Cosmology from String Theory
arXiv:0704.0928v3  [hep-ph]  26 Oct 2007
Cosmology from String Theory
Luis Anchordoqui,1 Haim Goldberg,2 Satoshi Nawata,1 and Carlos Nuñez3
1Department of Physics,
University of Wisconsin-Milwaukee, Milwaukee, WI 53201
2Department of Physics,
Northeastern University, Boston, MA 02115
3 Department of Physics,
University of Swansea, Singleton Park, Swansea SA2 8PP, UK
(Dated: April 2007)
Abstract
We explore the cosmological content of Salam-Sezgin six dimensional supergravity, and find a
solution to the field equations in qualitative agreement with observation of distant supernovae,
primordial nucleosynthesis abundances, and recent measurements of the cosmic microwave back-
ground. The carrier of the acceleration in the present de Sitter epoch is a quintessence field slowly
rolling down its exponential potential. Intrinsic to this model is a second modulus which is au-
tomatically stabilized and acts as a source of cold dark matter, with a mass proportional to an
exponential function of the quintessence field (hence realizing VAMP models within a String con-
text). However, any attempt to saturate the present cold dark matter component in this manner
leads to unacceptable deviations from cosmological data – a numerical study reveals that this
source can account for up to about 7% of the total cold dark matter budget. We also show that
(1) the model will support a de Sitter energy in agreement with observation at the expense of a
miniscule breaking of supersymmetry in the compact space; (2) variations in the fine structure
constant are controlled by the stabilized modulus and are negligible; (3) “fifth” forces are carried
by the stabilized modulus and are short range; (4) the long time behavior of the model in four
dimensions is that of a Robertson-Walker universe with a constant expansion rate (w = −1/3).
Finally, we present a String theory background by lifting our six dimensional cosmological solution
to ten dimensions.
http://arxiv.org/abs/0704.0928v3
I. GENERAL IDEA
The mechanism involved in generating a very small cosmological constant that satisfies
’t Hooft naturalness is one of the most pressing questions in contemporary physics. Re-
cent observations of distant Type Ia supernovae [1] strongly indicate that the universe is
expanding in an accelerating phase, with an effective de-Sitter (dS) constant H that nearly
saturates the upper bound given by the present-day value of the Hubble constant, i.e.,
H <∼ H0 ∼ 10−33 eV. According to the Einstein field equations, H provides a measure of
the scalar curvature of the space and is related to the vacuum energy density ρvac through
Friedmann’s equation, 3M2PlH
2 ∼ ρvac, where MPl ≃ 2.4 × 1018 GeV is the reduced Planck
mass. However, the “natural” value of ρvac coming from the zero-point energies of known
elementary particles is found to be at least ρvac ∼ TeV4. Substitution of this value of ρvac into
Friedmann’s equation yields H >∼ 10−3 eV, grossly inconsistent with the set of supernova
(SN) observations. The absence of a mechanism in agreement with ’t Hooft naturalness
criteria then centers on the following question: why is the vacuum energy needed by the
Einstein field equations 120 orders of magnitude smaller than any “natural” cut-off scale in
effective field theory of particle interactions, but not zero?
Nowadays, the most popular framework which can address aspects of this question is
the anthropic approach, in which the fundamental constants are not determined through
fundamental reasons, but rather because such values are necessary for life (and hence intel-
ligent observers to measure the constants) [2]. Of course, in order to implement this idea in
a concrete physical theory, it is necessary to postulate a multiverse in which fundamental
physical parameters can take different values. Recent investigations in String theory have
applied a statistical approach to the enormous “landscape” of metastable vacua present in
the theory [3]. A vast ensemble of metastable vacua with a small positive effective cosmo-
logical constant that can accommodate the low energy effective field theory of the Standard
Model (SM) have been found. Therefore, the idea of a string landscape has been used to
proposed a concrete implementation of the anthropic principle.
Nevertheless, the compactification of a String/M-theory background to a four dimen-
sional solution undergoing accelerating expansion has proved to be exceedingly difficult.
