Cosmological solutions in string theories

We obtain a large class of cosmological solutions in the toroidally compactified low-energy limits of string theories in D dimensions. We consider solutions a here a p-dimensional subset of the spatial coordinates, parametrizing a Bat space, a sphere, or a 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 no 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.