produce {s1, s2, . . . , sK}. Random linear coding is again performed on {s1, s2, . . . , sK} and we assume that the codebook for the pre-code is known to the decoder and does not need to be transmitted over the channel. At the receiver, decoding of the random linear code will be performed first using the coefficients αi, i = 1, 2, . . . ,K sent with each random linear combination. Subsequently the pre-code will be decoded. The use of the pre-code means that the system will be tolerant to some errors in the n symbols sent in each random linear combination. To account for the pre-code, we modify the throughput S expressed in (2) by multiplying it be Rpc, which results in E[N ] (1 − ǫ(n log2 q)). (6) To obtain the data rate R in bits per transmission, S is now scaled by k, the number of information symbols present in each packet. R = Sk log2 q (7) We can bound the erasure probability, and thus the through- put S and data rate R, using bounds on the error-correcting capabilities of the pre-code. Let t denote the number of symbol errors that the pre-code can correct. For a given code, t = ⌊1/2(dmin − 1)⌋ errors, where dmin is the minimum distance of the code. The erasure probability can be bounded as follows. 1− ǫ(n log2 q) i=t+1 Pr(i symbols in error) i=t+1 P iq(1− Pq)n−i ⌊1/2(d−1)⌋ P iq(1− Pq)n−i In the above expression, (a) holds with equality for the class of perfect codes [12], (b) follows by assuming that errors are independent, identically distributed ∼ Pq among the n symbols, and (c) holds for d ≤ dmin. We make use of the Gilbert-Varshamov bound to provide a lower bound on dmin as follows. d = inf dmin : dmin−2 (q − 1)i ≥ qn−k Lower bounds on the throughput and data rate are given as follows. SLB = E[N ] ⌊1/2(d−1)⌋ P iq(1− Pq)n−i (9) RLB = SLBk log2 q (10) For q → ∞ while all other parameters are fixed, if the modulation scheme has Pq → 1, then SLB, RLB → 0. In the limit of increasing symbols per packet n, we identify two different cases. First, if the pre-code rate Rpc is fixed (i.e, k → ∞ as n → ∞ with fixed ratio between the two) then SLB → Rpc (multiplied by K/E[N ] ≈ 1) and RLB increases without bound. On the other hand, if k is fixed (i.e., as n → ∞, Rpc → 0) then SLB, RLB → 0. We have computed numerical examples of SLB and RLB assuming QAM modulation with Pq as given in (5). In Fig. 3 we display the throughput and data rate as a function of q. Clearly, pre-coding allows the system to support higher q with non-zero throughput, although the throughput does eventually go to zero for large q. In comparing this result to Fig. 2, note that all other parameters are fixed and the pre-code rate Rpc = 1/2 results in a halving of the throughput and data rate. In Fig. 4 we have displayed the two different cases for performance as a function of increasing symbols per packet n. In Fig. 4(a), the rate Rpc is fixed at 1/2 and SLB → Rpc while RLB increases without bound. In this case, pre-coding ensures that the throughput and data rate increase with n. However, there is a caveat: the result in Fig. 4(a) requires that k → ∞, which means that the source must have an infinite supply of uncoded information to be transmitted. On the other hand, if the source has a finite amount of information to be transmitted, as shown in Fig. 4(b), then as n → ∞, the pre- code rate Rpc → 0 and the data rate also approaches zero. In summary, the use of a pre-code can improve performance on the erasure channel; indeed, pre-coding is the mechanism which allows us to represent a noisy channel as an erasure channel. However, pre-coding can only ensure that the data rate increases with packet length if there is an infinite supply of (uncoded) information at the source. This is a strong assumption for a network, where data often arrives in bursts. V. DISCUSSION The primary contribution of this work is the elucidation of the fact that the widely-used assumption of the erasure probability being constant as the packet length increases leads 1 2 3 4 5 6 Symbol alphabet size, log 1 2 3 4 5 6 Fig. 3. Bounds on the throughput SLB (solid line) and data rate RLB (dotted line) versus log of the symbol alphabet log q for QAM modulation, K = 80, n = 200, k = 100 (pre-code rate 1/2), and SNR per bit γb=3.5dB. to idealized performance that cannot be obtained in a practical scenario. We have shown that the assumption of increasingly long packets, either due to increasingly many symbols per packet or to an increasing alphabet size, can result in a data rate of zero for random linear coding if the erasure probability increases with packet length. While the use of a pre-code prior to performing random linear coding can improve performance, it can only ensure that the throughput increases with packet length if there is an infinite supply of data awaiting transmission at the source. We have also demonstrated that the overhead needed for random linear coding can have a devastating impact on the throughput. We have focused our attention on the erasure channel, but the tradeoff between packet length and throughput will arise in other channels, particularly in wireless channels where these effects can be exacerbated by fading. A number of techniques, such as pre-coding, can be employed to combat the adverse effects of the channel, but in making use of these techniques, the performance gains offered by network coding will be tempered. ACKNOWLEDGMENT This work is supported by the Office of Naval Research through grant N000140610065, by the Department of Defense under MURI grant S0176941, and by NSF grant CNS0626620. Prepared through collaborative participation in the Commu- nications and Networks Consortium sponsored by the U.S. Army Research Laboratory under The Collaborative Technol- ogy Alliance Program, Cooperative Agreement DAAD19-01- 2-0011. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U. S. Government. 0 500 1000 1500 2000 2500 Symbols per packet, n 0 500 1000 1500 2000 2500 (a) Fixed rate pre-code Rpc = 1/2. 400 600 800 1000 1200 1400 1600 1800 2000 2200 Symbols per packet, n 400 600 800 1000 1200 1400 1600 1800 2000 2200 (b) Variable pre-code rate, k = 400. Fig. 4. Bounds on the throughput SLB (solid line) and data rate RLB (dotted line) versus symbols per packet n for QAM modulation, K = 80, q = 8, and SNR per bit γb=3.5dB. REFERENCES [1] R. Ahlswede, N. Cai, S.-Y. R. Li and R. W. Yeung, “Network informa- tion flow,” IEEE Trans. Inform. Theory, vol. 46, no. 4, July 2000. [2] S.-Y. R. Li, R. W. Yeung, and N. Cai, “Linear network coding,” IEEE Trans. Inform. Theory, vol. 49, Feb 2003. [3] R. Koetter and M. Medard, “Beyond routing: an algebraic approach to network coding,” INFOCOM 2002. [4] S. Jaggi, P. Sanders, P. A. Chou, M. Effros, S. Egner, K. Jain and L. M. G. M. Tolhuizen, “Polynomial time algortihms for multicast network code construction,” IEEE Trans. Inform. Theory, vol. 51, no. 6, June 2005. [5] T. Ho, R. Koetter, M. Medard, D. R. Karger and M. Effros, “The benefits of coding over routing in a randomized setting,” ISIT 2003. [6] D. S. Lun, M. Medard, R. Koetter, and M. Effros, “On coding for reliable communication over packet networks,” submitted to IEEE Trans. Inform. Theory. [7] A. Eryilmaz, A. Ozdaglar and M. Medard, “On delay performance gains from network coding,” CISS, 2006. [8] A. F. Dana, R. Gowaikar, R. Palanki, B. Hassibi and M. Effros, “Capacity of wireless erasure networks,” IEEE Trans. Inform. Theory, vol. 52, no. 3, pp. 789-804, 2006. [9] B. Smith and S. Vishwanath, “Asymptotic transport capacity of wireless erasure networks,” Allerton conference, 2006. [10] B. Hong and A. Nosratinia, “Rate-constrained scalable video transmis- sion over the Internet,” IEEE Packet Video Workshop, 2002. [11] D. J. C. MacKay, “Fountain codes,” IEE Workshop on Discrete Event Systems, 1998. [12] J. G. Proakis, Digital Communications, Third edition, McGraw-Hill, Boston, 1995. Introduction Throughput of random linear coding Performance without pre-coding Performance with pre-coding Discussion References

0704.0832

Terrestrial and Habitable Planet Formation in Binary and Multi-star Systems

White Paper (Submitted to ExoPlanet Task Force) Terrestrial and Habitable Planet Formation in Binary and Multi-star Systems Authors Nader Haghighipour Institute for Astronomy and NASA Astrobiology Institute, University of Hawaii-Manoa Steinn Sigurdsson Department of Astronomy & Astrophysics and NASA Astrobiology Institute, Penn State University Jack Lissauer Space Science and Astrobiology Division, NASA-Ames Research Center Sean Raymond Center for Astrophysics and Space Astronomy, and Center for Astrobiology, University of Colorado Introduction The discovery of extrasolar planets during the past decade has confronted astronomers with many new challenges. The diverse and surprising dynamical characteristics of many of these objects have made scientists wonder to what extent the current theories of planet formation can be applied to other planetary systems. A major challenge of planetary science is now to explain how such planets were formed, how they acquired their unfamiliar dynamical state, whether there are habitable extrasolar planets, and how to detect such habitable worlds. In this respect, one of the most surprising discoveries is the detection of planets in binary star systems. Among the currently known extrasolar planet-hosting stars approximately 25% are members of binaries (Table 1). With the exception of the pulsar planetary system PSR B1620-26 (Sigurdsson et al. 2003; Richer et al. 2003; Beer et al. 2004), and possibly the system of HD202206 (Correia et al. 2005), planets in these binary systems revolve only around one of the stars. While the majority of these binaries are wide (i.e., with separations between 250 and 6500 AU, where the perturbative effect of the stellar companion on planet formation around the other star is negligible), the detection of Jovian-type planets in the two binaries γ Cephei (separation of 18.5 AU, see Hatzes et al. 2003) and GJ 86 (separation of 21 AU, see Els et al. 2001) have brought to the forefront questions on the formation of giant planets and the possibility of the existence of smaller bodies in moderately close binary and multiple star systems. Given that more than half of main sequence stars are members of binaries/multiples (Duquennoy & Mayor 1991; Mathieu et al. 2000), and the frequency of planets in binary/multiple systems is comparable to those around single stars (Bonavita & Desidera 2007), such questions have realistic grounds. At present, the sensitivity of the detection techniques does not allow routine discovery of Earth-sized objects around binary and multi-star systems. However, with the advancement of new techniques, and with the recent launch of CoRoT and the launch of Kepler in late 2008, the detection of more planets (possibly terrestrial-class objects) in such systems is on the horizon. Table 1- Binary and multi-star systems with extrasolar planets (Haghighipour 2006) Star Star Star Star HD142 (GJ 9002) HD3651 HD9826 (υAnd) HD13445 (GJ 86) HD19994 HD22049 (εEri) HD27442 HD40979 HD41004 HD75732 (55 Cnc) HD80606 HD89744 HD114762 HD117176 (70 Vir) HD120136 (τBoo) HD121504 HD137759 HD143761 (ρCrb) HD178911 HD186472 (16 Cyg) HD190360 (GJ 777 A) HD192263 HD195019 HD213240 HD217107 HD219449 HD219542 HD222404 (γCeph) HD178911 PSR B1257-20 PSR B1620-26 HD202206 See http://www.obspm.fr/planets for complete list of extrasolar planets with their corresponding references. Fig 1. The time of ejection, vs. the initial semimajor axis of an Earth-like planet in a co-planar arrangement in the γ Cephei system. The binary consists of a 1.59 solar-masses K1 IV subgiant as its primary (Fuhrmann 2003) and a probable red M dwarf with a mass of 0.41 solar-masses (Neuhauser et al 2007) as its secondary. The semimajor axis and eccentricity of the binary are 18.5 AU and 0.36, respectively (Hatzes et al. 2003). The primary star is host to a 1.7 Jupiter-masses object in an orbit with a semimajor axis of 2.13 AU and eccentricity of 0.12. The habitable zone of the primary is within 3 AU to 3.7 AU from this star (Haghighipour 2006). As shown here, the orbit of an Earth-sized object in the primary’s habitable zone is unstable. However, such an object can have a log-term stable orbit in distances closer to the primary star. Theoretical studies and numerical modeling of terrestrial and habitable planet formation in such dynamically complex environments are, therefore, necessary to gain fundamental insights into the prospects for life in such systems and have great strategic impact on NASA science and missions. Several lines of investigations are needed to ensure progress in understanding the formation of terrestrial and habitable planets in binary and multi-star systems. Fig 2. Results of simulations of the formation of Earth-like objects in the habitable zone of the primary of a binary star system. The stars of the binary are Sun-like and the primary is host to a Jupiter-sized object on a circular orbit at 5 AU. Simulations show the results for different values of the eccentricity and semimajor axis of the stellar companion (Haghighipour & Raymond 2007). As shown here, the orbital motion of the secondary star disturbs the orbit of the giant planet, which in turn affects the final assembly and water contents of the terrestrial objects. This figure also shows that binary systems with larger perihelia are more favorable for forming and harboring habitable planets. The quantities ab and eb represent the seimmajor axis and eccentricity of the binary. 1) Computational Modeling Extensive numerical studies are necessary to i) map the parameter-space of binary and/or multiple star systems to identify regions where giant and terrestrial planets can have long-term stable orbits, ii) simulate the collision and growth of planetesimals to form protoplanetary objects, iii) simulate the formation of planetesimals in circumbinary and circumstellar disks, iii) develop models of protoplanet disk chemistry that ensure delivery of water to terrestrial-class planets in the habitable zone, iv) simulate the interaction of planetary embryos and the late stage of terrestrial planet formation. The parameter-space is large and includes the masses and orbital parameters of the stars and planets. It is, therefore, necessary to develop a systematic approach, based on the results of Fig 3. Histograms of the number of final terrestrial planets formed in binary star systems with periastron distances of 5 AU (top), 7.5 AU (middle), and 10 AU (bottom). The color red corresponds to simulations in a binary in which the primary and secondary stars are 0.5 solar-masses. The color blue represents a binary with 1 solar-mass stars, and the color yellow corresponds to a binary with a 0.5 solar-masses primary and a 1 solar- mass secondary. The black line in the middle panel shows the results of simulations when the primary star is 1 solar-mass and the secondary is 0.5 solar-masses. As shown here, the typical number of final planets clearly increases in systems with larger stellar periastra, and also when the companion star is less massive than the primary (Quintana et al. 2007). current research, to avoid un-necessary simulations, particularly if the computational resources are limited. Current research has indicated that terrestrial-class planets can have long-term stable orbits as long as they are closer to their host stars and their orbits lie outside the influence zone of the giant planet of the system (figure 1, also see Holman & Wiegert 1999; David et al. 2003; Haghighipour 2006). This implies, in order for such systems to be habitable, the habitable zone of the planet-hosting star has to be considerably closer to it than orbit of its giant planet(s). Given that the location of the habitable zone is a function of the luminosity of a star, the above- mentioned criterion can be used to constrain stellar properties. Recent numerical simulations have also shown that (1) water-delivery is more efficient when the perihelion of the binary is large and the orbit of the giant planet is close to a circle (figures 2 and 3, also see Quintana et al. 2007, Haghighipour & Raymond 2007), and (2) habitable planets can form in the habitable zone of a star during the migration of giant planets (figure 4, see Raymond, Mandell & Sigurdsson, 2006). Since many stars are formed in clusters, their mutual interactions may change their orbital configurations and cause their giant planets to revolve around their host starts in un-conventional orbits. Theoretical studies are essential to identify systems capable of forming and harboring habitable planets. Fig 4. Habitable planet formation at presence of giant planet migration (Raymond, Mandell & Sigurdsson 2006). The system consists of a Sun-like star and a Jupiter-sized giant planet. The figure shows snapshots in time of the evolution of one simulation. Each panel plots the orbital eccentricity versus semimajor axis for each surviving body. The size of each body is proportional to its physical size (except for the giant planet, shown in black). The vertical "error bars" represent the sine of each body's inclination on the y-axis scale. The color of each dot corresponds to its water content (as per the color bar), and the dark inner dot represents the relative size of its iron core. For scale, the Earth's water content is roughly 10-3. As shown here, an Earth-like object can form in the habitable zone of the star while the giant planet migrates to closer distances. 2) Theoretical Analysis of Observation Data Recent observations of binary star systems, using Spitzer Space Telescope, show evidence of debris disks in these environments (Trilling et al. 2007). As shown by these authors, approximately 60% of their observed close binary systems (separation smaller than 3 AU) have excess in their thermal emissions, implying on-going collisions in their planetesimal regions. Future space-, air-, and ground-based telescopes such as ALMA, SOFIA and JWST will be able to detect more of such disks and will also be able to resolve their fine structures. Numerical simulations, similar to those for debris disks around single stars (Telesco et al. 2005), will be necessary in order to understand the dynamics of such planet-forming environments, and also identify the source of their disks features (e.g., embedded planets, and/or on-going planetesimals collision). Due to the complex nature of these systems, such numerical studies require more advanced computational codes, and more powerful computers. Developing theories of disk evolution in close binary systems is also essential. 3) Computational Resources Given the extent and complexity of simulations of planet formation in multi-star systems, and the high dimensionality of the parameter space of initial conditions, supports for developing computational resources with the primary focus of conducting numerical analysis of terrestrial planet formation are essential. Reliable simulations of collisional growth of planetesimals and planetary embryos require integration of the orbits of several hundred thousands of such objects. With the current technology, such simulations may take several months to a year to complete. It is therefore necessary to develop (i) faster integration routines, and (ii) major computational facilities with the primary focus of simulating terrestrial planet formation. Strategic Impact to NASA Missions Understanding terrestrial and habitable planet formation in binary and multiple star systems has implications for investigating the habitability of extra-solar planets. It ties directly into near future NASA missions, in particular Kepler, and JWST as well as complementary ongoing and planned NSF and privately funded surveys that include transit, and radial velocity. It is also closely coupled with the scientific aspect of the Space Exploration Vision and aligns with the 2006 NASA Science Program implementation of the Strategic sub-goal 3D: “Discover the origin, structure, evolution, and destiny of the universe and search for earth-like planets.” The strategic relevance to the NASA missions is in the prospects for detection of habitable Earth-like planets. Studies such as those presented here underlie hypotheses regarding the likelihood of the existence of such planets, the origin of life in the habitable zones of their host stars, and theories of evolution and persistence of life after initiation, at the presence of a stellar companion. Earth-like objects in and around binary star systems allow testing of theories of extrasolar habitability and origin of life. Prospects for testing of extrasolar life are intrinsically exciting and valuable to the NASA community and the public, and the systems to be explored, once found, provide calibration targets for future NASA missions. References Beer, M. E., King, A. P., & Pringle, J. E. 2004, MNRAS, 355, 1244 Bonavita, M., & Desidera, S. 2007, submitted to A&A (astro-ph/0703754) Correia, A. C. M., Udry, S., Mayor, M., Laskar, J., Naef, D., Pepe, F., Queloz, D., & Santos, N. C., 2005, A&A, 440, 751 David, E., Quintana, E. V., Fatuzzo, M., Adams, F. C., 2003, PASP, 115, 825 Duquennoy, A., & Mayor, M. 1991, A&A, 248, 485 Els, S. G., Sterzik, M. F., Marchis, F., Pantin, E., Endl, M., & Kurster, M. 2001, A&A, 370, L1 Fuhrmann, K. 2003, Astron.Nachr. 323, 392 Haghighipour, N. 2006, ApJ, 644, 543 Haghighipour, N., & Raymond, S. N., to appear in ApJ (astro-ph/0702706) Holman, M. J., & Wiegert, P. A. 1999, AJ, 117, 621 Hatzes, A. P., Cochran, W. D., Endl, M., McArthur, B., Paulson, D. B., Walker, G. A. H., Campbell, B., & Yang, S. 2003, ApJ, 599, 1383 Mathieu, R. D., Ghez, A. M., Jensen, E. L. N., & Simon, M. 2000, in Protostars and Planets IV, ed. V. Mannings, A. P. Boss, & S. S. Russell (Tucson: Univ. Arizona Press), 703 Neuhauser, R., Mugrauer, M., Fukagawa, M., Torres, G., Schmidt, T., 2007, A&A, 462, 777 Quintana, E. V., Adams, F. C., Lissauer, J. J., Chambers, J. E. 2007, ApJ, to appear in vol. 660 Raymond, S. N., Mandell, A. M., & Sigurdsson, S., 2006, Science, 313, 1413 Richer, H. B., Ibata, R., Fahlman, G., G., & Huber, M. 2003, ApJ, 597, L45 Sigurdsson, S., Richer, H. B., Hansen, B., M., Stairs, I. H., & Thorsett, S. E. 2003, Science, 301, 103 Telesco, C. M., Fisher, R. S., Wyatt, M. C., Dermott, S. F., Kehoe, T. J. J., Novotny. S., Mariñas, N., Radomski, J. T., Packham, C., De Buizer, J., Hayward, T. L., 2005, Nature, 433, 133 Trilling, D. E., Stansberry, J. A., Stapelfeldt, K. R., Rieke, G. H., Su, K. Y. L., Gray, R. O., Corbally, C. J., Bryden, G., Chen, C. H., Boden, A., Beichman, C. A. 2007, 658, 1289

0704.0833

Local well-posedness of nonlinear dispersive equations on modulation spaces

LOCAL WELL-POSEDNESS OF NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES ÁRPÁD BÉNYI AND KASSO A. OKOUDJOU Abstract. By using tools of time-frequency analysis, we obtain some improved local well-posedness results for the NLS, NLW and NLKG equations with Cauchy data in modulation spaces Mp,1 1. Introduction and statement of results The theory of nonlinear dispersive equations (local and global existence, regularity, scattering theory) is vast and has been studied extensively by many authors. Almost exclusively, the techniques developed so far restrict to Cauchy problems with initial data in a Sobolev space, mainly because of the crucial role played by the Fourier transform in the analysis of partial differential operators. For a sample of results and a nice introduction to the field, we refer the reader to Tao’s monograph [12] and the references therein. In this note, we focus on the Cauchy problem for the nonlinear Schrödinger equa- tion (NLS), the nonlinear wave equation (NLW), and the nonlinear Klein-Gordon equation (NLKG) in the realm of modulation spaces. Generally speaking, a Cauchy data in a modulation space is rougher than any given one in a fractional Bessel poten- tial space and this low-regularity is desirable in many situations. Modulation spaces were introduced by Feichtinger in the 80s [6] and have asserted themselves lately as the “right” spaces in time-frequency analysis. Furthermore, they provide an excellent substitute in estimates that are known to fail on Lebesgue spaces. This is not entirely surprising, if we consider their analogy with Besov spaces, since modulation spaces arise essentially replacing dilation by modulation. The equations that we will investigate are: (1) (NLS) i +∆xu+ f(u) = 0, u(x, 0) = u0(x), (2) (NLW ) −∆xu+ f(u) = 0, u(x, 0) = u0(x), (x, 0) = u1(x), (3) (NLKG) + (I −∆x)u+ f(u) = 0, u(x, 0) = u0(x), (x, 0) = u1(x), Date: October 30, 2018. 2000 Mathematics Subject Classification. Primary 35Q55; Secondary 35C15, 42B15, 42B35. Key words and phrases. Fourier multiplier, weighted modulation space, short-time Fourier trans- form, nonlinear Schrödinger equation, nonlinear wave equation, nonlinear Klein-Gordon equation, conservation of energy. http://arxiv.org/abs/0704.0833v1 2 Á. Bényi and K. A. Okoudjou where u(x, t) is a complex valued function on Rd×R, f(u) (the nonlinearity) is some scalar function of u, and u0, u1 are complex valued functions on R The nonlinearities considered in this paper will be either power-like (4) pk(u) = λ|u|2ku, k ∈ N, λ ∈ R, or exponential-like (5) eρ(u) = λ(e ρ|u|2 − 1)u, λ, ρ ∈ R. Both nonlinearities considered have the advantage of being smooth. The correspond- ing equations having power-like nonlinearities pk are sometimes referred to as alge- braic nonlinear (Schrödinger, wave, Klein-Gordon) equations. The sign of the coeffi- cient λ determines the defocusing, absent, or focusing character of the nonlinearity, but, as we shall see, this character will play no role in our analysis on modulation spaces. The classical definition of (weighted) modulation spaces that will be used through- out this work is based on the notion of short-time Fourier transform (STFT). For z = (x, ω) ∈ R2d, we let Mω and Tx denote the operators of modulation and transla- tion, and π(z) = MωTx the general time-frequency shift. Then, the STFT of f with respect to a window g is Vgf(z) = 〈f, π(z)g〉. Modulation spaces provide an effective way to measure the time-frequency concen- tration of a distribution through size and integrability conditions on its STFT. For s, t ∈ R and 1 ≤ p, q ≤ ∞, we define the weighted modulation space Mp,qt,s (Rd) to be the Banach space of all tempered distributions f such that, for a nonzero smooth rapidly decreasing function g ∈ S(Rd), we have ‖f‖Mp,qt,s = |Vgf(x, ω)|p < x >tp dx < ω >qs dω Here, we use the notation < x >= (1 + |x|2)1/2. This definition is independent of the choice of the window, in the sense that different window functions yield equivalent modulation-space norms. When both s = t = 0, we will simply write Mp,q = Mp,q0,0. It is well-known that the dual of a modulation space is also a modulation space, (Mp,qs,t )′ = M p′,q′ −s,−t, where p ′, q′ denote the dual exponents of p and q, respectively. The definition above can be appropriately extended to exponents 0 < p, q ≤ ∞ as in the works of Kobayashi [9], [10]. More specifically, let β > 0 and χ ∈ S such that suppχ̂ ⊂ {|ξ| ≤ 1} and k∈Zd χ̂(ξ − βk) = 1, ∀ξ ∈ Rd. For 0 < p, q ≤ ∞ and s > 0, the modulation space Mp,q0,s is the set of all tempered distributions f such that |f ∗ (Mβkχ)(x)|p dx < βk >sq NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 3 When, 1 ≤ p, q ≤ ∞ this is an equivalent norm on Mp,q0,s, but when 0 < p, q < 1 this is just a quasi-norm. We refer to [9] for more details. For another definition of the modulation spaces for all 0 < p, q ≤ ∞ we refer to [5, 15]. For a discussion of the cases when p and/or q = 0, see [4]. These extensions of modulation spaces have recently been rediscovered and many of their known properties reproved via different methods by Baoxiang et all [1], [2]. There exist several embedding results between Lebesgue, Sobolev, or Besov spaces and modulation spaces, see for example [11], [13]; also [1], [2]. We note, in particular, that the Sobolev space H2s coincides with M For further properties and uses of modulation spaces, the interested reader is referred to Gröchenig’s book [8]. The goal of this note is two fold: to improve some recent results of Baoxiang, Lifeng and Boling [1] on the local well-posedness of nonlinear equations stated above, by allowing the Cauchy data to lie in any modulation space Mp,10,s, p > dd+1 , s ≥ 0, and to simplify the methods of proof by employing well-established tools from time- frequency analysis. Ideally, one would like to adapt these methods to deal with global well-posedness as well. We plan to address these issues in a future work. In what follows, we assume that d ≥ 1, k ∈ N, d < p ≤ ∞, λ, ρ ∈ R and s ≥ 0 are given. With pk and eρ defined by (4) and (5) respectively, our main results are the following. Theorem 1. Assume that u0 ∈ Mp,10,s(Rd) and f ∈ {pk, eρ}. Then, there exists T ∗ = T ∗(‖u0‖Mp,10,s) such that (1) has a unique solution u ∈ C([0, T ∗],Mp,10,s(Rd)). Moreover, if T ∗ < ∞, then lim sup t→T ∗ ‖u(·, t)‖Mp,10,s = ∞. Theorem 2. Assume that u0, u1 ∈ Mp,10,s(Rd) and f ∈ {pk, eρ}. Then, there exists T ∗ = T ∗(‖u0‖Mp,10,s , ‖u1‖Mp,10,s) such that (2) has a unique solution u ∈ C([0, T ∗],Mp,10,s(Rd)). Moreover, if T ∗ < ∞, then lim sup t→T ∗ ‖u(·, t)‖Mp,10,s = ∞. Theorem 3. Assume that u0, u1 ∈ Mp,10,s(Rd) and f ∈ {pk, eρ}. Then, there exists T ∗ = T ∗(‖u0‖Mp,10,s , ‖u1‖Mp,10,s) such that (3) has a unique solution u ∈ C([0, T ∗],Mp,10,s(Rd)). Moreover, if T ∗ < ∞, then lim sup t→T ∗ ‖u(·, t)‖Mp,10,s = ∞. Remark 1. In Theorem 1 we can replace the (NLS) equation with the following more general (NLS) type equation: (7) (NLS)α i +∆α/2x u+ f(u) = 0, u(x, 0) = u0(x), for any α ∈ [0, 2] and p ≥ 1. The operator ∆α/2x is interpreted as a Fourier multiplier operator (with t fixed), ∆̂ x u(ξ, t) = |ξ|αû(ξ, t). This strengthening will become evident from the preliminary Lemma 1 of the next section. Remark 2. Theorems 1.1 and 1.2 of [1] are particular cases of Theorem 1 with p = 2 and s = 0. 4 Á. Bényi and K. A. Okoudjou 2. Fourier multipliers and multilinear estimates The generic scheme in the local existence theory is to establish linear and nonlinear estimates on appropriate spaces that contain the solution u. As indicated by the main theorems above, the spaces we consider here areMp,10,s, and we present the appropriate estimates in the lemmas below. In fact, we will need estimates on Fourier multipliers on modulation spaces. As proved in [3] and [7], a function σ(ξ) is a symbol of a bounded Fourier multiplier on Mp,q for 1 ≤ p, q ≤ ∞ if σ ∈ W (FL1, ℓ∞) (see the proofs of the following two lemmas for a definition of this space). As we shall indicate below, this condition can be naturally extended to give a sufficient criterion for the boundedness of the Fourier multiplier operator on Mp,q0,s for 0 < p, q ≤ ∞ and s ≥ 0. The notation A . B stands for A ≤ cB for some positive constant c independent of A and B. Lemma 1. Let σ be a function defined on Rd and consider the Fourier multiplier operator Hσ defined by Hσf(x) = σ(ξ) f̂(ξ) e2πξ·x dξ. Let χ ∈ S such that supp χ̂ ⊂ {|ξ| ≤ 1}. Let d ≥ 1, s ≥ 0, 0 < q ≤ ∞, and 0 < p < 1. If σ ∈ W (FLp, ℓ∞)(Rd), i.e., ‖σ‖W (FLp,ℓ∞) = sup ‖σ · Tβnχ‖FLp < ∞ for β > 0, then Hσ extends to a bounded operator on Mp,q0,s(Rd). Proof. We use the definition of the modulation spaces given by (6) (see also [9]). In particular, let χ ∈ S such that supp χ̂ ⊂ {|ξ| ≤ 1}, and define g ∈ S by ĝ = χ̂2. Denote g̃(x) = g(−x). For f ∈ S, β > 0, k ∈ Zd and x ∈ Rd we have: |Hσf ∗ (Mβkg̃)(x)| = |VgHσf(x, βk)| = |〈σf̂,M−xTβkĝ〉| = |〈σf̂,M−xTβkχ̂2〉| ≤ |F−1(σ · Tβkχ̂)| ∗ |F−1(f̂ · Tβkχ̂)|(x) ≤ |F−1(σ · Tβkχ̂)| ∗ |f ∗ (Mβkχ̃)|(x). Now, observe that supp σ · Tβkχ̂ ⊂ Γk := βk + {|ξ| ≤ 1} and supp f̂ · Tβkχ̂ ⊂ Γk. Moreover, by assumption we know that σ ∈ W (FLp, ℓ∞) and so F−1(σ · Tβkχ̂) ∈ Lp and f ∗(Mβkχ̃) ∈ Lp. Consequently, by [9, Lemma 2.6] we have the following estimate ‖Hσf ∗ (Mβkg̃)‖Lp ≤ C ‖F−1(σ · Tβkχ̂)‖Lp‖f ∗ (Mβkχ̃)‖Lp, where C is a positive constant that depends only on the diameter of Γk and p. Clearly, the diameter of Γk is independent of k, and this makes C a constant depending only on the dimension d and the exponent p. Therefore, for 0 < q ≤ ∞ we have ‖Hσf‖Mp,q0,s . sup ‖F−1(σ · Tβkχ̂)‖Lp ‖f‖Mp,q0,s = ‖σ‖W (FLp,ℓ∞) ‖f‖Mp,q0,s . NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 5 The result then follows from the density of S in Mp,q0,s for p, q < ∞; see [9, Theorem 3.10]. � We are now ready to state and prove the boundedness of Fourier multipliers that will be needed in establishing our main results. Lemma 2. Let d ≥ 1, s ≥ 0, and 0 < q ≤ ∞ be given. Define mα(ξ) = ei|ξ| 1 ≤ p ≤ ∞ and α ∈ [0, 2], then the Fourier multiplier operator Hmα extends to a bounded operator on Mp,q0,s(Rd). Moreover, If α ∈ {1, 2} and d < p ≤ ∞, then the Fourier multiplier operator Hmα extends to a bounded operator on M 0,s(R Proof. First, we prove the result when 1 ≤ p ≤ ∞, and 0 < q ≤ ∞. Let g ∈ S(Rd) and define χ ∈ S by χ̂ = g2. For f ∈ S, we have |VχHmαf(x, ξ)| mα(t) f̂(t) e 2πix·t χ̂(t− ξ) dt mα(t) Tξg(t) < t > s f̂(t) <ξ> <t>s <t−ξ>N < t− ξ > N g(t− ξ) e2πix·t dt mα(t) Tξg(t)φN(ξ, t) ̂< D >s f(t) TξgN(t) dt ∣∣∣∣F mα · Tξg φN(ξ, ·) ̂< D >s f TξgN ∣∣∣∣F(mα · Tξg) ∗ F2(φN(ξ, ·)) ∗ F( ̂< D >s f · TξgN)(−x) ∣∣∣∣, where N > 0 is an integer to be chosen later, gN(t) =< t > N g(t), φN(ξ, t) = <t>s <t−ξ>N , and < D > s is the Fourier multiplier defined by ̂< D >s f(ξ) =< ξ >s f̂(ξ). We also denote by Φ2,N (ξ, ·) := F2(φN(ξ, ·)) the Fourier transform in the second variable of φN(ξ, ·) 6 Á. Bényi and K. A. Okoudjou We can therefore estimate the weighted modulation norm of Hmαf as follows: ‖Hmαf‖Mp,q0,s |Vχf(x, ξ)|p dx < ξ >qs ∣∣∣∣F(mα · Tξg) ∗ Φ2,N (ξ, ·) ∗ F( ̂< D >s f · TξgN)(−x) ‖F−1(mα · Tξg)‖qL1 ‖Φ2,N(ξ, ·)‖ ‖F( ̂< D >s f · TξgN)‖qLp dξ ≤ sup ‖F−1(mα · Tξg)‖L1 sup ‖Φ2,N(ξ, ·)‖L1 ‖F−1( ̂< D >s f · TξgN)‖qLp dξ ≤ sup ‖F(mα · Tξg)‖L1 sup ‖Φ2,N (ξ, ·)‖L1 ‖f‖Mp,q0,s . Now, it follows from [3, Lemma 8] that, for α ∈ [0, 2], ‖F−1(mα · Tξg)‖L1 := ‖mα‖W (FL1,ℓ∞) < ∞. Moreover (see, for example, [13, Lemma 3.1] or [14, Lemma 2.1]), we can select a sufficiently large N > 0 such that ‖Φ2,N (ξ, ·)‖L1 ≤ |Φ2,N (ξ, x))|dx < ∞. Hence, using (8), we get ‖Hmαf‖Mp,q0,s ≤ Cα‖f‖Mp,q0,s . To prove the second part of the result we shall use Lemma 1. In particular, we need to show that for α ∈ {1, 2} and d < p < 1, mα ∈ W (FLp, ℓ∞). This, however, follows by straightforward adaptations of the proofs of [3, Theorems 9 and 11], which we leave to the interested reader. � In analogy to the proof of the previous lemma, we can prove the following weighted version of [3, Theorem 16]. Lemma 3. Let d ≥ 1, s ≥ 0, d < p ≤ ∞ and 0 < q ≤ ∞ be given, and let m(1)(ξ) = sin(|ξ|) |ξ| and m (2)(ξ) = cos(|ξ|), for ξ ∈ Rd. Then, the Fourier multiplier operators Hm(1) , Hm(2) can be extended as bounded operators on M A “smooth” version of Lemma 3 is obtained by replacing |ξ| with < ξ >. Lemma 4. Let d ≥ 1, s ≥ 0, d < p ≤ ∞ and 0 < q ≤ ∞ be given, and let m(ξ) = ei<ξ>, m(1)(ξ) = sin(<ξ>) and m(2)(ξ) = cos(< ξ >), for ξ ∈ Rd. Then, the Fourier multiplier operators Hm, Hm(1), Hm(2) can be extended as bounded operators on Mp,q0,s. NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 7 Proof. It is clear that m,m(1), m(2) are C∞(Rd) functions and that all their derivatives are bounded. Therefore, m,m(1), m(2) ∈ Cd+1(Rd) ⊂ M∞,1(Rd) ⊂ W (FL1, ℓ∞)(Rd) [8, 11]. Thus, for 1 ≤ p ≤ ∞, and 0 < q ≤ ∞ the result follows from [3] and Lemma 2. For d < p < 1 and 0 < q ≤ ∞, it can be showed that m,m(1), m(2) ∈ Cd+1(Rd) ⊂ W (FLp, ℓ∞)(Rd). Indeed, this follows from obvious modifications to the proof of the embedding Cd+1(Rd) ⊂ M∞,1(Rd) ⊂ W (FL1, ℓ∞)(Rd) [8, 11]. Furthermore, if we modify, for example, the multiplier m to mt(ξ) = e it<ξ>, t ∈ R, we have for < p ≤ 1 (9) ‖mt‖W (FLp,ℓ∞) ≤ (1 + |t|)d+1, and similar estimates hold for modified multipliers m t and m t . � Finally, we state a crucial multilinear estimate that will be used in our proofs. Although the estimate will be needed only in the particular case of a product of functions (see Corollary 1), we present it here in its full generality that applies to multilinear pseudodifferential operators. An m-linear pseudodifferential operator is defined à priori through its (distribu- tional) symbol σ to be the mapping Tσ from the m-fold product of Schwartz spaces S × · · · × S into the space S ′ of tempered distributions given by the formula Tσ(u1, . . . , um)(x) σ(x, ξ1, . . . , ξm) û1(ξ1) · · · ûm(ξm) e2πix·(ξ1+···+ξm) dξ1 · · · dξm,(10) for u1, . . . , um ∈ S. The pointwise product u1 · · ·um corresponds to the case σ = 1. Lemma 5. If σ ∈ M∞,10,s (R(m+1)d), then the m-linear pseudodifferential operator Tσ defined by (10) extends to a bounded operator from Mp1,q10,s ×· · ·×M pm,qm 0,s into M p0,q0 when 1 + · · ·+ 1 + · · ·+ 1 = m − 1 + 1 , and 0 < pi ≤ ∞, 1 ≤ qi ≤ ∞ for 0 ≤ i ≤ m. This result is a slight modification of [4, Theorem 3.1]. Its proof proceeds along the same lines, and therefore it is omitted here. Note that if σ ∈ M∞,10,s , and we pick u1 = · · · = um = u (some of them could be equal to ū since the modulation norm is preserved), p1 = · · · = pm = mp, 0 < p ≤ ∞, and q1 = · · · = qm = 1 we have (11) ‖Tσ(u, . . . , u)‖Mp,10,s . ‖u‖ Mmp,10,s . ‖u‖mMp,10,s , where we used the obvious embedding Mp,10,s ⊆ M 0,s . The notation A . B stands for A ≤ cB for some positive constant c independent of A and B. In particular, if we select σ = 1 (the constant function 1), then σ ∈ M∞,10,s ⊂ M∞,1, and we obtain Corollary 1. Let 0 < p ≤ ∞. If u ∈ Mp,10,s, then um ∈ M 0,s. Furthermore, ‖um‖Mp,10,s . ‖u‖ Mp,10,s 8 Á. Bényi and K. A. Okoudjou This is of course just a particular case of the more general multilinear estimate Mp0,q00,s ‖ui‖Mpi,qi0,s , where the exponents satisfy the same relations as in Lemma 1. When we consider the power nonlinearity f(u) = pk(u) = λ|u|2ku = λuk+1ūk, Corollary 1 becomes Corollary 2. Let 0 < p ≤ ∞. If u ∈ Mp,10,s, then pk(u) ∈ M 0,s. Furthermore, ‖pk(u)‖Mp,10,s . ‖u‖ Mp,10,s For a different proof of the estimate in Corollary 2, see [1, Corollary 4.2]. It is important to note that the previous estimate allows us to control the exponential nonlinearity eρ as well. Indeed, since eρ(u) = λ(e ρ|u|2 − 1)u = pk(u), if we now apply the modulation norm on both sides and use the triangle inequality, we arrive at Corollary 3. Let 0 < p ≤ ∞. If u ∈ Mp,10,s, then eρ(u) ∈ M 0,s. Furthermore, ‖eρ(u)‖Mp,10,s . ‖u‖Mp,10,s(e |ρ|‖u‖2 0,s − 1). 3. Proofs of the main results We are now ready to proceed with the proofs of our main theorems. We will only prove our results for the power nonlinearities f = pk, by making use of Corollary 2. The case of exponential nonlinearity f = eρ is treated similarly, by now employing Corollary 3. In all that follows we assume that u : [0, T )×Rd → C where 0 < T ≤ ∞ and that f(u) = pk(u) = λ|u|2ku. 3.1. The nonlinear Schrödinger equation: Proof of Theorem 1. We start by noting that (1) can be written in the equivalent form (13) u(·, t) = S(t)u0 − iAf(u) where (14) S(t) = eit∆, A = S(t− τ) · dτ. Consider now the mapping J u = S(t)u0 − i S(t− τ)(pk(u))(τ) dτ. It follows from Lemma 2 (see also [3, Corollary 18]) that ‖S(t)u0‖Mp,10,s ≤ C (t 2 + 4π2)d/4 ‖u0‖Mp,10,s , NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 9 where C is a universal constant depending only on d. Therefore, (15) ‖S(t)u0‖Mp,10,s ≤ CT ‖u0‖Mp,10,s , where CT = sup t∈[0,T ) C (t2 + 4π2)d/4. Moreover, we have S(t− τ)(pk(u))(τ) dτ Mp,10,s ‖S(t− τ)(pk(u))(τ)‖Mp,10,s dτ ≤ T CT sup t∈[0,T ] ‖pk(u)(t)‖Mp,10,s .(16) By using now Corollary 2, we can further estimate in (16) to get S(t− τ)(pk(u))(τ) dτ Mp,10,s . CT T ‖u(t)‖2k+1Mp,10,s . Consequently, using (15) and (17) we have (18) ‖J u‖ C([0,T ],Mp,10,s) ≤ CT (‖u0‖Mp,10,s + cT ‖u‖ Mp,10,s for some universal positive constant c. We are now in the position of using a stan- dard contraction argument to arrive to our result. For completeness, we sketch it here. Let BM denote the closed ball of radius M centered at the origin in the space C([0, T ],Mp,10,s). We claim that J : BM → BM , for a carefully chosen M . Indeed, if we let M = 2CT‖u0‖Mp,10,s and u ∈ BM , from (18) we obtain ‖J u‖ C([0,T ],Mp,10,s) + cCTTM 2k+1. Now let T be such that cCTTM 2k ≤ 1/2, that is, T ≤ T̃ (‖u0‖Mp,10,s ). We obtain ‖J u‖ C([0,T ],Mp,10,s) that is J u ∈ BM . Furthermore, a similar argument gives ‖J u− J v‖ C([0,T ],Mp,10,s) ‖u− v‖ C([0,T ],Mp,10,s) This last estimate follows in particular from the following fact: pk(u)(τ)− pk(v)(τ) = λ(u− v)|u|2k(τ) + λv(|u|2k − |v|2k)(τ). Therefore, using Banach’s contraction mapping principle, we conclude that J has a fixed point in BM which is a solution of (13); this solution can be now extended up to a maximal time T ∗(‖u0‖Mp,10,s ). The proof is complete. 10 Á. Bényi and K. A. Okoudjou 3.2. The nonlinear wave equation: Proof of Theorem 2. Equation (2) can be written in the equivalent form (19) u(·, t) = K̃(t)u0 +K(t)u1 − Bf(u) where (20) K(t) = sin(t −∆ , K̃(t) = cos(t −∆), B = K(t− τ) · dτ Consider the mapping J u = K̃(t)u0 +K(t)u1 − Bf(u). Recall that f = pk. If we now use Lemma 3 (see also [3, Corollary 21]) for the first two inequalities below and Corollary 2 for the last estimate, we can write ‖K̃(t)u0‖Mp,10,s ≤ CT‖u0‖Mp,10,s , ‖K(t)u1‖Mp,10,s ≤ CT‖u1‖Mp,10,s , ‖Bf(u)‖Mp,10,s ≤ cT CT‖u‖ Mp,10,s where c is some universal positive constant. The constants T and CT have the same meaning as before. The standard contraction mapping argument applied to J completes the proof. 3.3. The nonlinear Klein-Gordon equation: Proof of Theorem 3. The equiv- alent form of equation (3) is (22) u(·, t) = K̃(t)u0 +K(t)u1 + Cf(u) where now (23) K(t) = sin t(I−∆)1/2 (I−∆)1/2 , K̃(t) = cos t(I −∆) 1/2, C = K(t− τ) · dτ. Consider the mapping J u = K̃(t)u0 +K(t)u1 + Cf(u). Using Lemma 4 and the notations above, we can write ‖K̃(t)u0‖Mp,10,s ≤ CT‖u0‖Mp,10,s , ‖K(t)u1‖Mp,10,s ≤ CT‖u1‖Mp,10,s , ‖Cf(u)‖Mp,10,s ≤ cT CT‖u‖ Mp,10,s The standard contraction mapping argument applied to J completes the proof. NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 11 References [1] W. Baoxiang, Z. Lifeng, and G. Boling, Isometric decomposition operators, function spaces Eλp,q and applications to nonlinear evolution equations, J. Funct. Anal. 233 (2006), no. 1, 1–39. [2] W. Baoxiang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Diff. Equations 232 (2007), 36–73. [3] Á. Bényi, K. Gröchenig, K. A. Okoudjou, and L. G. Rogers, Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal. (2007), to appear. [4] Á. Bényi, K. Gröchenig, C. Heil, and K. Okoudjou, Modulation spaces and a class of bounded multilinear pseudodifferential operators, J. Operator Theory 54 (2005), no. 2, 387–399. [5] Y. V. Galperin, and S. Samarah, Time-frequency analysis on modulation spaces Mp,qm , 0 < p, q ≤ ∞, Appl. Comput. Harmon. Anal., 16 (2004), 1–18. [6] H. G. Feichtinger, Modulation spaces on locally Abelian groups, in: “ Proc. Internat. Conf. on Wavelets and Applications” (Radha, R.;Krishna, M.;Thangavelu, S. eds.), New Delhi Allied Publishers (2003), 1–56. [7] H. G. Feichtinger and G. Narimani, Fourier multipliers of classical modulation spaces, Appl. Comput. Harmon. Anal. 21 (2006), no. 3, 349–359. [8] K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston MA, 2001. [9] M. Kobayashi, Modulation spaces Mp,q for 0 < p, q ≤ ∞, J. Func. Spaces Appl. 4 (2006), no. 2, 329–341. [10] M. Kobayashi, Dual of modulation spaces, J. Func. Spaces Appl., to appear. [11] K. A. Okoudjou, Embeddings of some classical Banach spaces into the modulation spaces, Proc. Amer. Math. Soc., 132 (2004), no. 6, 1639–1647. [12] T. Tao, Nonlinear dispersive equations: Local and global analysis, CBMS Regional Conference Series in Mathematics, no. 106, American Mathematical Society, 2006 [13] J. Toft, Convolutions and embeddings for weighted modulation spaces, Advances in pseudo- differential operators, 165–186, Oper. Theory Adv. Appl. 155, Birkhauser, Basel, 2004. [14] J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus, II, Ann. Global Anal. Geom. 26 (2004), no. 1, 73–106. [15] H. Triebel, Modulation spaces on the euclidean n−space, Z. Anal. Anwendungen, 2 (1983), no. 5, 443–457. Árpád Bényi, Department of Mathematics, 516 High Street, Western Washington University, Bellingham, WA 98225, USA E-mail address : [email protected] Kasso A. Okoudjou, Department of Mathematics, University of Maryland, Col- lege Park, MD 20742, USA E-mail address : [email protected] 1. Introduction and statement of results 2. Fourier multipliers and multilinear estimates 3. Proofs of the main results 3.1. The nonlinear Schrödinger equation: Proof of Theorem ?? 3.2. The nonlinear wave equation: Proof of Theorem ?? 3.3. The nonlinear Klein-Gordon equation: Proof of Theorem ?? References

