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In the framework of a 3+1 scheme with an additional inert state, we considerthe thermalisation of sterile neutrinos in the early Universe taking intoaccount the full $4\times4$ mixing matrix. The evolution of the neutrino energydistributions is found solving the momentum-dependent kinetic equations withfull diagonal collision terms, as in previous analyses of flavour neutrinodecoupling in the standard case. The degree of thermalisation of the sterilestate 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 $\Deltam^2_{41}$ is discussed. Our results are relevant for fixing the contribution ofa fourth light neutrino species to the cosmological energy density, whose valueis very well constrained by the final Planck analysis. For the preferred regionof active-sterile mixing parameters from short-baseline neutrino experiments,we find that the fourth state is fully thermalised ($N_{\rm eff}\simeq 4$).

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