SUSXTH/98011
IEMFT174/98
Stabilizing the Dilaton in Superstring Cosmology
T. Barreiro^{1}^{1}1Email address: , B. de Carlos^{2}^{2}2Email address: B.deC and E. J. Copeland^{3}^{3}3Email address: E.J.C
Centre for Theoretical Physics, University of Sussex,
Falmer, Brighton BN1 9QH, United Kingdom
Abstract
We address the important issue of stabilizing the dilaton in the context of superstring cosmology. Scalar potentials which arise out of gaugino condensates in string models are generally exponential in nature. In a cosmological setting this allows for the existence of quasi scaling solutions, in which the energy density of the scalar field can, for a period, become a fixed fraction of the background density, due to the friction of the background expansion. Eventually the field can be trapped in the minimum of its potential as it leaves the scaling regime. We investigate this possibility in various gaugino condensation models and show that stable solutions for the dilaton are far more common than one would have naively thought.
I Introduction
Scalar fields in cosmology have been extensively studied over the past few years. One of the most intriguing areas in which they occur is Superstring theory, where the presence of the dilaton field is vital. Its vacuum expectation value (VEV) determines both the gauge and gravitational coupling constants of the low energy theory and also fixes the scale of supersymmetry (SUSY) breaking through the gravitino mass, . Therefore realistic models require a VEV of order one (in Planck units), and TeV.
Unfortunately, in string theory the dilaton potential is flat to all orders in perturbation theory, which of course means there is no way of obtaining a stable VEV for the field. This problem has to be overcome through some nonperturbative effect. The most promising possibility is through the formation of condensates of gaugino fields at an energy scale of around GeV [1]. The resulting scalar potential for the dilaton is then a combination of exponentials and polynomials in the field. A detailed investigation of these condensate models has demonstrated the need for at least two condensates to form if the dilaton potential is to develop a minimum at a realistic value although with a negative cosmological constant (these are the socalled “racetrack” models) [2].
An alternative proposal has recently been suggested as a method of obtaining a minimum for the dilaton field, and it has the advantage of relying on only one gaugino condensate [3, 4]. In this scenario the Kähler potential (which determines the kinetic terms of the dilaton in the action) requires string inspired nonperturbative corrections. A detailed analysis of these models [5] indicates that it is possible to have a minimum with zero or small positive cosmological constant. One additional positive feature that emerges is that the nonperturbative corrections can lead to a solution of the “moduli problem” for the dilaton [6] (fields with masses in the TeV range but which decay so slowly that they spoil nucleosynthesis), by giving it a huge mass.
Although attractive, both kind of models still have several problems associated with them. One is the difficulty of achieving inflation which was carefully studied by Brustein and Steinhardt [7] a few years ago. Taking a model of multiple gaugino condensation (as those studied in [2]) they argued that the kinetic energy associated with the dilaton field would dominate over its potential energy until (the canonically normalized field, related to the usual dilaton by ) would settle near a minimum of its potential. This obviously excludes inflation from happening, at least with the dilaton as the inflaton field, leaving the possibility of settling down to a minimum and then inflation being driven by other fields. However this second possibility also presented serious problems as the models they studied had a negative cosmological constant. Nonperturbative corrections to the Kähler potential can cure this latter problem, but in both cases the potentials are exponentially steep in the strong coupling regime. This would lead us to expect the dilaton to roll past the minimum rather than acquiring its VEV, which seems to be a major problem that superstring cosmology needs to address.
In this paper we turn our attention to the possibility that other matter fields rather than the dilaton drive the evolution of the Universe. Recent attention in cosmology has turned to the investigation of scaling solutions in models with exponential scalar field potentials [8, 9, 10]. These models are of particular interest because if the background dynamics are dominated by some matter source other than the field itself (i.e. radiation, dust, vacuum energy) then it is possible for the field to enter a scaling regime as it evolves down its potential. In this regime the friction term from the expansion of the Universe balances the kinetic energy of the field allowing it to enter this scaling era. Attractor solutions exist [8, 10] where the energy density in the field becomes a fixed fraction of the total energy density.
This intriguing behaviour can be applied to the case of the dilaton field arising from string theory. Under the assumption that it is evolving from somewhere in the strong coupling regime of its potential we show how, for a wide range of initial field values in the presence of a background dominated by a barotropic fluid, the dilaton enters a quasi scaling regime as it evolves down the potential, inspite of its steepness. This scaling behaviour eventually ends as the field enters the minimum of its potential, by which time it has slowed down sufficiently for it to simply oscillate about it, losing energy and eventually becoming fixed with a realistic VEV.