The obstruction to finding dS solutions in the low energy equations of String/M theory
is well known and summarized in the no-go theorem of [4]. This theorem states that in
a D-dimensional theory of gravity, in which (a) the action is linear in the Ricci scalar
curvature (b) the potential for the matter fields is non-positive and (c) the massless fields
have positive defined kinetic terms, there are no (dynamical) compactifications of the form:
ds2D = Ω
2(y)(dx2d+ ĝmndy
ndym), if the d dimensional space has Minkowski SO(1, d−1) or dS
SO(1, d) isometries and its d dimensional gravitational constant is finite (i.e., the internal
space has finite volume). The conclusions of the theorem can be circumvented if some of its
hypotheses are not satisfied. Examples where the hypotheses can be relaxed exist: (i) one
can find solutions in which not all of the internal dimensions are compact [5]; (ii) one may
try to find a solution breaking Minkowski or de Sitter invariance [6]; (iii) one may try to
add negative tension matter (e.g., in the form of orientifold planes) [7]; (iv) one can even
appeal to some intrincate String dynamics [8].
Salam-Sezgin six dimensional supergravity model [9] provides a specific example where
the no-go theorem is not at work, because when their model is lifted to M theory the
internal space is found to be non-compact [10]. The lower dimensional perspective of this,
is that in six dimensions the potential can be positive. This model has perhaps attracted
the most attention because of the wide range of its phenomenological applications [11]. In
this article we examine the cosmological implications of such a supergravity model during
the epochs subsequent to primordial nucleosynthesis. We derive a solution of Einstein field
equations which is in qualitative agreement with luminosity distance measurements of Type
Ia supernovae [1], primordial nucleosynthesis abundances [12], data from the Sloan Digital
Sky Survey (SDSS) [13], and the most recent measurements from the Wilkinson Microwave
Anisotropy Probe (WMAP) satellite [14]. The observed acceleration of the universe is
driven by the “dark energy” associated to a scalar field slowly rolling down its exponential
potential (i.e., kinetic energy density < potential energy density ≡ negative pressure) [15].
Very interestingly, the resulting cosmological model also predicts a cold dark matter (CDM)
candidate. In analogy with the phenomenological proposal of [16], such a nonbaryonic matter
interacts with the dark energy field and therefore the mass of the CDM particles evolves with
the exponential dark energy potential. However, an attempt to saturate the present CDM
component in this manner leads to gross deviations from present cosmological data. We
will show that this type of CDM can account for up to about 7% of the total CDM budget.
Generalizations of our scenario (using supergravities with more fields) might account for the
rest.
II. SALAM-SEZGIN COSMOLOGY
We begin with the action of Salam-Sezgin six dimensional supergravity [9], setting to
zero the fermionic terms in the background (of course fermionic excitations will arise from
fluctuations),
R− κ2(∂Mσ)2 − κ2eκσF 2MN −
e−κσ − κ
e2κσG2MNP
. (1)
Here, g6 = det gMN , R is the Ricci scalar of gMN , FMN = ∂[MAN ], GMNP = ∂[MBNP ] +
κA[MFNP ], and capital Latin indices run from 0 to 5. A re-scaling of the constants: G6 ≡ 2κ2,
φ ≡ −κσ and ξ ≡ 4 g2 leads to
R− (∂Mφ)2 −
eφ − G6
e−φF 2MN −
e−2φG2MNP
. (2)
The length dimensions of the fields are: [G6] = L
4, [ξ] = L2, [φ] = [g2MN ] = 1, [A
M ] = L
and [F 2MN ] = [G
MNP ] = L
Now, we consider a spontaneous compactification from six dimension to four dimension.
To this end, we take the six dimensional manifold M to be a direct product of 4 Minkowski
directions (hereafter denoted by N1) and a compact orientable two dimensional manifold N2
with constant curvature. Without loss of generality, we can set N2 to be a sphere S
2, or a
Σ2 hyperbolic manifold with arbitrary genus. The metric on M locally takes the form
ds26 = ds4(t, ~x)
2 + e2f(t,~x)dσ2, dσ2 =
r2c (dϑ
2 + sin2 ϑdϕ2) for S2
r2c (dϑ
2 + sinh2 ϑdϕ2) for Σ2 ,
where (t, ~x) denotes a local coordinate system in N1, rc is the compactification radius of N2.
We assume that the scalar field φ is only dependent on the point of N1, i.e., φ = φ(t, ~x).
We further assume that the gauge field AM is excited on N2 and is of the form
b cos ϑ (S2)
b cosh ϑ (Σ2) .
This is the monopole configuration detailed by Salam-Sezgin [9]. Since we set the Kalb-
Ramond field BNP = 0 and the term A[MFNP ] vanishes on N2, GMNP = 0. The field
strength becomes
F 2MN = 2b
2e−4f/r4c . (5)
Taking the variation of the gauge field AM in Eq. (2) we obtain the Maxwell equation
2f−φFMN
= 0. (6)
It is easily seen that the field strengths in Eq. (5) satisfy Eq. (6).
