2. THE PLANCK TIME

The Planck time in cosmology is certainly the most challenging epoch to study, as the framework in which it must be described is still lacking. Though there are several different approaches to this problem, I will limit myself to that of string/M-theory [5].

Much of the work on string cosmology has focussed on the problem of inflation, and I will return to that subject in the next section. The realization that all of the different string theories, together with 11-dimensional supergravity, can be related through dualities has had a major impact on how we view the Planck epoch in cosmology. It has even led to the ultimate question as to what we mean by the Planck epoch. In a standard 4-dimensional model, the gravitational action can be written as

(1)

where ² = 8 G_N = 8 / M_P². In the context of a four dimensional theory, there is nothing ambiguous here as the Planck mass, determined from Newton's constant is simply M_P = 1.2 x 10¹⁹ GeV. Even in the context of an ``old'' 10-dimensional string theory, the 10-dimensional gravitational constant, ²₁₀ = 8 / M₁₀⁸ is equivalent to its 4-dimensional counterpart if the size of the 6-dimensional compact space is Planck scale in extent, ie. M₁₀ = M_P. Of course, the gravitational action in string theory must be augmented by the presence of additional fields in the gravitational sector, most notably by the universal coupling of the string dilaton [6]. Restricting our attention to only the dilaton gravity action, we can write

(2)

Upon reduction to 4 dimensions, we recover a relatively simple dilaton gravity system, which for a fixed dilaton (fixed by some potential which is perhaps induced by supersymmetry breaking), is just Einstein gravity in 4 dimensions along with some higher derivative curvature corrections (not shown) and additional moduli fields (also not shown).

As noted above, M-theory is emerging as the single underlying theory capable of unifying all particle interactions [7]. Although our understanding of M-theory is still incomplete, its various low energy limits, and the links between them, are known. These are the consistent string theories and 11D supergravity, related by dualities. A consequence of these developments is that the dilaton can be viewed as another modulus field in 11D supergravity. This could have important consequences for cosmological applications. The troubles with implementing conventional inflationary scenarios in string theory arise because of the dilaton and its couplings to the other modes in the string spectrum (see below). In 11D supergravity, such couplings are absent, and thus some of the obstacles for inflation in dilaton-plagued string theories could perhaps be resolved by way of M-theory.

One of the key points in the application of M theory to phenomenology is the reconciliation of the bottom-up calculation of M_GUT ~ 10¹⁶ GeV with the string unification scale, which is close to the four-dimensional Planck mass scale M₄ ~ 10¹⁹ GeV. This is achieved by postulating a large fifth dimension ₅ >> M_GUT^-1, which is not felt by the gauge interactions, but causes the gravitational interactions to rise with energy much faster than in the conventional four dimensions [7]. If we assume that the compact 6-space (Calabi-Yau?) is of the size determined by the fundamental Planck scale, M₁₁ so that V₆ ~ M₁₁^-6, then the fundamental Planck scale, M₁₁ is related to the four dimensional Planck scale by M₁₁⁹V₆₅ ~ M₄². Or,

(3)

For ₅ >> M₁₁^-1, it is possible to achieve M₄ >> M₁₁ ~ M_GUT. In this type of scenario, one could expect that inflation should be considered within a five-dimensional framework.

Within this general five-dimensional framework, two favored ranges for the magnitude of ₅ can be distinguished. One is relatively close to M_GUT^-1: ₅^-1 ~ 10¹² to 10¹⁵ GeV, and the other could be as low as ₅^-1 ~ 1 TeV [8]. In this latter case, the large dimension is not necessarily the conventional fifth dimension of M theory. Indeed, in models studied in [8] the large dimension may be related to what is normally considered as one of the six ``small'' dimensions that is conventionally compactified a là Calabi-Yau. Of course the physics of a very large ( 1 mm) extra spatial dimension has received considerable attention lately [9, 5].

As a starting point therefore, one should consider the 11D supergravity action

(4)

where R is the scalar curvature of the 11D metric (terms involving the 3-form potential and its 4-form field strength are not shown). Reducing to 10D is done easily by assuming that the 11^th direction is compact. In this case, we can carry out Kaluza-Klein reduction of (4) to find

(5)

After a conformal rescaling g₁₀ = ₁₁^-1 g_s, and defining the dilaton by exp(2 / 3) = ₁₁, we find exactly the action given in eq. (2). This is precisely the effective action which describes the low energy limit of the IIA superstring (though to be sure of the identification, one would be required to keep track of the terms shown here as ^... ). It is easy to rewrite this action in the ten-dimensional Einstein frame, by a further conformal rescaling g_s = e^/2 g_E. The action (2) can be reduced further to make contact with type IIB and heterotic theories.

If the 6-space is of the fundamental scale, then as described above, the Universe will have passed through a phase where it can be effectively described by a 5-dimensional space time. A simple ansatz for the 5D metric is that of the FRW form

(6)

In this way, the dilaton expectation value <> is related to the scale factor c(t). In the specific case of Horava-Witten type compactifications on S¹ / Z₂, the Universe is described by two 4D branes at the end points of the line segment. This type of compactification is particularly well suited for a reduction of M-theory to heterotic string theory, where the matter and hidden sectors sit on opposing branes. The cosmology of these theories has been studied at length in [10].