SUSXTH94/37
SUSSEXAST94/62
hepth/9406216
Low energy effective string cosmology
E. J. Copeland^{a}^{a}a, Amitabha Lahiri^{b}^{b}b & David Wands^{c}^{c}c
School of Mathematical & Physical Sciences,
University of Sussex,
Brighton BN1 9QH.
U. K.
We give the general analytic solutions derived from the low energy string effective action for four dimensional FriedmannRobertsonWalker models with dilaton and antisymmetric tensor field, considering both long and short wavelength modes of the field. The presence of a homogeneous field significantly modifies the evolution of the scale factor and dilaton. In particular it places a lower bound on the allowed value of the dilaton. The scale factor also has a lower bound but our solutions remain singular as they all contain regions where the spacetime curvature diverges signalling a breakdown in the validity of the effective action. We extend our results to the simplest Bianchi I metric in higher dimensions with only two scale factors. We again give the general analytic solutions for long and short wavelength modes for the field restricted to the three dimensional space, which produces an anisotropic expansion. In the case of field radiation (wavelengths within the Hubble length) we obtain the usual four dimensional radiation dominated FRW model as the unique late time attractor.
1 Introduction
String inspired cosmology is currently attracting a great deal of attention. The most favored starting point in any analysis is the low energy string effective action from which the lowest order string betafunction equations can be derived [1]. These equations, for the closed string, consist of three long range fields, the dilaton , the KalbRamond field strength , and the graviton, all arising out of the massless excitation of the string. In addition there is a constant related to the central charge of the string theory which vanishes in the critical number of dimensions 10 or 26. The fact that only the massless excited state is used suggests that the effective action is not a valid description for probing the highest energies associated with string theory. However, we may hope that through the betafunction equations we are investigating physics associated with events from say the string scale down to the GUT scale. Such an approach has already been adopted by a number of authors [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].
Veneziano and his collaborators have emphasized the possible importance of the duality symmetry which characterizes the equations of string cosmology [5, 11, 13]. They point out the possibility that inflation can occur without relying on a potential energy density. Rather, the duality transforms, which relate the dilaton and the metric, lead to a possible inflationary mechanism. In [13] it is claimed that the inclusion of the field does not seem to change the underlying properties of these dualityrelated cosmologies, although they also point out possible problems with exiting inflation in these models. Cosmological solutions with a dilaton and a nontrivial field have been obtained by Tseytlin[9], for curved maximally symmetric spaces in string theory, with a nonzero central charge deficit. (See [9] for details and complete references).
In [12], the authors study the outcome of stringdominated cosmology in a four dimensional FriedmannRobertsonWalker (FRW) spacetime including a homogeneous field () as well as a nonzero critical charge deficit (some of the particular solutions were previously obtained by Tseytlin [8]). Their results are based on a phaseplane analysis. In this paper, we present the general analytic solutions, including both long and short wavelength solutions for the field, but setting , and then show how this can be extended to simple anisotropic models in higher dimensions.
In section 2 we introduce the low energy equations of motion in the string frame and demonstrate how they can be related via simple conformal transforms to equations in other frames. Section 3 concentrates on four dimensional FRW models and describes the complete solution for a homogeneous field, reproducing where applicable previous solutions found in the literature [9, 12]. We go on to describe radiation solutions in which the field has a spatial dependence on small scales (i.e. wavelengths much smaller than the Hubble length). Because the metric (and the dilaton) is homogeneous and isotropic we require the field energymomentum tensor to be homogeneous and isotropic on average, and demonstrate how at late times we recover the general relativistic results for radiation and curvature dominated models. In section 4 we consider the influence of homogeneous and radiation solutions of the field in cosmologies where the metric tensor is decomposed into the direct product of a four dimensional FRW metric and an dimensional metric. We give general analytic solutions for the simplest version of the dimensional Bianchi I models where there are just two scale factors. The results indicate how the field can produce an anisotropic expansion leading to one scale expanding whilst the other contracts. Finally in section 5 we summarize our main results.