In section II we introduce the concept of scaling solutions with exponential potentials. In section III we demonstrate how this can be successfully adapted to both the racetrack and modified Kähler potential models of gaugino condensates. Solutions are presented analytically and numerically showing how the dilaton field is stabilized in its minimum. We conclude in section IV.
Ii Scaling solutions with exponential potentials
In this section we will describe some general features concerning the cosmological evolution of scalar fields with exponential potentials. Let us consider a scalar field with a potential energy density given by , with and and constants, which is evolving in a FriedmannRobertsonWalker (FRW) Universe containing a fluid with barotropic equation of state , where is a constant (, for instance for radiation domination or for matter domination). The equations of motion for a spatially flat FRW model with Hubble parameter are
(1)  
subject to the constraint
(2) 
In the previous equations we have assumed that the only interaction between and the other matter fields is gravitational. In what follows we set but it can easily be reinstated. For exponential potentials the asymptotic behaviour of this system can be obtained analytically, and we shall focus on the solution for very steep potentials, namely , as these are of most interest in string theory. Let us first of all review the structure of the solutions which, as is well known [8], contain a latetime attractor solution. For that purpose it is useful to proceed as in [10], defining the variables and and using the logarithm of the scale factor as the time variable. The previous system Eq. (1) becomes
(3)  
where a prime denotes a derivative with respect to . In terms of these variables, the constraint Eq. (2) becomes or, in other words, for we have the bounds .
The evolution of the field in terms of can be seen in Fig. 1, for , and initial conditions , , and . After an initial increase driven by the initital velocity of the field, we see how the friction term (or, in other words, the expansion of the Universe) dominates and freezes at a constant value for a considerable amount of time, until the field reaches a scaling regime corresponding to the critical points and [10]. This behaviour can also be obtained by solving the system of equations Eq. (3) under the following assumptions:

Stage I (pre scaling regime): Since , and is decreasing, is very small in this early stage of the evolution and can be neglected in Eq. (3). The equation for now becomes
(4) and its solution is
(5) where is the initial condition for (at ). The solution for can be obtained by integrating Eq. (5) (recall that ), and is given by
(6) where is the initial value of the field. As increases this solution tends to a constant value given by
(7) Obviously for zero initial velocity, will reduce to .

Stage II (scaling regime): as mentioned above, the scaling regime is defined by constant (i.e. critical) values for and . We then obtain and
(8) where now we have an explicit dependence on the characteristics of the potential (i.e., and ).
It is important to note that the background evolution is being determined by the additional matter fields present, given by . It is not the field which is driving the evolution. The fact that the simultaneous evolution of the scalar field together with the background causes the former to reach a scaling regime, inspite of the steepness of its (exponential) potential, suggests that this could be also the case for the kind of potentials arising from SUSY breaking via gaugino condensation, where the superpotential depends exponentially on the dilaton field. In the next section we will apply these solutions to the gaugino condensation models studied in [2] and [5].
Iii Scaling solutions with gaugino condensates
iii.1 Two condensate potentials
Multiple gaugino condensation (or racetrack) models have been extensively studied in the literature [11, 2]. Essentially the idea is to consider a strong type interaction in the hidden sector of our effective supergravity (SUGRA) theory, which is governed by a nonsemisimple gauge group. The superpotential of such models will then be expressed in terms of a sum of exponentials which conspire to generate a local minimum for the dilaton. To be more precise, the scalar potential in any N=1 SUGRA model [12] is given by
(9) 
where is the Kähler potential, is the superpotential and the subindices , represent derivatives of these two functions with respect to the different fields. Given that we are interested in superstring derived models, and in particular in studying the hidden sector of the theory, both and will be dependent on the dilaton () and the moduli (, ) fields. In fact we know that in the case of orbifold compactifications the treelevel Kähler potential is given by:
(10) 
and we will restrict our study to the case of a hidden sector interaction governed by two gauge groups, SU() SU() under which we have and “quark” representations with Yukawa couplings to a set of singlet fields. For simplicity, we will assume an overall modulus and a generic singlet field for each of the gauge groups, and respectively. In this case, the superpotential is given by
(11) 
where , , , is the Dedekind function and the self coupling of the field is set to . [Note is not the same as , the number of efoldings defined earlier].
