1 Introduction

LPTENS–07/22, CPHT–RR025.0407

June 2007

{centering} Inflationary de Sitter Solutions
from Superstrings

Costas Kounnas and Hervé Partouche

Laboratoire de Physique Théorique, Ecole Normale Supérieure,
24 rue Lhomond, F–75231 Paris cedex 05, France
Costas.Kounnas@lpt.ens.fr

Centre de Physique Théorique, Ecole Polytechnique,
F–91128 Palaiseau, France
Herve.Partouche@cpht.polytechnique.fr

Abstract

In the framework of superstring compactifications with supersymmetry spontaneously broken, (by either geometrical fluxes, branes or else), we show the existence of new inflationary solutions. The time-trajectory of the scale factor of the metric , the supersymmetry breaking scale and the temperature are such that and remain constant. These solutions request the presence of special moduli-fields:
The universal “no-scale-modulus” , which appears in all effective supergravity theories and defines the supersymmetry breaking scale .
The modulus , which appears in a very large class of string compactifications and has a -dependent kinetic term. During the time evolution, remains constant as well, ( being the energy density induced by the motion of ).

The cosmological term , the curvature term and the radiation term are dynamically generated in a controllable way by radiative and temperature corrections; they are effectively constant during the time evolution.

Depending on , and , either a first or second order phase transition can occur in the cosmological scenario. In the first case, an instantonic Euclidean solution exists and connects via tunneling the inflationary evolution to another cosmological branch. The latter starts with a big bang and, in the case the transition does not occur, ends with a big crunch. In the second case, the big bang and the inflationary phase are smoothly connected.

  Research partially supported by the EU (under the contracts MRTN-CT-2004-005104, MRTN-CT-2004-512194, MRTN-CT-2004-503369, MEXT-CT-2003-509661), INTAS grant 03-51-6346, CNRS PICS 2530, 3059 and 3747, and ANR (CNRS-USAR) contract 05-BLAN-0079-01.
 Unité mixte du CNRS et de l’Ecole Normale Supérieure associée à l’Université Pierre et Marie Curie (Paris 6), UMR 8549.
 Unité mixte du CNRS et de l’Ecole Polytechnique, UMR 7644.

1 Introduction

In the framework of superstring and -theory compactifications, there are always moduli fields coupled in a very special way to the gravitational and matter sector of the effective four-dimensional supergravity. The gravitational and the scalar field part of the effective Lagrangian have the generic form

(1.1)

where is the metric of the scalar manifold and is the scalar potential of the supergravity. (We will always work in gravitational mass units, with GeV). What will be crucial in this work is the non-triviality of the scalar kinetic terms in the effective supergravity theories that will provide us, in some special cases, accelerating cosmological solutions once the radiative and temperature corrections are taken into account.

Superstring vacua with spontaneously broken supersymmetry [1] that are consistent at the classical level with a flat space-time define a very large class of “no-scale” supergravity models [2]. Those with spontaneous breaking deserve more attention. Some of them are candidates for describing (at low energy) the physics of the standard model and extend it up to  TeV energy scale. This class of models contains an enormous number of consistent string vacua that can be constructed either via freely acting orbifolds [1, 3] or “geometrical fluxes” [4] in heterotic string and type IIA,B orientifolds, or with non-geometrical fluxes [5] (e.g. RR-fluxes or else).

Despite the plethora of this type of vacua, an interesting class of them are those which are described by an effective “no-scale supergravity theory”. Namely the vacua in which the supersymmetry is sponaneously broken with a vanishing classical potential with undetermined gravitino mass due to at least one flat field direction, the “no-scale modulus . At the quantum level a non-trivial effective potential is radiatively generated which may or may not stabilize the “no-scale” modulus [2].

What we will explore in this work are the universal scaling properties of the “thermal” effective potential at finite temperature that emerges at the quantum level of the theory. As we will show in section 4, the quantum and thermal corrections are under control, (thanks to supersymmetry and to the classical structure of the “no-scale models”), showing interesting scaling properties.

In section 2, we set up our notations and conventions in the effective “no-scale” supergravities of the type IIB orientifolds with -branes and non-trivial NS-NS and RR three form fluxes and . We identify the “no-scale” modulus , namely the scalar superpartner of the Goldstino which has the property to couple to the trace of the energy momentum tensor of a sub-sector of the theory [6]. More importantly, it defines the field-dependence of the gravitino mass [2]

(1.2)

Other extra moduli that we will consider here are those with -dependent kinetic terms. These moduli appear naturally in all string compactifications [7]. We are in particular interested in scalars () which are leaving on -branes and whose kinetic terms scale as the inverse volume of the “no-scale” moduli space.

