The dynamics of general Bianchi IX model near the cosmological singularity

TL;DR

This paper investigates the behavior of the most general Bianchi IX cosmological model near singularity, challenging previous assumptions and identifying conditions under which asymptotic dynamics assumptions hold.

Contribution

It demonstrates that the assumption of persistent asymptotic dynamics near the singularity is generally incorrect, but becomes valid after a certain time depending on non-diagonality.

Findings

The assumption of continuous asymptotic dynamics is false in general.

A specific time t_0 exists after which the assumption holds.

The smaller the non-diagonality, the sooner the assumption becomes valid.

Abstract

Half a century ago, Belinsky and Khalatnikov proposed a generic solution of the Einstein equations near their cosmological singularity, based on a generalization of the homogeneous model of Bianchi type IX. Consideration of the evolution of the most general non-diagonal case of this model is greatly simplified if it is assumed that, when approaching the singularity t=0, it reduces to the so-called asymptotic dynamics, at which inequality (6) holds. It has been suggested that this inequality continues to be true from the moment of its first fulfilment up to the singularity of space-time. We analyze this assumption and show that it is incorrect in the general case. However, it is shown that there is always a time t_0, after which this assumption becomes true. The value of t_0 is the smaller the less is the degree of non-diagonality of the model. Some details of the behaviour of the non-diagonal homogeneous model of Bianchi type IX are considered at the stage of asymptotic dynamics of approaching the singularity.