INFLATION

It has been known for quite some time that it is very difficult to incorporate conventional inflationary scenarios based on (de Sitter expansion) into the low-energy limit of string theory [11]. The principal obstacle has been the fact that the low-energy dynamics of the theory contains massless scalar fields with non-minimal couplings to gravity whose coupling constants are precisely given by the conformal symmetry and/or the dualities of string theory. In an expanding universe, these fields roll during the course of the expansion as dictated by their equations of motion, consuming the available energy and hence decreasing the rate of expansion. That is, besides the Minkowski space solution, there are no other solutions to Einstein's equations (when the Gauss-Bonnet curvature squared terms are kept) of the dilaton-gravity system with constant curvature and a stationary dilaton. de Sitter solutions are possible if the dilaton sits in a potential minimum (with V 0) due to supersymmetry breaking effects, but in this case, the theory suffers a graceful exit problem.

Instead, one typically finds solutions where the scale factor of the universe grows as a power of time, with the power determined by the scalar coupling constants. Once the numerical values of these constants, fixed by string theory, are taken into account, it has been found that the resulting power laws are too slow to give an inflationary universe [12]. For example, if we ignore the compact 6-space of eq. (2) and include a cosmological term we have a 4D action

(7)

where now ₄^-2 = M₄² / 8. In the Einstein frame this becomes

(8)

In the Einstein frame, the solution to the equations of motion yield a scale factor which grows linearly in time [13]. However, this expansion turns out not to be physical - the corresponding scale factor in the string frame is constant. To resolve this quandary, one must compare the scale factor in each frame with some physical length scale, such as the Compton wavelength of a massive particle. Had we included a massive scalar field in the action (7), we would find that the scale factor relative to the Compton wavelength ~ m^-1 is also constant. However, in the Einstein frame, the mass term would also be dilaton dependent with mass e m, implying that the Compton wavelength also grows linearly in time ( ~ - ln t) and hence relative to a physical measure of length, the Universe according to (8) is also not expanding [12].

An interesting alternative to the standard inflationary picture in string theory is the pre-big bang scenario [14]. The solutions to the equations of motions in the string frame yield two distinct branches often labeled (+) and (-) corresponding to solutions which either evolve towards singularities and are singularity-free in the past, or evolve from singularities and are singularity-free in the future. One of the goals of the pre-big bang scenario is the connection of the expanding solutions in the two branches, in such a way that the (+)-branch chronologically precedes the (-)-branch and hence the cosmological singularity would be removed. It still remains to be seen if a coherent and fully consistent description of branch-changing can be found. In fact there are strong arguments showing the precise difficulty of a branch change [15] though some progress has been made to resolve this graceful exit problem [16]. It is interesting to note that in the string frame, both the expanding and contracting metrics are degenerate to a single Einstein frame metric, and that the only difference between the two subclasses of solutions is the sign of the dilaton field. In the context of a full 11D theory, there are several solutions which can be found [17] all degenerate with the known pre-big bang solutions.

There is of course no possibility in this contribution to review all of the work on inflation in the context of string theories and extra dimensions. However, before moving to the subject of baryogenesis, I will mention two possibilities in which extra dimensions aid the implementation of inflation. For other recent work see [18].

In the first [19], one makes use of the higher derivative curvature terms in the action. Among the first utilizations of higher-derivative curvature terms is the Starobinsky model [20], which is based on obtaining a self-consistent solution of Einstein's equations when they are modified to include one-loop quantum corrections to the stress-energy tensor T_µ. In its simplest form, the model is equivalent to a theory of gravity with an R² correction which can be written as [21]

(9)

It is well known that this theory is conformally equivalent to a theory of Einstein gravity plus a scalar field [22]. The potential for the resulting scalar is extremely flat for field values >> M₄ and has a minimum at = 0 with V( = 0) = 0. For large initial values of , one can recognize this as an excellent model for chaotic inflation [23].

In general, quantum corrections to the right-hand side of Einstein's equation in the absence of matter can be written as [24]

(10)

where k₂ and k₃ are constants that appear in the process of regularization. k₂ is related to the number of light spin states, which can be very large in variants of string theories based on M theory. On the other hand, the coefficient k₃ is independent of the number of light states. This term is equivalent to the variation of the R² term in the effective action. The theory admits a de Sitter solution which can be found from the 00 component of gravitational equation of motion [25]. Defining H' = 2880² / k₂ and M² = 2880² / k₃, and setting the spatial curvature k = 0, one finds [26]

(11)

where H is the Hubble parameter. The de Sitter solution corresponds to H = H' and of course = = 0.

In order to avoid the overproduction of gravitons there is a lower limit on the parameters k_{2, 3} [27, 26]: k₂ >> ~ 10¹⁰ implying the need for billions of spin degrees of freedom to be present. While this seems like an inordinately large number, it is possible to generate very large numbers of degrees of freedom in theories with extra dimensions. Particularly if N ~ M_{GUT 5} 10⁸ [19]. The bound for k₃ is k₃ 10⁹, corresponding to M 10¹⁴ GeV.