In section 3, we display the relevant gravitational, fields and thermal equations of motion in the context of a Friedman-Robertson-Walker (FRW) space-time. We actually generalize the mini-superspace (MSS) action by including fields with non-trivial kinetic terms and a generic, scale factor dependent, thermal effective potential.

In our analysis we restrict ourselves to the large moduli limit, neglecting non-perturbative terms and world-sheet instanton corrections , . On the other hand we keep the perturbative quantum and thermal corrections.

Although this study looks hopeless and out of any systematic control even at the perturbative level, it turns out to be manageable thanks to the initial no-scale structure appearing at the classical level (see section 4).

In section 5, we show the existence of a critical solution to the equations of motion that follows from the scaling properties derived in section 4. We have to stress here that we extremize the effective action by solving the gravitational and moduli equations of motion and do not consider the stationary solutions emerging from a minimization of the effective potential only. We find in particular that a universal solution exists where all scales evolve in time in a similar way, so that their ratios remain constant: const., const.. Along this trajectory, effective time-independent cosmological term , curvature term and radiation term are generated in the MSS action, characterizing the cosmological evolution.

Obviously, the validity of the cosmological solutions based on (supergravity) effective field theories is limited. For instance, in the framework of more fundamental theories such as string theory, there are high temperature instabilities occuring at , where is the Hagedorn temperature of order the mass of the first string excited state. To bypass these limitations, one needs to go beyond the effective field theory approach and consider the full string theory (or brane, M-theory,…) description. Thus, the effective solutions presented in this work are not valid anymore and must be corrected for temperatures above . Moreover, Hagedorn-like instabilities can also appear in general in other corners of the moduli space of the fundamental theory, when space-time supersymmetry is spontaneously broken.

Regarding the temperature scale as the inverse radius of the compact Euclidean time, one could conclude that all the internal radii of a higher dimensional fundamental theory have to be above the Hagedorn radius. This would mean that the early time cosmology should be dictated by a 10-dimensional picture rather than a 4-dimensional one where the internal radii are of order the string scale. There is however a loophole in this statement. Indeed, no tachyonic instability is showing up in the whole space of the moduli which are not involved in the spontaneous breaking of supersymmetry, as recently shown in explicit examples [8]. This leeds us to the conjecture that the only Hagedorn-like restrictions on the moduli space depend on the supersymmetry breaking. In our cosmological solutions, not only the temperature scale is varying, but also the supersymmetry breaking scale , which turns to be a moduli-dependent quantity. Based on the above statements, we expect that in a more accurate stringy description of our analysis, there should be restrictions on the temperature as well as the supersymetry breaking scale. This has been recently explicitly shown in the stringy examples considered in [8].

In section 6, our cosmological solutions are generalized by including moduli with other scaling properties of their kinetic terms.

Finally, section 7 is devoted to our conclusions and perspective for future work.

2 N=1 No-Scale Sugra from Type IIB Orientifolds

In the presence of branes and fluxes, several moduli can be stabilized. For instance, in “generalized” Calabi-Yau compactifications, either the Kälher structure moduli or the complex structure moduli can be stabilized according to the brane and flux configuration in type IIA or type IIB orientifolds [4, 9, 5, 6]. The (partial) stabilization of the moduli can lead us at the classical level to AdS like solutions, domain wall solutions or “flat no-scale like solutions”. Here we will concentrate our attention on the “flat no-scale like solutions”.

In order to be more explicit, let us consider as an example the type IIB orientifolds with -branes and non-trivial NS-NS and RR three form fluxes and . This particular configuration induces a well known superpotential that can stabilize all complex structure moduli and the coupling constant modulus [5, 4]. The remaining moduli “still remain flat directions at the classical level”, e.g. neglecting world-sheet instanton corrections and the perturbative and non-perturbative quantum corrections [5].

It is also well known by now that in the large limit the Kälher potential is given by the intersection numbers of the special geometry of the Calabi-Yau manifold and orbifold compactifications [10, 11]:

(2.3)

Thus, after the and moduli stabilization, the superpotential is effectively constant and implies a vanishing potential in all directions. The gravitino mass term is however non-trivial [2, 1, 4, 5, 11],

(2.4)

This classical property of “no-scale models” emerges from the cubic form of in the moduli and is generic in all type IIB orientifold compactifications with -branes and three form and fluxes [5, 4]. Keeping for simplicity the direction (for some constants ) and freezing all other directions, the Kälher potential is taking the well known structure [2],

(2.5)

This gives rise to the kinetic term and gravitino mass term,

(2.6)

where is a constant. Freezing Im and defining the field by

(2.7)

one finds a kinetic term

(2.8)

The choice normalizes canonically the kinetic term of the modulus .