0704.0834

P-adic arithmetic coding

Microsoft Word - P_v0_28.doc 4/5/2007 P-adic arithmetic coding. Anatoly Rodionov, Sergey Volkov Spectrum Systems, Inc Abstract A new incremental algorithm for data compression is presented. For a sequence of input symbols algorithm incrementally constructs a p-adic integer number as an output. Decoding process starts with less significant part of a p-adic integer and incrementally reconstructs a sequence of input symbols. Algorithm is based on certain features of p-adic numbers and p-adic norm. p-adic coding algorithm may be considered as of generalization a popular compression technique – arithmetic coding algorithms. It is shown that for p = 2 the algorithm works as integer variant of arithmetic coding; for a special class of models it gives exactly the same codes as Huffman’s algorithm, for another special model and a specific alphabet it gives Golomb-Rice codes. “For more than forty years I've been speaking in prose without even knowing it!” Moliere. Le Bourgeois Gentilhomme. Introduction Arithmetic coding algorithm in its modern version was published in Communications of ACM in June 1987 [Witten], but the authors, Ian Witten, Radford Neal and John Cleary, referred to [Abrahamson] as to “the first reference to what was to become the method of arithmetic coding”. So we may say that it is known “for more than forty years”. The algorithm now is a common knowledge – it was published in numerous textbooks (see for example [Salomon, Sayood]), some reviews were published [Bodden, Said], Dr. Dobb’s Journal popularized it [Nelson], wiki [wiki] contains an article about it, a lot of sources could be found on web... So why one more paper on this subject and what is this “p-adic arithmetic”? Let go back to the original idea of arithmetic coding. In arithmetic coding a message is represented as a subinterval [b, e) of union semi interval [0, 1). (We will give all definitions later) When a new symbol s comes a new subinterval [b(s), e(s)) of [b, e) is constructed. Common method to calculate a new subinterval is to divide a current interval into |A| (A is an alphabet, |A|- number of symbols) subintervals, each subinterval represents a symbol from A and has length proportional to probability of this symbol. For a new symbol s corresponding subinterval [b(s), e(s)) will be return by encoder. Thus encoding is a process of narrowing intervals (we will call them message intervals) starting from the union interval: [0, 1) ≡ [b0, e0), [b1, e1), [b2, e2), … , [bt, et) where 0 = b0, ≤ b1 ≤ b2 ≤ … ≤ bt 1 = e0, ≥ e 1 ≥ e 2 ≥ … ≥ e t All bi and ei are real numbers. A last constructed subinterval may be used as a final output, or any point x from last subinterval and message length. But usually a special symbol EOM (End Of Message), which does not belong to the alphabet, is used as termination symbol of a message. In this case only a point x can be used as coding result. Decoding is also a process on narrowing intervals. It starts with union interval and a point x inside it. Decoder finds a symbol by dividing current intervals into |A| subintervals and finds the one that contains point x, say [b1, e1). Corresponding to this interval symbol s1 is pushed into an output buffer; [b1, e1) is used as a new current interval. And so on until EOM symbol is received. 4/5/2007 But here is a problem – one have to use infinite precision real numbers to implement this algorithm and there is no such a thing like effective infinite precision real arithmetic. This problem was always considered as a technical one. Solution is simple - just use integers instead. There is a canonical implementation, first written in C [Witten], which was later reproduced in other languages, but no analysis of what happens to the algorithm after moving it from the field of real numbers to the ring of integer numbers was published. In this paper we introduce p-adic arithmetic coding which is based on mapping a message to a path on a p- tree (a tree with p outgoing branches at each vertex; we also assume that p is a prime number). This path is constructed as a common part of paths to the left and right edges of a subinterval [gl(s), gr(s)), where gl(s) and gr(s) are from a special equidistant grid G on [0, 1). This semi interval is constructed according to the same rules, as in real number arithmetic coding, but in contrast to it, the edges are not arbitrary real numbers, but belong to the grid. A path on a p-tree can be naturally presented as a p-adic integer number. p-adic distance proved to be a natural measure on paths – the longer a common part of two paths, the smaller p-adic distance between them. Function ordP, also known as p-adic logarithm, gives length of a common path. A path can be also identified by its final point on a grid. A grid point g can be represented by an integer index k from a finite integer ring as g=k*|G|-1(here |G| is number of elements in G). The crucial point of this algorithm is how we can calculate a path from an index and vice versa – an index from a path. IP (Index-Path) mapping, described in this article, presents an elegant and efficient way for this. Now we ready to give a brief sketch of how does the algorithm work. As initial step we have to define an alphabet A, a model M, an output buffer (it will contain a p-adic integer number B) and a grid G on [0, 1). Start with union coding semi interval represented by two indexes l=0 (left) and r=0 (right), and B=0. When a new symbol s comes, the model calculates a new subinterval [l(s), r(s)) (l and r are indexes from a finite integer ring, while l^ and r^ – paths presented as p-adic integer numbers). Using IP transformation (we use symbol ^ for this transformation) we can calculate p-adic representation of paths to these edges l(s)^ and r(s)^. If p-adic distance between them is equal to 1, we continue encoding using [l(s), r(s)) as a new current encoding interval. If the distance is less then 1, then l(s)^ and r(s)^ have a common path of length c=ordP(l(s)^, r(s)^). That means that path to any point inside [l(s), r(s)) have the same first (least significant) c digits as l(s)^. We can push this common path to an output buffer adding them as new most significant part of p-adic number B. We can also drop c least significant digit of l(s)^ and r(s)^. Both of these operations are possible, because p-adic numbers are read from left to right, i.e. less significant digit (those that are multiplied by less powers of p) are in the left part of buffer. This feature of p-adic integer numbers explains why p-adic arithmetic coding and decoding are incremental. Now we can continue encoding with truncated l(s)^ and r(s)^. To do this we must calculate new subinterval, corresponding to new paths. This also can be done using IP transformation. Encoding will continue using [(l(s)^)^, (r(s)^)^) as new current message interval from some grid. This procedure we will call PR rescaling. In the case of p=2 this is procedure is similar to well known E1/E2 rescaling [Bodden]. But PR rescaling gives a better insight of this mechanism, connects it with p-adic norm and can be used for any prime p. Moreover, PR rescaling is more accurate on boundaries and because of this the algorithm is able to reproduce Huffman codes for certain models. We will also generalize E3 rescaling, which is based on usual Archimedean norm (absolute value in this case), for any prime p. p-adic arithmetic coding algorithm generalizes not only arithmetic coding. For a special class of models, p- adic coding algorithm works exactly as Huffman’s algorithm [Huffman]. In this models weights of all symbols should be equal to p-n, where n some positive numbers and a sum of all weight is equal to 1. In other words, they are leaves of a Huffman code tree. For a special model and one symbol alphabet p-adic arithmetic coding reproduce Golomb-Rice codes [Golomb, Rice]. 4/5/2007 Definitions Alphabet Alphabet A - A non empty set of symbols ai. |A| - number of symbols in A. In most examples below 4 symbols alphabet [a, b, c, d] will be used. Other examples: binary alphabet [0, 1], 128 characters ASCII, alphabet of 256 different eight-bit characters. The last one is used in all tests. Even an alphabet containing only one symbol makes sense – as it will be shown, the algorithm creates exactly Golomb-Rice [Golomb, Rice] codes in this case. Message Message M – a sequence of symbols from alphabet A. M = (ao, a1, … , ai, … ,an) where ai belong to A. Example: (a, b, a, a, b, c, d, a). Semi interval [l, r) – includes l, but not r. Notation [,) means that the left point is included to the interval, while the right one is not. Below we always deal with subintervals of [0, 1). Grid Later we will subdivide [0, 1) into PN (P and N are natural numbers) semi intervals of equal length. Each has length P-N and can be identified by its left edge. These left edge points form a grid G(PN). Coordinate of a point of the grid with index k is evidently kP-N. We will use notation gk(N) for points from G(P Picture 1. Grid P2 (P=2) These indexes will play an important role in our discussion. If fact, all calculations will be done using indexes. The range of indexes is 0 ≤ k < PN In other words, indexes are nonnegative numbers modulo PN. Negative numbers are defined in this ring as -k = PN - k Weight interval Following the main idea of arithmetic coding let map alphabet A to a semi interval [0, 1), which we will refer as a weight interval. To do this enumerate symbols from A in any order (the order is not important in a sense that compression rate does not depend on it) and divide the interval in |A| semi intervals. Semi interval [wi, wi+1) corresponds to symbol ai. To make compression effective lengths of these intervals must be equal to probability of symbols in a message: | wi - wi+1 | = pi where pi – probability of symbol ai. 4/5/2007 Picture 2. Weight interval To define weight intervals we will use notation: { symbol0:[semininterval1), … , symbol|A|-1:[semininterval|A|-1) } For example: {a:[0, 0.5), b:[0.5, 0.75), c:[0.75, 0.875), d:[0.875,1)} Message interval Let fix a natural number N, a prime number P and create a grid G(PN). Messages will be mapped to semi intervals [l, r) of this interval. 0 ≤ l < r < 1 l, r belongs to G(PN). Arithmetic coding is just a process of narrowing a message interval. When a new symbol comes, a current message interval is divided in |A| subintervals proportional to weight interval and then a subinterval corresponding to a new symbol is selected as a new message interval. Thus starting with [0, 1) (empty message) interval we end up with a subinterval corresponding to the whole message. For example let see how a short message {a, b, a} may be coded (here we use weight interval from previous section): Picture 3. Message interval We may also present this in a table (using actual values of gk(N)): 4/5/2007 Message Semi interval Numerical value {} [ g0(0) , g0(0) ) [ 0 , 1 ) {a} [ g0(1) , g1(1) ) [ 0 , 1/2 ) {a, b} [ g2(3) , g3(3) ) [ 2/4 , 3/8 ) {a, b, a} [ g4(4) , g5(4) ) [ 4/8 , 5/8 ) An important difference from the original idea of real number arithmetic coding is that here we use only points from a grid as subintervals edges. Coding tree Consider a tower of grids G(P0) < G(P1) < G(P2) < … < G(Pn) … By construction, if a point belongs to G(Pn), then it belongs to G(Pn+1), G(Pn+2),…., G(Pk) (k>n). G(P0) consist only of one point. Now let construct a coding tree. Start with a root – which is evidently the only point from G(P0) - g0(0) . Then comes a first level – points from G(P1). Link the root g0(0) with points from G(P 1): g0(1), g1(1), … , gP-1(1). This gives us first level of the coding tree. Now we can continue. Let us assume that the tree build up to a level N. To create a new N+1 level, we have to: Construct a new grid G(Pn+1) as a new bottom level to the bottom (us usual, tree grows downwards). Link points from the last level (i.e. points from G(Pn)) to points from G(P n+1) according to the following rule: gk(n) link to points gk*P(n+1), g(k+1)*P(n+1), …, g(k+P-1)*P (n+1) 0 11/21/4 3/4 g0(1) g1(1) g0(2) g1(2) g2(2) g3(2) g0(3) g2(3) g4(3)g3(3) g5(3) g6(3) g7(3) g0(4) g1(4) g2(4) g3(4) g4(4) g5(4) g6(4) g7(4) g8(4) g9(4) g10(4) g11(4) g12(4) g13(4) g14(4) g15(4) g1(3) g0(0) Picture 4. Coding tree Here, as in most other illustrations, we use P=2 to simplify drawing. Now we can use the grid and the tree to code a simple message {a, b, a} using weights {a:[0, 1/2), b:[1/2, 3/4), c:[3/4, 7/8), d:[ 7/8, 1)} 4/5/2007 Picture 5. Paths on coding tree Message Semi interval Path {} [ g0(0) , g0(0) ) [ { g0(0) }, { g0(0) } ) {a} [ g0(1) , g1(1) ) [ { g0(0) , g0(1) } , { g0(0) , g1(1) } ) {a, b} [ g2(3) , g3(3) ) [ { g0(0), g0(1) , g1(2) , g2(3) }, { g0(0), g0(1) , g1(2) , g3(3) } ) {a, b, a} [ g4(4) , g5(4) ) [ { g0(0), g0(1) ,g1(2) ,g2(3),g4(4)}, { g0(0),g0(1) ,g1(2) ,g2(3),g5(4) } ) We need a more convenient way to refer to grid points and tree paths. Grid points can be easily represented as indexes, i.e. well known positive integer numbers, while for paths we will use p-adic integer numbers. Representation of paths as p-adic integer numbers From any point gk(n) we have P different links to a next (n+1) level. We can mark our next choice with a nonnegative integer number m 0 ≤ mj < P 0 ≤ j ≤ n Now we can represent any (final) paths on the coding tree as a vector M = {m0, m1, m2, … , mn} This vector can be mapped to a nonnegative number x x= m0P + m1P 1 + m2P 2+ … + mnP This mapping is evidently one to one. The number x may be considered as a p-adic integer number. These numbers are not well known among programmers. One can find an introduction in p-adic numbers in [Baker, Koblitz]. A very helpful way to visualize some unusual properties of p-adic mathematic may be found in [Holly]. The first coefficient m0 tells us to what top level subinterval of [0, 1) the point belongs. The next one m1 – to which subinterval of this interval the point belongs, and so on. 4/5/2007 Picture 6. p-adic representation of binary tree. Level Paths 0 {} 1 Paths {0} {1} p-adic number 0 1 2 Paths {0,0} {0,1} {1,0} {1,1} p-adic number 0 2 1 3 3 Path {0,0,0} {0,0,1} {0,1,0} {0,1,1} {1,0,0} {1,0,1} {1,1,0} {1,1,1} p-adic number 0 4 2 6 1 5 3 7 A tree for P=3 is shown in the next picture Picture 6. p-adic representation of a tree; P=3. 4/5/2007 Level Paths 0 {} 1 Paths {0} {1} {2} p-adic number 0 1 2 2 Paths {0,0} {0,1} {0,2 } {1,0} {1,1} {1,2} {2,0} {2,1} {2,2} p-adic number 0 3 6 1 4 7 2 5 8 In our algorithm we will use p-adic distance. This distance can be defined with the help of p-adic logarithm function, usually called as ordP. ordP(x) = max number r such that (x % P r) = 0; x ≠ 0 p-adic norm is |x|P = 1 / P**ordP(x) x ≠ 0 |0| = 0. and distance dP(x,y) = |x – y|P It can be shown [Koblitz] these are “real” norm and distance, i.e. all three axioms are valid for them. For two paths x and y x= m0P + m1P 1 + m2P 2+ … + mnP y= k0P + k1P 1 + k2P 2+ … + knP ordP(x-y) gives a number of common links. dP(x,y) have a very intuitive meaning – the greater number of common links have two paths, the closer they are in terms of p-adic distance. Index representation Now return to the first method of mapping paths – by end points. An end point belongs to a grid G(Pn), so it is defined by its index a which is just a plain nonnegative integer number. How this index is connected to the paths leading to this point? Let x be a path x= m0P + m1P 1 + m2P 2+ … + mnP 0 ≤ mj < P 0 ≤ j ≤ n Then the path x ends at a point g at level n g = m0P + m1P -2 + m2P -3 + … + mnP -n-1 We just negate powers and subtract 1. It can be proved by simple induction. Let a be an index corresponding to point g on the grid. g = aP-n-1 a= m0P + m1P n-1 + m2P n-2+ … + mnP We can rewrite a in usual form a= u0P + u1P 1 + u2P 2+ … + unP If we consider x and a as integer numbers then mapping is just reversing vectors of coefficients. This mapping we will call IP (Index-Path) mapping. 4/5/2007 We will introduce some useful notations in the next section and use this feature intensively in algorithms: we can perform ordinary integer arithmetic operations on indexes calculating new subinterval and immediately get paths to them. Mapping paths to points was considered in a more general form by S.V. Kozyrev [Kozyrev]. Following his notation we will use symbol ρ for it. The ρ mapping has a very important feature |ρ(x) - ρ(y)| ≤ |x – y|P were |ρ(x) - ρ(y)| is Archimedean (our usual) distance which in our case is absolute value. This means that if two paths are close to each other then corresponding end points are also close in our usual Archimedean norm. A proof of this can be found in [Kozyrev]. ρ mapping is not one to one, IP (Index-Path), mapping which deals with finite sums, is one to one mapping. Let, as previously a and g be a= m0P + m1P n-1 + m2P n-2+ … + mnP g = aP-n-1 To what subinterval of level 1 of [0, 1) the point belongs to? It depends on the value of m0 only. Straightforward calculations n-1 + m2P n-2+ … + mnP 0 ≤ (P-1)( P0 + P 1 + P2+ … + Pn-1) = (P-1)((Pn -1)/(P-1)) =(Pn -1) show that a sum of all less significant terms can’t move a point to another subinterval. With m0 fixed, we can conclude that m1 solely defines subinterval inside the second subinterval and so on. Let us continue with the example (with P=2) above by adding indexes of grid points to the table: Level Paths 0 {} 1 Paths {0} {1} p-adic number 0 1 Index 0 1 Points 0 1/2 2 Paths {0,0} {0,1} {1,0} {1,1} p-adic number 0 2 1 3 Index 0 1 2 3 Points 0 1/4 2/4 3/4 3 Paths {0,0,0} {0,0,1} {0,1,0} {0,1,1} {1,0,0} {1,0,1} {1,1,0} {1,1,1} p-adic number 0 4 2 6 1 5 3 7 Index 0 1 2 3 4 5 6 7 Points 0 1/8 2/8 3/8 4/8 5/8 6/8 7/8 As an illustration let us compare Archimedean and p-adic distances for the following three points g2(3) (or {0, 1, 0}), g3(3) ({0,1,1}) and g4(3) ({1, 0, 0}). Archimedean distance we all used to: Points (p-adic) 2 3 4 2 0 1/8 1/4 3 1/8 0 1/8 4 1/4 1/8 0 p-adic logarithm (ord) has values: 4/5/2007 Points (p-adic) 2 6 1 2 2 0 6 2 0 1 0 0 and p-adic distances are: Points (p-adic) 2 6 1 2 0 1/4 1 6 1/4 0 1 1 1 1 0 From this we may observe that the greater is a common path, the closer are points in p-adic norm. Operators ^ and [] Let us define operator ^ to transform points from index representation to path representation, or, in other words, from nonnegative integers modulo PN to p-adic integers, and back, from path to index representation. x = a^ It is convenient to rewrite a and x in form of scalar product. Consider N+1 element vectors MN and PN MN = (m0, m1, m2, … , mN) where 0 ≤ mj < P, 0 ≤ j ≤ N PN = (P 0, P1, P2, … , PN) Then x = m0P + m1P 1 + m2P 2+ … + mNP can be represented as scalar product of two vectors x = (MN • PN T – as usual, means operation of matrix transposition (i.e. changing rows to columns). While a= m0P + m1P N-1 + m2P N-2+ … + mNP a = (MN R • PN T) = (MN • (PN R )T) Here R means reverting elements of a vector. It is obvious that operator ^ is idempotent x^^ = a^ = x The important thing about this trivial operation is that we can perform arithmetic operation on points of a grid, and then immediately find a path to it by applying operator ^, and vice versa, for a given path we can find a corresponding grid point. It is convenient to define operator [] as coefficient in scalar representation: Let x be as previously x = m0P + m1P 1 + m2P 2+ … + mNP Then 4/5/2007 x[i] = mi It is easy to see that x[i] = x^[N-i] Mapping subintervals to paths Now any subinterval can be mapped to a pair of paths on a coding tree, provided that edge points of subintervals belong to some G(PN). We will use notation [ , ] for intervals presented as a pair of indexes, where l ≤ r and [l^, r^] – as a pair of paths and [ , ) for pairs paths to subintervals. A simple fact, just to note: if an interval [l, r] lies inside an interval [l1, r1], then dP(l1^, r1^) ≤ dP(l^, r^) In other words, paths to subinterval’s edges are closer then paths to enveloping interval. So, if an interval’s edges have a common path, then paths to edges of any subinterval have at least the same or even longer common paths. A length of common path can be calculated as ordP(r^ - l^). As an example see Picture 5. Let discuss in more details the rightmost semi interval, i.e. a subinterval which ends at point 1. This point has index equal to PN. Because we are working in the ring on integers numbers mod PN the index is equal to 0 in this ring. So path to 1 has the form {0, 0, … , 0}, an general form of the rightmost interval is [l, 0]. What is a p-adic length of a rightmost interval? By definition dP(l,0) = | 0 – l |P = | –l |P And common length is ordP(-l). What is a “negative” path in our case? Path is always a path to some point on a grid. We use indexes for representing them. So a negative path may be defined as a path to a point, represented by negated index. -l = (-(l^))^ By definition negative numbers in ring mod PN are l^ + -(l^) = 0 mod PN Common paths Consider common part of all paths to points of semi interval [l, r) (l and r are indexes here); both of them belongs to G(PN). All these paths end at corresponding points pi l ≤ pi ≤ r-1 (remember that r does not belong to subinterval). What is a common path to all these point? First consider length of a common path. To find it we may first find maximum p-adic distance among all pairs: max(|p^ - q^|P) ; l ≤ p < q ≤ r-1 We can use ultrametric feature of p-adic norm (see, for example, [Koblitz, Holly]): |x-y|P <= max(|x|P, |y| P) In our case we can use it as (|p^ - q^|P) = |(p^ - l^) + (l^ - q^)|P ≤ max(|(p^ - l^)|P , |(l^ - q^)|P) l ≤ p < q ≤ r-1 So all we need is to find max(|(pi^ - l^)|P) ; l < pi ≤ r-1 which, by construction, is: 4/5/2007 |(r -1)^ - l^|P Now we can calculate length of a common path. Special case l = r – 1 is important but trivial – the length here is simply a length of l. If l ≠ r – 1 then it is equal to ordP((r-1)^ - l^). Because function ordP is not defined for zero argument we introduce a function com, defined on p-adic numbers: comP,N(l, r) = N if ( l == r ) else ordP(r - l) If l, r belongs to G(PN) then length of common path is calculated as comP,N(l^, r^). Finally we have: Paths to points of semi interval [l, r) have a common path of length comP,N(l^, (r-1)^). Common path is a sub path of length comP,N(^, (r-1)^) of l^ starting from root. In the following table examples different intervals of level 2 from Picture 6 and their common paths are shown. l^ r^ l r r-1 l (r-1) - l com2,2( l^, r^ ) Common path {0, 0} {0, 1} 0 1 0 0 0 2 {0, 0} {0, 0} {1, 0} 0 2 1 0 2 1 {0} {0, 0} {1, 1} 0 3 2 0 1 0 {} {0, 0} {0, 0} 0 0 3 0 3 0 {} {0, 1} {1, 0} 1 2 1 2 0 2 {0, 1} {0, 1} {1, 1} 1 3 2 2 3 0 {} {0, 1} {0, 0} 1 0 3 2 1 0 {} {1, 0 } {1, 1} 2 3 2 1 1 2 {1, 0} {1, 0} {0, 0} 2 0 3 1 2 1 {1} {1, 1} {0, 0} 3 0 3 3 0 2 {1, 1} Rescaling based on P-adic distance (PR) Consider two paths x = {0, 0, 1} and y = {0, 1, 1} or, as p-adic numbers: x = 0*20 + 0*2 1 + 1*22 = 4 y = 0*20 + 1*2 1 + 1*22 = 6 or, as grid points x^*2-3 = (1*20 + 0*2 1 + 0*22)/8 = 1/8 y^*2-3 = (1*20 + 1*2 1 + 0*22)/8 = 3/8 Because all subintervals in coding process will be inside [1/8, 3/8) all subsequent intervals will be inside it (this is how coding works), that means that all paths to these subintervals will have a common part. We can calculate common path of x and y according to the procedure described above: y^ -1 = 1*20 + 1*2 1 + 0*22 = 6 (y^ -1)^ = 0*20 + 1*2 1 + 0*22 = 2 (y^ -1)^ - x = 2 And finally com2,3(x, y) = ord2(2) = 1 We can store this path as a vector of coefficients and proceed with remaining part. To make further descriptions shorter we introduce two operators: extracting and rescaling Extracting is a trivial operator - it creates a vector of the first j coefficients of x: 4/5/2007 x= m0P + m1P 1 + m2P 2+ … + mnP ext(x, j) = {m0, m1, m2, … , mj-1} if second argument is omitted, then all coefficients are extracted: ext(x) = {m0, m1, m2, … , mn} One more operation on vector representation: cut(x, n, m) removes m bit starting with position n and shrink the vector of coefficients. Rescaling is just omitting first j terms in x and removing common factor Pj, res(x, j) = mjP + mj+iP 1 + m j+2P 2+ … + mnP Why do we call it rescaling? Because we can continue with level n-j as a first level and do not care about previous steps. Let see what happens with corresponding index res(x, j)^ = mjP + mj+iP n-j+1 + m j+2P n-j+2+ … + mnP As an integer number it is smaller then the original one. Rescaling keeps numbers from growing and makes it possible to use computer’s integer arithmetic (not infinite precision) for calculations, which makes this algorithm robust. Continuing our example (do not forget remove common factor!) res(x,1) = (0*21 + 1*22) / 2 = 0*20 + 1*21 res(y,1) = (1*21 + 1*22) / 2 = 1*20 + 1*21 Indexes will be res(x,1)^ = 1 res(y,1)^ = 3 and grid points res(x,1)^*2-2 = 1/4 res(y,1)^*2-2 = 3/4 We can show how this works for weight interval {c:[0, 0.125), b:[ 0.125, 0.375), d:[0.375, 05), a:[0.5, 1)} and message {b}. 4/5/2007 Picture 7. Rescaling In arithmetic coding analogous procedure (see [Bodden]) is called E1/E2. We will call this rescaling PR (p-adic rescaling). Trivial, but important case is when an interval occupies a whole subinterval of level K. In this case x^ = y^- 1 and the interval can be rescale on full length to starting interval [0, 1). This fact will be used later in discussion how p-adic coding corresponds to Huffman algorithm. Lifting In all our previous considerations and examples we use grids of minimal level. We may as well fix a level deep enough to perform all calculations. In fact, adding or removing trailing zeros in path representation does not change p-adic representation of a point, but, of course, changes its index representation. We will call operation of adding or removing zeros lifting. The reason for this name is that on a picture it looks like moving points in vertical direction. There are several advantages of using fix level in calculations: • It may be easy and more efficient coded, especially for P=2. • Model may be unable to present results as numbers of current ring G(PN); in this case special procedure must be implemented for changing level on a model’s demand. Let x to be from G(PN). Lifting is a mapping x to G(PN+f) x = m0P + m1P 1 + m2P 2+ … + mNP lift(x, j) => x = m0P + m1P 1 + m2P 2+ … + mNP N + 0•PN+1+ … + 0•PN+j where j ≥ 0 Evidently as an integer number x does not change, but as an index it changes dramatically. Important, but trivial feature of lifting is that it does not change common paths. Lifting can be defined also for negative argument: lift(x, -j) => x = m0P + m1P 1 + m2P 2+ … + mN-jP N-j where j ≥ 0 4/5/2007 If last j coefficient were zero, negative lifting also does not change x as an integer number. To use negative lifting without changing results we need to know an order of the last non zero coefficient: lnz(x) = min( j: mk =0 for k>j) This function gives us the highest possible for x level. A semi interval [x y) may be positioned at level hpl(x,y) = max(lnz(x), lnz(y)) Our procedure for calculating common path length of a semi interval was defined for intervals at hpl level. To extend it for the case when an interval belongs to a fixed level we need first to lift it back to hpl. comP,N( x, y ) = N if ( x == y ) else ordP(x - y) comP,N( x, y ) = comP,hpl(x,y) (lift(x, hpl(x,y) - N ), lift(y, hpl(x,y) -N ) ) Fortunately we do not have to go in that complication. The reason for this is that lifting does not change number of common paths. Let explore previous example restricting all calculations to level 4. 0 11/21/4 3/4 {0} {1} {0, 0} {0, 1} {1, 0} {1, 1} {0, 0, 0} {0, 1, 0} {1, 0, 0}{0, 1, 1} {1, 0, 1} {1, 1, 0} {1, 1, 1} {0, 0, 0, 0} {0, 0, 0, 1} {0, 0, 1, 0} {0, 0, 1, 1} {0, 1, 0, 0} {0, 1, 0, 1} {0, 1, 1, 0} {0, 1, 1, 1} {1, 0, 0, 0} {1, 0, 0, 1} {1, 0, 1, 0} {1, 0, 1, 1} {1, 1, 0, 0} {1, 1, 0, 1} {1, 1, 1, 0} {1, 1, 1, 1} Before rescaling After rescaling {0, 0, 1} Picture 7a. Rescaling on level 4 Consider two paths x and y from the previous example, but fixed the level equal to 4. On this level x and y can be presented as {0, 0, 1, 0} and {0, 1, 1, 0} or, as p-adic numbers: x = 0*20 + 0*2 1 + 1*22 + 0*23 = 4 y = 0*20 + 1*2 1 + 1*22 + 0*23= 6 To determine common path length we need to calculate y^ -1 = 5 (y^ -1)^ = 10 (y^ -1)^ - x = 6 And finally 4/5/2007 com2,4(x, y) = ord2(6) = 1 After rescaling we have new x and y: x = 0*20 + 1*21 + 0*22 = 2 y = 1*20 + 1*21 + 0*22 = 3 But they belong to level 3. To return x and y back to level 4 lifting is needed: lift(x, 1) = 0*20 + 1*21 + 0*22 + 0*23 lift(y, 1) = 1*20 + 1*21 + 0*22 + 0*23 Finding the shortest path point When coding is over we can choose any paths to any point from a final semi interval as a result. But points from the same semi interval may and have different paths after dropping trailing zeros. Let take a simple example when a message finally ends with semi interval [g5(4), g10(4)). Because we can drop trailing zeros, point g8(4) is the best choice – after dropping trailing zeros it becomes g1(1). 0 11/21/4 3/4 g0(0) g0(1) g1(1) g0(2) g1(2) g2(2) g3(2) g0(3) g2(3) g4(3)g3(3) g5(3) g6(3) g7(3) g0(4) g1(4) g2(4) g3(4) g4(4) g5(4) g6(4) g7(4) g8(4) g9(4) g10(4) g11(4) g12(4) g13(4) g14(4) g15(4) g1(3) Picture 8. Shortest path point A shortest path point in a semi interval [l, r) can be defined as a point with minimum level. lv(x^) = max(i: mi ≠ 0) g = min(lv(x^): l ≤ x < r ) But let consider paths as integers. From this point of view a point with minimal path is just a minimal p- adic integer. So g = min(x^: l ≤ x < r) We can check this for our example: Point g5(4) g6(4) g7(4) g8(4) g9(4) g10(4) Path {0, 1, 0, 1} {0, 1, 1, 0} {0, 1, 1, 1} {1, 0, 0, 0} {1, 0, 0, 1} {1, 0, 1, 0} p-adic number 10 6 15 1 9 5 Index 5 6 7 8 9 10 4/5/2007 Model Model is just an abstraction for a set of functions. One function calculates new subinterval on a base of incoming symbol and current interval in a predefined grid. M.code(a, l, r) => lnew, rnew An other takes as arguments a point and current interval and returns a new subinterval and a symbol M.decode(g, l, r ) => lnew, rnew, a l, r, g belongs to grid G(PN), a – to alphabet A. Model operates with indexes from a ring of nonnegative integers modular PN, so we have three possible variants how one subinterval on a ring can be situated inside another: 0 ≤ l ≤ lnew < rnew ≤ r < P N r=0 ; 0 ≤ l ≤ lnew < rnew < P N r=0 ; rnew = 0; 0 ≤ l ≤ lnew < P N And, of course some technical things: initialization and taking care of end of message. M.init(A, P, N, x) Where A is alphabet, P, N – characteristics of grid G(PN), x – optional parameter, some auxiliary information, which may be used by a model for optimization. code and decode functions may update model, but they must do it in sync. Input and Output I (input) and O (Output) are abstracts for pushing and receiving information. To make notations short we introduce an ugly term P-bit, which means one of symbols 0, … P-1. For P=2 it is obviously a normal bit. Now let describe input and output operations. I.getC => returns next character from input stream or EOM (End Of Message) I.getB(n) => returns next n P-bit vector from input stream O.pushB(U ) – pushes all P-bits from vector U O.pushB(p, n) – pushes P-bit p n times O.pushC(a) – pushes a symbol to an output stream Algorithms Now we are in position to describe the p-adic coding algorithm. Main idea of this algorithm is the same as in arithmetic coding – a message is mapped to in interval on [0, 1). There two parts of the algorithm – encoding and decoding, but whatever we are doing the first step – initialize a model: M.init(A, N) Coding Start with an empty message – no symbols. An empty message is coded as [0, 1), empty path U = {} or as [0, 0). l, r = 0, 0 When a symbol a comes a = I.getC model calculates a new interval. 4/5/2007 l, r = M.code(a, l, r) Now calculate a common path length n = comP,N(l^, (r-1)^) If n > 0 we can push common path to an output O.pushB(ext (l^, n)) and do rescaling. l^, r^ = res(l^, n), res(r^, n) And we also need to lift rescaled values back to level N and convert to index representation. l, r = lift(l^, n)^, lift(r^, n)^ Now we can read a next symbol and repeat steps. Pseudo code M.init(A, P, N) l, r = 0, 0 while ( ( a = I.getC ) != EOM ) { l, r = M.code(a, l, r) n = comP,N(l^, (r-1)^) if ( n > 0 ) { O.pushB(ext(l^, n)) l, r = lift(res(l^, n), n)^, lift(res(r^, n), n)^ } //if } //while l, r = M.code(EOM, l, r) q = selectPoint(l, r) O.pushB(ext(q, lnz(q)) We do not specify here what selectPoint does. The only requirement is to return a grid point from final semi interval [l, r), but of cause, it’s a good idea to return a point with a shortest paths. As it follows from previous discussion, all we need is to find a minimal integer in p-adic representation. So, to select a point with minimal path we should define selectPoint( l, r ) = min(x^: l ≤ x < r) lnz used here not to push trailing zeros. Decoding Start with an empty message – no symbols. An empty message is coded as [0, 1), or as empty path U={} or as pair of indexes: l, r = 0, 0 As the first step read first N P-bits from an input stream and construct a number from the vector. We need also to transform a path we a getting from a stream, to a number, so we use operator ^. g= (I.getB(n) • PTN)^ where PTN is a vector PN = (P 0, P1, P2, … , PN) Model calculates a new interval and a symbol a M.decode(g, l, r ) => l, r, a Now, a is a new decoded symbol and can be pushed into a stream of decoded symbols 4/5/2007 O.pushC(a) Next, as in the coding algorithm, calculate common path length n = comP,N(l^, (r-1)^) If n > 0 we can drop common path and do rescaling. l, r, g = lift(res(l^, n), n)^, lift(res(r^, n), n)^, lift(res(g^, n))^ read additional n P-bits and recalculate g g = g + (I.getB(n) • PTn)^ Now we can repeat all steps. Pseudo code M.init(A, P, N) l, r = 0, 0 g = (I.getB(n) • PTN)^ while ( true ) { l, r, a = M.decode(g, l, r ) if ( a == EOM ) break O.pushC(a) n = comP,N(l^, (r-1)^) if ( n > 0 ) { l, r, g = lift(res(l^, n), n)^, lift(res(r^, n), n)^, lift(res(g^, n),n)^ g = g + (I.getB(n) • PTn)^ } //if } //while One important particular case – Huffman codes Now we are prepared to show that p-adic coding algorithm gives exactly the same codes as Huffman’s algorithm [Huffman] if a weight interval is prepared in a special way. Let as assume that for a given alphabet and symbol probabilities a Huffman code tree was constructed. For example: Symbol (s): a b C d e Codeword (h(s)): 000 001 10 01 11 Grid level (cl(s)): 3 3 2 2 2 Starting index in grid: 0 1 2 1 3 We can map the tree to weight interval using the same technique as we used for coding messages 4/5/2007 Picture 9. Mapping Huffman code tree to weight interval After lifting all intervals to highest grid (N=3): Symbol (s): a B c d e Codeword (lift(h(s),N-cl(s)): 000 001 100 010 110 Starting index in grid: 0 1 4 2 6 Algorithm of constructing weight intervals from a Huffman code tree for alphabet A is simple. Let cl(s) be a length of Huffman code of symbol s and N = max(cl(s)) among all s from A, h(s) - Huffman code of s, then symbols s occupies a semi interval starting at point with index lift(h(s),N-cl(s))^ and ending at starting point of a next symbol or 1. Constructed weight interval has an important property – all of subintervals occupy a whole grid interval of some level. It was shown above that in this situation left end right ends have an entire path in common; so PR rescaling will push all of it into an output and a next symbol will be coded starting with [0,1) interval. This proves that for this particular choice of weight interval p-adic coding works identical to Huffman’s algorithm. Another particular case – Golomb-Rice codes Surprisingly enough, but p-adic coding algorithm produces Golomb-Rice [Golomb, Rice] codes when supplied with single symbol alphabet and special model; no changes to algorithm itself are needed. If an alphabet contains only one symbol, the only information a message may contains is its length. So coding of a message is equivalent to coding of a natural number – the length. We will use symbol * to identify the only entry. The model is trivial: M.code(*, l, r) => l, r-1 M.code(EOM, l, r) => r-1, r M.decode(g, l, r ) => if (g == r-1) then l, r, EOM else l, r-1, * The algorithm will do all the work. Let start with P=2 and consider a grid 2N+1. Coding procedure starts with l = r = 0 4/5/2007 If a message is empty, we have to encode EOM. To do this we need to calculate r-1= 0 - 1, which is 2N+1-1 and return a path to 2N+1-1. This path consists of N+1 ones: {1, 1, … , 1, 1}. This is our new representation of zero. If a symbol comes, the model recalculates r and l: l = 0 r = 2N+1-1 If it was the only symbol in a message, then the model returns 2N+1-2, 2N+1-1, and a code is a path to a point with index 2N+1-2: {1, 1, … , 1, 0}. This procedure may be continued until a message’s length is less than 2N. At this point the model returns l = 0 r = 2N because comP,N(0^, (2 N -1)^) = 1 PR rescaling will be used; one 0 will be pushed to output buffer, l and r return to their initial values l = r = 0. The coder is in initial state and ready to receive a new symbol. Encoder stays almost without changes. We have defined selectPoint( l, r ) = l And drop lnz call in the last pushB operation to keep trailing zeros O.pushB(ext(q)) If a messages of length W comes W/2N zeros will be pushed in output buffer; the rest part of the output will contain a path to a point which index is 0 – (W%2N). After encoding EOM we have to move the point one step to the left. So finally index will be 0 – ((W%2N) + 1). For example, for N=3 we have: W code W Code 0 1111 8 01111 1 1110 9 01110 2 1101 10 01101 3 1100 11 01100 4 1011 12 01011 5 1010 13 01010 6 1001 14 01001 7 1000 15 01000 The codes look very much like Golomb-Rice codes. Indeed, they may be transformed to each other by replacing 1 with 0, and 0 with 1 - binary NOT. There is no magic in changing unary representation and delimiter – there is no difference between counting a number 0 of before first 1 and counting number of 1 before first 0. Transformation of the rest part – after delimiter, may be not that clear. In the ring of integers modular 2N 0 – (R +1) = (2N - 1) - R here R = (W%2N); R < 2N. In binary representation (2N – 1) is a vector U of N 1. Now NOT(U – R) = R This proves that after NOT transformation the rightmost part of codes transforms to W%2N. 4/5/2007 Any prime P can be used with this model. But this generalization does not look very promising. In fact, the reason why we discuss Huffman and Golomb-Rice codes here is to emphasize that the most popular entropy codes have a common base – they all maps messages to p-adic integer numbers. Rescaling based on Archimedean distance (AR) We were very ingenious when selecting most convenient for us weigh interval: {a:[0, 0.5), b:[0.5, 0.75), c:[0.75, 0.875), d:[0.875,1)} Yes, compression rate does not depend on an order of subintervals, but calculation and resulted codes do. Let shuffle the weigh interval: {b:[0, 0.25), a:[0.25, 0.75), c:[0.75, 0.875), d:[0.875,1)} Now subinterval a:[0.25, 0.75) covers the center point 1/2. Consider now a message containing only symbols a. It can be easily shown that left edge of message interval will be always less than 1/2, while the right one – greater. From p-adic point of view this means that ordp(l, r) is always zero and there is no common path and, as a sequence, rescaling will never happen. If we continue coding {a, a, … , a} we will end in integer overflow error or will be faced to use infinite precision arithmetic. To save our integer arithmetic from huge numbers we have to use the fact that Archimedean length in this case is less or equal to1/2. Picture 10. Coding {a, a, a} For P ≠ 2 situation is more complex. A semi interval can include any grid point 0 < n < P. In the following example (P = 3) an interval has Archimedean length 2/9, but p-adic length 1. 4/5/2007 Picture 11. Before rescaling Now let explore a case when a sub interval lies in the smallest interval of level 2, which includes a point of level 1 with index n. p-adic representation of left l and right r edges of such subinterval is. l = {n-1, P-1, …. } r = {n, 0, … } It’s Archimedean length is less or equal to 2/(P*P). We want to map it to a bigger interval, precisely to interval [n-1, n+1) from level 1. This can be done by a linear transformation: Y(X) = XP1 – nP0 +nP-1 Let’s consider how a semi interval defined in p-adic representation as l = m0P + m1P 1 + m2P 2+ … + mNP r = k0P + k1P 1 + k2P 2+ … + kNP transforms under this mapping. The first thing we need to do – to transform paths to points. We can do it by using IP transformation: a = m0P + m1P -2 + m2P -3+ … + mNP -N-1 b = k0P + k1P -2 + k2P -3+ … + kNP -N-1 Now we can apply linear transformation: Y(a) = (m0 - n)P + (m1 + n)P -1 + m2P -2+ … + mNP Y(b) = (k0 - n) P 0+ (k1 + n) P -1 + k2P -2+ … + kNP For this subinterval we have: m0 = n -1; m1 = P- 1 k0 = n; k1 = 0 Y(a) = 0P0 + (n - 1)P -1 + m2P -2+ … + mNP Y(b) = 0P0+ nP-1 + k2P -2+ … + kNP Rescaling will drop first zero terms. Reverting back from points to paths we can find how this transformation works on paths: Y(l) = (n - 1) P0 + m2P 1+ … + mnP Y( r ) = nP0 + k2P 1+ … + knP Or in vector representation: 4/5/2007 Y(l) = {n-1, P-1, …. } => {n -1, … } Y(r) = {n, 0, … } => {n, … } we just remove second (counting from the left) elements. It is also easy to verify that center point {n, 0, 0, …, 0} of this mapping is a stable point, i.e. Y maps it to itself Y : {n, 0, 0, …, 0} => {n, 0, …, 0} New interval [l, r) contains the stable point. Coming back to the example (here n=1) we can draw the picture after rescaling: Picture 11a. After rescaling We will refer this rescaling as AR. Important difference between AR and PR rescaling is that AR does not push anything in output buffer. It is convenient to invent a special predicate AR? for testing if AR rescaling can be applied for an interval. AR?(l, r, P) = (r[0] – l[0] == 1) AND (l[1] == P-1) AND (r[1] == 0) To continue coding we must remember the applied mapping, it can be done by storing only two parameters: n – a stable point and u – a number of times rescaling was applied. What may happen if we continue coding? 1. [l, r) are still contains n 1.1. value of AR? predicate is false 1.2. value of AR? predicate is true 2. [l, r) does not contain n 2.1. n lays to the right of r; toRight?( n, r) == true 2.2. n lays to the left of l; toLeft?( n, l) == true To test condition 2.1 and 2.2 we introduced two predicates toRight? and toLetf?. There predicates are suppose to receive a path as second argument, i.e. a number in p-adic integer number; first argument is an integer number toRight?(n, r) = ( r[0] < n ) OR ( r^ == n ) toLeft?(n, l) = l[0] ≥ n Now let discuss situations mentioned above: 1.1. This is the simplest case. We just continue coding. 1.2. Increase u: u = u +1; do AR rescaling and continue coding. 4/5/2007 2.1. This means that the whole interval lays in [{n-1, P-1, … , P-1}, {n, 0, … , 0}); where P-1 is added u times. Any subinterval from this interval has common path {n-1, P-1, … , P-1}, so we can now push this path into output and rescale l and r one more time removing fist digits. 2.2. This means that the whole interval lays in [{n, 0, … , 0, 1}, { n, 0, … , 0,1}); where 0 is added u times. Any subinterval from this interval has common path {n, 0, … , 0}, so we can now push this path into output and rescale l and r one more time removing fist digits. AR and PR rescaling procedures together guaranty that current coding interval will never be smaller than 2/P2-1/PN. This means that maximum value of indexes is 2PN-2-1. Algorithms revised Coding with AR A new feature here, comparing to the first variant of p-adic encoding algorithm, is that we need to track AR transformation. To do this we introduce two new variables sp and spn. • sp – stable point of AR; it is a point of level 1 and may be represented as a positive integer (not path) 0 < sp < P. • spn – number of times AR was applied. Some additional operations should be done at final step. First of all we need to check, as in the main loop, if the final interval is situated to the left or to the right of a stable point and, if this is the case, do necessary pushing and then proceed to usual final search for minimal point. If not and spn is not zero, then we are lucky and we already have a point from level 1 and all we need to do is just to push out sp. Pseudo code M.init(A, N) l, r = 0, 0 sp, spn = 0, 0 while ( ( a = I.getC ) != EOM ) { l, r = M.code(a, l, r) if ( spn ≠ 0 ) { if ( toLeft?(sp, l^) ) { O.pushB(sp, 1) O.pushB(0,spn) l, r = lift(res(l^, 1), 1)^, lift(res(r^, 1), 1)^ sp, spn = 0, 0 } //if if ( toRight?(sp, r^) ) { O.pushB(sp - 1, 1) O.pushB(P - 1, spn) l, r = lift(res(l^, 1), 1)^, lift(res(r^, 1), 1)^ sp, spn = 0, 0 } //if } //if // PR rescaling if ( spn == 0 ) { n = comP,N(l^, (r - 1)^) if ( n > 0 ) { O.pushB(ext(l^, n)) l, r = lift(res(l^, n), n)^, lift(res(r^, n), n)^ 4/5/2007 } //if } //AR rescaling while ( AR?(l^, r^) ) { sp = r^[0] if sp == 0 spn = spn + 1 l, r = lift(cut(l^,1,1),1)^, lift(cut(r^,1,1),1)^ } //while } //while l, r = M.code(EOM, l, r) if ( spn ≠ 0 ) { if ( toLeft?(sp, l^) ) { O.pushB(sp, 1) O.pushB(0, spn) l, r = lift(res(l^, 1), 1)^, lift(res(r^, 1), 1)^ sp, spn = 0, 0 } //if if ( toRight?(sp, r^) ) { O.pushB(sp - 1, 1) O.pushB(P - 1, spn) l, r = lift(res(l^, 1), 1)^, lift(res(r^, 1), 1)^ sp, spn = 0, 0 } //if } //if if (spn == 0) { q = selectPoint(l, r) O.pushB(ext(q, lnz(q)) } else { O.pushB(sp, 1) // we already have point of level 1 } //if //the End Decoding with AR AR rescaling is simpler for decoding process, because we do not care about pushing anything out and a final step is most simple – we just finish decoding. The only thing which is new is additional reading from an input stream. Pseudo code M.init(A, N) l, r = 0, 0 spn = sp = 0 g = (I.getB(N) • PTN)^ while ( true ) { l, r, a = M.decode(g, l, r ) if ( a == EOM ) break O.pushC(a) if ( spn ≠ 0 ) { if ( toLeft?(sp, l^) OR toRight?(sp, r^) ) { l, r, g = lift(res(l^, 1), 1)^, lift(res(r^, 1), 1)^ , lift(res(g^, 1), 1)^ g = g + (I.getB(1) • PT1)^ 4/5/2007 sp, spn = 0, 0 } //if } //if // PR rescaling n = comP,N(l^, (r-1)^) if ( n > 0 ) { l, r, g = lift(res(l^, n), n)^, lift(res(r^, n), n)^, lift(res(g^, n),n)^ g = g + (I.getB(n) • PTn)^ } //if // AR rescaling while ( AR?(l^, r^) ) { sp = r^[0] if sp == 0 spn = spn +1 l, r, g = lift(cut(l^,1,1),1)^, lift(cut(r^,1,1),1)^, lift(cut(g^,1,1),1)^ g = g + (I.getB(1) • PT1)^ } //while } //while //the End Of course, PT1 is just 1 and we can also omit ^ operator. The operation g = g + (I.getB(1) • PT1)^ can be replaced (in two places) by g = g + I.getB(1) Implementation We have implemented all algorithms and all tests in Ruby [Ruby] – a new popular interpreted, dynamically typed, pure object-oriented, scripting language. And Ruby proved to be very helpful. We would hardly be able to try so many variants and run innumerous tests in any other language. Now let discuss the practical case of P=2. All previous discussion remains valid – this is just a special case. This case has most important advantage – we can use real bits and binary vectors. This is extremely convenient. All algorithms remain the same. Only some small improvement can be done for AR rescaling. Because the only possible value for n is 1, there is no need to store it as spt. In case when toLeft? returns true we have to push 1 and a number of 0; if toRight? returns true we have to push 0 and a number of 1. Arithmetic coding We can see that arithmetic coding is just a special case of p-adic coding for P=2. All conditions expressed there as arithmetic operations can be done on bit level. In fact, many practical implementations use shifts instead. Let us examine E1 condition: mHigh < g_Half where g_Half = 0x40000000 This condition means that most significant bit in binary representation of mHigh must be 0. This is also true for gLow because mLow < mHigh. Reverting to paths we can see that both gLow and gHigh have 4/5/2007 most significant bits in p-adic representation are equal to 0, so p-adic distance is less than 1 and PR condition is fulfilled. However, in p-adic coding algorithm PR rescaling works for mHigh equal to g_Half. It is this small difference makes p-adic coding algorithm works exactly as Huffman algorithm for certain models. Arithmetic coding in this situation does not provide optimal compression (see discussion in [Bodden]). AR rescaling is similar to E3. AR? predicate is equivalent to (g_FisrtQuater <= mlow) AND (mHigh < g_ThirdQuater) We have implemented the same model as proposed in [Bodden] and get the same compression for all standard tests. Results Standard tests For testing we used Calgary/Canterbury text compression corpus – popular set of tests first discussed in [Bell]. It contains files bib, book1, book2, geo, news, obj1, obj2, paper1, paper2, paper3, paper4, paper5, paper6, pic, progc, progl, progp and trans. These files may be obtained from [Canterbury corpus]. Comparison with Arithmetic coding We used program codes published in [Bodden] to get results of arithmetic coding. This program adds additional 4 bytes to an output file; these 4 bytes are in most examples the only difference between arithmetic and p-adic coding results. In our test we use p-adic coding with P=2 and N= 31. Conclusion Tree is a well known and widely used data structure in computer science. Arithmetic, Huffman and Golomb-Rice coding are also well known and widely used for a long time algorithms. p-adic numbers, ultrametric spaces are not so popular in computer science; even for pure mathematic they are relatively new. Is there any connection between them? We hope that we have shown this connection and that this connection is quite natural and fundamental. A message, as sequences of symbols, may be considered as path on a tree. There are numerous ways to construct this mapping. It is quite fundamental and widely used way for presenting messages and is very popular in computer science and applications. On the other hand, trees are great models of p-adic numbers; many strange and unusual features of ultrametric spaces can be understood and visualized on trees [Holly]. This works also in the reverse direction – p-adic numbers is convenient tool for indexing paths and p-adic norm is a natural measure on trees. On the other hand, a message can be mapped to a subinterval of a unit interval – this what real number arithmetic algorithm does. While theoretically clear and simple, this method was never used in practice, because of its inefficiency due to problems with computer based real arithmetic. Integer arithmetic coding solved this problem by introducing some practical receipts how to use integer numbers, instead of real ones. The resulting algorithm proved to be efficient and robust and may be because of this fact no theoretical analysis has been done. Integer version looks pretty much like the original algorithm, but in fact, difference between them is considerable; while real number algorithm works on a field, its integer number variant deals with a finite ring. p-adic number coding algorithm explicitly works with numbers from the finite ring of positive integer numbers modular PN. These numbers, being mapped to a union interval, create an equidistance grid G(PN). The next step is to create a path from a root through grid points of upper levels (G(PK); k=0..N-1) to points of this grid. This construction creates a bridge between ultrametric space of tree paths and Archimedean space of grid points. Now we can identify any grid point not only by its index, but by a path, i.e. by some 4/5/2007 p-adic integer number; the reverse is also true – any path can be identify by its end point from the grid and as so by an index. This dualism is the real base of p-adic arithmetic coding algorithm. We found a simple and elegant way to transform paths to indexes and back. We called it IP transformation. As a transformation from paths to points IP transformation can be considered as Kozyrev’s transformation for finite paths, but IP transformation is reversible. p-adic arithmetic coding algorithm works as a bridge between two spaces – ultrametric space of paths and Archimedean space of grid points. Model calculates intervals with edge points on the grid and then IP transformation maps them to paths; if these paths are closed to each other as p-adic numbers, then common path is pushed to output buffer, these p-adic numbers are truncated, and IP transformation maps them back on grid. For P=2 PR rescaling works pretty much like E1/E2 rescaling but it has one small improvement. It is this improvement that makes it possible to show that for certain models and alphabet p-adic coding algorithm works as Huffman and Golomb-Rice algorithm. For P=2 and general models it works as arithmetic coding. So we may say that three most popular entropy coding algorithms can be considered as special cases of one algorithm - p-adic coding, working with different P, models and alphabets. This also gives an answer to the question in the begging of this paragraph - arithmetic, Huffman and Golomb-Rice coding algorithms maps messages to ultrametric space of p-adic numbers. They are “speaking in prose”! References 1. Abrahamson, N., "Information theory and coding”, McGraw-Hill, New York 1963. 2. Baker A.J., “Introduction to p-adic Numers and p-adic Analysis”, Department of Mathematics, University of Glasgow G12 8QW, Scotland 3. Bell, T.C., Witten, I.H. and Cleary, J.G., "Modeling for text compression", Computing Surveys 21(4): 557-591; December 1989. 4. Bodden Eric, Clasen Malte, Kneis Joachim, "Arithmetic Coding Revealed. A guided tour from theory to practice”, Translated and updated version, May, 2001. 5. Canterbury corpus, http://links.uwaterloo.ca/calgary.corpus.html . 6. Golomb S.W., "Run-length encoding”, IEEE Transactions on information Theory, IT-12:399- 401, July 1966. 7. Holly J.E.,”Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued fields”, Amer. Math. Monthly 108 (2001) 721-728 8. Huffman D.A., “A method for construction of minimum-redundancy codes ”, Proc. Inst. Radio Eng. 40, 9 (Sept. 1952), 1098-1101 9. Koblitz Neal, "p-adic numbers, p-adic analysis and zeta-functions”, Springer-Verlag, 1977. 10. Koc C.K., "A Tutorial on p-adic Arithmetic”, Electrical & Computer Engineering, Oregon State University, Corvallis, Oregon 97331, April 2002. 11. Kozyrev S.V., “Wavelet theory as p-adic spectral analysis”, Izvestiya: Mathematics 66:2 367-376 12. Moffat A, Neal R. M. and Witten I. H., “Arithmetic coding revised,” ACM Transactions on Information Systems, vol. 16, no. 3, pp. 256-294, 1998 13. Nelson Mark, "Arithmetic Coding + Statistical Modeling = Data Compression”, Dr. Dobb’s Journal, February, 1991. 14. Rice R.F., "Some Practical Universal Noiseless Coding Techniques”, Technical Report JPL Publication 79-22, JPL, March 1979. 15. Ruby, "Ruby”, http://www.ruby-lang.org/en/ 16. Said Amir, “Introduction to Arithmetic Coding – Theory and Practice”, Imagining Systems Laboratory, HP Laboratories, Palo Alto, HPL-2004-76, April 21,2004 17. Salomon David, "Data compression. The complete reference”, Springer-Verlag, 2004 18. Sayood Khalid, "Introduction to data compression. The complete reference”, Elsevier, 2006 19. wiki, "Arithmetic_coding”, http://en.wikipedia.org/wiki/Arithmetic_coding 20. Witten, I.H., Neal, R. and Cleary, J.G. (1987) “Arithmetic coding for data compression.” Communications of the ACM, 30(6), pp. 520-540, June. Reprinted in C Gazette 2(3) 4-25, December 1987

0704.0835

Compton X-ray and Gamma-ray Emission from Extended Radio Galaxies

Compton X-ray and γ-ray Emission from Extended Radio Galaxies C. C. Cheung1 Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA Abstract. The extended lobes of radio galaxies are examined as sources of X-ray and γ-ray emission via inverse Compton scattering of 3K background photons. The Compton spectra of two exemplary examples, Fornax A and Centaurus A, are estimated using available radio measurements in the ∼10’s MHz – 10’s GHz range. For average lobe magnetic fields of ∼ 0.3–1 µG, the lobe spectra are predicted to extend into the soft γ-rays making them likely detectable with the GLAST LAT. If detected, their large angular extents (∼1◦ and 8◦) will make it possible to “image” the radio lobes in γ-rays. Similarly, this process operates in more distant radio galaxies and the possibility that such systems will be detected as unresolved γ-ray sources with GLAST is briefly considered. Keywords: gamma-ray sources (astronomical); radiofrequency spectra; imaging; radiogalaxies PACS: 98.54.Gr,98.58.Fd INVERSE COMPTON "IMAGES" OF LARGE RADIO GALAXIES Inverse Compton (IC) scattering of the CMB is a mandatory process in synchrotron emitting sources. This emission becomes most prominent in regions of weaker B-field like the extended lobes of radio galaxies. Many such IC/CMB lobe X-ray sources are now known (e.g., Croston et al. 2005; Kataoka & Stawarz 2005) and we explore the possibility of the IC spectra extending into the γ-ray band. This is independent of possible γ-ray emission from the unresolved nuclei of radio galaxies, i.e., from the misaligned blazar (Sreekumar et al. 1999; Bai & Lee 2001; Foshini et al. 2005). The case of the nearby (D=18.6 Mpc) double-lobed radio galaxy, Fornax A was discussed in Cheung (2007). Radio flux density measurements down to ∼30 MHz (Isobe et al. 2006) were used to estimate the IC/CMB spectra of the lobes. Normalizing the IC spectra to the X-ray detections of the lobes (which indicate B∼1.5µG on average; Feigelson et al. 1995, Isobe et al. 2006), the presence of high frequency radio emission observed in the >∼ 10–90 GHz range with WMAP (with Fν ∝ ν−1.5) imply a detectable soft γ-ray signal. As this emission is not expected to be time variable, the LAT can simply integrate on this position during its normal scanning mode to test this prediction. Here, we similarly consider the case of Centaurus A which is only 3.5 Mpc away. It is long known to have structure extended over ∼8◦ in declination (Cooper et al. 1965, and references therein). We use the extensive compilation by Alvarez et al. (2000) of the various components of the radio source; Figure 1 shows a low resolution 408 MHz image from Haslam et al. (1982). The outer (degree-scale) giant lobes (GLs) visible in Figure 1 account for >∼ 2/3 of the total 408 MHz emission at ∼1000 Jy each; the arcmin-scale inner lobes (ILs) are only 3–4 times fainter than each GL. The northern GL was searched for such IC emission with ASCA data but the extended X-rays could not be uniquely attributed to such a process (Isobe et al. 2001). Repeating the analysis as for Fornax A, it appears that the extended components of Cen A will also emit γ-rays at a level detectable by GLAST. The various data from ∼10 MHz to 43 GHz are consistent with a single spectral index α=0.7. Since the luminosities of both the ILs and GLs are similar (within ∼20%), only the SEDs of the southern ones are plotted in Figure 1. Utilizing these radio measurements, the expected IC/CMB spectra for example B-field strengths are drawn. The integrated Compton Gamma-Ray Observatory (CGRO) COMPTEL detections of Cen A (Steinle et al. 1998) at ∼1021 Hz already limit B >∼ 1µG for both the northern and southern GLs (since they have similar radio spectra); a similar extrapolation for the ILs give B >∼ 0.3µG. Thermal emission will be a complicating factor at energies below ∼10 keV, so hard X-ray and soft γ-ray measure- ments are better suited for detecting the suspected IC/CMB emission. Additionally, since Fornax A and Cen A are 1 Jansky Postdoctoral Fellow. The National Radio Astronomy Observatory is operated by Associated Universities, Inc. under a cooperative agreement with the U.S. National Science Foundation. http://arxiv.org/abs/0704.0835v1 208.000 204.000 200.000 196.000 -38.000 -40.000 -42.000 -44.000 -46.000 -48.000 Right Ascension (J2000) GL (south) GL (north) Centaurus A 408 MHz 0.85 deg FIGURE 1. [Left] Radio image of Cen A at 0.85◦ resolution which is comparable to the angular resolution of GLAST/LAT. [Center] SEDs of the multiple components of the Cen A radio source with lines indicating Fν ∝ ν−0.7 spectra. The data points at > 1018 Hz are the integrated detections with CGRO with lines indicating the expected IC/CMB spectra of the southern giant lobe for different average B-fields. [Right] IC/CMB X-ray and γ-ray flux predictions for 17 of the highest-z radio galaxies discussed in the text. Typical Chandra “snapshots" are 5–10 ksec exposures so these sources are all expected to be easily detectable in the X-rays for the indicated field strengths. GLAST detections require electrons with γ >∼ 105 in a 1µG or smaller field which are optimistic. quite extended in the sky (∼1◦ and 8◦), if they are detected with GLAST, the contributions from the two lobes will be separable with the LAT making IC/CMB γ-ray “images” of these radio galaxies possible. These γ-ray images will appear most similar to radio maps at frequencies, ν >∼ 10 GHz; such radio maps of Cen A’s extended components have already been obtained by WMAP (Page et al. 2007, Fig. 2 therein) and are available for this comparison. THE HIGHEST-REDSHIFT RADIO GALAXIES Using the above examples as a guide, we can gauge the feasibility of detecting even more distant radio galaxies at the higher-energies. Utilizing the recent large compilation of bright z>2.5 radio sources by Carson et al. (2007), we consider the highest-redshift (z > 3.5) radio galaxies for illustration. The observed monochromatic Compton (X-ray, γ-ray) to synchrotron (radio) flux ratio for IC/CMB emission has a strong redshift dependence: for α=1, it is simply fc/ fs ≃ ucmb/uB ≃ 10(1+ z) 4δ 2/B2µG ( fν ≡ νFν , and δ is the Doppler factor which is set to 1). We use the NVSS (Condon et al. 1998) 1.4 GHz fluxes for fs. Most of the considered sources (13/17) are detected at 74 MHz in the VLSS database (Cohen et al. 2005) giving α74MHz−1.4GHz ∼ 0.9− 1.2, so the approximate relation is applicable. As in the nearby sources, these distant radio galaxies are expected to be IC/CMB X-ray sources unless B ≫10µG (Fig. 1). Chandra observations should easily detect this emission to constrain the lobe B-fields, and thus the lobe energetics. In one of the highest-redshift (z = 3.8) radio galaxies observed so far with Chandra (Scharf et al. 2003), it was necessary to remove the contribution from a bright nucleus (spatially) and extended IC emission from other sources of seed photons (by spectral fitting). Such X-ray observations can guide our determination of the expected level of (soft) γ-ray emission from the IC/CMB process; at the moment, the estimates (Fig. 1) are rather crude. REFERENCES 1. H. Alvarez, J. Aparici, J. May, & P. Reich, Astron. Astrophys. 355, pp. 863–872 (2000). 2. J. M. Bai & M. G. Lee, Astrophys. Journal Lett. 549, pp. L173–L177 (2001). 3. J. E. Carson, T. M. Arias, & C. C. Cheung, in preparation (2007). 4. C. C. Cheung, in The Central Engine of Active Galactic Nuclei, edited by L. C. Ho & J.-M. Wang, ASP Conf. Series, in press, arXiv:astro-ph/0612372 (2007). 5. A. S. Cohen, et al., in From Clark Lake to the Long Wavelength Array: Bill Erickson’s Radio Science, edited by N. Kassim et al., ASP Conf. Series 345, 299–303 (2005). 6. B. F. C. Cooper, R. M. Price, & D. J. Cole, Australian Journal of Physics 18, pp. 589–625 (1965). http://arxiv.org/abs/astro-ph/0612372 7. J. J. Condon, W. D. Cotton, E. W. Greisen, Q. F. Yin, R. A. Perley, G. B. Taylor, & J. J. Broderick, Astron. Journal 115, pp. 1693–1716 (1998). 8. J. H. Croston, et al., Astrophys. Journal 626, pp. 733–747 (2005). 9. E. D. Feigelson, S. A. Laurent-Muehleisen, R. I. Kollgaard, & E. B. Fomalont, Astrophys. Journal Lett. 449, pp. L149–L152 (1995). 10. L. Foschini et al., Astron. Astrophys. 433, pp. 515–518 (2005). 11. C. G. T. Haslam, C. J. Salter, H. Stoffel, & W. E. Wilson, Astron. Astrophys. Suppl. 47, pp. 1–142 (1982). 12. N. Isobe, K. Makishima, M. Tashiro, & H. Kaneda, in Particles and Fields in Radio Galaxies, edited by R. A. Laing & K. M. Blundell, ASP Conf. Series 250, pp. 394–399 (2001). 13. N. Isobe, K. Makishima, M. Tashiro, K. Itoh, N. Iyomoto, I. Takahashi, & H. Kaneda, Astrophys. Journal 645, pp. 256–263 (2006). 14. J. Kataoka, & Ł. Stawarz, Astrophys. Journal 622, pp. 797–810 (2005). 15. L. Page, et al., Astrophys. Journal, in press (2007). 16. C. Scharf, et al. Astrophys. Journal 596, pp. 105–113 (2003). 17. P. Sreekumar, D. L. Bertsch, R. C. Hartman, P. L. Nolan, & D. J. Thompson, Astroparticle Physics 11, pp. 221–223 (1999). 18. H. Steinle, et al. Astron. Astrophys. 330, pp. 97–107 (1998). Inverse Compton "Images" of Large Radio Galaxies The Highest-Redshift Radio Galaxies

