CPHT–RR079.0709, LPTENS–09/24, August 2009
Cosmological Phases of the String Thermal Effective Potential
F. Bourliot, J. Estes, C. Kounnas and H. Partouche
Centre de Physique Théorique, Ecole Polytechnique,
F–91128 Palaiseau cedex, France
Laboratoire de Physique Théorique,
Ecole Normale Supérieure,
24 rue Lhomond, F–75231 Paris cedex 05, France
In a superstring framework, the free energy density can be determined unambiguously at the full string level once supersymmetry is spontaneously broken via geometrical fluxes. We show explicitly that only the moduli associated to the supersymmetry breaking may give relevant contributions. All other spectator moduli give exponentially suppressed contributions for relatively small (as compared to the string scale) temperature and supersymmetry breaking scale . More concisely, for and , takes the form
We study the cosmological regime where and are below the Hagedorn temperature scale . In this regime, remains finite for any values of the spectator moduli . We investigate extensively the case of one spectator modulus corresponding to , the radius-modulus field of an internal compactified dimension. We show that its thermal effective potential admits five phases, each of which can be described by a distinct but different effective field theory. For late cosmological times, the Universe is attracted to a “Radiation-like evolution” with . The spectator modulus is stabilized either to the stringy enhanced symmetry point where , or fixed at an arbitrary constant . For arbitrary boundary conditions at some initial time, , may pass through more than one effective field theory phase before its final attraction.
partially supported by the ERC Advanced Grant 226371, ANR contract 05-BLAN-0079-02, CNRS PICS contracts 3747 and 4172, and the Groupement d’Intérêt Scientifique P2I.
Unité mixte du CNRS et de l’Ecole Polytechnique, UMR 7644.
Unité mixte du CNRS et de l’Ecole Normale Supérieure associée à l’Université Pierre et Marie Curie (Paris 6), UMR 8549.
String theory provides a framework to obtain a sensible theoretical description of the cosmological evolution of our Universe. Nowadays, it is the only known framework in which the quantum gravity effects are under control , at least for certain physically relevant cases. Following the stringy cosmological approach developed recently in Refs [2, 3, 4], the classical string vacuum is taken to be supersymmetric with a fixed amount of supersymmetries defined in flat space-time.
This initial choice does not give rise to any cosmological evolution. In the presence of supersymmetry, the quantum corrections to the gravitational background would lead to a flat space-time, or would modify it at most to Anti-de Sitter, domain walls or gravitational wave backgrounds respecting a time-like or light-like killing symmetry. The above cosmological obstructions are however physically irrelevant for two fundamental reasons:
Firstly, supersymmetry is broken in the real world, (at least spontaneously and not explicitly), at a characteristic supersymmetry breaking scale .
Secondly, in the case of thermal cosmologies, the supersymmetry is effectively (spontaneously) broken at the temperature scale .
Both the and supersymmetry breaking scales induce at the quantum level a non-trivial free energy density , which plays the role of an effective thermal potential that modifies the gravitational and field equations, giving rise to non-trivial cosmological solutions, as has been explicitly shown in Refs [2, 3, 5]. Both the supersymmetry breaking and finite temperature phenomena can be implemented in the framework of superstrings [6, 7, 8] by introducing non-trivial “fluxes” in the initially supersymmetric vacua. Furthermore, in the case where supersymmetry is spontaneously broken by “geometrical fluxes” [9, 10], the free energy is under control and is calculable at the full string level, free of any infrared and ultraviolet ambiguities [2, 3]. This is true, provided and are below a critical value close to the string mass scale, the so-called Hagedorn temperature [6, 7, 8, 11]. In the framework of stringy-thermal cosmologies, corresponds to very early times when we are facing non-trivial stringy singularities indicating a non-trivial phase transition at high temperatures [6, 8, 12, 13, 15] at time . In the literature, there are many speculative proposals concerning the nature of this transition [6, 8, 12, 13, 14, 15, 16].
