Billiard Representation for Multidimensional Cosmology with Intersecting p-branes near the Singularity

TL;DR

This paper models the near-singularity behavior of multidimensional cosmological theories with intersecting p-branes as billiard dynamics in hyperbolic space, analyzing conditions for finite-volume billiards and oscillatory cosmological behavior.

Contribution

It introduces a billiard representation for multidimensional cosmology with intersecting p-branes, linking geometric criteria to the finiteness and compactness of the billiard volume.

Findings

Finite-volume billiards correspond to oscillatory behavior near singularity.

Examples include Bianchi-IX and supergravity models with specific billiard geometries.

Chern-Simons term inclusion affects billiard confinement and dynamics.

Abstract

Multidimensional model describing the cosmological evolution of n Einstein spaces in the theory with l scalar fields and forms is considered. When electro-magnetic composite p-brane ansatz is adopted, and certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity is reduced to a billiard on the (N-1)-dimensional Lobachevsky space, N = n+l. The geometrical criterion for the finiteness of the billiard volume and its compactness is used. This criterion reduces the problem to the problem of illumination of (N-2)-dimensional sphere by point-like sources. Some examples with billiards of finite volume and hence oscillating behaviour near the singularity are considered. Among them examples with square and triangle 2-dimensional billiards (e.g. that of the Bianchi-IX model) and a 4-dimensional billiard in ``truncated'' D = 11 supergravity model (without the Chern-Simons term) are considered. It is shown that the inclusion of the Chern-Simons term destroys the confining of a billiard.