The Einstein-Vlasov system with a scalar field

We study the Einstein-Vlasov system coupled to a nonlinear scalar field with a nonnegative potential in locally spatially homogeneous spacetime, as an expanding cosmological model. It is shown that solutions of this system exist globally in time. When the potential of the scalar field is of an exponential form, the cosmological model corresponds to accelerated expansion. The system with an exponential potential shows the causal geodesic completeness of the spacetime towards the future. The asymptotic behavior of solutions of this system in the future time is analysed in various aspects, which shows power-law expansion.