Phase mixing and the Vlasov equation in cosmology

TL;DR

The paper analyzes the Vlasov equation on slowly expanding FLRW cosmologies with torus spatial topology, revealing a phase-mixing mechanism that accelerates the decay of the spatial density once the average is subtracted. It develops a robust, commuting-vector-field framework, supported by a dyadic physical-space localization, to derive precise decay rates for ρ−ρ̄ depending on the expansion rate q and the regularity of the initial data, including analytic and Sobolev classes. The results cover both the slowly expanding regime 0<q<1/2 and the degenerate radiation case q=1/2, with sharp (or near-sharp) polynomial or super-polynomial (even exponential in fractional powers of t or log t) enhancements, and they connect to broader phase-mixing/ Landau-damping phenomena in kinetic theory. The methods combine delicate operator identities (G_q, H_q), commutator algebra, and conservation laws, providing a framework that can accommodate non-compactly supported (analytic) data and potentially extend to Gevrey settings and related cosmological models.

Abstract

We consider the Vlasov equation on slowly expanding isotropic homogeneous tori, described by the Friedmann--Lemaître--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 $\exp(-μ(\log t)^ε)$ for some $μ,ε>0$) 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.

Figures (2)

• Figure 1: The phase mixing effect in the absence of expansion.
• Figure 2: The phase mixing effect in the presence of expansion. The support of a configuration initially localised around the origin may catch the expanding boundary if $0 \leq q \leq \frac{1}{2}$.

Theorems & Definitions (81)

• Theorem 1.1: Phase mixing for the Vlasov equation on slowly expanding FLRW spacetimes
• Theorem 1.2: Degenerate phase mixing for the Vlasov equation on the radiation FLRW spacetime
• Remark 1.3: Decay due to expansion and the case $q > \frac{1}{2}$
• Remark 1.4: The cases $q = \frac{1}{4}, \frac{1}{6}, \frac{1}{8},\ldots$
• Remark 1.5: The relativistic free transport equation $q = 0$
• Remark 1.6: Loss in analytic case compared to non-relativistic equation
• Remark 1.7: Gevrey regularity
• Remark 1.8: Estimates for derivatives
• Remark 1.9: Weak convergence to spatial average
• Remark 1.10: Non-relativistic limit