0704.0836

A matroid-friendly basis for the quasisymmetric functions

A Matroid-Friendly Basis for the Quasisymmetric Functions Kurt W. Luoto Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195, USA [email protected] October 26, 2018 Abstract A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the quasisymmetric functions associated to matroids by the Hopf algebra morphism F of Billera, Jia, and Reiner [3]. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F distinguishes isomorphism classes of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirrors the decomposability of its base polytope. An affirmative answer to the Hilbert basis question raised in [3] is given. 1 Introduction In this paper we construct a new Z-basis for the space of quasisymmetric functions, QSym and study its properties. For instance, we show that it has nonnegative structure constants, and that it behaves well with respect to the quasisymmetric functions associated to matroids by the Hopf algebra morphism Mat → QSym described by Billera, Jia, and Reiner [3]. We also answer in the affirmative a question regarding rank two matroids posed in [3, Question 7.10], and give an affirmative answer to [3, Question 7.12] in the case of rank two matroids. In [3], Billera, Jia, and Reiner describe an invariant for matroids in the form of a quasisymmetric function. They show that the mapping F : Mat → QSym is in fact a morphism of combinatorial Hopf algebras (given a suitable choice of character on Mat; see [1]), where Mat is the Hopf algebra of matroids introduced by Schmitt [15], and studied by Crapo and Schmitt [4], [5], [6], [7]. Billera, Jia, and Reiner show that, while the mapping F is not surjective over integer coefficients, it is surjective over rational coefficients. http://arxiv.org/abs/0704.0836v2 Our new basis for QSym is “matroid-friendly” in that it reflects the rank of loopless matroids as well as the size of the ground sets: for every 1 ≤ r ≤ n, there is a set Nnr of basis vectors such that for every loopless matroid M of rank r on an n-element ground set, F (M) ∈ span Nnr ; moreover, QSym decomposes as the direct sum of these subspaces. This provides us with a new product grading of QSym, according to matroid rank r. (The usual grading of QSym by degree corresponds to the size n of the matroid ground set.) Also, as with the monomial and fundamental bases of QSym, for every matroid M , F (M) has nonnegative coefficients in our basis. The paper has two main parts. The first part (Sections 2–4) presents the new basis and relevant background material. In Section 2, we recount background material from the literature regarding posets and quasisymmetric functions. In Section 3, we present a definition for our new basis for QSym by means of a construction, and highlight several of its important features. There we also prove that it is a Z-basis for QSym. In Section 4, we build necessary machinery regarding computing the quasisymmetric function associated to a labeled poset, in the form of alternative decompositions, and apply these tools to prove that the structure constants of the new basis are nonnegative. The second part, (Sections 5–7) discusses matroids and their quasisymmetric functions. In Section 5, we recall some of the concepts, terminology, and results from the paper [3], and prove our claims regarding the quasisymmetric functions of matroids vis-a-vis our new basis. In Section 6, we recall the context of [3, Section 7] regarding the relationship between decompositions of the quasisym- metric function associated to a matroid and decompositions of its matroid base polytope, and recall the statement of [3, Question 7.10] regarding the functions associated to rank two matroids. We develop a formula for the quasisymmetric function of a loopless rank two matroid in terms of the new basis, and apply it to show (1) that the morphism F : Mat → QSym distinguishes isomorphism classes of rank two matroids, (2) that the two types of decompositions mirror each other, i.e. an affirmative answer to [3, Question 7.12] for the case of rank two matroids, and (3) to give an affirmative answer to [3, Question 7.10]. In Section 7, we make additional observations regarding matroid functions and the new basis. We also compare the new basis with the other QSym bases discussed in Section 10 of [3], and sketch an alternate proof of the surjectivity of the map Mat → QSym over rational coefficients. 2 Preliminaries In this section we quote certain concepts, terminology, and facts from the litera- ture, as well as establish certain conventions which will be used in the remainder of the paper. 2.1 Compositions A composition α is a finite sequence of positive integers, i.e. α ∈ Pm for some m ∈ N. The number of parts of α, m, is the length of α, and denoted by ℓ(α). The weight of α = (α1, . . . , αm) is |α| = α1+ · · ·+αm. Included in our definition is the composition having no parts, which we denote by the (bold font) symbol 0. We have ℓ(0) = |0| = 0, the only composition with these properties. Note, for small examples where individual parts are less than 10, we will often write a composition as a sequence of digits, with no separating commas. For example, we may write (1, 5, 6, 3, 2, 3) as 156323 when the context is clear. We adopt a similar convention for the one-line notation of permutations in Sn when n < 10. There is a natural bijection between compositions of weight |α| = n and susbets of [n− 1] (where [n] = {1, 2, 3, . . . , n}), given by (α1, . . . , αm) ↔ {α1, α1 + α2, α1 + α2 + α3, . . . , α1 + · · ·+ αm−1}. We say that β is a refinement of α, or that β refines α (denoted β 4 α) if |α| = |β| and A ⊂ B, where A and B are the sets associated to α and β respectively. To any permutation π ∈ Sn there is an associated composition of weight n which we denote C(π) and whose parts give the lengths of successive increasing runs in the one-line notation of π. For example, for π = 934756218 ∈ S9, we have C(π) = 13212. In this paper, we mildly generalize the notion of a permutation to be any sequence of distinct positive integers. Given a set of positive integers X , we let S(X) denote the set of all permuations of all the elements of X . The run length operator C(π) extends to these general permutations in the obvious way. If X and Y are two sets of positive integers of the same cardinality n, then every bijection f : X → Y induces a mapping f : S(X) → S(Y ) given by f(x1, . . . , xn) = (f(x1), . . . , f(xn)). If f is an increasing function, then we have C(f(π)) = C(π) for every π ∈ S(X). 2.2 Well-known QSym bases The algebra of quasisymmetric functions QSym (or QSym(x) when we want to emphasize the variable set) forms a subring of the power series ring R[[x]] where x = (x1, x2, x3, ...) is a linearly ordered set of variables indexed by the positive integers, and R is a (fixed) commutative ring. In this paper we only deal with the cases where R is either Z or Q, assuming coefficients in Q unless otherwise stated. We often suppress the variables in our notation, writing simply f ∈ QSym rather than f(x) ∈ QSym(x). There are a number of well-known bases for QSym, all indexed by compo- sitions. (For the two considered here, see [10].) The best-known is the basis of monomial quasisymmetric functions, which here we denote {xα}. Given a composition α with ℓ(α) = k, xα is defined by xα := 1≤i1<i2<···<ik xα1i1 x · · ·xαkik . Another frequently used basis is the set {Lα} of fundamental quasisymmetric functions, defined by Lα := For example, L1 = x 1 is simply the degree one elementary symmetric function. We note that QSym as an algebra under the usual multiplication is graded by degree. For each of the bases described above, the set of basis elements indexed by all the compositions of a fixed weight n forms a basis for the homogeneous component of degree n, QSymn. Accordingly, dimQSymn = 2 2.3 Posets and P -partitions One of the early references to quasisymmetric functions is the paper of Ges- sel [10] (who built on the work of Stanley [17]), where they are related to P - partitions of labeled posets. Most of this material can also be found in Stanley [18]. In the following, we let ≤ denote the usual ordering on integers, and ≤P denote the partial order of a poset P . All posets we consider here are finite. We adopt a mild generalization of Gessel’s convention. We say that a labeled poset on n elements is a partial order on a set of n positive integers. These integers are referred to as the labels of the poset. Usually the set of labels is [n] = {1, 2, . . . , n}, but sometimes we make use of other labels. We often use the same symbol to refer to both the poset and its set of labels when the meaning is clear from context. Note. This convention differs from that used in [3]. There, a labeled poset consists of a pair (P, γ), where P is a poset on an arbitrary set of n elements, and γ is a labeling of P , that is a bijection between the elements of P and the set [n]. The notion is equivalent to Gessel’s. For our generalization, the labeling would be an injective function from the set of elements of the poset into the set P of positive integers. At times we find it convenient to write (P, γ) when we wish to discuss various labelings on the same underlying unlabeled poset. The following is not the actual definition used in [10] and [17], but rather is a formula developed by Stanley in [17]. We take it as our definition here. Definition 2.1. Let P be a labeled poset. Let L(P ) denote the Jordan-Hölder set of P, that is, the set of all permutations in S(P ) that are linear extensions of P . Then the quasisymmetric function of P is F (P ) := π∈L(P ) LC(π). (1) Remark 2.1. The function F (P ) depends only on the relative partial order of the labels at each covering relation of the poset and not on the absolute values of the labels themselves. Given labeled posets P defined on the set of labels A, and P ′ defined on the set of labels B, and a function f : A → B which is an isomorphism of their underlying unlabeled posets, then F (P ) = F (P ′) if for every covering relation (y covers x) in P we have x < y ⇐⇒ f(x) < f(y). 3 The new basis The main goal of this section is to define our new basis (see Definition 3.1 below) and to prove that it is in fact a Z-basis of QSym, that is to say, every quasisymmetric function that can be written in terms of either the standard monomial or fundamental basis using only integer coefficients can also be written in terms of the new basis using only integer coefficients. In Section 4.3, we prove the positivity of the structure constants for this new basis and the grading of QSym by composition rank. Following the notation of [3], given unlabelled posets P and Q, we denote by P ⊕Q their ordinal sum. The set of elements of P ⊕Q is the disjoint union of the elements of P and Q. All of the order relations of P and Q are retained, and in addition, x <P⊕Q y for all x ∈ P and y ∈ Q. As with all the well-known bases for QSym, the elements of the new basis are indexed by compositions. We denote the basis by {Nα}, where α ranges over all compositions. Definition 3.1. For a given composition α 6= 0, let Pα = A1 ⊕ · · · ⊕ Am be the graded poset on |α| elements, where m = ℓ(α) and Ai is an antichain on αi elements. Make Pα into a labeled poset by numbering the ranks in alternating fashion: first number the odd-ranked elements A2, A4, . . ., followed by the even- ranked elements A1, A3, . . .. We define N0 := 1, and for each α 6= 0, we define Nα := F (Pα) (see Equation (1)). Example 3.2. Let α = (1, 2, 2). Then P122 = {3} ⊕ {1, 2} ⊕ {4, 5}, L(P122) = {(31245), (31254), (32145), (32154)}, and N122 = L14 + L131 + L113 + L1121. Definition 3.3. Given a composition α = (α1, . . . , αk), the rank of α, denoted by r(α), is the sum of the odd-indexed parts of the composition. That is, r(α) := odd i αi = α1 + α3 + α5 + · · · . (2) We define N 00 := {N0} = {1}, and for 1 ≤ r ≤ n, Nnr := {Nα : |α| = n and r(α) = r}. (3) We also define the subspace V nr := span N r ⊂ QSymn. If we are working over a field of coefficients for QSym, then V nr may be viewed as a vector space, whereas if we are working over integer coefficients then we refer to the Z-span of Nnr and V nr is a Z-module. Theorem 3.4. The set of quasisymmetric functions {Nα}, as α ranges over all compositions, forms a Z-basis for QSym. Proof. We show that {Nα}|α|=n forms a basis for the homogeneous component QSymn for each nonnegative integer n. This is trivial for n = 0. For the general case, we prove the existence of a unitriangular transition matrix from {Nα}|α|=n of QSymn to the fundamental basis {Lα}|α|=n. Consider the following construction. Given a permutation ω ∈ Sn, let b(ω) ∈ 1{0, 1}n−1 be the n-digit binary word where the digits are given by 1 if i = 1 or ω(i− 1) < ω(i), 0 otherwise. Then define ρ(ω) to be the composition which gives the lengths of succes- sive runs of 1’s and 0’s in b(ω). For example, if ω = 184356729 ∈ S9 then b(ω) = 110011101, and ρ(ω) = 22311. Clearly one can determine the run-length composition C(ω) from ρ(ω) and vice versa. Given a composition α, let Pα be the labeled poset in Definition 3.1, and L(Pα) its set of linear extensions. Recall that by definition Nα := π∈L(Pα) LC(π). By the nature of the labeling on Pα, ρ(π) 4 α for all π ∈ L(Pα). Furthermore, there is a unique element π ∈ L(Pα) such that ρ(π) = α, namely the one in which all the labels of Ai are in ascending order if i is odd, and in descending order if i is even. Thus if we order the rows of the transition matrix (labeled by compositions α) and columns (labeled by compositions ρ(π)) in an arbitrary way that extends the partial refinement order 4, then the resulting matrix is unitriangular, and hence {Nα} is indeed a Z-basis for QSym. 4 Additional facts regarding F (P ) In this section we develop several additional facts regarding the quasisymmetric function F (P ) for labeled posets P , including an alternative way to decompose F (P ) for posets, the main idea being to partition L(P ). These facts, especially Lemmas 4.1 through 4.4, are key tools for the results in following sections. 4.1 Ordered partitions Consider a permutation π = (π1, . . . , πn) ∈ S(X), where |X | = n, and a compo- sition τ = (τ1, . . . , τk) with |τ | = n. We can “chop up”, or segment the one-line notation of π from left to right into k segments, where each respective segment si is a subsequence of consecutive elements of the one-line notation of length τi. We call the sequence of these segments s = (s1, . . . , sk) a segmentation of π of type τ (or induced by τ). Letting t0 = 0 and tj = i=1 τi be the j-th partial sum of the parts of τ , every permutation π ∈ S(X) has a unique segmentation sτ (π), whose segments, for 1 ≤ j ≤ k, are given by sj = (πtj−1+1, πtj−1+2, . . . , πtj ). An ordered partition K = (K1, . . . ,Kk) of a set X ⊂ P is a partitioning of X into non-empty, pairwise disjoint subsets called blocks, i.e. X = ⊔ki=1Ki, where the order of the blocks matters. Let τi = |Ki| for all i, and refer to the resulting composition τ(K) = (τ1, . . . , τk) as the type of K. Let K(X) denote the set of all ordered partitions of X . Every composition τ of weight n induces a mapping Kτ : S(X) → K(X) as follows. For every π ∈ S(X) there is a unique ordered partition Kτ (π), each of whose blocks Kj is the set of elements in the corresponding segment sj of the segmentation sτ (π). We abbreviate the inverse image K−1τ (K) as K −1(K) since, for a given or- dered partition K, the type τ , the set of elements X , and thus the permutation group S(X), can all be determined from K. Thus for an ordered partition K, we have K−1(K) := {π ∈ S(X) : Kτ(K)(π) = K}. (4) For example, K−1(({2, 7}, {5}, {1, 8})) = {27518, 27581, 72518, 72581}. The following lemma is simply an exercise in notation, so we omit its proof. Lemma 4.1. Let K be an ordered partition with k blocks. Let PK be the labeled poset PK = K1 ⊕ · · · ⊕Kk, where each Ki is regarded as an antichain. Then F (PK) = π∈K−1(K) LC(π). We say that an ordered partitionK = (K1, . . . ,Kk) is alternating if for every 1 ≤ i < k and for all x ∈ Ki and y ∈ Ki+1 we have x < y if i is even and x > y if i is odd. Lemma 4.2. Let K be an alternating ordered partition of type τ . Then F (PK) = Nτ . Proof. Each rank Ki of PK = K1 ⊕ · · · ⊕Kk (where ℓ(τ) = k) is an antichain. Hence F (PK) depends only on the relative ordering of the elements between adjacent ranks Ki and Ki+1 (see Remark 2.1). Since K is alternating, we can relabel its elements in each rank as we do in the construction of Pτ (as in Definition 3.1) and still maintain the same relative ordering between elements in adjacent ranks. Thus F (PK) = F (Pτ ) = Nτ . 4.2 Unordered partitions of X ⊂ P Let T = {T1, . . . , Tm} be an unordered partition of the set X ⊂ P. We say that an ordered partition K is a refinement of T if K, considered as an unordered partition, is a refinement of T . For every permutation π ∈ S(X), T induces a unique segmentation of π where each segment is contained in a block of T and this segmentation is least (coarsest), with respect to refinement, among all such segmentations. Corresponding to this segmentation there is a unique ordered partition KT (π), which clearly is is a refinement of T . We say that T induces the ordered partition KT (π) on π. Example 4.3. Let X = [9], T = { {1, 4}, {2, 6, 8, 9}, {3, 5, 7} }, π = 965412378. Then KT (π) = ({6, 9}, {5}, {1, 4}, {2}, {3, 7}, {8}). Let P be a labeled poset, and T an unordered partition of P . Define KP,T to be the set of induced ordered partitions KP,T := {KT (π) | π ∈ L(P )}. We say that T is antichain-inducing if for every ordered partition K ∈ KP,T , every block Ki of K is an antichain in P . Lemma 4.4. Let T be an antichain-inducing unordered partition of a labeled poset P . Then F (P ) = K∈KP,T F (PK). (5) We call this the decomposition of F (P ) with respect to T . Proof. By Lemma 4.1 it suffices to show that L(P ) = K∈KP,T K−1(K). The “⊂”-direction is trivial. Indeed, T induces some ordered partition on every permutation, and by definition KP,T includes all such partitions as permutations range over L(P ). Also, clearly K−1(K) ∩ K−1(J) = ∅ if K 6= J since KT is a well-defined map on L(P ), and so the union on the right is indeed a disjoint union. For the “⊃”-direction, let K ∈ KP,T . By definition of KP,T , there exists π ∈ L(P ) ∩ K−1(K). Let s = sτ(K)(π). Since T is antichain-inducing, the unordered set of elements Ki of each segment si is an antichain. It follows that if we form a new permutation π̂ by permuting the elements of si arbitrarily within si (and thus within π), we must also have that π̂ ∈ L(P ). Since this holds true for each segment of s, we have K−1(K) ⊂ L(P ). Remark. In the extreme case where T consists of all singleton sets, KT (π) is the list of singleton sets in the order specified by π, and K−1(KT (π)) = {π}. We can identify KT (π) with π itself, and similarly KP,T with L(P ), and the lemma is then equivalent to the formula (1). 4.3 Structure constants for the new basis Following the notation of [3] and [17], given labeled posets P and Q on sets X and Y respectively, we denote by P + Q any disjoint sum of the posets, constructed as follows. We first form the poset whose set of elements is the disjoint union of the sets of elements of P and Q, retaining all partial order relations of the two posets but adding no new relations. In order to ensure that all labels are distinct, we then relabel the elements in any fashion subject to the restriction that the resulting labels are all distinct and preserve the relative order of labels at all covering relations (see Remark 2.1). While the disjoint sum of the labeled posets is not uniquely defined, all disjoint sums so constructed will have the same quasisymmetric function. It is well-known and is easy to prove (see, for example, [10]) that F (P +Q) = F (P ) · F (Q). (6) We are now in a position to prove the nonnegativity of the structure constants for our new basis. Theorem 4.5. The quasisymmetric function algebra QSym is graded by the rank of the compositions indexing the basis {Nα}. Furthermore, the structure constants for {Nα} are nonnegative. That is, in the expansion NαNβ = cνα,βNν , all the constants cνα,β are nonnegative integers. Proof. We first prove the statement regarding structure constants. Since N0 = 1, the claim holds trivially if α = 0 or β = 0. Thus we assume α = (α1, . . . , αs) 6= 0 and β = (β1, . . . , βt) 6= 0. By (6) we have that NαNβ = F (Pα)F (Pβ) = F (Pα + Pβ). (7) We write Pα = A1 ⊕ · · · ⊕As and Pβ = B1 ⊕ · · · ⊕Bt, and identify the Ai and Bj subsets with their canonical inclusions in Pα + Pβ . We form a new poset Q by relabeling the elements of the Ai and Bj subsets while maintaining their ordering relations: first label the even-indexed Ai and Bj in order, with the numbers from [m], where m = |α|+ |β| − r(α)− r(β) and r(α) is the rank function from Definition 3.3, then label the odd-indexed Ai and Bj in order, with the numbers from {m + 1, . . . , |α| + |β|}. Since F (Pα + Pβ) depends only on the relative ordering of elements between adjacent ranks Ai and Ai+1 for 1 ≤ i < s and between adjacent ranks Bj and Bj+1 for 1 ≤ j < t, we have F (Pα + Pβ) = F (Q). (8) We consider the unordered partition T = {T1, T2} of Q given by odd i odd i , and T2 = even i even i Note that T is antichain-inducing, so we may apply Lemma 4.4: F (Q) = K∈KQ,T F (PK). (9) On the other hand, the labeling of Q implies that every ordered partition K ∈ KQ,T is alternating, so applying Lemma 4.2, we have F (Q) = K∈KQ,T Nτ(K). (10) Combining Equations (7) – (10) yields the positivity claim. In particular, cνα,β = |{K ∈ KQ,T : τ(K) = ν}|. To prove the statement regarding the grading of QSym by composition rank, we simply note that for every K ∈ KQ,T , we have r(τ(K)) = |T1| = r(α) + r(β). 5 Matroids This section begins the second part of the paper. Here we review some of the concepts, terminology, and results from [3], and prove our claims regarding the quasisymmetric functions of matroids vis-a-vis our new basis. For general background in matroid theory we refer the reader to standard texts such as Oxley’s [14]. We review several of the terms here. The direct sum of matroids M1 and M2, denoted M1⊕M2, has as its ground set the disjoint union E(M1 ⊕M2) = E(M1) ⊔ E(M2), and as its bases B(M1 ⊕M2) = {B1 ⊔B2 : B1 ∈ B(M1), B2 ∈ B(M2)}. A circuit is a minimal dependent set. If we declare two elements of a matroid to be equivalent if and only if they are both contained in some circuit, then the equivalence classes of elements are the components of the matroid. We say that the matroid is connected if it has only one component, and disconnected otherwise. A matroid is the direct sum of its components. 5.1 The quasisymmetric function of a matroid Billera, Jia, and Reiner [3] describe an invariant for isomorphism classes of matroids in the form of a quasisymmetric function. Rather than give the defi- nition from [3], we describe it in terms of a formula which is shown in [3] to be equivalent to the definition. Fix a matroid M , one of its bases B ∈ B(M), and let Bc = E(M)−B (the cobase of B). Define the poset PB on the ground set E(M) where e <PB e and only if e ∈ B, e′ ∈ Bc, and (B − e)∪ {e′} ∈ B(M). That is, e <PB e ′ if and only if swapping e′ for e in B yields another base in M . Thus the Hasse diagram of PB is a bipartite graph in which the elements of B are minimal elements of the poset and the elements of Bc are maximal elements. Note that if M has no loops, then in the Hasse diagram of PB , every element in B c has positive vertex degree. We say that a labeled poset is strictly labeled if for all x, y ∈ P we have that x <P y implies x > y. Similarly, a labeled poset is naturally labeled if for all x, y ∈ P , x <P y implies x < y. We apply a strict labeling to PB (any will do). The quasisymmetric function F (M) associated with M can be written as F (M) = B∈B(M) F (PB), (11) where F (PB) is the quasisymmetric function of the strictly labeled poset PB as defined in Definition 2.1. It was shown in [3] that the mapping F : Mat → QSym is in fact a morphism of combinatorial Hopf algebras, with a suitable choice of character on the algebra Mat. Here Mat is the Hopf algebra of matroids introduced by Schmitt [15] and studied by Crapo and Schmitt [4], [5], [6], [7]. The matroid algebra Mat has as its basis elements isomorphism classes of matroids. The product of two basis elements [M1] and [M2] in the algebra is given by [M1] · [M2] := [M1 ⊕ M2], where M1 ⊕M2 denotes the direct sum of matroids. Comultiplication in Mat is given by ∆([M ]) := A⊂E(M)[M |A]⊗ [M \A], where M |A is the restriction of M to A, and M \ A is the contraction of M by A. Under the morphism F we have that F (M1 ⊕M2) = F (M1) · F (M2). Billera, Jia, and Reiner also show that the mapping F , while not surjective over integer coefficients, is surjective over rational coefficients. The mapping F : Mat → QSym does not distinguish between loops and coloops. Indeed, let M ′ extend the matroid M by adding a loop ℓ, i.e. M ′ = M ⊕ {ℓ}, and let M ′′ extend the matroid M by adding a coloop c, i.e. M ′′ = M ⊕ {c}. Then F (M ′) = F (M ′′) = F (M) · L1. (12) Here L1 is the fundamental basis function indexed by the composition (1), which is the elementary symmetric function e1(x). Define an equivalence relation ∼ on isomorphism classes of matroids by [M1] ∼ [M2] if and only if one can obtain M1 from M2 by changing some number of loops to coloops or vice versa. Then by Equation (12) the mapping Mat → QSym factors through the quotient Mat → Mat/∼ → QSym. Accordingly, throughout most of our paper, we assume that, unless otherwise specified, our matroids have no loops; that is, out of each equivalence class in Mat/∼ we select the representative that has no loops when considering their images in QSym. 5.2 Expanding F (M) in the {N } basis Recall from Definition 3.3 that Nnr = {Nα : |α| = n, r(α) = r} and V spanNnr . In this subsection we may take our coefficient ring to be Z if we wish. Lemma 5.1. Let P be a strictly labeled poset on n elements of rank at most one and with r minimal elements. Then F (P ) ∈ V nr . Moreover the expan- sion of F (P ) in terms of the basis elements Nnr has only nonnegative integer coefficients. Proof. If P has rank 0, then P is an antichain and so r = n. Thus by labelling P with the elements of [n] and taking α = (n), we have F (P ) = F (Pα) = Nα = N(n) ∈ V Otherwise P has rank 1 and is not an antichain. Let T = {T1, T2} be the unordered partition of P in which T1 comprises the r minimal elements of P , and T2 the remaining elements. Note that some elements may be both minimal and maximal, and these will be placed in T1. Thus every element in T2 in the Hasse diagram of P has positive vertex degree. Since we are interested in computing F (P ), we may assume without loss of generality that P has a strict labeling which labels the elements of T1 = {n, n− 1, . . . , n− r+ 1} in arbitrary fashion, and the elements of T2 = {1, 2, . . . , n − r} in arbitrary fashion. Since T1 and T2 are themselves antichains, T is antichain-inducing, so by Lemma 4.4: F (P ) = K∈KP,T F (PK). Moreover, by the choice of labeling and the fact that every element in T2 has positive vertex degree, we have that every K ∈ KP,T is alternating. Lemma 4.2 then implies F (P ) = K∈KP,T Nτ(K). We also have that |τ(K)| = n and r(τ(K)) = odd i τi = |T1| = r, thus Nτ(K) ∈ V nr for every K ∈ KP,T . Hence F (P ) ∈ V r as claimed. Theorem 5.2. Let M be a loopless matroid of rank r on n elements. Then F (M) ∈ V nr . Moreover the expansion of F (M) in terms of the basis elements Nnr has only nonnegative integer coefficients. Proof. For a loopless matroid M , for every base B ∈ B(M), the base poset PB has r minimal elements out of a total of n elements, and rank at most one. The assertion then follows by Lemma 5.1, and the formula in Equation (11). We make a few observations here about the coefficients in the expansion of the quasisymmetric function F (M) of a matroid M in terms of our new basis. Given a quasisymmetric function q, define supp(q) = α : mα 6= 0 in the expansion q = We first note that for a (loopless) matroid M of rank r on n elements, the coefficient m(r,n−r) of Nα, where α = (r, n− r), is equal to the number of bases of M . Example 5.3. Let M = Ur,n be the uniform matroid of rank r on n elements. By definition, its bases are all the r-subsets of the ground set E(M), i.e. B(M) =( . Then every PB has a complete bipartite Kr,n−r graph for its Hasse diagram. Therefore F (Ur,n) = Nr,n−r, and supp(F (M)) = {(r, n− r)}. Thus the values of m(r,n−r) (how small) and |supp(F (M))| (how large) are, to some extent, measures of the degree to which M fails to be uniform. The coefficient mα where α = (r− 1, 1, 1, n− r− 1) also has a combinatorial interpretation. There is such an Nα term for every edge “missing” from the Hasse diagram of a base poset PB as compared to the complete bipartite graph Kr,n−r. In terms of matroid base polytopes, which are discussed in 6.1, the polytope Q(Ur,n−r) contains all possible vertices, namely , while the base polytope for a different matroid M of same rank and ground set size has only a subset of them, namely B(M). The coefficient mα is the number of edges in the 1-skeleton of Q(Ur,n−r) between the set of vertices B(M) and its complement( − B(M). Lemma 5.4. Let M be a matroid, possibly containing loops. Then the total number c of loops and coloops is given by c = max α∈supp(F (M)), ℓ(α) odd α̇, (13) where α̇ denotes the last part of the composition α. Proof. Since the morphism F : Mat → QSym factors through loop-coloop equiv- alence, F (M) = F (M ′) where M ′ is obtained from M by replacing all loops of M with coloops. Since M and M ′ both have the same total number of loops and coloops, without loss of generality, we assume that M has no loops. Consider a typical strictly labeled base poset PB of M, and antichain inducing partition {B,Bc} as in the proof of Lemma 5.1. We have α ∈ supp(F (M)) if and only if there is an induced ordered partition K of PB of type α. Now ℓ(α) is odd if and only if the last block of K is a subset of B, and in this case the elements in this block must be coloops. Thus c ≥ α̇. Conversely, there always exists an induced ordered partition of the poset, say of type α, which has all of the coloops of M in the last block, i.e. c = α̇, and this ordered partition will have odd length. The result follows. 6 Matroid base polytopes In this section we recall the context of [3, Section 7] regarding the relationship between decompositions of the quasisymmetric function associated to a matroid and decompositions of its matroid base polytope. In Subsection 6.2 we develop a formula for the quasisymmetric function of a loopless rank two matroid in terms of the new basis, and apply it to address [3, Question 7.12] and [3, Question 7.10]. 6.1 Matroid base polytopes and their decompositions The motivating context is to study the decompositions of the matroid base polytope Q(M) of a matroid M . This topic arises in the work of Lafforgue [12], [13], Kapranov [11, §1.2 – 1.4], and can be found in the work of Speyer [16]. If M is a matroid with |E(M)| = n, we define the matroid base polytope Q(M) by identifying E(M) with the set of standard basis vectors {ei} i=1 of R and declaring Q(M) := conv ei : B ∈ B(M) where B(M) is the set of bases of M . Useful facts about matroid base polytopes (see [9]), which we quote without proof, are: 1. If M has rank r, then Q(M) lies in the hyperplane {x ∈ Rn : i xi = r}. 2. There is an edge in Q(M) between vertices (bases) B1 and B2 if and only if there exist a pair of elements ei ∈ B1 and ej ∈ B2 such that B2 = (B1 − {ei}) ∪ {ej}. 3. Each face of a matroid base polytope is in turn the base polytope of some matroid. 4. The dimension of Q(M) is |E(M)| − s(M), where s(M) is the number of connected components of M . Billera, Jia, and Reiner define a matroid base polytope decomposition of Q(M) to be a decomposition Q(M) = Q(Mi), (14) where each Q(Mi) is also a matroid base polytope for some matroid, and for each i 6= j, the intersection Q(Mi) ∩ Q(Mj) = Q(Mi ∩ Mj) is a face of both Q(Mi) and Q(Mj). They call such a decomposition a split if t = 2. Billera, Jia, and Reiner show that the mapping F : Mat → QSym behaves as a valuation on matroid base polytopes. (See [2] for a discussion of valuations.) This implies that, given a matroid base polytope decomposition as in Equation (14), F (M) can be expressed in terms of the set of F (Mj) in an inclusion- exclusion fashion, where the Mj are the matroids of the faces of the constituent polytopes in the decomposition. For example, given a split Q(M) = Q(M1) ∪ Q(M2), we have F (M) = F (M1) + F (M2) − F (M1 ∩M2), where Q(M1 ∩M2) is necessarily a lower-dimensional face. Hence by Fact 4 above, the matroid M1 ∩M2 is disconnected, and so F (M1 ∩M2) can be expressed as a product. If we let m := d≥1 QSymd be the maximal ideal in the ring QSym, then F (M1∩M2) ∈ m 2. Therefore in the quotient space QSym/m2, we have F (M) = F (M1) + F (M2). In general, given a matroid base polytope decomposition as in Equation (14), there is an algebraic decomposition modulo m2 F (M) = F (Mi). (15) One of the open questions raised by Billera, Jia, and Reiner [3] is under what conditions the converse may hold; given a collection of matroids satisfying (15), what additional conditions are sufficient to conclude (14)? Note. So far we have ignored the distinction between the isomorphism class of a matroid on the one hand and a specific instance of that class on a given ground set on the other, since the quasisymmetric function of a matroid is invariant on the elements of the same isomorphism class. When discussing the existence of matroid base polytope decompositions, it is sometimes necessary to draw a distinction between the notions, as is done in the statement of Theorem 6.2 below. When this precision is necessary, we use the usual bracket notation [M ] to denote the isomorphism class of the matroid M . Given a ground set size n, the converse question is trivial for rank 0 and 1, and by matroid duality, for rank n and n − 1. One necessary condition they point out is that a specific set of matroids on a common ground set satisfying (14) must at least satisfy the condition B(Mi) ⊂ B(M) for all i, in which case they say that (15) is a weak image decomposition and that F (M) is weak image decomposable. They specifically ask, [3, Question 7.12] Does F (M) being weak image decomposable in QSym/m2 imply that Q(M) is decomposable? So far, general sufficient conditions are not known beyond the trivial ranks listed above. We claim that the converse ((15) ⇒ (14)) holds quite generally for rank two matroids, as shown in Section 6.2. By matroid duality, the converse also holds for matroids of corank two. As discussed in [3], the loopless rank two matroids are indexed, up to iso- morphism, by partitions having two or more parts, where there are as many parts as there are parellelism classes of elements in the matroid and the parts of the partition give the respective cardinalities of these classes. For this section we write Mλ to denote the loopless rank two matroid indexed by the partition λ. More generally, given a composition α, define Mα = Mλ where λ is the decreasing rearrangement of the parts of α. Kapranov [11, §1.3] gives a description of all decompositions of rank two matroid base polytopes. He shows [11, Lemma 1.3.14] that in rank two, all matroid base polytope decompositions arise from hyperplane splits. We provide some description here of the geometric situation, in our own words. Given the composition λ = (λ1, . . . , λm), with |λ| = n, set t0 = 0 and for 0 ≤ k ≤ m set i=1 λi be the k-th partial sum. Then the vertices of Q(Mλ) are precisely those 0/1-lattice points v lying in the hyperplane H = {x ∈ Rn : i xi = 2} subject to the restriction that tk+1∑ i=tk+1 vi ≤ 1 for all 0 ≤ k < m. If ℓ(λ) = 2, then Mλ = M(λ1,λ2) = U1,λ1 ⊕ U1,λ2 , where U1,n is the uniform matroid of rank 1 on n elements. It follows that if ℓ(λ) = 2, then dimQ(Mλ) = n− 2 and F (Mλ) ∈ m Supposing that ℓ(λ) = m > 3, choose index j such that 1 < j < m− 1. Let a = tj and b = n− tj , and define compositions µ = (a, b), α = (a, λj+1, . . . , λm), and β = (λ1, . . . , λj , b), all of which have weight n. Consider the hyperplane H ′ = {x ∈ Rn : i=1 xi = 1}. Then H ′ ∩ Q(Mλ) = Q(Mµ), giving us a hyperplane split Q(Mλ) = Q(Mα) ∪ Q(Mβ). It follows from the above that F (Mλ) = F (Mα) + F (Mβ) − F (Mµ), and F (Mλ) = F (Mα) + F (Mβ). We can summarize this in the following proposition. The relations given in the proposition remain true even if λ has only two or three parts, but in that case the resulting relations are trivial. Proposition 6.1. Let λ = (λ1, . . . , λt) be a composition with at least two parts. Let 1 ≤ s < t, a = i=1 λi, and b = i=s+1 λi. Consider compositions α = (a, λs+1, . . . , λt), β = (λ1, . . . , λs, b), and µ = (a, b). We then have F (Mλ) = F (Mα) + F (Mβ)− F (Mµ), and modulo m2, F (Mλ) = F (Mα) + F (Mβ). Moreover there is a split of matroid base polytopes Q(Mλ) = Q(Mα) ∪Q(Mβ). The splitting process can be repeated on the constituent matroid base polytopes until we have decomposed Q(Mλ) into the union of matroid base polytopes of type Q(Mα) where ℓ(α) = 3. Consequently, modulo m 2, F (Mλ) can be written as a positive sum F (Mλ) = F (Mi), where each Mi is a loopless rank 2 matroid indexed by a partition of length 3. In this setting, Billera, Jia, and Reiner , pose the following question: [3, Question 7.10] Fix n and consider the semigroup generated by F (M) within QSymn/m 2 as one ranges over all matroids M of rank 2 on n elements. Is the Hilbert basis for this semigroup indexed by those M for which λ(M) has exactly 3 parts? By repeated application of Proposition 6.1, the set {F (Mλ) : ℓ(λ) = 3} generates the semigroup in question, so the point of the question is whether this generating set is minimal, and whether distinct indices yield distinct functions. We prove that this is the case as a corollary of Theorem 6.2. 6.2 Results for rank two matroids In this section, we prove that the morphism F : Mat → QSym distinguishes isomorphism classes of rank two matroids and that decomposability of F (M) for a rank two matroid M implies decomposability of Q(M), as stated in the following theorem. Theorem 6.2. Let λ ⊢ n with ℓ(λ) ≥ 3, and let J be a multiset of partitions of n, all of length three or more, such that F ([Mλ]) = F ([Mµ]), (16) where [Mτ ] denotes the isomorphism class of (loopless) rank two matroids on n elements indexed by the partition τ . Then, taking the set of standard basis vec- tors of Rn as the common ground set, there exists a collection of representative matroids on this ground set, Mλ ∈ [Mλ] and Mµ ∈ [Mµ] for all µ ∈ J which form a decomposition of matroid base polytopes Q(Mλ) = Q(Mµ). (17) Before the main proof of this theorem, we establish some preliminary results. We begin by developing a formula for F (Mλ) in terms of the new basis {Nα}. We define the following quasisymmetric functions in V n2 = span{Nα : |α| = n, r(α) = 2}. For all 1 ≤ k ≤ n− 1 let T nk := N(2,n−2) + k − 1 N(1,j,1,n−2−j), (18) where we understand N(1,j,1,n−2−j) to be N(1,n−2,1) when j = n − 2. We also define quasisymmetric functions Unk := k(n− k)T Note that each of the sets {T nk } and {U k } forms a basis for the subspace V where we consider QSym to have rational coefficients. Lemma 6.3. Let Mλ be the rank two matroid on n elements indexed by the partition λ = (λ1, . . . , λm). Then F (Mλ) = Unλi . (19) Proof. We write c(λi) to denote the parallelism class of elements in Mλ cor- responding to the part λi. A typical base B ∈ B(Mλ) is B = {ei, ej}, where ei ∈ c(λi) and ej ∈ c(λj) are in distinct parallelism classes. The Hasse diagram of PB has two minimal elements, ei and ej. There are edges from ei to all elements of the cobase Bc = E(Mλ) − B except for the λj − 1 elements which are in the same parallelism class c(λj) as ej. Similarly, there are edges from ej to all elements of the cobase except for the λi − 1 elements which are in the same parallelism class c(λi) as ei. We can analyze F (PB) as in the proof of Lemma 5.1 by applying a strict labeling γ : E(Mλ) → [n] such that γ(ei) = n, γ(ej) = n − 1, and the cobase elements are arbitrarily labeled with {1, 2, . . . , n − 2}. We take T = {B,Bc} to be our antichain-inducing partition of (PB , γ). There is one induced ordered partition (of [n]) of type (2, n − 2), namely K = (B,Bc), classifying one set of permutations in L(PB , γ), and thus contributing one N(2,n−2) term to the expansion of F (PB). For each 1 ≤ k < λj , and for each k-set A ⊂ B c(λj), there is an induced ordered partition K = ({ej}, A, {ei}, B c −A) of type (1, k, 1, n − 2 − k) contributing a term N(1,k,1,n−2−k) to the expansion. Thus there are such terms N(1,k,1,n−2−k) corresponding to ordered partitions K of type (1, k, 1, n−2−k) withK1 = {ej}. Likewise there are such terms N(1,k,1,n−2−k) corresponding to ordered partitions K of type (1, k, 1, n− 2− k) with K1 = {ei}. All the Nα ∈ N 2 are of one of these types, and we know that the terms of F (PB) must lie in V 2 , so these are the only types appearing in the expansion for F (PB). There can be no other terms than these due to the order relations in PB. Thus F (PB) = N(2,n−2) + λi − 1 λj − 1 N(1,k,1,n−2−k). (20) Using Equation (18), we can rewrite this as F (PB) = T + T nλj . Finally, there are λiλj such bases B ∈ c(λi)× c(λj). Summing over all pairs of parallelism classes of the matroid yields the formula F (Mλ) = λi(n− λi)T Unλi . Next we develop a similar formula for F (Mλ) in QSymn/m 2. Our starting point is the following corollary. Corollary 6.4. Let a and b be positive integers such that a+ b = n. Then ab ·N(1,a−1) ·N(1,b−1) = U a + U Proof. Let λ = (a, b). ThenMλ = U1,a⊕U1,b, where U1,m is the uniform matroid of rank one onm elements. As discussed in Example 5.3, F (U1,m) = mN(1,m−1). Therefore by the Hopf algebra morphism, we have F (Mλ) = F (U1,a) · F (U1,b) = aN(1,a−1) · bN(1,b−1). On the other hand, by Lemma 6.3 we have F (Mλ) = U a + U b . Equating right hand sides yields the desired formula. Since QSym with respect to its product structure is graded by composition rank as well as degree, the vector subspace V n2 ∩m 2 is spanned by the vectors {N(1,a−1) ·N(1,b−1) : a+ b = n}. Thus a basis for V n2 ∩ m 2 is {Unk + U n−k : 1 ≤ k ≤ }. For expressing our formula for F (Mλ), we find it convenient to define vectors U k as follows: Unk = Unk if k < 0 if k = n −Unn−k if k > Thus the set {Unk : 1 ≤ k < } forms a basis (over rational coefficients) for V n2 /m 2. We have the immediate corollary of Lemma 6.3: Corollary 6.5. Let Mλ be the rank two matroid on n elements indexed by the partition λ = (λ1, . . . , λm). Then F (Mλ) = Unλi . (22) The next proposition provides a necessary step for the main result, but may be of interest in its own right. Proposition 6.6. Let M2 be the set of matroid isomorphism classes (including those with loops) of rank two matroids. Let Matc be the vector subspace of Mat spanned by the isomorphism classes of connected matroids, and let M2c be the set of matroid isomorphism classes of connected rank two matroids. Then the algebra morphism F : Mat → QSym is injective when restricted to M2, and the induced quotient map of vector spaces F : Matc → QSym/m 2 is injective when restricted to M2c. Proof. We show that we can recover the isomorphism class of the matroid from its respective function. Suppose we are given F (M) for a rank two matroid M . We know that F (M) is a non-zero homogeneous function of degree n = |E(M)|, and so we recover the size of the ground set. Clearly, n ≥ 2. It is possible that M may have loops or coloops. By Lemma 5.4 we can recover the total number s of loops and coloops of M from F (M) by Equation (13). If s = n, then M consists of two coloops and n − 2 loops. Otherwise s ≤ n− 2, and we may factor F (M) as F (M) = N(s) · F (M where (s) is the one-part composition of s, and M ′ is the matroid obtained from M by removing all loops and coloops. If now F (M ′) ∈ V n−s1 , we have M ′ ∼= U1,n−s and M has one coloop and s− 1 loops. Otherwise M has s loops, no coloops, F (M ′) ∈ V n−s2 , and M ′ is a loopless rank two matroid on n − s elements. So now without loss of generality, we assume that M has no loops or coloops and thus is isomorphic to Mλ for some λ ⊢ n. We expand F (M) as F (M) = k . (23) This expansion can be determined since the set of {Unk } form a basis of V 2 . Per Lemma 6.3, for each k, the coefficient tk is the number of parts of λ that are equal to k, and so we recover λ from F (Mλ). The argument for recoveringM from F (M) for a connected rank two matroid M is similar. Since M is connected, it has no loops or coloops, and so again M is isomorphic to Mλ for some λ ⊢ n with ℓ(λ) ≥ 3, where n is the degree of F (M). We expand F (M) = ⌊(n−1)/2⌋∑ k . (24) This expansion can be determined since the set {Unk : 1 ≤ k < } forms a basis for the subspace V n2 /m 2. Note that λ cannot have a pair of parts with values k and n − k. Using this fact together with Corollary 6.5, we see that if the coefficient tk is nonnegative, then λ has exactly tk parts with value k. From this we can determine all the parts of λ which are < n . Since λ cannot have more than one part ≥ n , this allows us to determine the remaining part of λ, if Proof of Theorem 6.2. We write A⊔B to denote the disjoint union of multisets A and B. Note that a partition may be considered to be a multiset of integers. We fix n > 2 and λ ⊢ n with ℓ(λ) ≥ 3, and proceed by induction on |J |. The base case |J | = 1 follows from Proposition 6.6. So we assume that the statement holds for |J | < m for some fixed m > 1. Suppose now that F ([Mλ]) = F ([Mµ]), (25) where |J | = m. Say that a pair of elements µ, ν ∈ J are matching if for some value 1 < k < n − 1 we have k ∈ µ and n − k ∈ ν. If µ, ν are a matching pair, then we can apply Proposition 6.1 to form a new relation of type (25) by replacing J with J ′ = (J − {µ, ν}) ⊔ {τ}, where τ = (µ ⊔ ν) − {k, n − k}. At the same time, Proposition 6.1 tells us that we also have a decomposition of base polytopes Q(Mτ ) = Q(Mµ) ∪ Q(Mν). Since |J ′| < m, we can apply our induction hypothesis, and we are done. It remains to show that there exists a matching pair in J . For a partition τ ⊢ n, define the multiset g(τ) = {τi : τi > 1, τi 6= Define multisets L = g(λ) and R = µ∈J g(µ). Per Corollary 6.5 we expand F (Mλ) = ℓ(λ)∑ Unλi , and we similarly expand each F (Mµ) on the right hand side of (25). Since the set {Unk : 1 ≤ k < } forms a basis for V n2 /m 2, with Unk = −U n−k, we conclude that L ⊆ R and that the parts in R − L can be matched into complementary pairs of the form (k, n− k). Since no partition in J can contain both parts of a complementary pair, there exists a matching pair in J if R− L 6= ∅. We are assuming that |J | ≥ 2, and that R and L contain all the parts not equal to n or 1 on the respective sides of (25). The parts equal to 1 on both sides must match since all of the partitions have at least three parts and hence no part equal to (n− 1). The only way to have R−L = ∅ is if there exist µ, ν ∈ J each of which contains a part equal to n , in which case they are matching. Thus in all cases, there exists a matching pair µ, ν ∈ J , and the result follows by induction. Now we can give an affirmative answer to [3, Question 7.10]. Corollary 6.7. For a fixed n, the Hilbert basis for the semigroup in QSym/m2 generated by the set S = {F (Mλ) : λ ⊢ n, ℓ(λ) ≥ 3} is indexed by those Mλ for which ℓ(λ) = 3. Proof. Let T = {F (Mλ) : λ ⊢ n, ℓ(λ) = 3}. It follows from Proposition 6.1 that for ℓ(λ) > 3, F (Mλ) is decomposable into a sum µ F (Mµ), where for all µ, ℓ(µ) < ℓ(λ). Hence T generates the same semigroup as S. As noted in [3, Section 7], if ℓ(λ) = 3, then Q(Mλ) is indecomposable. Theorem 6.2 then implies that F (Mλ) must also be indecomposable, so T is the minimal generating set, i.e. the Hilbert basis of the semigroup. By Proposition 6.6, distinct indexing partitions yield distinct images, establishing the claim. 7 Additional observations In this section we discuss additional aspects of our new basis, especially regard- ing the expansion of F (M) for a matroid M . 7.1 Matroid duality, loops, and coloops Although we describe the basis {Nα} as ‘matroid-friendly’, things are slightly less friendly when considering matroid duality in the presence of coloops. This is due to the fact, mentioned in Section 5.1, that the mapping F : Mat → QSym factors through the quotient Mat → Mat/∼ → QSym, where ∼ denotes loop- coloop equivalence. For example, a fact proved in [3] is that, for any matroid M , in terms of the monomial basis for QSym the following relationship holds: F (M) = α =⇒ F (M∗) = α∗ . (26) where α∗ is the reversal of α, obtained by writing the parts of α in reverse order. If M be is a matroid of rank r on n elements having no loops or coloops, then we have the analogous relationship F (M) = mαNα =⇒ F (M mαNα∗ . (27) However this relationship breaks down if M has loops or coloops. We showed in Theorem 5.2 that if M is a loopless matroid of rank r on n elements, then F (M) ∈ V nr . More generally, if M is of rank r on n elements and has exactly ℓ loops, then F (M) ∈ V nr+ℓ. Thus if M has exactly c coloops, then we have the duality relationship F (M) ∈ V nr =⇒ F (M ∗) ∈ V nn−r+c. 7.2 Comultiplication The matroid Hopf algebra is graded by matroid rank as well as ground set size. Let Wnr be the subspace of Mat spanned by the classes of matroids of rank r on n elements. Then Wnr ·W s ⊂ W r+s . For any matroid M and A ⊆ E(M), r(M) = r(M |A) + r(M/A). So comultiplication in Mat also respects these gradings. (For general background on Hopf algebras, see [8].) That is, ∆Wnr ⊆ a+b=n, s+t=r W as ⊗W . (28) One might wonder whether the standard comultiplication of the Hopf algebra QSym respects the grading by the rank function for our new basis, that is, whether ∆V nr ⊆ a+b=n, s+t=r V as ⊗ V . (29) This is not the case. For the simplest example, consider n = 2 and r = 1. We have N 00 = {N0} = {1} and N 1 = {N11} = {x 11}. Note that there is no Nm0 (or rather, Nm0 = ∅) for m > 0. The basis vectors corresponding to the right hand side of (29) are N11 ⊗N0 = x 11 ⊗ 1, and N0 ⊗N11 = 1⊗ x However, ∆N11 = ∆x 11 = x11 ⊗ 1 + x1 ⊗ x1 + 1⊗ x11, which clearly does not lie in the span of the above vectors. The failure of the comultiplication to respect the rank grading can be viewed as another artifact of loop-coloop equivalence under the morphism F , as evi- denced by the fact that the rank grading is respected by comultiplication in the quotient space corresponding to matroids with neither loops nor coloops. Let J ⊂ QSym be the ideal generated by degree one elements, i.e. by {N1} = {x Similarly, let I ⊂ Mat be the ideal generated by degree one elements, i.e. by {[U0,1], [U1,1]}. Both I and J are Hopf ideals in their respective Hopf algebras, hence Mat/I and QSym/J (with their naturally induced comultiplications) are Hopf algebras. Moreover, I = F−1(J), so F : Mat → QSym induces a sur- jective Hopf algebra morphism Mat/I → QSym/J . Note that a natural basis for Mat/I is the set of all matroid isomorphism classes that have neither loops nor coloops, while a natural basis for QSym/J is {Nα : ℓ(α) is even}. Taking appropriate images under the quotient map, the relation (29) holds in QSym/J . The duality formula (27) also holds in QSym/J . 7.3 Comparison with other QSym bases In the course of their proof in Section 10 of [3], the authors introduce two new Z-bases for QSym. They also compare their bases to another Z-basis due to Stanley [19]. Our new basis is different from these three, as evidenced by the report by those authors that all three of these bases have some negative structure con- stants, whereas our new basis does not. However, of the three, ours most closely resembles that of Stanley. Stanley’s basis element indexed by a composition α = (α1, . . . , αm) is F (P ) where, as with our basis, P = A1 ⊕ · · · ⊕ Am, is the ordered sum of antichains A1, · · · , Am on α1, . . . , αm elements respectively. However, Stanley applies a natural labeling to P , whereas we apply an alter- nating labeling to the ranks in the poset for our basis. 7.4 Surjectivity of the Hopf algebra morphism Billera, Jia, and Reiner devote [3, Section 10] to showing that the morphism F : Mat → QSym is surjective over rational coefficients. In this subsection we sketch one way to shorten their proof somewhat using our new basis. The reader will need to consult [3] to have the full context. Define an ordering on compositions as follows. To each composition α we assign the binary word b(α) that begins with α1 zeros followed by α2 ones, then α3 zeros, then α4 ones, etc. We then linearly order compositions according to their binary words: α < β if b(α) <lex b(β). In their proof, Billera, Jia, and Reiner make use of a novel basis for the quasisymmetric functions based on a family of posets {Rσ} of maximum rank one, indexed by binary words σ ∈ 0{0, 1}n−1, where n is the number of elements of the poset. We may equivalently index them using compositions of weight n, declaring Rα = Rb(α). We refer the reader to [3, Section 10] for the definition of this basis. Billera, Jia, and Reiner show, through a series of theorems that the set of {F (Rα)}, where the posets are strictly labeled, forms a Z-basis for QSym. Using our basis, one can show this more directly. We know from Lemma 5.1 that all β ∈ supp(F (Rα)) are of rank r(α) and weight |α|. It is not too hard to show that the largest β ∈ supp(F (Rα)) with respect to the above ordering is precisely α, and that the coefficient of Nα in the expansion of F (Rα) is one. Thus an array giving the expansion of all the {F (Rα)} of a fixed set size |α| = n in terms of {Nα} with rows and columns suitably ordered is unitriangular. Acknowledgments I wish to thank the referees for their helpful comments. I extend many thanks to Isabella Novik and Sara Billey for their encouragement and their many hours of proofreading and advice through several drafts of this manuscript. Thanks also go to Vic Reiner and Lou Billera for their clarification of parts of their paper and their helpful comments on this work. Special thanks go to Isabella Novik for bringing the paper of Billera, Jia, and Reiner to my attention, and to Vic Reiner for pointing me specifically to [3, Question 7.10], which was the starting point for my investigation. While working on this project the author was partially supported by a gradu- ate fellowship from VIGRE NSF Grant DMS-0354131. The seeds of this project were planted at the ICM in Madrid, 2006. The trip there was made possible by funds from NSF Grant DMS-9983797. References [1] M. Aguiar, N. Bergeron, and F. Sottile, Combinatorial Hopf al- gebras and generalized Dehn-Sommerville relations., Compos. Math., 142 (2006), pp. 1–30. [2] A. Barvinok, A Course in Convexity., Graduate Studies in Mathematics. 54. Providence, RI: American Mathematical Society., 2002. [3] L. J. Billera, N. Jia, and V. Reiner, A quasisymmetric function for matroids, arXiv:math.CO/0606646. [4] H. Crapo and W. Schmitt, Primitive elements in the matroid-minor Hopf algebra, arXiv:math.CO/0511033. [5] H. Crapo and W. Schmitt, A free subalgebra of the algebra of matroids., Eur. J. Comb., 26 (2005), pp. 1066–1085. [6] , A unique factorization theorem for matroids., J. Comb. Theory, Ser. A, 112 (2005), pp. 222–249. [7] , The free product of matroids., Eur. J. Comb., 26 (2005), pp. 1060– 1065. [8] S. Dăscălescu, C. Năstăsescu, and c. Raianu, Hopf Algebras. An Introduction., Pure and Applied Mathematics, Marcel Dekker. 235. New York, NY: Marcel Dekker., 2001. [9] I. Gel’fand and V. Serganova, Combinatorial geometries and torus strata on homogeneous compact manifolds., Russ. Math. Surv., 42 (1987), pp. 133–168. [10] I. M. Gessel, Multipartite P-partitions and inner products of skew Schur functions. Combinatorics and algebra, Proc. Conf., Boulder/Colo. 1983, Contemp. Math. 34, 289-301, 1984. [11] M. Kapranov, Chow quotients of Grassmannians. I. Gelfand, Sergej (ed.) et al., I. M. Gelfand Seminar. Providence, RI: American Mathematical Society. Adv. Sov. Math. 16(2), 29-110, 1993. [12] L. Lafforgue, Pavages des simplexes, schémas de graphes recollés et com- pactification des PGLn+1r /PGLr., Invent. Math., 136 (1999), pp. 233–271. [13] , Chirurgie des grassmanniennes., CRM Monograph Series. 19. Prov- idence, RI: American Mathematical Society., 2003. [14] J. G. Oxley, Matroid Theory., Oxford Graduate Texts in Mathematics 3. Oxford: Oxford University Press., 2006. [15] W. R. Schmitt, Incidence Hopf algebras., J. Pure Appl. Algebra, 96 (1994), pp. 299–330. [16] D. E. Speyer, A matroid invariant via the K-theory of the Grassmannian, arXiv:math.AG/0603551. [17] R. Stanley, Ordered Structures and Partitions, Memoirs Amer. Math. Soc., (1972). American Mathematical Society, Providence, RI. [18] , Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, MA, 1999. [19] R. P. Stanley, The descent set and connectivity set of a permutation., J. Integer Sequences, (2005). Introduction Preliminaries Compositions Well-known QSym bases Posets and P-partitions The new basis Additional facts regarding F(P) Ordered partitions Unordered partitions of X P Structure constants for the new basis Matroids The quasisymmetric function of a matroid Expanding F(M) in the {N} basis Matroid base polytopes Matroid base polytopes and their decompositions Results for rank two matroids Additional observations Matroid duality, loops, and coloops Comultiplication Comparison with other QSym bases Surjectivity of the Hopf algebra morphism

0704.0837

Bremsstrahlung Radiation At a Vacuum Bubble Wall

Bremsstrahlung Radiation At a Vacuum Bubble Wall Jae-Weon Lee∗ School of Computational Sciences, Korea Institute for Advanced Study, 207-43 Cheongnyangni 2-dong, Dongdaemun-gu, Seoul 130-722, Korea Kyungsub Kim and Chul H. Lee Department of Physics, Hanyang University, Seoul 133-791, Korea Ji-ho Jang Korea Atomic Energy Research Institute Yuseong, Daejeon 305-353, Korea When charged particles collide with a vacuum bubble, they can radiate strong electromagnetic waves due to rapid deceleration. Owing to the energy loss of the particles by this bremsstrahlung radiation, there is a non-negligible damping pressure acting on the bubble wall even when thermal equilibrium is maintained. In the non-relativistic region, this pressure is proportional to the velocity of the wall and could have influenced the bubble dynamics in the early universe. PACS numbers: 12.15.Ji, 98.80.Cq There have been many studies on cosmological roles of first-order phase transitions, which proceed by nucleations and collisions of vacuum bubbles[1]. For example, in electroweak baryo- genesis models[2] rapid bubble expansion can provide a non-equilibrium environment, which may result in asymmetry between matter and antimatter. Furthermore, in some inflation- ary models[3, 4, 5], the speed of expanding vacuum bubbles determines how long the infla- tion period lasts. To understand the bubble kinematics in a hot plasma, it is important to study particle scatterings at a moving bubble wall. To calculate the velocity of electro-weak bubbles[6, 7, 8, 9, 10, 11] and the CP violating charge transport rate by the wall available for baryogenesis[2], one should know the reaction force acting on the wall due to the scattered par- ticles, such as quarks and gauge bosons[12, 13]. (For a supersymmetric model see, for example, Ref. 14) At the first order cosmological phase transition, the false vacuum decays to the true vacuum, which has lower energy, by making a vacuum bubble. When it is created, the wall of the bubble is at rest. As the free energy difference between the inner and the outer parts of the bubble fuels the wall, the velocity of the wall increases to the light velocity unless there is a damping force. In the literature, it is generally believed that the non-trivial damping force is caused by a deviation of the particle population from a thermal equilibrium one. In this paper, we study the effect of bremsstrahlung radiations emitted by particles on the pressure acting on a bubble wall (not necessary electroweak bubbles) during cosmological first order phase transitions. The aim of this work is to show that, contrary to the usual arguments, the radiation damping could give a non-negligible pressure even when the particles maintain thermal equilibrium. Bremsstrahlung (braking radiation) is a radiation due to the acceleration or deceleration of a charged particle[15]. Entering a true vacuum through a bubble wall, particles interact with the wall and could acquire mass and be decelerated. For example, a fermion field ψ can get mass through the well-known Yukawa term gψ̄φψ = mψ̄ψ, where φ is a Higgs field. At this time, if the particle is charged electromagnetically, it can radiate strong electromagnetic waves due to the deceleration. Let us calculate the pressure from the scattering. For simplicity, we assume a linear profile for the bubble wall, i.e., gφ(x) ≡ m(x) = m0x/d when 0 < x < d. (See Fig.1.) and choose the coordinates of the rest frame of the bubble wall. ∗Electronic address: [email protected] http://arxiv.org/abs/0704.0837v1 mailto:[email protected] This approximation is good for the usual tanh profile of the wall. The radiation power of an accelerated particle is given by a relativistic version of the Larmor’s formula[16]: dErad , (1) where A = 2e2/3c3 ≃ 0.0611 in the natural units (~ = c = k = 1) and ~k = (kx, ky, kz) is the 3-momentum of a particle. We assume a situation where this classical description of bremsstrahlung is good enough. Also, assuming that the wall is planar and parallel to the y-z plane, we can treat the bubble as a 1-dimensional one along the x-axis. The energy, momentum, and mass of the particle satisfy the usual relation E2 ≡ m2(x) + ~k2(x). (2) Let us denote the x-component of the momentum (kx) as k from now on. Differentiating the above equation with time t and using dx/dt ≡ v and k = Ev, we get the force acting on the wall due to the particles , (3) which is the starting point of the pressure calculation[6]. However, if we also consider the energy carried away by the radiation Erad, then the total energy conserved is Etot ≡ E+Erad and the force and, hence, the pressure should be changed. From dEtot/dt = 0, we obtain = 0, (4) which has a solution for the force . (5) Up to O(A), one can expand the square root term and obtain . (6) The second term represents the radiation damping. Then, the total pressure due to the collisions of the particles in the plasma is given by[6] (2π)3 f(E(k)) , (7) where f(E) = (exp(βE) ± 1)−1 is a distribution function of fermions and bosons, respectively. First, let us briefly review the well-known results without radiation damping. When the mean velocity of the plasma fluid V relative to the wall (or the negative of the bubble wall velocity relative to the fluid ) is zero, the first term of Eq. (6) contributes dm2(x) (2π)3 eβE ± 1 = F (m0, T )− F (0, T ), (8) where F (φ, T ) is a free energy of φ at a temperature T = β−1. When V 6= 0, the distribution function is changed to f [γ(E − V k)] = eβγ(E−V k) ± 1 . (9) Here, γ = (1−V 2)− 2 . However, using the fact that the phase factor d3~k/E is a Lorentz invariant and changing the integration variable to k′ = γ(k − V E) and defining E′ ≡ γ(E − V k), one can find that the V dependency of P1 disappears [6]. From this, it is generally believed that to get non-trivial pressure on the wall, one needs to consider a non-equilibrium deviation of f [7]. Our work indicates this is not necessarily true for some phase transitions. To see this, consider the effect of the radiation (the second term of Eq. (6)). When V = 0, the term contributes to the pressure P2 = 2A dm(x) (2π)3 Ek(eβE ± 1) which also vanishes because the second integrand is an odd function of k. However, when V 6= 0, one can easily check that, due to the 1/k term, the V dependency survives even under the change of the integration variable. Thus, in this case, P2 = 2A dm(x) (2π)3 Ek(eβγ(E−V k) ± 1) dm(x) dx I2(x). (10) To be more concrete, let us calculate an approximate value of the integration when V ≪ 1 for fermions. In this case, we can expand f [γ(E − V k)] ≃ f(E) − V βkf(E)[f(E) − 1] = f(E) + V βkf2(E)exp(−βE). The integration of the first term gives zero, and the second term contributes I2 = V β (2π)3 f2(E)exp(−βE) ≃ (ln2)TV because (2π)3 f2(E)exp(−βE) ≃ (ln2)T 2 , (12) to lowest oder in (m/T )2 (See Ref. 7). Therefore, for the wall described in Fig. 1 the pressure by the radiation is (ln2)Am20T V, (13) which is comparable to the result of numerical integration of Eq. (10) for V ≪ 1, as shown in Fig. 2. (During the numerical study it is useful to change the measure from dkydkz to 2πEdE.) This pressure is proportional to the wall velocity up to the moderately relativistic case and exists even when the system is in a thermal equilibrium. During the electroweak phase transition, a particle’s electromagnetic charge is not definite, so the A value in Lamor’s formula can not be a constant. In this paper , however, to perform a rough calculation, we have assumed that A is a constant during the phase transition. For illustration of high temperature effects on electric charges, now we consider a Debye screening of electric charge by plasma during the phase transition, which is given by effective coupling αeff = α/(1 − 2α ln(k/Λ)/3π) ≃ 0.97α, where we used averaged momentum 〈k〉 ≃ 3T and Λ of order electron mass at the last approximation (see Eq. (42) of [17]). Thus, we obtain A = 0.0599 which is slightly smaller than the zero temperature value. We also plot the pressure with this A value. It is noteworthy that the pressure caused by the radiation damping (Eq. (13)) is of order O(α), which is bigger than the pressure due to a departure from thermal equilibriums[7, 18, 19] (O(α2))[18], and hence non-negligible. Here, α is the fine structure constant. Note also that the power of bremsstrahlung due to bubble walls is much stronger (O(α)) than that of ordinary bremsstrahlung of electrons colliding with ions in a plasma (O(α3))[20]. Since the electroweak phase transition is a complicated phenomenon, by no means is our work a full calculation of the pressure acting on the electroweak bubbles. The purpose of this paper is to present a general idea that radiation damping (although usually ignored in the many related works for bubble wall velocity calculations) could give rise to significant frictional forces even in thermal equilibrium states at some cosmological phase transitions. 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[17] R. A. Schneider, Phys. Rev. D66, 036003 (2002). [18] G. D. Moore, JHEP 0003, 006 (2000). [19] G. D. Moore and T. Prokopec, Phys. Rev. D 52, 7182 (1995). [20] S. Ichimaru, Basic Principles of Plasma Physics (W. A. Benjamin, Reading, MA., 1973). m ( x ) FIG. 1: The effective mass of the particle m(x) in the wall rest frame. 0.2 0.4 0.6 0.8 1 0.001 0.002 0.003 0.004 FIG. 2: The pressure by the radiation damping of fermions colliding with the linear bubble wall as a function of the wall velocity. The thick line shows numerical integration of Eq. (10) and the dotted line shows the approximate formula in Eq. (13). Here we set 1/d = 1 = m0 = T for simplicity. The dashed line represents the result with Debye screening of charge. ACKNOWLEDGEMENTS References