With this in mind, the Ricci scalar reduces to [17]
R[M ] = R[N1] + e
−2fR[N2]− 4✷f − 6(∂µf)2 , (7)
where R[M ], R[N1], and R[N2] denote the Ricci scalars of the manifolds M, N1, and N2;
respectively. (Greek indices run from 0 to 3). The Ricci scalar of N2 reads
R[N2] =
+2/r2c (S
−2/r2c (Σ2).
To simplify the notation, from now on, R1 and R2 indicate R[N1] and R[N2], respectively.
The determinant of the metric can be written as
g6 = e
2f√g4
gσ, where g4 = det gµν and
gσ is the determinant of the metric ofN2 excluding the factor e
2f . We define the gravitational
constant in the four dimension as
2πr2c
. (9)
Hence, by using the field configuration given in Eq. (4) we can re-write the action in Eq. (2)
as follows
e2f [R1 + e
−2fR2 + 2(∂µf)
2 − (∂µφ)2]−
e2f+φ − G6b
e−2f−φ
. (10)
Let us consider now a rescaling of the metric of N1: ĝµν ≡ e2fgµν and
ĝ4 = e
4f√g4. Such a
transformation brings the theory into the Einstein conformal frame where the action given
in Eq. (10) takes the form
R[ĝ4]− 4(∂µf)2 − (∂µφ)2 −
e−2f+φ −
e−6f−φ + e−4fR2
. (11)
The four dimensional Lagrangian is then
R− 4(∂µf)2 − (∂µφ)2 − V (f, φ)
, (12)
V (f, φ) ≡ ξ
e−2f+φ +
e−6f−φ − e−4fR2 , (13)
where to simplify the notation we have defined: g ≡ ĝ4 and R ≡ R[ĝ4].
Let us now define a new orthogonal basis, X ≡ (φ + 2f)/
G4 and Y ≡ (φ − 2f)/
so that the kinetic energy terms in the Lagrangian are both canonical, i.e.,
(∂X)2 − 1
(∂Y )2 − Ṽ (X, Y )
, (14)
where the potential Ṽ (X, Y ) ≡ V (f, φ)/G4 can be re-written (after some elementary algebra)
as [18]
Ṽ (X, Y ) =
G4X −R2e−
G4X +
. (15)
The field equations are
Rµν −
gµνR =
∂µX∂νX −
∂ηX ∂
∂µY ∂νY −
∂ηY ∂
− gµνṼ (X, Y )
, (16)
✷X = ∂X Ṽ , and ✷Y = ∂Y Ṽ . In order to allow for a dS era we assume that the metric takes
the form
ds2 = −dt2 + e2h(t)d~x 2, (17)
and that X and Y depend only on the time coordinate, i.e., X = X(t) and Y = Y (t). Then
the equations of motion for X and Y can be written as
Ẍ + 3ḣẊ = −∂X Ṽ (18)
Ÿ + 3ḣẎ = −∂Y Ṽ , (19)
whereas the only two independent components of Eq. (16) are
ḣ2 =
(Ẋ2 + Ẏ 2) + Ṽ (X, Y )
2ḧ+ 3ḣ2 =
(Ẋ2 + Ẏ 2) + Ṽ (X, Y )
. (21)
The terms in the square brackets in Eq. (15) take the form of a quadratic function of
G4 X . This function has a global minimum at e−
G4 X0 = R2 r
c/(2G6 b
2). Indeed, the
necessary and sufficient condition for a minimum is that R2 > 0, so hereafter we only
consider the spherical compactification, where e−
G4 X0 = M2Pl/(4πb
2). The condition for
the potential to show a dS rather than an AdS or Minkowski phase is ξb2 > 1. Now, we
expand Eq. (15) around the minimum,
Ṽ (X, Y ) =
(X −X0)2 +O
(X −X0)3
 , (22)
where
π brc
4πr2cb
(b2ξ − 1) . (24)
As shown by Salam-Sezgin [9] the requirements for preserving a fraction of supersymmetry
(SUSY) in spherical compactifications to four dimension imply b2ξ = 1, corresponding to
winding number n = ±1 for the monopole configuration. Consequently, a (Y -dependent)
dS background can be obtained only through SUSY breaking. For now we will leave open
the symmetry breaking mechanism and come back to this point after our phenomenological
discussion. The Y -dependent physical mass of the X-particles at any time is
MX(Y ) =
G4 Y/2
MX , (25)
which makes this a varying mass particle (VAMP) model [16], although, in this case, the
dependence on the quintessence field is fixed by the theory. The dS (vacuum) potential
energy density is
K . (26)
In general, classical oscillations for the X particle will occur for
MX > H =
G4ρtot
, (27)
where ρtot is the total energy density. (This condition is well known from axion cosmol-
ogy [19]). A necessary condition for this to hold can be obtained by saturating ρ with VY
from Eq. (26) and making use of Eqs. (23) to (27), which leads to ξb2 < 7. Of course, as
we stray from the present into an era where the dS energy is not dominant, we must check
at every step whether the inequality (27) holds. If the inequality is violated, the X-particle
ceases to behave like CDM.