2 String action
We shall take as our starting point the low energy dimensional string effective action [1]
(2.1) 
where and we adopt the sign conventions denoted (+++) by Misner, Thorne and Wheeler [14]. is the dilaton field determining the strength of the gravitational coupling and where .
The variation of this action with respect to the , and , respectively, yields the field equations
(2.2)  
(2.3)  
(2.4) 
where is the energymomentum tensor derived from the matter Lagrangian.
The effect of certain types of “stringy matter” has been considered elsewhere in the literature [11]. Specific schemes of compactification, not to mention the chosen gauge symmetries of the theory, will determine the behavior (and number) of both bosonic and fermionic matter fields. It is not inconceivable that in some ‘matterdominated era’ of the stringy epoch of the universe these matter fields will play a part in determining the cosmological evolution. In particular, the antisymmetric tensor may be considered a matter field, and as we shall see it plays a significant role in string cosmology. In favor of considering the possible effects of this field on the cosmology we shall ignore all other contributions from the matter Lagrangian.
The charge deficit is a constant proportional to for the bosonic string and in the heterotic or superstring. We will set in our analysis. This may well be necessary for a consistent theory either by choosing the appropriate number of spatial dimensions or due to cancellation with contributions from other matter fields. Even for nonzero this should be an increasingly good approximation at early times if the curvature and/or kinetic energy densities are large, , but we would need to consider its effect at late times in an expanding universe.
2.1 Conformal frames
These field equations are similar to those found in BransDicke gravity[15] with the BransDicke parameter . This is only strictly true in the absence of the field as in BransDicke gravity it is assumed that the energymomentum tensor of all fields (other than the BransDicke field, ) are minimally coupled to the metric . While the energymomentum tensor of other matter fields are assumed to be conserved with respect to this metric (the string metric), so that , we cannot define an energymomentum tensor solely in terms of the field and the string metric which is conserved independently of the dilaton. This is just a consequence of the equation of motion for (Eq. (2.3)) which has an explicit dependence upon .
is only minimally coupled in the conformally related metric
(2.5) 
which we might call the Bmetric, in which we find^{1}^{1}1Note that the 3form has a conformally invariant definition in terms of the potential , whereas its covariant form has indices raised by a particular metric.
(2.6) 
Another particularly useful metric to introduce is the Einstein metric[16]
(2.7) 
In this frame the action appears simply as the EinsteinHilbert action of general relativity in dimensions, while the dilaton appears simply as a matter field, albeit one interacting with the other matter fields,
(2.8)  
The corresponding field equations are then those for interacting fields in general relativity;
(2.9)  
(2.10)  
(2.11) 
where the terms on the righthand side of the Einstein equations correspond to
(2.12)  
(2.13)  
(2.14)  
(2.15) 
the energymomentum tensors for the matter, dilaton and fields and potential respectively. The total energymomentum must be conserved of course by the Ricci identity, but there are interactions between these four components. Henceforth, as remarked earlier, we shall set and to be zero.
3 Isotropic D=4 solutions
Firstly we will consider the behavior of dimensional homogeneous and isotropic cosmologies for which the most general metric is the FriedmannRobertsonWalker metric
(3.1)  
(3.2) 
in terms of the conformally invariant time coordinate, , with for spatially closed, flat or open models respectively. Just as we take the metric to be homogeneous we shall also assume that the dilaton has no spatial dependence, .
With the conformal transform to the Einstein frame gives a rescaled scale factor . The metric field equations are simplest in terms of the Einstein metric where we have the familiar constraint equation (the component of Eq. (2.9) with and )
(3.3) 
where prime denotes differentiation with respect to .
