Constraints on the sum of the neutrino masses in dynamical dark energy
  models with $w(z) \geq -1$ are tighter than those obtained in $\Lambda$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
$w(z)=w_0+w_a\,z/(1+z)$, and satisfying $w(z)\geq-1$ for all $z$. We perform a
Bayesian analysis and show that, within these models, the bounds on $M_{\nu}$
\textit{do not degrade} with respect to those obtained in the $\Lambda$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 $w(z)\geq-1$ (but not $w=-1$ for all $z$), the upper
limit on $M_{\nu}$ is tighter than the $\Lambda$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
$w(z)\geq-1$, all of which correspond to tighter limits on $M_{\nu}$ than the
$\Lambda$CDM limit. We find a 95\% confidence level (C.L.) upper bound of
$M_{\nu}<0.13\,\mathrm{eV}$. This bound can be compared with
$M_{\nu}<0.16\,\mathrm{eV}$ at 95\%~C.L., obtained within the $\Lambda$CDM
model, and $M_{\nu}<0.41\,\mathrm{eV}$ 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 $w(z)\geq-1$ 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)