In what follows, some combination of the parameters of the model will be determined by
fitting present cosmological data. To this end we assume that SM fields are confined to N1
and we denote with ρrad the radiation energy, with ρX the matter energy associated with
the X-particles, and with ρmat the remaining matter density. With this in mind, Eq. (19)
can be re-written as
Ÿ + 3H Ẏ = −∂Veff
, (28)
where Veff ≡ VY + ρX and H is defined by the Friedmann equation
H2 ≡ ḣ2 = 1
3M2Pl
Ẏ 2 + Veff + ρrad + ρmat
. (29)
(Note that the matter energy associated to the X particles is contained in Veff .)
It is more convenient to consider the evolution in u ≡ − ln(1+ z), where z is the redshift
parameter. As long as the oscillation condition is fulfilled, the VAMP CDM energy density
is given in terms of the X-particle number density nX [20]
ρX(Y, u) = MX(Y ) nX(u) = C e
G4Y/2 e−3u , (30)
where C is a constant to be determined by fitting to data. Along with Eq. (26), these define
for us the effective (u-dependent) VAMP potential
Veff(Y, u) ≡ VY + ρX = A e
G4Y + C e
G4Y/2 e−3u , (31)
where a A is just a constant given in terms of model parameters through Eqs. (22) and (24).
Hereafter we adopt natural units, MPl = 1. Denoting by a prime derivatives with respect
to u, the equation of motion for Y becomes
1− Y ′2/6
+ 3 Y ′ +
∂uρ Y
′/2 + 3 ∂Y Veff
= 0 , (32)
where ρ = Veff + ρrad + ρmat. Quantities of importance are the dark energy density
H2 Y ′2 + VY , (33)
generally expressed in units of the critical density (Ω ≡ ρ/ρc)
, (34)
and the Hubble parameter
3− Y ′2/2
. (35)
The equation of state is
H2 Y ′2
H2 Y ′2
. (36)
We pause to note that the exponential potential VY ∼ eλY/MPl , with λ =
2. Asymptotically,
this represents the crossover situation with wY = −1/3 [22], implying expansion at constant
velocity. Nevertheless, we will find that there is a brief period encompassing the recent past
(z <∼ 6) where there has been significant acceleration.
Returning now to the quantitative analysis, we take ρmat = Be
−3u and ρrad =
10−4 ρmat e
−u f(u) [21] where B is a constant and f(u) parameterizes the u-dependent
number of radiation degrees of freedom. In order to interpolate the various thresholds
appearing prior to recombination (among others, QCD and electroweak), we adopt a conve-
nient phenomenological form f(u) = exp(−u/15) [23]. We note at this point that solutions
of Eq. (32) are independent by an overall normalization for the energy density. This is also
true for the dimensionless quantities of interest ΩY and wY .
With these forms for the energy densities, Eq. (32) can be integrated for various choices
of A, B, and C, and initial conditions at u = −30. We take as initial condition Y (−30) = 0.
Because of the slow variation of Y over the range of u, changes in Y (−30) are equivalent
to altering the quantities A and C [24]. In accordance to equipartition arguments [24, 25]
we take Y ′(−30) = 0.08. Because the Y evolution equation depends only on energy density
ratios, and hence only on the ratios A : B : C of the previously introduced constants, we
may, for the purposes of integration and without loss of generality, arbitrarily fix B and
then scan the A and C parameter space for applicable solutions. In Fig. 1 we show a sample
qualitative fit to the data. It has the property of allowing the maximum value of X-CDM
FIG. 1: The upper panel shows the evolution of Y as a function of u. Today corresponds to
z = 0 and for primordial nucleosynthesis z ≈ 1010. We set the initial conditions Y (−30) = 0 and
Y ′(−30) = 0.08; we take A : B : C = 11 : 0.3 : 0.1. The second panel shows the evolution of ΩY
(solid line), Ωmat (dot-dashed line), and Ωrad (dashed line) superposed over experimental best fits
from SDSS and WMAP observations [13, 14]. The curves are not actual fits to the experimental
data but are based on the particular choice of the Y evolution shown in the upper panel, which
provides eyeball agreement with existing astrophysical observations. The lower panel shows the
evolution of the equation of state wY superposed over the best fits to WMAP + SDSS data sets
and WMAP + SNGold [14] . The solution of the field equations is consistent with the requirement
from primordial nucleosynthesis, ΩY < 0.045 (90%CL) [12], it also shows the established radiation
and matter dominated epochs, and at the end shows an accelerated dS era.
