  We introduce a cosmological invisible decay of the sterile neutrino with the
eV-scale mass indicated by short-baseline neutrino oscillation experiments in
order to allow its full thermalization in the early Universe. We show that the
fit of the cosmological data is practically as good as the fit obtained with a
stable sterile neutrino without mass constraints, which has been recently
considered by several authors for the explanation of the observed suppression
of small-scale matter density fluctuations and for a solution of the tension
between the Planck and BICEP2 measurements of the tensor to scalar ratio of
large-scale fluctuations. Moreover, the extra relativistic degree of freedom
corresponding to a fully thermalized sterile neutrino is correlated with a
larger value of the Hubble constant, which is in agreement with local
measurements.
