1 Introduction

LPTENS–07/50, CPHT–RR085.0707, October 2007

{centering} Thermal/quantum effects and
induced superstring cosmologies

Tristan Catelin-Jullien, Costas Kounnas
Hervé Partouche and Nicolaos Toumbas

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

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

Department of Physics, University of Cyprus,
Nicosia 1678, Cyprus
nick@ucy.ac.cy

Abstract

We consider classical superstring theories on flat four dimensional space-times, and where or supersymmetry is spontaneously broken. We obtain the thermal and quantum corrections at the string one-loop level and show that the back-reaction on the space-time metric induces a cosmological evolution. We concentrate on heterotic string models obtained by compactification on a torus and on orbifolds. The temperature and the supersymmetry breaking scale are generated via the Scherk-Schwarz mechanism on the Euclidean time cycle and on an internal spatial cycle respectively. The effective field theory corresponds to a no-scale supergravity, where the corresponding no-scale modulus controls the Susy-breaking scale. The classical flatness of this modulus is lifted by an effective thermal potential, given by the free energy. The gravitational field equations admit solutions where , and the inverse scale factor of the universe remain proportional. In particular the ratio is fixed during the time evolution. The induced cosmology is governed by a Friedmann-Hubble equation involving an effective radiation term and an effective curvature term , whose coefficients are functions of the complex structure ratio .

  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, ANR (CNRS-USAR) contract 05-BLAN-0079-01 and INTERREG IIIA Crete/Cyprus.
 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

It is important to develop a string theoretic framework for studying cosmology. The ultimate goal of this task is to determine whether string theory can describe basic features of our Universe. Despite considerable effort towards this direction over the last few years (see for example [1][10]), still very little is known about the dynamics of string theory in time-dependent, cosmological settings. The purpose of this work is to provide a new class of non-trivial string theory cosmological solutions, where some of the difficult issues can be explored and analyzed concretely.

At the classical string level, it seems difficult to obtain exact cosmological solutions [8]. Indeed, after extensive studies in the framework of superstring compactifications (with or without fluxes), the obtained results appear to be unsuitable for cosmology. In most cases, the classical ground states correspond to static Anti-de Sitter or flat backgrounds but not to cosmological ones. The same situation appears to be true in the effective supergravity theories. Naively, the results obtained in this direction lead to the conclusion that cosmological ground states are unlikely to be found in superstring theory.

From our viewpoint this conclusion cannot be correct for two reasons:
The first follows from the fact that already exact (to all orders in ) cosmological solutions exist, which are described by a two dimensional worldsheet conformal field theory based on a gauged Wess-Zumino-Witten model at negative level : , [2, 3, 4].
The second is that quantum and thermal corrections are neglected in the classical string/supergravity regime.

The first class of stringy cosmological models was studied recently in [5], where it was shown how to define a normalizable wave-function for this class of backgrounds, realizing the Hartle-Hawking no-boundary proposal [11] in string theory. Explicit calculable examples were given for small values of the level . As it was shown in [5], these models are intrinsically thermal with a temperature below but still close to the Hagedorn temperature. The disadvantage of small level , however, is the absence of a semi-classical limit with arbitrarily large, which prevents us from obtaining a clean geometrical picture and studying issues such as back-reaction and particle production in a straightforward way.

Another direction consists of studying the “quantum and thermal cosmological solutions,” which are generated dynamically at the quantum level of string theory [6, 12]. Although this study looks to be hopeless and out of any systematic control, it turns out that in certain cases the quantum and thermal corrections are under control thanks to the special structure of the underlying effective supergravity theory in its spontaneously broken supersymmetric phase. An effective field theory study has already been initiated in [6, 12]. (See also [13, 14].)

In order to see how cosmological solutions arise naturally in this context, consider the case of a supersymmetric flat string background. At finite temperature the thermal fluctuations produce a non-zero energy density that is calculable perturbatively at the full string level. The back-reaction on the space-time metric and on certain of the moduli fields gives rise to a specific cosmological evolution. For temperatures below the Hagedorn temperature, the evolution of the universe is known to be radiation dominated. (See for instance [15, 16] for some earlier work in this case and [16] for a review on string gas cosmology.)

More interesting cases are those where space-time supersymmetry is spontaneously broken at the string level either by geometrical [17] or non-geometrical fluxes . In the case where the geometrical fluxes are generated via freely acting orbifolds [18][23], the stringy quantum corrections are under control in a very similar way as the thermal ones. The back-reaction of the quantum and thermal corrections on the space-time metric and the moduli fields results in deferent kinds of cosmologies depending on the initial amount of supersymmetry .

