Asymptotic analysis of the Boltzmann equation for dark matter relic abundance

TL;DR

This paper develops an asymptotic approximation method for solving the Boltzmann equation to accurately determine dark matter relic abundance, accounting for resonance and threshold effects, and validates it against numerical solutions.

Contribution

It introduces a matched asymptotic approximation approach for the Boltzmann equation, improving accuracy in relic density calculations and extending applicability to related systems.

Findings

Asymptotic series accurately models relic abundance.

Resonance and threshold effects are negligible unless cross section is small.

Excellent agreement with numerical solutions in benchmark models.

Abstract

A solution to the Boltzmann equation governing the thermal relic abundance of cold dark matter is constructed by matched asymptotic approximations. The approximation of the relic density is an asymptotic series valid when the abundance does not deviate significantly from its equilibrium value until small temperatures. Resonance and threshold effects are taken into account at leading order and found to be negligible unless the annihilation cross section is negligible at threshold. Comparisons are made to previously attempted constructions and to the freeze out approximation commonly employed in the literature. Extensions to higher order matching is outlined, and implications for solving related systems are discussed. We compare our results to a numerical determination of the relic abundance using a benchmark model and find a fantastic agreement. The method developed also serves as a solution to a wide class of problems containing an infinite order turning point.