The Mixmaster cosmological metrics

TL;DR

This paper explores the Bianchi IX cosmological model, discussing Einstein equations, related ergodic properties, and the nature of solutions near singularities, contributing to understanding the Mixmaster universe's complex dynamics.

Contribution

It provides a detailed analysis of the Mixmaster model, including new insights into approximate solutions and the behavior of solutions near singularities.

Findings

No plausible case for finite Kasner epochs in better approximations

Connections between geodesic flows and cosmological solutions

Insights into ergodic properties of the model

Abstract

This paper begins with a short presentation of the Bianchi IX or ``Mixmaster'' cosmological model, and some ways of writing the Einstein equations for it. There is then an interlude describing how I came to a study of this model, and then a report of some mostly unpublished work from a Ph.\ D. thesis of D. M. (Prakash) Chitre relating approximate solutions to geodesic flows on finite volume negative curvature Riemannian manifolds, for which he could quote results on ergodicity. A final section restates studies of a zero measure set of solutions which in first approximation appear to have only a finite number of Kasner epochs before reaching the singularity. One finds no plausible case for such behavior in better approximations.