(about 7% of the total dark matter component) before the fits deviate unacceptably from
data.
It is worth pausing at this juncture to examine the consequences of this model for vari-
ation in the fine structure constant and long range forces. Specifically, excitations of the
electromagnetic field on N1 will, through the presence of the dilaton factor in Eq. (2), seem-
ingly induce variation in the electromagnetic fine structure constant αem = e
2/4π, as well
as a violation of the equivalence principle through a long range coupling of the dilaton to
the electromagnetic component of the stress tensor. We now show that these effects are
extremely negligible in the present model. First, it is easily seen using Eqs. (2) and (3)
together with Eqs. (8)-(15), that the electromagnetic piece of the lagrangian as viewed from
N1 is
Lem = −
G4X f̃ 2µν , (37)
where f̃µν denotes a quantum fluctuation of the electromagnetic U(1) field. (Fluctuations of
the U(1) background field are studied in the Appendix). At the equilibrium value X = X0,
the exponential factor is
G4X0 =
, (38)
so that we can identify the electromagnetic coupling (1/e2) ≃ M2Pl/b2. This shows that
b ∼ MPl. We can then expand about the equilibrium point, and obtain an additional factor
of (X − X0)/MPl. This will do two things [26]: (a) At the classical level, it will induce a
variation of the electromagnetic coupling as X varies, with ∆αem/αem ≃ (X − X0)/MPl;
(b) at the quantum level, exchange of X quanta will induce a new force through coupling to
the electromagnetic component of matter.
Item (b) is dangerous if the mass of the exchanged quanta are small, so that the force
is long range. This is not the case in the present model: from Eq. (22) the X quanta have
mass of O(MXMPl) ∼ MPl/(rcb), so that if rc is much less than O(cm), the forces will play
no role in the laboratory or cosmologically.
As far as the variation of αem is concerned, we find that ρX/ρmat = (C/B)e
2, so that
ρX ≃ 3× 10−120e−3uM4PleY/
X(X −X0)2eY
2M2Pl . (39)
This then gives,
〈(X −X0)2〉 ≡ ∆Xrms ≈ 10−60e−3u/2MPleY/(2
2)/MX . (40)
During the radiation era, Y ≃ const ≃ 0 (see Fig. 1), so that during nucleosynthesis
(u ≃ −23) ∆Xrms/MPl ≃ 10−45/MX , certainly no threat. It is interesting that such a small
value can be understood as a result of inflation: from the equation of motion for the X field,
it is simple to see that during a dS era with Hubble constant H , the amplitude ∆Xrms is
damped as e−3Ht/2. For 50 e-foldings, this represents a damping of 1032. In order to make the
numbers match (assuming a pre-inflation value ∆Xrms/MPl ∼ 1) an additional damping of
∼ 1013 is required from reheat temperature to primordial nucleosynthesis. With the e−3u/2
behavior, this implies a low reheat temperature, about 106 GeV. Otherwise, one may just
assume an additional fine-tuning of the initial condition on X .
As mentioned previously, the solutions of Eq. (32), as well as the quantities we are fitting
to (ΩY and wY ), depend only on the ratios of the energy densities. From the eyeball fit in
Fig. 1 we have, up to a common constant, ρordinary matter ≡ ρmat ∝ 0.3 e−3u and VY ∝ 11 e
We can deduce from these relations that
VY (now)
ρmat(now)
2Y (now) ≃ 36 e
2Y (now) . (41)
Besides, we know that ρmat(now) ≃ 0.3ρc(now) ≃ 10−120 M4Pl. Now, Eqs. (22) and (24) lead
VY (now) = e
2Y (now) M
8π r2c b
(b2ξ − 1) (42)
so that from Eqs. (41) and (42) we obtain
8π r2c b
(b2ξ − 1) ≃ 10−119 . (43)
It is apparent that this condition cannot be naturally accomplished by choosing large values
of rc and/or b. There remains the possibility that SUSY breaking [27] or non-perturbative
effects lead to an exponentially small deviation of b2ξ from unity, such that b2ξ = 1 +
O(10−119) [29]. Since a deviation of b2ξ from unity involves a breaking of supersymmetry,
a small value for this dimensionless parameter, perhaps (1 TeV/MPl)
2 ∼ 10−31, can be
expected on the basis of ’t Hooft naturalness. It is the extent of the smallness, of course,
which remains to be explained.
