A novel approach to quantifying the sensitivity of current and future cosmological datasets to the neutrino mass ordering through Bayesian hierarchical modeling

TL;DR

The paper develops a hierarchical Bayesian framework to constrain the neutrino mass sum $M_\nu$ without assuming a fixed mass ordering, by introducing a discrete hyperparameter $h_\mathrm{type}$ that encodes NH vs IH and by marginalizing over it. Using eight parameters (the six standard $\Lambda$CDM ones plus $m_\mathrm{light}$ and $h_\mathrm{type}$), the authors perform MCMC analyses with Planck CMB data and BAO, then forecast for COrE and DESI to assess the potential to determine the mass hierarchy. Current cosmology yields only mild sensitivity to the ordering, with small but notable NH preferences when BAO is included; future COrE+DESI can achieve a strong NH preference (e.g., about $9:1$) if $M_\nu\approx0.06$ eV, while $M_\nu\approx0.1$ eV remains challenging to distinguish cosmologically. The study also links cosmological constraints to neutrinoless double beta decay, showing how the marginalized hierarchy affects the allowed Majorana mass $m_{\beta\beta}$ and highlighting complementary prospects for probing neutrino properties across cosmology and laboratory experiments.

Abstract

We present a novel approach to derive constraints on neutrino masses from cosmological data, while taking into account our ignorance of the neutrino mass ordering. We derive constraints from a combination of current and future cosmological datasets on the total neutrino mass $M_ν$ and on the mass fractions carried by each of the mass eigenstates, after marginalizing over the (unknown) neutrino mass ordering, either normal (NH) or inverted (IH). The bounds take therefore into account the uncertainty related to our ignorance of the mass hierarchy. This novel approach is carried out in the framework of Bayesian analysis of a typical hierarchical problem. In this context, the choice of the neutrino mass ordering is modeled via the discrete hyperparameter $h_{type}$. The preference for either the NH or the IH scenarios is then encoded in the posterior distribution of $h_{type}$ itself. Current CMB measurements assign equal odds to the two hierarchies, and are thus unable to distinguish between them. However, after the addition of BAO measurements, a weak preference for NH appears, with odds of 4:3 from Planck temperature and large-scale polarization in combination with BAO (3:2 if small-scale polarization is also included). Forecasts suggest that the combination of upcoming CMB (COrE) and BAO surveys (DESI) may determine the neutrino mass hierarchy at a high statistical significance if the mass is very close to the minimal value allowed by oscillations, as for NH and $M_ν=0.06$ eV there is a 9:1 preference of NH vs IH. On the contrary, if $M_ν$ is of the order of 0.1 eV or larger, even future cosmological observations will be inconclusive. The unbiased limit on $M_ν$ we obtain with this innovative statistical strategy is crucial for ongoing and planned neutrinoless double beta decay searches.

Figures (7)

• Figure 1: One-dimensional probability posterior distribution of a selection of parameters analyzed in this work, for the combinations of current CMB and BAO datasets reported in the top legend. In the top panels, we report the posterior distributions for the sum of the neutrino masses $M_\nu=\Sigma_i m_i$, where the index $i=1,2,3$ runs over the three mass eigenstates $m_i$; the mass carried by the lightest eigenstate $m_{\mathrm{light}}$ and the Hubble constant $H_0$. In the bottom panels, we report the posterior distributions for the neutrino mass fractions $f_{\nu,i}=m_i/M_\nu$. The solid (dashed) lines are for CMB alone (CMB plus BAO) measurements. The vertical dashed lines in the bottom panels refer to the expected value of $f_{\nu,i}=1/3$ in the case of a fully degenerate mass spectrum. All the posterior shown in this figure also take into account information from oscillation measurements.
• Figure 2: One-dimensional probability posterior distribution of a selection of parameters analyzed in this work, for the combination of datasets reported in the figure. In the top panels, we report the posterior distributions for the sum of the neutrino masses $M_\nu=\Sigma_i m_i$, where the index $i=1,2,3$ runs over the three mass eigenstates $m_i$; the mass carried by the lightest eigenstate $m_{\mathrm{light}}$ and the Hubble constant $H_0$. In the bottom panels, we report the posterior distributions for the neutrino mass fractions $f_{\nu,i}=m_i/M_\nu$. The solid (dashed) lines refer to COrE (COrE plus DESI) forecasted MCMC results. The vertical dashed lines in the bottom panels refer to the expected value of $f_{\nu,i}=1/3$ in the case of a fully degenerate mass spectrum. All the posterior shown in this figure also take into account information from oscillation measurements.
• Figure 3: Two dimensional 68% and 95% probability contours in the $M_\nu-m_{\beta\beta}$ plane, for current cosmological (Planck TT, TE, EE + BAO) and neutrino oscillation data. The contours are calculated by marginalizing over $h_\mathrm{type}$, so they take into account the uncertainty on the mass ordering. The contours trace two distinct regions of large probability, that are more clearly visible in Fig.\ref{['fig:densmbbmnu']}, roughly corresponding to the portion of parameter space preferred by each of the two hierarchies (see the main text for details). The orange horizontal bands correspond to the 90% upper bounds on $m_{\beta\beta}$ obtained from KamLAND-Zen KamLAND-Zen:2016pfg, for different assumptions for the values of the nuclear matrix elements that enter into the calculation of $m_{\beta\beta}$.
• Figure 4: Density plot of the two-dimensional posterior in the $M_\nu-m_{\beta\beta}$ plane, for current cosmological (Planck TT, TE, EE + BAO) and neutrino oscillation data. Darker colors correspond to higher probability regions. The plot shows that the posterior is bimodal.
• Figure 5: Two dimensional 68% and 95% probability contours in the $M_\nu-m_{\beta\beta}$ plane, for future cosmological (COrE+DESI) and neutrino oscillation data, considering the NH scenario as the nature's choice and $M_{\nu}=0.06\,\,\mathrm{eV}$ (left panel) or $M_{\nu}=0.1\,\,\mathrm{eV}$ (right panel). The contours are calculated by taking into account the uncertainty on the mass ordering, without fixing a priori the mass hierarchy. The two regions that can be inferred in both figures (even if not completely isolated) correspond to the two mass orderings. The horizontal bands correspond to the 90% upper bounds on $m_{\beta\beta}$ obtained from KamLAND-Zen KamLAND-Zen:2016pfg (orange) and to those expected from a future nEXO-like experiment (green) nEXO (assuming a vanishing Majorana mass), for different assumptions for the values of the nuclear matrix elements that enter into the calculation of $m_{\beta\beta}$.