Exploring neutrino mass and mass hierarchy in the scenario of vacuum energy interacting with cold dark matter

We investigate the constraints on total neutrino mass in the scenario of vacuum energy interacting with cold dark matter. We focus on two typical interaction forms, i.e., $Q=\beta H\rho_{\rm c}$ and $Q=\beta H\rho_{\Lambda}$. To avoid the occurrence of large-scale instability in interacting dark energy cosmology, we adopt the parameterized post-Friedmann approach to calculate the perturbation evolution of dark energy. We employ observational data, including the Planck cosmic microwave background temperature and polarization data, baryon acoustic oscillation data, a JLA sample of type Ia supernovae observation, direct measurement of the Hubble constant, and redshift space distortion data. We find that, compared with those in the $\Lambda$CDM model, much looser constraints on $\sum m_{\nu}$ are obtained in the $Q=\beta H\rho_{\rm c}$ model, whereas slightly tighter constraints are obtained in the $Q=\beta H\rho_{\Lambda}$ model. Consideration of the possible mass hierarchies of neutrinos reveals that the smallest upper limit of $\sum m_{\nu}$ appears in the degenerate hierarchy case. By comparing the values of $\chi^2_{\rm min}$, we find that the normal hierarchy case is favored over the inverted one. In particular, we find that the difference $\Delta \chi^2_{\rm min} \equiv \chi^2_{\rm IH; min}-\chi^2_{\rm NH; min}> 2$ in the $Q=\beta H\rho_{\rm c}$ model. In addition, we find that $\beta=0$ is consistent with the current observations in the $Q=\beta H\rho_{\rm c}$ model, and $\beta < 0$ is favored at more than the $1\sigma$ level in the $Q=\beta H\rho_{\Lambda}$ model.