III. THE STRING CONNECTION
We now briefly comment on how the six dimensional solution derived above reads in
String theory. To this end, we use the uplifting formulae developed by Cvetic, Gibbons and
Pope [10]; we will denote with the subscript “cgp” the quantities of that paper and with
“us” quantities in our paper. Let us more specifically look at Eq. (34) in Ref. [10], where
the authors described the six dimensional Lagrangian they uplifted to Type I String theory.
By simple inspection, we can see that the relation between their variables and fields with
the ones we used in Eq. (2) is φ|cgp = −2φ|us, F2|cgp =
G6F2|us, H3|cgp =
G6/3G3|us, and
ḡ2|cgp = ξ/(8G6)|us. Our six dimensional background is determined by the (string frame)
metric ds26 = e
− dt2 + e2hdx23 + r2c dσ22
, the gauge field Fϑϕ = −b sin ϑ, and the t-
dependent functions h(t), f(t) =
G4 (X − Y )/4, and φ(t) =
G4 (X + Y )/2. Identifying
these expressions with those in Eqs. (47), (48) and (49) of Ref. [10] one obtains a full Type
I or Type IIB configuration, consisting of a 3-form (denoted by F3),
8G6 sinh ρ̂ cosh ρ̂
ξ cosh2 2ρ̂
dρ̂ ∧
b cos ϑdϕ
b cosϑdϕ
2G6b√
ξ cosh 2ρ̂
sinϑdθ ∧ dϕ ∧
cosh2 ρ̂
b cosϑdϕ
− sinh2 ρ̂
dβ +
b cosϑdϕ
 , (44)
a dilaton (denoted by φ̂)
e2φ̂ =
cosh(2ρ̂)
, (45)
and a ten dimensional metric that in the string frame reads
ds2str = e
φ ds26 + dz
dρ̂2 +
cosh2 ρ̂
cosh 2ρ̂
b cosϑdϕ
sinh2 ρ̂
cosh 2ρ̂
dβ +
b cosϑdϕ
 , (46)
where ρ̂, z, α, and β denote the four extra coordinates. It is important to stress that though
the uplifted procedure decribed above implies a non-compact internal manifold, the metric
in Eq. (46) can be interpreted within the context of [7] (i.e., 0 ≤ ρ̂ ≤ L, with L ≫ 1 an
infrared cutoff where the spacetime smoothly closes up) to obtain a finite volume for the
internal space and consequently a non-zero but tiny value for G6.
IV. CONCLUSIONS
We studied the six dimensional Salam-Sezgin model [9], where a solution of the form
Minkowski4×S2 is known to exist, with a U(1) monopole serving as background in the two-
sphere. This model circumvents the hypotheses of the no-go theorem [4] and then when lifted
to String theory can show a dS phase. In this work we have allowed for time dependence
of the six-dimensional moduli fields and metric (with a Robertson-Walker form). Time
dependence in these fields vitiates invariance under the supersymmetry transformations.
With these constructs, we have obtained the following results:
(1) In terms of linear combinations of the S2 moduli field and the six dimensional dilaton,
the effective potential consists of (a) a pure exponential function of a quintessence field
(this piece vanishes in the supersymmetric limit of the static theory) and (b) a part which
is a source of cold dark matter, with a mass proportional to an exponential function of the
quintessence field. This presence of a VAMP CDM candidate is inherent in the model.
(2) If the monopole strength is precisely at the value prescribed by supersymmetry, the
model is in gross disagreement with present cosmological data – there is no accelerative
phase, and the contribution of energy from the quintessence field is purely kinetic.
However, a miniscule deviation of O(10−120) from this value permits a qualitative match
with data. Contribution from the VAMP component to the matter energy density can be
as large as about 7% without having negative impact on the fit. The emergence of a
VAMP CDM candidate as a necessary companion of dark energy has been a surprising
aspect of the present findings, and perhaps encouraging for future exploration of
candidates which can assume a more prominent role in the CDM sector.
