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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 $Λ$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 $Λ$CDM+$r$+$ν_s$ model in this paper. Compared to the $Λ$CDM+$r$ model, there are two extra parameters, $N_{\rm eff}$ and $m_{ν,{\rm sterile}}^{\rm eff}$, in the $Λ$CDM+$r$+$ν_s$ 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 $Λ$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 $Λ$CDM+$r$+$ν_s$ 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 $Λ$CDM+$r$+$ν_s$ model $n_s=0.999\pm 0.011$, $r=0.21^{+0.04}_{-0.05}$, $N_{\rm eff}=3.95\pm 0.33$ and $m_{ν,{\rm sterile}}^{\rm eff}=0.51^{+0.12}_{-0.13}$ eV. Thus, our results prefer $ΔN_{\rm eff}>0$ at the 2.7$σ$ level and a nonzero mass of sterile neutrino at the 3.9$σ$ level.