Constraints on the sum of the neutrino masses in dynamical dark energy models with $w(z)\ge -1$ are tighter than those obtained in $\mathrm{\Lambda }$CDM

We explore cosmological constraints on the sum of the three active neutrino masses $M_{\nu }$ in the context of dynamical dark energy (DDE) models with equation of state (EoS) parametrized as a function of redshift $z$ by $w(z)=w_0+w_{a}z\mathrm{/}(1+z)$, and satisfying $w(z)\ge -1$ for all $z$. We perform a Bayesian analysis and show that, within these models, the bounds on $M_{\nu }$ $\mathit{\text{do}}\mathit{\text{not}}\mathit{\text{degrade}}$ with respect to those obtained in the $\mathrm{\Lambda }$CDM case; in fact the bounds are slightly tighter, despite the enlarged parameter space. We explain our results based on the observation that, for fixed choices of $w_0,w_a$ such that $w(z)\ge -1$ (but not $w=-1$ for all $z$), the upper limit on $M_{\nu }$ is tighter than the $\mathrm{\Lambda }$CDM limit because of the well-known degeneracy between $w$ and $M_{\nu }$. The Bayesian analysis we have carried out then integrates over the possible values of $w_0$-$w_a$ such that $w(z)\ge -1$, all of which correspond to tighter limits on $M_{\nu }$ than the $\mathrm{\Lambda }$CDM limit. We find a 95% confidence level (C.L.) upper bound of $M_{\nu }<0.13\mathrm{e}\mathrm{V}$. This bound can be compared with $M_{\nu }<0.16\mathrm{e}\mathrm{V}$ at 95%~C.L., obtained within the $\mathrm{\Lambda }$CDM model, and $M_{\nu }<0.41\mathrm{e}\mathrm{V}$ at 95%~C.L., obtained in a DDE model with arbitrary EoS (which allows values of $w<-1$). Contrary to the results derived for DDE models with arbitrary EoS, we find that a dark energy component with $w(z)\ge -1$ is unable to alleviate the tension between high-redshift observables and direct measurements of the Hubble constant $H_0$. Finally, in light of the results of this analysis, we also discuss the implications for DDE models of a possible determination of the neutrino mass hierarchy by laboratory searches. (abstract abridged)