In general R² corrections to the action do not appear in 5 (11) dimensions. Therefore the first correction is of order R⁴. Like the curvature squared correction, the action

(12)

can be conformally transformed to a 5D Einstein theory with an additional scalar field () and a potential U() [28]. Making the KK reduction to 4D, we find in addition to , a second scalar field () which is the modulus of the 5th dimension and is related to the dilaton. The resulting potential takes the form [19]

(13)

The dilaton potential here, as in most string descendent models, is problematic unless it is fixed by an additional potential. In the absence of a dilaton potential, this model will not inflate. The remaining potential (of ) differs from that in the R² model. As one can see it is no longer flat, but rather takes the shape similar to that of the double well potential. As a result, chaotic inflation [23] with a large initial value of is impossible here. Nevertheless there may remain the possibility to realize inflationary expansion in this model by using the potential energy around the local maximum, V(_m), as in the topological inflation scenario of Linde and Vilenkin [29]. In this scenario, if the scalar field (x) is randomly distributed initially with a large dispersion, some part of the universe will roll to = 0, while in other parts it will run away to infinity. Between any two such regions there will appear domain walls, containing a large energy density, ~ V(_m). If the wall is thicker than the Hubble radius of this energy density, there will exist a sufficiently large quasi-homogeneous region, filled with large potential energy, where inflationary expansion naturally sets in.

It can be shown [19] that the potential (13) does satisfy the conditions for topological inflation. In addition, there is no problem with the graceful exit. However, the spectrum of density fluctuations can be calculated and the spectral index is given by n_s = 1 - m² / H_m², where m² and H_m² are the curvature (of the potential) and Hubble parameter determined at the maximum. In this model, n_s = 3/8. Furthermore, the magnitude of density fluctuations requires H_m ~ M₅ ~ 10^-14M₄ which are probably unacceptably small.

These problems can be remedied by either the inclusion of an R² correction to the action, or by a large tower of KK states. The resolution of the problem can be traced to the magnitude of the Hubble parameter, which is determined by H² ~ U() / M₄². In the presence of a large number of degrees of freedom, quantum corrections modify this relation along the line of eq. (11) in which case H is driven to H' and need not be exceptionally small. It suffices now that H, M₅ ~ 10^-5M₄ with ₅^-1 ~ O(100) GeV. In this case, the spectral index is 0.95 and the magnitude of fluctuations is acceptably small.

The second possibility [30, 31], also makes use of the tower of KK states. The idea of using multiple fields to drive inflation where the parameters of the theory would normally not lead to sufficient inflation is called assisted inflation [32]. For example, it is well known that scalar fields with exponential potentials of the form V() = e^-, lead to power law expansion with the cosmological scale factor growing as R(t) ~ t^p and p = 2 / ². Density fluctuations are no longer scale invariant but scale as |ð / (k)|² ~ k^{n - 1} with n = 1 - 2/(p - 1). Sufficient inflation along with n 1, requires p to be large. In [32], it was shown that a system of scalar fields each with exponential potentials, (even if the individual powers, p_i, are not sufficiently large to generate inflation) has an attractor solution in which the universe power-law expands with a power given by p = p_i.

Assistance, can in fact be generalized to other types of potentials [30]. For example, we can consider a general field theory of multiple, self-interacting scalar fields of the form

(14)

This system consists of N completely equivalent, decoupled scalar fields. As a result, the Lagrangian can be written as

(15)

where

(16)

The resulting theory describes a single scalar field with the same type of self-interactions compared to the fields in the original theory. However, these self-interactions are considerably weaker since both of the coupling constants now scale with the number of scalar fields N. As a result, as the number of scalar fields that we include in the theory becomes larger, the coupling constants become smaller and the corresponding fine-tuning becomes milder. Thus the same basic idea expounded in [32, 33] carries over very simply to chaotic inflation [23] based on a quartic potential. While ₄ must still be of order 10^-12, the fundamental coupling in the theory ₄ can now be much larger if N is large. In addition, the problem associated with chaotic inflation [34], large ( >> M_p) initial field values and the necessary fine tuning of non-renormalizable interactions, is cured [31]. Because of the rescaling in eq. (16), in a theory of multiple fields, chaotic inflation will operate, even though none of the fields have expectation values greater than M_P.

In [30, 31], it was suggested that the source of the multiple fields is the KK reduction from a higher dimensional theory to 4D. For example, the reduction of a 5D theory results in N ₅ M₅ fields. Though these fields are not decoupled as in the example above, and hence assistance is not guaranteed [30, 35], attractor solutions do exist and assistance is generated primarily due to the rescaling of the fields (from 5D to 4D) which is necessary due to the dimensional reduction. To see this, consider the action

(17)

When KK-reduced to 4D, G₅ -> G₄₅ and canonical 4D scalar fields must be defined by = sqrt[2₅] and hence the quartic potential becomes ⁴ / 4! with = / 2₅ M₅ = / N.