Billiard representation for multidimensional multi-scalar cosmological model with exponential potentials

TL;DR

This paper analyzes a multidimensional cosmological model with multiple scalar fields and exponential potentials, showing that for more than one scale factor the solutions tend to Kasner-like behavior near singularities, while for a single scale factor, oscillations occur.

Contribution

It introduces a billiard representation for the dynamics of the model near singularities, extending previous approaches to multidimensional scalar field cosmologies.

Findings

For n > 1, solutions exhibit Kasner-like asymptotic behavior.

For n = 1, finite-volume billiards lead to oscillatory behavior.

The billiard approach simplifies understanding near-singularity dynamics.

Abstract

Multidimensional cosmological-type model with n Einstein factor spaces in the theory with l scalar fields and multiple exponential potential is considered. The dynamics of the model near the singularity is reduced to a billiard on the (N-1)-dimensional Lobachevsky space H^{N-1}, N = n+l. It is shown that for n > 1 the oscillating behaviour near the singularity is absent and solutions have an asymptotical Kasner-like behavior. For the case of one scale factor (n =1) billiards with finite volumes (e.g. coinciding with that of the Bianchi-IX model) are described and oscillating behaviour of scalar fields near the singularity is obtained.