In this work we restrict attention to four-dimensional backgrounds with initial or space-time supersymmetry, obtained by toroidal compactification of the heterotic superstring on and -orbifolds. The spontaneous breaking of supersymmetry is implemented via freely acting orbifolds (as in [18][23]). The quantum and thermal corrections are determined simultaneously by considering the Euclidean version of the model where all coordinates are compactified: (for the four-dimensional space-time part) (for the internal manifold). Apart from being interesting in their own right, these examples may give us useful hinds on how to handle the phenomenologically more relevant cases. The cases will be studied elsewhere.

The thermal corrections are implemented by introducing a coupling of the space-time fermion number to the string momentum and winding numbers associated to the Euclidean time cycle . The breaking of supersymmetry is generated by a similar coupling of an internal -symmetry charge to the momentum and winding numbers associated to an internal spatial cycle , e.g. the coordinate cycle.

We stress here, that the thermal and supersymmetry breaking couplings correspond to string theoretic generalizations of Scherk-Schwarz compactifications. Two very special mass scales appear both associated with the breaking of supersymmetry. These are the temperature scale and the supersymmetry breaking scale , with and the radii of the Euclidean time cycle, , and of the internal spatial cycle, , respectively. The initially degenerate mass levels of bosons and fermions split by an amount proportional to or , according to the charges and . This mass splitting is the signal of supersymmetry breaking and gives rise to a non-trivial free energy density, which incorporates simultaneously the thermal corrections and quantum corrections due to the supersymmetry breaking boundary conditions along the spatial cycle .

At weak coupling, the free energy density can be obtained from the one-loop Euclidean string partition function [20][22]. The perturbative string amplitudes are free of the usual ultraviolet ambiguities that plague a field theoretic approach towards quantum gravity and cosmology. For large enough , , the Euclidean system is also free of tachyons – the presence of tachyons would correspond to infrared instabilities, driving the system towards a phase transition [21][23]. Therefore, the corresponding energy density and pressure can be determined unambiguously, and we can use them as sources in Einstein’s equations to obtain non-trivial cosmological solutions. This perturbative approach breaks down near the initial space-like singularity. We speculate whether this breakdown of perturbation theory can be associated with an early universe phase transition.

The paper is organized as follows. Section 2 is mainly a review, where we also fix most of our notations and conventions. We first consider the four-dimensional heterotic string models at finite temperature. We obtain the one-loop thermal partition function at the full string level, and then we discuss the effective field theory limit at large radius . We also review the analogous computation of the one loop string partition function at zero temperature and in the case where Susy-breaking boundary conditions are placed along the internal spatial cycle , [18][23]. In the large radius limit, the Einstein frame effective potential is proportional to the fourth power of the gravitino mass scale, and it can be positive or negative depending on the choice of the Susy-breaking operator .

In section 3, we consider the case where thermal and quantum corrections due to the supersymmetry breaking are present simultaneously. For the simplest choice , the corresponding one-loop string partition function is invariant under the exchange, manifesting the underlying temperature/gravitino mass scale duality of the models. This duality is broken by the other allowable choices for the Susy-breaking operator , which we classify for both the and the orbifold cases.

In the large radii limit, the pressure consists of two pieces: the purely thermal part which scales as , with the coefficient being the number of all massless boson/fermion pairs in the initially supersymmetric theory, and another potential-like piece which scales as and with the coefficient being positive or negative depending on the choice of the operator . In both pieces, the rest of the dependence on the scales and can be expressed neatly in terms of non-holomorphic Eisenstein series of order whose variable is the complex structure-like ratio . In addition, we incorporate the effects of small, continuous Wilson line deformations in our computation. Wilson lines along any of the internal spatial cycles, other than , introduce new mass scales, and pieces proportional to and arise in the effective thermodynamic quantities.

In section 4 we present our ansatz for the induced cosmological solutions. These are homogeneous and isotropic cosmologies for which the Susy-breaking scales and as well as the inverse of the scale factor evolve the same way in time, and so the ratio of any two of these quantities is constant. The form of this ansatz is dictated by the scaling properties of the effective energy density and pressure. The compatibility of the gravitational field equations with the equation of motion of the scalar modulus controlling the size of the gravitino mass scale fixes the ratio . By solving the compatibility equations numerically, we find that in the absence of Wilson lines along , non-trivial four dimensional solutions exist when is negative and the ratio is small enough. These conditions are satisfied by various models we describe explicitly in the paper. When we include Wilson lines along , the value of the ratio for some of the solutions can be large or small, and so we can have models with a hierarchy for the scales and .

