Asymptotic Analysis of the Boltzmann Equation for Dark Matter Relics

TL;DR

This paper conducts an asymptotic analysis of Boltzmann equations related to dark matter relics, revealing long-term behaviors and effects of dilaton fields on dark matter abundance using boundary-layer and delta expansion techniques.

Contribution

It introduces novel asymptotic methods to analyze Boltzmann equations in dark matter physics, including cases from general relativity and string cosmology.

Findings

Characterization of freeze-out and post-freeze-out behavior

Demonstration of power-law decay of dark matter abundance

Calculation of dilaton effects on dark matter relics

Abstract

This paper presents an asymptotic analysis of the Boltzmann equations (Riccati differential equations) that describe the physics of thermal dark-matter-relic abundances. Two different asymptotic techniques are used, boundary-layer theory, which makes use of asymptotic matching, and the delta expansion, which is a powerful technique for solving nonlinear differential equations. Two different Boltzmann equations are considered. The first is derived from general relativistic considerations and the second arises in dilatonic string cosmology. The global asymptotic analysis presented here is used to find the long-time behavior of the solutions to these equations. In the first case the nature of the so-called freeze-out region and the post-freeze-out behavior is explored. In the second case the effect of the dilaton on cold dark-matter abundances is calculated and it is shown that there is a large-time power-law fall off of the dark-matter abundance. Corrections to the power-law behavior are also calculated.