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

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 ($N_{\rm eff}\simeq 4$).