The other extra moduli that we will consider are those with -dependent kinetic terms. We are in particular interested to the scalars whose kinetic terms scale as the inverse volume of the -moduli. For one of them, , one has

(2.9)

Moduli with this scaling property appear in a very large class of string compactifications. Some examples are:
All moduli fields leaving in the parallel space of -branes [5, 4].
All moduli coming from the twisted sectors of -orbifold compactifications in heterotic string [7], after non-perturbative stabilization of by gaugino condensation and flux-corrections [12].

Our analysis will also consider other moduli fields with different scaling properties, namely those with kinetic terms of the form:

(2.10)

with weight and

3 Gravitational, Moduli and Thermal Equations

In a fundamental theory, the number of degrees of freedom is important (and actually infinite in the context of string or M-theory). However, in an effective field theory, an ultraviolet cut-off set by the underlying theory determines the number of states to be considered. We focus on cases where these states include the scalar moduli fields and , with non-trivial kinetic terms given by

(3.11)

In this Lagrangian, the “” denote all the other degrees of freedom, while the effective potential depends on and the renormalization scale . We are looking for gravitational solutions based on isotropic and homogeneous FRW space-time metrics,

(3.12)

where is a 3-dimensional compact space with constant curvature , such as a sphere or an orbifold of hyperbolic space. This defines an effective one dimensional action, the so called “mini-super-space” (MSS) action [13, 14, 15, 16].

A way to include into the MSS action the quantum fluctuations of the full metric and matter degrees of freedom (and thus taking into account the back-reaction on the space-time metric), is to switch on a thermal bath at temperature [14, 15, 16]. In this way, the remaining degrees of freedom are parameterized by a pressure and a density , where are the non-vanishing masses of the theory. Note that and have an implicit dependence on , through the mass defined in eq. (1.2) [6]. The presence of the thermal bath modifies the effective MSS action, including the corrections due to the quantum fluctuations of the degrees of freedom whose masses are below the temperature scale . The result, together with the fields and , reads

(3.13)

where a “dot” denotes a time derivation. is a gauge dependent function that can be arbitrarily chosen by a redefinition of time. We will always use the gauge , unless it is explicitly specified.

The variation with respect to gives rise to the Friedman equation,

(3.14)

where .

The other gravitational equation is obtained by varying the action with respect to the scale factor :

(3.15)

In the literature, the last term is frequently taken to be zero. However, this is not valid due to the dependence of on , when this scale is chosen appropriately as will be seen in section 3. We thus keep this term and will see that it plays a crucial role in the derivation of the inflationary solutions under investigation.

We find useful to replace eq. (3.15) by the linear sum of eqs. (3.14) and (3.15), so that the kinetic terms of and drop out,

(3.16)

The other field equations are the moduli ones,

(3.17)

and

(3.18)

The last equation (3.18) can be solved immediately,

(3.19)

where is a positive integration constant. It is important to stress here that we insist to keep in eq. (3.17) both terms and that are however usually omitted in the literature. The first term vanishes only under the assumption that all masses are taken to be -independent, while the absence of the second term assumes a trivial kinetic term. However, both assumptions are not valid in string effective supergravity theories ! (See section 3.)

Finally, we display for completeness the total energy conservation of the system,

(3.20)

Before closing this section, it is useful to derive some extra useful formulas that are associated to the thermal system. The integrability condition of the second law of thermodynamics reaches, for the thermal quantities and ,

(3.21)

The fact that these quantities are four-dimensional implies

(3.22)

Then, the second eq. (3.22) together with the eq. (3.21) implies [6]:

(3.23)

Among the non-vanishing , let us denote with “hat-indices” the masses that are -independent, and with “tild-indices” the masses that have the following -dependence:

(3.24)

for some constants . Then, utilizing eq. (3.23), we obtain a very fundamental equation involving the modulus field [6],

(3.25)

where and are the contributions to and associated to the states with -dependent masses . The above equation (3.25) clearly shows that the modulus field couples to the (sub-)trace of the energy momentum tensor associated to the thermal system [6] , of the states with -dependent masses defined in eq. (3.24). We return to this point in the next section.