These kind of models were thoroughly studied in [2], so let us summarize their main features: the presence of the function, imposed by the requirement of target space modular symmetry and, in general, the dependence of the potential ensures the presence of a minimum for (in Planck units) [13], independently of the particular gauge groups and/or matter representations (provided the dilaton acquires a VEV); also there exist minima in the direction if both condensates have opposite phases. Finally it was shown the existence of many examples for which there is a minimum in the direction at the phenomenologically acceptable value (remember that ), with a reasonable ( TeV) gravitino mass but always with a negative value of the potential energy. A typical example is shown in Fig. 2(a), which perfectly illustrates the problem that Brustein and Steinhardt pointed out in their paper: the steepness of the dilaton potential, which would prevent the field from settling down at its (negative) minimum, instead allowing it to run over the tiny maximum towards infinity.
We would like to study the evolution of the dilaton field (more precisely of its real part) before it settles to the minimum. We have already seen that a single exponential scalar potential can be solved analytically leading to scaling solutions. However there are major differences between this and the more realistic gaugino condensate models. First, the dilaton couples not only gravitationally but also directly to the matter fields. For simplicity, and to avoid making any assumptions about specific models, we will neglect this effect throughout this paper. Second, in the case of two condensates, the superpotential Eq. (11) contains two different exponentials of , therefore the scalar potential Eq. (9) will have all the different terms coming from . In particular there will be a mixture of exponential and polynomial terms in the dilaton. And finally, the differential equation for written in Eq. (1) is meant to be obeyed by a canonically normalised field, and is not. As it is well known, in the SUGRA Lagrangian the kinetic terms for scalar fields are given by , which in the case of introduces an extra factor of . Therefore, the correct procedure in order to study the evolution of the dilaton in an expanding Universe would be to solve the system in terms of the canonically normalised field , with or, alternatively, to modify Eq. (1) accordingly in order to account for the non canonical kinetic terms.
Fortunately these problems can be overcome. First we note that, even though the two condensates in Eq. (11) have to be carefully finetuned to produce a minimum for at the right value, for most of the evolution towards such a minimum only one of them (the one with a smaller value) will dominate. For simplicity, we also keep the matter fields constant at their minimum value during the evolution of . Therefore in the region we are studying, the superpotential can be approximated by, for example, the first condensate; moreover it is also easy to show that, in Eq. (9), the term proportional to dominates among those within the brackets. In conclusion, our scalar potential can be very well represented in the region before the minimum by the following expression
(12) 
which corresponds to considering only the dominant term mentioned above and setting everywhere but in the exponential. The result of such an approximation can be seen in Fig. 2(a), represented by the dashed line (with, as everywhere in this article, ), and it is good enough to justify our use of Eq. (12) when studying the evolution of away from the minimum.
Concerning the second problem, that of not being a canonically normalised field, we start by showing the exact numerical result of the evolution of the field, plotted in Fig. 2(b) in terms of versus , for a radiation dominated Universe (i.e. ) and initial conditions and . The different lines correspond to different initial conditions for . It is remarkable how the behaviour in the first stages of the evolution is very similar to that shown in Fig. 1 for a pure exponential potential, the difference appearing after efoldings and due to the presence of a minimum in this case. Depending on the initial position of the field it may or may not fall in the minimum, but what is clear is that if the field reaches the scaling regime the former will certainly happen, and that occurs, as we can see, for a very wide range of values of , contradicting the general belief that only for a very narrow range of initial values, all around the minimum, would the dilaton settle at its minimum. In fact the top curve of Fig. 2(b), which is the only one that does not end up at the minimum, corresponds to an asymptotic value of the field beyond the maximum of the potential.
We have checked that the scaling solution shown in Fig. 2(b) is very well represented by Eq. (8) in the case of the potential being given by Eq. (12), i.e. and . That is, as if were a canonically normalised field. This approximate solution corresponds to the thick solid line shown in Fig. 2(b).