In dimensions the field equation of motion, Eq. (2.10), is solved by the Ansatz,
(3.4) 
where is the antisymmetric 4form (obeying ) and the integrability condition, , becomes the new equation of motion for
(3.5) 
Thus evolves as a massless scalar field coupled to the dilaton (except in the Bframe where ). The same interaction appears in the dilaton equation of motion (Eq. (2.11), with ) as
(3.6) 
Thus we have two interacting scalar fields whose energymomentum tensors are given by
(3.7)  
(3.8) 
These equations simplify considerably in two cases.
3.1 Homogeneous solution
Thus far we have allowed for the possibility of a spatial dependence of the field. However as we have already restricted ourselves to considering a homogeneous metric and dilaton field this will only be consistent with choosing a source that is at least homogeneous on average. Indeed as far as we are aware, the only case that has been considered to date[12] is that of a strictly isotropic field, where , and thus .
In this case the equation of motion for the field (Eq. (3.5)) becomes
(3.9) 
which is easily integrated to give where is a nonnegative constant. It is this kinetic energy density of the field that drives the dilaton. Note that back in the string frame this solution corresponds to . Thus it will dominate any charge deficit in the dilaton equation of motion, Eq. (2.4), as .
Because both and are functions only of time, we can define a new scalar field where
(3.10) 
The energymomentum tensor in the Einstein frame is then simply that for this single minimally coupled field
(3.11) 
The equation of motion for a homogeneous minimally coupled field is
(3.12) 
which can be integrated to give , where is a positive constant. (We will not consider the trivial case where .) Thus in the Einstein frame we have an isotropic perfect stiff fluid whose energy density .
The constraint Eq.(3.3) for the Einstein scale factor, , can then be integrated to give[17]
(3.13) 
where we define
(3.14) 
We emphasize that in the Einstein frame the scale factor evolves in a wholly unremarkable fashion[10]. We have a singularity at with and the models expand away from it for or collapse towards it for . Only closed models can turn around, and these recollapse at . There are no bounce solutions. Notice also that the behavior of the Einstein scale factor is independent of , and thus is the same in vacuum, i.e. without the presence of the field, as it is with the field.
Combining the first integrals for and the field with the definition of we also have
(3.15) 
This too can be integrated to give
(3.16) 
where is an integration constant. Note that the evolution in the presence of the antisymmetric tensor field is quite distinct from the vacuum behavior. In particular there is a lower bound on the dilaton, . By contrast, there are two distinct branches in vacuum where the dilaton is either monotonically increasing or decreasing.
We can use this to recover the scale factor in the string frame
(3.17) 
The evolution of the scale factor and dilaton in different cases is shown in Figs. (1–4). The singular behavior of the conformal factor, , at the initial singularity in the Einstein frame produces cosmologies which have a minimum nonzero value for the scale factor in the string frame. In the presence of the field, diverges both as (or as for ) and as , so all models “bounce”, although they are still singular in the sense that the Ricci curvature diverges.
Note that the vacuum solutions again exhibit two distinct branches corresponding to and either monotonically increasing or decreasing. The models correspond simply to powerlaw solutions shown in Fig. 1. We can write the solutions in terms of the proper time in the string frame by integrating . The two solutions are then

and ;

and .
These two branches, corresponding to a decelerated or accelerated scale factor, are related by the duality transformation[18] and .
For the solution corresponds to the powerlaw vacuum solutions at early and late times and we find that it smoothly interpolates between the accelerated and decelerated branches. (See Fig. 2.) In some sense then it could be described as a “selfdual” solution.

as , we have and as ;

as , we have and as .
This is precisely the behavior found by Goldwirth and Perry[12] in their phaseplane analysis. The vacuum solutions correspond to particular solutions (here corresponding to the limiting behavior where is either infinite or zero) found previously[8].
The solutions in spatially curved models approach the flat space results only near . At late times in open () models (Fig. 3) and thus the dilaton becomes frozenin at a fixed value as the curvature dominates the evolution and and we approach the Einstein result. In closed models (Fig. 4) where as , the scale factor diverges in a finite proper time. Thus although these models bounce, and therefore must undergo a period of inflation () in the string frame, they still become curvature dominated at late times.
