Updated results on neutrino mass and mass hierarchy from cosmology

Current upper bounds on the sum of 3 active neutrino masses, $\sum m_{\nu}$ from analyses of cosmological data in the backdrop of $\Lambda\textrm{CDM}+\sum m_{\nu}$ model are close to the minimum sum of neutrino masses required by the inverted hierarchy, which is around 0.1 eV. However, these analyses are usually done with the assumption of degenerate masses, which is not a good approximation any more since the bounds are strong enough that the neutrino mass-squared splittings can no longer be considered negligible. In this work we update the bounds on $\sum m_{\nu}$ from latest publicly available cosmological data while explicitly considering particular neutrino mass hierarchies. In the minimal $\Lambda\textrm{CDM}+\sum m_{\nu}$ model with Planck 2015 TT,TE,EE, BAO, Planck 2018 lensing, and a Gaussian prior on the reionization depth $\tau=0.0506\pm0.0086$ from Planck 2018 lowE data, we find that at 95\% C.L. the bounds are: $\sum m_{\nu}<0.122$ eV (degenerate), $\sum m_{\nu}<0.145$ eV (normal), $\sum m_{\nu}<0.170$ eV (inverted); i.e., the bounds vary significantly across the different mass orderings. Also, we find that the normal hierarchy is mildly preferred to the inverted: $\Delta \chi^2 \equiv \chi^2_{\textrm{NH}}- \chi^2_{\textrm{IH}} = -3.70$ (best-fit). In this paper we also provide bounds on $\sum m_{\nu}$ considering different hierarchies in various extended cosmological models: $\Lambda\textrm{CDM}+\sum m_{\nu}+r$, $w\textrm{CDM}+\sum m_{\nu}$, $w_0 w_a \textrm{CDM}+\sum m_{\nu}$, $w_0 w_a \textrm{CDM}+\sum m_{\nu}$ with $w(z)\geq -1$, $\Lambda \textrm{CDM} + \sum m_{\nu} + \Omega_k$, and $\Lambda \textrm{CDM} + \sum m_{\nu} + A_{\textrm{Lens}}$. We do not find any strong evidence of normal hierarchy over inverted hierarchy from looking at the $\chi^2$ values in the extended models either (abstract abridged).