Some exact results on the Belinski-Khalatnikov-Lifshitz scenario

TL;DR

This paper provides exact results on the BKL scenario near cosmological singularities, proving their inevitability, describing the approach dynamics, and analyzing oscillatory behaviors with explicit solutions.

Contribution

It offers new exact solutions and insights into the chaotic oscillations and asymptotic behavior of the universe near singularities within the BKL framework.

Findings

Singularity is an inevitable beginning or end of the universe.

Approaching the singularity requires infinite time, with no finite-time singularities.

Oscillations near the singularity are chaotic and have sawtooth shapes in logarithmic variables.

Abstract

The well-known Bielinski-Khalatnikov-Lifshitz (BKL) scenario for the universe near the cosmological singularity is supplemented with a few exact results following from the BKL asymptotic of the Einstein equations: (1) The cosmological singularity is proved to be an inevitable beginning or end of the universe as described by these equations. (2) Attaining the singularity from shrinking initial conditions requires infinite time parameter $\tau$; no singularity of any kind may occur in a finite $\tau$. (3) The previously found exact solution [P.G. and W. Piechocki, Eur. Phys. J. C 82:216 (2022)] is the only asymptotic with well-defined proportions between the directional scale factors which have been appropriately compensated against indefinite growth of anisotropy. In all other cases, the universe undergoes oscillations of Kasner type, which reduce the length scales to nearly zero in some directions, while largely extending it in the others. Together with instability of the exact solution [op. cit.], it makes the approach to the singularity inevitably chaotic. (4) Reduced equations are proposed and explicitly solved to describe these oscillations near their turning points. In logarithmic variables, the oscillations are found to have sawtooth shapes. A by-product is a quadric of kinetic energy, a simple geometric tool for all this analysis.