The Futures of Bianchi type VII0 cosmologies with vorticity

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We use expansion-normalised variables to investigate the Bianchi type VII₀ model with a tilted γ-law perfect fluid. We emphasize the late-time asymptotic dynamical behaviour of the models and determine their asymptotic states. Unlike the other Bianchi models of solvable type, the type VII₀ state space is unbounded. Consequently we show that, for a general non-inflationary perfect fluid, one of the curvature variables diverges at late times, which implies that the type VII₀ model is not asymptotically self-similar to the future. Regarding the tilt velocity, we show that for fluids with γ<4/3 (which includes the important case of dust, γ=1) the tilt velocity tends to zero at late times, while for a radiation fluid, γ=4/3, the fluid is tilted and its vorticity is dynamically significant at late times. For fluids stiffer than radiation (γ>4/3), the future asymptotic state is an extremely tilted spacetime with vorticity.