0704.0838

Universal Source Coding for Monotonic and Fast Decaying Monotonic Distributions

Universal Source Coding for Monotonic and Fast Decaying Monotonic Distributions∗ Gil I. Shamir Department of Electrical and Computer Engineering University of Utah Salt Lake City, UT 84112, U.S.A e-mail: [email protected]. Abstract We study universal compression of sequences generated by monotonic distributions. We show that for a monotonic distribution over an alphabet of size k, each probability parameter costs essentially 0.5 log(n/k3) bits, where n is the coded sequence length, as long as k = o(n1/3). Otherwise, for k = O(n), the total average sequence redundancy is O(n1/3+ε) bits overall. We then show that there exists a sub-class of monotonic distributions over infinite alphabets for which redundancy of O(n1/3+ε) bits overall is still achievable. This class contains fast decaying distributions, including many distributions over the integers and geometric distributions. For some slower decays, including other distributions over the integers, redundancy of o(n) bits overall is achievable, where a method to compute specific redundancy rates for such distributions is derived. The results are specifically true for finite entropy monotonic distributions. Finally, we study individual sequence redundancy behavior assuming a sequence is governed by a monotonic distribution. We show that for sequences whose empirical distributions are monotonic, individual redundancy bounds similar to those in the average case can be obtained. However, even if the monotonicity in the empirical distribution is violated, diminishing per symbol individual sequence redundancies with respect to the monotonic maximum likelihood description length may still be achievable. Index Terms: monotonic distributions, universal compression, average redundancy, indi- vidual redundancy, large alphabets, patterns. ∗Supported by NSF Grant CCF-0347969. Part of the material in this manuscript was accepted for presentation in the IEEE International Symposium on Information Theory, Nice, France, June, 2007. http://arxiv.org/abs/0704.0838v1 1 Introduction The classical setting of the universal lossless compression problem [5], [8], [9] assumes that a se- quence xn of length n that was generated by a source θ is to be compressed without knowledge of the particular θ that generated xn but with knowledge of the class Λ of all possible sources θ. The average performance of any given code, that assigns a length function L(·), is judged on the basis of the redundancy function Rn (L,θ), which is defined as the difference between the expected code length of L (·) with respect to (w.r.t.) the given source probability mass function Pθ and the nth-order entropy of Pθ normalized by the length n of the uncoded sequence. A class of sources is said to be universally compressible in some worst sense if the redundancy function diminishes for this worst setting. Another approach to universal coding [29] considers the individual sequence redundancy R̂n (L, x n), defined as the normalized difference between the code length obtained by L(·) for xn and the negative logarithm of the maximum likelihood (ML) probability of the sequence xn, where the ML probability is within the class Λ. We thereafter refer to this negative logarithm as the ML description length of xn. The individual sequence redundancy is defined for each sequence that can be generated by a source θ in the given class Λ. Classical literature on universal compression [5], [8], [9], [23], [29] considered compression of sequences generated by sources over finite alphabets. In fact, it was shown by Kieffer [15] (see also [13]) that there are no universal codes (in the sense of diminishing redundancy) for sources over infinite alphabets. Later work (see, e.g., [21], [25]), however, bounded the achievable redundancies for identically and independently distributed (i.i.d.) sequences generated by sources over large and infinite alphabets. Specifically, while it was shown that the redundancy does not decay if the alphabet size is of the same order of magnitude as the sequence length n or greater, it was also shown that the redundancy does decay for alphabets of size o(n). 1 While there is no universal code for infinite alphabets, recent work [20] demonstrated that if one considers the pattern of a sequence instead of the sequence itself, universal codes do exist in the sense of diminishing redundancy. A pattern of a sequence, first considered, to the best of our knowledge, in [1], is a sequence of indices, where the index ψi at time i represents the order of first occurrence of letter xi in the sequence x n. Further study of universal compression of patterns [20], [21], [26], [28] provided various lower and upper bounds to various forms of redundancy in universal 1For two functions f(n) and g(n), f(n) = o(g(n)) if ∀c,∃n0, such that, ∀n > n0, f(n) < cg(n); f(n) = O(g(n)) if ∃c, n0, such that, ∀n > n0, 0 ≤ f(n) ≤ cg(n); f(n) = Θ(g(n)) if ∃c1, c2, n0, such that, ∀n > n0, c1g(n) ≤ f(n) ≤ c2g(n). compression of patterns. Another related study is that of compression of data, where the order of the occurring data symbols is not important, but their types and empirical counts are [30]-[31]. This paper considers universal compression of data sequences generated by distributions that are known a-priori to be monotonic. Hence, the order of probabilities of the source symbols is known in advance to both encoder and decoder and can be utilized as side information to improve universal compression performance. Monotonic distributions are common for distributions over the integers, including the geometric distribution and others. Such distributions do occur in image compression problems (see, e.g., [18], [19]), and in other applications that compress residual signals. A specific application one can consider for the results in this paper is compression of the list of last or first names in a given city of a given population. One can usually find some monotonicity for such a distribution in the given population, which both encoder and decoder may be aware of a-priori . For example, the last name “Smith” can be expected to be much more common than the last name “Shannon”. Another example is the compression of a sequence of observations of different species, where one has prior knowledge which species are more common, and which are rare. Finally, one can consider compressing data for which side information given to the decoder through a different channel gives the monotonicity order. Unlike compression of patterns, Foster, Stine, and Wyner, showed in [10] that there are no universal block codes in the standard sense for the complete class of monotonic distributions. The main reason is that there exist such distributions, for which much of the statistical weight lies in symbols that have very low probability, and most of which will not occur in a given sequence. Thus, in practice, even though one has the prior knowledge of the monotonicity of the distribution, this monotonicity is not necessarily retained in an observed sequence. Therefore, actual coding can be very similar to compressing with infinite alphabets, and the additional prior knowledge of the monotonicity is not very helpful in reducing redundancy. Despite that, Foster, Stine, and Wyner demonstrated codes that obtained universal per-symbol redundancy of o(1) as long as the source entropy is fixed (i.e., neither increasing with n nor infinite). However, instead of considering redundancy in the standard sense, the study of monotonic distributions resorted to studying relative redundancy , which bounds the ratio between average assigned code length and the source entropy. This approach dates back to work by Elias [7], Rissanen [22], and Ryabko [24]. The work in [10] studied coding sequences (or blocks) generated by i.i.d. monotonic distributions, and designed codes for which the relative block redundancy could be (upper) bounded. Unlike that work, the focus in [7], [22], and [24] was on designing codes that minimize the redundancy or relative redundancy for a single symbol generated by a monotonic distribution. Specifically, in [22], minimax codes, which minimize the relative redundancy for the worst possible monotonic distribution over a given alphabet size, were derived. In [24], it was shown that redundancy of O(log log k), where k is the alphabet size, can be obtained with minimax per-symbol codes. Very recent work [16] considered per-symbol codes that minimize an average redundancy over the class of monotonic distributions for a given alphabet size. Unlike [10], all these papers study per-symbol codes. Therefore, the codes designed always pay non-diminishing per-symbol redundancy. A different line of work on monotonic distributions considered optimizing codes for a known monotonic distribution but with unknown parameters (see [18], [19] for design of codes for two-sided geometric distributions). In this line of work, the class of sources is very limited and consists of only the unknown parameters of a known distribution. In this paper, we consider a general class of monotonic distributions that is not restricted to a specific type. We study standard block redundancy for coding sequences generated by i.i.d. monotonic distributions, i.e., a setting similar to the work in [10]. We do, however, restrict ourselves to smaller subsets of the complete class of monotonic distributions. First, we consider monotonic distributions over alphabets of size k, where k is either small w.r.t. n, or of O(n). Then, we extend the analysis to show that under minimal restrictions of the monotonic distribution class, there exist universal codes in the standard sense, i.e., with diminishing per-symbol redundancy. In fact, not only do universal codes exist, but under mild restrictions, they achieve the same redundancy as obtained for alphabets of size O(n). The restrictions on this subclass imply that some types of fast decaying monotonic distributions are included in it, and therefore, sequences generated by these distributions (without prior knowledge of either the distribution or of its parameters) can still be compressed universally in the class of monotonic distributions. The main contributions of this paper are the development of codes and derivation of their upper bounds on the redundancies for coding i.i.d. sequences generated by monotonic distributions. Specifically, the paper gives complete characterization of the redundancy in coding with monotonic distributions over “small” alphabets (k = o(n1/3)) and “large” alphabets (k = O(n)). Then, it shows that these redundancy bounds carry over (in first order) to fast decaying distributions. Next, a code that achieves good redundancy rates for even slower decaying monotonic distributions is derived, and is used to study achievable redundancy rates for such distributions. Lower bounds are also presented to complete the characterization, and are shown to meet the upper bounds in the first three cases (small alphabets, large alphabets, and fast decaying distributions). The lower bounds turn out to result from lower bounds obtained for coding patterns. The relationship to patterns is demonstrated in the proofs of those lower bounds. Finally, individual sequences are considered. It is shown that under mild conditions, there exist universal codes w.r.t. the monotonic ML description length for sequences that contain the O(n) more likely symbols, even if their empirical distributions are not monotonic. The outline of this paper is as follows. Section 2 describes the notation and basic definitions. Then, in section 3, lower bounds on the redundancy for monotonic distributions are derived. Next, in Section 4, we propose codes and upper bound their redundancy for coding monotonic distribu- tions over small and large alphabets. These bounds are then extended to fast decaying monotonic distributions in Section 5. Finally, in Section 6, we consider individual sequence redundancy. 2 Notation and Definitions Let xn = (x1, x2, . . . , xn) denote a sequence of n symbols over the alphabet Σ of size k, where k can go to infinity. Without loss of generality, we assume that Σ = {1, 2, . . . , k}, i.e., it is the set of positive integers from 1 to k. The sequence xn is generated by an i.i.d. distribution of some source, determined by the parameter vector θ = (θ1, θ2, . . . , θk), where θi is the probability of X taking value i. The components of θ are non-negative and sum to 1. The distributions we consider in this paper are monotonic. Therefore, θ1 ≥ θ2 ≥ . . . ≥ θk. The class of all monotonic distributions will be denoted by M. The class of monotonic distributions over an alphabet of size k is denoted by Mk. It is assumed that prior to coding xn both encoder and decoder know that θ ∈ M or θ ∈ Mk, and also know the order of the probabilities in θ. In the more restrictive setting, k is known in advance and it is known that θ ∈ Mk. We do not restrict ourselves to this setting. In general, boldface letters will denote vectors, whose components will be denoted by their indices in the vector. Capital letters will denote random variables. We will denote an estimator by the hat sign. In particular, θ̂ will denote the ML estimator of θ which is obtained from xn. The probability of xn generated by θ is given by Pθ (x = Pr (xn | Θ = θ). The average per-symbol2 nth-order redundancy obtained by a code that assigns length function L(·) for θ is Rn (L,θ) EθL [X n]−Hθ [X] , (1) where Eθ denotes expectation w.r.t. θ, and Hθ [X] is the (per-symbol) entropy (rate) of the source 2In this paper, redundancy is defined per-symbol (normalized by the sequence length n). However, when we refer to redundancy in overall bits, we address the block redundancy cost for a sequence. (Hθ [X n] is the nth-order sequence entropy of θ, and for i.i.d. sources, Hθ [X n] = nHθ [X]). With entropy coding techniques, assigning a universal probability Q (xn) is identical to designing a uni- versal code for coding xn where, up to negligible integer length constraints that will be ignored, the negative logarithm to the base of 2 of the assigned probability is considered as the code length. The individual sequence redundancy (see, e.g., [29]) of a code with length function L (·) per sequence xn is R̂n (L, x {L (xn) + log PML (xn)} , (2) where the logarithm function is taken to the base of 2, here and elsewhere, and PML (x n) is the probability of xn given by the ML estimator θ̂Λ ∈ Λ of the governing parameter vector Θ. The negative logarithm of this probability is, up to integer length constraints, the shortest possible code length assigned to xn in Λ. It will be referred to as the ML description length of xn in Λ. In the general case, one considers the i.i.d. ML. However, since we only consider θ ∈ M, i.e., restrict the sequence to one governed by a monotonic distribution, we define θ̂M ∈ M as the monotonic ML estimator. Its associated shortest code length will be referred to as the monotonic ML description length. The estimator θ̂M may differ from the i.i.d. ML θ̂, in particular, if the empirical distribution of xn is not monotonic. The individual sequence redundancy in M is thus defined w.r.t. the monotonic ML description length, which is the negative logarithm of PML (x xn | Θ = θ̂M ∈ M The average minimax redundancy of some class Λ is defined as R+n (Λ) = min Rn (L,θ) . (3) Similarly, the individual minimax redundancy is that of the best code L (·) for the worst sequence R̂+n (Λ) = min {L (xn) + logPθ (xn)} . (4) The maximin redundancy of Λ is R−n (Λ) = sup w (dθ)Rn (L,θ) , (5) where w(·) is a prior on Λ. In [5], it was shown that R+n (Λ) ≥ R−n (Λ). Later, however, [6], [11], [24] the two were shown to be essentially equal. 3 Lower Bounds Lower bounds on various forms of the redundancy for the class of monotonic distributions can be obtained with slight modifications of the proofs for the lower bounds on the redundancy of coding patterns in [14], [20], [21], and [26]. The bounds are presented in the following three theorems. For the sake of completeness, the main steps of the proofs of the first two theorems are presented in appendices, and the proof of the third theorem is presented below. The reader is referred to [14], [20], [21], [25] and [26] for more details. Theorem 1 Fix an arbitrarily small ε > 0, and let n → ∞. Then, the nth-order average max- imin and minimax universal coding redundancies for i.i.d. sequences generated by a monotonic distribution with alphabet size k are lower bounded by R−n (Mk) ≥ log n + k−1 log πe log k , for k ≤ πn1−ε )1/3 · (1.5 log e) · n(1−ε)/3 , for k > πn1−ε )1/3 . (6) Theorem 2 Fix an arbitrarily small ε > 0, and let n→ ∞. Then, the nth-order average universal coding redundancy for coding i.i.d. sequences generated by monotonic distributions with alphabet size k is lower bounded by Rn (L,θ) ≥ log n − k−1 log 8π log k , for k ≤ 1 1.5 log e 2π1/3 · n(1−ε)/3 , for k > 1 )1/3 (7) for every code L(·) and almost every i.i.d. source θ ∈ Mk, except for a set of sources Aε (n) whose relative volume in Mk goes to 0 as n→ ∞. Theorems 1 and 2 give lower bounds on redundancies of coding over monotonic distributions for the class Mk. However, the bounds are more general, and the second region applies to the whole class of monotonic distributions M. As in the case of patterns [20], [26], the bounds in (6)-(7) show that each parameter costs at least 0.5 log(n/k3) bits for small alphabets, and the total universality cost is at least Θ(n1/3−ε) bits overall for larger alphabets. Unlike the currently known results on patterns, however, we show in Section 4 that for k = O(n) these bounds are achievable for monotonic distributions. The proofs of Theorems 1 and 2 are presented in Appendix A and in Appendix B, respectively. Theorem 3 Let n → ∞. Then, the nth-order individual minimax redundancy for i.i.d. sequences with maximal letter k w.r.t. the monotonic ML description length with alphabet size k is lower bounded by R̂+n (Mk) ≥ log n log e 23/12 log k , for k ≤ e5/18 (2π)1/3 · n1/3 e5/18 (2π)1/3 (log e) · n1/3 , for n > k > e (2π)1/3 · n1/3 (log e) · n1/3 , for k ≥ n. Theorem 3 lower bounds the individual minimax redundancy for coding a sequence believed to have an empirical monotonic distribution. The alphabet size is determined by the maximal letter that occurs in the sequence, i.e., k = max {x1, x2, . . . , xn}. (If k is unknown, one can use Elias’ code for the integers [7] using O(log k) bits to describe k. However this is not reflected in the lower bound.) The ML probability estimate is taken over the class of monotonic distributions, i.e., the empirical probability (standard ML) estimate θ̂ is not θ̂M in case θ̂ does not satisfy the monotonicity that defines the class M. While the average case maximin and minimax bounds of Theorem 1 also apply to R̂+n (Mk), the bounds of Theorem 3 are tighter for the individual redundancy and are obtained using individual sequence redundancy techniques. Proof of Theorem 3: Using Shtarkov’s normalized maximum likelihood (NML) approach [29], one can assign probability Q (xn) yn Pθ̂M maxθ′∈M Pθ′ (x yn maxθ′∈M Pθ′ (y to sequence xn. This approach minimizes the individual minimax redundancy, giving individual redundancy of R̂n (Q,x maxθ′∈M Pθ′ (x Q (xn) Pθ′ (y to every xn, specifically achieving the individual minimax redundancy. It is now left to bound the logarithm of the sum in (10). For the first two regions, we follow the approach used in Theorem 2 in [21] for bounding the redundancy for standard compression of i.i.d. sequences over large alphabets, but adjust it to monotonic distributions. Alternatively, one can derive the same bounds following the approach used for bounding the individual minimax redundancy of patterns in proving Theorem 12 in [20]. Let nℓx = (nx(1), nx(2), . . . , nx(ℓ)) denote the occurrence counts of the first ℓ letters of the alphabet Σ in xn. For ℓ = k, i=1 nx(i) = n. Now, following (10), nR̂+n (Mk) ≥ log yn:θ̂(yn)∈M ≥ log ny(1), . . . , ny(ℓ) ny(i) )ny(i) ≥ log ny(1), . . . , ny(k) ny(i) )ny(i) ≥ log ek/12 · (2π)k/2 nx(i) ≥ log k − 1 ek/12 ≥ k − 1 + k log e23/12√ −O (log k) (11) where (a) follows from including only sequences yn that have a monotonic empirical (i.i.d. ML) distribution in Shtarkov’s sum. Inequality (b) follows from partitioning the sequences yn into types as done in [21], first by the number of occurring symbols ℓ, and then by the empirical distribution. Unlike standard i.i.d. distributions though, monotonicity implies that only the first ℓ symbols in Σ occur, and thus the choice of ℓ out of k in the proof in [21] is replaced by 1. Like in coding patterns, we also divide by ℓ! because each type with ℓ occurring symbols can be ordered in at most ℓ! ways, where only some retain the monotonicity. (Note that this step is the reason that step (b) produces an inequality, because more than one of the orderings may be monotonic if equal occurrence counts occur.) Except the division by ℓ!, the remaining steps follow those in [21]. Retaining only the term ℓ = k yields inequality (c). Inequality (d) follows from Stirling’s bound 2πm · ≤ m! ≤ 2πm · · exp . (12) Then, (e) follows from the relation between arithmetic and geometric means, and from expressing the number of types as the number of ordered partitions of n into k parts k − 1 . Finally, (f) follows from applying (12) again and by lower bounding k − 1 The first region in (8) results directly from (11). The behavior is similar to patterns as shown in [1] for this region. As mentioned in [20], to obtain the second region, the bound is maximized by retaining ℓ̂ = n1/3e5/18 /(2π)1/3 instead of k in step (c) of (11), for every k ≥ ℓ̂. The bounds obtained are equal to those obtained for patterns because the first step (a) in (11) discards all the sequences whose contributions to Shtarkov’s sum are different between patterns and monotonic distributions. A similar step is effectively done deriving the bounds for patterns. The difference is that in the case of patterns, components of Shtarkov’s sum are reduced, but all are retained in the sum, while here, we omit components from the sum, corresponding to sequences with non- monotonic i.i.d. ML estimates. The analysis in [20] that also attains the second region of the bound in (8) is still valid here. It differs from the steps taken above by lower bounding a pattern probability by a larger probability than the ML i.i.d. probability corresponding to the pattern. The bound used in the derivation of Theorem 12 in [20] adds a multiplicative factor to each pattern probability which equals the number of sequences with the same pattern and an equal i.i.d. ML probability. However, this similar effect is included in Shtarkov’s sum for monotonic distributions since all these sequences do have a corresponding i.i.d. ML estimate which is monotonic, and are thus not omitted by step (a) of the derivation. The analysis in [14] yields the third region of the bound in (8), since, for k ≥ n, R̂+n (Mk) = Ψ(yn) 1.5n1/3 log e log n , (13) where Ψ(yn) is the pattern of the sequence yn. Inequality (a) holds because each pattern cor- responds to at least one sequence whose ML probability parameter estimates are ordered, i.e., θ̂i ≥ θ̂i+1,∀i, where the most probable index represents i = 1, the second most probable index i = 2, and so on. Note that the sum element on the right hand side is for a probability of a sequence, not a pattern, but the sum is over all patterns. The left hand side also includes sequences for which the probabilities are unordered. Furthermore, exchanging the letters that correspond to two indices with the same occurrence count will not violate monotonicity. Thus the inequality follows. Step (b) in (13) is taken from [14], where the sum on the left hand side was shown to equal the right hand side. This was true when summing over all patterns with up to n indices, thus requiring k ≥ n. Note that this requirement does not mean that n distinct symbols must occur in xn, only that the maximal symbol in xn is n or greater. This concludes the proof of Theorem 3. � 4 Upper Bounds for Small and Large Alphabets In this section, we demonstrate codes that asymptotically achieve the lower bounds for θ ∈ Mk and k = O(n). We begin with a theorem that shows the achievable redundancies, and devote the remainder of the section to describing the codes and deriving upper bounds on their redundancies. The theorem is stated assuming no initial knowledge of k. The proof first considers the setting where k is known, and then shows how the same bounds are achieved even when k is unknown in advance, but as long as it satisfies the conditions. Theorem 4 Fix an arbitrarily small ε > 0, and let n → ∞. Then, there exist a code with length function L∗ (·) that achieves redundancy Rn (L ∗,θ) ≤ (1 + ε) k−1 n(logn)2 , for k ≤ n1/3, (1 + ε) (log n) log k n1/3−ε , for n1/3 < k = o(n), (1 + ε) 2 (log n) 2 n1/3 , for n1/3 < k = O(n), for i.i.d. sequences generated by any source θ ∈ Mk. Slightly tighter bounds are possible in the first and second regions and between them. The bounds presented, however, are inclusive for each of the regions. Note that the third region con- tains the second, but if k = o(n), a tighter bound is possible in the second region. The code designed to code a sequence xn is a two part code [23] that quantizes a distribution that minimizes the cost, and uses it to code xn. The total redundancy cost consists of the cost of describing the quantized distribution and the quantization cost. The second is bounded through the quantized true distribution of the sequence, which cannot result in lower cost than that of the chosen dis- tribution (which minimizes the cost). In order to achieve the low costs of the lower bound, the probability parameters are quantized non-uniformly, where the smaller the probability the finer the quantization. This approach was used in [25] and [26] to obtain upper bounds on the redundancy for coding over large alphabets and for coding patterns, respectively. The method used in [25] and [26], however, is insufficient here, because it still results in too many quantization points due to the polynomial growth in quantization spacing. Here, we use an exponential growth as the parameters increase. This general idea was used in [28] to improve an upper bound on the redundancy of coding patterns. Here, however, we improve on the method presented in [28]. Another key step in the proof here is the fact that since both encoder and decoder know the order of the probabilities a-priori , this order need not be coded. It is sufficient to encode the quantized probabilities of the monotonic distribution, and the decoder can identify which probability is associated with which symbol using the monotonicity of the distribution. Proof of Theorem 4: We start with k ≤ n1/3 assuming k is known. Let β = 1/(log n) be a parameter (note, that we can choose other values). Partition the probability space into J1 = ⌈1/β⌉ intervals, n(j−1)β , 1 ≤ j ≤ J1. (15) Note that I1 = [1/n, 2/n), I2 = [2/n, 4/n), . . . , Ij = [2 j−1/n, 2j/n). Let kj = |θi ∈ Ij| denote the number of probabilities in θ that are in interval Ij. In interval j, take a grid of points with spacing . (16) Note that to complete all points in an interval, the spacing between two points at the boundary of an interval may be smaller. There are ⌈log n⌉ intervals. Ignoring negligible integer length constraints (here and elsewhere), in each interval, the number of points is bounded by |Ij | ≤ , ∀j : j = 1, 2, . . . , J1, (17) where | · | denotes the cardinality of a set. Let the grid τ = (τ1, τ2, . . .) = , . . . , , . . . be a vector that takes all the points from all intervals, with cardinality = |τ | ≤ 1 ⌈log n⌉ . (19) Now, let ϕ = (ϕ1, ϕ2, . . . , ϕk) be a monotonic probability vector, such that ϕi = 1, ϕ1 ≥ ϕ2 ≥ · · · ≥ ϕk ≥ 0, and also the smaller k−1 components of ϕ are either 0 or from τ , i.e., ϕi ∈ (τ ∪ {0}), i = 2, 3, . . . , k. One can code xn using a two part code, assuming the distribution governing xn is given by the parameter ϕ. The code length required (up to integer length constraints) is L (xn|ϕ) = log k + LR(ϕ)− log Pϕ (xn) , (20) where log k bits are needed to describe how many letter probabilities are greater than 0 in ϕ, and LR(ϕ) is the number of bits required to describe the quantized points of ϕ. The vector ϕ can be described by a code as follows. Let k̂ϕ be the number of nonzero letter probabilities hypothesized by ϕ. Let bi denote the index of ϕi in τ , i.e., ϕi = τbi . Then, we will use the following differential code. For ϕ we need at most 1 + log b + 2 log(1 + log b bits to code its index in τ using Elias’ coding for the integers [7]. For ϕi−1, we need at most 1 + log(bi−1 − bi + 1) + 2 log[1 + log(bi−1 − bi + 1)] bits to code the index displacement from the index of the previous parameter, where an additional 1 is added to the difference in case the two parameters share the same index. Summing up all components of ϕ, and taking b k̂ϕ+1 LR(ϕ) ≤ k̂ϕ − 1 + log (bi − bi+1 + 1) + 2 log [1 + log (bi − bi+1 + 1)] ≤ (k − 1) + (k − 1) log B1 + k − 1 + 2(k − 1) log log B1 + k − 1 + o(k) = (1 + ε) k − 1 n (log n) . (21) Inequality (a) is obtained by applying Jensen’s inequality once on the first sum, twice on the second sum utilizing the monotonicity of the logarithm function, and by bounding k̂ϕ by k and absorbing low order terms in the resulting o(k) term. Then, low order terms are absorbed in ε, and (19) is used to obtain (b). To code xn, we choose ϕ which minimizes the expression in (20) over all ϕ, i.e., L∗ (xn) = min L (xn|ϕ) △= L (xn|ϕ̂) . (22) The pointwise redundancy for xn is given by nRn (L ∗, xn) = L∗ (xn) + log Pθ (x n) = log k + L∗R (ϕ̂) + log Pθ (x Pϕ̂ (x . (23) Note that the pointwise redundancy differs from the individual one, since it is defined w.r.t. the true probability of xn. To bound the third term of (23), let θ′ be a quantized still monotonic version of θ onto τ , i.e., θ′i ∈ (τ ∪ {0}), i = 2, 3, . . . , k, where if θi > 0 ⇔ θ′i > 0 as well. Define the quantization error, δi = θi − θ′i. (24) The quantization is performed from the smallest parameter θk to the largest, where monotonicity is retained, as well as minimal absolute quantization error. This implies that θi will be quantized to one of the two nearest grid points (one smaller and one greater than it). It also guarantees that |δ1| ≤ ∆ , where j2 is the index of the interval in which θ2 is contained, i.e., θ2 ∈ Ij2 . Now, since θ′ is included in the minimization of (22), we have, for every xn, L∗ (xn) ≤ L xn|θ′ , (25) and also nRn (L ∗, xn) ≤ log k + LR + log Pθ (x Pθ′ (x . (26) Averaging over all possible xn, the average redundancy is bounded by nRn (L ∗,θ) = log k + EθL R (ϕ̂) + Eθ log Pθ (X Pϕ̂ (X ≤ log k + EθLR ′)+ Eθ log Pθ (X Pθ′ (X . (27) The second term of (27) is bounded with the bound of (21), and we proceed with the third term. Eθ log Pθ (X Pθ′ (X θi log θ′i + δi ≤ n(log e) θ′i + δi = n(log e) ≤ k log e+ 2(log e)k kj · njβ ≤ 5(log e)k. (28) Equality (a) is since the argument in the logarithm is fixed, thus expectation is performed only on the number of occurrences of letter i for each letter. Representing θi = θ i + δi yields equation (b). We use ln(1+x) ≤ x to obtain (c). Equality (d) is obtained since all the quantization displacements must sum to 0. The first term of inequality (e) is obtained under a worst case assumption that θi ≪ 1/n for i ≥ 2. Thus it is quantized to θ′i = 1/n, and the bound |δi| ≤ 1/n is used. The second term is obtained by separating the terms into their intervals. In interval j, the bounds θ′i ≥ n(j−1)β/n, and |δi| ≤ knjβ/n1.5 are used, and also nβ = 2. Inequality (f) is obtained since j ≤ 2n. (29) Inequality (29) is obtained since k1 ≤ n, k2 ≤ (n− k1)/2, k3 ≤ (n− k1)/4− k2/2, and so on, until kJ1 ≤ 2J1−1 2J1−ℓ j ≤ 2n. (30) The reason for these relations are the lower limits of the J1 intervals that restrict the number of parameters inside the interval. The restriction is done in order of intervals, so that the used probabilities are subtracted, leading to the series of equations. Plugging the bounds of (21) and (28) into (27), we obtain, nRn (L ∗,θ) ≤ log k + (1 + ε) k − 1 n (log n) + 5(log e)k 1 + ε′ ) k − 1 n (log n) , (31) where we absorb low order terms in ε′. Replacing ε′ by ε normalizing the redundancy per symbol by n, the bound of the first region of (14) is proved. We now consider the larger values of k, i.e., n1/3 < k = O(n). The idea of the proof is the same. However, we need to partition the probability space to different intervals, the spacing within an interval must be optimized, and the parameters’ description cost must be bounded differently, because now there are more parameters quantized than points in the quantization grid. Define the jth interval as n(j−1)β , 1 ≤ j ≤ J2, (32) where J2 = ⌈2/β⌉ = ⌈2 log n⌉. Again, let kj = |θi ∈ Ij| denote the number of probabilities in θ that are in interval Ij. It could be possible to use the intervals as defined in (15), but this would not guarantee bounded redundancy in the rate we require if there are very small probabilities θi ≪ 1/n. Therefore, the interval definition in (15) can be used for larger alphabets only if the probabilities of the symbols are known to be bounded. Define the spacing in interval j as , (33) where α is a parameter to be optimized. Similarly to (17), the interval cardinality here is |Ij| ≤ 0.5 · nα, ∀j : j = 1, 2, . . . , J2, (34) In a similar manner to the definition of τ in (18), we define η = (η1, η2, . . .) = , . . . , , . . . . (35) The cardinality of η is = |η| ≤ 0.5 · nα ⌈2 log n⌉ ≤ nα ⌈log n⌉ . (36) We now perform the encoding similarly to the small k case, where we allow quantization to nonzero values to the components of ϕ up to i = n2. (This is more than needed but is possible since η1 = 1/n 2.) Encoding is performed similarly to the small k case. Thus, similarly to (27), we nRn (L ∗,θ) ≤ 2 log n+ EθLR ′)+ Eθ log Pθ (X Pθ′ (X , (37) where the first term is due to allowing up to k̂ = n2. Since usually in this region k ≥ B2 (except the low end), the description of vectors ϕ and θ′ is done by coding the cardinality of |ϕi = ηj | and |θ′i = ηj |, respectively, i.e., for each grid point the code describes how many letters have probability quantized to this point. This idea resembles coding profiles of patterns, as done in [20]. However, unlike the method in [20], here, many probability parameters of symbols with different occurrences are mapped to the same grid point by quantization. The number of parameters mapped to a grid point of η is coded using Elias’ representation of the integers. Hence, in a similar manner to (21), 1 + log |θ′i = ηj |+ 1 + 2 log 1 + log |θ′i = ηj |+ 1 ≤ B2 +B2 log k +B2 + 2B2 log log k +B2 + o (B2) (1 + ε)(log n) log k nα, for nα < k = o(n), (1 + ε)(1 − α) (log n)2 nα, for nα < k = O(n). The additional 1 term in the logarithm in (a) is for 0 occurrences, (b) is obtained similarly to step (a) of (21), absorbing all low order terms in the last term. To obtain (c), we first assume, for the first region, that knε ≫ B2 (an assumption that must be later validated with the choice of α). Then, low order terms are absorbed in ε. The extra nε factor is unnecessary if k ≫ B2. The second region is obtained by upper bounding k without this factor. It is possible to separate the first region into two regions, eliminate this factor in the lower region, and obtain a more complicated, yet tighter, expression in the upper region, where k ∼ Θ(n1/3). Now, similarly to (28), we obtain Eθ log Pθ (X Pθ′ (X ≤ n(log e) ≤ O(1) + 2 log e n1+2α ≤ 4(log e)n1−2α +O(1). (39) The first term of inequality (a) is obtained under the assumption that k = O(n), θ′i ≥ 1/n2, and |δi| ≤ 1/n2. For the second term |δi| ≤ njβ/n2+α, and θ′i ≥ n(j−1)β/n2. Inequality (b) is obtained in a similar manner to inequality (f) of (28), where the sum is shown similarly to be 2n2. Summing up the contributions of (38) and (39) in (37), it is clear that α = 1/3 minimizes the total cost (to first order). This choice of α also satisfies the assumption of step (c) in (38). Using α = 1/3, absorbing all low order terms in ε and normalizing by n, we obtain the remaining two regions of the bound in (14). It should be noted that the proof here would give a bound of O(n1/3+ε) up to k = O(n4/3). If the intervals in (15) were used for bounded distributions, the coefficients of the last two regions will be reduced by a factor of 2. Additional manipulations on the grid η may reduce the coefficients more (see, e.g., [28]). The proof up to this point assumes that k is known in advance. This is important for the code resulting in the bound for the first region because the quantization grid depends on k. Specifically, if in building the grid, k is underestimated, the description cost of ϕ increases. If k is overestimated, the quantization cost will increase. Also, if the code of the second region is used for a smaller k, a larger bound than necessary results. To solve this, the optimization that chooses L∗ (xn) is done over all possible values of k (greater than or equal to the maximal symbol occurring in xn), i.e., every greater k in the first region, and the construction of the code for the other regions. For every k in the first region, a different construction is done, using the appropriate k to determine the spacing in each interval. The value of k yielding the shortest code word is then used, and O(log n) additional bits are used at the prefix of the code to inform the decoder which k is used. The analysis continues as before. This does not change the redundancy to first order, giving all three regions of the bound in (14), even if k is unknown in advance. This concludes the proof of Theorem 4. � 5 Upper Bounds for Fast Decaying Distributions This section shows that with some mild conditions on the source distribution, the same redundancy upper bounds achieved for finite monotonic distributions can be achieved even if the monotonic distribution is over an infinite alphabet. The key observation that allows this is that a distribution that decays fast enough will result in only a small number of occurrences of unlikely letters in a sequence. These letters may very likely be out of order, but since there are very few of them, they can be handled without increasing the asymptotic behavior of the coding cost. More precisely, fast decaying monotonic distributions can be viewed as if they have some effective bounded alphabet size, where occurrences of symbols outside this limited alphabet are rare. We present two theorems and a corollary that show how one can upper bound the redundancy obtained when coding with some unknown distribution. The first theorem provides a slightly stronger bound (with smaller coefficient) even for k = O(n), where the smaller coefficient is attained by improved bounding, that more uniformly weights the quantization cost for minimal probabilities. In the weaker version of the results presented here, if the distribution decays slower and there are more low probability symbols, the redundancy order does increase due to the penalty of identifying these symbols in a sequence. However, we show, consistently with the results in [10], that as long as the entropy of the source is finite, a universal code, in the sense of diminishing redundancy per symbol, does exist. We begin with stating the two theorems and the corollary, then the proofs are presented. The section is concluded with three examples of typical monotonic distributions over the integers, to which the bounds are applied. 5.1 Upper Bounds We begin with some notation. Fix an arbitrary small ε > 0, and let n→ ∞. Define m △= mρ as the effective alphabet size, where ρ > ε. (Note that ρ = (logm)/(log n).) Let Rn(m) log n , for m = o ρ+ ε− 1 (log n) n1/3, otherwise. Theorem 5 I. Fix an arbitrarily small ε > 0, and let n → ∞. Let xn be generated by an i.i.d. monotonic distribution θ ∈ M. If there exists m∗, such that, nθi log i = o [Rn (m∗)] , (41) then, there exists a code with length function L∗(·), such that Rn (L ∗,θ) ≤ (1 + ε) Rn (m∗) (42) for the monotonic distribution θ. II. If there exists m∗ for which ρ∗ = o n1/3/(log n) , such that, θi log i = o(1), (43) then, there exists a universal code with length function L∗(·), such that Rn (L ∗,θ) = o(1). (44) Theorem 5 implies that if a monotonic distribution decays fast enough, its effective alphabet size does not exceed O(nρ), and, as long as ρ is fixed, bounds of the same order as those obtained for finite alphabets are achievable. Specifically, very fast decaying distributions, although over infinite alphabets, may even behave like monotonic distributions with o symbols. The condition in (41) merely means that the cost that a code would obtain in order to code very rare symbols, that are larger than the effective alphabet size, is negligible w.r.t. the total cost obtained from other, more likely, symbols. Note that for m = n, the bound is tighter than that of the third region of Theorem 4, and a constant of 5/9 replaces 2/3. The second part of the theorem states that if the decay is slow, but the cost of coding rare symbols is still diminishing per symbol, a universal code still exists for such distributions. However, in this case the redundancy will be dominated by coding the rare (out of order) symbols. This result leads to the following corollary: Corollary 1 As n → ∞, sequences generated by monotonic distributions with Hθ(X) = O(1) are universally compressible in the average sense. Corollary 1 shows that sequences generated by finite entropy monotonic distributions can be com- pressed in the average with diminishing per symbol redundancy. This result is consistent with the results shown in [10]. While Theorem 5 bounds the redundancy decay rate with two extremes, a more general theorem can be used to provide some best redundancy decay rate that a code can be designed to adapt to for some unknown monotonic distribution that governs the data. As the examples at the end of this section show, the next theorem is very useful for slower decaying distributions. Theorem 6 Fix an arbitrarily small ε > 0, and let n → ∞. Let xn be generated by an i.i.d. monotonic distribution θ ∈ M. Then, there exists a code with length function L∗(·), that achieves redundancy nRn (L ∗,θ) ≤ (1 + ε) · α,ρ:ρ≥α+ε · (ρ+ 2α) (ρ− α) (log n)2nα + 5(log e)n1−2α + θi log i for coding sequences generated by the source θ. We continue with proving the two theorems and the corollary. Proof : The idea of the proof of both theorems is to separate the more likely symbols from the unlikely ones. First, the code determines the point of separation m = nρ. (Note that ρ can be greater than 1.) Then, all symbols i ≤ m are considered likely and are quantized in a similar manner as in the codes for smaller alphabets. Unlike bounded alphabets, though, a more robust grid is used here to allow larger values of m. Coding of occurrences of these symbols uses the quantized probabilities. The unlikely symbols are coded hierarchically. They are first merged into a single symbol, and then are coded within this symbol, where the full cost of conveying to the decoder which rare symbols occur in the sequence is required. Thus, they are presented giving their actual value. As long as the decay is fast enough, the average cost of conveying these symbols becomes negligible w.r.t. the cost of coding the likely symbols. If the decay is slower, but still fast enough, as the case described in condition (43), the coding cost of the rare symbols dominates the redundancy, but still diminishing redundancy can be achieved. In order to determine the best value of m for a given sequence, all values are tried and the one yielding the shortest description is used for coding the specific sequence xn. Let m ≥ 2 determine the number of likely symbols in the alphabet. For a given m, define θi, (46) as the total probability of the remaining symbols. Given θ, m and Sm, a probability P (xn|m,Sm,θ) nx(i) · Snx(x>m)m · nx(i) nx(x > m) )nx(i) , (47) can be computed for xn, where nx(i) is the occurrence count of symbol i in x n, and nx(x > m) is the count of all symbols greater than m in xn. This probability mass function clusters all large symbols (with small probabilities) greater than m into one symbol. Then, it uses the ML estimate of each of the large symbols to distinguish among them in the clustered symbol. For every m, we can define a quantization grid ξm for the first m probability parameters of θ. The idea is similar to that used for all probability parameters in the proof of Theorem 4. If m = o(n1/3), we use ξm = τm, where τm is the grid defined in (18) where m replaces k. Otherwise, we can use the definition of η in (35). However, to obtain tighter bounds for large m, we define a different grid for the larger values of m following similar steps to those in (32)-(36). First, define the jth interval as n(j−1)β nρ+2α nρ+2α , 1 ≤ j ≤ Jρ, (48) where ρ = (logm)/(log n) as defined above, α is a parameter, and β = 1/(log n) as before. Within the jth interval, we define the spacing in the grid by nρ+3α . (49) As in (34), |Ij | ≤ 0.5 · nα, ∀j : j = 1, 2, . . . , Jρ, (50) and the total number of intervals is Jρ = ⌈(ρ+ 2α) log n⌉ . (51) Similarly to (35), ξm is defined as ξm = (ξ1, ξ2, . . .) = nρ+2α nρ+2α nρ+3α , . . . , nρ+2α nρ+2α nρ+3α , . . . . (52) The cardinality of ξm is thus = |ξm| ≤ 0.5 · nα ⌈(ρ+ 2α) log n⌉ . (53) An mth order quantized version θ′m of θ is obtained by quantizing θi, i = 2, 3, . . . ,m onto ξm, such that θ′i ∈ ξm for these values of i. Then, the remaining cluster probability Sm is quantized into S′m ∈ [1/n, 2/n, . . . , 1]. The parameter θ′1 is constrained by the quantization of the other parameters. Quantization is performed in a similar manner as before, to minimize the accumulating cost and retain monotonicity. Now, for any m ≥ 2, let ϕm be any monotonic probability vector of cardinality m whose last m− 1 components are quantized into ξm, and let σm ∈ [1/n, 2/n, . . . , 1] be a quantized estimate of the total probability of the remaining symbols, such that i=1 ϕi,m+σm = 1, where ϕi,m is the ith component of ϕm. If m, σm and ϕm are known, a given x n can be coded using P (xn|m,σm,ϕm) as defined in (47), where σm replaces Sm, and the m components of ϕm replace the first m com- ponents of θ. However, in the universal setting, none of these parameter are known in advance. Furthermore, neither the symbols greater than m nor their conditional ML probabilities are known in advance. Therefore, the total cost of coding xn using these parameters requires universality costs for describing them. The cost of universally coding xn assigning probability P (xn|m,σm,ϕm) to it thus requires the following five components: 1) m should be described using Elias’ representation with at most 1+ ρ log n+2 log(1+ ρ log n) bits. 2) The value of σm in its quantization grid should be coded using log n bits. 3) The m components of ϕm require LR (ϕm) (which is bounded below) bits. 4) The number cx(x > m) of distinct letters in x n greater than m is coded using log n bits. 5) Each letter i > m in xn is coded. Elias’ coding for the integers using 1 + log i+ 2 log(1 + log i) bits can be used, but to simplify the derivation we can also use the code, also presented in [7], that uses no more than 1 + 2 log i bits to describe i. In addition, at most log n bits are required for describing nx(i) in x n. For n→ ∞, m≫ 1, and ε > 0 arbitrarily small, this yields a total cost of L (xn|m,σm,ϕm) ≤ − log P (xn|m,σm,ϕm) + LR (ϕm) + [(1 + ε)ρ+ cx(x > m) + 2] log n +cx(x > m) + 2 i>m,i∈xn log i, (54) where we assume m is large enough to bound the cost of describing m by (1 + ε)ρ log n. The description cost of ϕm for m = o(n 1/3) is bounded by LR (ϕm) ≤ (1 + ε) using (21), where m replaces k. The (log n)2 factor in (21) can be absorbed in ε since we limit m to o(n1/3), unlike the derivation in (21). For larger values of m, we describe symbol probabilities of ϕm in the grid ξm in a similar manner to the description of O(n) symbol probabilities in the grid η. Similarly to (38), we thus have LR(ϕm) ≤ Bρ +Bρ log nρ +Bρ + 2Bρ log log nρ +Bρ + o (Bρ) ≤ (1 + ε) (ρ+ 2α) (ρ+ ε− α) (log n)2nα (56) where to obtain inequality (a), we first multiply nρ by nε in the numerator of the argument of the logarithm. This is only necessary for ρ→ α to guarantee that nρ+ε ≫ Bρ. Substituting the bound on Bρ from (53), absorbing low order terms in the leading ε, yields the bound. A sequence xn can now be coded using the universal parameters that minimize the length of the sequence description, i.e., L∗ (xn) = min σm′∈[ ,...,1] ϕm′ :ϕi∈ξm′ ,i≥2 xn|m′, σm′ ,ϕm′ xn|m,S′m,θ′m , (57) where θ′m and S m are the true source parameters quantized as described above, and the inequality holds for every m. Note that the maximization on m′ should be performed only up to the maximal symbol the occurs in xn. Following (54)-(57), up to negligible integer length constraints, the average redundancy using L∗(·) is bounded, for every m ≥ 2, by nRn (L ∗,θ) = Eθ [L ∗ (Xn) + log Pθ (X Xn | m,S′m,θ′m + logPθ (X ≤ Eθ log Pθ (X Xn | m,S′m,θ′m ) + LR Pθ (i ∈ Xn) log i +(1 + ε) [EθCx (X > m) + ρ+ 2] log n (58) where (a) follows from (57), and (b) follows from averaging on (54) with σm = S m, and ϕm = θ where the average on cx(x > m) is absorbed in the leading ε. Expressing Pθ (x n) as Pθ (x nx(i)  · Snx(x>m)m · )nx(i) , (59) and defining δS = Sm − S′m, the first term of (58) is bounded, for the upper region of m, by Eθ log Pθ (X Xn | m,S′m,θ′m ) ≤ Eθ Nx(i) log θ′i,m +Nx (X > m) log Nx(i) log θi/Sm Nx(i)/Nx(X > m) ≤ n · θi log θ′i,m + nSm log ≤ n(log e) θ′i,m ≤ (log e) · n · nρ nρ+2α + 2(log e)n1−ρ−4α · jβ + log e ≤ 5(log e)n1−2α + log e, (60) where (a) is since for the third term, the conditional ML probability used for coding is greater than the actual conditional probability assigned to all letters greater than m for every xn. Hence, the third term is bounded by 0. For the other terms expectation is performed. Inequality (b) is obtained similarly to (28) where quantization includes the first m components of θ and the parameter Sm. Then, inequality (c) follows the same reasoning as step (a) of (39). The first term bounds the worst case in which all nρ symbols are quantized to 1/nρ+2α with |δi| ≤ 1/nρ+2α. The second term is obtained where θ′i,m ≥ n(j−1)β/nρ+2α and |δi| ≤ njβ/nρ+3α for θi ∈ Ij , and kj = |θi ∈ Ij| as before. The last term is since S′m ≥ 1/n and |δS | ≤ 1/n. Finally, (d) is obtained similarly to step (b) of (39), where as in (29), jβ ≤ 2nρ+2α. For m = o(n1/3), the same initial steps up to step (b) in (60) are applied, and then the remaining steps in (28) are applied to the left sum with m replacing k, yielding a total quantization cost of 5(log e)m+ log e. To bound the third and fourth terms of (58), we realize that Pθ (i ∈ Xn) = 1− (1− θi)n ≤ nθi. (61) Similarly, EθCx(X > m) = Pθ (i ∈ Xn) ≤ nSm. (62) Combining the dominant terms of the third and fourth terms of (58), we have Pθ (i ∈ Xn) log i+ (1 + ε)EθCx(X > m) log n Pθ (i ∈ Xn) [2 log i+ (1 + ε) log n] 1 + ε Pθ (i ∈ Xn) log i 1 + ε θi log i (63) where (a) is because EθCx(X > m) = i>m Pθ (i ∈ Xn), (b) is because for i > m = nρ, log i > ρ log n, and (c) follows from (61). Given ρ > ε for an arbitrary fixed ε > 0, the resulting coefficient above is upper bounded by some constant κ. Summing up the contributions of the terms of (58) from (28), (55), and (63), absorbing low order terms in a leading ε′, we obtain that for m = o(n1/3), nRn (L ∗,θ) ≤ 1 + ε′ ) m− 1 θi log i. (64) For the second region, substituting α = 1/3, and summing up the contributions of (60), (56), and (63) to (58), absorbing low order terms in ε′, we obtain nRn (L ∗,θ) ≤ (1 + ε′) ρ+ ε′ − (log n) n1/3 + κn θi log i. (65) Since (64)-(65) hold for every m > nε, there exists m∗ for which the minimal bound is obtained. To bound the redundancy, we choose this m∗. Now, if the condition in (41) holds, then the second term in (64) and (65) is negligible w.r.t. the first term. Absorbing it in a leading ε, normalizing by n, yields the upper bound of (42), and concludes the proof of the Part I of Theorem 5. For Part II of Theorem 5, we consider the bound of the second region in (65). If there exists ρ∗ = o n1/3/(log n) for which the condition in (43) holds, then both terms of (65) are of o(n), yielding a total redundancy per symbol of o(1). The proof of Theorem 5 is concluded. � To prove Corollary 1, we use Wyner’s inequality [32], which implies that for a finite entropy monotonic distribution, θi log i = Eθ [logX] ≤ Hθ [X] . (66) Since the sum on the left hand side of (66) is finite if Hθ[X] is finite, there must exist some n0 such θi log i = o(1). Let n > n0, then for m ∗ = n and ρ∗ = 1, condition (43) is satisfied. Therefore, (44) holds, and the proof of Corollary 1 is concluded. � We now consider only the upper region in (58) with parameters α and ρ taking any valid value. (The code leading to the bound of the upper region can be applied even if the actual effective alphabet size is in the lower region.) We can sum up the contributions of (60), (56), and (63) to (58), absorbing low order terms in ε. Equation (56) is valid without the middle ε term as long as ρ ≥ α + ε. Since, in the upper region of m, i ≥ m is large enough, Elias’ code for the integers can be used costing (1 + ε) log i to code i, with ε > 0 which can be made arbitrarily small. Hence, the leading coefficient of the bound in (63) can be replaced by (1 + ε)(1 + 1/ρ). This yields the expression bounding the redundancy in (45). This expression applies to every valid choice of α and ρ, including the choice that minimizes the expression. Thus the proof of Theorem 6 is concluded. � 5.2 Examples We demonstrate the use of the bounds of Theorems 5 and 6 with three typical distributions over the integers. We specifically show that the redundancy rate of O n1/3+ε bits overall is achievable when coding many of the typical monotonic distributions, and, in fact, for many distributions faster convergence rates are achievable with the codes provided in proving the theorems above. The assumption that very few unlikely symbols are likely to appear in a sequence generated by a monotonic distribution, which is reflected in the conditions in (41) and (43), is very realistic even in practical examples. Specifically, in the phone book example, there may be many rare names, but only very few of them may occur in a certain city, and the more common names constitute most of any possible phone book sequence. 5.2.1 Fast Decaying Distributions Over the Integers Consider the monotonic distributions over the integers of the form, , i = 1, 2, . . . , (67) where γ > 0, and a is a normalization coefficient that guarantees that the probabilities over all integers sum to 1. It is easy to show by approximating summation by integration that for some m→ ∞, Sm ≤ (1 + ε) θi log i ≤ (1 + ε) a logm . (69) For m = nρ and fixed ρ, the sum in (41) is thus O n1−ργ log n , which is o n1/3(log n)2 for every ρ ≥ 2/(3γ). Specifically, as long as γ ≤ 2 (slow decay), the minimal value of ρ required to guarantee negligibility of the sum in (41) is greater than 1/3. Using Theorem 5, this implies that for γ ≤ 2, the second (upper) region of the upper bound in (42) holds with the minimal choice of ρ∗ = 2/(3γ). Plugging in this value in the second region of (40) (i.e., in (42)) yields the upper bound shown below for this region. For γ > 2, 2/(3γ) < 1/3. Hence, (41) holds for m∗ = o . This means that for the distribution in (67) with γ > 2, the effective alphabet size is o , and thus the achievable redundancy is in the first region of the bound of (42). Thus, even though the distribution is over an infinite alphabet, its compressibility behavior is similar to a distribution over a relatively small alphabet. To find the exact redundancy rate, we balance between the contributions of (55) and (63) in (58). As long as 1 − ργ < ρ, condition (41) holds, and the contribution of small letters in (63) is negligible w.r.t. the other terms of the redundancy. Equality, implying ρ∗ = 1/(1 + γ), achieves the minimal redundancy rate. Thus, for γ > 2, nRn (L ≤ (1 + ε) a(2ρ∗ + 1) ∗γ log n+ (1− 3ρ∗) log n = (1 + ε) 1+γ log n (70) where the first term in (a) follows from the bounds in (63) and (69), with m = nρ , and the second term from that in (55), and (b) follows from ρ∗ = 1/(1 + γ). Note that for a fixed ρ∗, the factor 3 in the first term can be reduced to 2 with Elias’ coding for the integers. The results described are summarized in the following corollary: Corollary 2 Let θ ∈ M be defined in (67). Then, there exists a universal code with length function L∗(·) that has only prior knowledge that θ ∈ M, that can achieve universal coding redundancy Rn (L ∗,θ) ≤ (1 + ε) 1 1 + 1 + ε− 1 n1/3(logn)2 , for γ ≤ 2, (1 + ε) 1+γ logn , for γ > 2. Corollary 2 gives the redundancy rates for all distributions defined in (67). For example, if γ = 1, the redundancy is O n1/3(log n)2 bits overall with coefficient 2/9. For γ = 3, O(n1/4 log n) bits are required. For faster decays (greater γ) even smaller redundancy rates are achievable. 5.2.2 Geometric Distributions Geometric distributions given by θi = p (1− p)i−1 ; i = 1, 2, . . . , (72) where 0 < p < 1, decay even faster than the distribution over the integers in (67). Thus their effective alphabet sizes are even smaller. This implies that a universal code can have even smaller redundancy than that presented in Corollary 2 when coding sequences generated by a geometric distribution (even if this is unknown in advance, and the only prior knowledge is that θ ∈ M). Choosing m = ℓ · log n, the contribution of low probability symbols in (63) to (58) can be upper bounded by θi (log i+ log n) ≤ 2n(1− p)m log n+O (n(1− p)m logm) = 2n1+ℓ log(1−p)(log n) +O n1+ℓ log(1−p) log log n where (a) follows from computing Sm using geometric series, and bounding the second term, and (b) follows from substituting m = ℓ log n and representing (1 − p)ℓ logn as nℓ log(1−p). As long as ℓ ≥ 1/(− log(1− p)), the expression in (73) is O(log n), thus negligible w.r.t. the redundancy upper bound of (42) with m∗ = ℓ∗ log n = (log n)/(− log(1 − p)). Substituting this m∗ in (42), we obtain the following corollary: Corollary 3 Let θ ∈ M be a geometric distribution defined in (72). Then, there exists a universal code with length function L∗(·) that has only prior knowledge that θ ∈ M, that can achieve universal coding redundancy Rn (L ∗,θ) ≤ 1 + ε −2 log(1− p) · (log n) . (74) Corollary 3 shows that if θ parameterizes a geometric distribution, sequences governed by θ can be coded with average universal coding redundancy of O (log n)2 bits. Their effective alphabet size is O(log n), implying that larger symbols are very unlikely to occur. For example, for p = 0.5, the effective alphabet size is log n, and 0.5(log n)2 bits are required for a universal code. For p = 0.75, the effective alphabet size is (log n)/2, and (log n)2/4 bits are required by a universal code. 5.2.3 Slow Decaying Distributions Over the Integers Up to now, we considered fast decaying distributions, which all achieved the O(n1/3+ε/n) redun- dancy rate. We now consider a slowly decaying monotonic distribution over the integers, given i (log i) , i = 2, 3, . . . , (75) where γ > 0 and a is a normalizing factor (see, e.g., [12], [27]). This distribution has finite entropy only if γ > 0 (but is a valid infinite entropy distribution for γ > −1). Unlike the previous distributions, we need to use Theorem 6 to bound the redundancy for coding sequences generated by this distribution. Approximating the sum with an integral, the order of the third term of (45) θi log i = O (logm)γ . (76) In order to minimize the redundancy bound of (45), we define ρ = nℓ. For the minimum rate, all terms of (45) must be balanced. To achieve that, we must have α+ 2ℓ = 1− 2α = 1− γℓ. (77) The solution is α = γ/(4 + 3γ), and ℓ = 2/(4 + 3γ). Substituting these values in the expression of (45), with ρ = nℓ, results in the first term in (45) dominating, and yields the following corollary: Corollary 4 Let θ ∈ M be defined in (75) with γ > 0. Then, there exists a universal code with length function L∗(·) that has only prior knowledge that θ ∈ M, that can achieve universal coding redundancy Rn (L ∗,θ) ≤ (1 + ε) 3γ+4 (log n)2 . (78) Due to the slow decay rate of the distribution in (75), the effective alphabet size is much greater here. For γ = 1, for example, it is nn . This implies that very large symbols are likely to appear in xn. As γ increases though, the effective alphabet size decreases, and as γ → ∞, m → n. The redundancy rate increases due to the slow decay. For γ ≥ 1, it is O n5/7(log n)2/n . As γ → ∞, since the distribution tends to decay faster, the redundancy rate tends to the finite alphabet rate n1/3(log n)2/n . However, as the decay rate is slower γ → 0, a non-diminishing redundancy rate is approached. Note that the proof of Theorem 6 does not limit the distribution to a finite entropy one. Therefore, the bound of (78) applies, in fact, also to −1 < γ ≤ 0. However, for γ ≤ 0, the per-symbol redundancy is no long diminishing. 6 Individual Sequences In this section, we first show that individual sequences whose empirical distributions obey the monotonicity constraints can be universally compressed as well as the average case. We then study compression of sequences whose empirical distributions may diverge from monotonic. We demonstrate that under mild conditions, similar in nature to those of Theorems 5 and 6, redundancy that diminishes (slower than in the average case) w.r.t. the monotonic ML description length can be obtained. However, these results are only useful when the monotonic ML description length diverges only slightly from the (standard) ML description length of a sequence, i.e., the empirical distribution of a sequence only mildly violates monotonicity. Otherwise, the penalty of using an incorrect monotone model overwhelms the redundancy gain. We begin with sequences that obey the monotonicity constraints. Theorem 7 Fix an arbitrarily small ε > 0, and let n→ ∞. Let xn be a sequence for which θ̂ ∈ M, i.e., θ̂1 ≥ θ̂2 ≥ . . .. Let k = k̂ be the number of letters occurring in xn. Then, there exists a code L∗ (·) that achieves individual sequence redundancy w.r.t. θ̂M = θ̂ for xn which is upper bounded R̂n (L ∗, xn) ≤ (1 + ε) k−1 n(logn) , for k ≤ n1/3, (1 + ε) (log n) log k n1/3−ε , for n1/3 < k = o(n), (1 + ε) 1 (log n) 2 n1/3 , for n1/3 < k = O(n). Note that by the monotonicity constraint, the number of symbols k̂ occurring in xn also equals to the maximal symbol in xn. Since, in the individual sequence case, this maximal symbol defines the class considered and also to be consistent with Theorem 3, we use k to characterize the alphabet size of a given sequence. (The maximal symbol in the individual sequence case is equivalent to the alphabet size in the average case.) Finally, since θ̂ is monotonic, θ̂M = θ̂. Proof of Theorem 7: The result in Theorem 7 follows directly from the proof of Theorem 4. Both regions of the proof apply here, where instead of quantizing θ to θ′, we quantize θ̂ to θ̂ similar manner, and do not need to average over all sequences. In fact, instead of using any general ϕ̂ to code xn, we can use θ̂ without any additional optimizations, where log n bits describe k. The description costs of θ̂ are almost the same as those of θ′. The factor 2 reduction in the last region is because it is sufficient here to replace n2 by n in the denominators of (32). This is because for every occurring symbol θ̂′i ≥ 1/n and δi ≤ 1/n, thus the first term of step (a) in (39) holds with the new grid, and B2 in (36) reduces by a factor of 2. The quantization costs bounded in (28) and (39) are thus bounded similarly, where θ̂ replaces θ and θ̂ replaces θ′. This results in the bounds in (79) and concludes the proof of Theorem 7. � If one a-priori knows that xn is likely to have been generated by a monotonic distribution, the case considered in Theorem 7 is with high probability the typical one. However, a typical sequence can also be one for which θ̂ 6∈ M, where θ̂ mildly violates the monotonicity. In the pure individual sequence setting (where no underlying distribution is assumed but some monotonicity assumption is reasonable for the empirical distribution of xn), one can still observe sequences that have empirical distributions that are either monotonic or slightly diverge from monotonic. Coding for this more general case can apply the methods described in Section 5 to the individual sequence case. If the divergence from monotonicity is small, one may still achieve bounds of the same order of those presented in Theorem 7 with additional negligible cost of relaying which symbols are out of order. The next theorem, however, provides a general upper bound in the form of the bounds of Theorems 5 and 6 for the individual sequence redundancy w.r.t. the monotonic ML description length, as defined in (10). We begin, again, with some notation. Recall the definition of an effective alphabet size m = nρ (where ρ = (logm)/(log n).) Now, use this definition for a specific individual sequence xn. Let R̂n(m) log n , m ≤ n1/3, m log n , n1/3 < m = o ( minα<ρ ρ+1+α (ρ− α) (log n)2 nα + 3(log e)n1−α , otherwise. Theorem 8 Fix an arbitrarily small ε > 0, and let n→ ∞. Then, there exists a code with length function L∗(·), that achieves individual sequence redundancy w.r.t. the monotonic ML description length of xn (as defined in (10)) bounded by R̂n (L ∗, xn) ≤ 1 + ε R̂n (nρ) + i>nρ,i∈xn log i for every xn. Theorem 8 shows that if one can find a relatively small effective alphabet of the symbols that occur in xn, and the symbols outside this alphabet are small enough, xn can be described with diminishing per-symbol redundancy w.r.t. its monotonic ML description length. This implies that as long as the occurring symbols are not too large, there exist a universal code w.r.t. a monotonic ML distribution for any such sequence xn. This is unlike standard individual sequence compression w.r.t. the i.i.d. ML description length. Specifically, if the effective alphabet size is O(n), and only a small number of symbols which are only polynomial in n occur, the universality cost is n(log n)2) bits overall, which gives diminishing per-symbol redundancy of O((log n)2/ This redundancy is much better than what can be achieved in standard compression. The penalty, of course, is when the empirical distribution of an individual sequence diverges significantly away from a monotonic one. While the monotonic redundancy can be made diminishing under mild conditions, there is a non-diminishing divergence cost by using the monotonic ML description length instead of the ML description length in that case. This implies that one should compress a sequence as generated by a monotonic distribution only if the total description length required to code xn as such is shorter than the total description length required to code xn with standard methods. As shown in the proof of Theorem 8, one prefix bit can inform the decoder which type of description is used. Theorem 8 shows that as long as the effective alphabet size is polynomial in n, α = 0.5 optimizes the third region of the upper bound, thus yielding the rate shown above, unless very large symbols occur in xn. For small effective alphabets (the first region), there is no redundancy gain in using the monotonic ML description length over the ML description length. The reason, again, is that the bound is obtained for cases where the actual empirical distribution of a sequence may not be monotonic. One can still use an i.i.d. ML estimate w.r.t. only the effective alphabet, if the additional cost of symbols outside this alphabet is negligible, to better code such sequences. Theorem 8 also shows that if a very large symbol, such as i = an; a > 1, occurs in xn, xn cannot be universally compressed even w.r.t. its monotonic ML description length. This is because it is impossible to avoid the cost of (1+ε) log i = (1+ε)n log a bits to describe this symbol to the decoder. The bound above and its proof below give a very powerful method to individually compress sequences that have an almost monotonic empirical distribution but may have some limited disorder, for which the monotonic ML description length diverges only negligibly from the ML description length. Proof of Theorem 8: The proof follows the same steps as the proof of Theorems 5 and 6. Each value of m is tested and the best one is chosen, where the same coding costs described in the mentioned proof are computed for each m. In addition, one can test the cost of coding xn using the description lengths for both θ̂ and θ̂M. Then, one bit can be used to relay which ML estimator is used. If θ̂ is used, the codes for coding individual sequences over large alphabets in either [21] or [25] can be used. In the first region in (81), the bound in [25] is obtained since log P (xn) ≥ log P for every xn. This bound yields smaller redundancy for this region than that obtained using θ̂M if θ̂M 6= θ̂. It implies that for small alphabets, if xn does not have an empirical monotonic distribution, it is better coded, even in terms of universal coding redundancy, using standard universal compression methods without taking advantage of a monotonicity assumption. For the other two regions, we start with a lemma. Lemma 6.1 Let θ̂M = θ̂1,M, θ̂2,M, . . . , θ̂k,M be the monotonic ML estimator of θ from xn, i.e., θ̂1,M ≥ θ̂2,M ≥ · · · ≥ θ̂k,M, where k = max {x1, x2, . . . , xn}. Then, θ̂k,M ≥ . (82) Lemma 6.1 provides a lower bound on the minimal nonzero probability component of the monotonic ML estimator. This bound helps in designing the grid of points used to quantize the monotonic ML distribution of xn, while maintaining bounded quantization costs. The proof of Lemma 6.1 is in Appendix C. For m in the second region, we cannot use the grid in (18). The reason is that, here, the quantization cost is affected by both θ̂ and θ̂M. This is unlike the average case, where the av- erage respective vectors merge. To limit the quantization cost for very small probabilities, using Lemma 6.1, the minimal grid point must be 1/n2 or smaller. To make the quantization cost neg- ligible w.r.t. the cost of describing the quantized ML, the ratio ∆j/ϕi,M between the spacing in interval j, and a quantized version ϕi,M of θ̂i,M in the jth interval, must be O(m/n). Hence, using the same methodology of the proof of Theorems 5 and 6, we define the jth interval for an effective alphabet m = nρ = o ( n) as Îj = n(j−1)β , 1 ≤ j ≤ Ĵρ. (83) The spacing in the jth interval is . (84) This gives a total of B̂ρ ≤ log n (85) quantization points. Using the same methodology as in (21), this yields a representation cost of LR (ϕm) ≤ (1 + ε)m log where ϕm is the quantized version of θ̂M in which only the firstm components of θ̂M are considered. Using the quantization with the grid defined in (83)-(86) in a code similar to the one used in the proof of Theorems 5 and 6, the individual quantization cost is given by P (xn|m,S′m,ϕm) θ̂i log θ̂i,M + log e ≤ n(log e) + log e ≤ (log e) · n ·mn+ (log e) · n · mn + log e = 3m(log e) + log e. (87) where (a) follows the same steps as in (60), (b) follows from ln(1+x) ≤ x, and then x ≤ |x|, where = θ̂i,M − ϕi,m, and (c) follows from Lemma 6.1 and the definition of Îj in (83) (for the worst case first term, |δi| ≤ 1/n2 and ϕi,m ≥ 1/(mn)), from (84) and (83) (the second term), and since θ̂i = 1. The only additional non-negligible cost of coding sequences using a code as defined in the proof of Theorems 5 and 6 for a given m is the cost of coding all symbols i > m that occur in xn. Using a similar derivation to (54), with Elias’ asymptotic code for the integers, this yields an additional cost of (1 + ε) (1 + 1/ρ) i>nρ,i∈xn log i code bits. Combining all costs, absorbing low order terms in ε, and normalizing by n, yields the second region of the bound in (81). Note that this bound also applies to the first region, but in that region, a tighter bound is obtained by using a code that uses the standard i.i.d. ML estimator θ̂. This is because very fine quantization is needed to offset the cost of mismatch between θ̂ and θ̂M. This quantization requires higher description costs than the description of a quantized type of a sequence when using standard compression. (This is not the case when θ̂ obeys the monotonicity, as in Theorem 7. Even if θ̂ does not obey monotonicity in the upper regions of the bound, this is not the case.) For the last region of the bound, we follow the same steps above as was done for the upper region of the bound in Theorem 5 with a parameter α. The intervals are chosen, again, to guarantee bounded quantization costs. Hence, Îj = n(j−1)β nρ+1+α nρ+1+α , 1 ≤ j ≤ Ĵρ. (88) The spacing in the jth interval is nρ+1+2α . (89) This gives a total of B̂ρ ≤ 0.5nα ⌈(ρ+ 1 + α) log n⌉ (90) quantization points. Using the same methodology as in (56), this yields a representation cost of LR (ϕm) ≤ (1 + ε) ρ+ 1 + α (ρ+ ε− α) (log n)2nα. (91) Similarly to (87), P (xn|m,S′m,ϕm) ≤ (log e) nρ+1+α + (log e)2n1−α + log e = 3(log e)n1−α + log e (92) where (a) follows from similar steps to (a)-(c) of (87). Using Lemma 6.1, ϕi,m ≥ 1/nρ+1 and |δi| ≤ 1/nρ+1+α, leading to the first term. Bounding |δi| ≤ njβ/nρ+1+2α and ϕi,m ≥ n(j−1)β/nρ+1+α leads to the second term. Note that as before, m is used here in place of k, because using an ef- fective alphabet m, all greater symbols are packed together as one symbol, and the additional cost to describe them is reflected in an additional term. Adding this additional term with an identical expression to that in the lower regions, absorbing low order terms in ε, and normalizing by n, yields the third region of the bound in (81). Since the bound holds for every α and every ρ > α, it can be optimized to give the values that attain the minimum, concluding the proof of Theorem 8. � 7 Summary and Conclusions Universal compression of sequences generated by monotonic distributions was studied. We showed that for finite alphabets, if one has the prior knowledge of the monotonicity of a distribution, one can reduce the cost of universality. For alphabets of o(n1/3) letters, this cost reduces from 0.5 log(n/k) bits per each unknown probability parameter to 0.5 log(n/k3) bits per each unknown probability parameter. Otherwise, for alphabets of O(n) letters, one can compress such sources with overall redundancy of O(n1/3+ε) bits. This is a significant decrease in redundancy from O(k log n) or O(n) bits overall that can be achieved if no side information is available about the source distribution. Redundancy of O(n1/3+ε) bits overall can also be achieved for much larger alphabets including infinite alphabets for fast decaying monotonic distributions. Sequences generated by slower decaying distributions can also be compressed with diminishing per-symbol redundancy costs under some mild conditions and specifically if they have finite entropy rates. Examples for well-known monotonic distributions demonstrated how the diminishing redundancy decay rates can be computed by applying the bounds that were derived. Finally, the average case results were extended to individual sequences. Similar convergence rates were shown for sequences that have empirical monotonic distributions. Furthermore, universal redundancy bounds w.r.t. the monotonic ML description length of a sequence were also derived for the more general case. Under some mild conditions, these bounds still exhibit diminishing per-symbol redundancies. Appendix A – Proof of Theorem 1 The proof follows the same steps used in [25] and [26] to lower bound the maximin redundancies for large alphabets and patterns, respectively, using the weak version of the redundancy-capacity theorem [5]. This version ties between the maximin universal coding redundancy and the capacity of a channel defined by the conditional probability Pθ (x n). We define a set ΩMk of points θ ∈ Mk. Then, show that these points are distinguishable by observing Xn, i.e., the probability that Xn generated by θ ∈ ΩMk appears to have been generated by another point θ ′ ∈ ΩMk diminishes with n. Then, using Fano’s inequality [3], the number of such distinguishable points is a lower bound on R−n (Mk). Since R+n (Mk) ≥ R−n (Mk), it is also a lower bound on the average minimax redundancy. The two regions in (6) result from a threshold phenomenon, where there exists a value km of k that maximizes the lower bound, and can be applied to all Mk for k ≥ km. We begin with defining ΩMk . Let ω be a vector of grid components, such that the last k − 1 components θi, i = 2, . . . , k, of θ ∈ ΩMk must satisfy θi ∈ ω. Let ωb be the bth point in ω, and define ω0 = 0 and 2(j − 1 , b = 1, 2, . . . . (A.1) Then, for the bth point in ω, . (A.2) To count the number of points in ΩMk , let us first consider the standard i.i.d. case, where there is no monotonicity requirement, and count the number of points in Ω, which is defined similarly, but without the monotonicity requirement (i.e., ΩMk ⊆ Ω). Let bi be the index of θi in ω, i.e., θi = ωbi . Then, from (A.1)-(A.2) and since the components of θ are probabilities, ωbi = θi ≤ 1. (A.3) It follows that for θ ∈ Ω, b2i ≤ n1−ε. (A.4) Hence, since the components bi are nonnegative integers, = |Ω| ≥ n1−ε⌋ n1−ε−b22 · · · n1−ε− i=2 b n1−ε−x22 · · · n1−ε− i=2 x dxk · · · dx3dx2 (A.5) where Vk−1 is the volume of a k − 1 dimensional sphere with radius , (a) follows from monotonic decrease of the function in the integrand for all integration arguments, and (b) follows since its left hand side computes the volume of the positive quadrant of this sphere. Note that this is a different proof from that used in [25]-[26] for this step. Applying the monotonicity constraint, all permutations of θ that are not monotonic must be taken out of the grid. Hence, = |ΩMk | ≥ k! · 2k−1 , (A.6) where dividing by k! is a worst case assumption, yielding a lower bound and not an equality. This leads to a lower bound equal to that obtained for patterns in [26] on the number of points in ΩMk . Specifically, the bound achieves a maximal value for km = πn1−ε/2 and then decreases to eventually become smaller than 1. However, for k > km, one can consider a monotonic distribution for which all components θi; i > km, of θ are zero, and use the bound for km. Distinguishability of θ ∈ ΩMk is a direct result of distinguishability of θ ∈ Ω, which is shown in Lemma 3.1 in [25], i.e., there exits an estimator Θ̂g(X n) ∈ Ω for which the estimate θ̂g satisfies limn→∞ Pθ θ̂g 6= θ = 0 for all θ ∈ Ω. Since this is true for all points in Ω, it is also true for all points in ΩMk ⊆ Ω, where now, θ̂g ∈ ΩMk . Assuming all points in ΩMk are equally probable to generate Xn, we can define an average error probability Pe Θ̂g(X n) 6= Θ θ∈ΩMk θ̂g 6= θ /MMk . Using the redundancy-capacity theorem, nR−n [Mk] ≥ C [Mk → Xn] ≥ I[Θ;Xn] = H [Θ]−H [Θ|Xn] = logMMk −H [Θ|X ≥ (1− Pe) (logMMk)− 1 ≥ (1− o(1)) logMMk , (A.7) where C [Mk → Xn] denotes the capacity of the respective channel and I[Θ;Xn] is the mutual information induced by the joint distribution Pr (Θ = θ) · Pθ (Xn). Inequality (a) follows from the definition of capacity, equality (b) from the uniform distribution of Θ in ΩMk , inequality (c) from Fano’s inequality, and (d) follows since Pe → 0. Lower bounding the expression in (A.6) for the two regions (obtaining the same bounds as in [26]), then using (A.7), normalizing by n, and ab- sorbing low order terms in ε, yields the two regions of the bound in (6). The proof of Theorem 1 is concluded. � Appendix B – Proof of Theorem 2 To prove Theorem 2, we use the random-coding strong version of the redundancy-capacity theorem [17]. The idea is similar to the weak version used in Appendix A. We assume that grids ΩMk of points are uniformly distributed over Mk, and one grid is selected randomly. Then, a point in the selected grid is randomly selected under a uniform prior to generate Xn. Showing distinguishability within a selected grid, for every possible random choice of ΩMk , implies that a lower bound on the cardinality of ΩMk for every possible choice is essentially a lower bound on the overall sequence redundancy for most sources in Mk. The construction of ΩMk is identical to that used in [26] to construct a grid of sources that generate patterns. We pack spheres of radius n−0.5(1−ε) in the parameter space defining Mk. The set ΩMk consists of the center points of the spheres. To cover the space Mk, we randomly select a random shift of the whole lattice under a uniform distribution. The cardinality of ΩMk is lower bounded by the relation between the volume of Mk, which equals (as shown in [26]) 1/[(k− 1)!k!], and the volume of a single sphere, with factoring also of a packing density (see, e.g., [2]). This yields eq. (55) in [26], MMk ≥ (k − 1)! · k! · Vk−1 n−0.5(1−ε) · 2k−1 , (B.1) where Vk−1 n−0.5(1−ε) is the volume of a k−1 dimensional sphere with radius n−0.5(1−ε) (see, e.g., [2] for computation of this volume). For distinguishability, it is sufficient to show that there exists an estimator Θ̂g(X n) ∈ ΩMk such that limn→∞ PΘ Θ̂g(X n) 6= Θ = 0 for every choice of ΩMk and for every choice of Θ ∈ ΩMk . This is already shown in Lemma 4.1 in [25] for a larger grid Ω of i.i.d. sources, which is constructed identically to ΩMk over the complete k−1 dimensional probability simplex. Therefore, by the monotonicity requirement, for every ΩMk , there exists such Ω, such that ΩMk ⊆ Ω. Since Lemma 4.1 in [25] holds for Ω, it then must also hold for the smaller grid ΩMk . Note that distinguishability is easier to prove here than for patterns because Θ̂g(X n) is obtained directly form Xn and not from its pattern as in [26]. Now, since all the conditions of the strong random- coding version of the redundancy-capacity theorem hold, taking the logarithm of bound in (B.1), absorbing low order terms in ε, and normalizing by n, leads to the first region of the bound in (7). More detailed steps follow those found in [26]. The second region of the bound is handled in a manner related to the second region of the bound of Theorem 1. However, here, we cannot simply set the probability of all symbols i > km to zero, because all possible valid sources must be included in one of the grids ΩMk to generate a complete covering of Mk. As was done in [26], we include sources with θi > 0 for i > km in the grids ΩMk , but do not include them in the lower bound on the number of grid points. In- stead, for k > km, we bound the number of points in a km-dimensional cut of Mk for which the remaining k− km components of θ are very small (and insignificant). This analysis is valid also for k > n. Distinguishability for k > km is shown for i.i.d. non-monotonically restricted distributions in the proof of Lemma 6.1 in [26]. As before, it carries over to monotonic distributions, since as before, for each ΩMk , there exists an unrestricted corresponding Ω, such that ΩMk ⊆ Ω. The choice of km = 0.5(n 1−ε/π)1/3 gives the maximal bound w.r.t. k. Since, again, all conditions of the strong version of the redundancy-capacity theorem are satisfied, the second region of the bound is obtained. Again, more detailed steps can be found in [26]. This concludes the proof of Theorem 2. � Appendix C – Proof of Lemma 6.1 For cardinality k, we consider the largest component of θ̂M; θ̂1,M, as the constraint component, i.e., θ̂1,M = 1− i=2 θ̂i,M. For any given probability parameter ϕ of cardinality k with ϕ1 > 0, we Pϕ (x n) = ϕ nx(1) 1 (1− ϕ1) n−nx(1) · 1− ϕ1 )nx(i) △ nx(1) 1 (1− ϕ1) n−nx(1) nx(i) i (C.1) where we recall that nx(i) is the occurrence count of i in x n. Therefore, maximization of Pϕ (x w.r.t. ϕ1 is independent of the maximization over ϑi; i > 1, and is obtained for ϕ1 = θ̂1 = nx(1)/n. Since for all i > 1, θ̂1,M ≥ θ̂i,M, θ̂1,M can thus only increase from θ̂1 by the monotonicity constraint. (Note that the monotonicity constraint implies a water filling [3] optimization to achieve θ̂M.) Hence, θ̂1,M ≥ nx(1)/n. Now, using the result above, we show that the derivative of lnPϕM (x n) w.r.t. ϕk,M is positive for ϕk,M < 1/(kn) and a monotonic ϕM. A component of a parameter vector ϕM, which is monotonic, can be expressed as ϕi,M = ϕ′ℓ, ϕ ℓ ≥ 0. (C.2) Hence, ∂ lnPϕM (x ∂ϕk,M ϕ1,M=θ̂1,M ∂ lnPϕM (x ϕ1,M=θ̂1,M nx(i) − (k − 1)nx(1) θ̂1,M knx(k) − knx(1) = 0 (C.3) where (a) follows from ϕk,M being the smallest nonzero component of ϕM, (b) is since by (C.2), ϕ′k is included in all terms, and ϕ1,M = 1− ϕi,M = 1− (i− 1)ϕ′i − (k − 1)ϕk,M, (C.4) where the last equality follows from (C.2), (c) follows by omitting all terms of the sum except i = k, from the assumption that ϕk,M < 1/(nk) ≤ θ̂k/k, and since θ̂1,M ≥ nx(1)/n = θ̂1, and (d) follows since its left hand side is 0 for the (i.i.d.) ML parameter values. Hence, PϕM (x n) must increase, with ϕ1,M taking its optimal value, for all ϕM for which ϕk,M < 1/(nk), and the maximum is thus achieved for θ̂k,M ≥ 1/(nk). � References [1] J. Åberg, Y. M. Shtarkov, and B. J. M. Smeets, “Multialphabet coding with separate alphabet description,” in Proceedings of Compression and Complexity of Sequences, pp. 56-65, Jun. 1997. [2] J. H. Conway, N. J. A. 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Introduction Notation and Definitions Lower Bounds Upper Bounds for Small and Large Alphabets Upper Bounds for Fast Decaying Distributions Upper Bounds Examples Fast Decaying Distributions Over the Integers Geometric Distributions Slow Decaying Distributions Over the Integers Individual Sequences Summary and Conclusions – Proof of Theorem ?? – Proof of Theorem ?? – Proof of Lemma ??