4 Effective Potential and Thermal Corrections

In order to find solutions to the coupled gravitational and moduli equations discussed in the previous section, it is necessary to analyze the structure of the scalar potential and the thermal functions , . More precisely, we have to specify their dependence on and . Although this analysis looks hopeless in a generic field theory, it is perfectly under control in the string effective no-scale supergravity theories.

Classically the potential is zero along the moduli directions and . At the quantum level, it receives radiative and thermal corrections that are given in terms of the effective potential [11], , and in terms of the thermal function, . Let us consider both types of corrections.

4.1 Effective Potential

The one loop effective potential has the usual form [11, 17],

(4.26)

where is the classical part, which vanishes in the string effective “no-scale” supergravity case. An ultraviolet cut-off is introduced and stands for the renormalization scale.

(4.27)

is a sum over the -th power of the mass eigenvalues. In our notations, the index is referring to both massless and massive states (with eventually -dependant masses).The weights account for the numbers of degrees of freedom and the statistics of the spin particles.

The quantum corrections to the vacuum energy with the highest degree of ultraviolet divergence is the term, whose coefficient is equal to the number of bosonic minus fermionic degrees of freedom. This term is thus always absent in supersymmetric theories since they possess equal numbers of bosonic and fermionic states.

The second most divergent term in eq. (4.26) is the contribution proportional to . In the spontaneously broken supersymmetric theories, it is always proportional to the square of the gravitino mass-term ,

(4.28)

The coefficient is a field independent number. It depends only on the geometry of the kinetic terms of the scalar and gauge manifold, and not on the details of the superpotential [17, 11]. This property is very crucial in our considerations.

The last term has a logarithmic behavior with respect to the infrared scale and is independent of the ultraviolet cut-off . Following the infrared regularization method valid in string theory (and field theory as well) adapted in ref. [18], the scale is proportional to the curvature of the three dimensional space,

(4.29)

where is a numerical coefficient chosen appropriately according to the renormalization group equation arguments. Another physically equivalent choice for is to be proportional to the temperature scale, . The curvature choice (4.29) looks more natural and has the advantage to be valid even in the absence of the thermal bath.

Modulo the logarithmic term, the can be expanded in powers of gravitino mass ,

(4.30)

Including the logarithmic terms and adding the quadratic contribution coming from the , we obtain the following expression for the effective potential organized in powers of :

(4.31)

where

(4.32)

for constant coefficients and , (). These contributions satisfy

(4.33)

The logarithmic dependence in the effective potential can be derived in the effective field theory by considering the Renormalization Group Equations (RGE). They involve the gauge couplings, the Yukawa couplings and the soft-breaking terms [19, 11]. These soft-breaking terms are usually parameterized by the gaugino mass terms , the soft scalar masses , the trilinear coupling mass term and the analytic mass term , [19, 11]. However, what will be of main importance in this work is that all soft breaking mass terms are proportional to [17, 11].

4.2 Thermal Potential

For bosonic (or fermionic) fluctuating states of masses (or ) in thermal equilibrium at temperature , the general expressions of the energy density and pressure are

(4.34)

where

(4.35)

and .

There are three distinct sub-sectors of states:
The sub-sector of bosonic and fermionic massless states. From eqs. (4.34) and (4.35), their energy density and pressure satisfy

(4.36)

In particular, we have and .
The sub-sector of states with non vanishing masses independent of .
Consider the bosons and fermions whose masses we denote by are below T. The energy density and pressure associated to them satisfy

(4.37)
(4.38)

where the ’s are non-vanishing positive constants. In particular, one has .
For the masses above , the contributions of the particular degrees of freedom are exponentially suppressed and decouple from the thermal system. We are not including their contribution.
The sub-sector with non vanishing masses proportional to . Its energy density and pressure satisfy

(4.39)

as was shown at the end of section 2. This identity is also valid for the massless system we consider in case .

According to the scaling behaviors with respect to and , we can separate

(4.40)

where

(4.41)

and are the sums of the contributions of the massless states (case ), the parts of and (case ), and and (case ),

(4.42)
(4.43)

while and arise from the parts of and (case ):

(4.44)

5 Critical Solution

The fundamental ingredients in our analysis are the scaling properties of the total effective potential at finite temperature,

(5.45)

Independently of the complication appearing in the radiative and temperature corrected effective potential, the scaling violating terms are under control. Their structure suggests to search for a solution where all the scales of the system, , and , remain proportional during their evolution in time,

(5.46)

Our aim is thus to determine the constants and in terms of in eq. (3.19), , and the computable quantities , , in string theory, such that the equations of motion for , and the gravity are satisfied. On the trajectory (5.46), the contributions , defined in eq. (4.32) satisfy

