Strong Bayesian Evidence for the Normal Neutrino Hierarchy

TL;DR

The paper tackles whether the neutrino mass hierarchy is normal or inverted by computing Bayesian evidence for both options using oscillation mass splittings and cosmological bounds on the sum of neutrino masses $\Sigma m_\nu$. It advances the methodology by analyzing the full three-dimensional mass space $(m_1,m_2,m_3)$ within a hierarchical prior framework, marginalising over hyperparameters $\mu$ and $\sigma$ to control prior sensitivity. The results show a strong preference for the normal hierarchy, with Bayes factors around $K\approx 31$–$42$ (and potentially higher with additional cosmological data), and provide posterior estimates for the individual masses ($m_1\approx3.8$ meV, $m_2\approx8.8$ meV, $m_3\approx50.4$ meV) and $\Sigma$ (approximately $64$ meV, with IH favored values near $112$ meV). These findings have important implications for neutrinoless double-beta decay prospects and for planning future cosmology experiments, while highlighting the crucial role of prior choice and data combination in Bayesian hierarchical model selection.$

Abstract

The configuration of the three neutrino masses can take two forms, known as the normal and inverted hierarchies. We compute the Bayesian evidence associated with these two hierarchies. Previous studies found a mild preference for the normal hierarchy, and this was driven by the asymmetric manner in which cosmological data has confined the available parameter space. Here we identify the presence of a second asymmetry, which is imposed by data from neutrino oscillations. By combining constraints on the squared-mass splittings with the limit on the sum of neutrino masses of $Σm_ν< 0.13$ eV, and using a minimally informative prior on the masses, we infer odds of 42:1 in favour of the normal hierarchy, which is classified as "strong" in the Jeffreys' scale. We explore how these odds may evolve in light of higher precision cosmological data, and discuss the implications of this finding with regards to the nature of neutrinos. Finally the individual masses are inferred to be $m_1 = 3.80^{+26.2}_{-3.73} \, \text{meV}, m_2 = 8.8^{+18}_{-1.2} \, \text{meV}, m_3 = 50.4^{+5.8}_{-1.2} \, \text{meV}$ ($95\%$ credible intervals).

Figures (8)

• Figure 1: A visualisation of the heavily reduced parameter space available in the case of the inverted neutrino mass hierarchy, relative to the normal case. The red dash-dot contours illustrate constraints on the mass splittings, as imposed by neutrino oscillation experiments (broadened to show $10\sigma$ uncertainties for visualisation purposes). The solid black line corresponds to the combination of a cosmological upper bound on the sum of the neutrino masses $\Sigma < 0.12\, \text{eV}$ with the measurement of $\Delta m_{12}^2$. The diagonal dashed line demarcates the two hierarchies. The colouring of the shaded areas represents the amount of parameter space available in the third dimension, $\Delta (\log m_2)$.
• Figure 2: Marginal likelihoods of the hyperparameters $\mu$ and $\sigma$, as defined in equation (\ref{['eq:hyperbayes']}). In each panel the same data is being used, namely the two measurements of the mass splittings as given in equations (\ref{['eq:m13']}) and (\ref{['eq:m12']}), and an upper bound is imposed on the sum of the neutrino masses $\Sigma < 0.15$ (95% C.L.). The left hand panel corresponds to the case of a normal neutrino mass hierarchy, while the right hand panel reflects the inverted ordering scheme. Note that a logarithmic colour scheme is employed in order to enhance the visibility of the right hand panel. The ratio of the two integrated regions yields a Bayes factor, $K$, of 31. Results for other values of $\Sigma$ are presented in Table \ref{['tab:evidence']}.
• Figure 3: As Fig. \ref{['fig:musigma']} but without the cosmological upper bound imposed on the sum of the neutrino masses. Instead we impose the laboratory constraint $\Sigma < 6.9\,$eV, yielding odds of $2.6:1$. Comparison with Fig. \ref{['fig:musigma']} shows how cosmology has disproportionately eliminated the space available for the inverted hierarchy.
• Figure 4: An illustration of how the evidence ratio evolves across the $\mu - \sigma$ plane, due to data from neutrino oscillations. Notice that for broader distributions, $\log \sigma>0$, we consistently find evidence values of $K \simeq 30$. This limiting case is analogous to that of our earlier investigation in Section \ref{['sec:simplelogprior']}, since a large value of $\sigma$ ensures that the lightest mass is likely to be very much smaller than the other two.
• Figure 5: Exploring our sensitivity to the choice of hyperprior. The above panels are in a similar format to Fig. \ref{['fig:musigma']}, but here we show the posterior distribution on the hyperparameters when imposing an informative prior of the form $\pi(\sigma) \propto \sigma^{-2}$. This enforces a bias in favour of closely clustered neutrino masses, which leads to a slight weakening of the evidence ratio, yielding odds of $17:1$ for $\Sigma < 0.15 \,$ eV (95% C.L.), and $33:1$ for $\Sigma < 0.12 \,$ eV (95% C.L.).