3.2 Radiation solution,
It is also possible to consider cases where the field does have a spatial dependence. Because our metric (and dilaton) is homogeneous and isotropic we will require that the field energymomentum tensor is also homogeneous and isotropic on average. It is natural (in flat space) to decompose any field into Fourier modes, where is a spatial comoving threevector. We see then that the preceding case corresponds to the long wavelength mode where . The other case in which we can solve the equation of motion is where where effects of spacetime curvature can be neglected and the usual Minkowski spacetime result holds.
Specifically we find that, in the Bframe, for , the equation of motion for reduces to
(3.18) 
Thus, for a single short wavelength mode, in the limit , we have where is a constant. This corresponds to an energymomentum tensor for the field in the Einstein frame
(3.19) 
where is a null 4vector with . Clearly this is not isotropic for a single wavevector, , but if we consider an isotropic distribution of wavevectors we find
(3.20)  
(3.21) 
the usual result for radiation in a FRW universe, where .
Note that the fluid is tracefree, and thus conformally invariant. This means that the energymomentum of the radiation is conserved in all conformal frames and so the field is decoupled from the dilaton which appears as a minimally coupled scalar field in the Einstein frame. Thus the first integral of its equation of motion, for a homogeneous , gives
(3.22) 
Just like the field for homogeneous , the energymomentum of the field behaves like a stiff fluid, with
(3.23)  
(3.24) 
Once again the constraint Eq.(3.3), now for two noninteracting fluids, one radiation, one stiff, can be integrated[17] to give
(3.25) 
using the time coordinate defined in Eq. (3.14). (As in the homogeneous case, this is just the familiar behavior for a FRW universe in general relativity with matter obeying the strong energy condition and thus singular for all models at .) This in turn allows the equation for , Eq. (3.22), to be integrated, yielding and thus, via the conformal transformation, the scale factor in the string frame.
(3.26)  
(3.27) 
where we have introduced
(3.28) 
Notice how the evolution of the dilaton field is now similar to that in the vacuum case except it is determined by the function rather than . At early times () is proportional to but, unlike the vacuum case, as and so the field becomes frozen in at late times in the flat model as well as the open model (where and thus ). Thus we recover the late time general relativistic results for radiation and curvature dominated models respectively.
In common with the vacuum solutions, there are two distinct branches (Fig. 5) with the dilaton monotonically increasing from zero (the decelerated branch) or decreasing from infinity (the accelerated branch) at . For the evolution is essentially identical to that in the vacuum case with two branches where as when , but when . Thus we have no “bounce” solution interpolating between the vacuum branches.
4 Anisotropic solutions
Having investigated the homogeneous and radiation solutions for the field in four dimensions, we return to Eqs. (2.8)–(2.14) in the Einstein frame where we will consider a metric tensor in dimensions that can be decomposed into the direct product form
(4.1) 
where we let run from to and run from to . The scale factors and thus refer to the space and space split respectively. We have chosen these spaces to be spatially flat, thus this is a Bianchi I metric. The procedure we outline here could also be applied to the general dimensional Bianchi I metric, but we restrict our discussion here to a twoscale factor model to avoid introducing too many degrees of freedom. Eq. (4.1) is of course related to the original string metric through the conformal transformation Eq. (2.7).
The Einstein evolution equations, Eq. (2.9), for the two scale factors written in terms of and are then
(4.2)  
(4.3) 
plus the constraint equation
(4.4) 
where , and (no sum over or ). Note that any isotropic stiff fluid for which makes no contribution to the righthandside of the evolution equations, entering only into the constraint equation.
