Probing the Neutrino Mass Hierarchy beyond $\Lambda$CDM Model

Taking the neutrino oscillation data into consideration, a dimensionless parameter $\Delta = (m_3-m_1)/(m_3+m_1)$ is adopted to parameterize the three neutrino mass eigenstates and the normal (positive $\Delta$) or inverted (negative $\Delta$) mass hierarchies in three typical cosmological models. Using the currently available cosmic observational data, several Markov Chain Monte Carlo chains are obtained with uniform priors on the free parameters at first. Applying importance sampling the results are compared with three new priors, i.e., logarithmic prior on $|\Delta|$, linear and logarithmic priors on $\Sigma m_\nu$. It turns out that the three new priors increase the upper limits of neutrino mass, but do not change the tendency towards different model's preference for different hierarchies, i.e., the normal hierarchy tends to be favored by $\Lambda$CDM and $w$CDM, which, however, disappears in the $w_0 w_a$CDM model. In addition, the almost symmetrical contours in the $w-\Delta$, $w_0-\Delta$, $w_a-\Delta$ planes indicate that the normal and inverted hierarchy have strong degeneracy. Finally, we perform a Bayesian model comparison analysis, finding that flat linear prior on $\Delta$ and $w_0 w_a$CDM are the most preferred prior and model, respectively.