# Diffusion-limited Relic Particle Production

## TL;DR

This paper investigates how finite diffusion lengths in the early universe affect relic particle abundances, deriving a modified Boltzmann equation and showing that diffusion can significantly alter freeze-out and freeze-in outcomes, especially for large cross sections or masses.

## Contribution

The authors derive a modified Boltzmann equation accounting for finite diffusion lengths and analyze its impact on relic particle production, providing approximate solutions with high accuracy.

## Key findings

- Finite diffusion length increases relic abundance in freeze-out scenarios.
- Finite diffusion length decreases relic abundance in freeze-in scenarios.
- Effects are significant for large scattering cross sections or particle masses.

## Abstract

We examine the thermal evolution of particle number densities in the early universe when the particles have a finite diffusion length. Assuming that annihilations are impossible when the mean separation of the particles is larger than their diffusion length, we derive a version of the Boltzmann equation for freeze out in this scenario and an approximate solution, accurate to better than 2\%. The effect of a finite diffusion length is to increase the final relic freeze-out abundance over its corresponding value when diffusion effects are ignored. When diffusion is limited only by scattering off of the thermal background, and the annihilation cross section is bounded by unitarity, a significant effect on the freeze-out abundance requires a scattering cross section much larger than the annihilation cross section. A similar effect is demonstrated when the relic particles are produced via the freeze-in mechanism, but in this case the finite diffusion length is due to the scattering of particles that annihilate into the relic particle of interest. For freeze in, the effect of a finite diffusion length is to reduce the final relic particle abundance. The effects of a finite diffusion length are most important when the scattering cross section or the relic mass are very large. While we have not found a particularly compelling example where this would affect previous results, with the current interest in new dark matter candidates it could become an important consideration.