(5.47)

and

(5.48)

Also, the contributions of and in in eq. (3.19) conspire to give a global dependence,

(5.49)

Finally, the sums over the full towers of states with -dependent masses behave in and as pure constants, (see eqs. (4.42) and (4.43)),

(5.50)
(5.51)

As a consequence, using eqs. (4.33) and (4.39), the -equation (3.17) becomes,

(5.52)

On the other hand, using eq. (4.33), the gravity equation (3.16) takes the form

(5.53)

The compatibility of the -equation and the gravity equation along the critical trajectory implies an identification of the coefficients of the monomials in . The constant terms determine in term of

(5.54)

which amounts to fixing ,

(5.55)

The quadratic terms determine the parameter :

(5.56)

Finally, the quartic terms relate to the integration constant appearing in ,

(5.57)


At this point, our choice of anzats (5.46) and constants , allows to reduce the differential system for , and the gravity to the last equation. We thus concentrate on the Friedman equation (3.14) in the background of the critical trajectory ,

(5.58)

The dilatation factor in front of can be absorbed in the definition of , and , once we take into account eqs. (5.54), (5.56) and (5.57),

(5.59)

where

(5.60)
(5.61)

and

(5.62)

We note that for , is positive. In that case, the constraint (5.57) allows us to choose a lower bound for the arbitrary constant , so that is large enough to have . This means that the theory is effectively indistinguishable with that of a universe with cosmological constant , uniform space curvature , and filled with a thermal bath of radiation coupled to gravity. This can be easily seen by considering the Lagrangian

(5.63)

and the metric anzats (3.12), with a 3-space of constant curvature . In the action, one can take into account a uniform space filling bath of massless particles by adding a Lagrangian density proportional to (see [14, 15, 16]) in the MSS form. One finds

(5.64)

whose variation with respect to gives (5.59). Actually, the thermal bath interpretation is allowed as long as , since the term is an energy density. However, in the case under consideration, the effective can be negative due to the contribution of the effective potential. The general solution of the effective MSS action of eq. (5.64) with , and was recently investigated in ref. [16]. It amounts to a thermally deformed de Sitter solution, while the pure radiation case where , was studied in ref. [6]. In the latter case, the time trajectory (5.46) was shown to be an attractor at late times, giving rise to a radiation evolving universe with

(5.65)

Following ref. [16], the general case with and gives rise to cosmological scenarios we summarize here. Depending on the quantity

(5.66)

a first or second order phase transition can occur:

(5.67)

i) The case
There are two cosmological evolutions connected by tunnel effect:

(5.68)

and

(5.69)

where

(5.70)

The “cosh”-solution corresponds to a deformation of a standard de Sitter cosmology, with a contracting phase followed at by an expanding one. The “sinh”-solution describes a big bang with a growing up space till , followed by a contraction till a big crunch arises. The two evolutions are connected in Euclidean time by a -Gravitational Instanton

(5.71)

The cosmological scenario is thus starting with a big bang at and expands up to , following the “sinh”-evolution. At this time, performing the analytic continuation reaches (5.71), (where is chosen in the range111It is also possible to consider the instantons associated to the ranges , , see [16]. ). At , a different analytic continuation to real time exists, , that gives rise to the inflationary phase of the “cosh”-evolution, for ,

Figure 1: A first order phase transition can occur. The two cosmological evolutions and are connected by an instanton . The universe starts with a big bang at and expands till along the solution . Then, the scale factor can either contract, or jump instantaneously and enter into the inflationary phase of .

(see fig. 1).

There are thus two possible behaviors when is reached. Either the universe carries on its “sinh”-evolution and starts to contract, or a first order transition occurs and the universe enters into the inflationary phase of the “cosh”-evolution. In that case, the scale factor jumps instantaneously from to at ,

(5.72)

An estimate of the transition probability is given by

(5.73)

where is the Euclidean action computed with the instanton solution (5.71), for . Actually, following refs. [15, 16], one has:

(5.74)

where and are the complete elliptic integrals of first and second kind, respectively, and

(5.75)


ii) The case
There is a cosmological solution,

(5.76)

where

(5.77)

As in the previous case, it starts with a big bang. However, the behavior evolves toward the inflationary phase in a smooth way, (see fig. 2).

Figure 2: A second order phase transition occurs. The universe starts with a big bang at and evolves smoothly toward the inflationary phase.

The transition can be associated to the inflection point arising at , where ,