Direct comparison of sterile neutrino constraints from cosmological data, $$\nu _{e}$$νe disappearance data and $$\nu _{\mu } \rightarrow \nu _{e} $$νμ→νe appearance data in a $$3+1$$3+1 model

We present a quantitative, direct comparison of constraints on sterile neutrinos derived from neutrino oscillation experiments and from Planck data, interpreted assuming standard cosmological evolution. We extend a $$1+1$$ model, which is used to compare exclusion contours at the 95% Cl derived from Planck data to those from $$\nu _{e}$$-disappearance measurements, to a $$3+1$$ model. This allows us to compare the Planck constraints with those obtained through $$\nu _{\mu }\rightarrow \nu _{e}$$ appearance searches, which are sensitive to more than one active-sterile mixing angle. We find that the cosmological data fully exclude the allowed regions published by the LSND, MiniBooNE and Neutrino-4 collaborations, and those from the gallium and rector anomalies, at the 95% Cl. Compared to the exclusion region from the Daya Bay $$\nu _{e}$$-disappearance search, the Planck data are more strongly excluding above $$|\Delta m^{2}_{41}|\approx 0.1\,\mathrm {eV}^{2}$$ and $$m_\mathrm {eff}^\mathrm {sterile}\approx 0.2\,\mathrm {eV}$$, with the Daya Bay exclusion being stronger below these values. Compared to the combined Daya Bay/Bugey/MINOS exclusion region on $$\nu _{\mu }\rightarrow \nu _{e}$$ appearance, the Planck data is more strongly excluding above $$\Delta m^{2}_{41}\approx 5\times 10^{-2}\,\mathrm {eV}^{2}$$, with the exclusion strengths of the Planck data and the Daya Bay/Bugey/MINOS combination becoming comparable below this value.