Cosmological Solutions in String Theories

TL;DR

The authors address cosmological evolution within string/M-theory by constructing a broad set of exact, time-dependent solutions in $D$-dimensional toroidally-compactified theories, where a $p$-dimensional expanding space coexists with a contracting internal space. The dynamics reduce to solvable $1$-scalar Liouville equations or $SL(N+1,R)$ Toda systems, enabling explicit expressions for the metric and dilatonic fields; many solutions can be oxidised to $D=10$ string theory or $D=11$ M-theory, revealing higher-dimensional origins and the potential for dynamical compactification. The paper systematically develops one-scalar, dyonic, multi-charge, and $SL(N+1,R)$ Toda cosmologies, analyzes their cosmological characteristics in $D=11$ and $D=10$, and demonstrates how dimensional reduction and oxidation connect lower- and higher-dimensional pictures. These results provide a constructive framework for exploring inflationary behavior and dynamical compactification within string/M-theory and offer exact solutions that can seed further phenomenological investigations.

Abstract

We obtain a large class of cosmological solutions in the toroidally-compactified low energy limits of string theories in $D$ dimensions. We consider solutions where a $p$-dimensional subset of the spatial coordinates, parameterising a flat space, a sphere, or an hyperboloid, describes the spatial sections of the physically-observed universe. The equations of motion reduce to Liouville or $SL(N+1,R)$ Toda equations, which are exactly solvable. We study some of the cases in detail, and find that under suitable conditions they can describe four-dimensional expanding universes. We discuss also how the solutions in $D$ dimensions behave upon oxidation back to the $D=10$ string theory or $D=11$ M-theory.