(3) In our model, the exponential potential VY ∼ eλY/MPl , with Y the quintessence field
and λ =
2. The asymptotic behavior of the scale factor for exponential potentials
eh(t) ≈ t2/λ2 , so that for our case h ≈ ln t, leading to a conformally flat Robertson-Walker
metric for large times. The deviation from constant velocity expansion into a brief
accelerated phase in the neighborhood of our era makes the model phenomenologically
viable. In the case that the supersymmetry condition (b2ξ = 1) is imposed, and there is
neither radiant energy nor dark matter except for the X contribution, we find for large
times that the scale parameter eh(t) ≈
t, so that even in this case the asymptotic metric
is Robertson-Walker rather than Minkowski. Moreover, and rather intriguingly, the scale
parameter is what one would find with radiation alone [28].
In sum, in spite of the shortcomings of the model (not a perfect fit, requirement of a tiny
deviation from supersymmetric prescription for the monopole embedding), it has provided
a stimulating new, and unifying, look at the dark energy and dark matter puzzles.
Acknowledgments
We would like to thank Costas Bachas and Roberto Emparan for valuable discussions.
The research of HG was supported in part by the National Science Foundation under Grant
No. PHY-0244507.
V. APPENDIX
In this appendix we study the quantum fluctuations of the U(1) field associted to the
background configuration. We start by considering fluctuations of the background field A0M
in the 4 dimensional space, i.e,
AM → A0M + ǫ aM , (47)
where A0M = 0 if M 6= ϕ and aM = 0 if M = ϑ, ϕ. The fluctuations on A0M lead to
FMN → F 0MN + ǫ fMN . (48)
Then,
MN = gML gNP [F 0MNF
LP + ǫ F
MN fLP + ǫ
2fMN fLP ] . (49)
The second term vanishes and the first and third terms are nonzero because F 0MN 6= 0 in
the compact space and fMN 6= 0 in the 4 dimensional space. If the Kalb-Ramond potential
BNM = 0, then the 3-form field strength can be written as
GMNP = κA[M FNP ] =
[AM FNP + AP FMN −AN FMP ] . (50)
Now we introduce notation of differential forms, in which the usual Maxwell field and field
strenght read
A1 = AMdx
M and F2 = FMN dx
M ∧ dxN ; (51)
respectively. (Note that dxM ∧ dxN is antisymmetrized by definition.) With this in mind
the 3-form reads
G3 = κA1 ∧ F2 = κAMFNP dxM ∧ dxN ∧ dxP . (52)
Substituting Eqs. (47) and (48) into Eq. (52) we obtain
G3 = κ
(A0M + ǫaM )(F
NP + ǫfNP ) dx
M ∧ dxN ∧ dxP
. (53)
The background fields read
A01 = b cosϑ dϕ, F
2 = −b sinϑ dϑ ∧ dϕ , (54)
and the fluctuations on the probe brane become
a1 = aµdx
µ, f2 = fdx
µ ∧ dxν , with f = ∂µaν − ∂µaν . (55)
All in all,
= A0ϕF
ϑϕ dϕ ∧ dϑ ∧ dϕ+ ǫA0ϕfµν dϕ ∧ dxµ ∧ dxν + ǫF 0ϑϕaµ dϑ ∧ dϕ ∧ dxµ
+ ǫ2aµfζνdx
µ ∧ dxζ ∧ dxν . (56)
Using Eq. (54) and the antisymmetry of the wedge product, Eq. (56) can be re-written as
b cos ϑfµνdϕ ∧ dxµ ∧ dxν − baµ sinϑdϑ ∧ dϕ ∧ dxµ + ǫaµfζνdxµ ∧ dxζ ∧ dxν
. (57)
From the metric
ds2 = e2αdx24 + e
2β(dϑ2 + sin ϑ2dϕ2) (58)
we can write the vielbeins
ea = eαdxa, eϑ = eβdϑ, eϕ = eβ sinϑdϕ,
dxa = e−αea, dϑ = e−βeϑ, dϕ =
eϕ (59)
where β ≡ f+ln rc. (Lower latin indeces from the beginning of the alphabet indicate coordi-
nates associted to the four dimensional Minkowski spacetime with metric ηab.) Substituting
into Eq. (57) we obtain
cos ϑ
sin ϑ
e−2α−βfabe
ϕ ∧ ea ∧ eb − be−α−2βaaeϑ ∧ eϕ ∧ ea + ǫe−3αaafcbea ∧ ec ∧ eb
, (60)
where fab = ∂aab − ∂baa. Because the three terms are orthogonal to each other straightfor-
ward calculation leads to
G23 = κ
2ǫ2(b2 cot2 ϑ e−4α−2βf 2ab + b
2e−2α−4βa2a) +O(ǫ4) . (61)
Then, the 5th term in Eq. (2) can be written as
SG3 = −
e4α+2β
dϑdϕ sinϑ
κ2ǫ2b2 cot2 ϑe−4α−2β
f 2ab
κ2ǫ2b2e−2α−4β
, (62)
whereas the contribution from the 4th term in Eq. (2) can be computed from Eq. (49)
