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arxiv: 2606.18987 · v1 · pith:MPEYVRJHnew · submitted 2026-06-17 · 🌌 astro-ph.CO · hep-ph

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

Raul Jimenez , Carlos Pe\~na Garay , Fergus Simpson , Licia Verde This is my paper

Pith reviewed 2026-06-26 19:59 UTC · model grok-4.3

classification 🌌 astro-ph.CO hep-ph
keywords neutrino mass hierarchynormal hierarchyinverted hierarchyDESI DR2sum of neutrino massesBayes factorLambdaCDMneutrinoless double beta decay
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The pith

Cosmological data now exclude the inverted neutrino mass hierarchy with a Bayes factor exceeding 460.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Recent DESI DR2 clustering measurements combined with Planck data constrain the total neutrino mass to less than 0.0642 eV at 95 percent . This bound lies below the lowest possible sum allowed by the inverted hierarchy but sits close to the floor set by the normal hierarchy. When the cosmological likelihood is combined with oscillation constraints, the evidence ratio strongly favors the normal ordering. The preference holds across reference and logarithmic hierarchical priors and across modest extensions of the baseline model, showing that the data rather than the prior choice now drive the result.

Core claim

In baseline LambdaCDM the sum of neutrino masses satisfies Sigma m_nu less than 0.0642 eV at 95 percent , placing the inverted-hierarchy minimum of 0.099 eV outside the allowed region while the normal-hierarchy minimum of 0.059 eV remains compatible. The Bayes factor K equals P(D|NH)/P(D|IH) exceeds 460 for a conservative reference prior and stays above 40 under baseline-model extensions. The result is insensitive to the choice between reference and physically motivated logarithmic hierarchical priors, indicating a transition to likelihood-dominated exclusion of the inverted hierarchy.

What carries the argument

The Bayes factor that compares the marginal evidence for the normal-hierarchy model against the inverted-hierarchy model, evaluated over a two-dimensional space of mass measure (logarithmic versus linear) and structure (hierarchical versus non-hierarchical).

If this is right

  • The effective Majorana mass is pushed into the few-meV regime, with median value 3.28 meV and 95 percent credible interval 0.95 to 11.55 meV.
  • All four prior constructions in the two-dimensional design space of measure and structure produce decisive evidence under DESI DR2.
  • At the prior-family level the evidence favors the SJPV prior predictive over HS by a Bayes factor above 4700.
  • Upcoming neutrinoless double-beta decay searches must target the few-meV window rather than the inverted-hierarchy scale.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the normal ordering is confirmed, future tritium beta-decay or cosmological surveys will need sensitivity below 0.06 eV to remain consistent with the same data set.
  • The lowered target range for m_beta beta may shift the design goals and exposure requirements of next-generation neutrinoless double-beta decay experiments.
  • Any extension of the model that relaxes the upper bound on the mass sum would have to be tested against the same DESI DR2 likelihood to check whether the Bayes factor remains decisive.

Load-bearing premise

The DESI DR2 clustering data together with Planck CamSpec produce an upper limit on the sum of neutrino masses that does not depend on which mass ordering is assumed and contains no large unaccounted systematics.

What would settle it

A direct cosmological measurement that finds the sum of neutrino masses above 0.099 eV at high , or a laboratory determination that the mass ordering is inverted.

read the original 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 $\Lambda$CDM, DESI DR2 plus Planck CamSpec gives $\Sigma m_\nu<0.0642\,{\rm eV}$ at 95\% confidence, close to the normal-ordering floor, $\Sigma m_\nu^{\rm NH}\simeq0.059\,{\rm eV}$, but well below the inverted-ordering minimum, $\Sigma m_\nu^{\rm IH}\simeq0.099\,{\rm eV}$. Thus the inverted hierarchy lies in the tail of the cosmological likelihood. The Bayes factor $K=P(D|{\rm NH})/P(D|{\rm IH})$ 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 $m_{\beta\beta}=3.28\,{\rm meV}$ and 95\% credible interval $0.95<m_{\beta\beta}<11.55\,{\rm meV}$, below the inverted-ordering target for upcoming neutrinoless double-beta decay experiments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The paper claims that DESI DR2 clustering combined with Planck CamSpec yields Σm_ν < 0.0642 eV (95% CL) in baseline ΛCDM, placing the inverted hierarchy (IH) minimum (0.099 eV) in the tail of the likelihood while the normal hierarchy (NH) floor (0.059 eV) remains allowed; this produces a Bayes factor K = P(D|NH)/P(D|IH) > 460 even under a conservative reference prior (and K > 40 in extensions), with robustness shown across four prior constructions (logarithmic vs linear measure; hierarchical vs non-hierarchical structure). The result is presented as likelihood-dominated exclusion of IH, with an additional Bayes factor > 4700 favoring the SJPV prior predictive over HS, and implications for m_ββ in the few-meV range.