A way to bypass the Hagedorn transition ambiguities was proposed in Ref. . It consists of assuming the emergence of () large space-like directions for times , describing the ()-dimensional space of the Universe, and possibly some internal space directions of an intermediate size characterizing the scale of the spontaneous breaking of supersymmetry via geometrical fluxes [9, 10]. Within these assumptions, the ambiguities of the “Hagedorn transition exit at ” can be parameterized, for , in terms of initial boundary condition data at . In this way, the intermediate cosmological era , i.e. after the “Hagedorn transition exit” and before the electroweak symmetry breaking phase transition at , was extensively studied in Ref.  in the case of . An output of the present analysis is that the cosmological “radiation-like” evolution found in Refs [2, 3, 5, 17] generalizes to a “Radiation-like Dominated Solution” (RDS) in -dimensional space-time,
and is unique at late times in certain physically relevant supersymmetry breaking schemes. As a necessary and sufficient consistency requirement, we note that in this intermediate cosmological regime , the smallness of the space-time curvature scales, and , the dilaton and the evolving radii scales, , ,, , are guaranteed to be small (), thanks to the “attractor mechanism” towards the RDS in late cosmological times. In particular, they are all decreasing at late cosmological times and so our quasi-static approximation becomes better and better as time passes. In addition, our perturbative approximation becomes better and better as time progress due to the falling of the dilaton.
We point out that the evolution is radiation-like in the sense that the external space-time’s evolution is identical to that of a radiation dominated universe. However, in our case the expansion is driven not only by radiation, but also by the coherent motion of the supersymmetry breaking modulus . We would also like to stress that a key result of  was that the is actually an attractor of the dynamics. The important consequence of this attractor, is to wash out the dependence of the cosmological evolution on the choice of initial boundary conditions, which were used to parameterize our ignorance of the physics involved in the Hagedorn transition. In this paper we show that the attractor naturally extends to the higher-dimensional .
Although this analysis was done in the framework of initial vacua with supersymmetry, the claim is that it will still be valid in more realistic models with initial supersymmetry . We would like to stress here that the limitation in the infrared regime follows from the appearance in the low energy effective field theory of a new scale, namely the “infrared renormalization group invariant transmutation scale ”, at which the supersymmetric standard model Higgs (mass) becomes negative, (no-scale radiative breaking of [19, 20]). is irrelevant as long as ; however it becomes relevant and stops the evolution when at i.e. when the electroweak breaking phase transition takes place. Although the physics for is of main importance in particle physics and in inflationary cosmology at , it will not be examined in this work. The main reason for us is its strong dependence on the initial vacuum data which screens interesting universality properties. We therefore work in the intermediate cosmological era or , i.e. after the Hagedorn phase transition and before the electroweak one. In this regime the transmutation scale can be consistently neglected and, furthermore, the Hagedorn transition ambiguities are taken into account in terms of initial boundary conditions (IBC) after the “Hagedorn transition exit” at . This scenario gives a dynamical explanation of the smallness of the supersymmetry breaking scale as compared to the string or Planck scales. Indeed, extrapolating the up to the low energy regime where one finds, (thanks to the attractor mechanism), that the natural value of the supersymmetry breaking scale is naturally small and around the electroweak phase transition, independently of its initial value at early cosmological times.
The only known supersymmetry breaking mechanism that can be unambiguously adapted at the string perturbative level, is the one we consider here where supersymmetry is spontaneously broken via “geometrical fluxes”. This choice implies the existence of at least one relatively large compact dimension (the one which is associated to supersymmetry breaking). This is by far not in contradiction with experimental results both in particle physics and cosmology. For several years, the possibility of “large extra dimensions” has attracted the attention of the particle physics community; the future data analysis at the LHC and elsewhere includes searches for signals indicating the existence of large extra dimensions at scales , which is the characteristic prediction of supersymmetry breaking via geometrical fluxes.
Many other choices for supersymmetry breaking exist. However in most cases it is not well known, in our days, the precise stringy corrections to all orders in that are necessary in order to study the intermediate cosmological regime. In all other supersymmetry breaking mechanisms we are forced to work in the effective supergravity framework. Our results may help to make more general stringy approaches possible in the future. There are indications that our results can be converted to other string vacua, utilizing string/string and M-theory dualities. In particular the geometrical fluxes are mapped to other types of fluxes, for instance the three form R-R and NS-NS fluxes in type IIB - orientifolds. In this respect, the exact stringy approach in the intermediate cosmological regime gives us profound (non-perturbative) information via M-theory and string dualities.
