# Thermalisation of sterile neutrinos in the early Universe in the 3+1 scheme with full mixing matrix

## TL;DR

This paper investigates how sterile neutrinos thermalize in the early Universe within a 3+1 mixing scheme, analyzing their impact on cosmological energy density and comparing with observational constraints.

## Contribution

It provides a detailed calculation of sterile neutrino thermalization using full mixing matrices and momentum-dependent kinetic equations, extending previous analyses.

## Key findings

- Sterile neutrinos are fully thermalized for preferred mixing parameters.
- The effective number of neutrinos, N_eff, approaches 4 in the preferred parameter space.
- Results are consistent with Planck constraints on cosmological neutrino energy density.

## Abstract

In the framework of a 3+1 scheme with an additional inert state, we consider the thermalisation of sterile neutrinos in the early Universe taking into account the full $4\times4$ mixing matrix. The evolution of the neutrino energy distributions is found solving the momentum-dependent kinetic equations with full diagonal collision terms, as in previous analyses of flavour neutrino decoupling in the standard case. The degree of thermalisation of the sterile state is shown in terms of the effective number of neutrinos, $N_{\rm eff}$, and its dependence on the three additional mixing angles ($\theta_{14}$, $\theta_{24}$, $\theta_{34}$) and on the squared mass difference $\Delta m^2_{41}$ is discussed. Our results are relevant for fixing the contribution of a fourth light neutrino species to the cosmological energy density, whose value is very well constrained by the final Planck analysis. For the preferred region of active-sterile mixing parameters from short-baseline neutrino experiments, we find that the fourth state is fully thermalised ($N_{\rm eff}\simeq 4$).