Having solved the compatibility equations, the time-dependence of the system is governed solely by the familiar Friedmann-Hubble equation. There is a radiation term, , whose coefficient is positive in our examples. An effective curvature term, , can be generated by turning on Wilson line deformations. The sign of can be a priori positive or negative, depending on the model. When we turn on the kinetic terms of some of the extra flat moduli, we generate an additional term that scales as (with positive).

In section 5, we solve the Friedmann-Hubble equation for the various possible cases, and we elaborate on the properties of the cosmological solutions:
When , we have standard hot big bang cosmologies with an intermediate radiation dominated era. The late time behavior is governed by the spatial curvature of the models.
We also consider a priori possible exotic models characterized by . A big bang occurs when . The cosmological evolution always ends with a big crunch when . The case however is more interesting. It involves either a first or second order phase transition between the big bang cosmology and a linearly expanding universe. The first case corresponds to a tunneling effect involving a gravitational instanton, while the transition is smooth in the second case. If the first order transition does not occur, the universe ends in a big crunch.
We finish with our conclusions and directions for future research.

2 Thermal and quantum corrections in heterotic backgrounds

Our starting point is the class of four dimensional string backgrounds obtained by toroidal compactification of the heterotic string on and orbifolds. Initially the amount of space-time supersymmetry is for the case of compactification on the torus and for the orbifold compactifications, and the four dimensional space-time metric is flat. Space-time supersymmetry is then spontaneously broken by introducing Scherk-Schwarz boundary conditions on an internal spatial cycle and/or by thermal corrections. Due to the supersymmetry breaking, the one-loop string partition function is non-vanishing, giving rise to an effective potential. Our aim is to determine the back-reaction on the initially flat metric and moduli fields.

At the one-loop level, the four dimensional string frame effective action is given by

(2.1)

where is the dilaton field and the ellipses stand for the kinetic terms of other moduli fields (to be specified later). At zero temperature, the effective potential can be obtained from the one-loop Euclidean string partition function as follows:

(2.2)

with the Euclidean volume. The absence of a dilaton factor multiplying the potential term in the action is due to the fact that this arises at the one loop level.

At finite temperature, the one-loop Euclidean partition function determines the free energy density and pressure to this order

(2.3)

The subscript indicates that these densities are defined with respect to the string frame metric. The relevant Euclidean amplitude incorporates simultaneously the thermal corrections and quantum corrections which arise from the spontaneous breaking of supersymmetry and which are present even at zero temperature.

In order to determine the back-reaction of the (thermal and/or) quantum corrections, it is convenient to work in the Einstein frame where there is no mixing between the metric and the dilaton kinetic terms. We define as usual the complex field ,

(2.4)

where is the axion field. Then after the Einstein rescaling of the metric, the one loop effective action becomes:

(2.5)

where is the metric on the scalar field manifold , which is parameterized by various compactification moduli including the field . This manifold includes also the main moduli fields , which are the volume and complex structure moduli of the three internal -cycles respectively. We notice that in the Einstein frame the effective potential, , is rescaled by a factor , where . Taking this rescaling into account, we have

(2.6)

This relation will be crucial for our work later on. (We will always work in gravitational mass units, with GeV).

Keeping only the main moduli fields , their kinetic terms are determined in terms of the Kälher potential [24, 25]:

(2.7)

with . The classical superpotential depends on the way supersymmetry is broken. Generically string backgrounds with spontaneously broken supersymmetry are flat at the classical level due to the no-scale structure of the effective supergravity theory [25]. Once the thermal and/or quantum corrections are taken into account, we obtain in some cases interesting cosmological solutions.

2.1 Heterotic supersymmetric backgrounds at finite temperature

In order to fix our notations and conventions, we first consider the case of an exact supersymmetric background at finite temperature [21][23]. For definiteness we choose the heterotic string with maximal space-time supersymmetry (). All nine spatial directions as well as the Euclidean time are compactified on a ten dimensional torus. At zero temperature, the Euclidean string partition function is zero due to space-time supersymmetry. At finite temperature however the result is a well defined finite quantity. Indeed, at genus one the string partition function is given by:

(2.8)

where is a shifted Narain lattice (which we specify more precisely below). The non-vanishing of the partition function is due to the non-trivial coupling of the lattice to the spin structures . Here, the argument is zero for space-time bosons and one for space-time fermions. The spin/statistics connection and modular invariance require that the unshifted sub-lattice of the Euclidean time cycle

(2.9)

be replaced as follows:

(2.10)

Redefining

(2.11)

where are integers defined modulo , and introducing the notation for a shifted lattice,

(2.12)

the thermal partition function takes the form:

(2.13)

Defining and and using the Jacobi identity

(2.14)

we obtain

(2.15)

The temperature in string frame is given by .