In addition to generalizing the results of  to arbitrary dimension, we analyze the time behavior of the spectator moduli not participating in the breaking of supersymmetry. Following Refs [2, 3, 5], one can show that only the supersymmetry breaking moduli and can give a relevant contribution to the free energy density . Intuitively, all other moduli are either attracted and stabilized to the “stringy” extended gauge symmetry points, close to the string scale , or are effectively frozen to an arbitrary value such that and , giving rise to exponentially suppressed contributions:
One point of the present paper is to explicitly verify this intuition for the moduli coming from the spectator tori. Indeed, the supersymmetry breaking moduli generate a non-trivial potential for the spectator moduli and freeze them as expected. Considering the effect on a single spectator modulus () in the case of the heterotic string, the thermal effective potential admits five distinct phases, each of which can be described by a different effective field theory. The interesting result is that by using a string theory framework and working at the full string level, we are able to link together, within a single framework, the different effective field theories. Furthermore, the main result of this paper is to derive at the string perturbative level, (however exact in ), the full string free energy as a functional of the supersymmetry breaking moduli , the string coupling constant modulus and “spectator moduli” , for a certain class of string vacua where the spontaneous breaking of supersymmetry is induced by geometrical fluxes.
The form of the potential is sketched in Fig. 1 and the phases are summarized as follows ():
I. Higgs phase: With
This phase contains the stringy extended symmetry point at the self-dual point . The appropriate effective field theory description of this phase is in terms of a -dimensional theory of gravity coupled to an gauge theory. is dynamically stabilized at this point. Such a notion of moduli stabilization, has been studied in the literature before [21, 22, 13]. Here we demonstrate such moduli stabilization in the context of the heterotic superstring by an explicit computation of the effective potential.
II. Flat potential phase: With
Here, the appropriate effective field theory description is in terms of a -dimensional theory of gravity coupled to an gauge field.
III. Higher-dimensional phase: With
For macroscopic values of the spectator modulus , the appropriate effective field theory description is the -dimensional theory of gravity. The modulus becomes the component of the metric, and the evolution is attracted to that of an RDS in dimensions.
One may also consider the case in which the radius is still internal. For large enough values of so that we can neglect terms of order and , the evolution is attracted to an RDS for a long period of time. However, at late times and always catch and the solution is ultimately attracted to the RDS of phase II.
IV. Dual flat potential phase: With
The effective theory description is T-dual to that of phase II. The light degrees of freedom are the winding modes instead of the Kaluza-Klein momenta of phase II.
V. Dual higher-dimensional phase: With
This phase is T-dual to phase III. Its properties are derived from the ones of phase III under the replacement and phase II IV. In particular, the dual effective field theory is a -dimensional theory of gravity, with .
The above different phases of a common string setting cannot be described in the context of a single field theory. This is due to the necessary presence of the string winding modes. In a field theory framework, only the phases II and III (or IV and V) can be described by a common field theory. The winding modes are particularly important for the stabilization of the modulus in phase I at the extended symmetry point, and furthermore for the description of the T-dual phases IV and V. In contrast to field theory, string theory naturally interpolates between these various phases, due to the generation of an effective potential in the presence of temperature and spontaneous supersymmetry breaking.
For each phase, there exists an RDS as in (1.1). We show that these solutions are stable against small perturbations and that for arbitrary IBC close to an RDS, the cosmological evolution is attracted to this RDS. In , the spectator moduli were taken to be frozen and it was shown that under this hypothesis the RDS is a global attractor. Taking into account the existence of gravitational friction for an expanding universe, we expect in the present work with dynamical spectator moduli such as , the results of  to generalize so that the evolution is always attracted to the RDS of one of the five phases.
In the case of type II strings, the phase I does not exist perturbatively, so that the phases II and IV combine into a single plateau. This is due to the lack of massless states necessary to enhance the to . However, by heterotic-type II duality, we expect such an enhancement to exist non-perturbatively, so that all five phases should exist at the non-perturbative level. The effects correspond to the addition of branes whose separation is governed by the spectator moduli.
The organization of the paper is as follows. In section 2, we discuss in more details the specific setup analyzed in this paper. The thermal effective potential is given and it is shown to have the five phases discussed above. In section 3, we show that an RDS solution exists in each phase. In addition, we show the stability of the solutions against small perturbations. In section 4, we briefly discuss the role of non-perturbative objects in the type II string theory. In section 5, we summarize our results and discuss further avenues of research. In appendix A, the thermal partition functions for the heterotic and type II strings are presented, together with their asymptotic properties to be used in the different phases. The gravity and field equations are given for each effective field theory phase in appendix B.
