Level spacing distribution for the prototype of the Bianchi IX model

TL;DR

This paper investigates quantum chaos in the Bianchi IX cosmological model by analyzing eigenvalue level spacing distributions, revealing signs of chaos through Brody's distribution with level repulsion, using an approximation valid near the singularity.

Contribution

It introduces an approximation method for solving the eigenvalue problem of the Bianchi IX model's Hamiltonian and analyzes the resulting level spacing distribution for signs of chaos.

Findings

Level spacing distribution fits Brody's distribution with β≈0.3

Evidence of level repulsion indicating chaos

Approximation valid near the cosmic singularity

Abstract

Our results concern quantum chaos of the vacuum Bianchi IX model. We apply the equilateral triangle potential well approximation to the potential of the Bianchi IX model to solve the eigenvalue problem for the physical Hamiltonian. Such approximation is well satisfied in vicinity of the cosmic singularity. Level spacing distribution of the eigenvalues is studied with and without applying the unfolding procedure. In both cases, the obtained distributions are qualitatively described by Brody's distribution with the parameter $\beta\approx0.3$, revealing some sort of the level repulsion. The observed repulsion may reflect chaotic nature of the classical dynamics of the Bianchi IX universe. However, full understanding of this effects will require examination of the Bianchi IX model with the exact potential.