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