2 Effective thermal potential in superstrings with spontaneously broken supersymmetry
In the presence of temperature, the one-loop partition function of both the heterotic and type II superstrings is non-vanishing and yields the one-loop effective potential at finite temperature. In addition, spontaneous supersymmetry breaking is induced by the presence of geometrical fluxes[2, 3] along the internal cycles of the background manifold. We introduce these fluxes via the generalization to the context of string theory of Scherk-Schwarz  compactifications in field theory[24, 25]. They induce further contributions to the one-loop partition function, which persist even at zero temperature[2, 3]. Due to various ways of introducing the fluxes, there are multiple supersymmetry breaking configurations for the same initially supersymmetric background. In [2, 3], such one-loop thermal effective potentials were derived in the limit of small temperature and small supersymmetry breaking scale for the heterotic and type II superstrings compactified on and orbifolds. The partition functions were calculated for small but otherwise arbitrary temperature and supersymmetry breaking scale, while the remaining moduli were taken to be frozen close to the string scale.
We want to relax the latter hypothesis and examine the behavior of the spectator moduli in the presence of temperature and supersymmetry breaking. In appendix A, we compute the partition functions for the heterotic (A.16) and type II (A.21) cases. The background manifold is of the form (or in the orbifold models), where is the compact Euclidean time circle and the torus involves the geometrical fluxes which generate the breaking of supersymmetry. The spectator moduli are not participating in the breaking of supersymmetry. We take both the temperature and supersymmetry breaking scales to be small, while allowing the spectator moduli to remain arbitrary. This enables us to study the resulting effects of the effective thermal potential on the moduli.
For simplicity, we specialize to the following 10-dimensional Euclidean background that contains:
The Euclidean time direction, with radius which determines the temperature .
The directions, which are taken to be very large and form, together with the time, a -dimensional space-time.
The circle , with arbitrary radius. For small , is considered as part of the internal compactified space. For macroscopic , however, becomes part of a space-time of dimension . By macroscopic we mean that we can probe it with current experiments. is the only “spectator” radius whose dynamics is taken into account.
The circle involved in the spontaneous breaking of supersymmetry. We take it to be along the compact direction , with radius .
The remaining compact directions, with radii . They are taken to be fixed close to the string scale. (In the orbifold models, the factor spans the directions . Its dynamics are consider in the companion paper .)
Utilizing the general expressions associated with the heterotic and type II partition functions given in appendix A, we can easily obtain the ones associated to the background chosen above, namely
where or . The heterotic (type II) models admit a supersymmetry characterized by 16 or 8 (32 or 16) supercharges, which are spontaneously broken by the “stringy Scherk-Schwarz compactifications” in the directions 9 and 0[2, 3]. The scales of supersymmetry breaking and temperature are characterized by and , respectively. We take and to be large, but still much smaller than the radii of the torus so that we have the following inequality
As long as is smaller than the size of the external space , it is more convenient to express the effective field theory action in terms of fields, which have a natural interpretation in dimensions. We are interested in isotropic and homogeneous backgrounds. More specifically, we take the gauge fields to be pure gauge and the remaining scalar fields and the space-time metric to depend only on time. This will have the advantage that after such a reduction, the different effective field theories will be describable within a single framework. The backgrounds we will consider are non-trivial for the -dimensional metric , the -dimensional dilaton , and the moduli fields. However, since we allow to vary arbitrarily in size, it may become of the order of the external radii, so that should be considered as a part of a -dimensional space-time. In this case the effective action is naturally expressed in terms of redefined fields and space-time metric in dimensions.
In appendix B, the dimensional reduction from 10 dimensions to dimensions is carried out explicitly and the resulting action in Einstein frame is given in (B.7). The case we are considering here has , (where in the toroidal models and in the orbifold ones) and . The resulting action is
where we have defined the normalized fields,
The remaining moduli are taken to be fixed close to the string scale.
The source is a pressure equal to , the free energy density (see Eq. (B.12)). It is the opposite of the one-loop effective potential at finite temperature and is related to the one-loop partition function as,
where is the -dimensional Euclidean volume (in string frame) and is the partition function computed in appendix A. In the next sub-section we shall give the exact form of in terms of a convenient set of variables.
2.1 Specific form of the effective thermal potential
For the heterotic case, the partition function is given in Eq. (A.16) which can be re-written in the following convenient form
For the type II case, Eq. (A.21), there is no contribution .
