Direct comparison of sterile neutrino constraints from cosmological
  data, $\nu_{e}$ disappearance data and $\nu_{\mu}\rightarrow\nu_{e}$
  appearance data in a $3+1$ 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 exclusions 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 regions 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.