Since our aim is the study of induced cosmological solutions in dimensions, we consider the case for which the radii of three spatial directions are very large: . In this case the three dimensional spatial volume factorizes

(2.16)

Using the expression for the shifted lattice we obtain:

(2.17)

where is the four dimensional space-time volume, the free energy density and the pressure in string frame.

Before we proceed further, we make some comments:
The sector gives zero contribution. This is due to the fact that we started with a supersymmetric background.
In the odd winding sector, , the partition function diverges when is between the Hagedorn radius and its dual : . The divergence is due to a winding state that is tachyonic when takes values in this range, and it signals a phase transition around the Hagedorn temperature [21][23]. In this paper we study the regime , where there is no tachyon and the odd winding sector is exponentially suppressed. The high temperature regime and the cosmological consequences of the phase transition will be examined in future work [32].
When , the contributions of the oscillator states are also exponentially suppressed, provided that the moduli parameterizing the internal lattice are of order unity.

2.2 The effective field theory in the large limit

As we already mentioned, the sector of the theory gives exponentially suppressed contributions of order . Also, the sector vanishes due to supersymmetry. Thus for large , only the sector contributes significantly. Using the identity:

(2.18)

and neglecting the sectors, we may replace

(2.19)

in the integral expression for . For each lattice term we decompose the contribution in modular orbits: and . For , the integration over the fundamental domain is equivalent with the integration over the whole strip but with . The contribution is integrated over the fundamental domain. Now the contribution of cancels the one of , and we are left with the integration over the whole strip:

(2.20)

The integral over imposes the left-right level matching condition. The left-moving part contains the ratio

(2.21)

which implies that the lowest contribution is at the massless level. Thus after the integration over , the partition function takes the form

(2.22)

where denotes the multiplicity of the mass level and is the multiplicity of the massless level. Changing the integration variable by setting , we have:

(2.23)

Now the second term in the parenthesis is exponentially suppressed when the masses are of order (or close) to the string oscillator mass scale. This will be the case when all of the internal radii and the Wilson-line moduli of the lattice are of order unity. For this specific case, the partition function simplifies to

(2.24)

where is the number of the massless boson/fermion pairs in the theory. The free energy density and pressure in string frame are given by

(2.25)

In the Einstein frame, energy densities are rescaled by a factor as in Eq. (2.6). Thus the pressure and free energy density in this frame are given by

(2.26)

where is the proper temperature in the Einstein frame. This result is expected from the effective field theory point of view. When only massless states are thermally excited, the field theory expression for the pressure is given by

(2.27)

where and are the numbers of massless bosonic and fermionic degrees of freedom respectively. When , as in a supersymmetric theory, we recover Eq. (2.26).

2.3 Spontaneous breaking of supersymmetry at zero temperature

In this case we consider the same class of heterotic models, but now the breaking of supersymmetry arises due to the coupling of the space-time fermion number to the momentum and winding quantum numbers of an internal spatial cycle [18][23]. Since the temperature is taken to be zero, the spin structures do not couple to the quantum numbers of the Euclidean time cycle which will be taken to be very large. We also consider the case where three additional spatial directions are large. Following similar steps to the purely thermal case, the partition function is given by

(2.28)

where now is a five dimensional volume and the lattice parameterizes the internal space. Here also, the sectors give exponentially suppressed contributions , and the sector vanishes due to supersymmetry. The rest of the steps can be repeated as in the derivation above to find

(2.29)

which after the change of variables gives

(2.30)

For of order unity, this simplifies to

(2.31)

with .

This result was expected from the effective field theory point of view. Indeed in a theory with spontaneously broken supersymmetry, the one loop effective potential receives a non-zero contribution proportional to the mass super-trace , which in turn is proportional to the fourth power of the gravitino mass. The super-traces vanish for . In the example of supersymmetry breaking we examined above, the masses of the states are shifted according to their spin. For initially massless states, the mass after supersymmetry breaking becomes :

(2.32)

This shows that the string frame gravitino mass is of order and thus . Including the contributions from all Kaluza-Klein states, one obtains the result given in formula (2.31). We obtain for the string frame effective potential:

(2.33)

In the Einstein frame, we have – see Eq. (2.6) – so that

(2.34)

where , and is the gravitino mass scale in the Einstein frame.

