Massive Fermi Gas in the Expanding Universe

TL;DR

This paper analyzes the evolution of a decoupled ideal Fermi gas in an expanding universe, deriving new formulas for its spectrum and phase space density, with applications to cosmic neutrino background predictions.

Contribution

It introduces a novel expansion of the Fermi-Dirac integral and provides a comprehensive model for the spectrum evolution of a decoupled Fermi gas in cosmology.

Findings

Derived expressions for effective temperature and chemical potential during expansion.

Compared phase space density evolution to real temperature and chemical potential cases.

Estimated the relic velocity of non-thermal cosmic neutrino background.

Abstract

The behavior of a decoupled ideal Fermi gas in a homogeneously expanding three-dimensional volume is investigated, starting from an equilibrium spectrum. In case the gas is massless and/or completely degenerate, the spectrum of the gas can be described by an effective temperature and/or an effective chemical potential, both of which scale down with the volume expansion. In contrast, the spectrum of a decoupled massive and non-degenerate gas can only be described by an effective temperature if there are strong enough self-interactions such as to maintain an equilibrium distribution. Assuming perpetual equilibration, we study a decoupled gas which is relativistic at decoupling and then is red-shifted until it becomes non-relativistic. We find expressions for the effective temperature and effective chemical potential which allow us to calculate the final spectrum for arbitrary initial conditions. This calculation is enabled by a new expansion of the Fermi-Dirac integral, which is for our purpose superior to the well-known Sommerfeld expansion. We also compute the behavior of the phase space density under expansion and compare it to the case of real temperature and real chemical potential. Using our results for the degenerate case, we also obtain the mean relic velocity of the recently proposed non-thermal cosmic neutrino background.