Bianchi Cosmologies: A Tale of Two Tilted 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.

16 pages, 1 figure, to appear in CQG

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