Superstring theory continues to be our leading candidate for a quantum theory of gravity. However, the theory is cast within a framework that leaves the most imaginative physicists yearning for a more tangible theory. In particular, String theory demands the presence of additional dimensions, which are usually taken to be on the order of the Planck length. So the question arises, why are these so different than the usual four space-time dimensions to which we are accustomed?
Brandenberger and Vafa addressed this question in the late 1980’s in the context of t-duality and string winding modes. Assuming the background space is homogeneous and toroidal, they showed that not only could t-duality eliminate the big-bang singularity, but it could also explain the dimensionality of space-time. Strings can wrap the toroidal background in winding or anti-winding states. When these strings collide they form momentum (free) states and the background is no longer constrained from expansion. Lacking the exact equations of string theory, they showed from a simple counting argument that three dimensions was the maximum number that could expand under this process.
This argument was further investigated in the context of Dilaton cosmology. A gas of string winding modes is considered, which gives rise to a confining potential in the equations of motion. This result is in agreement with numerical simulations invoking the classical Einstein equations with a pure background metric.
In the years that followed, we witnessed the advent of M-theory and the introduction of higher degrees of freedom. If this dimensionality argument is to hold, it must do so for these higher degrees of freedom, such as d-branes. In recent work, it was shown that if we consider d-branes in 11 dimensional M-theory compactified on S1, giving a 10-dimensional Type II-A string theory that the argument will still hold. This investigation again relies on a homogenous toroidal background, initially in thermal equilibrium.
We propose to further investigate the dimensionality of space-time in a more general context. All previous investigation has relied on a homogenous, toroidal background in thermal equilibrium. The introduction of inhomogeneities into the theory not only challenges the argument for dimensionality, but also promises to offer insight into the important link between string theory and cosmology. What would the effects of these inhomogeneities be in the early universe and how do they affect the dimensionality arguments?
The research will be conducted both analytically and numerically. Analytic considerations will include that of a brane gas in the context of Dilaton cosmology. As the universe undergoes cosmological inflation inhomogeneities will be “smeared out” to prevent grouping. Then the analysis will be continued by obtaining an equation of state and introduce an effective action for the theory. The examination of the growth and behavior of these inhomogeneities in the context of string theory cosmology will follow. Numerical simulations will be made in the context of 4+1 dimensional gravity, with a Dilaton field and matter sources.
This work will be conducted in corroboration with Robert Brandenberger of Brown University, Brian Greene of Columbia University, and Saul Tenkolsky of Cornell University. Numerical simulations will be conducted at the Brown University Supercomputing Center.