Search for sterile neutrinos in a universe of vacuum energy interacting with cold dark matter

We investigate the cosmological constraints on sterile neutrinos in a universe in which vacuum energy interacts with cold dark matter by using latest observational data. We focus on two specific interaction models, $Q=\beta H{\rho }_v$ and $Q=\beta H{\rho }_c$. To overcome the problem of large-scale instability in the interacting dark energy scenario, we employ the parametrized post-Friedmann (PPF) approach for interacting dark energy to do the calculation of perturbation evolution. The observational data sets used in this work include the Planck 2015 temperature and polarization data, the baryon acoustic oscillation measurements, the type-Ia supernova data, the Hubble constant direct measurement, the galaxy weak lensing data, the redshift space distortion data, and the Planck lensing data. Using the all-data combination, we obtain $N_{\mathrm{eff}}<3.522$ and $m_{\nu ,\mathrm{sterile}}^{\mathrm{eff}}<0.576\mathrm{eV}$ for the $Q=\beta H{\rho }_v$ model, and $N_{\mathrm{eff}}=3.204_{-0.135}^{+0.049}$ and $m_{\nu ,\mathrm{sterile}}^{\mathrm{eff}}=0.410_{-0.330}^{+0.150}\mathrm{eV}$ for the $Q=\beta H{\rho }_c$ model. The latter indicates $\Delta N_{\mathrm{eff}}>0$ at the 1.17$\sigma$ level and a nonzero mass of sterile neutrino at the 1.24$\sigma$ level. In addition, for the $Q=\beta H{\rho }_v$ model, we find that $\beta =0$ is consistent with the current data, and for the $Q=\beta H{\rho }_c$ model, we find that $\beta >0$ is obtained at more than 1$\sigma$ level.