We will adopt the simplest extension of our Ansatz for the field,
(4.5) 
where
Note that whereas this Ansatz included all the solutions in dimensions, this represents only one of many degrees of freedom for the field in the case. An antisymmetric field in the higherdimensional space corresponds to a whole range of fields in the dimensional reduced theory[19]. At a classical level we can obtain selfconsistent results if these other fields are set to zero initially, though in a full quantum treatment we would have to consider precisely how the massive modes of these fields are excited in the extra dimensions. Our choice of Ansatz makes any dependence of the function on the space coordinates, , irrelevant as this cannot affect and so we need consider only . A similar approach of demanding the field live only in three space dimensions was considered in [4] who found numerically that in the Einstein frame the effect was to drive that space to a large size while the other spaces remained of order the Planck scale.
Using (4.5) the integrability condition becomes the equation of motion for ;
(4.6) 
We can decompose the field into its Fourier components, so that the equation of motion for the homogeneous function is
(4.7) 
The final equation of motion is that for the dilaton in the Einstein frame, Eq.(2.11), driven by giving
(4.8) 
Note that having chosen all the fields to be independent of the coordinates on the space we could replace the dimensional action by an effective theory written in terms of the four dimensional part of the metric in the string frame
(4.9) 
where the effective dilaton is given by
(4.10) 
and the scale factors of the extra dimensions just act as massless (moduli) fields. However to emphasize the dynamical evolution of the space we shall treat this scale factor on an equal footing with that of the space. Note that the effective Einstein frame would not be the same as the dimensional Einstein frame due to the modified dilaton.
4.1 Homogeneous solution
The equation of motion for for longwavelength modes (where ) becomes
or
(4.11) 
where is a positive constant. From Eq. (2.14) and making use of Eqs. (4.5) and (4.11) we have
Hence in Eq. (2.14) we obtain
(4.12)  
(4.13)  
(4.14) 
which means that the field acts like an anisotropic fluid satisfying
Note that, just as in the case, in the string frame is inversely proportional to the square of the volume of the 3space ().
There is also a contribution to the energymomentum from the dilaton field given by Eq. (2.13),
(4.15)  
(4.16) 
which means that the field acts like an isotropic stiff fluid,
and thus, as remarked in the preceding section, drops out of the evolution equations for and .
We can substitute the energymomentum tensors into Eqs. (2.11), (4.2) and (4.3) to obtain the following equations of motion,
while the constraint equation becomes
(4.17) 
Notice how the presence of the field on the righthandside of these evolution equations tends to drive and in a positive direction, but drives negative. Thus it produces shear and an anisotropic expansion.
Introducing a new time coordinate through
(4.18) 
we obtain first integrals for the equations of motion,
(4.19)  
(4.20)  
(4.21) 
and
where etc, and and are constants of integration. In fact at least one of these is redundant as the origin of the variable is clearly arbitrary as is only defined by the differential relation in Eq.(4.18). Henceforth we shall take so that at . We can solve for by differentiating Eq. (4.21) and substituting in for from Eq. (4.19) to obtain
(4.22) 
where is another integration constant related to the others via the constraint Eq. (4.17).
(4.23) 
There is another important constraint which emerges. From Eqs. (4.21) and (4.22), by demanding that the combination remains nonnegative we obtain
(4.24) 
which means that the allowed range of is bounded by , where are the roots of the above expression;
(4.25) 
and we have introduced Clearly solutions only exist for .
We solve Eq. (4.22) to obtain
(4.26) 
where is a constant and
(4.27) 
We obtain similar solutions for and using Eqs. (4.20), (4.19) and (4.26)
(4.28)  
(4.29) 
where and are constants (with ), and being given by
(4.30)  
(4.31) 
It is clear from Eqs. (4.26), (4.28) and (4.29) that the behavior of and is similar in each case, the differences emerging in the exponents of the . We need to know how these solutions appear in the string frame as this is where the original theory has emerged from. This is straightforward to do. Using Eqs. (2.7) we see that the scale factors in the string frame are given by