0704.0839

Moduli spaces of rational tropical curves

MODULI SPACES OF RATIONAL TROPICAL CURVES GRIGORY MIKHALKIN Abstract. This note is devoted to the definition of moduli spaces of rational tropical curves with nmarked points. We show that this space has a structure of a smooth tropical variety of dimension n − 3. We define the Deligne-Mumford compactification of this space and tropical ψ-class divisors. This paper gives a detailed description of the moduli space of tropical rational curves mentioned in [4]. The survey [4] was prepared under rather sharp time and volume constraints. As a result the coordinate presentation of this moduli space from [4] contains a mistake (it was over-simplified). In this paper we’ll correct the mistake and give a detailed description on M0,5 as our main example. 1. Introduction: smooth tropical varieties In this section we follow the definitions of [5] and [4]. The underlying algebra of tropical geometry is given by the semifield T = R∪{−∞} of tropical numbers. The tropical arithmetic operations are “a + b” = max{a, b} and “ab” = a + b. The quotation marks are used to distinguish between the tropical and classical operations. With respect to addition T is a commutative semigroup with zero “0T” = −∞. With respect to multiplication T× = T r {0T} ≈ R is an honest commutative group with the unit “1T” = 0. Furthermore, the addition and multiplication satisfy to the distribution law “a(b+c)” = “ab+ac”, a, b, c ∈ T. These operations may be viewed as a result of the so-called dequantization of the classical arithmetic operations that underlies the patchworking construction, see [3] and [8]. These tropical operations allow one to define tropical Laurent poly- nomials. Namely, a tropical Laurent polynomial is a function f : Rn → f(x) = “ j” = max (aj + jx), where jx denotes the scalar product, x ∈ (T×)n ≈ Rn, j ∈ Zn and only finitely many coefficients aj ∈ T are non-zero (i.e. not −∞). http://arxiv.org/abs/0704.0839v1 2 GRIGORY MIKHALKIN Affine-linear functions with integer slopes (for brevity we call them simply affine functions) form an important subcollection of all Laurent polynomials. Namely, these are such functions f : Rn → R that both f and “1T ” = −f are tropical Laurent polynomials. We equip Tn ≈ [−∞,∞)n with the Euclidean topology. Let U ⊂ Tn be an open set. Definition 1.1. A continuous function f : U → T is called regular if its restriction to U ∩ Rn coincides with a restriction of some tropical Laurent polynomial to U ∩ Rn. We denote the sheaf of regular functions on Tn with O (or sometimes OTn to avoid confusion). Any subset X ⊂ T n gets an induced regular sheaf OX by restriction. For our purposes we restrict our attention only to the case when X is a polyhedral complex, i.e. when X is the closure of a union of convex polyhedra (possibly unbounded) in Rn such that the intersection of any number of such polyhedra is their common face. We say that X is an k-dimensional polyhedral complex if it is obtained from a union of k-dimensional polyhedra. These polyhedra are called the facets of X . Let V ⊂ X be an open set and f ∈ OX(V ) be a regular function in V . A point x ∈ V is called a “zero point” of f if the restriction of ” = −f toW ⊂ V is not regular for any open neighborhoodW ∋ x. Note that it may happen that x is a “zero point” for φ : U → T, but not for φ|X∩U . It is easy to see that if X is a k-dimensional polyhe- dral complex then the “zero locus” Zf of f is a (k − 1)-dimensional polyhedral subcomplex. To each facet of Zf we may associate a natural number, called its weight (or degree). To do this we choose a “zero point” x inside such a facet. We say that x is a “simple zero” for f if for any local de- composition into a sum (i.e. the tropical product) of regular function f = “gh” = g+h on V near x we have either g or h affine (i.e. without a “zero”). We say that the weight is l if f can be locally decomposed into a tropical product of l functions with a simple zero at x. A regular function f allows us to make the following modification on its domain V ⊂ X ⊂ Tn. Consider the graph Γf ⊂ V × T ⊂ T It is easy to see that the “zero locus” Γ̄f ⊂ V × T of the (regular) function “y+ f(x)” (defined on V ×T), where x is the coordinate on V and y is the coordinate on T, coincides with the union MODULI SPACES OF RATIONAL TROPICAL CURVES 3 of Γf and the undergraph UΓf,Z = {(x, y) ∈ V × T | x ∈ Zf , y ≤ f(x)}. Furthermore, the weight of a facet F ⊂ Γ̄f is 1 if F ∈ Γf (recall that as V is an unweighted polyhedral complex all the weights of its facets are equal to one) and is the weight of the corresponding facet of Zf if F ∈ UΓf,Z . We view Γ̄f as a “tropical closure” of the set-theoretical graph Γf . Note that we have a map Γ̄f → V . We set Ṽ = Γ̄f to be the result of the tropical modification µf : Ṽ → V along the regular function f . The locus Zf is called the center of tropical modification. The weights of the facets of Ṽ supplies us with some inconvenience as they should be incorporated in the definition of the regular sheaf OṼ on Ṽ . Namely, the affine functions defined by OṼ on a facet of weight w should contain the group of functions that come as restrictions to this facet of the affine functions on Tn+1 as a subgroup of index w. Sometimes one can get rid of the weights of Ṽ by a reparameteriza- tion with the help of a map V̄ → Ṽ that is given by locally linear maps in the corresponding charts. Indeed, the restriction of µg : V̄ → Ṽ to a facet is locally given by a linear function between two k-dimensional affine-linear spaces defined over Z. If its determinant equals to w then the push-forward of OV̄ supplies an extension of OṼ required by the weights. Note however that if w > 1 then the converse map is not defined over Z and thus is not given by elements of OṼ . Tropical modifications give the basic equivalence relation in Tropical Geometry. It can be shown that if we start from Tk and do a number of tropical modifications on it then the result is a k-dimensional polyhe- dral complex Y ⊂ Tn that satisfies to the following balancing property (cf. Property 3.3 in [4] where balancing is restated in an equivalent way). Property 1.2. Let E ⊂ Y ∩ RN be a (k − 1)-dimensional face and F1, . . . , Fl be the facets of Y adjacent to F whose weights are w1, . . . , wl. Let L ⊂ RN be a (N−k)-dimensional affine-linear space with an integer slope and such that it intersects E. For a generic (real) vector v ∈ RN the intersection Fj ∩ (L + v) is either empty or a single point. Let ΛFj ⊂ Z N be the integer vectors parallel to Fj and ΛL ⊂ Z N be the integer vectors parallel to L. Set λj to be the product of wj and the index of the subgroup ΛFj +ΛL ⊂ Z N . We say that Y ⊂ Tn is balanced if for any choice of E, L and a small generic v the sum j | Fj∩(L+v)6=∅ 4 GRIGORY MIKHALKIN is independent of v. We say that Y is simply balanced if in addition for every j we can find L and v so that Fj ∩ (L + v) 6= ∅, ιL = 1 and for every small v there exists an affine hyperplane Hv ⊂ L such that the intersection Y ∩ (L + v) sits entirely on one side of Hv + v in L + v while the intersection Y ∩ (Hv + v) is a point. Definition 1.3 (cf. [5],[4]). A topological space X enhanced with a sheaf of tropical functions OX is called a (smooth) tropical variety of dimension k if for every x ∈ X there exist an open set U ∋ x and an open set V in a simply balanced polyhedral complex Y ⊂ TN such that the restrictions OX |U and OY |V are isomorphic. Tropical varieties are considered up to the equivalence generated by tropical modifications. It can be shown that a smooth tropical variety of dimension k can be locally obtained from Tk by a sequence of tropical modifications centered at smooth tropical varieties of dimension (k−1). This follows from the following proposition. Proposition 1.4. Any k-dimensional simply balanced polyhedral com- plex X ⊂ Rn can be obtained from Tk by a sequence of consecutive trop- ical modifications whose centers are simply balanced (k−1)-dimensional polyhedral complexes. Proof. We prove this proposition inductively by n. Without the loss of genericity we may assume that X is a fan, i.e. each convex polyhedron of X is a cone centered at the origin. The base of the induction, when n = k, is trivial. If n > k let us take a (n− k)-dimensional affine-linear subspace L ⊂ Rn given by Property 1.2. Choose a linear projection λ : Rn → Rn−1 defined over Z and such that ker(λ) is a line contained in L. The image λ(X) ⊂ Rn−1 is a k-dimensional polyhedral complex since L is transversal to some facets of X . We claim that λ|X : X → λ(X) is a tropical modification once we identify Rn and Rn−1×R. The center of this modification is the locus Zf = {x ∈ R n−1 | dim(λ−1(x) ∩X) > 0}. Here we use the dimension in the usual topological sense. Note that the (k − 1)-dimensional complex Zf ⊂ R n−1 is simply balanced, existence of the needed (n− k)-dimensional affine-linear spaces follows from the fact that X ⊂ Rn is simply balanced. MODULI SPACES OF RATIONAL TROPICAL CURVES 5 To justify our claim we note that near any point x ∈ Zf the sub- complex Y ⊂ X obtained as the (Euclidean topology) closure of X r λ−1(Zf) is a (set-theoretical) graph of a convex function. This, once again, follows from the fact that X ⊂ Rn is simply balanced, this time applied to the points in the facets on X r Y . Thus it gives a regular tropical function f and it remains only to show that the the weight of any facet of E ⊂ Zf is 1. But this follows, in turn, from the balancing condition at λ−1(E) ∩ Y . � 2. Tropical curves and their moduli spaces The definition of tropical variety is especially easy in dimension 1. Tropical modifications take a graph into a graph (with arbitrary va- lence of its vertices) and the tropical structure carried by the sheaf OX amounts to a complete metric on the complement of the set of 1-valent vertices of the graph X (cf. [5], [6], [1]). Thus, each 1-valent vertex of a tropical curve X is adjacent to an edge of infinite length. A tropical modification allows one to contract such an edge or to attach it at any point of X other than a 1-valent vertex. If we have a finite collection of marked points on X then by passing to an equivalent model if needed we may assume that the set of marked points coincides with the set of 1-valent vertices. (Of course, if X is a tree then we have to have at least two marked points to make such assumption.) The genus of a tropical curve X is dimH1(X). Let Mg,n be the set of all tropical curves X of genus g with n distinct marked points. Fixing a combinatorial type of a graph Γ with n marked leaves defines a subset UΓ ⊂ Mg,n consisting of marked tropical curves with this combinatorics. A length of any non-leaf edge of Γ defines a real-valued function on UΓ. Such functions are called edge-length functions. To avoid difficulties caused by self-automorphisms of X from now on we restrict our attention to the case g = 0. Definition 2.1. The combinatorial type of a tropical curve X is its equivalence class up to homeomorphisms respecting the markings. Combinatorial types partite the set M0,n into disjoint subsets. The edge-length functions define the structure of the polyhedral cone RM≥0 in each of those subsets (as the lengths have to be positive). The number M here is the number of the bounded (non-leaf) edges in X . By the Euler characteristic reasoning it is equal to n − 3 if X is (1- and) 3-valent, it is smaller if X has vertices of higher valence. Furthermore, any face of the polyhedral cone RM≥0 coincides with the cone corresponding to another combinatorial type, the one where we 6 GRIGORY MIKHALKIN contract some of the edges of X to points. This gives the adjacency (fan-like) structure on M0,n, so M0,n is a (non-compact) polyhedral complex. In particular, it is a topological space. Theorem 1. The set M0,n for n ≥ 3 admits the structure of an (n−3)- dimensional tropical variety such that the edge-length functions are reg- ular within each combinatorial type. Furthermore, the space M0,n can be tropically embedded in RN for some N (i.e. M0,n can be presented as a simply balanced complex). Proof. This theorem is trivial for n = 3 as M0,3 is a point. Otherwise, any two disjoint ordered pairs of marked points can be used to define a global regular function on M0,n with values in R = T ×. Namely, each such ordered pair defines the oriented path on the tropical curve X connecting the corresponding marked points. These paths can be embedded. Since the two pairs of marked points are disjoint the intersection of the two corresponding paths has to have finite length. We take this length with the positive sign if the orientations agree and with the negative sign otherwise. This defines a function on M0,n. We call such functions the double ratio functions. Take all possible disjoint pairs of marked points and use them as coordinates for our embedding ι : M0,n → R where N is the number of all possible decompositions of n into two disjoint pairs. The theorem now follows from the following two lemmas. Note that, strictly speaking, each coordinate in RN depends not only on the choice of two disjoint pairs of marked points but also on the order of points in each pair. However, changing the order in one of the pairs only reverses the sign of the double ratio. Taking an extra coordinate for such a change of order would be redundant. Indeed, for any balanced complex Y ⊂ RN and any affine-linear function λ : RN → R with an integer slope the graph of λ is a balanced complex in RN+1 isomorphic to the initial complex Y . Lemma 2.2. The map ι is a topological embedding. Proof. First, let us prove that ι is an embedding. The combinatorial type of X is determined by the set of the coordinates that do not vanish on X . Indeed, any non-leaf edge E of the tree X separates the leaves (i.e. the set of markings) into two classes corresponding to the components of XrE. Let us take a coordinate in Rn that corresponds MODULI SPACES OF RATIONAL TROPICAL CURVES 7 to four marking points (union of the two disjoint pairs) such that two of these points belong to one class and two to the other class. We call such a coordinate an E-compatible coordinate. Note that an E-compatible coordinate vanishes on X if and only if the pairs of markings defined by the coordinate agree with the pairs defined by the classes. This observation suffices to reconstruct the combinatorial type of X . Furthermore, the length of E equals to the minimal non-zero abso- lute value of the E-compatible coordinates. This implies that ι is an embedding. � Lemma 2.3. The image ι(M0,n) is a simply balanced complex in R Proof. This is a condition on codimension 1 faces of M0,n. First we shall check it for the case n = 4. There are three ways to split the four marking points into two disjoint pairs. Accordingly, there are three combinatorial types of 3-valent trees with three marked leaves. Thus our space M0,4 is homeomorphic to the tripod, or the “interior” of the letter Y , see Figure 1. Each ray of this tripod correspond to a combinatorial type of a 3-valent tree with 4 leaves while the vertex correspond to the 4-valent tree. Figure 1. The tropical moduli space M0,4 and its points on the corresponding edges. 8 GRIGORY MIKHALKIN Up to the sign we have the total of three double ratios for n = 4. Let us e.g. take those defined by the following ordered pairs: {(12), (34)}, {(13), (24)} and {(14), (23)} Each is vanishing on the corresponding ray of the tripod. Let us parameterize each ray of the tripod by its only edge-length t ≥ 0 and compute the corresponding map to R3. We have the following embeddings on the three rays t 7→ (0, t, t), t 7→ (t, 0,−t), t 7→ (−t,−t, 0). The sum of the primitive integer vectors parallel to the resulting direc- tions is 0 and thus ι(M0,4) is balanced. In the case n > 4 the codimension 1 faces of M0,n correspond to the combinatorial types of X with a single 4-valent vertex. Near a point inside of such face F the space M0,n looks like the product of M0,4 and R n−4. The factor Rn−4 comes from the edge-lengths on F (its combinatorial type has n−4 bounded edges) while the factor M0,4 comes from perturbations of the 4-valent vertex (which result in a new bounded edge in one of the three possible combinatorial types of the result). We have a well-defined map from the union U of the F -adjacent facets to F by contracting the new edge to a point. Note that the edge-length functions exhibit F as the positive quadrant in Rn−4. Fur- thermore, in the combinatorial type of F we may choose 4 leaves such that contracting all other leaves will take place outside of the 4-valent vertex (see Figure 2). This contraction defines a map U → M0,4. Figure 2. One of the possible contractions of a tree with a 4-valent vertex to the tree corresponding to the origin O ∈ M0,4. The lemma now follows from the observation that the resulting de- composition into M0,4 × R n−4 agrees with the double ratio functions. Indeed, note that the complement of the 4-valent vertex for a curve in the combinatorial type F is composed of four components. If the double ratio is such that its four markings are in one-to-one correspon- dence with these components then at U it coincides with sum of the pull-back of the corresponding double ratio in F with the pull-back of the corresponding double ratio in M0,4. If one of the four components MODULI SPACES OF RATIONAL TROPICAL CURVES 9 is lacking a marking from the double ratio ρ then ρ|U coincides with the corresponding pull-back from F . � Remark 2.4. The functions Zxi,xj from [4] do not define regular func- tions on M0,n, contrary to what is written in [4]. These functions were a result of an erroneous simplification of the double ratio functions. But these functions cannot be regular as they are always positive and Proposition 5.12 of [4] is not correct. Even the projectivization of the embedding is not a balanced complex already for M0,5. One should use the (non-simplified) double ratios instead. Clearly, the space M0,n is non compact. However it is easy to com- pactify it by allowing the lengths of bounded edges to assume infinite values. Let M0,n be the space of connected trees with n (marked) leaves such that each edge of this tree is assigned a length 0 < l ≤ +∞ so that each leaf has length necessarily equal to +∞. Corollary 2.5. The space M0,n is a smooth compact tropical variety. To verify that M0,n is smooth near a point x at the boundary ∂M0,n = M0,n rM0,n we need to examine those double ratios that are equal to ±∞ at x. There we use only those signs that result in −∞ do that the map takes values in TN . Remark 2.6. Note that the compactification M0,n ⊃ M0,n corresponds to the Deligne-Mumford compactification in the complex case as under the 1-parametric family collapse of a Riemannian surface to a tropical curve the tropical length of an edge corresponds to the rate of growth of the complex modulus of the holomorphic annulus collapsing to that edge. Furthermore, similarly to the complex story the infinite edges de- compose a tropical curve into components (where the non-leaf edges are finite). Any tropical map from an infinite edge which is bounded would have to be constant and thus the image would have to split as a union of several tropical curves in the target. Such decompositions were used by Gathmann and Markwig in their deduction of the tropical WDVV equation in R2, see [1]. 3. Tropical ψ-classes Note that we do have the forgetting maps ftj : M0,n+1 → M0,n 10 GRIGORY MIKHALKIN for j = 1, . . . , n + 1 by contracting the leaf with the j-marking. This map is sometimes called the universal curve. Each marking k 6= j defines a section σk of ftj. The conormal bundle to σk defines the ψk- class in complex geometry (to avoid ambiguity we take j = n+1). This notion can be adapted to our tropical setup. Recall that so far our choice of tropical models in their equivalence class was such that the leaves of the tropical curves were in 1-1 cor- respondence with the markings. For this choice we have the images σk(M0,n) contained in the boundary part of M0,n+1. This presenta- tion is compatible with the point of view when we think about line bundles in tropical geometry to be given by H1(X,O×). Here X is the base of the bundle and O× is the sheaf of “non-vanishing” tropi- cal regular functions. Such functions are given in the charts to RN by affine-linear functions with integer slopes, see [6]. (Recall that T× = R is an honest group with respect to tropical multiplication, i.e. the classical addition.) However, the following alternative construction allows one to obtain the ψ-classes more geometrically (as we’ll illustrate in an example in the next section). This approach is based on contracting the leaves marked by number k. The canonical class of a tropical curve is supported at its vertices, namely we take each vertex with the multiplicity equal to its valence minus 2, cf. [6]. Furthermore, the cotangent bundle near a 3-valent vertex point can be viewed as a neighborhood of the origin for the line given by the tropical polynomial “x + y + 1T” in R 2, so the +1 self- intersection of the line gives the required multiplicity for the canonical class at any 3-valent vertex. Thus we can use the intersections with the corresponding codimension 1 faces inM0,n to define the ψ-classes there. In other words, tropical ψ-classes will be supported on the (n − 4)- dimensional faces in M0,n. Namely, for a ψk-class we have to collect those codimension 1 faces in M0,n whose only 4-valent vertex is adjacent to the leaf marked by k. After a contraction of this leaf we get a 3-valent vertex, thus the multiplicity of every face in a ψ-divisor is 1. We arrive to the following definition. Definition 3.1. The tropical ψk-divisor Ψk ⊂ M0,n is the union of those (n−4)-dimensional faces that correspond to tropical curves with a 4-valent vertex adjacent to the leaf marked by k, k = 1, . . . , n. Each such face is taken with the multiplicity 1. Proposition 3.2. The subcomplex Ψk is a divisor, i.e. satisfies the balancing condition. MODULI SPACES OF RATIONAL TROPICAL CURVES 11 Proof. Recall that the balancing condition is a condition at (n − 5)- dimensional faces. In M0,n there are two types of such faces, one corresponding to tropical curves with two 4-valent vertices and one corresponding to a tropical curve with a 5-valent vertex. Near the faces of the first type the moduli space M0,n is locally a product of two copies of M0,4 and R n−5. The Ψ-divisor is a product of Rn−5, one copy of M0,4 and the central (3-valent) point in the other copy of M0,4 (this is the point corresponding to the 4-valent vertex adjacent to the leaf marked by k). Thus the balancing condition holds trivially in this case. Near the faces of the second type the moduli space M0,n is locally a product of M0,5 and R n−5. As in the proof of Theorem 1 each double ratio decomposes to the sum of the corresponding double ration inM0,5 (perhaps trivial if two of the markings for the double ratio correspond to the same edge adjacent to the 5-valent vertex) and an affine-linear function in Rn−5. Thus it suffices to check only the balancing condition for the Ψ-divisors in M0,5. This example is considered in details in the next section. The balancing condition there follows from Proposition 4.1. � Conjecturally, the tropical Ψ-divisors are limits of some natural rep- resentatives of the divisors for the complex ψ-classes under the collapse of the complex moduli space onto the corresponding tropical moduli space M0,n. Note that our choice for the tropical Ψ-divisor is not con- tained in the boundary ∂M0,n ⊂ M0,n (cf. the calculus of the complex boundary classes in [2]), but comes as a closure of a divisor in M0,n. 4. The space M0,5 We have already described the moduli space M0,4 as the tripod of Figure 1. It has only one 0-dimensional face O ∈ M0,4. This point (considered as a divisor) coincides with the divisors Ψ1 = Ψ2 = Ψ3 = Ψ4. The description of M0,5 is somewhat more interesting. There are 15 combinatorial types of 3-valent trees with 5 marked leaves. If we forget about the markings there is only one homeomor- phism class for such a curve (see Figure 3). To get the number of non-isomorphic markings we take the number all possible reordering of vertices (equal to 5! = 120) and divide by 23 = 8 as there is an 8-fold symmetry of reordering. Indeed there is one symmetry interchanging the left two leaves, one interchanging the right two leaves and the cen- tral symmetry around the central leave of the 3-valent tree on top of Figure 3. 12 GRIGORY MIKHALKIN 1 15 5 Figure 3. Adjunction of combinatorial types corre- sponding to the quadrant connecting the rays (45) and (12). (25) (13) (15) (23) Figure 4. The link of the origin in M0,5. MODULI SPACES OF RATIONAL TROPICAL CURVES 13 Thus the space M0,5 is a union of 15 quadrants R ≥0. These quad- rants are attached along the rays which correspond to the combinato- rial types of curves with one 4-valent vertex. Such curves also have one 3-valent vertex which is adjacent to two leaves and the only bounded edge of the curve, see the bottom of Figure 3. Such combinatorial types are determined by the markings of the two leaves emanating from the 3-valent vertex. Thus we have a total of = 10 of such rays. The two boundary edges of the quadrant correspond to contractions of the bounded edges of the combinatorial type as shown on Figure 3. The global picture of adjacency of quadrants and rays is shown on Figure 4 where the reader may recognize the well-known Petersen graph, cf. the related tropical Grassmannian picture in [7]. Vertices of this graph correspond to the rays of M0,5 while the edges correspond to the quadrants. Thus the whole picture may be interpreted as the link of the only vertex O ∈ M0,5 (the point O corresponds to the tree with a 5-valent vertex adjacent to all the leaves). To locate the Ψk-divisor we recall that the kth leaf has to be adjacent to a 4-valent vertex if it appears in Ψk. This means that Ψk consists of 6 rays that are marked by pairs not containing k. Proposition 4.1. The subcomplex Ψk ⊂ M0,5 is a divisor. Proof. Since the whole M0,5 is S5-symmetric it suffices to check the balancing condition only for Ψ1. The embedding M0,5 ⊂ R N is given by the double ratios, so it suffices to check that for each double ratio function the sum of its gradients on the six rays of Ψ1 vanishes. If the double ratio is determined by two pairs disjoint from the mark- ing 1, e.g. by {(23), (45)} then its restriction onto the six rays of Ψ1 is the same as its restriction to the three rays M0,4 taken twice and thus balanced. Namely its gradient is 1 on the rays (24) and (35); −1 on the rays (25) and (34); and 0 on the rays (23) and (45). If the four markings of the double ratio contain the marking 1 then thanks to the symmetry we may assume that the double ratio is given by {(12), (34)}. It vanishes on the rays (34), (35), (45) and (25); it has gradient +1 on the ray (24) and the gradient −1 on the ray (23). Once again, the balancing condition holds. � As our final example of the paper we would like to describe explicitly the universal curve ft5 : M0,5 → M0,4. This is presented on Figure 5. Once again, we interpret the Peterson graph as the link L of the vertex O ∈ M0,5. Similarly, the link of the 14 GRIGORY MIKHALKIN Figure 5. The three fibers and four sections of the universal curve ft5 : M0,5 → M0,4. origin in M0,4 consists of three points. Thus L is the union of the fibers of ft5 (away from a neighborhood of infinity) over these three points and four copies of a neighborhood of the origin in M0,4 corresponding to the four sections σ1, σ2, σ3 and σ4 of the universal curve. Figure 5 depicts the fibers in L with solid lines and the sections with dashed lines. Acknowledgements. I am thankful to Valery Alexeev and Kristin Shaw for discussions related to geometry of tropical moduli spaces. My research is supported in part by NSERC. References [1] Gathmann, A., Markwig, H., Kontsevich’s formula and the WDVV equations in tropical geometry, http://arxiv.org/abs/math.AG/0509628. [2] Keel, S., Intersection theory of moduli space of stable N -pointed curves of genus zero, Transactions of the AMS 330 (1992), 545–574. [3] Litvinov, G. L., The Maslov dequantization, idempotent and tropical mathe- matics: a very brief introduction. In Idempotent mathematics and mathemat- ical physics, Contemp. Math., 377, Amer. Math. Soc., Providence, RI, 2005, 1–17. [4] Mikhalkin, G., Tropical Geometry and its application, to appear in the Pro- ceedings on the ICM-2006, Madrid; http://arxiv.org/abs/math/0601041. [5] Mikhalkin, G., Tropical Geometry, book in preparation. http://arxiv.org/abs/math.AG/0509628 http://arxiv.org/abs/math/0601041 MODULI SPACES OF RATIONAL TROPICAL CURVES 15 [6] Mikhalkin, G., Zharkov, I., Tropical curves, their Jacobians and Theta func- tions, http://arxiv.org/abs/math/0612267. [7] Speyer, D., Sturmfels, B., The tropical Grassmannian. Adv. Geom. 4 (2004), no. 3, 389–411. [8] Viro, O. Ya., Dequantization of real algebraic geometry on logarithmic paper. In European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math., 201, Birkhäuser, Basel, 2001, 135–146. Department of Mathematics, University of Toronto, 40 St George St, Toronto ON M5S 2E4 Canada http://arxiv.org/abs/math/0612267 1. Introduction: smooth tropical varieties 2. Tropical curves and their moduli spaces 3. Tropical -classes 4. The space M0,5 References

0704.0841

Difermion condensates in vacuum in 2-4D four-fermion interaction models

Difermion condensates in vacuum in 2-4D four-fermion interaction models Bang-Rong Zhou† College of Physical Sciences, Graduate School of the Chinese Academy of Sciences, Beijing 100049, China In any four fermion (denoted by q) interaction models, the couplings of (qq)2-form can always coexist with the ones of (q̄q)2-form via the Fierz transformations. Hence, even in vacuum, there could be interplay between the condensates 〈q̄q〉 and 〈qq〉. Theoretical anal- ysis of this problem is generally made by relativistic effective potentials in the mean field approximation in 2D, 3D and 4D models with two flavor and Nc color massless fermions. It is found that in ground states of these models, interplay between the two condensates mainly depend on the ratio GS/HS for 2D and 4D case or GS/HP for 3D case, where GS , HS and HP are respectively the coupling constants in a scalar (q̄q), a scalar (qq) and a pseudoscalar (qq) channel. In ground states of all the models, only pure 〈q̄q〉 condensates could exist if GS/HS or GS/HP is bigger than the critical value 2/Nc, the ratio of the color numbers of the fermions entering into the condensates 〈qq〉 and 〈q̄q〉. Below it, differences of the models will manifest themselves. In the 4D Nambu-Jona-Lasinio (NJL) model, as GS/HS decreases to the region below 2/Nc, one will first have a coexistence phase of the two condensates then a pure 〈qq〉 con- densate phase. Similar results come from a renormalized effective potential in the 2D Gross- Neveu model, except that the pure 〈qq〉 condensates could exist only if GS/HS = 0. In a 3D Gross-Neveu model, when GS/HP < 2/Nc, the phase transition similar to the 4D case can arise only if Nc > 4, and for smaller Nc, only a pure 〈qq〉 condensate phase exists but no coexistence phase of the two condensates happens. The GS −HS (or GS − HP ) phase diagrams in these models are given. The results deepen our understanding of dynamical phase structure of four-fermion inter- action models in vacuum. In addition, in view of absence of difermion condensates in vacuum of QCD, they will also imply a real restriction to any given two-flavor QCD-analogous NJL model, i.e. in the model, the derived smallest ratio GS/HS via the Fierz transformations in the Hartree approximation must be bigger than 2/3. The project supported by the National Natural Science Foundation of China under Grant No.10475113. Electronic mailing address: [email protected] http://arxiv.org/abs/0704.0841v3 I. MAIN RESULTS We have researched interplay between the fermion(q)-antifermion (q̄) condensates 〈q̄q〉 and the difermion condensates 〈qq〉 in vacuum in 2D, 3D and 4D four-fermion interaction models with two flavor and Nc color massless fermions. It is found that the ground states of the systems could be in different phases shown in the following GS − HS and GS − HP phase diagrams [Fig.(a)–Fig.(d)], where GS — coupling constant of scalar (q̄q) 2 channel HS — coupling constant of scalar color Nc(Nc − 1) −plet (qq)2 channel (4D, 2D case) HP — coupling constant of pseudoscalar color Nc(Nc − 1) −plet (qq)2 channel (3D case) Λ — Euclidean Momentum cutoff of loop integrals (4D,3D case) (σ1, 0) — pure 〈q̄q〉 phase (0,∆1) — pure 〈qq〉 phase (σ2,∆2) — mixed phase with both 〈q̄q〉 and 〈qq〉 Fig.(a)-Fig.(d) (pages 3-6) 3 4D NJL Model y=GSΛ 2/π2 y =2x/ Nc (σ1, 0) (σ2,Δ2) 1/Nc (0 ,Δ1) R 0 1/2 x =HSΛ R: y=x/[1+(Nc-2)x] Fig. (a) 4 2D GN Model y=GS /π y=2x /Nc (σ1, 0) (σ2 ,Δ2) x=HS /π (0,Δ1) Fig. (b) 3D GN Model, Nc≤4 y=GSΛ/π y=2x/ Nc (σ1, 0) 1/4Nc (0,Δ1) 0 1/8 x=HPΛ/π Fig. (c) 3D GN Model, Nc≥5 y=GSΛ/π y =2x/ Nc (σ1, 0) 1/4Nc (σ2,Δ2) (0,Δ1) 0 1/8 x =HPΛ/π Fig. (d) Main Conclusions 1. In all the models, pure 〈q̄q〉 phase happens if ) > 2 (also GS must be large enough in 3D and 4D model). 2. The phases with condensates 〈qq〉, including pure 〈qq〉 phase and mixed phase with 〈q̄q〉 and 〈qq〉, arise only if ) < 2 3. In 3D Gross-Neveu model, no mixed phase with 〈q̄q〉 and 〈qq〉 exists for Nc ≤ 4. II. Motive and general approach • In any four-fermion interaction model [1, 2], the couplings of (qq)2-form and (q̄q)2-form can always coexist via the Fierz transformations, hence there must be interplay between the condensates 〈q̄q〉 and 〈qq〉 in ground state of the system. • In the vacuum, despite of absence of net fermions, based on a relativistic quantum field theory, it is possible that the condensates 〈qq〉 and 〈q̄q̄〉 are generated simultaneously. • The mean field approximation has been taken. In this case, we have used the Fierz transformed four-fermion couplings in the Hartree approximation to avoid double counting [3]. • In selecting the couplings of (qq)2-form, we always simulate SU(Nc) gauge interaction, where two fermions are attractive in the antisymmetric Nc(Nc − 1) -plet. • Euclidean momentum cutoffs in 3D and 4D models have been used so as to maintain Lorentz invariance of effective potentials in the vacuum. • In massless fermion limit, all the discussions can be made analytically. • The coupling constants GS and HS (or HP ) are viewed as independent parameters. III. 4D Nambu-Jona-Lasinio model With 2 flavors and Nc color massless fermions, the Lagrangian L = q̄iγµ∂µq +GS [(q̄q) 2 + (q̄iγ5~τq) 2] +HS (q̄iγ5τ2λAq C)(q̄C iγ5τ2λAq), (1) where the fermion fields q are in the doublet of SUf (2) and the Nc-plet of SUc(Nc), i.e. i = 1, · · · , Nc, (2) http://arxiv.org/abs/0704.0841v3 qC is the charge conjugate of q and ~τ = (τ1, τ2, τ3) are the Pauli matrices acting in two-flavor space. The matrices λA run over all the antisymmetric generators of SUc(Nc). Assume that the four-fermion interactions can lead to the scalar condensates 〈q̄q〉 = φ (3) with all the Nc color fermion entering them, and the scalar color Nc(Nc − 1) 2 -plet difermion and di-antifermion condensates (after a global SUc(Nc) transformation) 〈q̄Ciγ5τ2λ2q〉 = δ, 〈q̄iγ5τ2λ2q C〉 = δ∗, (4) with only two color fermions enter them. The corresponding symmetry breaking is that SUfL(2) ⊗ SUfR(2) → SUf (2), SUc(Nc) → SUc(2), and a ”rotated” electric charge UQ̃(1) and a ”rotated” quark number U ′q(1) leave unbroken. It should be indicated that in the case of vacuum, the Goldstone bosons induced by spontaneous breaking of SUc(Nc) could be some combinations of difermions and di-antifermions. Define that σ = −2GSφ, ∆ = −2HSδ, ∆ ∗ = −2HSδ ∗. (5) With standard technique and a 4D Euclidean momentum cutoff Λ [4], we obtain the relativistic effective potential V4(σ, |∆|) = 2 + 2|∆|2)Λ2 − (Nc − 2) −(σ2 + |∆|2)2 σ2 + |∆|2 . (6) The ground states of the system, i.e. the minimum points of V4(σ, |∆|), will be at (σ, |∆|) = (0, ∆1) (σ2, ∆2) (σ1, 0) , 0 ≤ 1 + (Nc − 2) 1 + (Nc − 2) , (7) Eq.(7) gives the phase diagram Fig.(a) of the 4D NJL model. IV. 2D Gross-Neveu model The Lagrangian is expressed by L = q̄iγµ∂µq +GS [(q̄q) 2 + (q̄iγ5~τq) 2] +HS(q̄iγ5τSλAq C)(q̄Ciγ5τSλAq), (8) All the denotations are the same as ones in 4D NJL model, except that in 2D space-time , γ1 = = −C, γ5 = γ and τS = (τ0 ≡ 1, τ1, τ3) are flavor-triplet symmetric matrices. It is indicated that the product matrix Cγ5τSλA is antisymmetric. Assume that the four-fermion interactions could lead to the scalar quark-antiquark conden- sates 〈q̄q〉 = φ, (9) which will break the discrete symmetries χD : q(t, x) → γ5q(t, x), P1 : q(t, x) → γ1q(t,−x), and that the coupling with HS can lead to the scalar color Nc(Nc − 1) -plet difermion con- densates and the scalar color anti- Nc(Nc − 1) -plet di-antifermion condensates (after a global transformation in flavor and color space) 〈q̄C iγ51fλ2q〉 = δ, 〈q̄iγ51fλ2q C〉 = δ∗ (10) which will break discrete symmetries Zc (center of SUc(3)) and Z (center of SUf (2)), besides χD and P1. Noting that in a 2D model, no breaking of continuous symmetry needs to be considered on the basis of Mermin-Wagner-Coleman theorem [5]. The model is renormalizable. In the space-time dimension regularization approach, we can write down the renormalized L in D = 2− 2ε dimension space-time by the replacements GS → GSM 2−DZG, HS → HSM 2−DZH , with the scale parameter M , the renormalization constants ZG and ZH . In addition, the γ L will become 2D/2 × 2D/2 matrices. Define the order parameters σ = −2GSM 2−DZGφ, ∆ = −2HSM 2−DZHδ, (11) which will be finite if ZG and ZH are selected so as to cancel the UV divergences in φ and δ. In the minimal substraction scheme, ZG = 1− 2NcGS , ZH = 1− . (12) By similar derivation to the one made in Ref.[6], the corresponding renormalized effective po- tential in the mean field approximation up to one-loop order becomes V2(σ, |∆|) = σ2 + |∆|2 + (Nc − 2) ln σ2 + |∆|2 , M̄2 = 2πe−γM2, (13) where γ is the Euler constant. The ground states of the system i.e. the minimal points of V2(σ, |∆|) will be at (σ, |∆|) = (0, ∆1) (σ2, ∆2) (σ1, 0) GS/HS = 0 0 < GS/HS < 2/Nc GS/HS > 2/Nc Eq.(14) gives the phase diagram Fig.(b) of 2D GN model. In 2D case, the GS -HS phase structure has the following feature: 1. The pure 〈qq〉 phase (0,∆1) could appear only if GS/HS = 0; 2. Formations of the condensates do not call for that the coupling constant GS and HS have some lower bounds. V. 3D Gross-Neveu model The Lagrangian is expressed by L = q̄iγµ∂µq +GS [(q̄q) 2 + (q̄~τq)2] +HP (q̄τ2λAq C)(q̄Cτ2λAq), (15) where γµ(µ = 0, 1, 2) are taken to be 2× 2 matrices , γ1 = , γ2 = It is noted that the product matrix Cτ2λA is antisymmetric, and since without the ”γ5” matrix, the only possible color Nc(Nc − 1) -plet difermion interaction channel is pseudoscalar one. The condensates 〈q̄q〉 will break time reversal symmetry T : q(t, ~x) → γ2q(−t, ~x), special parity P1 : q(t, x 1, x2) → γ1q(t,−x1, x2), special parity P2 : q(t, x 1, x2) → γ2q(t, x1,−x2). The difermion condensates 〈q̄Cτ2λ2q〉 (after a global rotation in the color space) will break SUc(Nc) → SUc(2) and leave a ”rotated” electrical charge U (1) and a ”rotated” fermion number U ′q(1) unbroken. It also breaks parity P : q(t, ~x) → γ0q(t,−~x) and this shows pseudoscalar feature of the difermion condensates. Define the order parameters in the 3D GN model σ = −2GS〈q̄q〉, ∆ = −2HP 〈q̄ Cτ2λ2q〉, (16) on bases of the same method used in Ref.[7], we find out the effective potential in the mean field approximation V3(σ, |∆|) = 2 + 2|∆|2)Λ 6σ2|∆|+ 2|∆|3 + (Nc − 2)σ 3 + 2θ(σ − |∆|)(σ − |∆|)3 , (17) where Λ is a 3D Euclidean momentum cutoff. The ground states of the system correspond to the least value points of V3(σ, |∆|) which will respectively be at (σ, |∆|) = (0,∆1), (0,∆1), (σ2,∆2), (σ1, 0), , for Nc ≤ 4 for Nc > 4 , for all Nc Eq.(18) gives the GS −HP phase diagrams Fig.(c) and Fig.(d) of the 3D GN model. VI. Summary • Present research deepens our theoretical understanding of the four-fermion interaction models: 1. Even in vacuum, it is possible that the difermion condensates are generated as long as the coupling constants of the difermion channel are strong enough (bigger than zero or some finite values). 2. Interplay between the condensates 〈q̄q〉 and 〈qq〉 mainly depends on GS/HS (or GS/HP ), the ratio of the coupling constants of scalar fermion-antifermion channel and scalar (or pseudoscalar ) difermion channel. 3. In all the discussed 2-flavor models, if GS/HS (GS/HP ) > 2/Nc, the ratio of the color numbers of the fermions entering into the condensates 〈qq〉 and 〈q̄q〉, (and also with sufficiently large GS in 4D and 3D model), then only pure 〈q̄q〉 condensates phase may exist. Below 2/Nc, (and also with sufficiently large HS or HP in 4D or 3D model), one will always first have a mixed phase with condensates 〈q̄q〉 and 〈qq〉, then a pure 〈qq〉 condensate phase, except that in the 3D GN model, no the mixed phase appears when Nc ≤ 4. • In view of absence of 〈qq〉 condensates in vacuum of QCD, the result here also implies a real restriction to any given two-flavor QCD-analogue NJL model: in such model, the derived smallest ratio GS/HS via the Fierz transformation in the Hartree approximation must be bigger than 2/3 [4]. [1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345; 124 (1961) 246. [2] D.J. Gross and A. Neveu, Phys. Rev. D 10 (1974) 3235. [3] M. Buballa, Phys. Rep. 407 (2005) 205. [4] Zhou Bang-Rong, Commun. Theor. Phys. 47 (2007) 95. [5] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133; S. Coleman, Commun. Math. Phys. 31 (1973) 259. [6] Zhou Bang-Rong, Commun. Theor. Phys. 47 (2007) 520. [7] Zhou Bang-Rong, Commun. Theor. Phys. 47 (2007) 695. Main results References

