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.
