The chaotic behavior of the Bianchi IX model under the influence of quantum effects

TL;DR

This paper investigates how quantum effects influence the chaotic dynamics of the Bianchi IX cosmological model, revealing that quantum fluctuations weaken but do not eliminate chaos.

Contribution

It introduces a quantum framework for Bianchi IX, extending classical phase space with quantum fluctuations, and applies dynamical systems methods to analyze chaos.

Findings

Quantum effects reduce chaos strength compared to classical model

The model remains chaotic despite quantum influences

Fractal analysis shows diminished chaos intensity

Abstract

A quantum analysis of the vacuum Bianchi IX model is performed, focusing in particular on the chaotic nature of the system. The framework constructed here is general enough for the results to apply in the context of any theory of quantum gravity, since it includes only minimal approximations that make it possible to encode the information of all quantum degrees of freedom in the fluctuations of the usual anisotropy parameters. These fluctuations are described as canonical variables that extend the classical phase space. In this way, standard methods for dynamical systems can be applied to study the chaos of the model. Two specific methods are applied that are suitable for time-reparameterization invariant systems. First, a generalized version of the Misner-Chitre variables is constructed, which provides an isomorphism between the quantum Bianchi IX dynamics and the geodesic flow on a suitable Riemannian manifold, extending, in this way, the usual billiard picture. Secondly, the fractal dimension of the boundary between points with different outcomes in the space of initial data is numerically analyzed. While the quantum system remains chaotic, the main conclusion is that its strength is considerably diminished by quantum effects as compared to its classical counterpart.