The late-time behaviour of vortic Bianchi type VIII Universes

We use the dynamical systems approach to investigate the Bianchi type VIII models with a tilted $\gamma$-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 ($1<\gamma <2$), the fluid will in general tend towards a state of extreme tilt. For dust ($\gamma =1$), or for fluids less stiff than dust ($0<\gamma <1$), we show that the fluid will in the future be asymptotically non-tilted. Furthermore, we show that for all $\gamma \ge 1$ the universe evolves towards a vacuum state but does so rather slowly, $\rho \mathrm{/}{H}^2\propto 1\mathrm{/}\ln t$.