0704.0842

Oscillation bands of condensates on a ring: Beyond the mean field theory

Oscillation bands of condensates on a ring: Beyond the mean field theory C. G. Bao Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Collisions, Lanzhou 730000, P. R. China The State Key Laboratory of Optoelectronic Materials and Technologies, Zhongshan University, Guangzhou, 510275, P.R. China Abstract: The Hamiltonian of a N-boson system confined on a ring with zero spin and repulsive interaction is diagonalized. The excitation of a pair of p-wave-particles rotating reversely appears to be a basic mode. The fluctuation of many of these excited pairs provides a mechanism of oscillation, the states can be thereby classified into oscillation bands. The particle correlation is studied intuitively via the two-body densities. Bose-clustering originating from the symmetrization of wave functions is found, which leads to the appearance of 1-, 2-, and 3-cluster structures. The motion is divided into being collective and relative, this leads to the establishment of a relation between the very high vortex states and the low-lying states. After the experimental realization of the Bose-Einstein condensation1, various condensates confined under dif- ferent circumstances have been extensively studied the- oretically and experimentally. Mostly, the condensates are considered to be confined in a harmonic trap. Con- densates trapped by periodic potential have also been studied due to the appearance of optical lattices.2 It is believed that the appearance of condensates confined in particular geometries is possible. Experimentally, the particle interactions can now be tuned from very weak to very strong,3−8 it implies that the particle correlation may become important. Theoretically, to respond, go- ing beyond the mean field Gross-Pitaevskii (GP) theory is desirable, and the condensates confined in particular geometries are also deserved to be considered. Along this line, in addition to the ground state, the yrast states have been studied both analytically and numerically.9−17 The condensation on a ring has also been studied recently.12 The present paper is also ded- icated to the N−boson systems confined on a ring with weak interaction, its scope is broader and covers the whole low-lying spectra. A similar system has been in- vestigated analytically by Lieb and Liniger16,17. How- ever, the emphasis of their papers is different from the present one, which is placed on analyzing the structures of the excited states to find out their distinctions and sim- ilarities, and to find out the modes of excitation. Based on the analysis, an effort is made to classify the ex- cited states. Traditionally, the particle correlation and its effect on the geometry of N−boson systems is a topic scarcely studied if N is large. In this paper, the corre- lation is studied intuitively so as the geometric features inherent in the excited states can be understood. Tradi- tionally, a separation between the collective and internal motions is seldom to be considered if N is large. In this paper such a separation is made and leads to the estab- lishment of a relation between the vortex states and the low-lying states. It is assumed that the N identical bosons confined on a ring have mass m, spin zero, and square-barrier inter- action. The ring has a radius R, N is given at 100, 20 and 10000. Let G = ~2/(2mR2) be the unit of energy. The Hamiltonian then reads H = − i<j Vij (1) where θi is the azimuthal angle of the i-th boson. Vij = Vo if |θj − θi| ≤ θrange , or = 0 otherwise. Let φk = eikθ/ 2π be a single particle state, −kmax ≤ k ≤ kmax is assumed. The N−body normalized basis functions in Fock-representation are |α〉 ≡ |n−kmax , · · · nkmax〉, where nj is the number of bosons in φj , nj = N, njj = L, the total angular momentum. Then, H is diagonalized in the space spanned by |α〉, the low- lying spectrum together with the eigen-wave-functions, each is a linear combination of |α〉, are thereby obtained. Let Kα = 2 be the total kinetic energy of an |α〉 state. Evidently, those |α〉 with a large Kα are negligi- ble for low-lying states. Therefore, one more constraint Kα ≤ Kmax is further added to control the number of |α〉. In this procedure, the crucial point is the calcula- tion of the matrix elements of H . This can be realized by using the fractional parentage coefficients18 (refer to eq.(6) below). Numerical results are reported as follows. This paper concerns only the cases with weak interac- tion. Firstly, let Vo = 1, θrange = 0.025, and N = 100. This is corresponding to γ = 0.00157 , where γ is in- troduced by Lieb and Liniger to measure the strength of interaction,16,17 this is shown later. When kmax and Kmax are given at a number of values, the associated eigen-energiesEj of the first, fifteenth, and sixteenth L = 0 eigen-states are listed in Table I. When (kmax,Kmax) is changed from (3, 50) to (5,60), the total number of |α〉 is changed from 2167 to 8890. Table I demonstrates that the great increase of basis functions does not lead to a remarkable decrease of eigen-energies. Thus the conver- gency is qualitatively satisfying even for the higher states. http://arxiv.org/abs/0704.0842v1 TABLE I: Eigen-energies Ej (the unit is G) of the L = 0 states. The first row is (kmax,Kmax), the first column is the serial number of states j. Vo = 1, θrange = 0.025, and N = 100 are given. (3,50) (4,50) (4,60) (5,60) 1 39.109 39.090 39.090 39.078 15 53.645 53.616 53.616 53.613 16 54.822 54.800 54.800 54.790 In the following the choice kmax = 4 and Kmax = 50 are adopted, this limitation leads to a 3254-dimensional space. Thereby the resultant data have at least three effective figures, this is sufficient for our qualitative pur- pose. N=100, L=0 FIG. 1: The spectrum of L = 0 states, the unit of energy is G = ~2/(2mR2). N = 100, Vo = 1, and θrange = 0.025 are assumed, they are the same for Fig.1 to Fig.5. The levels in a column constitute an oscillation band, the levels in bold line are doubly degenerate. The low-lying spectrum is given in Fig.1, where the lowest fourteen levels are included. Twelve of them can be ascribed into three bands, in each band the levels are distributed equidistantly, this is a strong signal of harmonic-like oscillations. From now on the labels Ψ and E(L,Z,i) are used to denote the wave function and en- ergy of the i-th state of the Z-th band (Z=I, II, III,· · ·). It turns out that the excitation of a pair of particles both in p-wave but rotating reversely, namely, one par- ticle in φ1 while the other one in φ−1, is a basic mode, the pair is called a basic pair in the follows. A number of such basic pairs might be excited. When 2j particles are in basic pairs while the remaining N − 2j particles are in φ0, the associated |α〉 is written as |P (j)〉. For all the states of the I−band, we found Ψ(0)I,i is mainly a linear combination of |P (j)〉 together with a small component denoted by ∆I,i, i.e., I,i = (0,I,i) (j)〉+∆I,i (2) where ∆I,i is very small as shown in Table II, while the coefficients C (L,Z,i) j arise from the diagonalization. Thus the basic structure of the I−band is just a fluctuation of many of the basic pairs. TABLE II: The weights of ∆Z,i of the bands with L = 0 i I−band II−band III−band 1 0.009 0.017 0.040 2 0.012 0.030 0.056 3 0.021 0.061 0.088 4 0.035 0.028 5 0.055 0.035 6 0.079 0.106 For lower states, C (0,Z,i) j would be very small if j is larger, e.g., for the ground state, C (0,I,1) 0 = 0.968 and C (0,I,1) j ≥2 ≈ 0, it implies that the excitation of many pairs is not probable. It also implies that the ground state wave function obtained via mean-field theory might be a good approximation. However, for higher states, many pairs would be excited. E.g., for the third state of the I−band, C(0,I,3)j = 0.052, 0.406, 0.681,−0.528, and 0.245 when j is from 0 to 4, it implies a stronger fluctuation. When a |α〉 has not only 2j particles in the basic pairs, but also m particles in φk, while the remaining particles in φ0, then it is denoted as |(k)mP (j)〉 (where k = ±1 are allowed) Similarly, we can define |(k1)m1(k2)m2P (j)〉, and so on. For all the states of the II−band, we found II,i = (0,II,i) [ |(2)1(−1)2P (j)〉 ±|(−2)1(1)2P (j)〉] + ∆II,i where both the + and − signs lead to the same energy, thus the level is two-fold degenerate. Again, all the ∆II,i are very small as shown in Table II, thus the fluctua- tion of basic pairs is again the basic structure. However, the II−band is characterized by having the additional 3-particle-excitation (one in d-wave and two in p-wave). For all the states of the third band, we found III,i (0,III,i) j |(2)1(−2)1P (j)〉+∆III,i (4) Thus, the III−band contains, in addition to the fluc- tuation of basic pairs, a more energetic pair with each particle in d-wave. It was found that the spacing E(0,Z,i+1) − E(0,Z,i) inside all the bands are nearly the same, they are ∼3.15. This arises because they have the same mechanism of oscillation, namely, the fluctuation of basic pairs. When the energy goes higher, more oscillation bands can be found. The two extra levels in Fig.1 at the right are the band-heads of higher bands. Incidentally, the band-heads of the above three bands are dominated by |P (0)〉, |(2)1(−1)2P (0)〉 ± |(−2)1(1)2P (0)〉 and |(2)1(−2)1P (0)〉, respectively, and their kinetic energies Kα = 0, 6, and 8. Among all the basis functions with L = 0 and without basic pairs, these three are the lowest three. This explains why the band- heads are dominated by them. Once a band-head is fixed, the corresponding oscillation band would grow up via the fluctuation of basic pairs. The particle correlations can be seen intuitively by ob- serving the two-body densities ρ2(θ1, θ2) = dθ3 · · · dθN Ψ(L)∗Z,i Ψ Similar to the calculation of the matrix elements of interaction, the above integration can be performed in coordinate space by extracting the particles 1 and 2 from |α〉 by using the fractional parentage coefficients18, namely, |α〉 = nk(nk − 1)/N(N − 1)φk(1)φk(2)|αk〉 ka,kb (ka 6=kb) nkankb/N(N − 1)φka(1)φkb(2)|αkakb〉 where |αk〉 is different from |α〉 by replacing nk with nk− 2, |αkakb〉 is different from |α〉 by replacing nka and nkb with nka − 1 and nkb − 1, respectively. 0 60 120 180 240 300 360 3 (c) III-band 3 (b) II-band I-band FIG. 2: ρ2 as functions of θ2 for the I ( a), II ( b), and III (c) bands of L = 0 states, θ1 = 0 is given. The labels i of the states Ψ are marked by the curves. ρ2 gives the spatial correlation between any pair of particles as shown in Fig.2. For the ground state Ψ ρ2 is flat implying that the correlation is weak. However, it is a little larger when the two particles are opposite to each other (θ1 = 0 and θ2 = π). It implies the existence of a weak correlation which is entirely ignored by the mean field theory. Thus, even the interaction adopted is weak and even for the ground state, there is still a small revision to the mean field theory. For higher states of the I−band, the fluctuation of basic pairs becomes stronger. Due to the fluctuation, the particles tend to be close to each other to form a single cluster. This tendency is clearly shown in Fig.2a. For the first state of the III − band, Ψ(0) III,1 has two peaks in ρ2 implying a 2-cluster structure. It arises from the two d-wave paticles inherent in the band. The fea- ture of Ψ II,1 is lying between Ψ I,1 and Ψ III,1. For all higher states of every band, due to the strong fluctuation of basic pairs, all the particles tend to be close to each other as shown in 2b and 2c. To understand the physics why the particles tend to be close to each other, let us study the most important basis state |P (j)〉. By inserting |P (j)〉 into eq.(5) to replace and by using (6), ρ2 reads ρ2(θ1, θ2) = (2π)2N(N−1) [N(N − 1)− j(4N − 6j) +4j(N − 2j)(1 + cos(θ1 − θ2)) + 4j2 cos2(θ1 − θ2)] Where there are four terms at the right, the non- uniformity arises from the third and fourth terms. The third term causes the particles to be close to each other to form a single cluster, while the fourth term causes the two-cluster clustering. When j is small, the fourth term can be neglected, and the particles tend to form a single cluster. However, when j ≈ N/2 , the third term can be neglected, and the particles tend to form two clusters. It is noted that, if the symmetrization were dropped, the density contributed by |P (j)〉 would be uni- form. The appearance of the clustering originates from the symmetrization of the bosonic wave functions, there- fore it can be called as bose-clustering. For L = 1 states, the lowest energy E(1,I,1) is higher than E(0,I,1) by 1.606, but lower than E(0,I,2). Thus Ψ is the true first excited state of the system. A number of oscillation bands exist as well, the wave functions of the lowest six bands are found as (1,I,i) j |(1)1P (j)〉+∆I,i (1,II,i) j |(2)1(−1)1P (j)〉+∆II,i III,i (1,III,i) j |(−2)1(1)3P (j)〉+∆III,i IV,i = (1,IV,i) j |(2)1(−2)1(1)1P (j)〉+∆IV,i V,i = (1,V,i) j |(3)1(−1)2P (j)〉+∆V,i V I,i = (1,V I,i) j |(3)1(−2)1P (j)〉+∆V I,i Where the weights of all the ∆Z,i ≤ 0.1 if i ≤ 4. Thus, just as the above L = 0 case, all the bands have the common fluctuation of basic pairs, but each band has a specific additional few-particle excitation. The ener- gies of the band-heads from I to V I are 40.70, 45.42, 48.58, 50.14, 52.04, and 53.58 respectively. Further- more, the spacing ∼3.15 found above is found again for all these bands due to having the same mechanism of os- cillation. The I−band is similar to the above I−band with L = 0 but having an additional single p-wave ex- citation, the ρ2 of them are one-one similar. Similarly, the ρ2 of the IV−band is one-one similar to those of the above III− band with L = 0. The ρ2 of the II and III−bands are both similar to those of the above II−band with L = 0. However, the V and V I−bands are special due to containing the f-wave excitation, the ρ2 of their band-heads exhibit a 3-cluster structure as shown in Fig.3. When the energy goes even higher, more higher oscillation bands will appear. For the above six L = 1 bands, their band-heads are dominated by the |α〉 with Kα = 1, 5, 7, 9, 11, and 13. Obviously, a higher Kα leads to a higher band. 0 60 120 180 240 300 360 FIG. 3: ρ2 for selected L = 1 states. θ1 = 0, the (Z, i) labels are marked by the curves. In general, all the low-lying states can be classified into oscillation bands. For all the lower bands disregarding L, it was found that each band-head is dominated by a basis function containing a specific few-particle exci- tation but not containing any basic pairs. The energy order of the bands is determined by the magnitudes of Kα associated with the dominant basis function |α〉 of the band-heads. Once a band-head stands, an oscillation band will grow up from the band-head simply via the fluc- tuation of basic pairs. For examples, for L = 2 states, the dominant |α〉 of the band-heads of the four lowest oscil- lation bands are |(1)2P (0)〉, |(2)1P (0)〉, |(−2)1(1)4P (0)〉, and |(3)1(−1)1P (0)〉 with Kα = 2, 4, 8, and 10, respec- tively. For L = 3 states, the dominant |α〉 of the band-heads of the three lowest bands are |(1)3P (0)〉, |(2)1(1)1P (0)〉, and |(3)1P (0)〉, with Kα = 3, 5, and 9, respectively. Since the p-, d-, and f-wave appear successively, these band-heads exhibit 1-cluster, 2-cluster, and 3-cluster structures, re- spectively, as shown in Fig.4. 0 60 120 180 240 300 360 III,1ρ FIG. 4: ρ2 for the band-heads of L = 3 states, θ1 = 0. Furthermore, a −L state can be derived from the corresponding L state simply by changing every k to −k, i.e., change the components |(k1)m1(k2)m2P (j)〉 to |(−k1)m1(−k2)m2P (j)〉, and so on. Therefore Ψ(−L)Z,i = )∗, and E(−L,Z,i) = E(L,Z,i). 0 2 4 6 8 Yrast line N=100, Vo=1, θrange=0.025 FIG. 5: Energies of the yrast states with L = 0 to 10. Let us study the yrast states Ψ I,1 , each is the lowest one for a given L. The energies of them are plotted in Fig.5, their wave functions are found as I,1 = (L,I,1) j |(1)LP (j)〉+∆LI,1 (9) where ∆LI,i is very small. When L is small, the fluc- tuation of basic pairs is small, and the yrast states are dominated by the j = 0 component |(1)LP (0)〉.When L is larger, the weight of the |(1)LP (0)〉 component becomes smaller. E.g., when L = 0, 2, 4, and 10, the weights of |(1)LP (0)〉 are 0.94, 0.84, 0.75, and 0.54, respectively. Evidently, the energy going up linearly in the yrast line in Fig.5 is mainly due to the linear increase of the number of p-wave particles. N=20, L=0 III IV V FIG. 6: The spectrum of the L = 0 states with N = 20, Vo = 1, and θrange = 0.025. Refer to Fig.1 When N = 20, all the above qualitative features re- main unchanged. Examples are given in Fig.6 and 7 to be compared with Fig.1 and 4. Nonetheless, the decrease of N implies that the particles have a less chance to meet each other, thus the particle correlation is expected to be weaker. Quantitatively, it was found that (i) The spac- ing of adjacent oscillation levels becomes smaller, it is now ∼2.2 to replace the previous 3.15 (ii) The fluctu- ation becomes weaker. E.g., the weights of |P (j)〉 of the Ψ I,3 state are 0.01, 0.95, and 0.03 for j = 1, 2, and 3, respectively, while these weights would be 0.16, 0.46, 0 60 120 180 240 300 360 0.20 N=20, L=3 III,1 FIG. 7: The same as Fig.4 but with N = 20 and 0.28 if N = 100. (iii) When N becomes small, the geometric features would become explicit. E.g., for the 3-cluster structure, the difference between the maximum and minimum of ρ2 is ∼0.007 in Fig.4, but ∼ 0.033 in Fig.7. The decrease of Vo or θrange was found to cause an effect similar to the decrease of N , the spectra would remain qualitatively unchanged. Quantitatively, when Vo is changed from 1 to 0.1, the spacing inside a band is changed from ∼ 3.15 to ∼ 2.14, and the fluctuation becomes much weaker as expected. In what follows we study the vortex states. For an arbitrary Lo ≤ N/2, the spectra of the Ψ(N−Lo)Z,i and Z,i states are found to be identical 15, except the former shifts upward as a whole by N − 2Lo, namely, E(N−Lo,Z,i) = E(Lo,Z,i) +N − 2Lo (10) Furthermore, their ρ2 are found to be identical. Let us define an operator X so that the state X |α〉 is related to |α〉 by changing every ki in |α〉 to −ki + 1, i.e., φki(θ) to φ−ki+1(θ) = e iθφ−ki(θ). We further found from the numerical data that (N−Lo) Z,i = holds exactly. In fact, X causes a reversion of rota- tion of each particle plus a collective excitation. It does not cause any change in particle correlation, therefore ρ2 remains exactly unchanged. Thus the L large states, including the vortex states L = N , can be known from the L small states. The underlying physics of this finding is the separabil- ity of the Hamiltonian (it is emphasized that the sepa- rability is exact as can be proved by using mathematical induction). Let θcoll = θi/N, which describes a col- lective rotation. Then H = − 1 +Hint, where Hint describes the relative (internal) motions and does not de- pend on θcoll. Accordingly, E (L,Z,i) = L2/N + E (L,Z,i) int , the former is for collective and the latter is for relative (internal) motions. The eigen-states can be thereby sep- arated as Ψ Z,i = eiLθcoll ψ (L,Z,i) int . The feature of the internal states ψ (L,Z,i) int has been studied in [19]. Where it was found that, for an arbitrary Lo (N+Lo,Z,i) int = ψ (Lo,Z,i) With these in mind, eq.(10) and (11) can be derived as follows. From the separability (N−L) ei(N−L)θcoll ψ (N−L,Z,i) int (13) X acts on a wave function with L, from the defini- tion of X, L should be changed to −L and an additional factor eiθj = eiN θcoll should be added, thus Z,i = ei(N−L)θcoll ψ (−L,Z,i) Due to (12), the right hand sides of (13) and (14) are equal, thereby (11) is proved. Furthermore, since ψ (−L,Z,i) int = (ψ (L,Z,i) int ) ∗, the internal energy E (N−L,Z,i) int = E (−L,Z,i) int = E (L,Z,i) int . Therefore, E(N−L,Z,i) − E(L,Z,i) = (N − L)2/N − L2/N = N − 2L. This recovers eq.(10), the energy difference arises purely from the difference in collective rotation. If the particles are tightly confined on the ring, rapidly rotating state with a large L = JN − Lo would exist, where J is an integer. Their spectra would remain the same but shift upward by J(JN − 2Lo) from the spec- trum with L = Lo, while Ψ (JN−Lo) Z,i = Z,i , where XJ changes each φki to φ−ki+J . Thus the rapidly rotat- ing states have the same internal structure as the corre- sponding lower states but have a much stronger collective rotation. When N increases greatly while Vo or θrange decreases accordingly, the qualitative behaviors remain unchanged. E.g., when N = 10000 and Vo = 0.01 (θrange remains unchanged), the spectrum and the wave functions are found to be nearly the same as the case N = 100 and Vo = 1, except that the spectrum has shifted upward nearly as a whole by 3939. This is again a signal that, for weak interaction and for the ground states, the mean- field theory is a good approximation. It is noted that the confinement by a ring is quite differ- ent from a 2-dimensional harmonic trap. In the latter, the energy of a particle in the lowest Landau levels is proportional to its angular momentum k. However, for the rings, it is proportional to k2. Consequently, higher partial waves are seriously suppressed and the p-wave ex- citation becomes dominant. For a harmonic trap it was found in [10,11] that d- and f-wave excitations are more important than the p-wave excitation when L is small. This situation does not appear in our case. When the zero-range interaction Vij = gδ(θi − θj) is adopted, The results are nearly the same with those from the square-barrier interaction if the parameters are re- lated as g = 2Voθrange (in this choice both interactions have the same diagonal matrix elements). For an exam- ple, a comparison is made in Table III. The high simi- larity between the two sets of data imply that the above findings are also valid for zero-range interaction. TABLE III: Eigen-energies of the four lowest L = 0 states for a system with N = 100 and with zero-range interaction Vij = 0.05δ(θi − θj) (the unit of energy is G as before). The weights of the j = 0 components of these states are also listed. The corresponding results from square-barrier inter- action with Vo = 1, and θrange = 0.025 are given in the parentheses. (L,Z, i) E(L,Z,i) (C (L,Z,i) (0, I, 1) 39.0900 (39.0902) 0.9370 (0.9371) (0, I, 2) 42.2982 (42.2983) 0.0510 (0.0510) (0, I, 3) 45.4733 (45.4733) <0.02 (<0.02) (0, II, 1) 47.0076 (47.0074) 0.8372 (0.8373) The numerical results from using zero-range interac- tion can be compared with the exact results from solving integral equations by Lieb and Liniger [16,17]. The vari- ables γ and e(γ) introduced in [16] are related to those of this paper as γ = gπ/N and e(γ) = 4π2E/N3 (the unit of E is G). However, this paper concerns mainly the case of weak interaction, say, g ≤ 0.05,or γ ≤ 0.00157 (other- wise, the procedure of diagonalization would not be valid due to the cutoff of the space). Nonetheless, even γ is as large as 0.5 (g = 15.9) the evolution of the ground state energy with N = 100 against γ obtained via di- agonalization coincide, in the qualitative sense, with the exact results quite well . This is shown in Fig.8 to be compared with Fig.3 of [16], where γ is ranged from 0 to 10. In Fig.8, the constraint γ < e(γ) is recovered. Furthermore, when γ is small, e(γ) against γ appears as a straight line. 0.0 0.1 0.2 0.3 0.4 0.5 FIG. 8: e(γ) = 4π2E/N3 against γ = gπ/N . N is given at 100 and E is the ground state energy calculated from the diagonalization in the unit G. In summary, a detailed analysis based on the numerical data of N−boson systems on a ring with weak interac- tion has been made. The main result is the discovery of the basic pairs, which exist extensively in all the excited states and dominates the low-lying spectra. The fluc- tuation of basic pairs provides a common mechanism of oscillation, the low-lying states are thereby classified into oscillation bands. Each band is characterized by having its specific additional excitation of a few particles. Since the mechanism of oscillation is common, the level spac- ings of different bands are nearly equal in a spectrum. To divide the motion into being collective and relative provides a better understanding to the relation between the higher and lower states. The very high vortex states with L ≈ N can be understood from the correspond- ing low-lying states because they have exactly the same internal states. The particle correlation has been intuitively studied. particle densities are found to be in general non-uniform, bose-clustering originating from the symmetrization of wave functions is found, which leads to the appearance of one, two, and three clusters. This phenomenon would become explicit and might be observed if N is small. Acknowledgment: The support by NSFC under the grants 10574163 and 90306016 is appreciated. REFERENCES 1, D.M. Stamper-Kurn, M.R. Andrews, A.P. Chikkatur, S. Inouye, H.-J. Miesner, J. Stenger, and W. Ketterle, Phys. Rev. Lett. 80, 2027 (1998) 2, B.P. Anderson and M.A. Kasevich, Science 281, 1686 (1998) 3, J.L. Roberts, et al, Phys. Rev. Lett. 81, 5159 (1998) 4, J. 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0704.0843

Kadowaki-Woods Ratio of Strongly Coupled Fermi Liquids

Kadowaki-Woods Ratio of Strongly Coupled Fermi Liquids Takuya Okabe Faculty of Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu 432-8561,Japan (Dated: November 28, 2018) On the basis of the Fermi liquid theory, the Kadowaki-Woods ratio A/γ2 is evaluated by using a first principle band calculation for typical itinerant d and f electron systems. It is found as observed that the ratio for the d electron systems is significantly smaller than the normal f systems, even without considering their relatively weak correlation. The difference in the ratio value comes from different characters of the Fermi surfaces. By comparing Pd and USn3 as typical cases, we discuss the importance of the Fermi surface dependence of the quasiparticle transport relaxation. PACS numbers: 71.10.Ay, 71.18.+y, 71.20.Be, 71.27.+a, 72.15.-v It is widely known as a universal feature of heavy fermion systems that there holds the Kadowaki-Woods (KW) relation A/γ2 ≃ 1 × 10−5µΩ cm(mol K/mJ)2 be- tween the electronic specific heat coefficient γ of C = γT and the coefficient A of the resistivity ρ = AT 2 in the clean and low temperature limit.[1] According to the Fermi liquid theory, this is interpreted as an indication of the fact that A is squarely proportional to quasiparti- cle mass enhancement due to strong electron correlation. On the other hand, transition metal systems are reported since before to obey a similar relation with a more than an order of magnitude smaller value of A/γ2.[2, 3] In view of the observation that there seems to exist several types of systems in this regard, the recent finding by Tsujii et al.[4] is quite impressive that many Yb-based compounds show the KW ratio A/γ2 as small as the transition met- als. Kontani derived the small ratio as a result of the large orbital degeneracy of the the 4f13 state of trivalent Yb by applying the dynamical mean field approximation to a periodic Anderson model of an orbitally degenerate f electron states coupled with a single conduction band.[5] To discuss the KW ratio A/γ2 and the many-body mass enhancement effect, a simple model is usually adopted at the cost of neglecting material specific individ- ual factors. In the present work, we are interested in such an effect as caused by a system-dependent factor, that is, the Fermi surface dependence of quasiparticle current re- laxation. The system should have a large enough Fermi surface relative to the Brillouin zone boundary in order for the quasiparticle current to dissipate effectively into an underlying lattice through mutual quasiparticle scat- terings. In other words, the effectiveness of the trans- port relaxation may depend on the size and shape of the Fermi surface. To investigate this point definitely, we discuss the quasiparticle transport by taking account of the momentum dependence of quasiparticle scattering on the basis of realistic band structures. This has been ham- pered so far by a task required for not so simple Fermi surfaces of many band systems as could be simply mod- elled analytically. In terms of fairly realistic energy bands obtained from a first principle calculation, we evaluate those quantities which are not affected severely by the electron correlation effect. The theory in use is essen- tially within the phenomenological Fermi liquid theory described by renormalized quantities, and unlike a model calculation no bare microscopic quantities appear explic- itly. Schematic results using simple abstract models have been given before, in which a tight binding square lattice model and a two-band model are investigated.[6, 7, 8] For the ratio A/γ2 we make use of the expression, = 21.3αFa [µΩ (mol K/mJ) ], (1) which corresponds to Eq. (4.11) in Ref. 7 where we set a = 4Å for the lattice constant. In what follows we sub- stitute a calculated value for a. Below we follow how to derive αF , where α is a coupling constant, and F is a factor determined by the Fermi surface. Following a microscopic analysis of the quasiparticle transport with vertex corrections properly taken into account,[9] we may derive a phenomenological linearized Boltzmann equation.[7] Generalizing the theory to take a many-band effect into account, in the low temperature T → 0 we end up with the equation ∗Electronic address: [email protected] vipµ = (πT ) pp′kρ p′+k(l pµ + l p′µ − l p′+kµ − l p−kµ), (2) where vipµ and ρ p = δ(µ − ε p) are the velocity compo- nent and the local density of state of the renormalized http://arxiv.org/abs/0704.0843v2 mailto:[email protected] (mass-enhanced) quasiparticle with the crystal momen- tum p in the i-th band. The superscripts i and j are the band indices, while the subscript µ = x, y, z are Carte- sian coordinates. In the right hand side of Eq. (2), the 2nd to 4th terms in the parenthesis represent vertex cor- rections in the microscopic formulation. In terms of the solution lipµ, which physically represents stationary devi- ation of the Fermi surface in an applied electric field Eµ, the conductivity is given by σ ≡ σµ = 2e pµ, (3) The above equations (2) and (3) correspond to Eqs. (3.10) and (3.15) of Ref. 8 respectively. We may suppress the index µ (= x) in Eq. (3) as we discuss the cubic systems in what follows. Instead of solving the simultaneous matrix equations (2) exactly, we use trial functions for lipµ as commonly ap- plied in a variational principle formulation of the trans- port problems.[10] Assuming lipµ ∝ e |vipµ| we obtain αijci,j ρ2|vx| , (4) where ci,j = k1,k2,k3,k4 k1+k2=k3+k4 ρik1ρ ρik4(e −eik4) 2/4ρiρj , ρ|vx| ≡ ρip|v px|. (6) We define coupling constants αij = ρiρj〈W ij〉/π, where p, is the density of states of the i-th band at the Fermi level and 〈W ij〉 denotes the quasiparticle scat- tering probability W pp′k averaged over the momenta p, p and k. As the double sum in (2), dominated by Umk- lapp processes, covers a complicated shaped phase space over the Fermi surface, it is generally a good approxi- mation to take W pp′k out of the momentum sum as an averaged quantity. The total density of states ρ = is substituted for γ = 2π2ρ/3. In heavy fermion systems, the momentum dependence pp′k could be generally neglected, for the quasiparti- cle scattering W pp′k is primarily caused by strong on-site Coulomb repulsion U . Then we can make an order of magnitude estimate of αii in terms of Landau parame- ters F and F . For an anisotropic Fermi liquid, as in an isotropic case, one can derive that the charge and spin susceptibilities are given by χic = 2ρi/(1+F ) and χis = 2ρi/(1 + F ), respectively. Thus, for the systems in which charge fluctuations are suppressed, χic → 0, we obtain F ≫ 1. On the other hand, in terms of A /(1 +F ), one obtains a rough estimate of the cou- pling αii = 1 )2 + 1 . There- fore, under the normal condition that the spin enhance- ment is moderate, (1+F )−1 ∼ 1, αii should universally stay around a constant of an order of unity.[7] This corre- sponds to the condition to make the Wilson ratioRW = 2 in the impurity model.[11, 12] We discuss a normal state that the system is well away from critical instabilities, around which A/γ2 will be strongly enhanced at vari- ance with experimental results under consideration.[13] We evaluate F numerically for α = αij = 1 to obtain A/γ2, and investigate the Fermi surface dependence. It is noted that the factor F is determined by the shape and extent of the Fermi surfaces relative to the Brillouin zone boundary. Microscopically, the mass enhancement due to the many-body effect is represented by the ω- derivative of the electron self-energy Σ(q, ω), or by the renormalization factor zip as ρ p = ρ 0,p/z p, where ρ 0,p is a bare density of states. It is easily checked that the factor z cancels in F when zip is independent of i. Oth- erwise, in case that a dominant contribution to the re- sistivity comes from an electron-correlated main band, then the other bands may be neglected and A/γ2 be- comes independent of z of the main band. As we see below numerically, it is found indeed that F is domi- nated by a few scattering channels within a main band or two. Hence, we elaborate on a numerical estimate of F on the basis of a realistic band calculation reproducing reliable Fermi surfaces of relevant bands, even if it may not take account of local many-body correlation effects fully enough for the renormalized quantities like ρi and vip to be separately compared with experiments. As a matter of course, we must exclude the extreme case in which strong correlation modifies electron states around the Fermi level qualitatively from those of a band calcu- lation. We apply our theory to those itinerant electron systems in which correlation strength is not negligible but not so strong. To calculate F for some typical cubic d and f itiner- ant electron systems in the fcc and Cu3Au structures, we have performed ab initio band calculations within den- sity functional theory using the plane wave pseudopoten- tial code VASP with the Perdew-Wang 1991 generalized gradient approximation to the exchange correlation func- tional Exc.[14, 15, 16, 17] By minimizing the total energy we obtain the lattice constant a, which is accurate enough to be used in Eq. (1). To evaluate F numerically, we have to broaden the delta function ρip = δ(µ − ε p) by ∆ to pick up electron states around the Fermi level. The width ∆ of the order of real temperature should be decreased as the number of the k-points is increased until we confirm to have a con- vergent result. For the number L of subdivisions along re- TABLE I: Calculated results. a (Å) ρ|vx| a F N A/γ2 b USn3 4.60 3.1 4.0 3 0.39 UIn3 4.61 4.9 1.6 3 0.16 UGa3 4.24 3.9 2.5 3 0.23 Pd 3.86 7.4 0.23 3 0.019 Pt 3.91 8.4 0.15 4 0.012 aIn unit of a = 1. bIn unit of [10−5 µΩ cm (mol K/mJ)2]. ciprocal lattice vectors, band calculations are performed with Lband ∼ 50, from which we obtain the band energies εik on the finer k-mesh of L ∼ 200 by interpolation. As the four-fold k-sum in the numerator of Eq. (4), especially for the most important terms coming from the main d or f correlated bands, constitutes the most time consuming part of the calculation, we have to reduce the numerical task by some symmetry considerations not only on the cubic symmetry of the quasiparticle states, but on the relative directions of the four momentum vectors of the scattering quasiparticle states and the x-direction of the current flow. The reduction is particularly effective for the intra-band scatterings i = j. The calculated results are shown in Table I, where F and A/γ2 for α = αij = 1 are shown along with the lattice constant a, the number N of metallic bands con- tributing to the resistivity, and ρ|vx| defined in Eq. (6). We find that our results explain well the experimental tendency of an order of magnitude small values of the ratio A/γ2 for the transition metal systems. As for the absolute values of the ratio, our results are a few times smaller than observed evenly, but the accuracy of this order should not be taken seriously here. Among other things, the results indicate that different characters of the Fermi surfaces play an important role. To show the relative contribution to the resistivity from relevant bands, relative magnitudes of ci,j in the nu- merator of Eq. (4) are shown for Pd and USn3 in Figs. 1 and 2, respectively. For Pd, the contribution to F comes from the 4th to 6th bands, among which dominant is the 5th hole band of the 3d character. Similarly, the 5th band contributes majorly not only to ρ, i.e., ρ5 ≃ 5.4ρ4 ≃ 12ρ6, but to ρ|vx| in Eq. (6). On the other hand, for USn3, while the 14th heavy electron band plays a central role, the 12th and 13th hole bands also make non-negligible contributions through the inter-band scatterings. Hence, as the first point to note, numerical importance of the inter-band contributions makes F large in the f electron system. This is partly because ρi for i = 12, 13, 14 are comparable with each other, namely, ρ14 ≃ 2ρ13 ≃ 3ρ12. Moreover, it is remarked that the large and nearly spher- ical shape of the Fermi surfaces are essential too. As the second point to note, the importance of the Fermi surface geometry can be understood within a single band model by comparing contribution from the main band. We find that c5,5/ρ 5 = 0.097 for Pd is an order of magnitude 6 4 FIG. 1: cij (i, j = 4, 5, 6) for Pd. The contribution from the 5th band is dominant for the resistivity. 14 12 FIG. 2: cij (i, j = 12, 13, 14) for USn3. The interband con- tribution with the 14th band is important too. smaller than c14,14/ρ 14 = 0.93 for USn3. The difference comes from the different characters of the Fermi surfaces. According to an elementary formula σ = e2ρv2τ = e2ρvl, the conductivity σ depends on ρv as well as l. In this context, the mean free path l is not a single particle property determined by a lifetime of the particle state, but it is the transport property which characterizes how efficiently the total electric current decays into a lattice system, e.g., in our case, through mutual Umklapp scat- tering processes between the current carriers. In partic- ular, regardless of interaction, electrons in free space will not have resistivity.[9] Thus, to evaluate the transport property l correctly, it is crucial to take account of the momentum dependence of the scattering states and their conservation modulo the reciprocal lattice vectors. Note that ρ|vx| defined in Eq. (6) is related to the sur- face area S of the Fermi surfaces, as ρdε = Sdk⊥/(2π) Hence, ρ|vx| too is independent of the mass renormaliza- tion z as F is, and for free electrons we obtain ρ|vx| ∝ ∝ n2/3. One can see a correlation between F and ρ|vx| in Table I. In fact, Pd and Pt have twice as large ρ|vx| as the uranium compounds. The difference can- not be simply explained by the difference in the Fermi surface volume n. It is caused by the fact that the f - electron systems have the nearly isotropic Fermi surfaces while the d-electron systems have complicated ones with FIG. 3: The intersection of the Fermi surfaces of Pd d-hole states with the (111̄) plane. FIG. 4: The intersection of the Fermi surfaces of USn3 with the (100) plane relatively large area compared to their total volume, as indicated in Figs. 3 and 4. The different characters of the surfaces affect not only the single particle quantity ρ|vx| but also the transport property of the total cur- rent relaxation. As the order of magnitude difference in F is not explained merely by ρ|vx|, we have to have re- sort to the other factor, that is, the transport property depending on the Fermi surfaces. It originates from the detailed k-dependence of the scattering states, as repre- sented in ci,j , or by the phase space volume available for all possible scattering channels under strict restrictions of energy and momentum conservations. Thus our quan- titative analysis concludes the important effect on the quasiparticle transport due to the shape and complexity of the Fermi surfaces. In summary, we evaluated the Kadowaki-Woods ratio A/γ2 of some itinerant d and f electron systems numeri- cally on the basis of the Fermi liquid theory using quasi- particle Fermi surfaces obtained by band calculations. In a single framework, we find the d electron systems have smaller ratio than the f systems, as observed, and among others we pointed out an important effect to the transport coefficient A originating from a commonly ne- glected specific feature depending on the characters of the Fermi surfaces. The effect is not understood fully as a single-particle property of interacting systems, but we stress the importance of the phase space restriction due to momentum conservation in two-body scattering processes to dissipate a total electric current. In short, to realize effective dissipation, the system should have a large and regular shaped Fermi surface. In future we will examine that the Fermi-surface dependent efficiency of mutual quasiparticle scatterings may depend on a type of transport current to be relaxed. Acknowledgment The author is grateful to N. Fujima, S. Kokado and T. Hoshino for providing assistance in the numerical cal- culations. He also acknowledges computational resources offered from YITP computer system in Kyoto University. [1] K. Kadowaki and S. B. Woods, Solid State Commun. 58, 507 (1986). [2] M. J. Rice, Phys. Rev. Lett. 20, 1439 (1968). [3] K. Miyake, T. Matsuura, and C. M. Varma, Solid State Commun. 71, 1149 (1989). [4] N. Tsujii, H. Kontani, and K. Yoshimura, Phys. Rev. Lett. 94, 057201 (2005). [5] H. Kontani, J. Phys. Soc. Jpn. 73, 515 (2004). [6] T. Okabe, J. Phys. Soc. Jpn. 67, 2792 (1998). [7] T. Okabe, J. Phys. Soc. Jpn. 67, 4178 (1998). [8] T. Okabe, J. Phys. Soc. Jpn. 68, 2721 (1999). [9] K. Yamada and K. Yosida, Prog. Theor. Phys. 76, 621 (1986). [10] J. M. Ziman, Electrons and Phonons (Clarendon Press, Oxford, 1960). [11] P. Nozières, J. Low. Temp. Phys. 17, 31 (1974). [12] K. Yosida and K. Yamada, Prog. Theor. Phys. 53, 1286 (1975). [13] T. Takimoto and T. Moriya, Solid State Commun. 99, 457 (1996). [14] G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996). [15] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). [16] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). [17] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 (1992).

0704.0844

Structure of Strange Dwarfs with Color Superconducting Core

Structure of Strange Dwarfs with Color Superconducting Core Masayuki Matsuzaki∗ and Etsuchika Kobayashi Department of Physics, Fukuoka University of Education, Munakata, Fukuoka 811-4192, Japan Abstract We study effects of two-flavor color superconductivity on the structure of strange dwarfs, which are stellar objects with similar masses and radii with ordinary white dwarfs but stabilized by the strange quark matter core. We find that unpaired quark matter is a good approximation to the core of strange dwarfs. PACS numbers: 95.30.-k ∗[email protected] http://arxiv.org/abs/0704.0844v1 mailto:[email protected] Witten made a conjecture that the absolute ground state of quantum chromodynamics (QCD) is not 56Fe but strange quark matter, which is a plasma composed of almost equal number of deconfined u, d, and s quarks [1]. Although this conjecture has been neither confirmed nor rejected, if this is true, since deconfinement is expected in high density cores of compact stars, there could exist stars that contain strange quark matter converted from two-flavor quark matter via weak interaction. Strange quark stars whose radii are about 10 km, with or without thin nuclear crust, have long been investigated. Glendenning et al. proposed a new class of compact stars containing strange quark matter and thick nuclear crust ranging from a few hundred to ten thousand km [2, 3, 4]. They named them the strange dwarfs because their radii correspond to those of white dwarfs. Alcock et al. discussed the mechanism that the strange quark core supports the cruct [5]. Since the mass of s quark is larger than those of u and d, strange quark matter is positively charged. In order to electrically neutralize the core, electrons are bound to the surface of the core. They estimated that the thickness of this electric dipole layer is a few hundred fm. Then this layer can support a nuclear crust. Although Alcock et al. considered only thin crusts, Glendenning et al. considered thick crusts up to about ten thousand km. Very recently, Mathews et al. identified eight candidates of strange dwarfs from observed data [6]. A theoretical facet whose importance in nuclear physics was recognized later is color superconductivity in quark matter. At asymptotically high density, the color-flavor locking (CFL) is believed to be the ground state [7]. At realistic densities, however, the two-flavor color superconductivity (2SC) is thought to be realized even when electric neutrality is imposed if the coupling constant is strong [8]. Thus, in the present paper, we discuss effects of the 2SC phase in the strange quark matter core on the structure of strange dwarfs. In order to determine the structure of compact stars, we solve the general relativistic Tolman-Oppenheimer-Volkoff (TOV) equation, dp(r) Gǫ(r)M(r) 4πr3p(r) M(r)c2 2GM(r) , (1) M(r) = 4π ǫ(r′) r′2dr′, (2) for the pressure p(r), the energy density ǫ(r), and the mass enclosed within the radius r, M(r). Here G is the gravitational constant and c is the speed of light. The equation is closed when an equation of state (EOS), a relation between p and ǫ, is specified. In the present case, strange dwarfs are composed of the strange quark matter core and the nuclear crust. Accordingly two parameters, the pressure at the center and at the core-crust boundary, must be specified to integrate the TOV equation. The latter must be equal or less than that corresponds to the nucleon drip density ǫdrip. Otherwise neutrons drip and gravitate to the core. In the present calculation we take a pcruct calculated from ǫcrust = ǫdrip. We assume zero temperature throughout this paper. As for the EOS of the quark core, we adopt the MIT bag model without any QCD corrections (see Ref. [4], for example). For unpaired free quark matter, p = −B + µfkFf µ2f − m4f ln µf + kFf , (3) ǫ = B + µfkFf µ2f − m4f ln µf + kFf , (4) where mf , kFf , and µf = m2f + k Ff are the mass, the Fermi momentum, and the chemical potential of quarks of each flavor, respectively, and f runs u, d, and s. Hereafter we put c = h̄ = 1. The quantity B is the bag constant. The effect of color superconductivity is incorporated as a chemical potential dependent effective bag constant. In the 2SC case [9], Beff = B − ∆2(µ)µ2, (5) where ∆(µ) is the quark pairing gap as a function of a chemical potential µ, whose relation to µf is specified later. The pairing gap is obtained as a function of the Fermi momentum by solving the gap equation [10] ∆(kF) = − v̄(kF, k) E ′(k) k2dk, (6) E ′(k) = (Ek − EkF) 2 + 3∆2(k), (7) with kF = kFu = kFd, Ek = k2 +m2q , and mq = mu = md. The one gluon exchange pairing interaction is given by v̄(p, k) = − pkEpEk 2EpEk + 2m q + p 2 + k2 +m2E (p+ k)2 +m2E (p− k)2 +m2E 6EpEk − 6m q − p 2 − k2 m2E = 2, (8) where p and k are the magnitudes of 3-momenta. The running coupling constant is given by [11] q2max+q q = p− k, qmax = max{p, k}. (9) As for the EOS of the crust, we adopt the tabulated one for β-equilibrium nuclear matter of Baym, Pethick, and Sutherland [12] (BPS) conforming to Refs. [2, 3, 4]. The positively charged strange quark matter in the core is simply approximated by µ = µu = µd = µs. Quark masses are given by mu = md = 10 MeV, ms = 150 MeV. The bag constant is chosen to be B1/4 = 160 MeV. Parameters entering into the pairing interaction are q2c = 1.5Λ QCD and ΛQCD = 400 MeV. The nucleon drip density is ǫdrip = 4.3×10 11 g/cm3. 26 28 30 32 34 36 38 log� (J/m free quark 2SC quark FIG. 1: Equations of state of free and 2SC quark matter and β-equilibrium nuclear matter. The latter is tabulated in Refs. [12] and [4]. The adopted EOS is displayed in Fig. 1. The logarithm is to base 10 throughout this paper. The quark matter EOS describes the core and the BPS EOS describes the crust. At the boundary, the pressure is common whereas the energy density jumps discontinuously. In order to obtain the EOS for 2SC matter, the pairing gap must be calculated at each kF beforehand. This is shown in Fig. 2 left. The effective bag constant determined by the pairing gap is shown in Fig. 2 right. The resulting 2SC EOS is included in Fig. 1. Figure 3 presents the mass-radius relation obtained by integrating the TOV equation with a fixed pcruct, determined from ǫcrust = ǫdrip, and various central pressures. This result can 200 300 400 500 600 700 800 µ (MeV) 200 300 400 500 600 700 800 µ (MeV) FIG. 2: Left: color superconducting pairing gap and right: effective bag constant, as functions of the quark chemical potential. 0 1 2 3 4 5 logR (km) FIG. 3: Mass-radius relation of strange dwarfs and white dwarfs. be classified into three regions. The first region (larger central pressures), almost vertical curve at around R ∼ 10 km, describes strange stars with thin crusts. In this region, color superconductivity makes the maximum mass and radius larger because the pairing gap reduces the bag constant and consequently the energy density decreases and the pressure increases. This is consistent with another calculation with the CFL phase [13]. The second region, horizontal at around M/Msun ∼ 10 −2, and the third region, vertical at around R ∼ 104 km up to the maximum mass, correspond to strange dwarfs. In the second region, color superconducting quark cores support slightly larger masses than unpaired free quark cores. In the third region, effect of color superconductivity is negligible. In Fig. 3, The mass-radius relation of ordinary white dwarfs without quark matter cores calculated by adopting the BPS EOS is also shown although it is known that the BPS EOS is not very suitable for white dwarfs. As the central pressure decreases, the quark matter core shrinks (Fig. 4 left) and eventually strange dwarfs reduce to ordinary white dwarfs. When their masses are the same, the former is more compact than the latter (see also Fig. 5 right) because of the gravity of the core. Mathews et al. paid attention to this difference in the mass-radius relation and classified the observed data of dwarfs [6]. According to their work, eight of them are classified into strange dwarfs. 28 29 30 31 32 33 34 35 logp0 (J/m 28 29 30 31 32 33 34 35 logp0 (J/m FIG. 4: Left: core radius and right: mass of strange dwarfs, as functions of the central pressure. 14 15 16 log�0 (g/cm -1 0 1 2 3 4 logr (km) SD(free) FIG. 5: Left: mass of strange dwarfs as a function of the central energy density. Right: energy profile of a strange dwarf with M/Msun = 0.465 and that of a white dwarf with M/Msun = 0.466. Figure 4 right indicates that strange dwarfs, in particular those of 103 km < R < 104 km, are realized in a very narrow range of the central pressure. This is reflected in the density of calculated points. During this rapid structure change from the second to the third region, the core radius almost does not change, see Fig. 4 left. Figure 5 left also graphs M/Msun as Fig. 4 right but as a function of the central energy density. The difference between these two figures at the low pressure/energy density side can be understood from the quark matter EOS in Fig. 1 such that the pressure decreases steeply at the lowest energy density. Figure 5 left indicates that strange dwarfs have central energy densities just below the lowest stable compact strange stars and several orders of magnitude larger than those of ordinary white dwarfs. This is clearly demonstrated in Fig. 5 right. To summarize, we have solved the Tolman-Oppenheimer-Volkoff equation for strange dwarfs with ǫcrust = ǫdrip and a wide range of the central pressure. We have examined effects of the two-flavor color superconductivity in the strange quark matter core in a simplified manner. The obtained results indicate that, aside from a slight increase of the minimum mass, effect of color superconductivity is negligible in the mass-radius relation. This is consistent with the conjecture given in Ref. [6]. As a function of the central energy density, however, strange dwarfs are realized at slightly lower energy densities than the unpaired free quark case reflecting the effect on the equation of state. Recently Usov discussed that electric fields are also generated on the surface of the color-flavor locked matter [14]. This suggests that strange dwarfs with color-flavor locked cores might also be possible although this is expected only at relatively high densities. Since the pairing gap enters into the calculation only through the effective bag constant, aside from a possible slight change in chemical potentials, it can surely be expected that the effect of color-flavor locking does not differ much from that of the two-flavor color superconductivity. In conclusion, unpaired quark matter is a good approximation to the core of strange dwarfs. Another aspect that might be affected by color superconductivity is the cooling [15]. This is beyond the scope of the present study. [1] E. Witten, Phys. Rev. D 30 (1984), 272. [2] N. K. Glendenning, Ch. Ketter and F. Weber, Phys. Rev. Lett. 74 (1995), 3519. [3] N. K. Glendenning, Ch. Ketter and F. Weber, Astrophys. J. 450 (1995), 253. [4] N. K. Glendenning, Compact Stars (Springer, New York, 1996). [5] C. Alcock, E. Farhi and A. Olinto, Astrophys. J. 310 (1986), 261. [6] G. J. Mathews, I. -S. Suh, B. O’Gorman, N. Q. Lan, W. Zech, K. Otsuki and F. Weber, J. Phys. G 32 (2006), 747. [7] K. Rajagopal and F. Wilczek, Phys. Rev. Lett. 86 (2001), 3492. [8] H. Abuki and T. Kunihiro, Nucl. Phys. A 768 (2006), 118. [9] M. Alford and K. Rajagopal, J. High Energy Phys. 06 (2002), 031. [10] M. Matsuzaki, Phys. Rev. D 62 (2000), 017501. [11] K. Higashijima, Prog. Theor. Phys. Suppl. 104 (1991), 1. [12] G. Baym, C. Pethick and P. Sutherland, Astrophys. J. 170 (1971), 299. [13] G. Lugones and J. E. Horvath, Astron. and Astrophys. 403 (2003), 173. [14] V. V. Usov, Phys. Rev. D 70 (2004), 067301. [15] O. G. Benvenuto and L. G. Althaus, Astrophys. J. 462 (1996), 364. References

0704.0845

Information entropic superconducting microcooler

Information entropic superconducting microcooler A. O. Niskanen,1, 2 Y. Nakamura,1, 3, 4 and J. P. Pekola5 1CREST-JST, Kawaguchi, Saitama 332-0012,Japan 2VTT Technical Research Centre of Finland, Sensors, PO BOX 1000, 02044 VTT, Finland 3NEC Fundamental Research Laboratories, Tsukuba, Ibaraki 305-8501, Japan 4The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan 5Low Temperature Laboratory, Helsinki University of Technology, PO BOX 3500, 02015 TKK, Finland (Dated: October 25, 2018) We consider a design for a cyclic microrefrigerator using a superconducting flux qubit. Adiabatic modulation of the flux combined with thermalization can be used to transfer energy from a lower temperature normal metal thin film resistor to another one at higher temperature. The frequency selectivity of photonic heat conduction is achieved by including the hot resistor as part of a high frequency LC resonator and the cold one as part of a low-frequency oscillator while keeping both circuits in the underdamped regime. We discuss the performance of the device in an experimentally realistic setting. This device illustrates the complementarity of information and thermodynamic entropy as the erasure of the quantum bit directly relates to the cooling of the resistor. PACS numbers: 74.50.+r,85.80.Fi,03.67.-a For the purpose of quantum computing, the coher- ence properties of superconducting quantum bits (qubits) should be optimized by decoupling them from all noise sources as well as possible. However, many interesting experiments can be envisioned also when the decoupling is far from perfect. One such experiment closely related to coherence optimization is using a qubit as a spectrom- eter [1, 2, 3] for the environmental noise by monitoring the effect of the environment on the quantum two-level system. Here we focus on the opposite phenomenon, i.e. the effect of a qubit on the environment. Recently a superconducting flux qubit [4, 5] with a quite small tun- neling energy from the point of view of quantum com- puting was cooled using sideband cooling and a third level [6] from about 400 mK down to 3 mK. Motivated by this experiment we consider the possibility of using a single quantum bit as a cyclic refrigerator for environ- mental degrees of freedom. The utilized heat conduction mechanism is photonic which was recently studied also in experiment [7]. Besides the possible practical uses, the device is interesting physically as it directly illus- trates the connection between information entropy and thermodynamical entropy. For related superconducting high-frequency cooler concepts see eg. Refs. [8, 9]. Here we study a flux qubit coupled inductively to two different loops shown in Fig. 1a. In loop j (j = 1, 2) we have a resistor Rj in series with an inductance Lj and a capacitance Cj . These form two damped harmonic os- cillators. The resistors are in general at different tem- peratures T1 and T2. The coupling of the qubit to these two admittances Y1 and Y2 is assumed to be sufficiently large to dominate the relaxation of the qubit. This as- sumption can be easily validated by e.g. increasing the mutual inductance. The flux qubit is an otherwise su- perconducting loop except for three or four Josephson junctions with suitably picked parameters. In particu- FIG. 1: (color online) Principle of the flux-qubit cooler. (a) Layout of the circuit. (b) Energy band diagram. (c) Schematic of the cooling cycle in the qubit temperature- entropy plane. lar one of the junctions is made smaller than others to form a two-level system. When biased close to half of the flux quantum Φ0 = h/2e, the qubit can be described (in persistent current basis) by the Hamiltonian H/~ = −1 (∆σx + εσz) (1) where σx and σz are Pauli matrices, ~ε = 2Ip(Φ−Φ0/2) is the flux-tunable energy bias and Φ is the controllable flux threading the qubit loop. Away from Φ = Φ0/2 the eigenstates have the persistent currents ±Ip circulating in the loop. The tunneling energy ~∆ results in an an- http://arxiv.org/abs/0704.0845v1 ticrossing at Φ = Φ0/2 and there the energy eigenstates do not carry average current. The resonant angular fre- quency of the qubit is ω = ε2 +∆2. Consider the ideal cycle shown in Fig. 1b-c where the bias of the flux qubit is swept slowly (slower than ∆/2π) between two extreme values ε1 and ε2 corresponding to two different energy level separations ~ω1 and ~ω2. Let us further assume that ωj ≈ ωLCj and Qj ≫ 1, where ωLCj = 1/ LjCj and Qj = Lj/Cj/Rj. This choice guarantees that the qubit mainly couples to resistor R1 (R2) at bias point 1 (2). The cooling cycle consists of steps O, P, Q and R. First in step O the qubit has the angular frequency ω2 and is allowed to thermalize. Be- cause of the bandwidth limitations imposed by the reac- tive elements, the qubit tends to thermalize with resistor R2 to temperature T2. In the next step P the flux bias is adiabatically changed to point 1 such that the level popu- lations do not change but the energy eigenstates do. The sweep is assumed to be however faster than relaxation. In point 1 the angular frequency is reduced to ω1. Be- cause the level populations and therefore the Boltzmann factors do not change the qubit must now be at lower temperature T̃2 given by T̃2 = T2ω1/ω2 in order to com- pensate for the change of the qubit splitting. Note that the quantum mechanical adiabaticity implies also ther- modynamical adiabaticity: while the energy eigenbasis changes the level populations and thus also entropy do not change. In step Q the qubit is allowed to thermalize to temperature T1 which results in heating of the qubit and in cooling of resistor 1 if T̃2 < T1. At this point the ideally pure quantum state of the qubit gets erased and information stored is lost. The entropy of the qubit increases, but locally the entropy of resistor 1 decreases such that one can say that some information is “stored” in the resistor as it cools but naturally with some loss. Finally in step R the qubit is adiabatically shifted back to frequency ω2 which results in heating of the qubit to the effective temperature T̃1 = T1ω2/ω1 which is assumed to be higher than T2. The excess energy is dumped to admittance 2 when the cycle starts again from the be- ginning. Note that due to the condition T̃2 < T1 resistor 1 can never be cooled below T2ω1/ω2. Since there is no isothermal stage in the above cycle the present device is not even in principle a Carnot cooler but rather an Otto-type device.[10] The density matrix of the qubit with the resonant an- gular frequency ω at temperature T (β = (kBT ) −1) is given by ρeq(β, ε) = . (2) Using this the cooling power and the efficiency of the ideal cycle in Fig. 1c can be easily calculated. It is given by the area of the shaded region in the entropy-temperature plane below points P and Q. In principle one could solve for the effective temperature of the qubit along the line between points P and Q as a function of entropy given by S = −kBTr(ρ ln ρ). Alternatively, we can simply note that the expectation value of the energy stored in the qubit in point P is EP = Tr(ρeq(β2ω2/ω1, ε1)H1) while after relaxation we have EQ = Tr(ρeq(β1, ε1)H1), where H1 = H(ε1) is the Hamiltonian at point 1. We thus get for the ideal cooling power P/f = EQ − EP = −β1~ω1 e−β1~ω1 + 1 − ~ω1e −β2~ω2 e−β2~ω2 + 1 ≤ ~ω1 where f is the pump frequency. The cooling power achieves the maximum value of ~ω1f/2 when the ther- mal population in step O (and P) is small and when the population in step Q is large, i.e. when β2~ω2 ≫ 1 and β1~ω1 ≪ 1. Naturally a practical device has to be designed to fulfill the first condition always, in which case the smallest achievable temperature is on the or- der of ~ω1/kB below which the cooling power decreases rapidly. The dynamic range could be made wider by a tunable ∆ which can be achieved by splitting the small- est junction into a dc SQUID geometry. Another figure of merit is the ratio η of the heat removed from resis- tor 1 divided by the heat added to resistor 2. It can be obtained as the ratio of the shaded area divided by the sum of the hatched area and the shaded area, i.e., η = (EQ−EP )/(ER−EO) where EO = Tr(ρeq(β2, ε2)H2) and ER = Tr(ρeq(β1ω1/ω2, ε2)H2). This simplifies neatly to η = ω1/ω2 < 1 which is in harmony with the second law of thermodynamics. For more quantitative analysis we have to consider the details of the relaxation rates due to the baths. The Golden Rule transition rates due to resistor j are given ↓,↑ = |〈0|dH/dΦ|1〉|2M2j S I (±ωj) M2j S I(±ωj) (4) where the positive sign corresponds to relaxation. The total thermalization rate is Γ ↑ + Γ ↓. Here the unsymmetrized noise spectrum is given by I(ω) = e−iωt〈δIj(0)δIj(t)〉dt 2~ωReYj(ω) 1− exp(−βj~ω) . (5) where ReYj(ω) = R j /[1+Q − ωLCj )2] is the real part of admittance of circuit j. The total relaxation rate is thus 2(Ip∆Mj) 2 coth 1 +Q2j − ωLCj ) . (6) To model the behavior of the device we utilize the Bloch master equation (see e.g Ref. [11]) given in our case by ~̇M = − ~B× ~M−Γ1th( ~M‖− ~MT1)−Γ2th( ~M‖− ~MT2)−Γ2 ~M⊥, where ~M = Tr(~σρ) is the “magnetization” of the qubit, and ~B = ∆~x + ε~z is the fictitious magnetic field. Note however that the z-component of ~B and ~M do correspond to real magnetic field and magnetization, respectively. In Eq. (7) ~M‖ and ~M⊥ are the components of the magnetiza- tion parallel and perpendicular to ~B, respectively. These are explicitly ~M‖ = (∆Mx + εMz)(∆~x + ε~z) (8) ~M⊥ = ε2Mx−∆εMz ~x+My~y + 2Mz−∆εMx ~z. (9) Here ~MT stands for the ε-dependent equilibrium mag- netization of a qubit at temperature T given explicitly ~MT = and Γ2 = (Γ th + Γ th)/2 + Γϕ is the dephasing rate. The possibility of pure dephasing at the rate Γϕ has been included. In the simulation we neglect pure dephasing due to the intentionally large dominating thermalization rate. Equation (7) describes relaxation towards instan- taneous equilibrium with two competing rates due to two different thermal baths. Equations of this type are usually used in the stationary case, but for driving fre- quencies slower than ∆/~ it should be also valid. As is obvious from Eq. (7), the qubit actually tends to re- lax towards an effective ε-dependent equilibrium mag- netization (Γ1th ~MT1 + Γ ~MT2)/(Γ th + Γ th) at the rate To illustrate the practical potential of the device we show in Fig. 2 the simulated cooling power with si- nusoidal driving of ǫ(t) compared to the ideal case along with the actual loop in the entropy temperature plane. The heat flow Pj from resistor j to the qubit is simply obtained by integrating the product of the thermalization rate and the energy deficit, i.e., Pj = ∫ 1/f [Tr(ρeq(βj , ǫ(t))H)− Tr(ρ(t)H)] . The den- sity matrix ρ(t) = 1 ~M(t) · ~σ is solved numerically using the Bloch equation (system is followed over a few periods until it has converged to the limit cycle). We see that the actual simulated behavior does not significantly deviate at low f from the ideal behavior and that cooling pow- ers on the order of fW can be achieved with reasonable sample parameters. The oscillatory behavior at high f is interpreted as Landau-Zener interference [12, 13]. However, the cooling power has to be compared with realistic heat loads to evaluate the utility of the flux qubit cooler. On one hand, resistor 1 is subject to heat load 0 ln 2 (a) (c) (b) (d) 0 0.5 1 1.5 f (GHz) 0 ln 2 0 0.5 1 1.5 f (GHz) FIG. 2: (color online) Example of the simulated cooling power with ω1/2π = ∆/2π = 5 GHz (ǫ = 0 GHz), ω2/2π = 20.62 GHz (ǫ = 20 GHz), Q1 = Q2 = 10, ωj = ωLCj and 2(Ip∆Mj) 2/(Rj~ωj) = 20 × 10 9s−1. This can be achieved e.g. with Ip = 200 nA, M1 = 29 pH, M2 = 59 pH and R1 = R2 = 1 Ω. The driving is sinusoidal. (a) The solid line illustrates the path in the T − S plane for the ideal cy- cle described in the text while the dashed (dotted) line is a result of simulation for f = 0.05 GHz (f = 1 GHz) with T1 = T2 = 0.3 × ~ω2/kB ≈ 300 mK. (b) Simulated cooling power vs. f for the same temperatures as in (a) is shown with the dashed line while the solid line is the ideal result of Eq. 3 (c-d) Same as (a-b) but with T1 = 0.5× T2 ≈ 150 mK. The cooling threshold at 0.14 GHz in (d) is caused by finite Q-factor. from the phonons of the substrate on which the device rests. On the other hand, resistor 2 should be coupled well enough to phonon bath such that the unavoidable work done on it does not raise T2 excessively. The heat flow between the electron system of resistor j and the phonon system is given by Pel−ph = ΣV (T j −T 5ph) where Vj is the volume of resistor j and Σ is typically on the order of 109 Wm−3K−5. Thus resistor 1 needs to have a sufficiently small volume while resistor 2 should be large enough physically in order to serve as a heat sink. In ad- dition the photonic heat conduction between the resistors due to temperature gradient may in principle contribute also. Following an analysis similar to Ref. [14], the heat flow from admittance Y2(ω) to Y1(ω) can be written as ReY1(ω)ReY2(ω)(n2(ω)− n1(ω)) where nj(ω) = [exp(βj~ω − 1)]−1 are the boson occu- pation factors and M is the mutual inductance between the loops. For detuned high-Q resonators the photonic heat conduction turns out to be quite negligible. For instance for the values of Fig. 3 with M = 5 pH and R1 = R2 = 1 Ω we get only Pγ = 2 × 10−18 W even 0 0.5 1 1.5 f (GHz) FIG. 3: (color online) Equilibrium temperature as a function of pump frequency for three different phonon bath temper- atures. The temperature of resistor 1 (volume 10−21m3) is shown with dashed line while the temperature of resistor 2 (volume 10−18m3) is shown with solid line. The bath tem- peratures Tph ≈ T2 from top to bottom are 0.3 × ~ω2/kB, 0.2 × ~ω2/kB and 0.1 × ~ω2/kB. Otherwise the parameters are like in Fig. 2. if T1 = 0 K and T2 = 300 mK. Figure 2 illustrates the calculated equilibrium temperature versus operation fre- quency obtained numerically by finding the balance be- tween the dominating phononic heat conduction and the integrated cooling power. We see that almost a factor of 2 reduction of T1 is possible with realistic parameters. In practice the drop of T1 can be measured e.g. using an additional SINIS thermometer, in which resistor 1 will serve as the normal metal N. Its reading is sensitive to the electronic temperature of N only, and self-heating can be made very small. The resistors should be made out of thin film normal metal such as copper or gold with typ- ically sub 1 Ω square resistance. Volume can be picked freely. To get the resonant frequencies and quality factor as above we need L1 = 320 pH, C1 = 3.2 pF, L2 = 80 pH and C2 = 0.8 pF which are also realistic. For the inductor one may use either Josephson or the kinetic in- ductance of superconducting wire while the capacitance values are similar to those in typical flux qubits [2]. To satisfy the conditions of the above numerical example we need quite large mutual inductances which however can be easily achieved using e.g. kinetic inductance [15]. The strong driving requires also rather large inductance be- tween the microwave line and the qubit, which should not result in uncontrolled relaxation. For instance, Mmw=5 pH coupling to the control line is acceptable as it would result in at most 3 × 107 s−1 relaxation rate assuming a 50 Ω environment at 0.3 K. This choice will not degrade the performance of the device significantly since driving is much faster. Yet sufficiently strong driving can be achieved with a modest 3 µA ac current. Fabrication process will require most likely three lithography steps. In conclusion, we have described a method of using a superconducting flux qubit driven strongly at microwave frequency to cool an external metal resistor. Here we con- sidered LC resonators to achieve the required frequency selectivity but a coplanar wave-guide resonator or a me- chanical oscillator could be used in principle, too. We demonstrated by a numerical example that it is possible to observe the associated temperature decrease experi- mentally. This effect is directly related to the loss of information and thus to the increase of entropy of the quantum bit. J.P.P thanks NanoSciERA project ”NanoFridge” of EU for financial support. [1] O. Astafiev, Yu. A. Pashkin, Y. Nakamura, T. Ya- mamoto, and J. S. Tsai, Phys. Rev. Lett. 93, 267007 (2004). [2] F. Yoshihara, K. Harrabi, A. O. Niskanen, Y. Nakamura, J.S. Tsai, Phys. Rev. Lett. 97, 167001 (2006). [3] P. Bertet, I. Chiorescu, G. Burkard, K. Semba, C. J. P. M. Harmans, D.P. DiVincenzo, and J.E. Mooij, Phys. Rev. Lett. 95, 257002 (2005). [4] J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H. van der Wal, S. Lloyd, Science 285, 1036 (1999). [5] I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, Science 299, 1869 (2003). [6] S. O. Valenzuela, W. D. Oliver, D. M. Berns, K. K. Berggren, L. S. Levitov, and T. P. Orlando, Science 314, 1589 (2006). [7] M. Meschke, W. Guichard, and J. P. Pekola, Na- ture(London) 444, 187 (2006). [8] J. Hauss, A. Fedorov, C. Hutter, A. Shnirman, and G. Schön, cond-mat/0701041. [9] J. P. Pekola, F. Giazotto, and O. P. Saira, Phys. Rev. Lett. 98, 037201 (2007). [10] H.T. Quan, Y.X. Liu, C. P. Sun, and Franco Nori, quant-ph/0611275. [11] Yu. Makhlin, G. Schön, and A. Shnirman, in New Di- rections in Mesoscopic Physics (Towards Nanoscience), edited by R. Fazio, V. F. Gantmakher, and Y. Imry (Kluwer, 2003), p. 197; cond-mat/0309049. [12] M. Sillanpää, T. Lehtinen, A. Paila, Yu. Makhlin, and P. Hakonen, Phys. Rev. Lett. 96, 187002 (2006). [13] W. D. Oliver, Y. Yu, J. C. Lee, K. K. Berggren, L. S. Levitov, and T. P. Orlando, Science 310, 1653 (2005). [14] D. R. Schmidt, R. J. Schoelkopf, and A. N. Cleland, Phys. Rev. Lett. 93, 045901 (2004). [15] A. O. Niskanen, K. Harrabi, F. Yoshihara, Y. Nakamura, and J. S. Tsai, Phys. Rev. B 74, 220503(R) (2006). http://arxiv.org/abs/cond-mat/0701041 http://arxiv.org/abs/quant-ph/0611275 http://arxiv.org/abs/cond-mat/0309049

