The late-time behaviour of vortic Bianchi type VIII Universes

Abstract

We use the dynamical systems approach to investigate the Bianchi type VIII models with a tilted $γ$-law perfect fluid. We introduce expansion-normalised variables and investigate the late-time asymptotic behaviour of the models and determine the late-time asymptotic states. For the Bianchi type VIII models the state space is unbounded and consequently, for all non-inflationary perfect fluids, one of the curvature variables grows without bound. Moreover, we show that for fluids stiffer than dust ($1<γ<2$), the fluid will in general tend towards a state of extreme tilt. For dust ($γ=1$), or for fluids less stiff than dust ($0<γ< 1$), we show that the fluid will in the future be asymptotically non-tilted. Furthermore, we show that for all $γ\geq 1$ the universe evolves towards a vacuum state but does so rather slowly, $ρ/H^2\propto 1/\ln t$.