On billiard approach in multidimensional cosmological models

TL;DR

This paper reviews the billiard approach in multidimensional cosmological models, exploring its mathematical representation, conditions for different asymptotic behaviors, and connections to hyperbolic Kac-Moody algebras, with implications for understanding cosmological dynamics.

Contribution

It provides a comprehensive overview of the billiard approach in multidimensional cosmology, including new conditions for Kasner-like and oscillating behaviors and examples linked to hyperbolic Kac-Moody algebras.

Findings

Formulated conditions for Kasner-like and oscillating behaviors.

Presented examples related to hyperbolic Kac-Moody algebras.

Connected billiard dynamics with cosmological models and algebraic structures.

Abstract

A short overview of the billiard approach for cosmological-type models with n Einstein factor-spaces is presented. We start with the billiard representation for pseudo-Euclidean Toda-like systems of cosmological origin. Then we consider cosmological model with multicomponent "perfect-fluid" and cosmological-type model with composite branes. The second one describes cosmological and spherically-symmetric configurations in a theory with scalar fields and fields of forms. The conditions for appearance of asymptotical Kasner-like and oscillating behaviors in the limit \tau \to +0 and \tau \to + \infty (where \tau is a "synchronous-type" variable) are formulated (e.g. in terms of inequalities on Kasner parameters). Examples of billiards related to the hyperbolic Kac-Moody algebras E_{10}, AE_3 and A_{1,II} are given.