0704.0846

PI degree parity in q-skew polynomial rings

PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS HEIDI HAYNAL Abstract. For k a field of arbitrary characteristic, and R a k-algebra, we show that the PI degree of an iterated skew polynomial ring R[x1; τ1, δ1] · · · [xn; τn, δn] agrees with the PI degree of R[x1; τ1] · · · [xn; τn] when each (τi, δi) satisfies a qi-skew relation for qi ∈ k× and extends to a higher qi-skew τi-derivation. We confirm the quantum Gel’fand-Kirillov conjecture for various quantized coordinate rings, and calculate their PI degrees. We extend these results to completely prime factor algebras. 1. Introduction Presented here is a new technique for analyzing skew polynomial rings satisfying a poly- nomial identity with an eye toward discovering their PI degrees. It combines and extends the methods of Jøndrup [21] and Cauchon [5], who introduced techniques of “deleting derivations” in skew polynomial rings, by means of which they showed that some proper- ties of certain types of iterated skew polynomial ring A = k[x1][x2; τ2, δ2] · · · [xn; τn, δn] are determined by the corresponding ring A′ = k[x1][x2; τ2] · · · [xn; τn]. Jøndrup’s re- sults imply that A and A′ have the same PI degree under certain hypotheses, including characteristic zero for the base field. Cauchon developed an algorithm that gives an isomorphism between certain localizations of A and A′, but this requires a qi-skew condition on each (τi, δi) with qi not a root of unity, which usually precludes A from satisfying a polynomial identity. We relax the restrictions placed on the base field and its chosen scalars by Jøndrup and Cauchon, respectively, by introducing the notion of a higher q-skew τ -derivation. If we “twist” the multiplication in the (commutative) coordinate ring of affine, symplec- tic, or Euclidean n-space over a field k, we get a (noncommutative) quantized coordinate ring which has the structure of an iterated skew polynomial ring with coefficients in k. This structure is also exhibited in the quantized Weyl algebras and in the quantized coordinate ring of n×n matrices over k. Letting A represent one of these k-algebras, the quantum Gel’fand-Kirillov conjecture asserts that FractA is isomorphic to the quotient division ring of a quantum affine space over a purely transcendental extension of k. For 1991 Mathematics Subject Classification. 16R99; 16S36; 81R50; 16P40. Key words and phrases. noncommutative rings; skew polynomial rings; quantum algebras. This research will form a part of the author’s PhD dissertation at the University of California at Santa Barbara. http://arxiv.org/abs/0704.0846v1 2 HEIDI HAYNAL more information on the quantum Gel’fand-Kirillov conjecture and proofs of conditions under which the result holds, see [1] [7] [23] [28] [32] [33]. We will confirm some of these cases in a new way. The first section sets up the conventions under which we work, including definitions and an established result concerning the PI degree of quantum affine space. We assume that the reader has some familiarity with the subject, so we do not give an exhaustive collection of definitions. A comprehensive discussion of any unfamiliar terms can be found in [16] [4] and [27]. In the second section we define higher τ -derivations and give necessary and sufficient conditions for their existence. Of particular interest are higher τ -derivations which satisfy a q-skew relation. In the third section we present a structure theorem for a localization of q-skew polynomial rings. This extends the work of Cauchon [5], and the calculations are simplified by the presence of higher q- skew τ -derivations. In the fourth section we deal with the structure of iterated skew polynomial rings. Sometimes it is advantageous to rearrange the order in which the indeterminates appear, so we establish a sufficient condition that allows such reordering. The main theorem there asserts that if A is an iterated q-skew polynomial ring with certain higher τ -derivations, then there is a finitely generated Ore set T ⊆ A such that AT−1 is isomorphic to a localization of a much “nicer” iterated skew polynomial ring. In the fifth section, we use the tools developed in the previous sections to confirm certain cases of the quantum Gel’fand-Kirillov conjecture and to find the PI degree of some quantized coordinate rings and quantized Weyl algebras. In the last section, we follow up with a structure theorem for completely prime factors of iterated skew polynomial rings. We also present an open question which, if answered positively, would show that the quantum Gel’fand-Kirillov conjecture holds for certain of the prime factor algebras we study. Throughout, k will denote a field of arbitrary characteristic, q ∈ k a nonzero ele- ment. The following assumptions apply to all skew polynomial rings that we will con- sider: • all coefficient rings are k-algebras • all automorphisms are k-algebra automorphisms • all skew derivations are k-linear • in all skew polynomial rings R[x; τ, δ], τ is an automorphism, not just an endo- morphism. To say that R[x; τ, δ] is a q-skew polynomial ring means that the auomorphism and skew derivation satisfy the relation δτ = qτδ. The reader will note that this is opposite to Cauchon’s conventions, but it matches the presentation in [10] and others. To say that δ is locally nilpotent means that for every r ∈ R there is an integer nr ≥ 0 such that δnr(r) = 0, and δp(r) 6= 0 for p < nr. Such nr is called the δ-nilpotence index of r. The PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 3 symbol N refers to the set of positive integers. For a real number m we use the notation ⌊m⌋ in section five to indicate the integer part of m. Definition 1.1. We say that two rings R and S exhibit PI degree parity when these two conditions are satisfied: (1) R is a PI ring if and only if S is a PI ring, (2) PIdegR = PIdegS. For a field k and multiplicatively antisymmetric λ ∈ Mn(k), the corresponding mul- tiparameter quantum affine space is the k-algebra Oλ(kn) with generators x1, . . . , xn and relations xixj = λijxjxi for all i, j. The corresponding multiparameter quantum torus is the k-algebra Oλ((k×)n) given by generators x±11 , . . . , x±1n and the same rela- tions. The multiplicative set generated by x1, . . . , xn in Oλ(kn) is a denominator set, and Oλ((k×)n) is a localization of Oλ(kn) with respect to this set. In this paper we’ll show that iterated skew polynomial algebras covering a large class of standard examples have PI degree parity with Oλ(kn) for an appropriately chosen λ. To find out what that PI degree may be, we utilize a result of De Concini and Procesi. In [8, Proposition 7.1], they establish the following formula for calculating the PI degree of a quantum affine space Oλ(kn). Their assumption of characteristic zero from [8, Section 4] is not used in this result. Theorem 1.2. [De Concini - Procesi] Let λ = (λij) be a multiplicatively antisymmetric n× n matrix over k. (1) The quantum affine space Oλ(kn) is a PI ring if and only if all the λij are roots of unity. In this case, there exist a primitive root of unity q ∈ k× and integers aij such that λij = q aij for all i, j. (2) Suppose λij = q aij for all i, j, where q ∈ k is a primitive ℓth root of unity and the aij ∈ Z. Let h be the cardinality of the image of the homomorphism n (aij)−−−−−−→ Zn π−−−−→ (Z/ℓZ)n where π denotes the canonical epimorphism. Then PI-deg (Oλ(kn)) = 2. Higher q-Skew τ-Derivations Before the featured definition, a brief discussion of a tool used to study q-skew polyno- mial rings is needed. Having the q-skew relation δτ = qτδ in place allows us to group terms of the same degree when we do skew polynomial arithmetic. The means to do this are provided by the q-Liebnitz rules. 4 HEIDI HAYNAL Definition 2.1. For an indeterminate t, and integers n ≥ m ≥ 0, we define the following polynomial functions: (m)t = t m−1 + tm−2 + · · ·+ t + 1 (1) (m)!t = (m)t(m− 1)t · · · (1)t, and (0)!t = 1 (2)( (n)!t (m)!t(n−m)!t The expressions are called the t-binomial coefficients, or Gaussian polynomials. The t-binomial coefficients have properties similar to those of the regular binomial coefficients. Two that will be useful for this work are:( = 1 for all n ≥ 0 (4) + tn−m for all 0 < m < n Proofs for these identities may be found in combinatorics texts such as [39]. When we evaluate the t-binomial coefficients at t = q, we obtain the q-binomial coefficients that we need for studying q-skew polynomial rings. As shown in [10, Section 6], the following q-Liebnitz rules hold for any q-skew polynomial ring R[x; τ, δ]: δn(rs) = τn−iδi(r)δn−i(s) for all r, s ∈ R and n = 0, 1, 2, ... xnr = τn−iδi(r)xn−i for all r ∈ R and n = 0, 1, 2, ... Now, taking a cue from the study of Schmidt differential operator rings, for instance [25], we define a sequence of k-linear maps that allows us to broaden the class of rings for which we may derive results like those of Jøndrup and Cauchon. Definition 2.2. A higher q-skew τ -derivation (h.q-s.τ -d.) on a k-algebra R is a sequence d0, d1, d2, . . . of k-linear operators on R such that d0 is the identity dn(rs) = τn−idi(r)dn−i(s) for all r, s ∈ R and all n diτ = q iτdi for all i. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 5 If a sequence of k-linear maps satisfies the first two conditions, we refer to it as a higher τ -derivation. We abbreviate the sequence {di}∞i=0 usually as just {di}. A h.q-s.τ -d is locally nilpotent if for all r ∈ R, there exists an integer n ≥ 0 such that di(r) = 0 for all i ≥ n, and dp(r) 6= 0 for p < n. In this case, n is called the d-nilpotence index of r. A h.q-s.τ -d is iterative if didj = di+j for all i, j. This implies that the di commute with each other. A q-skew τ -derivation δ on R extends to a h.q-s.τ -d. if there is a h.q-s.τ -d {di} on R with d1 = δ. For example, consider the k-algebra with two generators x and y, and one relation xy − qyx = 1, where q ∈ k×. We’ll assume that q 6= 1 and recognize this algebra as a q-skew polynomial ring k[y][x; τ, δ] with τ(y) = qy and δ(y) = 1, commonly known as a quantized Weyl algebra and denoted A 1(k). If q is not a root of unity, then the (i)!q comprise an iterative higher q-skew τ -derivation that extends δ on k[y]. The prop- erties of a higher q-skew τ -derivation follow directly from the fact that δ is a q-skew τ -derivation and the first q-Liebnitz rule. This particular h.q-s.τ -d. is also locally nilpotent because yn−i when i ≤ n, 0 when i > n. Proposition 2.3. Let {di} be a sequence of k-linear maps on a k-algebra R with d0 = idR, and let R[[x; τ −1]] be the skew power series ring where τ is a k-linear automor- phism of R, the coefficients are written on the right of the variable x, and rx = xτ(r) for all r ∈ R. (a) Then {di} is a higher τ -derivation on R if and only if the map Ψ : R → R[[x; τ−1]] given by r 7→ i=0 x idi(r) is a ring homomorphism. (b) Extend τ to an automorphism of R[[x; τ−1]] such that τ(x) = xq. Assume that {di} is a higher τ -derivation. Then the sequence {di} is a h.q-s.τ -d. if and only if this diagram is commutative: R[[x; τ−1]] // R[[x; τ−1]] 6 HEIDI HAYNAL Proof. (a) Suppose {di} is a higher τ -derivation on R. Consider any r, s ∈ R. It is clear that Ψ is additive and Ψ(1) = 1. Applying the definition 2.2 gives Ψ(rs) = xidi(rs) = τ i−mdm(r)di−m(s) Power series multiplication, with rx = xτ(r), gives Ψ(r)Ψ(s) = xidi(r) )( ∞∑ xidi(s) τ i−mdm(r)di−m(s) So Ψ preserves products. Therefore, Ψ is a ring homomorphism. To demonstrate the other implication, suppose Ψ is a ring homomorphism. Then Ψ(r)Ψ(s) = Ψ(rs) implies that dn(rs) = i=0 τ n−idi(r)dn−i(s) for all r, s ∈ R. There- fore, {di} is a higher τ -derivation. (b) Suppose that {di} is a h.q-s.τ -d. Then the relations diτ = qiτdi imply that τΨ(r) = i=0 x iqiτdi(r) = i=0 x idi(τ(r)) = Ψτ(r), for all r ∈ R. Now if the diagram is commutative, then comparing the coefficients of τΨ(r) = i=0 x iqiτdi(r) and Ψτ(r) = i=0 x idi(τ(r)) for all r ∈ R yields that diτ = q iτdi. � Remark 2.4. If {di} is locally nilpotent on R, we observe that claims analogous to the proposition can be made for the map Ψ : R → R[x; τ−1]. Proposition 2.5. Let {di} be a h.q-s.τ -d. on a k-algebra R, where τ is an automor- phism, and let S be a right denominator set in R with τ(S) = S. Then {di} can be uniquely extended to a h.q-s.τ -d. on RS−1. Proof. It has been established that τ and d1 extend uniquely to RS −1 by τ(rs−1) = τ(r)τ(s)−1 and d1(rs −1) = d1(r)s −1 − τ(rs−1)d1(s)s−1 in [10, Lemma 1.3]. Suppose that {di} extends to a h.q-s.τ -d. on RS−1. For r ∈ R and s ∈ S, we apply dn to the equation r1−1 = (rs−1)(s1−1) to get dn(r)1 −1 = dn (rs−1)(s1−1) τn−jdj(rs −1)dn−j(s1 = τn(rs−1)dn(s)1 −1 + · · ·+ dn(rs−1)s1−1. This implies that dn(rs −1) = dn(r)− τn−jdj(rs −1)dn−j(s) So we have uniqueness in case of existence. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 7 To show existence, let Ψ : R → R[[x; τ−1]] be the map defined in Proposition 2.3, and let φ : R[[x; τ−1]] → RS−1[[x; τ−1]] be the natural map. Consider the composite map Φ = φΨ : R → RS−1[[x; τ−1]]. For any s ∈ S, the constant term of Φ(s) is a unit. So we may inductively solve for the coefficients of an inverse for Φ(s) in RS−1[[x; τ−1]]. Details, as in [37, 1.2], are left to the reader. Hence, Φ extends to a ring homomorphism Φ′ : RS−1 → RS−1[[x; τ−1]] such that Φ′(rs−1) = Φ(r)Φ(s)−1, and we consider the diagram: RS−1[[x; τ−1]] // RS−1[[x; τ−1]] // RS−1 where τ has been extended to an automorphism of RS−1[[x; τ−1]] as in Proposition 2.3. Since Φ(r) = i=0 x idi(r)1 −1, and {di} is a h.q-s.τ -d. on R, we have τΦ(r) = xiqiτdi(r)1 1−1 = Φτ(r) for all r ∈ R. It follows directly that τΦ′(rs−1) = Φ′τ(rs−1). So, indeed, the diagram is commutative. Define a sequence {di} on RS−1 such that di(t) equals the coefficient of xi in Φ′(t) for all t ∈ RS−1. Then by Proposition 2.3 we conclude that this sequence is a h.q-s.τ -d. on RS−1 extending {di} on R. � Lemma 2.6. Let A be a k-algebra, B ⊆ A a k-subalgebra generated by {b1, b2, . . . }, τ a k-linear automorphism of A, and {di} a higher τ -derivation on A. If di(bj) ∈ B and τ(bj) ∈ B, for all i, j ∈ N, then di(B) ⊆ B for all i. Proof. First, observe that τ(bj) ∈ B for all j implies that τ(B) ⊆ B. Since the di are k-linear maps, it suffices to check monomials in the bj , using induction on their length. Suppose, inductively, that for integers m ≥ 1 and 1 ≤ ℓ ≤ m − 1, we have di(bj1 · · · bjℓ) ∈ B for all i and all j1, . . . , jℓ. Then using the product rule for h.q-s.τ -d. gives dn(bj1 · · · bjm) = τn−1di(bj1 · · · bjm−1)dn−i(bjm) ∈ B for all n and all j1, . . . , jm, by the induction hypothesis. � 8 HEIDI HAYNAL Lemma 2.7. Let A be a k-algebra with a set {xj} of generators, τ an automorphism of A, and {di} a h.q-s.τ -d. on A. If {di} is locally nilpotent for all xj, then {di} is locally nilpotent on A. Proof. It suffices to check monomials in the xj because the di are k-linear maps. We proceed by using induction on the length of such monomials. For a given xn, let i(n) be its nilpotence index, so di(xn) = 0 for all i ≥ i(n). Suppose inductively that for n ≥ 2, all integers ℓ with 1 ≤ ℓ ≤ n− 1, and all choices of j1, . . . , jℓ, there exists an integer m such that di(xj1 · · ·xjl) = 0 for all i ≥ m. For instance, m = i(j1)+ · · ·+ i(jℓ) will suffice, although the d-nilpotence index of xj1 · · ·xjℓ may be less than this sum. Then, for p ≥ m+ i(jn), we have dp(xj1 · · ·xjn) = τ p−idi(xj1 · · ·xjn−1)dp−i(xjn) = 0, completing the induction. � Consider again the quantized Weyl algebra A 1(k). In case q is an ℓ-th root of unity, the dℓ given in (7) would be undefined due to the occurrence of a zero denominator. However, realizing A 1(k) as a factor of a quantized Weyl algebra over k[t ±1] allows us to define a h.q-s.τ -d. on A 1(k) nonetheless. The k[t ±1]-algebra At1(k[t ±1]) has generators x and y and one relation xy− tyx = 1. This is a t-skew polynomial ring k[t±1][y][x; τ̄ , δ̄] where τ̄ (y) = ty, τ̄(t) = t, δ̄(y) = 1, and δ̄(t) = 0. Note that δ̄i(yn) = (n)!t (n−i)!t yn−i when i ≤ n 0 when i > n implying that δ̄i k[t±1][y] ⊆ (i)!tk[t±1][y]. So the assignment d̄i = (i)!t defines an iterative, locally nilpotent h.t-s.τ̄ -d. {d̄i} on k[t±1][y]. Now, the relation xy − tyx = 1 is equivalent to the relation xy − qyx = 1 modulo 〈t − q〉. Hence we k[t±1] /〈t− q〉 ∼= Aq1(k). When q is an ℓth root of unity, we have δ̄ℓ k[t±1][y] ⊆ 〈t−q〉k[t±1][y]. Nonetheless, the h.t-s.τ̄ -d. {d̄i} on k[t±1][y] induces a h.q-s.τ -d. {di} on k[y], also iterative and locally nilpotent, with d1 = δ. Note that even though δ ℓ = 0 in this algebra, we have di(y i) = 1 for all i. This phenomenon is not unique to the quantized Weyl algebras. The conditions that drive it are codified in the following theorem. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 9 Theorem 2.8. Let R be a k-algebra and R[x; τ, δ] a q-skew polynomial ring where q ∈ k, q 6= 1. Suppose there exists a torsion-free k[t±1]-algebra R and R[x; τ̄ , δ̄] a t-skew polynomial ring such that R/〈t− q〉R ∼= R, with τ̄ and δ̄ reducing to τ and δ. Suppose further that δ̄i(R) ⊆ (i)!tR for all i. Then δ extends to an iterative h.q-s.τ -d. {di} on R. If δ̄ is locally nilpotent, then so is {di}. If q is not a root of unity, then di = δ (i)!q all i. If q is a primitive ℓth root of unity, then di = (i)!q for i < ℓ. Proof. The assumption δ̄i(R) ⊆ (i)!tR for all i implies that the sequence of maps d̄i = (i)!t make up a well-defined iterative h.t-s.τ̄ -d. on R, and also implies that δ̄ℓ(R) ⊆ 〈t − q〉R because (ℓ)t ≡ (ℓ)q = 0 modulo 〈t − q〉. Since τ̄ and δ̄ reduce to τ and δ modulo 〈t− q〉, we have an isomorphism R/〈t− q〉[x; τ̄ , δ̄] ∼= R[x; τ, δ] whereby {d̄i} induces an iterative h.q-s.τ -d. {di} on R. The reduction of the maps from R to R also implies the remaining results. � We will find that all of the conditions assumed above are satisfied by the common quantized coordinate rings and related examples, which will be discussed in a subsequent section. 3. The τ-Derivation Removing Homomorphism Following the pattern in [5], let A = R[x; τ, δ], and suppose that δ is locally nilpotent. Set S = {xn | n ∈ N ∪ {0}} ⊂ A. Lemma 3.1. The set S is a denominator set in A. Proof. Clearly, S is a multiplicative set inA. And, since S contains only regular elements of A, it is left and right reversible. It remains to show that S is an Ore set. Let a = i=0 rix i be an element of A with rn 6= 0. For each ri in the expression of a, and each mi ≥ 0, we have xmiri = τmi−jδj(ri)x = a′ix+ δ mi(ri) for some a i ∈ A. Since δ is locally nilpotent, we may choose mi to be the δ-nilpotence index of ri to conclude that xmiri = a ix for some a i ∈ A. Set ma = max{mi | 0 ≤ i ≤ n}. Then for each ri, we have x mari = ãix, and hence x maa = ãx for some ã ∈ A. Now suppose, inductively, that for a given a ∈ A and xp ∈ S we can find elements xma ∈ S and ā ∈ A such that xmaa = āxp, say ā = i=0 r̄ix i. We know that there 10 HEIDI HAYNAL exists an element xmā such that xmā ā = a′x for some a′ ∈ A. So, xmaa = āxp implies xmā+maa = a′xp+1, completing the induction. Hence, for any a ∈ A and s ∈ S, we have Sa∩As 6= ∅. So S is a left Ore set in A. We see that S is a right Ore set by applying the same argument to Aop = Rop[x; τ−1,−δτ−1]. � Suppose also that the derivation δ extends to an iterative, locally nilpotent higher q- skew τ -derivation {di} on R and that q 6= 1. Denote  = AS−1 = S−1A, the localization of A with respect to S, and define a map f : R −→  by f(r) = n(n+1) 2 (q − 1)−ndnτ−n(r)x−n, noting that {di} is locally nilpotent and that q − 1 is invertible. If q is not a root of unity and {di} is obtained from a q-skew τ -derivation δ as in (7), the formula for f can be rewritten as f(r) = n(n+1) (q − 1)−n (n)!q δnτ−n(r)x−n. The rewritten formula matches the one presented in [5, Section 2] when q is replaced by q−1 to account for the difference between δτ = qτδ (used here) and τδ = qδτ (used in [5]). We will show that f is a homomorphism and that the the multiplication in imf is made simpler than that in A by removing the derivation, as seen in the following. Proposition 3.2. If r ∈ R, then xf(r) = f x in Â. Proof. Using the hypothesis that {di} is iterative, we compute that xf(r) = n(n+1) 2 (q − 1)−nxdnτ−n(r)x−n n(n+1) 2 (q − 1)−n −n(r)x+ d1dnτ −n(r) n(n+1) 2 (q − 1)−nq−ndnτ−n+1(r)x−n+1 n(n+1) 2 (q − 1)−n(n+ 1)qdn+1τ−n(r)x−n n(n+1) 2 (q − 1)−nq−ndnτ−n(τ(r))x−n+1 n(n−1) 2 (q − 1)−n+1(n)qdnτ−n(τ(r))x−n+1 PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 11 = τ(r)x n(n+1) 2 (q − 1)−nq−n + q n(n−1) 2 (q − 1)−n+1(n)q −n(τ(r))x−n+1 = τ(r)x (q − 1)−n 2 + q 2 (qn − 1) −n(τ(r))x−n+1 = τ(r)x+ (q − 1)−nq n(n+1) 2 dnτ −n(τ(r))x−n+1 n(n+1) 2 (q − 1)−ndnτ−n(τ(r))x−n x = f which gives the result. � From Proposition 3.2, it follows by routine induction that xmf(r) = f τm(r) xm ∀m ∈ Z. (8) This is what we need in order to show that our map is indeed a k-algebra homomor- phism. Proposition 3.3. The map f : R −→  is a k-algebra homomorphism. Proof. It is immediate that f is k-linear (τ and {di} are k-linear), and that f(1) = 1. We’ll show that f is multiplicative. If r, s ∈ R, then using Prop. 3.2, f(r)f(s) = i(i+1) 2 (q − 1)−idiτ−i(r)x−if(s) i(i+1) 2 (q − 1)−idiτ−i(r)f(τ−i(s))x−i i≥0, j≥0 i(i+1)+j(j+1) 2 (q − 1)−(i+j)diτ−i(r)djτ−(i+j)(s)x−(i+j). For n ∈ N, the coefficient of x−n in the sum above is i≥0, j≥0, i+j=n i(i+1)+j(j+1) 2 (q − 1)−ndiτ−i(r)djτ−n(s) (n−p)2+p2+n 2 (q − 1)−ndn−pτ p−n(r)dpτ−n(s) 12 HEIDI HAYNAL n(n+1) 2 (q − 1)−n qp(p−n)dn−pτ pτ−n(r)dpτ −n(s) n(n+1) 2 (q − 1)−n τ pdn−p(τ −n(r))dp(τ −n(s)) n(n+1) 2 (q − 1)−ndn(τ−n(r)τ−n(s)) n(n+1) 2 (q − 1)−ndnτ−n(rs), computed by putting p = j and using the second condition in the Definition 2.2. In summary, f(r)f(s) = n=0 q n(n+1) 2 (q − 1)−ndnτ−n(rs)x−n = f(rs). � Proposition 3.4. (1) The map f extends uniquely to an algebra homomorphism, also denoted f , of R[y; τ ] to  satisfying f(y)=x. (2) The extended homomorphism is injective. Proof. (1) This result follows from Proposition 3.2 and the universal property of Ore extensions. (2) Let P = pmy m + · · ·+ p1y + p0 be a nonzero element of R[y; τ ], where each pi ∈ R, m ≥ 0, pm 6= 0. Then f(P ) = f(pm)xm + · · ·+ f(p1)x+ f(p0). Since f(pi) = n(n+1) 2 (q − 1)−ndnτ−n(pi)x−n ∈ AS−1, we know that there exists an integer l ≥ 0 such that each f(pi)xl is a nonzero element of A of positive degree l (in x) whenever pi 6= 0. (Because {di} is locally nilpotent, we may choose an l large enough.) It follows that f(P )xl is a nonzero element of  of degree m+ l, hence f(P ) 6= 0. � Definition 3.5. The algebra homomorphism f : R[y; τ ] −→  = AS−1 is called the derivation removing homomorphism. The image of f , call it A′, is the subalgebra of  = AS−1 generated by x and f(R), and is isomorphic (as an algebra) to R[y; τ ] by the derivation removing homomorphism f . Observe that A′ contains the multiplicative system S = {xn | n ∈ N ∪ {0}}. Since equation (8) holds and f(y) = x, the elements of this set are normal in A′. Hence, S satisfies the (two-sided) Ore condition in A′. The elements of S are regular in A′ because they are regular in Â, and thus: Proposition 3.6. A′S−1 = AS−1 Proof. We have A′S−1 ⊆ AS−1 because A′ = im(f) ⊆ AS−1. To show the other inclusion, it suffices to show that R ⊆ A′S−1. (This suffices because A is built up from PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 13 R by x, x2, . . . . So if R ⊆ A′S−1, then AS−1 ⊆ A′S−1.) Consider any r ∈ R and let ℓ be the d-nilpotence index of r. We show that r ∈ A′S−1 with an induction argument on ℓ. If ℓ ≤ 1, then d1(r) = 0, whence f(r) = r ∈ A′ ⊆ AS−1. If ℓ ≥ 2, we write f(r) = r + −n, with rn = q n(n+1) 2 (q − 1)−ndnτ−n(r) ∈ R. We’ll show that n=1 rnx −n ∈ A′S−1 in order to conclude that r ∈ A′S−1, because f(r) − n=1 rnx −n = r. That is, we need to show that each rn ∈ A′S−1. Suppose, inductively, that for any element r̃ ∈ R with d-nilpotence index m such that m < ℓ, we have r̃ ∈ A′S−1. Note that for n ∈ {1, . . . , ℓ}, we have dℓ−n(rn) = q n(n+1) 2 (q − 1)−n −n(r) = q n(n+1) 2 (q − 1)−n q−nℓτ−ndℓ(r) = 0 because dℓ(r) = 0 by hypothesis. Hence, by the induction hypothesis, each rn ∈ A′S−1 for 1 ≤ n ≤ ℓ− 1. It follows that r = f(r)− n=1 rnx −n also belongs to A′S−1. � This equality of quotient rings reveals that if A is a PI ring, then PIdegA = PIdegA′ = PIdegR[y; τ ], with the second equality arising from the derivation removing homomorphism f . This recovers the result of Jøndrup [21] without the assumption that k has characteristic zero. We summarize the results of this section in the following theorem. Theorem 3.7. Let k be a field, R a k-algebra and A = R[x; τ, δ] a q-skew polynomial ring in which δ extends to a locally nilpotent, iterative h.q-s.τ -d. {di} on R for some q ∈ k×, q 6= 1. Let S be the Ore set in A generated by x, and define a map f : R −→ AS−1 by f(r) = n=0 q n(n+1) 2 (q − 1)−ndnτ−n(r)x−n. Then f is a k-algebra homomorphism, and it extends to an injective homomorphism f : R[y; τ ] −→ AS−1 sending y to x. Furthermore, the extension f : R[y±1; τ ] −→ AS−1 is an isomorphism. So there is PI degree parity between A and R[y; τ ]. Moreover, if R is a noetherian domain, then FractA ∼= FractR[y; τ ]. 14 HEIDI HAYNAL 4. Main Theorem In the case where A is an iterated skew polynomial ring, we would like to apply re- peatedly the method presented above to remove all of the derivations and compare the resulting Ore localizations. We must first establish some facts about the behavior of h.q-s.τ -d. when the variables adjoined to the coefficient ring are rearranged, and about iterated localization. The results of these lemmas will ensure that after the induction step in the proof of the main theorem we are left with a ring to which the method of the preceding section applies. The first parts of the following lemmas hold in a broader class of skew polynomial rings and also when the q-skew condition is imposed. The final parts assert that h.q-s.τ -d. are preserved when rearranging of the variables is permissible. Lemma 4.1. Let S = R[x; τ, δ], A = R[x; τ, δ][y; σ], and  = R[x; τ, δ][y±1; σ], where σ(R) = R and σ(x) = λx for some λ ∈ k×. (1) Then A = R[y; σ′][x; τ ′; δ′], and  = R[y±1; σ′][x; τ ′; δ′], where σ′ = σ , τ ′ = τ , = δ, τ ′(y) = λ−1y, and δ′(y) = 0. (2) If (τ, δ) is q-skew, then so is (τ ′, δ′). (3) Suppose further that δ extends to a h.q-s.τ -d. {di} on R, and that σdi = λidiσ for all i. Then the τ ′-derivation δ′ extends to a h.q-s.τ ′-d. {d′i} on R[y±1; σ′] such that the restrictions of the d′i to R coincide with di, and d i(y) = 0 for all i ≥ 1. Moreover, {d′i} restricts to a h.q-s.τ ′-d. on R[y; σ′]. (a) If {di} is iterative, then {d′i} is iterative. (b) If {di} is locally nilpotent, then {d′i} is locally nilpotent. Proof. (1) Routine details omitted so as not to try the patience of the reader. (2) Suppose that (τ, δ) is q-skew on R. We’ll check that the two τ ′-derivations τ ′−1δ′τ ′ and qδ′ agree on R[y±1; σ′]. It suffices to check their agreement on a set of generators, R ∪ {y, y−1}. It is clear that τ ′−1δ′τ ′(r) = qδ′(r) for all r ∈ R. Since δ′(y) = 0, they agree on {y, y−1} as well. So (τ ′, δ′) is q-skew. (3) Define a sequence of maps d′i : R[y ±1; σ′] → R[y±1; σ′] by di(rj)y Clearly these are k-linear maps, d′i(r) = di(r) for all r ∈ R; also d′i(y) = di(1)y = 0 for i ≥ 1, and d′0 is the identity on R[y±1; σ′]. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 15 Because δ extends to {di} on R, we get d1(rj)y δ(rj)y j = δ′ for all rj ∈ R. So d′1 = δ′ on R[y±1; σ′]. Now, for integers j,m, n, and elements r, s ∈ R, (ryj)(sym) = d′n rσj(s)yj+m rσj(s) τn−idi(r)dn−i(σ j(s))yj+m τn−idi(r)y jσ−jdn−i(σ j(s))ym τn−idi(r)y jλ−j(n−i)dn−i(s)y (τ ′)n−i di(r)y d′n−i(sy (τ ′)n−id′i(ry j)d′n−i(sy So {d′i} satisfies the product rule for a higher τ -derivation on R[y±1; σ′]. Furthermore, τ ′d′i = τ ′ di(rj)y τdi(rj)λ −jyj, and d′iτ = d′i τ(rj)λ diτ(rj)λ τdi(rj)λ −jyj, giving the q-skew relation d′iτ ′ = qiτ ′d′i on R[y ±1; σ′]. It follows directly from the definition of the maps {di} that their restrictions to the k-subalgebra R[y; σ′] also exhibit the properties of definition 2.2. 16 HEIDI HAYNAL If {di} is iterative on R, then d′ℓd′i(rym) = d′ℓ di(r)y = dℓdi(r)y dℓ+i(r)y d′ℓ+i(ry m) for all r ∈ R, m ∈ Z, and non-negative integers ℓ, i. Hence, {d′i} is iterative on R[y±1; σ′]. Suppose that {di} is locally nilpotent on R. By Lemma 2.7 we need only check that {d′i} is locally nilpotent on R ∪ {y, y−1}, a set of generators for R[y±1; σ′]. This is clear because d′i(r) = di(r) for all r ∈ R, and d′i(y) = 0 for all i by construction. � Lemma 4.2. Let A = R[x1; τ1, δ1][x2; τ2, δ2] · · · [xn; τn, δn][y; σ],  = R[x1; τ1, δ1][x2; τ2, δ2] · · · [xn; τn, δn][y±1; σ], where σ(R) = R, and for all i ∈ {1, . . . , n}, σ(xi) = λixi for some nonzero λi ∈ k. Let Aj = R[x1; τ1; δ1][x2; τ2, δ2] · · · [xj ; τj, δj ] for j = 1, 2, . . . , n, and A0 = R. (1) Then A = R[y; σ∗][x1; τ 1][x2; τ 2] · · · [xn; τ ′n, δ′n],  = R[y±1; σ∗][x1; τ 1][x2; τ 2] · · · [xn; τ ′n, δ′n], where σ∗ = σ , τ ′i = τi, δ = δi, τ i(y) = λ i y, and δ i(y) = 0 for all 1 ≤ i ≤ n and j ≤ i− 1. (2) If (τi, δi) is qi-skew for any 1 ≤ i ≤ n, then (τ ′i , δ′i) is also qi-skew. (3) Suppose that each δi extends to an h.qi-s.τi-d. {di,p}∞p=0, and that σdi,p = λ i di,pσ on Ai−1 for all i and p. Then each δ i extends to a h.qi-s.τ i -d. {d′i,p}∞p=0 on the algebra R〈y, y−1, x1, . . . , xi−1〉, where d′i,p coincides with di,p on Aj, for j < i, and d′i,p(y) = 0 for p ≥ 1. Moreover, {d′i,p} restricts to a h.qi-s.τ ′i -d. on R〈y, x1, . . . , xi−1〉. (a) If {di,p} is iterative for any 1 ≤ i ≤ n, then {d′i,p} is iterative. (b) If {di,p} is locally nilpotent for any 1 ≤ i ≤ n, then {d′i,p} is locally nilpotent. Proof. (1) The condition σ(xi) = λixi for all i implies that σ(Ai) = Ai. We will use induction on n to prove the result. Lemma 4.1 proves the case n = 1. Suppose the result holds for all m < n, and consider A = An−1[xn; τn, δn][y; σ]. Application of Lemma 4.1, and then the induction hypothesis, gives A = An−1[xn; τn, δn][y; σ] = An−1[y; σ ′][xn; τ = R[x1; τ1, δ1] · · · [xn−1; τn−1, δn−1][y; σ′][xn; τ ′n, δ′n] = R[y; σ∗][x1; τ 1] · · · [xn; τ ′n, δ′n], PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 17 with the desired conditions met by the automorphisms and derivations, completing the induction. Similarly,  = R[y±1; σ∗][x1; τ 1] · · · [xn; τ ′n, δ′n]. (2) Consider the two τ ′i -derivations τ i and qiδ i on the ring R[y±1; σ∗][x1; τ 1] · · · [xi−1; τ ′i−1, δ′i−1] for 1 ≤ i ≤ n. Since (τi, δi) is q-skew, it is clear that these two τ ′i derivations agree on Ai−1. And since δ i(y) = 0 for all i = 1, . . . , n, these two τ i -derivations agree on a full set of generators of R[y±1; σ∗][x1; τ 1] · · · [xi−1; τ ′i−1, δ′i−1]. Hence, δ′iτ ′i = qiτ ′iδ′i. (3) Suppose the result holds for the algebra R[x1; τ1, δ1] · · · [xn−1; τn−1, δn−1][y±1; σ]. Then Lemma 4.1 may be applied, with An−1 providing the coefficients, to get An−1[xn; τn, δn][y ±1; σ] = An−1[y ±1; σ′][xn; τ where δ′n extends to a h.qn-s.τ n-d. {d′n,p} on An−1[y±1]. The induction hypothesis gives the result. � Definition 4.3. For a k-algebra A and a, b ∈ A, we say that a and b scalar commute if there is an element α ∈ k× such that ab = αba. We may also say that a and b α-commute. In the following two lemmas, we let D denote the division ring of fractions for the noetherian domain A. When comparing localizations of A, we identify them as subrings of D. Lemma 4.4. Let A be a noetherian domain, S ⊆ A \ {0} an Ore set. Let T be an Ore set in AS−1 \ {0} with S ⊆ T . (1) Then there exists an Ore set T̃ ⊆ A\{0} with S ⊆ T̃ such that AT̃−1 = (AS−1)T−1. (2) Suppose A is a k-algebra and S is generated by s1, . . . , sn satisfying sisj = γijsjsi for all i, j and some γij ∈ k×. Further suppose that T is generated by S ∪ t for some t ∈ AS−1 that satisfies sit = λitsi for all i and some λi ∈ k×. Then there exist a cyclic Ore set T̂ ⊆ A \ {0} and an (n + 1)-generator Ore set Ŝ ⊆ A \ {0} such that S ⊆ Ŝ, and (AS−1)T−1 = AT̂−1 = AŜ−1. Proof. (1) Consider T ∩A, the subset in T of elements with a denominator of 1. Clearly, this is a multiplicative set in A which contains S. Set T̃ = T ∩A. Let a ∈ T̃ and α ∈ A. Then a ∈ T , and since α ∈ AS−1, there exist b′ ∈ T and β ′ ∈ AS−1 such that aβ ′ = αb′. By [16, 10.2], there exist y ∈ S, and b, β ∈ A such that β ′ = βy−1 and b′ = by−1; hence, aβy−1 = αby−1 in AS−1. It follows that aβ = αb in A. So T̃ satisfies the right Ore condition in A, and the left Ore condition by symmetry. By the universal property, AT̃−1 ∼= (AS−1)T−1. As subrings of D, we have AT̃−1 = (AS−1)T−1. 18 HEIDI HAYNAL (2) The generating element t has the form t = ā(sm11 s 2 · · · smnn )−1 for some mi ∈ N, and ā ∈ A. For any si ∈ S, we have siā(s 2 · · · smnn )−1 = λiā(sm11 sm22 · · · smnn )−1si = µλiāsi(sm11 sm22 · · · smnn )−1, where µ is a product of powers of the γij. So ā scalar commutes with the genera- tors of S via the relations siā = µλiāsi. Let Ŝ be the multiplicative set generated by ā, s1, . . . , sn in A, and T̂ the multiplicative set generated by ās1s2 · · · sn in A. Recall that (AS−1)T−1 = AT̃−1, where T̃ = T ∩ A from part (1). From the scalar com- muting relations it follows that any element at̃−1 ∈ AT̃−1 may be written in the form b(ās1, · · · sn)−m for some m ∈ N ∪ {0}, b ∈ A, or the form cā−ℓn+1s−ℓ11 · · · s−ℓnn , for ℓj ∈ N ∪ {0}, c ∈ A. So we conclude that Ŝ and T̂ are Ore sets in A and that (AS−1)T−1 = AT̂−1 = AŜ−1. � Lemma 4.5. Let A be a noetherian domain, S1 ⊆ A \ {0} an Ore set, and for integers j = 2, . . . , n let Sj be an Ore set in ((AS 1 ) · · · )S−1j−1 \ {0} with Sj−1 ⊆ Sj. (1) Then there exists an Ore set T ⊆ A \ {0} such that AT−1 = (((AS−11 )S−12 ) · · · )S−1n . (2) Suppose A is a k-algebra, S1 is generated by s1, and for j = 2, . . . , n, Sj is generated by Sj−1 ∪ {sj}, where sisj = γijsjsi for some multiplicatively antisymmetric matrix (γij) ∈ Mn(k×). Then there are a cyclic Ore set T̂ ⊆ A and an n-generator Ore set Ŝ ⊆ A such that S1 ⊆ Ŝ, and ((AS−11 )S−12 ) · · ·S−1n = AT̂−1 = AŜ−1. Proof. (1) The proof proceeds by induction on n. The case n = 1 is covered in the lemma above. Suppose that for all j ≤ n− 1 there exists an Ore set Tj ⊆ A \ {0} such that AT−1j = (((AS 2 ) · · · )S−1j . Then the equality AT−1n−1 = (((AS 2 ) · · · )S−1n−1 identifies an Ore set Tn ⊆ AT−1n−1 \ {0} such that (AT−1n−1)T n = (((AS 2 ) · · ·S−1n−1)S−1n . Furthermore, Lemma 4.4 implies the existence of an Ore set T ⊆ A \ {0} such that AT−1 = (AT−1n−1)T n = (((AS 2 ) · · ·S−1n−1)S−1n . (2) Suppose, inductively, that there exist (i) a cyclic Ore set T̂n−1 ⊆ A \ {0} generated by s1ā2 · · · ān−1 (ii) an (n− 1)-generator Ore set Ŝn−1 ⊆ A \ {0} with S1 ⊆ Ŝn−1 and generators s1, ā2, ā3, . . . , ān−1 (iii) the āi scalar commute with s1 and with each other (iv) ((AS−11 )S 2 ) · · ·S−1n−1 = AT̂−1n−1 = AŜ−1n−1 as subrings of D. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 19 Then sn = ān(s1ā2 · · · ān−1)−r for some ān ∈ A and r ∈ N. Using the relations sisj = γijsjsi, routine calculations show that the āi scalar commute with the sj , and also with each other, for all i, j. Let T̂ be the multiplicative set generated by s1ā2 · · · ān, and let Ŝ be the multiplicative set generated by s1, ā2, ā3, . . . , ān. Then ((AS−11 )S 2 ) · · ·S−1n = (AT̂−1n−1)S−1n = AT−1 from part (1). Using Lemma 4.4, we con- clude that T̂ and Ŝ are Ore sets in A and that AT−1 = AT̂−1 = AŜ−1. � In the proof of the main theorem, we will use without mention the facts gathered here. For greater details on these statements, see [16, 10X, 10Y] and [10, 1.4]. (1) Given a noetherian ring A and a normal element x ∈ A, the multiplicative set generated by x is an Ore set. (2) The multiplicative set generated by a nonempty family of right Ore sets is right (3) Let A = R[x; τ, δ], and S a right denominator set in R such that τ(S) = S. Then S is a right denominator set in A and the identity map on AS−1 extends to an isomorphism of AS−1 onto (RS−1)[x; τ, δ] sending x1−1 to x. Note that if A is a k-algebra, τ , δ are k-linear, and τ(k×S) = k×S, then the result holds because S is a denominator set if and only if k×S is a denominator set. Theorem 4.6. Let R be a k-algebra and noetherian domain, A = R[x1; τ1, δ1] · · · [xn; τn, δn], where each τi is a k-linear automorphism of R〈xi, . . . , xi−1〉 such that τi(xj) = λijxj for all i, j with 1 ≤ j < i ≤ n and some λij ∈ k×, and where each δi is a k-linear τi- derivation. Assume that there exist elements qi ∈ k× with qi 6= 1 such that δiτi = qiτiδi, and that δi extends to a locally nilpotent, iterative h.qi-s.τi-d. on R〈xi, . . . , xi−1〉 for i = 1, . . . , n. (1) Then there exists an Ore set T ⊆ A generated by n elements of A such that AT−1 ∼= R[y±11 ; τ1][y±12 ; τ ′2] · · · [y±1n ; τ ′n] where τ ′i |R = τi and τ ′i(yj) = λijyj for all i, j with 1 ≤ j < i ≤ n (2) There is PI degree parity between A and R[y1; τ1][y2; τ 2] · · · [yn; τ ′n]. Moreover, these algebras have isomorphic division rings of fractions. Proof. (a) Suppose, inductively, that we have R[x1; τ1, δ1][y 2 ; τ2] · · · [y±1n ; τ ′n] ∼= AS−12 where the restriction of τ ′i to R〈x1〉 coincides with τi, τ ′i(ym) = λimym for 2 ≤ i ≤ n and 1 < m < i, and S2 is an Ore set in A generated by n − 1 elements from A. Then by 20 HEIDI HAYNAL Lemma 4.2 AS−12 ∼= R[y±12 ; τ ′′2 ] · · · [y±1n ; τ ′′n ][x1; τ ′1, δ′1] (9) where the restrictions of τ ′1 and δ 1 to R coincide with τ1 and δ1, τ 1(yj) = λ j1 yj, δ 1(yj) = 0, and τ ′′i coincides with the restriction of τi to R〈y2, . . . , yi−1〉 for 2 ≤ i ≤ n. Observe that by Lemmas 4.2 and 2.7 we also have δ′1τ 1 = q1τ 1, and that δ 1 extends to a locally nilpotent iterative h.q1-s.τ -d. on R〈y±12 , . . . , y±1n 〉. Then applying the derivation removing homomorphism to the right hand side of (9) gives an isomorphism (AS−12 )T ∼= R[y±12 ; τ ′2] · · · [y±1n ; τ ′n][y±11 ; τ ′1] where T1 ⊆ AS−12 is an Ore set generated by one element of AS−12 . Then Lemma 4.5 and a reordering of variables shows the existence of an Ore set T ⊆ A, generated by n elements of A, such that AT−1 ∼= R[y±11 ; τ1][y±12 ; τ ′2] · · · [y±1n ; τ ′n]. (2) This follows from part (1). � Corollary 4.7. Let A = k[x1; τ1, δ1] · · · [xn; τn, δn] with the hypotheses as in Theorem 4.6. Set λ = (λij). Then (1) A and Oλ(kn) have isomorphic division rings of fractions. (2) A is a PI-algebra if and only if all the λij are roots of unity, in which case A and Oλ(kn) have the same PI degree. In general, identification of the generators for the Ore set T in Theorem 4.6 is very cumbersome. To illustrate the computations on a fairly short iterated skew polynomial ring, we consider the multiparameter second quantized Weyl algebra A 2 (k). Here, Q = (q1, q2) ∈ (k×)2, qi 6= 1 for all i, and Γ = (γij) ∈ M2(k×) with γii = 1 and γ21 = γ 12 . The algebra A 2 (k) may be presented as an iterated skew polynomial ring of the form k[y1][x1; τ2, δ2][y2; τ3][x2; τ4, δ4], where the τi are k-linear automorphisms and the δ2i are k-linear τ2i-derivations such that τ2(y1) = q1y1, δ2(y1) = 1 τ3(y1) = γ 12 y1 τ3(x1) = γ12x1 τ4(y1) = q1γ12y1, δ4(y1) = 0 τ4(x1) = q 1 γ21x1, δ4(x1) = 0 τ4(y2) = q2y2, δ4(y2) = (q1 − 1)y1x1 + 1. For greater detail about this algebra, the reader is referred to [1], [23], [12], and [15]. Routine computations show that the pair (τ2, δ2) is a q1-skew derivation and that (τ4, δ4) is a q2-skew derivation. To show that δ2 and δ4 are locally nilpotent, it suffices to check PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 21 for local nilpotence on a set of generators. Given their definitions, this is accomplished by verifying their action on powers of y1 and y2: δi2(y 1 ) = (n)!q1 (n−i)!q1 yn−i1 i ≤ n 0 i > n δi4(y 2 ) = (n)!q2 (n−i)!q2 [δ4(y2)] iyn−i2 i ≤ n 0 i > n Using Theorem 2.8 we have a h.q1-s.τ2-d. {d2,i} extending δ2, and a h.q2-s.τ4-d. {d4,i} extending δ4, both of which are iterative and locally nilpotent. Let S2 ⊆ AQ,Γ2 (k) be the multiplicative set generated by x2. The derivation removing homomorphism induces an isomorphism Φ : k[y1][x1; τ2, δ2][y2; τ3][z 2 ; τ4] −→ A 2 (k)S whose action on generators is given by y1 7→ y1 x1 7→ x1 z2 7→ x2 y2 7→ y2 + (q2 − 1)−1 (q1 − 1)y1x1 + 1 x−12 . For simplicity, label the domain of Φ as BZ−1. Let X1 ⊆ BZ−1 be the Ore set generated by z2 and x1. Applying the derivation removing homomorphism to BZ −1 induces an isomorphism Ψ : k[y1][z 1 ; τ2][y2; τ3][z 2 ; τ 4] −→ (BZ−1)X−11 whose action on generators is given by z1 7→ z1 z2 7→ z2 y2 7→ y2 y1 7→ y1 + (q1 − 1)−1x−11 . The derivation removing homomorphism need not be employed again to achieve the result. Through iterated localization we find that there is an Ore set T ⊆ AQ,Γ2 (k) such 2 (k)T −1 ∼= k[y±11 ][x±11 ; τ2][y±12 ; τ3][x±12 ; τ4] and T is generated by the four elements x2, x1, y2x2(q2 − 1) + y1x1(q1 − 1) + 1, and y1x1(q1 − 1) + 1. Note that we recover the result of [22, Theorem 5]. 22 HEIDI HAYNAL 5. Examples We will demonstrate how each of the following k-algebras satisfies all the conditions of Theorem 2.8. Then Corollary 4.7 is applied to obtain an isomorphism of quotient division rings (thereby confirming the quantum Gel’fand-Kirillov conjecture) and PI degree parity with a multiparameter quantum affine space. When calculating the PI degree of a quantum affine space, we encounter an antisymmetric, or skew-symmetric, integral matrix. As proved in [30, Theorem IV.1], such a matrix is congruent to a matrix in skew normal form. Theorem 5.1. [Newman] Let A be a skew-symmetric matrix of rank r which belongs to Mn(R), where the commutative principal ideal domain R is not of characteristic 2. Then r = 2s and A is congruent to a matrix in block diagonal form  −h1 0 0 −h2 0 . . . 0 −hs 0  where hi | hi+1, 1 ≤ i ≤ s− 1. The same result, in the language of alternating bilinear forms, can be found in [3, Section 5.1]. The matrix S in Theorem 5.1 is clearly equivalent to the more familiar Smith normal form, diag(h1, h1, h2, h2, . . . , hs, hs, 0, 0, . . . , 0), where the diagonal entries are the in- variant factors of the matrix A. In the examples that follow, we outline the operations necessary to obtain the Smith normal form. Definition 5.2. Let A = k[x1; τ1, δ1] · · · [xn; τn, δn] and A′ = k[x1; τ1] · · · [xn; τn] be iterated skew polynomial rings. (1) If there exists Q = (q1, . . . , qn) ∈ (k×)n such that δiτi = qiτiδi for i = 1, . . . , n, then A is called an iterated Q-skew polynomial ring. (2) If there exist λji ∈ k× such that τj(xi) = λjixi for all i < j, then set λij = λ−1ji and λii = 1 for all i. We call Λ = (λij) ∈ Mn(k×) the matrix of relations for A′. Lemma 5.3. Let C be a commutative k-algebra, A a C-algebra, B ⊆ A a C-subalgebra generated by {b1, b2, . . . }. Let τ be a C-algebra automorphism of A, and δ a u-skew τ -derivation on A for some unit u ∈ C. If τ(bj) ∈ B and δn(bj) ∈ (n)!uB for all j, n, then δn(B) ⊆ (n)!uB for all n. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 23 Proof. Note that τ(bj) ∈ B for all j implies that τ(B) ⊆ B and hence we have (j)!uB ⊆ (j)!uB for all j. Suppose that for integers m ≥ 1 and 1 ≤ ℓ ≤ m − 1, we have δi(bj1 · · · bjℓ) ∈ (i)!uB for all i, and all choices of j1, . . . , jℓ. Then δn(bj1 · · · bjm) = τn−iδi(bj1 · · · bj(m−1))δ n−i(bjm) (i)!u(n− i)!uB ⊆ (n)!uB for all n and all j1, . . . , jm by induction. � For a first family of examples, we take odd-dimensional quantum Euclidean spaces. The even-dimensional ones will be covered in Example 5.4. 5.1. The coordinate ring of odd-dimensional quantum Euclidean space; Oq(ok2n+1). For q ∈ k×, assuming q has a (fixed) square root q1/2 ∈ k, the k-algebra Oq(ok2n+1) may be presented as an iterated skew polynomial ring k[w][y1; σ1][x1; τ1, δ1] · · · [yn; σn][xn; τn, δn] with automorphisms σi, τi and derivations δi defined by σi(w) = q −1w all i τi(w) = qw all i σi(yj) = q −1yj j < i σi(xj) = q −1xj j < i τi(yj) = qyj i 6= j τi(xj) = qxj j < i τi(yi) = yi all i δi(w) = δi(xj) = δi(yj) = 0 j < i δi(yi) = (q 1/2 − q3/2)w2 + (1− q2) yℓxℓ all i. Quantum Euclidean spaces have been studied since 1990 when they were introduced by Reshetikhin et al. in [36]. The three-dimensional case has applications to the structure of space-time at small distances. Musson simplified the original set of relations in [29], and Oh further simplified them, renaming the generators ω, xi, yi in [31]. Here, we have made a change to Oh’s variables, yi 7→ qiyi, to obtain the relations in our presentation of Oq(ok2n+1). Routine computations show that τ−1i δiτi(yi) = q −2δi(yi) for all i, and so we conclude that each (τi, δi) is a q −2-skew derivation. We may present the analogous k[t±1]-algebra 24 HEIDI HAYNAL Ot(ok[t±1]2n+1) as an iterated skew polynomial ring with coefficient ring k[t±1] and generators w, yi, xi for i = 1, . . . , n, k[t±1][w][y1; σ̄1][x1; τ̄1, δ̄1] · · · [yn; σ̄n][xn; τ̄n, δ̄n] where the automorphisms and derivations are defined analogously to those of the algebra Oq(ok2n+1) with t ∈ k[t±1] replacing q ∈ k×. So each (τ̄i, δ̄i) is a t−2-skew derivation. It is immediate that Ot(ok[t±1]2n+1)/〈t− q〉 ∼= Oq(ok2n+1) with each τ̄i and δ̄i reducing to τi and δi respectively. Let Aj denote the k[t ±1]-subalgebra generated by w, ym, xm for m < j, and yj. To show that δ̄ij(Aj) ⊆ (i)!t−2Aj, we apply Lemma 5.3 noting that δ̄ij(yj) has been given for i = 1 and is zero for i > 1. So, by Theorem 2.8, each δi in our presentation of Oq(ok2n+1) extends to an iterative, locally nilpotent h.q−2-s.τi-d. on an appropriate subalgebra. Then Corollary 4.7 gives FractOq(ok2n+1) ∼= FractOB(k2n+1), where the matrix of relations is  1 q q−1 q q−1 · · · q q−1 q−1 1 1 q q−1 · · · q q−1 q 1 1 q q−1 · · · q q−1 q−1 q−1 q−1 1 1 · · · q q−1 q q q 1 1 · · · q q−1 . . . q−1 q−1 q−1 q−1 q−1 · · · 1 1 q q q q q · · · 1 1  If q ∈ k× is a root of unity, we may assume without loss of generality that it is a primitive rth root of unity. Then the powers of q from the matrix B become the entries of a (2n+ 1)× (2n+ 1) integer matrix  0 1 −1 1 −1 · · · 1 −1 −1 0 0 1 −1 · · · 1 −1 1 0 0 1 −1 · · · 1 −1 −1 −1 −1 0 0 · · · 1 −1 1 1 1 0 0 · · · 1 −1 . . . −1 −1 −1 −1 −1 · · · 0 0 1 1 1 1 1 · · · 0 0  Now, PIdegOq(ok2n+1) can be computed from Theorem 1.2(2) using the matrix B′. The cardinality of the image will not be changed if we first perform some row reductions on B′. Letting N = 2n+ 1, n > 2, we manipulate the rows as follows. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 25 • For i = 2, 4, 6, . . . , N − 1, replace row i with row i + row (i+ 1). • For i = N,N − 2, N − 4, . . . , 5, replace row i with row i − row (i− 2). • Replace row 5 with row 5 − row 1. • For i = 2, 4, 6, . . . , N − 5, replace row i with row i − 2row (i+ 5). • Multiply the even numbered rows, except row 2n− 2, by −1. The resulting matrix has 2n pivots and one zero row. We put the rows in this order 3, 1, 5, 7, 2, 9, 4, 11, 6, 13, . . . , 2i, 2i+ 7, . . . , N,N − 5, N − 3, N − 1 to place the pivots on the main diagonal and the zero row in the last position. Then we have a matrix of this form  1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 1 −1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 4 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 1 1 ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 4 ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 . . . ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 1 1 ∗ ∗ 0 0 0 0 0 0 0 0 0 4 ∗ ∗ 0 0 0 0 0 0 0 0 0 0 2 −2 0 0 0 0 0 0 0 0 0 0 0 0  The diagonal entries of this echelon matrix do not yet reveal the size of its image because the pivot in row three does not divide all of the (suppressed) entries in its row when n ≥ 3. So more row reduction is needed. First replace row 3 with row 3 + row(4i+ 2). For n even and j = 5, 7, 9, . . . , 2n− 3, replace row j as follows: for j = 4p+ 1, p ≥ 1, use row j + i=p+1 2 · row(4i) + row(2n); for j = 4p+ 3, p ≥ 1, use row j + i=p+1 2 · row(4i+ 2). 26 HEIDI HAYNAL For n odd and j = 5, 7, 9, . . . , 2n− 5, replace row j as follows: for j = 4p+ 1, p ≥ 1, use row j + i=p+1 2 · row(4i) + 2 · row(2n); for j = 4p+ 3, p ≥ 1, use row j + i=p+1 2 · row(4i+ 2) + row(2n). Then add row(2n) to row(2n − 3), and add 2·row(2n) to row(2n − 1). For integers 4 ≤ j ≤ 2n − 1, with j 6≡ 2(mod 4), add (−1)jcol 3 to col j. Subtract col(2n + 1) from col 3; add row 3 to row(2n − 2); and subtract 2·row 3 from row(2n). The result is an upper echelon matrix in which each pivot divides all the nonzero entries in its row. So it is trivial to diagonalize by column operations. The Smith normal form for n odd is diag(1, 1, . . . , 1, 4, 4, . . . , 4, 0) with n+1 ones and n− 1 fours. The Smith normal form for n even is diag(1, 1, . . . , 1, 2, 2, 4, 4, . . . , 4, 0) with n ones, two twos, and n − 2 fours. For the cases n = 1, 2, the row-reduced matrices are, respectively, 1 0 0 0 1 −1 0 0 0  1 0 0 1 −1 0 1 −1 1 −1 0 0 2 −2 2 0 0 0 2 −2 0 0 0 0 0  Hence we have, for all n > 0, PIdegOq(ok2n+1) = rn, r odd rn/2⌊ ⌋, r even, r /∈ 4Z rn/2n−1, r ∈ 4Z 5.2. The multiparameter quantized Weyl algebras; AQ,Γn (k). For a fixed n-tuple Q = (q1, . . . , qn) ∈ (k×)n and Γ = (γij) a multiplicatively antisymmetric n × n matrix over k, the algebra AQ,Γn (k), studied in [23] and [26], may be presented as an iterated skew polynomial ring k[y1][x1; τ1, δ1][y2; σ2][x2; τ2, δ2] · · · [yn; σn][xn; τn, δn] PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 27 where the automorphisms and derivations are defined by σi(yj) = γjiyj j < i σi(xj) = γijxj j < i τi(yj) = qjγjiyj j < i τi(xj) = q j γijxj j < i τi(yi) = qiyi all i δi(xj) = δi(yj) = 0 j < i δi(yi) = 1 + (qℓ − 1)yℓxℓ all i. Routine computations show that τ−1i δiτi(yi) = qiδi(yi) for all i, and so we conclude that each (τi, δi) is a qi-skew derivation. We may present the k[t 1 , . . . , t n ]-algebra AT,Γn (k[t 1 , . . . , t n ]) as an iterated skew polynomial ring k[t±11 , . . . , t n ][y1][x1; τ̄1, δ̄1][y2; σ̄2][x2; τ̄2, δ̄2] · · · [yn; σ̄n][xn; τ̄n, δ̄n] where the automorphisms and derivations are defined analogously to those of AQ,Γn (k) with ti ∈ k[t±11 , . . . , t±1n ] replacing qi ∈ k. So each (τ̄i, δ̄i) is a ti-skew derivation. It is immediate that AT,Γn (k[t 1 , . . . , t n ])/〈t1 − q1, . . . , tn − qn〉 ∼= AQ,Γn (k) with each τ̄i and δ̄i reducing to τi and δi respectively. Let Aj denote the k[t 1 , . . . , t n ]-subalgebra generated by ym, xm for m < j, and yj. To show that δ̄ij(Aj) ⊆ (i)!tjAj, it suffices to check δ̄ij(yj) by Lemma 5.3. But this is given by definition for i = 1 and is zero for i > 1. So, by Theorem 2.8, each δi in our presentation of AQ,Γn (k) extends to an iterative, locally nilpotent h,qi-s.τi-d. on the appropriate subalgebra. Then Corollary 4.7 gives FractAQ,Γn (k) ∼= FractOΛ(k2n), where the 2n× 2n matrix of relations Λ is comprised of 2× 2 blocks Bii = 1 q−1i , for all i; Bij = γji q i γji γij qiγij , for i < j; Bij = γji γij qjγji q j γij , for i > j. If γij and qi are roots of unity for all i, j, then OΛ(k2n) is a PI algebra. Assuming that γij is an r ij root of unity and that qi is an r i root of unity, we let r = lcm{rij, ri | i, j = 1, . . . , n}. 28 HEIDI HAYNAL Then there exists a primitive rth root of unity q ∈ k and integers bi, bij such that qi = qbi and γij = q bij for i, j = 1, . . . , n. The powers of this q from the matrix Λ give a 2n× 2n integer matrix Λ′ comprised of 2× 2 blocks B′ii = 0 −bi , for all i; B′ij = bji bji − bi bij bij + bi , for i < j; B′ij = bji bij bj + bji bij − bj , for i > j. Then PIdegAQ.Γn (k) can be computed using the matrix Λ ′ in Theorem 1.2 (2). Consider the single parameter case, denoted Aqn(k), where qi = q for all i, and γij = 1 for i < j, relegating the σi to identity maps. Assuming that q is a primitive r th root of unity, then δi(y i ) = 0 and τi(y i ) = y i for all i, implying that y i is central. The definition of the τi, along with the q-Liebnitz rule, implies that x i is central for all i. So the algebra Aqn(k) is a finitely generated module over the central subring k[y i , x 1, . . . , y n]. To find the PI degree in this case, the integer matrix becomes  0 −1 0 −1 . . . 0 −1 1 0 0 1 0 1 0 0 0 −1 0 −1 1 −1 1 0 0 1 . . . 0 0 0 0 . . . 0 −1 1 −1 1 −1 . . . 1 0  which is seen to have a trivial kernel after these row reductions: • Replace row 2n with row 2n− row (2n− 2)− row (2n− 3) • For j = n−1, n−2, . . . , 2, replace row 2j with row 2j−row (2j−2)−row (2j−3) • Rearrange the rows to order 2, 1, 4, 3, 6, 5 . . . , 2n, 2n− 1. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 29 The resulting matrix has the form  0 −1 ∗ . . . 0 1 0  thus verifying that PIdegAqn(k) = r 5.3. The multiparameter coordinate ring of quantum n×nmatrices; Oλ,p Mn(k) The multiparameter coordinate ring of quantum n×n matrices was introduced by Artin, Schelter, and Tate in [2]. The k-algebra Oλ,p Mn(k) is defined by generators xij for i, j = 1, . . . , n and relations xℓmxij = pℓipjmxijxℓm + (λ− 1)pℓiximxlj (ℓ > i, m > j) λpℓipjmxijxℓm (ℓ > i, m ≤ j) pjmxijxℓm (ℓ = i, m > j), where λ ∈ k× and p = (pij) ∈ Mn2(k×) is multiplicatively antisymmetric. It can also be presented as an iterated skew polynomial ring k[x11][x12; τ12] · · · [xij ; τij, δij ] · · · [xnn; τnn, δnn] where each τℓm and δℓm is k-linear and satisfies τℓm(xij) = pℓipjmxij when ℓ > i and m 6= j λpℓipjmxij when ℓ > i and m = j pjmxij when ℓ = i and m > j, δℓm(xij) = (λ− 1)pℓiximxℓj when ℓ > i and m > j 0 otherwise. Routine computations show τ−1ℓm δℓmτℓm(xij) = λ −1δℓm(xij) as in [9, Section 5], and so we conclude that each (τℓm, δℓm) is a λ −1-skew derivation. We may present the k[t±1]- algebra Ot,p Mn(k[t as an iterated skew polynomial ring with generators xij for i, j = 1, . . . , n k[t±1][x11][x12, τ̄12] · · · [xij ; τ̄ij, δ̄ij ] · · · [xnn; τ̄nn, δ̄nn] 30 HEIDI HAYNAL where the automorphisms and derivations are defined analogously to those of the algebra Mn(k) with t ∈ k[t±1] replacing λ ∈ k. So each (τ̄ℓm, δ̄ℓm) is a t−1-skew derivation. It is immediate that Mn(k[t /〈t− λ〉 ∼= Oλ,p Mn(k) with each τ̄ℓm and δ̄ℓm reducing to τℓm and δℓm respectively. Let A−ℓm denote the k[t ±1]-subalgebra generated by the xij with (i, j) < (ℓ,m) in the lexicographic order. Lemma 5.3 allows us to to verify that δ̄sℓm(A ℓm) ⊆ (s)!t−1(A−ℓm) by checking only that δ̄sℓm(xij) is contained in A ℓm. This is immediate from the formula for δ̄ℓm given above. Thus, by Theorem 2.8, each δℓm in our presentation of Oλ,p Mn(k) extends to an iterative, locally nilpotent h.λ−1-s.τℓm-d. on the appropriate k-subalgebra. Then Corollary 4.7 gives FractOλ,p Mn(k) ) ∼= FractOΛ(kn where the matrix of relations Λ = (bij) ∈ Mn2(k) is comprised of n× n blocks Bii =  1 p21 p31 · · · pn1 p12 1 p32 · · · pn2 p13 p23 1 · · · pn3 . . . p1n p2n p3n · · · 1  for all i, Bij =  λ−1pij pijp21 pijp31 · · · pijpn1 λ−1pijp12 λ −1pij pijp32 · · · pijpn2 λ−1pijp13 λ −1pijp23 λ −1pij · · · pijpn3 . . . λ−1pijp1n λ −1pijp2n λ −1pijp3n · · · λ−1pij  , for i < j, Bij =  λpij λpijp21 λpijp31 · · · λpijpn1 pijp12 λpij λpijp32 · · · λpijpn2 pijp13 pijp23 λpij · · · λpijpn3 . . . pijp1n pijp2n pijp3n · · · λpij  , for i > j. If λ and pij are roots of unity for all i, j, then OΛ(kn ) is a PI algebra. In this case we may assume that λ is an sth root of unity and that pij is an r ij root of unity, and let r = lcm{s, rij | i, j = 1, . . . , n}. Then there exists a primitive rth root of unity q ∈ k and integers b, bij such that λ = q b and pij = q bij . The powers of this q from the matrix PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 31 Λ provide entries for an n2 × n2 integer matrix Λ′ made up of n× n blocks B′ii =  0 b21 b31 · · · bn1 b12 0 b32 · · · bn2 b13 b23 0 · · · bn3 . . . b1n b2n b3n · · · 0  for all i, B′ij =  bij − b bij + b21 bij + b31 · · · bij + bn1 bij + b12 − b bij − b bij + b32 · · · bij + bn2 bij + b13 − b bij + b23 − b bij − b · · · bij + bn3 . . . bij + b1n − b bij + b2n − b bij + b3n − b · · · bij − b  , for i < j, B′ij =  bij + b bij + b21 + b bij + b31 + b · · · bij + bn1 + b bij + b12 bij + b bij + b32 + b · · · bij + bn2 + b bij + b13 bij + b23 bij + b · · · bij + bn3 + b . . . bij + b1n bij + b2n bij + b3n · · · bij + b  , for i > j. Then PIdegOλ,p Mn(k) can be calculated using Λ′ in Theorem 1.2 (2). The single parameter quantized coordinate ring of n × n matrices, Oq(Mn(k)), is de- fined over k analogously to Oλ,p(Mn(k)), but with relations that are recovered by setting λ = q−2 and pij = q for all i > j. When k has characteristic zero and q is a primitive m root of unity for m odd, Jakobsen and Zhang found in [20] that PIdegOq(Mn(k)) = m n(n−1) 2 by using De Concini’s and Procesi’s tool given in Theo- rem 1.2. This result is reproved in [19] using results of De Concini and Procesi and also Jøndrup’s work from [21]. Now we can recover PIdegOq(Mn(k) without the assumption that k has characteristic zero. In the single parameter case of n × n quantum matrices, the matrix that we use to calculate the PI degree is  An In In In · · · In −In An In In · · · In −In −In An In · · · In −In −In −In −In · · · An  32 HEIDI HAYNAL where  0 1 1 1 · · · 1 −1 0 1 1 · · · 1 −1 −1 0 1 · · · 1 −1 −1 −1 · · · −1 0  is n× n and In is the n× n identity matrix. For any n, the characteristic polynomial of An is the sum of the terms of degree ≡ n (mod 2) in the binomial expansion of (x+1)n, so in fact χn(x) = (x+1)n+ 1 (x− 1)n. But there is also a recursion formula for the characteristic polynomial for n ≥ 3 given χn(x) = χn−1(x)(x+ 1)− (x− 1)n−1, which will be useful in the linear algebra that follows. We will perform the following row reductions on the rows of blocks of Λ′. For ease of notation, we’ll denote the jth row of blocks as BRj , the interchange of BRi and BRj as BRi ↔ BRj , and the addition of a multiple of BRi to BRj as MBRi + BRj 7→ BRj , where M ∈ Mn(Z). • BR1 ↔ BRn. • −InBR1 7→ BR1. • For i = 2, . . . , n− 1, BR1 +BRi 7→ BRi. • BRn − AnBR1 7→ BRn. This yields the matrix  In In In In · · · −An 0 An + In 2In 2In · · · In −An 0 0 An + In 2In · · · In −An . . . 0 0 0 · · · An + In In −An 0 In −An In −An In −An · · · In + A2n  which can be reduced further by n − 2 block row operations, each of which produces one zero block in the nth row. We list the first three here along with the resulting (n, n) block. • (An + In)BRn − (In − An)BR2 7→ BRn : A3n + 3An • (An + In)BRn + (In −An)2BR3 7→ BRn : A4n + 6A2n + In • (An + In)BRn − (In − An)3BR4 7→ BRn : A5n + 10A3n + 5An PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 33 In general, the block row operations that we need to perform in order to obtain a block upper triangular matrix are: • For i = 2, . . . , n− 1, (An + In)BRn + (−1)i−1(In − An)i−1BRi 7→ BRn. These row operations are justified when m is odd because An + In is invertible in Mn(Z/mZ) in that case, as will be shown below. After applying this step to the i th row, the (n, n) block is χi+1(An). So the resulting block upper triangular matrix is  In In In In · · · −An 0 An + In 2In 2In · · · In −An 0 0 An + In 2In · · · In −An . . . 0 0 0 · · · An + In In −An 0 0 0 0 · · · χn(An)  where χn(An) is the n× n zero matrix. Each block on the diagonal is An + In =  1 1 1 1 · · · 1 −1 1 1 1 · · · 1 −1 −1 1 1 · · · 1 −1 −1 −1 −1 · · · 1  which can be row reduced just by adding row 1 to rows 2 through n to yield the matrix   1 1 1 1 · · · 1 0 2 2 2 · · · 2 0 0 2 2 · · · 2 . . . 0 0 0 0 · · · 2  In particular this shows that An + In is invertible in Mn(Z/mZ) for m odd. Hence Λ′ can be reduced through row operations to an upper triangular n2 × n2 matrix with 2n− 2 ones, (n− 1)(n− 2) twos, and n zeroes on the diagonal. Assuming that q ∈ k is a primitive mth root of unity, and recalling Theorem 1.2, the cardinality of the image in (Z/mZ)n is mn 2−n if m is odd. Thus we conclude that PIdegOqMn(k) = m n(n−1) recovering the result of Jakobsen and Zhang [20] in characteristic zero. By similar methods, one can show that PIdegOqMn(k) = m n(n−1) (n−1)(n−2) 2 when m is even. For details on this result see [20] or [17]. 5.4. The algebra K n,Γ (k), which generalizes the coordinate rings of even- dimensional quantum Euclidean space and quantum symplectic space. For P = (p1, . . . , pn) and Q = (q1, . . . , qn) in (k ×)n with pi 6= qi for all i = 1, . . . , n, and 34 HEIDI HAYNAL Γ = (γij) ∈ Mn(k×) multiplicatively antisymmetric, the k-algebra KP,Qn,Γ (k) introduced in [18] is defined by generators xi, yi for i = 1, . . . , n and relations yiyj = γijyjyi all i, j xixj = qip j γijxjxi i < j xiyj = pjγjiyjxi i < j xiyj = qjγjiyjxi i > j xiyi = qiyixi + (qℓ − pℓ)yℓxℓ all i. This algebra may be presented in the form of an iterated skew polynomial ring k[y1][x1; τ1][y2; σ2][x2; τ2, δ2] · · · [yn; σn][xn; τn, δn] where the automorphisms τi, σi and derivations δi are defined by σi(yj) = γijyj j < i σi(xj) = p i γjixj j < i τi(yj) = qjγjiyj j < i τi(xj) = q j piγijxj j < i τi(yi) = qiyi all i δi(xj) = δi(yj) = 0 j < i δi(yi) = (qℓ − pℓ)yℓxℓ all i. Routine computations show that τ−1i δiτi(yi) = qip i δi(yi) for all i, and so we conclude that each (τi, δi) is a qip i -skew derivation. For ease of notation we now shall let k = k[t±11 , . . . , t n , u 1 , . . . , u n ] with T = (t1, . . . , tn) ∈ k and U = (u1, . . . , un) ∈ k. We may present the k-algebra K n,Γ (k) as an iterated skew polynomial ring k[y1][x1; τ̄1][y2; σ̄2][x2; τ̄2, δ̄2] · · · [yn; σ̄n][xn; τ̄n, δ̄n] where the automorphisms and derivations are defined analogously to those of K n.Γ (k) with ti replacing pi and ui replacing qi. Let I ⊆ KT,Un,Γ (k) be the ideal generated by the 2n monomials ti − pi, ui − qi for i = 1, . . . , n. It is immediate that n,Γ (k)/I ∼= KP,Qn,Γ (k), with each τ̄i, δ̄i, σ̄i reducing to τi, δi, σi respectively. Let Aj denote the subalgebra of K n,Γ (k) generated by ym, xm for m < j and yj. To show that δ̄ij(Aj) ⊆ (i)!ujt−1j Aj , it suffices to check that δ̄ j(yj) is an element of (i)!ujt−1j PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 35 by Lemma 5.3. This is given for i = 1 by the formula for δ̄j and is zero for i > 1. So, by Theorem 2.8, each δi in our presentation of K n,Γ (k) extends to an iterative, locally nilpotent h.qip i -s.τi-d. on the appropriate subalgebra. Then Corollary 4.7 gives FractK n,Γ (k) ∼= FractOΛ(k2n) where the 2n× 2n matrix of relations Λ = (Bij) is comprised of 2× 2 blocks Bii = 1 q−1i , for all i; Bij = γij q i γji pjγji qip j γij , for i < j; Bij = γij p i γij qjγji q j piγij , for i > j. If the qi, pi and γi are all roots of unity, then OΛ(k2n) is a PI algebra. Suppose qi is an rthi root of unity, pi is an s i root of unity, and γij is an r ij root of unity for all i, j. Let r = lcm{ri, si, γij | i, j = 1, . . . , n}. Then there extsis a primitive rth root of unity q ∈ k and integers bi, ci, bij such that qi = q bi , pi = q ci, and γij = q bij for all i, j. The powers of q from the matrix Λ provide the entries for an integer matrix Λ′ comprised of 2 × 2 blocks B′ii = 0 −bi , for all i; B′ij = bij bji − bi bji + cj bi + bij − cj , for i < j; B′ij = bij bij − ci bji + bj bij + ci − bj , for i > j. Then PIdegK n,Γ (k) can be calculated using Λ ′ in Theorem 1.2 (2). The coordinate ring of quantum Euclidean 2n-space over k, Oq(ok2n), is formed by setting qi = 1, pi = q −2 for all i, and γij = q −1 for i < j in the parameters Q, P , and Γ 36 HEIDI HAYNAL (see [18], Example 2.6). Then the integer matrix, Λ′, is  0 0 −1 1 −1 1 . . . −1 1 0 0 −1 1 −1 1 . . . −1 1 1 1 0 0 −1 1 −1 1 −1 −1 0 0 −1 1 ... 1 1 1 1 0 0 −1 −1 −1 −1 0 0 . . . 1 1 1 1 1 . . . 0 0 −1 −1 −1 −1 −1 . . . 0 0  We perform the following row reductions that preserve the size of the image of the homomorphism Z2n −→ Z2n given by Λ′: • For j = 2n, 2n− 1, 2n− 2, . . . , 4, replace row j with row j + row (j − 1) • Replace row 2 with row 2− row 1 • Replace the (new) row 5 with row 5 + row 1 • For j = 4, 6, 8, . . . , 2n− 4, replace row j with row j + 2row (j + 3) • For n ≥ 4, rearrange the rows to order 3, 1, 5, 7, 4, 9, 6, 11, . . . , 2i, 2i+ 5, . . . , 2n− 4, 2n− 2, 2, 2n. The resulting matrix has the form  0 −1 1 0 2 ∗ 0 1 1 . . . 0 1 1 0 0 4 0 −2 2  When n is even, the pivot in the third row does not divide all the entries in its row, so more elementary row and column operations are needed before it becomes clear that the matrix can be diagonalized. By a method similar to that used in Example 5.1, suppressed here in the interest of saving space but listed explicitly in [17], we obtain the Smith normal form diag(1, 1, . . . , 1, 4, 4, . . . , 4, 0, 0) with n ones and n− 2 fours when n PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 37 is even; and diag(1, 1, . . . , 1, 2, 2, 4, 4, . . . , 4, 0, 0), with n− 1 ones and n− 3 fours when n is odd. Thus we have PIdegOq(ok2n) = rn−1, r odd rn−1/2⌊ ⌋, r even /∈ 4Z rn−1/2n−2, r ∈ 4Z . (10) The low-dimensional cases do not fit the same pattern, but the matrices for the cases n = 2 and n = 3 are readily transformed to 1 1 0 0 0 0 −1 1 0 0 0 0 0 0 0 0  and  1 1 0 0 −1 1 0 0 −1 1 −1 1 0 0 0 2 0 0 0 0 0 0 −2 2 0 0 0 0 0 0 0 0 0 0 0 0  respectively. Therefore, formula (10) holds for all n ≥ 2. As a specific case ofK n,Γ (k), quantum symplectic spaceOq(sp(k2n)) is formed by setting qi = q −2 and pi = 1 for all i, and γij = q for i < j (see [18], Example 2.4). With these parameters, the 2n× 2n integer matrix Λ′ is  0 2 1 1 1 1 . . . 1 1 −2 0 −1 −1 −1 −1 . . . −1 −1 −1 1 0 2 1 1 1 1 −1 1 −2 0 −1 −1 −1 −1 −1 1 −1 1 0 2 ... −1 1 −1 1 −2 0 . . . −1 1 −1 1 −1 1 . . . 0 2 −1 1 −1 1 −1 1 . . . −2 0  We perform the following row reductions that preserve the size of the image of the homomorphism Z2n −→ Z2n given by Λ′: • For j = 2n, 2n− 1, . . . , 4, replace row j with row j − row (j − 1) • Replace row 2 with −(row 2− 2row 3 + row 1) • For j = 4, 6, 8, . . . , 2n− 2, replace row j with row j + 2row (j + 1) • For n ≥ 3, order the rows 3, 1, 5, 2, 7, 4, 9, . . . , 2j, 2j + 5, . . . , 2n− 4, 2n, 2n− 2. 38 HEIDI HAYNAL This yields a matrix whose image is more easily measured:  0 1 1 ∗ . . . 0 0 4 −2 −2  But the pivot in row 2 is problematic because it does not always divide the other entries in its row. With further elementary row and column operations, full details of which can be found in [17], we can bring this matrix into Smith normal form diag(1, 1, . . . , 1, 4, 4, . . . , 4) with n ones and n fours when n is even; or the form diag(1, 1, . . . , 1, 2, 2, 4, 4, . . . , 4) with n − 1 ones, two twos, and n − 1 fours when n is odd. For n = 1, 2, the row reduced matrices are, respectively, −1 1 0 2 0 1 1 1 0 0 −4 −4 0 0 0 −4  . Hence we have, for all n, PIdegOq(sp(k2n)) = rn, r odd rn/2⌊ ⌋, r even, r /∈ 4Z rn/2n, r ∈ 4Z 6. Prime Factor Localizations In this section we present a structure theorem for completely prime factors of iterated skew polynomial rings analogous to the main theorem of section four. Applying this result to the algebras studied in section five, we’d like to strengthen it to the form of the quantum Gel’fand-Kirillov conjecture. Recall that the assumptions about skew polynomial rings from section one are still in effect. Theorem 6.1. Let A = R[x; τ, δ], where R is noetherian and δτ = qτδ for some q ∈ k×. Assume that δ extends to a locally nilpotent, iterative h.q-s.τ -d., {di}, on R. Let P ∈ specA be completely prime. Then PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 39 (1) there exists a cyclic Ore set S in A/P such that (A/P )S−1 ∼= R[y; τ ]/Q Y −1 for some completely prime Q ∈ specR[y; τ ] and cyclic Ore set Y , (2) FractA/P ∼= FractR[y; τ ]/Q. Proof. The completely prime ideal P naturally satisfies one of two cases: x ∈ P or x /∈ P . If x ∈ P , then xA ⊆ P and Ax ⊆ P . So the relation xr = τ(r)x + δ(r) implies that δ(r) ∈ P for all r ∈ R. Hence, there is a completely prime ideal I ∈ R such that A/P ∼= R/I ∼= R[y; τ ]/(I + 〈y〉). In this case, we can take S = Y = {1} and localize. If x /∈ P , then xi /∈ P for all i ∈ N ∪ {0} because A/P is a domain. Letting S = {1, x, x2, . . . }, which is a known denominator set in A, we have P ∩ S = ∅. Since extension and contraction provide inverse bijections between the sets specAS−1 and {I ∈ specA | I ∩ S = ∅}, we know that P e ∈ specAS−1. From Theorem 3.7, we have AS−1 ∼= R[y±1; τ ], a localization of R[y; τ ]. So there is a completely prime ideal Q̄⊳R[y±1; τ ] such that AS−1/P e ∼= R[y±1; τ ]/Q̄. Setting Y = {1, y, y2, . . . , }, contraction to R[y; τ ] gives a completely prime ideal Q, where Q∩ Y = ∅, such that R[y±1; τ ]/Q̄ is isomorphic to (R[y; τ ]/Q)Y −1. The canonical projection π : AS−1 −→ (A/P )S−1 gives AS−1/P e ∼= (A/P )S−1. Thus (A/P )S−1 ∼= R[y; τ ]/Q Y −1. � Theorem 6.2. Let R be a noetherian k-algebra, and let A = R[x1, τ1, δ1] · · · [xn; τn, δn] be an iterated skew polynomial ring where, for j < i and λij ∈ k×, τi(xj) = λijxj, and δi is a qi-skew τi-derivation, qi 6= 1, which extends to a locally nilpotent, iterative h.qi-s.τi-d. {di,p}∞p=0 on R[x1; τ1, δ1] · · · [xi−1; τi−1, δi−1] for all i. Let A′ = R[y1; τ ′1][y2; τ ′2] · · · [yn; τ ′n] where τ ′i(yj) = λijyj for all i with j < i and the same units λij as above. Let P be a completely prime ideal in A. Then (1) there exists a finitely generated Ore set Sn in A/P such that (A/P )S n is isomorphic Y −1n for some completely prime ideal Q ⊆ A′ and finitely generated Ore set (2) FractA/P ∼= FractA′/Q. Proof. The case n = 1 has been established in Theorem 6.1. Suppose the result holds for the case n− 1, and let An−1 = R[x1, τ1, δ1] · · · [xn−1; τn−1, δn−1] ⊆ A. Then we have A = An−1[xn; τn, δn]. If xn ∈ P , then as in Theorem 6.1 there is a completely prime ideal I ⊆ An−1 such that A/P ∼= An−1/I ∼= An−1[yn; τ ′n]/(I + 〈yn〉). The induction hypothesis and Lemma 4.2 imply that An−1[yn; τ n]/(I + 〈yn〉) S−1 ∼= Y −1 for some finitely generated Ore sets S and Y . Hence there is a finitely generated Ore set Sn in A such that (A/P )S ∼= (A′/Q)Y −1. If xn /∈ P , let Sn = {1, xn, x2n, . . . } ⊆ A and Yn = {1, yn, y2n, . . . } ⊆ An−1[yn; τn]. Then from the single-variable result, it follows that ( An−1[yn; τ n]/Q̄ Y −1n , (11) 40 HEIDI HAYNAL for a completely prime ideal Q̄ ⊆ An−1[yn; τ ′n]. From Lemma 4.2, we have An−1[yn; τ n] = R[yn; τ n][x1; τ 1] · · · [xn−1; τ ′n−1, δ′n−1], which is an iterated skew polynomial ring in n − 1 variables over the coefficient ring R[yn; τ n] that satisfies the current assumptions. So, we apply the induction hypothesis and rearrange variables to obtain An−1[yn; τn]/Q̄ Y −1n R[yn; τ n][y1; τ 1] · · · [yn−1; τ ′n−1]/Q R[y1; τ 1][y2; τ 2] · · · [yn; τ ′n]/Q for a completely prime ideal Q ⊆ R[y1; τ ′1][y1; τ ′1] · · · [yn; τ ′n] and a denominator set Z ⊆ R[y1; τ ′1][y1; τ ′1] · · · [yn; τ ′n]/Q. This, along with isomorphism (11) gives the re- sult. � When R is replaced by k, we have the following result. Corollary 6.3. Let A = k[x1, τ1, δ1] · · · [xn; τn, δn], where τi(xj) = λijxj and δiτi = qiτiδi, qi 6= 1, for λij , qi ∈ k× and all i with j < i. Assume that each δi extends to a locally nilpotent, iterative h.qi-s.τi-d. {di,m}∞m=0 on the subalgebra k[x1; τ1, δ1] · · · [xi−1; τi−1, δi−1]. Let P be a completely prime ideal in A and set λii = 1 and λji = λ ij . Then for λ = (λij) ∈ Mn(k), and an appropriate completely prime ideal Q ⊆ Oλ(kn), we have FractA/P ∼= FractOλ(kn)/Q. We summarize how this applies to the k-algebras of quantized coordinate type. Corollary 6.4. Let A be any of the examples discussed in sections 5.1 - 5.4, and let P be a completely prime ideal of A. Then there exist a positive integer N , a multiplicatively antisymmetric N ×N matrix λ over k, and a completely prime ideal Q ∈ Oλ(kN) such that FractA/P ∼= FractOλ(kN)/Q. To complete the question posed by the corollary, one might ask how far the quantum Gel’fand-Kirillov conjecture extends to prime factor algebras. For instance: Question 6.5. Find conditions under which we can conclude that for any positive in- teger n, multiplicatively antisymmetric matrix λ ∈ Mn(k×), and completely prime ideal Q ∈ specOλ(kn), we have FractOλ(kn)/Q ∼= FractOp(Km) for some field extension K ⊇ k, integer m ≤ n, and m×m matrix p over K. The case n = 1 is trivial. When n = 2 and Q contains x1 or x2, then FractOλ(k2)/Q is isomorphic either to FractOp(k(y)) where p = (1), or to k itself. In fact, for any n, if Q is generated by a subset S of {x1, . . . , xn}, then the result holds, with p the submatrix of λ formed by deleting the ith row and column for xi ∈ S, and K = k. When xi /∈ Q PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 41 for all i, answering the question fully will likely require different methods depending on the presence of roots of unity among the λij. A positive answer in the generic case has been provided in the proof of [13, Theorem 2.1]: Theorem 6.6. [Goodearl - Letzter] Let k be a field, λ = (λij) a multiplicatively anti- symmetric n× n matrix over k×, and Λ the subgroup of k× generated by the λij. If Λ is torsionfree, then all of the prime ideals Q of Oλ(kn) are completely prime. In their proof, they showed that FractOλ(kn)/Q ∼= FractOp(Km), and identified K as the quotient field of a commutative domain embedded in the center of Oλ((k×)n)/Q′, where Q′ is the prime ideal in Oλ((k×)n) induced by localization. Quantum affine space is included in a class called quantum solvable algebras by A. N. Panov. The main theorem of [34, Section 3], states that when the group generated by the λij is torsionfree, then FractOλ(kn)/Q is isomorphic to the quotient division ring of a quan- tum torus. The main theorem of [35, Section 3], allows roots of unity and states that when Q satisfies the extra condition of being stable under a certain set of derivations, then FractOλ(kn)/Q is isomorphic to the quotient division ring of a quantum torus. Cauchon’s work may also be specialized to apply to quantum affine space when the group generated by the λij is torsionfree. The result of [5, Theorem 6.1.1], indicates that FractOλ(kn)/Q is isomorphic to FractOp(Km) which specializes to this result. But the division ring of real quaternions provides an example showing that Question 6.5 needs to have some conditions imposed. Note that H ∼= Oλ(R3)/Q, where λ = 1 −1 −1 −1 1 −1 −1 −1 1  , andQ = 〈x21 + 1, x22 + 1, x23 + 1〉. Therefore, we cannot obtain the desired isomorphism of quotient division rings in this case, illustrating the necessity of an extra condition such as the one imposed by Panov in [35]. acknowledgments The author thanks her dissertation advisor, Ken Goodearl, for his direction that was so freely given in many inspiring discussions. References [1] J. Alev and F. Dumas, Sur le corps de fractions de certaines algèbres quantiques, J. Algebra 170 (1994), 229-265 [2] M. Artin, W. Schelter, and J. 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Smith, Quantum groups: An introduction and survey for ring theorists, in Noncommutative Rings (S. Montgomery and L. W. Small, eds.), pp131-178, MSRI Publ. 24, Springer-Verlag, Berlin (1992) [39] R. P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole, Monterey, CA, Department of Mathematics, University of California, Santa Barbara, California 93106 E-mail address : [email protected] 1. Introduction 2. Higher q-Skew -Derivations 3. The -Derivation Removing Homomorphism 4. Main Theorem 5. Examples 5.1. The coordinate ring of odd-dimensional quantum Euclidean space; Oq (o k2n+1) 5.2. The multiparameter quantized Weyl algebras; AnQ, (k) 5.3. The multiparameter coordinate ring of quantum n n matrices; O, bold0mu mumu pppppp(to.Mn(k))to. 5.4. The algebra Kn, P, Q (k), which generalizes the coordinate rings of even-dimensional quantum Euclidean space and quantum symplectic space 6. Prime Factor Localizations acknowledgments References

