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Taking into account the mass splittings between three active neutrinos, weinvestigate impacts of dark energy on constraining the total neutrino mass$\sum m_{\nu}$ by using recent cosmological observations. We consider twotypical dark energy models, namely, the $w$CDM model and the holographic darkenergy (HDE) model, which both have an additional free parameter compared withthe $\Lambda$CDM model. We employ the Planck 2015 data of CMB temperature andpolarization anisotropies, combined with low-redshift measurements on BAOdistance scales, type Ia supernovae, Hubble constant, and Planck lensing.Compared to the $\Lambda$CDM model, our study shows that the upper limit on$\sum m_{\nu}$ becomes much looser in the $w$CDM model while much tighter inthe HDE model. In the HDE model, we obtain the $95\%$ CL upper limit $\summ_{\nu}<0.105~\textrm{eV}$ for three degenerate neutrinos. This might be themost stringent constraint on $\sum m_{\nu}$ by far and is almost on the vergeof diagnosing the neutrino mass hierachies in the HDE model. However, thedifference of $\chi^2$ is still not significant enough to distinguish theneutrino mass hierarchies, even though the minimal $\chi^2$ of the normalhierarchy is slightly smaller than that of the inverted hierarchy.

Impacts of dark energy on weighing neutrinos: mass hierarchies considered