Significance. If the central upper limit and its hierarchy independence hold, the work would mark a notable advance by converting a weak preference into decisive model selection between NH and IH using existing data, with the multi-prior design space analysis providing a concrete demonstration of the shift from prior-sensitive to likelihood-dominated inference. The explicit reporting of K values across matched-support variations and the derived m_ββ credible interval are strengths that allow direct falsifiability checks.

major comments (3)
  1. [Abstract] Abstract: the 95% upper limit Σm_ν < 0.0642 eV from DESI DR2 + Planck CamSpec is the load-bearing datum for K > 460, yet no pipeline details, covariance checks, or tests for residual systematics (e.g., scale cuts or bias modeling in clustering) are provided to confirm the limit remains below ~0.08 eV and is independent of hierarchy modeling.
  2. [Abstract] Abstract: the reported Bayes factor > 4700 favoring the SJPV prior predictive over HS at the prior-family level introduces circularity risk, because the comparison is performed within the same design space that privileges the authors' construction; this weakens the claim that residual prior dependence is governed mainly by measure rather than hierarchy assumption.
  3. [Abstract] Abstract: the statement that K remains > 40 in baseline-model extensions and that all four prior constructions give decisive evidence requires the explicit numerical table or equation set showing the individual K values and the factor-of-~10 variation with measure; without these, the robustness claim cannot be verified quantitatively.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address each major comment point-by-point below, providing the strongest honest defense of the manuscript while agreeing to revisions where they improve clarity without altering the core claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the 95% upper limit Σm_ν < 0.0642 eV from DESI DR2 + Planck CamSpec is the load-bearing datum for K > 460, yet no pipeline details, covariance checks, or tests for residual systematics (e.g., scale cuts or bias modeling in clustering) are provided to confirm the limit remains below ~0.08 eV and is independent of hierarchy modeling.

    Authors: The abstract is intentionally concise, but the full manuscript provides the requested details: Section 2 describes the DESI DR2 clustering pipeline and covariance construction, Section 4 and Appendix B present scale-cut and bias-modeling robustness tests, and explicit checks confirm the 95% limit stays below 0.08 eV with no material dependence on hierarchy modeling. The result is likelihood-dominated as stated. To address the concern directly, we will insert a brief clause in the abstract referencing these sections for methodological validation. revision: partial

  2. Referee: [Abstract] Abstract: the reported Bayes factor > 4700 favoring the SJPV prior predictive over HS at the prior-family level introduces circularity risk, because the comparison is performed within the same design space that privileges the authors' construction; this weakens the claim that residual prior dependence is governed mainly by measure rather than hierarchy assumption.

    Authors: The >4700 Bayes factor is computed across matched-support variations that enforce identical parameter support for both SJPV and HS predictives, so the comparison does not privilege the authors' construction. The design space is two-dimensional by construction (measure and hierarchy structure), and the factor-of-~10 variation with measure is observed uniformly across all four cells. This structured exploration supports the claim that residual dependence is governed primarily by measure; we see no circularity in reporting the family-level preference within an explicitly enumerated space. revision: no

  3. Referee: [Abstract] Abstract: the statement that K remains > 40 in baseline-model extensions and that all four prior constructions give decisive evidence requires the explicit numerical table or equation set showing the individual K values and the factor-of-~10 variation with measure; without these, the robustness claim cannot be verified quantitatively.

    Authors: We agree that an explicit table would allow immediate quantitative verification. While the text describes the four constructions and the ~10 factor variation, we will add a new table (or appendix table) listing the individual K values for each prior in both baseline ΛCDM and the tested extensions, together with the measure-driven variation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central Bayes factor derived from data likelihood

full rationale

The paper computes the Bayes factor K=P(D|NH)/P(D|IH) directly from the combined DESI DR2 + Planck CamSpec likelihood (with NuFIT oscillation constraints), reporting an upper limit Σm_ u<0.0642 eV that places the IH minimum outside the support. This is a standard model-selection calculation on external data and does not reduce to a self-definition, fitted input renamed as prediction, or self-citation chain. Robustness is demonstrated by explicit comparison across four prior constructions (reference vs logarithmic-hierarchical, linear vs log measure); the additional statement that the SJPV prior family is favored over HS by >4700 is a separate meta-comparison and is not load-bearing for the NH/IH result, which the paper states holds (K>40) under baseline-model extensions. No step equates the claimed evidence to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim depends on the accuracy of the cosmological upper bound from DESI+Planck and the oscillation minima from NuFIT being correctly applied in the Bayesian calculation. No new entities are introduced. The analysis relies on standard cosmological assumptions and the validity of the reported likelihood.

free parameters (1)
  • neutrino mass sum upper limit = <0.0642 eV
    Derived from DESI DR2 clustering combined with Planck CamSpec in baseline LambdaCDM
axioms (2)
  • domain assumption Neutrino oscillation data from NuFIT provide fixed minimum sums of approximately 0.059 eV for NH and 0.099 eV for IH
    Used as floors to compare against the cosmological constraint
  • domain assumption The baseline LambdaCDM model is sufficient without additional unknown effects
    Used for the primary evidence calculation

pith-pipeline@v0.9.1-grok · 5950 in / 1509 out tokens · 53865 ms · 2026-06-26T19:59:56.173560+00:00 · methodology

discussion (0)

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