$N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ in low-scale seesaw models versus the lightest neutrino mass

We evaluate the contribution to $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ of the extra sterile states in low-scale Type I seesaw models (with three extra sterile states). We explore the full parameter space and find that at least two of the heavy states always reach thermalisation in the Early Universe, while the third one might not thermalise provided the lightest neutrino mass is below ${\mathcal O}(10^{-3}$eV). Constraints from cosmology therefore severely restrict the spectra of heavy states in the range 1eV- 100 MeV. The implications for neutrinoless double beta decay are also discussed.