yielding
SF2 = −
η42πe
2β−φG6ǫ
2f 2ab
2f−φr2cǫ
2f 2ab . (63)
Thus,
SG3 + SF2 = −
f 2ab +
, (64)
where the four dimensional effective coupling and the effective mass are of the form
= 4 ǫ2
πe2f−φr2c +
κ2b2e−2φ
dϑdϕ sinϑ cot2 ϑ
→ ∞ (65)
πκ2b2ǫ2e2α−2β−2φ . (66)
For the moment we let
dϑdϕ sinϑ cot2 ϑ = N , where eventually we set N → ∞. Now
to make quantum particle identification and coupling, we carry out the transformation
aa → gâa [30]. This implies that the second term in the right hand side of Eq. (64) vanishes,
yielding
fab = ∂a(gâb)− ∂b(gâa) = ∂ag âb − ∂bg âa + g ∂aâb − g ∂bâa = gf̂ab + â ∧ dg (67)
and consequently to leading order in N
f 2ab =
[g2f̂ 2ab + (â ∧ dg)2 + 2 g âb f̂ab ∂ag] . (68)
If the coupling depends only on the time variable,
f 2ab → f̂ 2ab +
â2a + 2
âi f̂
ti (69)
where ġ = ∂tg and lower latin indices from the middle of the alphabet refer to the brane
space-like dimensions. If we choose a time-like gauge in which at = 0, then the term
(ġ/g) âi f̂
ti can be written as (1/2)(ġ/g)(d/dt)(âi)
2, which after an integration by parts
gives −(1/2)[(d/dt)(ġ/g)]â2i ; with g ∼ e−φ, the factor in square brackets becomes −φ̈. Since
G4(X + Y ), the rapidly varying Ẍ will average to zero, and one is left just with the
very small Ÿ , which is of order Hubble square. For the term (ġ/g)2(ai)
2, the term (Ẋ)2
also averages to order Hubble square, implying that the induced mass term is of horizon
size. These “paraphotons” carry new relativistic degrees of freedom, which could in turn
modify the Hubble expansion rate during Big Bang nucleosynthesis (BBN). Note, however,
that these extremely light gauge bosons are thought to be created through inflaton decay
and their interactions are only relevant at Planck-type energies. Since the quantum gravity
era, all the paraphotons have been redshifting down without being subject to reheating, and
consequently at BBN they only count for a fraction of an extra neutrino species in agreement
with observations.
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[28] This comes from a behavior Y ≃ −
2u (compatible with the equations of motion), when
combined with the e−3u in Eq. (30).
[29] Before proceeding, we remind the reader that the requirements for preserving a fraction of
SUSY in spherical compactifications to four dimensions imply b2ξ = 1, corresponding to
the winding number n = ±1 for the monopole configuration. In terms of the Bohm-Aharonov
argument on phases, this is consistent with usual requirement of quantization of the monopole.
The SUSY breaking has associated a non-quantized flux of the field supporting the two sphere.
In other words, if we perform a Bohm-Aharonov-like interference experiment, some phase
change will be detected by a U(1) charged particle that circulates around the associated Dirac
string. The quantization of fluxes implied the unobservability of such a phase, and so in our
cosmological set up, the parallel transport of a fermion will be slightly path dependent. One
possibility is that the non-compact ρ coordinate (in the uplift to ten dimensions, see Sec. III) is
the direction in which the Dirac string exists. Then the cutoff necessary on the physics at large
ρ will introduce a slight (time-dependent) perturbation on the flux quantization condition. We
are engaged at present in exploring possibilities along this line.
[30] This is because the definition of the propagator with proper residue for correct Feyman rules
in perturbation theory, and therefore also the couplings, needs to be consistent with the form
of the Hamiltonian =
k ω(k)a
kak, with [a, a
†] = 1. This in turn implies that the kinetic term
in the Lagrangian has the canonical form, (1/4)f̂2ab, with the usual expansion of the vector
field aa.
