Tilt of primordial gravitational wave spectrum in a universe with
  sterile neutrinos
  In this work, we constrain the spectral index $n_t$ of the primordial
gravitational wave power spectrum in a universe with sterile neutrinos by using
the Planck temperature data, the WMAP 9-year polarization data, the baryon
acoustic oscillation data, and the BICEP2 data. We call this model the
$\Lambda$CDM+$r$+$\nu_s$+$n_t$ model. The additional massive sterile neutrino
species can significantly relieve the tension between the Planck and BICEP2
data, and thus can reduce the possible effects of this tension on the fit
results of $n_t$. To constrain the parameters of sterile neutrino, we also
utilize the Hubble constant direct measurement data, the Planck
Sunyaev-Zeldovich cluster counts data, the Planck CMB lensing data, and the
cosmic shear data. We find that due to the fact that the BICEP2 data are most
sensitive to the multipole $\ell\sim150$ corresponding to $k\sim0.01$
Mpc$^{-1}$, there exists a strong anticorrelation between $n_t$ and $r_{0.002}$
in the BICEP2 data, and this further results in a strongly blue-tilt spectrum.
However, a slightly red-tilt tensor power spectrum is also allowed by the
BICEP2 data in the region with larger value of $r_{0.002}$. By using the full
data sets, we obtain $m_{\nu,{\rm{sterile}}}^{\rm{eff}}=0.48^{+0.11}_{-0.13}$
eV, $N_{\rm{eff}}=3.73^{+0.34}_{-0.37}$, and $n_t=0.96^{+0.48}_{-0.63}$ for the
$\Lambda$CDM+$r$+$\nu_s$+$n_t$ model.
