Bianchi Cosmologies: A Tale of Two Tilted fluids

TL;DR

This paper analyzes Bianchi type VI$_0$ cosmological models with two tilted fluids using dynamical systems, identifying late-time attractors and bifurcations, and exploring the effects of fluid stiffness and tilt on cosmic evolution.

Contribution

It provides a detailed dynamical systems analysis of tilted two-fluid Bianchi VI$_0$ models, identifying equilibrium points and bifurcations that determine late-time behavior.

Findings

Stiffest fluid becomes insignificant if $ ext{γ}<6/5$.

Bifurcation occurs at $ ext{γ}=6/5$, affecting tilt behavior.

Chaotic behavior is suppressed near the initial singularity for stiff fluids.

Abstract

We use a dynamical systems approach to study Bianchi type VI$_0$ cosmological models containing two tilted $\gamma$-law perfect fluids. The full state space is 11-dimensional, but the existence of a monotonic function simplifies the analysis considerably. We restrict attention to a particular, physically interesting, invariant subspace and find all equilibrium points that are future stable in the full 11-dimensional state space; these are consequently local attractors and serve as late-time asymptotes for an open set of tilted type VI$_0$ models containing two tilted fluids. We find that if one of the fluids has an equation of state parameter $\gamma<6/5$, the stiffest fluid will be dynamically insignificant at late times. For the value $\gamma=6/5$ there is a 2-dimensional bifurcation set, and if both fluids are stiffer than $\gamma=6/5$ both fluids will have extreme tilt asymptotically. We investigate the case in which one fluid is extremely tilting in detail. We also consider the case with one stiff fluid ($\gamma=2$) close to the initial singularity, and find that the chaotic behaviour which occurs in general Bianchi models with $\gamma<2$ is suppressed.