In the heterotic case, is generically suppressed except when is close to its self-dual point, where an enhancement of the gauge group occurs. The contribution for generic can be written in two equivalent forms
Observe that in the ratio , the dependence drops out. The expression for gets simplified drastically once it is written in terms of the complex structure modulus ,
In terms of the independent variables , the pressure takes the factorized form
with and defined in (2.7). Furthermore, can be written in terms of functions with natural interpretations either in or dimensions. (In Eq. (A.16), the first case corresponds to and the second to .) In the heterotic case, the two equivalent forms for are:
with the functions defined below. For the type II case, one simply takes . Note that is an even function of , as follows from T-duality . In this expression, is the number of massless boson/fermion pairs of states in the originally supersymmetric background, for generic . is the number of additional ones at the enhanced gauge symmetry point. The value of is given by the sum over the pairs, with each pair weighted by a sign. The distribution of signs depends on the specific supersymmetry breaking configuration and can yield a negative . For the heterotic models we consider, one has
The definitions of the various functions appearing in Eq. (2.13) are given by
where are modified Bessel functions of the second kind. The remaining functions with lower index are related to those with lower index by duality transformations ():
We will first focus on the dynamics of the modulus , and so we consider the behavior of at fixed , and . Note that when the functions and can be neglected, the pressure only depends on two quantities, and .
2.2 The five heterotic effective field theory phases
Considering the effect on a single spectator modulus in the case of the heterotic string, the thermal effective potential admits five distinct phases corresponding to different effective field theories. The form of the potential is sketched in Fig. 1 and the phases are summarized as follows:
I: Higgs phase
This phase contains the stringy extended symmetry point at the self-dual point . The appropriate effective field theory description is in terms of a -dimensional theory of gravity coupled to an gauge field. We will see that the modulus can be stabilized at the self-dual point which turns out to be the minimum of the effective thermal potential. Indeed, considering the expression of in (2.13), the functions and/or are of the same order as and , while and are exponentially small due to the behavior of the modified Bessel functions . In particular, one has at the origin the following behavior,
This is precisely the form obtained in , when the dynamics of was ignored i.e. was taken to be stabilized close to the string scale. In Eq. (2.18), the contribution of and is of the same form since both contributions come from massless states when is at the enhanced gauge symmetry point . Due to the fact that and are positive, the extremum of at is always a minimum.
II: The flat potential phase
For this range of the modulus, there exists a description in terms of a -dimensional theory of gravity coupled to a gauge field. Note that the range of this region grows as and increase. Modulo exponentially suppressed terms , the potential for the modulus is flat. We will see that for certain IBC, the modulus may be frozen to an arbitrary value on this plateau, due to the gravitational friction of the expanding universe. The exponentially suppressed terms are irrelevant in this phase and cannot modify this behavior.
In this range (2.19), the contributions of and , as well as and are exponentially small compared to and , so that is independent of . The pressure reproduces the result in dimensions for massless boson/fermion pairs in the originally supersymmetric model (as opposed to phase I which has massless pairs). Physically, we are away from the enhanced symmetry point and so the previous states are no longer massless. More concretely, along the plateau we have
Either sign of this quantity is allowed when . Indeed, considering the large behavior, (2.20) implies
where (and for later use) is a constant,
and we see that may take any value.
III: Higher-dimensional phase
For large values of the spectator modulus , the appropriate effective field theory description is the -dimensional theory of gravity. The modulus becomes the component of the string frame metric, .
All contributions of in and/or are substantial, and the behavior of is better understood in terms of its second expression in (2.13). In addition, and are exponentially small. In particular, for and (which are both )
2, one has (2.24)
where the term is power-like subdominant. In this expression, we neglect terms that are exponentially small in or . The appearance of the functions and confirms that it is more natural to consider the system in dimensions. This is the case since, in this limit, the circle of radius is very large. In Fig. 1, the exponential growth of when is decreasing. This is always the case when but for is only true when i.e. is small enough. This can be seen by considering the large limit of (2.24),
where is defined in (2.22). We will see that the attraction to the RDS implies to evolve such that the potential for ends by being exponentially decreasing. The RDS will then correspond to a run away behavior . The RDS is stable only when the subdominant term can be neglected. This is always true when is macroscopic, since we restrict our study to temperatures above the electroweak scale i.e. not very large compared to the internal space and in particular.
If is internal and the subdominant term is not negligible, we will find that the universe is attracted back to phase II, where becomes static and the evolution becomes that of an RDS. Finally we note that at early times, where and are well above the electro-weak scale, it is possible to have internal while keeping negligible for a large amount of time. In this case at early times the evolution is initially attracted to an RDS. However, at late times will always become relevant and the evolution always ends in an RDS.
IV: T-dual flat potential phase
The effective theory description is T-dual to the phase II where the light degrees of freedom are the winding modes instead of the Kaluza-Klein momentum modes of the phase II.
