The evolution of a spatially homogeneous and isotropic universe filled with a collisionless gas

TL;DR

This paper reviews the evolution of a homogeneous, isotropic universe filled with a collisionless gas, deriving the general distribution functions, and analyzing the universe's dynamics including curvature effects and a cosmological constant.

Contribution

It provides a detailed derivation of the most general collisionless, isotropic distribution functions in a Friedmann-Robertson-Walker universe and analyzes their impact on cosmic evolution.

Findings

Universes with collisionless gases have a singular origin.

Flat or hyperbolic universes expand forever with diluting energy density.

Spherical universes reach maximum expansion and recollapse.

Abstract

We review the evolution of a spatially homogeneous and isotropic universe described by a Friedmann-Robertson-Walker spacetime filled with a collisionless, neutral, simple, massive gas. The gas is described by a one-particle distribution function which satisfies the Liouville equation and is assumed to be homogeneous and isotropic. Making use of the isometries of the spacetime, we define precisely the homogeneity and isotropicity property of the distribution function, and based on this definition we give a concise derivation of the most general family of such distribution functions. For this family, we construct the particle current density and the stress-energy tensor and consider the coupled Einstein-Liouville system of equations. We find that as long as the distribution function is collisionless, homogenous and isotropic, the evolution of a Friedmann-Robertson-Walker universe exhibits a singular origin. Its future development depends upon the curvature of the spatial sections: spatially flat or hyperboloid universes expand forever and this expansion dilutes the energy density and pressure of the gas, while a universe with compact spherical sections reaches a maximal expansion, after which it reverses its motion and recollapses to a final crunch singularity where the energy density and isotropic pressure diverge. Finally, we analyze the evolution of the universe filled with the collisionless gas once a cosmological constant is included.