Constraints on the sum of the neutrino masses in dynamical dark energy models with $w(z) \geq -1$ are tighter than those obtained in $Λ$CDM

TL;DR

This work assesses how allowing non-phantom dynamical dark energy with a CPL redshift dependence, $w(z)=w_0+w_a\frac{z}{1+z}$ and $w(z)\ge -1$, affects cosmological bounds on the sum of neutrino masses $M_\nu$. Using Planck CMB TT, BAO, and SNeIa data (with and without small-scale polarization) and marginalizing over $w_0,w_a$ under non-phantom priors, the authors find that $M_\nu$ bounds tighten slightly compared to ΛCDM: $M_\nu<0.13$ eV (base) and $<0.11$ eV (pol) for NPDDE, versus $<0.16$ eV (base) and $<0.13$ eV (pol) for ΛCDM. The physical explanation centers on preserving the angular sound horizon scale $\Theta_s=r_s/D_A$; in NPDDE, a higher late-time $E(z)$ necessitates lower $H_0$ and/or $M_\nu$, weakening the $M_\nu$–$H_0$ degeneracy and tightening the bound. The analysis also yields a mild preference for the normal neutrino mass ordering, with posterior odds $\sim 2:1$ (base) and $\sim 3:1$ (pol); if the ordering is inverted, the results imply phantom or non-standard physics would be required under NPDDE. Overall, the study reveals a nontrivial connection between dark-energy dynamics and neutrino properties, with implications for future tests of the mass hierarchy and the nature of cosmic acceleration.

Abstract

We explore cosmological constraints on the sum of the three active neutrino masses $M_ν$ 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 $M_ν$ \textit{do not degrade} with respect to those obtained in the $Λ$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 $M_ν$ is tighter than the $Λ$CDM limit because of the well-known degeneracy between $w$ and $M_ν$. 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 $M_ν$ than the $Λ$CDM limit. We find a 95\% confidence level (C.L.) upper bound of $M_ν<0.13\,\mathrm{eV}$. This bound can be compared with $M_ν<0.16\,\mathrm{eV}$ at 95\%~C.L., obtained within the $Λ$CDM model, and $M_ν<0.41\,\mathrm{eV}$ 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)

Figures (6)

