Sterile neutrinos help reconcile the observational results of primordial
  gravitational waves from Planck and BICEP2
  We show that involving a sterile neutrino species in the $\Lambda$CDM+$r$
model can help relieve the tension about the tensor-to-scalar ratio $r$ between
the Planck temperature data and the BICEP2 B-mode polarization data. Such a
model is called the $\Lambda$CDM+$r$+$\nu_s$ model in this paper. Compared to
the $\Lambda$CDM+$r$ model, there are two extra parameters, $N_{\rm eff}$ and
$m_{\nu,{\rm sterile}}^{\rm eff}$, in the $\Lambda$CDM+$r$+$\nu_s$ model. We
show that in this model the tension between Planck and BICEP2 can be greatly
relieved at the cost of the increase of $n_s$. However, comparing with the
$\Lambda$CDM+$r$+$dn_s/d\ln k$ model that can significantly reduce the tension
between Planck and BICEP2 but also makes trouble to inflation due to the large
running of the spectral index of order $10^{-2}$ produced, the
$\Lambda$CDM+$r$+$\nu_s$ model is much better for inflation. By including a
sterile neutrino species in the standard cosmology, besides the tension with
BICEP2, the other tensions of Planck with other astrophysical data, such as the
$H_0$ direct measurement, the Sunyaev-Zeldovich cluster counts, and the galaxy
shear data, can all be significantly relieved. So, this model seems to be an
economical choice. Combining the Planck temperature data, the WMAP-9
polarization data, and the baryon acoustic oscillation data with all these
astrophysical data (including BICEP2), we find that in the
$\Lambda$CDM+$r$+$\nu_s$ model $n_s=0.999\pm 0.011$, $r=0.21^{+0.04}_{-0.05}$,
$N_{\rm eff}=3.95\pm 0.33$ and $m_{\nu,{\rm sterile}}^{\rm
eff}=0.51^{+0.12}_{-0.13}$ eV. Thus, our results prefer $\Delta N_{\rm eff}>0$
at the 2.7$\sigma$ level and a nonzero mass of sterile neutrino at the
3.9$\sigma$ level.
