Since the last Snowmass meeting, tremendous progress has been made in understanding the non-perturbative structure of string theory and the theory into which it has evolved - M-theory. These recent advances may be at last opening the door for a serious approach to analyzing the earliest times in the universe. Since the early universe is a hot, dense, highly energetic place and time, we expect a non-perturbative understanding of quantum effects in gravity to be essential to an analysis of cosmology in such an environment. For those who believe our basic framework for addressing these questions is in place, the recent explosion of interest in extra-dimensional physics should act as a cautionary tale. Solid tests of our cosmological model go back only as far as the epoch of primordial nucleosynthesis, at which the temperature of the universe was still only a few MeV. If we are to gain a quantitative understanding of earlier epochs, it will be necessary to develop a consistent approach to theoretical analyses of the early universe, and innovative new ideas for probing the nature of space and time at these early times. While most ideas in this field are wildly speculative at the present, we nevertheless present some interesting first steps here.
A. Finite Temperature String / M-Theory and Cosmology
If we accept string / M-Theory as the correct theory of everything, then it must hold the key to the most fundamental problems in cosmology. Perhaps the most obvious modification to the standard equations of cosmology is that gravity is no longer described purely by general relativity. In particular, Einstein's theory is modified by the appearance of a dilaton field, related to the compactification of the theory, and by higher derivative terms in the action, at nonzero order in the string coupling constant. There have been a number of attempts to use these modifications to address the origin of the hot big bang phase of the universe, the issue of the initial singularity, and even the origin of the number of macroscopic dimensions we observe.
In the Brandenberger-Vafa scenario [Brandenberger and Vafa(1989), Tseytlin and Vafa(1992)], it is assumed that the fundamental physics respects a T-duality, interchanging large and small radii of a toroidal compactification. This has two particularly interesting implications for cosmology. First, since the small radius of the universe limit is equivalent to the large radius limit, there is no big-bang singularity in the usual general relativistic sense. Second, the dynamics of string winding modes constrains the number of dimensions that may become macroscopic. Imposing T-duality on the modified Einstein equations results in string winding around a particular direction preventing the expansion of that dimension. In the early universe, at high temperatures, it is argued that one should think of a gas of strings (and branes in an extended picture [Alexander et al.(2000), Brandenberger et al.(2001)]) and their modes. In more than three spatial dimensions, string winding modes cannot annihilate, and hence only as many as three dimensions may decompactify in this picture. Although there are a number of problems with this picture (such as the need for non-trivial one-cycles to wrap around, which do not exist in typical Calabi-Yau compactifications) this provides an interesting possiblity for explaining features of the universe that are inaccessible to our lower energy effective theories. In general, a careful analysis of the implications of finite temperature effects in string theory seems an interesting research avenue to be pursued.