We stress here that the one loop effective potential depends only on the gravitino mass scale, which in turn depends only on the product of the , and moduli. This suggests to freeze all moduli and keep only the diagonal combination

(2.35)

The Kälher potential of the diagonal modulus , (with ), takes the well known structure[25]

(2.36)

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

(2.37)

Freezing and defining the field by

(2.38)

one finds the kinetic term

(2.39)

The choice normalizes canonically the kinetic term of the modulus . The potential for this particular model is:

(2.40)

Observe that in this simple model the sign of the potential is negative. As we now explain, we can construct models with a positive potential, but with the rest of the dependence on the modulus being the same. All we have to do is to couple the momentum and winding numbers of the Scherk-Schwarz cycle not only to the space-time fermion number but also to another internal charge. For example consider the heterotic string on and instead of coupling just to , we couple to , where denotes the charge of an representation decomposed in terms of ones, and similarly for . These charges take half integer values for the spinorial representations and integer values for the others. The initial Susy-breaking co-cycle gets modified as follows

(2.41)

where as before the argument is one for space-time fermions and zero for space-time bosons, and for the spinorial representations of and for the adjoint representations. This operation breaks explicitly the gauge group to . Proceeding in similar way as in the previous example, one finds:

(2.42)

where

(2.43)

In the previous example only positive signs appear in the above formula since there is no coupling of the Scherk-Schwarz lattice quantum numbers to the charges, giving the value . The reversing of sign for some representations indicates that it is for the bosons that the masses are shifted and not for the fermions in the corresponding multiplet.

We note that in the case, we cannot change the left-multiplicity since all of the left-moving R-charges are equivalent as required by symmetry. This however is not true for the and cases. Consider for instance the class of supersymmetric backgrounds obtained by compactifying the heterotic string on a orbifold (e.g. the -orbifold limit of the CY-compactification). In this class of models (see for instance [27]) four internal supercoordinates are twisted and the corresponding four internal R-charges are half-shifted. The Euclidean partition function is given by

(2.44)

Here is the contribution of two internal coordinates111In our notations, the space-time coordinates are , while the internal ones are . () and -right moving world-sheet bosons . Before supersymmetry breaking, the corresponding -lattice is unshifted. stands for the contribution of four internal coordinates () all of which are -twisted by , and -right moving world-sheet bosons which can be -twisted breaking part of the initial gauge group. The -function terms come from the contribution of the left-moving world-sheet fermions. Four of them are -twisted by . The contribution associated to the space-time bosons is when , while the one associated to the space-time fermions is when .

From the above supersymmetric partition function, the thermal partition function is obtained in a way similar to the example, by the following replacement of the Euclidean time sub-lattice:

(2.45)

In the case of Scherk-Schwarz spontaneous supersymmetry breaking, the partition function can be obtained by a similar replacement of the internal coordinate lattice, either by utilizing the same operator

(2.46)

or by utilizing an R-symmetry operator associated to one of the twisted complex planes

(2.47)

These are in fact the only two possibilities involving left-moving R charges since all others are equivalent choices. However, many other choices exist by utilizing parity-like operators involving the right moving gauge charges , as in the explicit example of spinorial representations we gave above:

(2.48)

In the next section we examine representative examples in the case where thermal and spontaneous Susy breaking operations are present simultaneously.

3 Thermal and spontaneous breaking of Susy

The most interesting situation for cosmological applications is the case where spontaneous supersymmetry breaking and thermal corrections are taken into account simultaneously.

3.1 Untwisted sector

The untwisted sector of the case, in Eq. (2.44),222For (even windings), the sector gives zero net contribution due to the identity . has an structure and thus all choices for the left R-symmetry operators are equivalent. The quantum numbers of the Euclidean time cycle and the internal -cycle are coupled to the spin structures in the same way. After performing the Jacobi theta-function identity the partition function becomes:

(3.49)

The factor of is due to the orbifolding of the theory.

Proceeding as in the simpler examples before and neglecting the and sectors for large , the non-zero contributions to the partition function occur when . Assuming also that all other moduli are of order unity, the only non-exponentially suppressed contributions come from the zero mass left- and right-levels. We obtain

(3.50)

which after the change of variables gives