We turn our attention to explaining this scaling behaviour of the non canonical dilaton field by trying to solve Eq. (1) analytically for this case. In terms of the normalised dilaton, , the equations to solve are
(13)  
where and are the and of Eq. (3). Note that the presence of a more involved potential requires the replacement . We can proceed as in the case of the pure exponential and solve for the two different stages defined before. Solving for Stage I is trivial, as the key point in this regime is to neglect any dependence on the potential and therefore the solution (for ) is identical to that in Eq. (6). Then
(14) 
where , with given by Eq. (7). However solving for the scaling regime of Stage II is much more complicated. To begin with, the form of the potential and the range of values of we are interested in guarantees that , and therefore we can solve for , obtaining the usual result , once again indicating that the background fields are determining the evolution of the Universe. As for the other two equations, let us rewrite them in terms of and a new variable (note that ). Again in the same approximation of small , we have for the two first equations
(15) 
Obviously, for the scaling regime observed in Fig. 2(b) (i.e. Stage II) we must have . The second equation in Eq. (15) will give us then the expected solution for , analogous to the pure exponential case
(16) 
whereas from the first equation we find that
(17) 
that is, does not seem to reach a critical value but instead has a dependence on . However, given the size of () and the range of values we are considering for (between 0.3 and 2), the deviation of from its expected critical value, given by , is not going to be significant, as it is obvious from Fig. 2(b). In any case let us compute this correction which modifies the pure scaling result by a factor of . Our ansatz is then
(18) 
Substituting into Eq. (17) and using the definition of () and the solution for we obtain
(19) 
which is indeed a very small numerical correction to the standard result (in fact ), and the only noticeable effect of having solved for a scalar field with an exponential potential but non minimal kinetic terms, a very encouraging result.
The pure exponential approximation will eventually break down as the dilaton approaches its minimum, . As can be seen from Fig. 2(b), if the field is in its scaling regime, it will not have enough energy to go over the maximum of the potential. Instead, it will oscillate around the minimum with an exponentially damped amplitude, settling down quickly to its final value. A simple estimation of the number of efoldings needed to reach the minimum can then be obtained from the scaling solution Eq. (18) by equating to . One obtains,
(20) 
(where we have ignored the correction as this result is accurate enough). Therefore once we have defined the example we are working with and the background, we will have . In fact, with this result we can also calculate
(21) 
which, remarkably enough, is independent of .
We have estimated and in a radiation dominated Universe () for a number of hidden sector gauge groups with and TeV, obtaining the almost invariant result
(22) 
Assuming a radiation dominated Universe, where , this implies GeV. The invariance of this result can be more or less understood by rewriting Eq. (20) in terms of the gravitino mass and the gauge group factors of the two condensates. The requirement of the model to be phenomenologically viable fixes most of those parameters leading to a constant value for and, therefore, and .
From these results it is possible to estimate the range of initial conditions that will eventually lead to a stable dilaton. As long as the field enters the scaling solution before reaching its minimum, we can ensure that it will be stabilized. For this to happen, a suficient condition is to take an initial value between the scaling solution at (i.e. the constant term in Eq. (18)) and , and an initial velocity such that the asymptotic solution in Eq. (14), is smaller than . Namely we obtain the following bounds for and ,
(23)  
Some examples can be seen in Fig. 2(b). Note that this region gets smaller for decreasing , as expected. There will also be a few initial conditions outside these limits that will still lead to a stable dilaton, such as initial values between the maximum and the minimum, values close to the lower bound on or allowing for negative initial velocities. Still, even if we do not take these into account, it is clear that there is a sizable region in parameter space that allows the dilaton to evolve to its minimum and stay there.
A further possibility is to consider an inflationary scenario. This will correspond to the case of a very small (and changing) . Note that it is the barotropic fluid that is producing inflation, so that we are not considering any kind of dilaton driven inflation. We can see from Eq. (18) that for the scaling solution is practically horizontal. It would take a large amount of efoldings for this solution to reach the minimum of the potential. Furthermore, for some models of inflation, would be rapidly changing and one does not expect the field to follow the scaling solution exactly in these cases [10]. Nevertheless, a few efoldings of inflation can open up a large region of parameter space. Unless the energy density is completely dominated by the dilaton, the almost constant of an inflationary scenario will quickly freeze the field at a constant value that will then lead to a scaling solution during reheating (). The bound for in Eq. (23) still applies, but is maximized for . One then expects most of the region of parameter space to evolve to a stabilized dilaton if there is an initial small period of inflation.
iii.2 One condensate with Knp
In this section we will perform the same analysis of the cosmological evolution of the dilaton field for a different class of models proposed more recently [3, 4, 5]. These consist of a single condensate which gets stabilized by the presence of nonperturbative corrections to the Kähler potential, which form has been suggested in [14]. Therefore we are dealing now with a scalar potential given again by Eq. (9), where the superpotential is simply
(24) 
with for a SU(N) group and . The Kähler potential is now more involved, , where is defined by Eq. (10) (with an overall modulus) and the nonperturbative correction is parameterised as
(25) 
that is in terms of three constants , , and , the first of which just determines the value of at the minimum. Therefore this description is effectively made in terms of only and , which are positive numbers. It has been shown [5] that for a wide range of values of both parameters it is possible to generate a minimum at with zero cosmological constant. The shape of these potentials is again very similar to that of the two condensate models.