• Figure 1: One-dimensional posterior probabilities of the sum of the three active neutrino masses $M_{\nu}$ (in eV) for three cases: the $w_0w_a$CDM generic DDE case which allows for values of $w$ both smaller than or larger than $-1$ (in blue), the $\Lambda\mathrm{CDM}$ case (in black), and the non-phantom dynamical dark energy (NPDDE) model with $w(z)\geq-1$ (in red). Results have been obtained using a Bayesian analysis that marginalizes over all applicable $w_0, w_a$ values, and are shown for the two dataset combinations employed in this work as described at the beginning of Sec. \ref{['sec:method']}: solid for base (using CMB, BAO, and SN data), dashed for pol (also including CMB polarization at small scales). The vertical black dotted-dashed line corresponds to the minimal mass of $M_{\nu,\min} \approx 0.1\, {\rm eV}$ allowed by neutrino oscillation data within the inverted ordering.
• Figure 2: Two-dimensional probability contours in the $H_0-M_{\nu}$ plane. The blue contours are obtained for the $\Lambda$CDM model, the red contours are for a dynamical dark energy model with EoS parametrized by Eq. (\ref{['cpl']}) and satisfying $w(z)\geq-1$ (NPDDE), and the grey contours are for a generic dark energy model with EoS parametrized by Eq. (\ref{['cpl']}). The green band indicates the 68% C.I. for $H_0$ from direct measurements of the Hubble Space Telescope Riess:2016jrrRiess:2011yx. The horizontal dashed line corresponds to $M_{\nu,\mathrm{min}}\simeq0.1\,\mathrm{eV}$, the minimal value for the sum of the neutrino masses allowed in the inverted ordering scenario by neutrino oscillation data. When moving from the $\Lambda$CDM contours (blue) to models with $w(z)\geq-1$ (red), the shifts of $H_0$ and $M_{\nu}$ to smaller values are evident. These shifts are necessary to keep the angular scale of the sound horizon at recombination $\Theta_{\rm s}$ fixed, see discussion in the main text. It is also clear that the $M_{\nu}$-$H_0$ degeneracy is weakened when moving from $\Lambda$CDM to models with $w(z)\geq-1$ (NPDDE). For further information, see discussion in main text concerning the $M_{\nu}$-$H_0$ correlation coefficient, which is reduced from $-0.43$ ($\Lambda$CDM) to $-0.14$ (NPDDE). The tension between direct measurements of $H_0$ and cosmological estimates is not resolved by a dark energy component with $w(z)\geq-1$. The tension is partially alleviated by a generic dark energy component which can access the $w(z)<-1$ region (grey contours). The contour regions are obtained for the base dataset combination of CMB, BAO and SNeIa data, with no CMB small scale polarization data. Similar considerations apply to the contours derived from the combination which also includes small scale CMB polarization data.
• Figure 3: ${\cal E}(z)$, defined in Eq. (\ref{['ez']}), quantifies the difference in the normalized expansion rate $H(z)/H_0$ between a dynamical dark energy model with equation of state $w(z)\geq-1$ (NPDDE) and a $\Lambda$CDM model. The quantity ${\cal E}(z)$ is plotted for sample cosmologies with $w_0=-0.95,\,w_a=-0.05$ (blue curve), $w_0=-0.9,\,w_a=-0.1$ (green curve), $w_0=-0.8,\,w_a=-0.2$ (red curve), and $w_0=-1,\,w_a=0$ (black curve, $\Lambda$CDM, where ${\cal E}(z)=0$). We have fixed $\Omega_{m,0}=0.3$ and $\Omega_{\text{DE},0}=0.7$. The negative ${\cal E}(z)$ indicates that the normalized expansion rate is higher in the NPDDE model compared to $\Lambda$CDM. The four vertical dashed lines indicate the redshift of the four BAO measurements we consider in this work: 6dFGS (cyan), SDSS MGS (orange), BOSS DR11 LOWZ (purple), and BOSS DR11 CMASS (green). The grey shaded band refers to the redshift coverage of the JLA Supernovae Ia sample. Thus the measurements considered in this work probe the redshift range in which the dip in ${\cal E}(z)$ is most prominent.
• Figure 4: One-dimensional posterior probabilities of the sum of the three active neutrino masses $M_{\nu}$ (in eV) for a selection of cosmological models with $w_0$ and $w_a$fixed, described in Sec. \ref{['subsec:comment']}. Models a)-d) have $w_0$ and $w_a$ fixed to values satisfying the condition $w(z)\geq-1$, and are represented by the dashed light blue, dashed purple, dashed yellow, and dashed red curves respectively. Models e) and f) have $w_0$ and $w_a$ fixed to values not satisfying the condition $w(z)\geq-1$, and are represented by the dashed dark blue and dashed green curves respectively. The $\Lambda\mathrm{CDM}$ result corresponds to the solid black line. The region where $w(z)\geq-1$ is satisfied is shaded in green and labeled "Non-phantom"; conversely, the region where $w(z)\geq-1$ is not satisfied is shaded in pink and labeled "Phantom". It is clear that the bounds on $M_{\nu}$ for models where $w_0$ and $w_a$ are fixed to values satisfying $w(z)\geq-1$ are always tighter than the $\Lambda\mathrm{CDM}$ bound. Therefore, a Bayesian analysis marginalizing over the range of $w_0$, $w_a$ values satisfying $w(z)\geq-1$ is expected to obtain a bound on $M_{\nu}$ which is slightly tighter than the $\Lambda\mathrm{CDM}$ one, as shown by the results in Sec. \ref{['subsec:bounds']}.
• Figure 5: $68\%$ C.L. (dark blue/red) and $95\%$ C.L. (light blue/red) joint posterior distributions in the $M_{\nu}$-$w_0$-$w_a$ plane, along with their marginalized posterior distributions from the base dataset, for the $w_0w_a$CDM (blue contours) and NPDDE models (red contours). The marginalized posterior distributions appearing along the diagonal are normalizable probability distributions and hence in arbitrary units. The sharp cuts in the red posteriors are due to the hard NPDDE priors [see Eq. \ref{['snpdeprior']}].