The late-time behaviour of tilted Bianchi type VIII universes in presence of diffusion

We apply the dynamical systems approach to ever-expanding Bianchi type VIII cosmologies filled with a tilted $\gamma$-fluid undergoing velocity diffusion on a scalar field. We determine the future attractors and investigate the late-time behaviour of the models. We find that at late times the normalized energy density $\Omega$ tends to zero, while the scalar potential $\Phi$ approaches 1 and dominates the evolution. Moreover, we demonstrate that in presence of diffusion fluids with $\gamma<3/2$, which includes physically important cases of dust $(\gamma=1)$ and radiation $(\gamma=4/3)$, are asymptotically non-tilted; the velocity of the fluid with $\gamma=3/2$ tends to a constant value $0<\bar{V}<1$; and stiffer fluids evolve towards a state of extreme tilt. Finally, we show that diffusion significantly reduces the decay rates of energy density for dust and fluids stiffer than dust $(\gamma \geq 1)$; for example, at $\gamma=4/3$ (radiation) we obtain $\rho/H^2 \propto e^{-3H_0 t}$ at late times, while $\rho/H^2 \propto e^{-4H_0 t}$ when diffusion is absent.