# Asymptotic analysis of the Boltzmann equation for dark matter relic abundance

**Authors:** Logan A. Morrison, Hiren H. Patel, Jaryd F. Ulbricht  
**arXiv:** [2009.04012](https://arxiv.org/abs/2009.04012)  
**Date:** 2020-09-08  
**Summary page:** https://www.summarizepaper.com/en/paper/2009.04012v1/

## Key points

- Authors present a solution to the Boltzmann equation for dark matter relic abundance
- Utilize matched asymptotic approximations to construct an approximation of relic density
- Consider resonance and threshold effects, determining their negligible impact under certain conditions
- Compare approach to previous constructions and freeze-out approximations, highlighting effectiveness and accuracy
- Validate results through comparison with numerical determination using benchmark model
- Method offers solution for problems characterized by infinite order turning points, showcasing versatility across scientific disciplines

## Summary

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In their paper titled "Asymptotic Analysis of the Boltzmann Equation for Dark Matter Relic Abundance," authors Logan A. Morrison, Hiren H. Patel, and Jaryd F. Ulbricht present a solution to the Boltzmann equation governing the thermal relic abundance of cold dark matter. The authors utilize matched asymptotic approximations to construct an approximation of the relic density, represented as an asymptotic series valid under conditions where the abundance remains close to its equilibrium value until reaching low temperatures. The study considers resonance and threshold effects at leading order, determining them to be negligible unless the annihilation cross section approaches zero at threshold values. By comparing their approach to previous constructions and commonly used freeze-out approximations in literature, the authors highlight its effectiveness and accuracy. Additionally, they outline extensions to higher-order matching techniques and discuss implications for solving related systems within this framework. The authors validate their results by comparing them to a numerical determination of relic abundance using a benchmark model, demonstrating excellent agreement between theory and observation. Notably, this work has broader applicability beyond dark matter relic abundance calculations. The developed method offers a solution for a wide range of problems characterized by infinite order turning points, showcasing its versatility and potential impact across various scientific disciplines. Overall, this study provides valuable insights into refining calculations related to dark matter relic abundance through advanced asymptotic analysis techniques, making significant contributions to theoretical physics and cosmology. : Utilizing matched asymptotic approximations to construct an approximation of the relic density. : Governing the thermal relic abundance of cold dark matter. : The abundance of dark matter particles in the universe. : A method used to approximate solutions to differential equations with multiple scales. : The study of fundamental laws and principles governing the universe.