0704.0847

Semi-spheroidal Quantum Harmonic Oscillator

Semi-spheroidal Quantum Harmonic Oscillator D. N. Poenaru,1, 2, ∗ R. A. Gherghescu,1, 2 A. V. Solov’yov,1 and W. Greiner1 1Frankfurt Institute for Advanced Studies, J. W. Goethe Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany 2 Horia Hulubei National Institute of Physics and Nuclear Engineering (IFIN-HH), P.O. Box MG-6, RO-077125 Bucharest-Magurele, Romania (Dated: November 15, 2018) A new single-particle shell model is derived by solving the Schrödinger equation for a semi- spheroidal potential well. Only the negative parity states of the Z(z) component of the wave function are allowed, so that new magic numbers are obtained for oblate semi-spheroids, semi-sphere and prolate semi-spheroids. The semi-spherical magic numbers are identical with those obtained at the oblate spheroidal superdeformed shape: 2, 6, 14, 26, 44, 68, 100, 140, ... The superdeformed prolate magic numbers of the semi-spheroidal shape are identical with those obtained at the spherical shape of the spheroidal harmonic oscillator: 2, 8, 20, 40, 70, 112, 168 ... PACS numbers: 03.65.Ge, 21.10.Pc, 31.10.+z, The spheroidal harmonic oscillator have been used in various branches of Physics. Of particular interest was the famous single-particle Nilsson model [1] very success- ful in Nuclear Physics and its variants [2, 3, 4] for atomic clusters. Major spherical-shells N = 2, 8, 20, 40, 58, 92 have been found [2] in the mass spectra of sodium clus- ters of N atoms per cluster, and the Clemenger’s shell model [3] was able to explain this sequence of spherical magic numbers. In the present paper we would like to write explicitly the analytical relationships for the energy levels of the spheroidal harmonic oscillator and to derive the corre- sponding solutions for a semi-spheroidal harmonic oscil- lator which may be useful to study atomic cluster de- posited on planar surfaces. For spheroidal equipotential surfaces, generated by a potential with cylindrical symmetry the states of the va- lence electrons were found [3] by using an effective single- particle Hamiltonian with a potential Mω20R 2 + δ 2 + δ In order to get analytical solutions we shall neglect an additional term proportional to (l2 − 〈l2〉n). We plan to include in the future such a term which needs a numerical solution. K. L. Clemenger introduced the deformation δ by expressing the dimensionless two semiaxes (in units of the radius of a sphere with the same volume, R0 = 1/3, where rs is the Wigner-Seitz radius, 2.117 Å for Na [5, 6]) as 2 + δ ; c = 2 + δ The spheroid surface equation in dimensionless cylindri- cal coordinates ρ and z is given by = 1 (3) where a is the minor (major) semiaxis for prolate (oblate) spheroid and c is the major (minor) semiaxis for prolate (oblate) spheroid. Volume conservation leads to a2c = 1. One can separate the variables in the Schrödinger equation, HΨ = EΨ, written in cylindrical coordinates. As a result the wave function [7, 8] may be written as Ψ(η, ξ, ϕ) = ψmnr (η)Φm(ϕ)Znz (ξ) (4) where each component of the wave function is ortonor- malized leading to Φm(ϕ) = e 2π (5) ψ(η) = Nmnrη |m|/2e−η/2L nr (η) Nmnr = α⊥(nr+|m|)! in which η = R20ρ 2/α2⊥ and the quantum numbers m = (n⊥−2i) with i = 0, 1, ... up to (n⊥−1)/2 for an odd n⊥ or to (n⊥− 2)/2 for an even n⊥. Lmn (x) is the associated Laguerre polynomial and the constant α⊥ = h̄/Mω⊥ has the dimension of a length. Znz(ξ) = Nnze −ξ2Hnz (ξ) Nnz = π2nznz!) 1/2 (7) where ξ = R0z/αz, αz = h̄/Mωz, and the main quan- tum number n = n⊥ + nz = 0, 1, 2, .... The eigenvalues are En = h̄ω⊥(n⊥ + 1) + h̄ωz(nz + 1/2) (8) The parity of the Hermite polynomials Hnz (ξ) is given by (−1)nz meaning that the even order Hermite poly- nomials are even functions H2nz(−ξ) = H2nz (ξ) and the odd order Hermite polynomials are odd functions H2nz+1(−ξ) = −H2nz+1(ξ). There is a recurrence rela- tionship 2zHn = Hn+1+2nHn−1. One hasH0 = 1, H1 = http://arxiv.org/abs/0704.0847v1 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 (spheroidal deformation) -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 (spheroidal deformation) FIG. 1: LEFT: Spheroidal harmonic oscillator energy levels in units of h̄ω0 vs. the deformation parameter δ. Only 6 major shells (N = 0, 1, 2, ..., 5) have been considered. Each level is labeled by n, n⊥ quantum numbers and is (2n⊥+2)-fold degenerate. The labels are 0, 0; 1, 0, 1, 1; 2, 0, 2, 1, 2, 2; 3, 0, 3, 1, 3, 2, 3, 3; 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, etc. RIGHT: Semi-spheroidal harmonic oscillator energy levels in units of h̄ω0 vs. the deformation coordinate δ. Only 9 major shells (N = 0, 1, 2, ..., 8) have been considered. Each level is labeled by n, n⊥ quantum numbers (with nz = n− n⊥ = 1, 3, 5, ... and is (2n⊥ + 2)-fold degenerate. The labels are 1, 0; 2, 1; 3, 2, 3, 0; 4, 3, 4, 1; 5, 4, 5, 2, 5, 0; 6, 5, 6, 3, 6, 1, etc. The semi-spherical magic numbers are identical with those obtained at the oblate spheroidal superdeformed shape (δ = −2/3): 2, 6, 14, 26, 44, 68, 100, 140, ... 2z, H2 = 4z 2−2, H3 = 8z3−12z, H4 = 16z4−48z2+12, H5 = 32z 5 − 160z3 + 120z, etc. In units of h̄ω0 the eigenvalues, ǫ = E/(h̄ω0), are given (2− δ)1/3(2 + δ)2/3 For a prolate spheroid, δ > 0, at n⊥ = 0 the energy level decreases with deformation except for n = 0, but when n⊥ = n it increases. For a given prolate deformation and z0=0.5 z1=1.5 z2=2.5 z3=3.5 z4=4.5 z5=5.5 FIG. 2: LEFT: Harmonic oscillator potential V = V (ξ), the wave functions Znz = Znz (ξ) for nz = 0, 1, 2, 3, 4, 5 and the corresponding contributions to the total energy levels ǫz nz = Enz/h̄ωz = (nz + 1/2) for spherical shapes, δ = 0. ξ = h̄/Mωz. RIGHT: The similar functions for a semi- spherical harmonic oscillator potential. Only negative parity states are retained which are vanishing at ξ = 0 where the potential wall is infinitely high . a maximum energy ǫm, there are nmin closed shells and other levels for high-order shells up to nmax: nmin = (2− δ)1/3(2 + δ)2/3ǫm − 2 + δ nmax = (2− δ)1/3(2 + δ)2/3ǫm − 2− δ (11) and similar formulae for oblate deformations, δ < 0. The low lying energy levels for the six shells (main quantum number n = 0, 1, 2, 3, 4, 5) can be seen in figure 1. Each level, labelled by n⊥, n, may accomodate 2n⊥ + 2 parti- cles. One has 2 (n⊥+1) = (n+1)(n+2) nucleons in a completely filled shell charcterized by n, and the total number of states of the low-lying n + 1 shells is n=0(n+1)(n+2) = (n+1)(n+2)(n+3)/3 leading to the magic numbers 2, 8, 20, 40, 70, 112, 168... for a spheri- cal shape. Besides the important degeneracy at a spher- ical shape (δ = 0), one also have degeneracies at some superdeformed shapes, e.g. for prolate shapes at the ra- tio c/a = (2 + δ)/(2 − δ) = 2 i.e. δ = 2/3. More details may be found in the Table I. The first five shells can reproduce the experimental magic numbers mentioned above; in order to describe the other shells Clemenger introduced the term proportional to (l2 − 〈l2〉n). Let us consider a particular shape (half of an oblate or prolate spheroid) of a semi-spheroidal cluster deposited on a surface with the z axis perpendicular on the sur- face and the ρ axis in the surface plane. Then the semi- 20 40 60 80 100 120 140 20 40 60 80 100 120 140 20 40 60 80 100 120 140 160 20 40 60 80 100 120 140 = - 1 = - 2/3 = - 0.4 = - 0.8/3 = 2/3 = 0.4 = 0.8/3 FIG. 3: Variation of shell corrections with N for Na clusters. TOP: δ = 0. The semi-spherical magic numbers are identical with those obtained at the oblate spheroidal superdeformed shape: 2, 6, 14, 26, 44, 68, 100, 140, ... For a prolate superdeformed (δ = 2/3) shape the magic numbers are identical with those obtained at the spherical shape: 2, 8, 20, 40, 70, 112, 168, ... Other magic numbers are given in Table I. Oblate and prolate shapes are considered on the left-hand side and right-hand side, respectively. spheroidal surface equation is given by (a/c)2(c2 − z2) z ≥ 0 0 z < 0 The radius of the semi-sphere obtained for the defor- mation δ = 0 is Rs, given by the volume conservation, (1/2)(4πR3s/3) = 4πR 0/3, leading to Rs = 2 1/3R0. We shall give ρ, z, a, c in units of Rs instead of R0. Accord- ing to the volume conservation, a2cR3s/2 = R 0 so that a2c = 1. Other kind of shapes obtained from a spheroid by removing less or more than its half (as in the liquid drop calculations [9]) will be considered in the future; in this case it is not possible to obtain analytical solutions. The new potential well we have to consider in order to solve the quantum mechanical problem is shown in the right-hand side of the figure 2. The potential along the symmetry axis, Vz(z), has a wall of an infinitely large height at z = 0, and concerns only positive values of z ∞ z = 0 MR2sω 2/2 z ≥ 0 (13) In this case the wave functions should vanish in the origin, where the potential wall is infinitely high, so that only negative parity Hermite polynomials (nz odd) should be taken into consideration. From the energy levels given in figure 1 we have to select only those corresponding to this condition. In this way the former lowest level with n = 0, n⊥ = 0 should be excluded. From the two leveles with n = 1 we can retain the level with n⊥ = 0 i.e. nz = 1. This will be the lowest level for the semi-spherical harmonic oscillator and will accomodate 2n⊥+2 = 2 atoms. From the three levels with n = 2 only the one with nz = n⊥ = 1 with 2n⊥ + 2 = 4 degeneracy is retained so that the first two magic numbers at spherical shape (δ = 0) are now 2 followed by 6, etc. Some deformed magic numbers may be found in the Table I and as position of minima in Fig. 3. Each level, labelled by n⊥, n, may accomodate 2n⊥+2 particles. When n is an odd number, one should only have even n⊥ in order to select the odd nz = n − n⊥. The contribution of the shells with odd n to the semi- spherical magic numbers will be neven (2n⊥ + 2) = (n+ 1)2 leading to the sequence 2, 8, 18 for n = 1, 3, 5. The con- tribution of the shells with even n to the semi-spherical TABLE I: TOP: Deformed magic numbers of the spheroidal harmonic oscillator. BOTTOM: Deformed magic numbers of the semi-spheroidal harmonic oscillator. OBLATE PROLATE δ a/c Magic numbers δ a/c Magic numbers −0.8/3 17/13 2, 8, 18, 20, 34, 38, 58, 64, 92, 100, 136, 148, ... 0.8/3 13/17 2, 8, 20, 22, 42, 46, 76, 82, 124, 134 ... −0.4 1.5 2, 6, 8, 14, 18, 28, 34, 48, 58, 76, 90, 114, 132, ... 0.4 2/3 2, 8, 10, 22, 26, 46, 54, 66, 84, 96, 114, 138, 156, ... −2/3 2 2, 6, 14, 26, 44, 68, 100, 140, ... 2/3 0.5 2, 4, 10, 16, 28, 40, 60, 80, 110, 140, ... −1 3 2, 6, 12, 22, 36, 54, 78, 108, 144, 1 1/3 4, 12, 18, 24, 36, 48, 60, 80, 100, 120, 150, ... −0.8/3 17/13 2, 6, 12, 22, 26, 36, 42, 56, 64, 82, 92, 114, 126, 154, ... 0.8/3 13/17 2, 6, 8, 14, 18, 28, 34, 48, 58, 76, 90, 114, 132, ... −0.4 1.5 2, 6, 12, 22, 36, 54, 78, 108, 144, 0.4 2/3 2, 8, 18, 20, 34, 38, 50, 58, 64, 80, 92, 100, ... −2/3 2 2, 6, 12, 20, 32, 48, 68, 92, 122, 158, ... 2/3 0.5 2, 8, 20, 40, 70, 112, 168, ... −1 3 2, 6, 20, 30, 42, 58, 78, 102, 130, 1 1/3 2, 8, 10, 14, 22, 26, 46, 54, 66, 84, 96, 114, 138, 156, ... magic numbers will be (2n⊥ + 2) = n(n+ 2) which gives the sequence 4, 12, 24 for n = 2, 4, 6. This should be interlaced with the preceding one so that the magic numbers will be 2, 2+4 = 6, 6+8 = 14, 14+12 = 26, 26+18 = 44, 44+24 = 68, as shown at the right-hand side of the Fig. 1. The equation (9) from the harmonic oscillator, in units of h̄ω0 is still valid, but one should only allow the values of n and n⊥ for which nz = n−n⊥ ≥ 1 are odd numbers. The ortonormalization condition of the Znz component of the wave function became Zn′z(z)Znz(z)dz = δn′znz (16) with nz = 1, 3, 5, ..., n for odd n and nz = 1, 3, 5, ..., n− 1 for even n. Consequently the normalization factor is times the preceding one Znz(ξ) = 2Nnze −ξ2Hnz(ξ) Nnz = π2nznz!) 1/2 (17) For a nucleus with mass number A the shell gap is given by h̄ω00 = 41A 1/3 MeV. For an atomic cluster [10] the single-particle shell gap is given by h̄ω0(N) = 13.72 eV Å rsN1/3 which is 3.0613N−1/3 eV in case of Na clusters. Since we consider solely monovalent elements, N in this eq. is the number of atoms and t denotes the electronic spillout for the neutral cluster according to [10]. The shell correction energy, δU [11], in figure 3 shows minima at the oblate and prolate magic numbers given in the lower part of the table I. The striking result is that the superdeformed prolate magic numbers of the semi-spheroidal shape are identical with those obtained at the spherical shape of the spheroidal harmonic oscil- lator. We expect that this kind of symmetry will not be present anylonger for the Hamiltonian including the term proportional to (l2−〈l2〉n) and/or the more complex equipotential surface we shall study in the future. ∗ [email protected] [1] S. G. Nilsson, Det Kongelige Danske Videnskabernes Sel- skab (Dan. Mat. Fys. Medd.) 29 (1955). [2] W. D. Knight, K. Clemenger, W. A. de Heer, W. A. Saunders, M. Y. Chou, and M. L. Cohen, Phys. Rev. Lett. 52, 2141 (1984). [3] K. L. Clemenger, Phys. Rev. B 32, 1359 (1985). [4] S. M. Reimann, M. Brack, and K. Hansen, Z. Phys. D 28, 235 (1993). [5] M. Brack, Phys. Rev. B 39, 3533 (1989). [6] C. Yannouleas and U. Landman, Phys. Rev. B 51, 1902 (1995). [7] A. J. Rassey, Phys. Rev. 109, 949 (1958). [8] D. Vautherin, Phys. Rev. C 7, 296 (1973). [9] V. V. Semenikhina, A. G. Lyalin, A. V. Solov’yov, and W. Greiner, to be published (2007). [10] K. L. Clemenger, Ph. D. Dissertation (1985), University of California, Berkeley. [11] V. M. Strutinsky, Nuclear Physics, A 95, 420 (1967). mailto:[email protected]

0704.0848

Growing Perfect Decagonal Quasicrystals by Local Rules

Growing Perfect Decagonal Quasicrystals by Local Rules Hyeong-Chai Jeong Department of Physics, Sejong University, Seoul 143-747, Korea, Asia Pacific Center for Theoretical Physics, POSTECH, Pohang 790-784, Korea, and Department of Physics, Princeton University, Princeton, NJ 08540, USA A local growth algorithm for a decagonal quasicrystal is presented. We show that a perfect Penrose tiling (PPT) layer can be grown on a decapod tiling layer by a three dimensional (3D) local rule growth. Once a PPT layer begins to form on the upper layer, successive 2D PPT layers can be added on top resulting in a perfect decagonal quasicrystalline structure in bulk with a point defect only on the bottom surface layer. Our growth rule shows that an ideal quasicrystal structure can be constructed by a local growth algorithm in 3D, contrary to the necessity of non-local information for a 2D PPT growth. The announcement of icosahedral phase of alloys in 1984 [1] posed many puzzles. The first question was what kind of arrangement of atoms could produce Bragg peaks with a rotational symmetry forbidden to crystals. The quasiperiodic translational order was proposed im- mediately as a candidate, and such materials began to be called quasiperiodic crystals or quasicrystals for short [2]. However, the appearance of quasicrystals brings new puz- zles: why and how the atoms can arrange themselves to have such order, and especially, how quasicrystals can grow with perfect quasiperiodic order has been a dilemma since it seemingly requires non-local information while atomic interactions in metallic alloys are generally con- sidered to be short ranged. There are currently two alternative pictures to de- scribe quasicrystals: energy-driven perfect quasiperi- odic quasicrystals and entropy-driven random-tiling qua- sicrystals. Accordingly, two alternative scenarios for the growth [3] of quasicrystals exist: matching-rule based, energy-driven growth, and finite-temperature entropy- driven growth [4]. A major criticism for the former ap- proach has been that no local growth rules can produce a perfect quasicrystalline structure in 2D [5, 6]. Here, we show how to overcome this obstacle in a 3D quasicrystals. Penrose tiling [7] has been a basic template for describ- ing formation and structure of ideal quasicrystals. It can be constructed from fat and thin rhombi with arrowed edges shown in Fig. 1(a). The infinite tiling consistent with arrow-matching rules is the Penrose tiling but they do not guarantee the growth of a perfect Penrose tiling (PPT) from a finite seed. Successive “legal” (obeying the arrow-matching rules) additions of tiles to the surface of the already existing legal patch of tiles can produce de- fects. They usually occur after only a handful of tiles are added, and hence the arrow-matching rules cannot explain the long-range quasicrystalline order engendered by growth kinetics. There has been a great amount of discussion and a number of debates on the possibility of local growth al- gorithm for a PPT [3, 4, 5, 6, 8, 9, 10]. The debates par- tially emerge from a different assumption on the growth processes at the surface, uniform growth and preferential growth. In the former, the growth occurs at any surface site with the same attaching probability, while it occurs with different attaching probabilities in the latter. In 1988, Penrose proved that a PPT cannot be grown by local rules with uniform growth by showing that “decep- tions” are unavoidable [5] where a deception is a legal patch which cannot be found in a PPT [5, 6]. In the same year, Onoda et al. introduced a preferential growth algorithm which can avoid deception by local rules called “vertex rules” [3]. However, vertex rule growth stops at a “dead surface” and non-local information or arbi- trarily small growth rates are required to be an infinite PPT. Yet, their growth algorithm is believed to provide methods to grow the most ideal quasicrystalline struc- tures with local information. If an initial seed contains a special kind of defect, called “decapod” [11], Onoda et al. showed that the seed can be grown to an almost PPT (whose only defect is the initial decapod defect) [3]. A point defect in a 2D tiling growth usually implies a line defect in a 3D decagonal tiling growth. If we apply a solid-on-solid type growth [12] so that a layer copies configuration of the one below, we get decagonal tiling consisting of identical layers with a decapod defect at the center of each layer. This line defect has been consid- ered to be a minimum imperfection for the 3D decagonal quasicrystal structure from the local growth algorithm. In this letter, we consider the growth of decagonal quasicrystals and present a local growth algorithm for 3D decagonal tiling which consists of PPT layers except the bottom layer. We use the two well known results of the Onoda et al. study on planar decagonal quasicrys- tal growth [3]. 1. A local growth method around a “cartwheel decapod” leads to dead surfaces, from which further growth of a PPT requires non-local information. 2. Infinite local growth is possible if it starts from an “active decapod defect” but the resultant tiling contains the defect and is not a PPT. By combining an active decapod defect in the bottom layer with a cartwheel de- capod in the second layer and using the information of the underneath configuration, we make the growth con- tinue beyond dead surfaces in the second, and subsequent layers. The bottom surface layer has a point defect but we can consider the overall structure as that of a per- fect decagonal quasicrystal since deviations from the bulk http://arxiv.org/abs/0704.0848v1 FIG. 1: (Color online) (a) Fat and skinny Penrose tiles with arrows. (b) The eight ways of surrounding a vertex in a PPT. (c) Dead surfaces encountered when a tiling is grown by Rule- L from a cartwheel decagon. See text for details. (d) A de- capod tiling. Ten semi-infinite worms meet at the decapod decagon at the center. From the yellow seed tiles, an infinite tiling (decapod tiling) can be grown by Rule-L. layer structure are natural for the surface layer even for ideal crystal materials. Let us first discuss the growth rule in a 2D Penrose tiling. For the arrow-matching rule growth, a deception can be made as few as three tiles [5]. Since the growth process does not allow tiles to be removed, a deception (which is not a part of a PPT) cannot grow to a PPT, and we need a growth rule which allows no deceptions of any size for a PPT growth. We can avoid three tile deceptions by introducing a more restricted growth rule which allows only correct (subset of a PPT) three tile patches. However, the new growth rule can make a de- ception in a larger scale, for example, a three tile de- ception of inflated [11] tiles. Since a deception can be made in all scales of multiply inflated tile sizes [11], it is unavoidable for a local growth rule. The absence of local rules for perfect tiling growth seems to be the case for general aperiodic tilings in 2D [6]. Based on this ob- servation, Penrose even speculates that there may be a non-local quantum-mechanical ingredient to quasicrystal growth [5, 13]. Onoda et al. proposed “vertex rules” which avoid an encounter of deceptions [3]. Here, a tile can be added only to a “forced edge” which admits only one way of adding a tile for its end vertices to be consistent with any of the eight PPT vertex configurations shown in Fig. 1(b). In this Letter, their vertex rules will be called “Rule- L” and used for the “lateral”-direction growth (for 3D decagonal tilings). The problem of Rule-L is that the growth stops at a finite size patch called a “dead surface” which consists of unforced edges. There are special kinds of point-like defects, called “de- capod” defects [11], which can be an ideal seed to grow an almost PPT without encountering a dead surface [3]. A decapod is a decagon with single arrowed edges. Since there are 10 arrows, each of which can take two indepen- dent orientations, there are 210 combinations of states. After eliminating rotations and reflections, we get 62 dis- tinct decapods. We can tile inside the decagon legally for only one decapod, the cartwheel case and the rest of the 61 decapods are called decapod defects. One no- table property of the decapods is that the outside of the decagon region can be legally tiled for all 62 cases. This can be easily understood from the fact that six semi- infinite worms and two infinite worms meet at the cen- ter cartwheel decagon [shown by green tiles in 1(c)] in a cartwheel tiling. If we remove the tiles in the cen- ter cartwheel decagon, two infinite worms become four semi-infinite worms, and we have ten semi-infinite worms which start at the perimeter of the center decagon. A de- capod defect tiling is formed by flipping one or more of these ten semi-infinite worms. The arrows on the worm perimeter will still fit except a mismatch at the decapod decagon perimeter. Figure 1(d) shows an example ob- tained by flipping the worm denoted by red hatched tiles. Among 61 decapod defects, there are 51 “active” deca- pod defects which have at least three consecutive arrows of the same orientation on their decagon perimeter [14]. One can show that a patch containing an active decapod defect is never enclosed by a dead surface [15]. Our 3D growth rules are constructed by observing that a cartwheel PPT and a decapod tiling can be differ- ent only in ten semi-infinite worms. Consider two layer growth from a (two layer) seed that contains a cartwheel decagon [yellow tiles in Fig. 1(c)] and a decapod defect [yellow tiles in Fig. 1(d)] at the upper and the lower lay- ers respectively. If each layer grows with Rule-L inde- pendently, the growth of the upper layer would stop at the red-purple-blue dead surface while the lower layers grow indefinitely. Now, we introduce a vertical growth rule so that a tile can be added at the dead surface of the upper layer properly. Note that the basic tiles for 3D decagonal tiling are rhombus prisms which have top and bottom faces as well as side faces. By vertical growth, we mean attaching tiles on the surface layer such that the bottom faces of the attached tiles contact with the top faces of the surface layer tiles, while lateral growth means attaching tiles to side faces at the perimeters. We propose a vertical growth rule, “Rule-V” with which the lateral growth rule, Rule-L produces a PPT on a decapod tiling. If a tile in a flipped worm of the decapod tiling (the lower layer) is copied by a vertical growth, a defect on the upper layer is inevitable. Our Rule-V is designed to avoid such a case and allow to attach tiles vertically only on the “sticky” top faces, blue-circled fat tiles in S3 and S4 configurations shown in Fig. 2(a) [16]. They always form D-hexagons indicated by dotted lines. Such S4 S3 (a) (b) (c) (d) (e) FIG. 2: (Color online) (a) Sticky sites, on which a tile can be attached vertically, are indicated with blue circles. (b) Upper layer configuration of a (two layer) seed. It contains a cartwheel decagon and ten hexagons attached to the decagon. The tiles denoted by blue circles are sticky site and the tile denoted by X is the nucleate site. (c) Lower layer configu- ration of the (two layer) seed. It contains an active decapod decagon with five consecutive arrows of the same orientation. (d) Sticky sites on a decapod tiling. All sticky sites are out- side of the ten semi-infinite worms. (e) Dead surfaces which contain the center cartwheel decagon in a cartwheel tiling. The two crossing infinite worms of the cartwheel tiling always pass the two 72 degree corners of dead surfaces. D-hexagons can lie only at the end of worms since the other (uncircled) tiles in S3 or S4 configurations prevent formation hexagons next to the circled tiles. Therefore, the sticky sites can be located only outside or at the ends of the semi-infinite worms as illustrated in Figs. 2(b)-(e). One can further show that sticky sites are strictly outside of the semi-infinite worm if it is flipped since the flipping makes the vertices at the end be illegal (and therefore they cannot be S3 or S4). Therefore, no sticky sites are in the flipped worm, and hence Rule-V does not intro- duce a defect or deception for the layer that grows on a decapod tiling. Now we show that Rule-V is enough for the upper layer to grow beyond the dead surfaces when the growth starts from a proper seed. Our seed consists of two layer finite patches which include a cartwheel decagon and an active decapod decagon at the upper and the lower layers, respectively. Figures 2(b) and (c) show an example. The upper layer seed [Fig. 2(b)] consists of a cartwheel decagon and 10 D-hexagons. It covers all ends of the semi-infinite worms in the lower decapod seed [Fig. 2(c)] which consists of a decapod decagon and 10 D- hexagons [17]. Let us first consider the properties of dead surfaces which contain the upper layer seed. By apply- ing inflations to a cartwheel tiling, one can show that the dead surfaces, which contain the center cartwheel decagon, have two 72 degree corners and each corner is passed by an infinite worm [green worms in Fig. 2(e)] of the cartwheel tiling [10, 11]. The D-hexagon at the 72- degree corner forces the next two hexagons just outside of the dead surface (in the infinite worm direction) to be D and Q. These two hexagons force a cartwheel decagon to form just outside of the corner as illustrated by (red and purple) dashed lines in Fig. 2(e). A Q-decagon (de- noted by yellow tiles) in the dashed cartwheel decagon forces a tile just outside of the 72 degree corner [denoted by the solid green circles in Fig. 2(e)] to be sticky. We call these sticky sites as “launching” sites. The exact position of a launching site depends on the orientation of the corner [10] but the patch can grow by Rule-L for both cases. The position of the launching site determines the orientation of the worm along the side lines of the dead surface making the edges at the dead surface be- come forced. Hence, the upper layer would grow to infi- nite by Rule-L if Rule-V guarantees tiles at the launching sites. This is the case when it grows on an active deca- pod tiling obtained by flipping a semi-infinite worm [18] of a cartwheel tiling as shown in Fig. 2(d). Since neither crossing infinite-worms are flipped [compare Figs. 2(d) and (e)], the underneath tiles of the launching sites will be always the sticky sites of the decapod tiling and tiles at the launching sites are guaranteed by Rule-V on the decapod tiling. For the completeness of the 3D decagonal quasicrystal growth, we need to provide the rule for the nucleation of an island (seed) from the third layer. The physical pro- cess of the nucleation of an island on a PPT would be sim- ilar to that of a perfect crystal surface. High quality qua- sicrystals are grown when they grow slowly, or in other words, when the chemical potential of bulk quasicrystal is slightly less than that of the fluid phase. Therefore, adatoms or “adtiles” on a terrace would be unstable and probably diffuse on the terrace until they evaporate (i.e., go back to the fluid phase) or attach to preferential sites (forced or sticky sites) [12]. We believe that the chemical potentials of the forced sites are less than those of sticky sites and adtiles attach to forced sites for most cases. However, when the terrace forms a dead surface (or part of a dead surface), it is not easy for an adatom to find a forced site and it would attach on a sticky site, especially to a launching site whose chemical potential is expected to be lower than that of an isolated sticky site. Note that both forced and launching sites are at the perimeters of terraces and become irrelevant to adtiles on the middle of the terraces as they grow sufficiently large. It is then conceivable that two or more adtiles meet on a terrace and begin to form a new patch of the next layer before they arrive at the perimeter. With this physical process in mind, we allow a nucleation process in our growth algorithm. The nucleation of an island can happen in cooperation of a quite large cluster of tiles. We choose the “cartwheel seed”, a cartwheel decagon and the 10 D-hexagons arranged as Fig. 2(b), as such a cluster and introduce a nucleation site on it. The site X in the figure is called a “nucleate site” if its lateral neighboring tiles form a cartwheel seed and if it has a underneath tile [19]. When a nucleate site is selected, we create a cartwheel seed on it. Let us summarize our growth mechanism for decago- nal quasicrystals. It consists of three processes: lateral growth by Rule-L, vertical growth by Rule-V, and the is- land nucleation (seed formation) for the new layer. Algo- rithmically, it is realized by the following steps: 1. Start with a two layer seed whose upper and lower layers con- tain a cartwheel decagon and an active decapod decagon, respectively. 2. Randomly choose a surface site. Check if it is a sticky, nucleate, or unsticky site when it is a top face. For a side face, check if it is a forced or un- forced site. 3. Perform the vertical growth, nucleation, or lateral growth if the chosen site is a sticky, nucleate, or forced site, respectively. Do nothing for unsticky (top face) or unforced (side face) site. For simplicity, we have chosen the unit attaching prob- ability for all sticky, forced and nucleate sites. In real material, they probably have different attaching proba- bilities due to difference in their chemical potentials and attaching kinetics. We think that the attaching probabil- ities are different even among the forced sites (and among the sticky sites) since they depend on the local configu- rations. However, the nucleation probability would be much smaller than that of the attaching probability of a tile in any case since the former demands a coopera- tion of many tiles. Slow process of nucleation implies a layer by layer growth for a perfect decagonal quasicrys- tal [20]. It is beyond the scope of this Letter to predict the growth kinetics of real quasicrystals since it requires knowing atomic cluster structures corresponding to each type of tiles as well as the kinetic parameters of atomic attachment of real materials. Our growth algorithm has a couple of limitations. First, it can produce only one kind of PPT, a cartwheel tiling. Second, the seed must include a decapod defect. However, a decapod defect may form under quite general conditions. It is believed that every possible hole sur- rounded by an arrow-matched Penrose tiling is equivalent to a decapod hole [11]. The bottom layer, which may grow under structurally different environment, is natu- ral place to have such defect. Our algorithm shows that PPT is possible from the second layer if the defect can be surrounded by legal tiles. Once a PPT layer begins to form on the second layer, our growth algorithm produces PPT layers easily from the third layer. We hope that the present work stimulates studies on 3D growth rules for real quasicrystals. We would like thank P. J. Steinhardt and M. Rechts- man for valuable comments. This work was supported by the Korea Research Foundation Grant (KRF-2005- 015-C00169). [1] D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. 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Gardner, Sci. Am. 236, 110 (1977). [12] H.-C. Jeong and E. D. Williams, Surface Science Reports 34, 171 (1999). [13] R. Penrose, Emperor’s New Mind (Oxford University Press, New York, 2002), p. 640ff. [14] T. Dotera, H.-C. Jeong, and P. J. Steinhardt, in Methods of structural analysis of modulated structures and qua- sicrystals, edited by et. al.. J. M. Perez-Mato (World Scientific, Singapore, 1991), pp. 660–663. [15] K. Ingersent, Ph.D. thesis, University of Pennsylvania, 1990. [16] We introduce the concept of sticky sites as a mathemat- ical devise to avoid the mistakes of copying a tile in a flipped worm but it may mimic the real growth process of quasicrystals. Recently, Fournée et al. observed “traps” or sticky sites on which adatoms are easily captured for some quasicrystal surfaces [21]. [17] Covering of all ends of the semi-infinite worms further ensures that all sticky sites in the decapod tiling stay outside of the semi-infinite worms. [18] From the Fig 2(b), one can see that an active decapod tiling is obtained by flipping a semi-infinite worm in a cartwheel tiling while flipping a half of an infinite worm results in an inactive decapod tiling. [19] The requirement of the underneath tiles is added to en- sure that the nucleation happens from the third layer. In real growth, this requirement may not be needed. Nu- cleation is likely to happen only on the top layer (hence from the third layer) which can be large enough to wait the slow necleation process. [20] The sticky sites (Fig. 2(a)) are formed only on a compact cluster of tiles. Therefore, the vertical growth by Rule-V, which produces isolated tiles, cannot be continue more than one layer height without nucleation process. [21] V. Fournée et al., Phys. Rev. B 67, 033406 (2003). http://arxiv.org/abs/cond-mat/9903074