Nonlinear dynamics from the relativistic Boltzmann equation in the Friedmann-Lema\^itre-Robertson-Walker spacetime

TL;DR

This paper investigates the nonlinear relativistic Boltzmann equation in an expanding FLRW universe, revealing decoupling of modes, transient distribution tails, and differences in equilibration dynamics across approximation methods.

Contribution

It demonstrates the complete decoupling of non-hydrodynamic modes in FLRW spacetime and analyzes the limitations of common approximation methods in capturing distribution tails.

Findings

Non-hydrodynamic modes decouple from hydrodynamic ones.

Transient tails with nontrivial momentum dependence are observed.

Relaxation time approximation over-simplifies the approach to equilibrium.

Abstract

The dissipative dynamics of an expanding massless gas with constant cross section in a spatially flat Friedmann-Lema\^itre-Robertson-Walker (FLRW) universe is studied. The mathematical problem of solving the full nonlinear relativistic Boltzmann equation is recast into an infinite set of nonlinear ordinary differential equations for the moments of the one-particle distribution function. Momentum-space resolution is determined by the number of non-hydrodynamic modes included in the moment hierarchy, i.e., by the truncation order. We show that in the FLRW spacetime the non-hydrodynamic modes decouple completely from the hydrodynamic degrees of freedom. This results in the system flowing as an ideal fluid while at the same time producing entropy. The solutions to the nonlinear Boltzmann equation exhibit transient tails of the distribution function with nontrivial momentum dependence. The evolution of this tail is not correctly captured by the relaxation time approximation nor by the linearized Boltzmann equation. However, the latter probes additional high-momentum details unresolved by the relaxation time approximation. While the expansion of the FLRW spacetime is slow enough for the system to move towards (and not away from) local thermal equilibrium, it is not sufficiently slow for the system to actually ever reach complete local equilibrium. Equilibration is fastest in the relaxation time approximation, followed, in turn, by kinetic evolution with a linearized and a fully nonlinear Boltzmann collision term.