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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 $r$ betweenthe Planck temperature data and the BICEP2 B-mode polarization data. Such amodel is called the $\Lambda$CDM+$r$+$\nu_s$ model in this paper. Compared tothe $\Lambda$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. Weshow that in this model the tension between Planck and BICEP2 can be greatlyrelieved 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 tensionbetween Planck and BICEP2 but also makes trouble to inflation due to the largerunning of the spectral index of order $10^{-2}$ produced, the$\Lambda$CDM+$r$+$\nu_s$ model is much better for inflation. By including asterile neutrino species in the standard cosmology, besides the tension withBICEP2, the other tensions of Planck with other astrophysical data, such as the$H_0$ direct measurement, the Sunyaev-Zeldovich cluster counts, and the galaxyshear data, can all be significantly relieved. So, this model seems to be aneconomical choice. Combining the Planck temperature data, the WMAP-9polarization data, and the baryon acoustic oscillation data with all theseastrophysical 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 $m_{\nu,{\rm sterile}}^{\rmeff}=0.51^{+0.12}_{-0.13}$ eV. Thus, our results prefer $\Delta N_{\rm eff}>0$at the 2.7$\sigma$ level and a nonzero mass of sterile neutrino at the3.9$\sigma$ level.

Sterile neutrinos help reconcile the observational results of primordial gravitational waves from Planck and BICEP2