V: T-dual higher-dimensional phase
This phase is the T-dual of phase III. The light degrees of freedom are the winding modes. has the same form as in case III, after one transforms . The effective field theory is naturally described in space-time dimensions, with in string frame.
We have to stress here that the above phases arise from a common string setting but cannot be described in the context of a single field theory, due to the lack of the string winding modes. In field theory, only the phases II and III (or their T-dual IV and V) can be described by a single field theory. In phase I, the string winding modes play a particularly important role for the stabilization of the radius at the extended symmetry point, as shown in Sect. 3.1. In contrast to field theory, string theory naturally interpolates between these various phases.
2.3 The type II field theory phases
The perturbative type II structure of can be derived from the heterotic one by taking , so that phase I is now equivalent to phases II and III. The local minimum of at is not present anymore and there is a single plateau I II IV (see Fig. 1). In phase III (or V), when and (which are both ), the function in type II is identical to the heterotic one given in Eq. (2.24). Thus, in type II, the field admits flat potential phases II and IV in -dimensions, and the higher-dimensional phases III and V in dimensions. Again the higher-dimensional phases III and V are stable only to the extent that we can neglect the contribution in Eq. (2.24), which is always valid for macroscopic values of . However, when is internal, the final evolution is always attracted to the RDS of phase II or IV. In addition, during the times for which may be ignored, the universe admits an earlier evolution well approximated by an RDS. Since there is no enhancement of , the “ Higgs phase” does not exist perturbatively for type II theories. However, by heterotic-type II duality, we expect non-perturbative effects which may enhance the and imply an “ phase” I. The non-perturbative effects can correspond to the addition of branes whose separation is governed by the spectator modulus. This will be discussed in more details in Sect. 4. Alternatively, branes wrapped on a vanishing cycle whose size is fixed by the spectator modulus provide another dual type II set up.
3 Radiation-like dominated solutions (RDS) of the Universe and stabilization of the spectator moduli
In section 2.2, five distinct phases for the thermal effective potential were identified for the heterotic string. Here, we analyze in detail the behavior of the system in the first three phases. The behavior of phases IV and V is found from that of phases II and III by T-duality and we do not consider them explicitly. We show that the radius can be constant, either at the minimum of the potential in phase I or at any value along the flat region II. In phase III, initially increases along with the expansion of the space-time. When the quantities and can be neglected, this evolution continues and is well described by an RDS. For values of which are internal, is always caught by and and after which the evolution is attracted back to phase II. A distinct RDS exists in regions I, II and IV, while in region III and V there exists an RDS for macroscopic values of .
Next we show that these cosmological evolutions are stable against small perturbations. In particular, for phase I, the spectator modulus is stabilized at the self-dual point, while for phases III (and V) it becomes part of the space-time metric. For phases II (and IV) the spectator modulus is weakly stabilized due to the presence of gravitational friction arising from the expansion of the universe. From these results, one expects in general that when the dynamics of all spectator moduli are taken into account, the radii that are not dynamically decompactified are (weakly) stabilized at scales smaller than the ones characterizing the temperature and supersymmetry breaking.
For the type II string case described in Sect. 2.3, there is no phase I, due to the absence of the heterotic enhancement at the self dual point. The remaining type II phases are identical to the heterotic ones.
3.1 Case I: Higgs phase
In this case, the radius is naturally interpreted as a scalar Higgs field for an gauge group coupled to gravity in dimensions. As mentioned before, our analysis is restricted to field configurations which are isotropic and homogeneous. We thus look for extrema of the action (2.5) whose metric, temperature and scalars satisfy the ansatz
where denotes the partial derivative with respect to . Given solutions to the scalar equations of motion, we may always find corresponding solutions to the Einstein equations. We therefore focus first on solving the scalar equations. Their reduction on the ansatz (3.28) is given in Eqs. (B.35), (B.38) and (B.39) and summarized here as
where is defined in Eq. (B.33) and is a function which vanishes when all first and second derivatives in its arguments vanish. We have reparameterized our fields in terms of the scale factor so that time-derivatives have been replaced with -derivatives denoted as .
Radiation-like dominated solution
To start off, we note that the fact the model is invariant under the T-duality implies is an even function of , so that the first derivative of with respect to vanishes at . Thus, is a solution to Eq. (3.32). Next, from Eq. (2.18), the source is independent of at , so that . As a consequence, Eq. (3.31) is solved for any constant , and we find that remains a modulus.
It is convenient to introduce the quantities and , which are related to the pressure and energy density at as