Phase mixing and the Vlasov equation in cosmology

We consider the Vlasov equation on slowly expanding isotropic homogeneous tori, described by the Friedmann--Lema\^itre--Robertson--Walker cosmological spacetimes. For expansion rate $t^q$, with $0< q<\frac{1}{2}$ (excluding certain exceptional values), we show that the spatial density decays at the rate $t^{-6q}$ and that, when the spatial average is removed, the density decays at an enhanced rate due to a phase mixing effect. This enhancement is polynomial for Sobolev initial data and super-polynomial, but sub-exponential, for real analytic initial data. We further show that, when the expansion rate is the borderline $t^{\frac{1}{2}}$ -- the rate which describes a radiation filled universe -- a degenerate phase mixing effect results in a logarithmic enhancement for Sobolev initial data and a super-logarithmic enhancement (in fact, a gain of $\exp(-\mu(\log t)^{\epsilon})$ for some $\mu,\epsilon>0$) for analytic initial data. The proof is based on a collection of commuting vector fields, and certain combinatorial properties of an associated collection of differential operators. The vector fields are not explicit, but are shown to have good properties when $t$ is large with respect to the momentum support of the solution. A physical space dyadic localisation is employed to treat non-compactly supported (in particular, non-trivial real analytic) but suitably decaying solutions.