Another feature of this ansazt for the nonperturbative corrections is that it is very well approximated in the region by the following expression:
(26) 
and therefore the scalar potential is given by
(27) 
In this second example, even though we have a single condensate to start with, the dependence of the potential upon is much more complicated, as can be expected from the form of the nonperturbative corrections to , given by Eq. (26). To be able to write Eq. (27) in the usual exponential form we set everywhere except in the exponents for which we use a linear fit. We obtain
(28) 
that is a pure exponential, where the exponent depends on the value of . As it was mentioned before, for a given hidden gauge group, there exist a series of values of (, ) for which has a minimum at with zero cosmological constant. That means that the cosmological evolution of a particular hidden sector interaction will be different depending on which values of the pair (, ) we are considering. This can be clearly appreciated in Fig. 3 where we plot the cosmological evolution of the dilaton field versus the number of efoldings for the same gauge group, SU(5), and initial conditions , , , .
Depending on the value of (the corresponding is fixed in order to have a zero cosmological constant at the minimum given by ) we see that the field will fall into the minimum provided that this minimum is not too finetuned. This is determined by the magnitude of , as was described in [5], the smaller it is the bigger the value of (and the amount of finetuning in the potential) is to set at the minimum. Therefore in this particular case we see how for the minimum becomes too finetuned and is unable to stop the field from rolling past the maximum. Unfortunately these small solutions are precisely the ones which correspond to a largest hierarchy between the dilaton and the gravitino masses [5], and therefore provide us with a solution to the “moduli problem” for this field. However, for the range of values of which give a satisfactory stabilization of the dilaton we can generate ratios between those two masses of up to which would, in most cases, be more than enough to solve the abovementioned problem. Changing the value of would correspond to a change in the minimal value of from which onwards the dilaton would fall in the minimum. That is, a bigger (smaller) value of corresponds to a bigger (smaller) value of the minimal , and therefore to a smaller (bigger) hierarchy between the dilaton and gravitino masses.
A similar analysis to the one performed in the previous section for the two condensates model can be done here concerning the analytical solutions of the evolution equations (for which we use Eq. (28)). Solving for Stage I is identical as before (see Eq. (14)), as the system of differential equations given by Eq. (3) does not depend on the characteristics of the potential in this regime. For Stage II we can obtain the analogous of Eq. (18) and in this case it is defined by and . Expressions for the number of efoldings needed to reach the minimum, as well as for , are also obtained by replacing in Eqs. (20,21) the current expressions for and . Once again, the fact that we are imposing a consistent phenomenology (i.e., and TeV) at low energies implies that the numerical values of and are very similar to those obtained in the previous section. In particular we can express as
(29) 
For the example shown in Fig. 3, namely , , we obtain, to a very good approximation, the almost invariant value which can be translated into .
Iv Conclusions
We have examined the issue of the cosmological evolution of the dilaton field in gaugino condensation models of SUSY breaking. By studying the behaviour of a scalar field with an exponential potential, we have been able to obtain analytic expressions for its time evolution in the presence of the Hubble parameter and a dominating background consisting of other matter fields. It turns out that under such circumstances the field tends to enter a scaling regime defined by constant values of both the potential and the velocity of the field relative to the expansion of the Universe, provided that this potential is steep enough.
Encouraged by these promising results we turned to apply them to the dilaton field in two particular models of SUSY breaking: multiple gaugino condensation (or racetrack) and one condensate with nonperturbative corrections to the Kähler potential. However two major differences arise with respect to the ideal case we had dealt with before: the shape of these potentials is not exactly exponential and the dilaton field is not canonically normalised. We have shown that these two problems can be easily overcome, and we have found very accurate analytic approximations to explain our numerical results. In both cases it is possible, contrary to the general belief, to stabilize the dilaton at its minimum for a large range of initial conditions (i.e., initial values for its position and velocity with ) despite of the steepness of the potentials in their strong coupling regime. A previous small period of inflation (i.e., ) opens up even more the region of parameter space leading to a stable dilaton. Moreover, the number of efoldings required to do so and the value of the Hubble parameter at the minimum seem to be fixed by the requirement of having a successful phenomenology at low energies.
Acknowledgements
The authors thank Andrew Liddle for interesting discussions. The work of T. B. was supported by JNICT (Portugal). The work of B. dC. and E. C. was supported by PPARC.
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