Bounds on light sterile neutrino mass and mixing from cosmology and laboratory searches

We provide a consistent framework to set limits on properties of light sterile neutrinos coupled to all three active neutrinos using a combination of the latest cosmological data and terrestrial measurements from oscillations, $\beta$-decay and neutrinoless double-$\beta$ decay ($0\nu\beta\beta$) experiments. We directly constrain the full $3+1$ active-sterile mixing matrix elements $|U_{\alpha4}|^2$, with $\alpha \in ( e,\mu ,\tau )$, and the mass-squared splitting $\Delta m^2_{41} \equiv m_4^2-m_1^2$. We find that results for a $3+1$ case differ from previously studied $1+1$ scenarios where the sterile is only coupled to one of the neutrinos, which is largely explained by parameter space volume effects. Limits on the mass splitting and the mixing matrix elements are currently dominated by the cosmological data sets. The exact results are slightly prior dependent, but we reliably find all matrix elements to be constrained below $|U_{\alpha4}|^2 \lesssim 10^{-3}$. Short-baseline neutrino oscillation hints in favor of eV-scale sterile neutrinos are in serious tension with these bounds, irrespective of prior assumptions. We also translate the bounds from the cosmological analysis into constraints on the parameters probed by laboratory searches, such as $m_\beta$ or $m_{\beta \beta}$, the effective mass parameters probed by $\beta$-decay and $0\nu\beta\beta$ searches, respectively. When allowing for mixing with a light sterile neutrino, cosmology leads to upper bounds of $m_\beta < 0.09$ eV and $m_{\beta \beta} < 0.07$ eV at 95\% C.L, more stringent than the limits from current laboratory experiments.