From Evidence to Evident: Decisive Cosmological Evidence for the Normal Neutrino Mass Hierarchy

Abstract

Cosmological data have reached the precision needed to turn the neutrino mass ordering from a weak Bayesian preference into a decisive model-selection test. We compute the evidence for the Normal and Inverted Hierarchies by combining DESI DR2 clustering with NuFIT oscillation data. In baseline $\hat{I}›$CDM, DESI DR2 plus Planck CamSpec gives $\hat{I}\pounds m_{\hat{I}}½<0.0642\mathrm{eV}$ at 95\% confidence, close to the normal-ordering floor, $\hat{I}\pounds m_{\hat{I}}{½}^{\mathrm{NH}}\simeq 0.059\mathrm{eV}$, but well below the inverted-ordering minimum, $\hat{I}\pounds m_{\hat{I}}{½}^{\mathrm{IH}}\simeq 0.099\mathrm{eV}$. Thus the inverted hierarchy lies in the tail of the cosmological likelihood. The Bayes factor $K=P(D|\mathrm{NH})/P(D|\mathrm{IH})$ exceeds $460$ even for a conservative reference prior, and remains strong, $K>40$, in baseline-model extensions. We show that this result is robust to the choice between a reference prior and a physically motivated logarithmic hierarchical prior, marking the transition from {\em prior-sensitive evidence} to {\em likelihood-dominated exclusion} of the inverted hierarchy within standard cosmology. Embedding these priors in the two-dimensional design space of measure (logarithmic versus linear in mass) and structure (hierarchical versus non-hierarchical), we find that all four prior constructions give decisive evidence under DESI DR2, with residual prior dependence governed mainly by the measure -- a factor $\sim 10$ in $K$ -- rather than by the hierarchy assumption. At the prior-family level, the evidence favors the SJPV prior predictive over HS by a Bayes factor above $4,700$ across each matched-support variation tested. The favored normal ordering pushes the effective Majorana mass to the few-meV regime, with median $m_{{\hat{I}}^2{\hat{I}}^2}=3.28\mathrm{meV}$ and 95\% credible interval $0.95<m_{{\hat{I}}^2{\hat{I}}^2}<11.55\mathrm{meV}$, below the inverted-ordering target for upcoming neutrinoless double-beta decay experiments.