The Bianchi IX attractor

TL;DR

This paper analyzes the asymptotic behavior of Bianchi type IX cosmological models near singularities, revealing convergence to attractors and unbounded curvature invariants in most cases.

Contribution

It provides a detailed analysis of the asymptotic dynamics of Bianchi IX spacetimes, including convergence to attractors and curvature behavior, extending understanding of singularity approaches.

Findings

Solutions converge to points or attractors near singularities.

In the stiff fluid case, all Bianchi class A solutions converge to a point.

Most solutions exhibit unbounded curvature invariants in incomplete directions.

Abstract

We consider the asymptotic behaviour of spatially homogeneous spacetimes of Bianchi type IX close to the singularity (we also consider some of the other Bianchi types, e. g. Bianchi VIII in the stiff fluid case). The matter content is assumed to be an orthogonal perfect fluid with linear equation of state and zero cosmological constant. In terms of the variables of Wainwright and Hsu, we have the following results. In the stiff fluid case, the solution converges to a point for all the Bianchi class A types. For the other matter models we consider, the Bianchi IX solutions generically converge to an attractor consisting of the closure of the vacuum type II orbits. Furthermore, we observe that for all the Bianchi class A spacetimes, except those of vacuum Taub type, a curvature invariant is unbounded in the incomplete directions of inextendible causal geodesics.