# Derivation of the sterile neutrino Boltzmann equation from quantum kinetics

## TL;DR

This paper derives a Boltzmann equation for sterile neutrino production from quantum kinetics using a relaxation-time approximation, accurately capturing oscillations, decoherence, and plasma effects, with implications for cosmological models.

## Contribution

It provides a new, direct derivation of the sterile neutrino Boltzmann equation from quantum kinetics, improving understanding of the underlying approximations and dynamics.

## Key findings

- The relaxation-time approximation accurately reproduces oscillation and decoherence dynamics.
- Numerical illustrations show the semiclassical equation's successes and limitations.
- Brief discussion on including sterile neutrino interactions enhances the model's scope.

## Abstract

An extensive, growing body of work has been penned on cosmologies that include one or more sterile neutrinos. Early entries in the literature formulated a Boltzmann-like equation describing sterile-neutrino production in a way that bypasses the numerical tracking of high-frequency complex phases, and meticulous quantum-kinetic analyses shortly thereafter put the formula on firmer ground. A new and more direct derivation is given here, showing that the equation follows almost immediately from a quantum relaxation-time approximation and an expansion in the mixing angle. Besides reproducing the desired result, the relaxation ansatz captures to a high degree of accuracy the interlaced dynamics of oscillations, decoherence, and plasma repopulation. Successes and limitations of the semiclassical equation are illustrated numerically and are shown to reflect the accuracy of the approximations employed in the derivation. The inclusion of interactions among the sterile neutrinos is also briefly addressed.