# Analytic expressions for the background evolution of massive neutrinos and dark matter particles

## TL;DR

This paper derives exact analytic formulas for the background evolution of massive neutrinos and dark matter particles, and validates them against numerical models, enhancing precision in cosmological computations.

## Contribution

It introduces precise analytic expressions for neutrino and dark matter background properties, improving upon existing numerical methods in cosmological modeling.

## Key findings

- Analytic expressions agree within 0.0001% with CLASS code.
- Modifications do not affect computational performance.
- Provides limits at relativistic and late times for neutrino equations of state.

## Abstract

We provide exact analytic expressions for the density, pressure, average number density and pseudo-pressure for massive neutrinos and generic dark matter particles, both fermions and bosons. We then focus on massive neutrinos and we compare our analytic expressions with the numerical implementation in the CLASS Boltzmann code. We find that our modifications including the exact analytic expressions are in agreement to better than $10^{-4}\%$ with the default CLASS implementation in the estimation of the CMB power spectrum; our modifications do not have an impact on the performance of the code. We also provide several specific limits of our expressions at the relativistic regime, but also at late times for the neutrino equation of state.