Revisiting the impact of neutrino mass hierarchies on neutrino mass constraints in light of recent DESI data

Laura Herold; Marc Kamionkowski

REVIEW 1 major objections 4 minor 7 cited by

Revisiting the impact of neutrino mass hierarchies on neutrino mass constraints in light of recent DESI data

T0 review · 1 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The tight DESI-era upper limits on the sum of neutrino masses do not change when the degenerate-mass assumption is replaced by the normal or inverted hierarchy, once the oscillation-motivated lower bound on the total mass is imposed.

desk verdict A careful, honest robustness check confirming the degenerate-mass approximation holds for Planck+DESI once the oscillation lower bound is imposed; the main caveat is the excluded ACT/PR4 lensing data. read the letter →

arxiv 2412.03546 v3 pith:MB6HAOFE submitted 2024-12-04 astro-ph.CO hep-ph

Laura Herold , Marc Kamionkowski This is my paper · ORCID

classification astro-ph.COhep-ph

keywords neutrinomassesmasshierarchycosmicmicrowavebackgroundbaryonacousticoscillationsDESIprofilelikelihoodssumofcosmologicalconstraints

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Recent DESI baryon acoustic oscillation data combined with Planck CMB data have produced the tightest cosmological upper limits yet on the sum of neutrino masses. Those analyses usually approximate the three neutrino species as having equal masses (the degenerate-mass, DM, approximation), whereas oscillation experiments allow two physical orderings, normal (NH) and inverted (IH), each with a minimum total mass. This paper asks whether the tight DESI-era limits survive when the physically motivated NH or IH is used instead of the DM shortcut. The Bayesian upper limits do loosen, from $M_\mathrm{tot} < 0.086\,\mathrm{eV}$ under DM to $M_\mathrm{tot} < 0.13\,\mathrm{eV}$ under NH and $M_\mathrm{tot} < 0.16\,\mathrm{eV}$ under IH, but the frequentist analysis shows all hierarchies fit the data equally well. The entire loosening is traced to the lower mass floors imposed by oscillation data rather than to the mass-splitting physics, so the DM approximation remains a good stand-in once those floors are imposed.

What carries the argument

The central object is the total neutrino mass $M_\mathrm{tot} = m_1 + m_2 + m_3$ together with the fixed mass-squared splittings $\Delta m^2_{21}$ and $\Delta m^2_{3\ell}$ from oscillation experiments, which express $M_\mathrm{tot}$ in terms of the lightest neutrino mass for NH and IH and generate the floors $M_\mathrm{tot}\gtrsim 0.06\,\mathrm{eV}$ (NH) and $M_\mathrm{tot}\gtrsim 0.1\,\mathrm{eV}$ (IH). The argument separates prior effects from physical modeling by comparing Bayesian MCMC posteriors with frequentist profile likelihoods, using a boundary-corrected construction of frequentist confidence intervals to handle the physical boundary at $M_\mathrm{tot}=0$. The control run that carries the argument is the DM approximation run with the NH or IH floor imposed, which matches the full NH/IH posteriors and isolates the lower bound as the cause of the loosened constraints.

What would settle it

Recompute the full NH and IH analyses including Planck PR4 and ACT lensing; if the difference between the NH/IH upper limits and the DM-with-prior limits grows beyond roughly 0.02 eV, or if the relative goodness of fit across hierarchies changes, the robustness claim fails. A second check is a future data combination that pushes the 95% upper limit below 0.06 eV, where the NH floor itself would dominate the posterior and the DM approximation would need revalidation.

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Extended reading notes

Core claim

The central claim is that the three-degenerate-masses approximation remains a valid stand-in for the normal and inverted hierarchies for Planck data and for Planck+DESI data, provided the analysis enforces the lower bound on the total neutrino mass implied by oscillation experiments: $M_\mathrm{tot} \gtrsim 0.06\,\mathrm{eV}$ for NH and $M_\mathrm{tot} \gtrsim 0.1\,\mathrm{eV}$ for IH. For Planck+DESI, the DM approximation gives a Bayesian 95% upper limit of $M_\mathrm{tot} < 0.086\,\mathrm{eV}$, while the full NH and IH analyses give $M_\mathrm{tot} < 0.13\,\mathrm{eV}$ and $M_\mathrm{tot} < 0.16\,\mathrm{eV}$, respectively; the corresponding frequentist limits are $0.07$, $0.07$, and $0.06\,\mathrm{eV}$, and all orderings fit the data comparably well. The decisive control experiment is to impose the NH or IH lower bound on the DM approximation itself, which reproduces the full NH and IH posteriors. The paper concludes that the looser NH and IH bounds are an effect of the oscillation-motivated prior floor, not of the ordering's physical predictions.

Load-bearing premise

The Planck+DESI conclusion depends on the omission of the Planck PR4 and ACT lensing datasets, which the DESI baseline analysis includes; the paper expects but does not test that these datasets would not change the likelihood shape across the different neutrino mass orderings.

Editorial extensions

If this is right

The quoted Planck+DESI upper limits on the neutrino mass sum can be interpreted with the degenerate-mass approximation without needing a full NH/IH treatment, as long as the corresponding lower bound is placed on the prior.
The one-massive-plus-two-massless approximation fits the data worse than DM, NH, or IH and should be avoided in parameter inference.
The looser NH and IH bounds shift the inferred Hubble constant downward relative to the DM case, worsening the Hubble tension.
Bayesian and frequentist upper limits differ by roughly 10-40% in this near-boundary regime, and the paper expects that gap to shrink as data improve.
Imposing the NH or IH lower bound on the DM approximation reproduces the full NH or IH posterior, so the two approaches become interchangeable in practice.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If future runs include Planck PR4 and ACT lensing, the hierarchy-dependence of the likelihood could in principle change; the robustness result should be re-checked with those datasets before being quoted as final.
A push of the upper limit below 0.06 eV would bring the NH floor itself into play, making the hierarchy choice a shape effect rather than a prior effect and invalidating the DM-with-prior shortcut without a dedicated check.
The same prior-versus-modeling separation demonstrated here, comparing profile likelihoods with physically motivated priors imposed on a simplified model, could be applied to other near-boundary cosmological parameters such as curvature or the dark-energy equation of state.
A direct extension is to run the exact NH/IH treatment on the DESI full-shape likelihood once computational cost drops; agreement with DM-with-prior would close the loophole left by the BAO-only analysis.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a careful, honest robustness check. It confirms that the degenerate-mass approximation used in DESI neutrino mass analyses is adequate for Planck+DESI data, as long as you impose the lower bound on Mtot from oscillation experiments. That was suspected from forecasts and older Planck analyses, but this is the first explicit test with the actual 2024 DESI BAO data.

What it does well: the question is sharply posed, and the analysis is thorough. They use both Bayesian posteriors and frequentist profile likelihoods, with public code and data. The key comparison, Figure 5, is convincing: imposing the NH or IH lower bound on the degenerate-mass model reproduces the full NH/IH posteriors. The frequentist analysis shows all neutrino models fit the data similarly well, so the looser bounds (0.13 eV for NH, 0.16 eV for IH) are driven by the prior boundary, not by a hierarchy-dependent fit. The paper also correctly warns against the one-massive-neutrino approximation, which is a worse fit.

Soft spots: the main one is the dataset. They use Planck PR3 plus DESI BAO, excluding the ACT and Planck PR4 lensing data that appear in the DESI baseline result. They say they expect that won't change conclusions, and that's plausible — their profile likelihoods are nearly identical across hierarchies in the relevant Mtot range, and the difference is dominated by the lower bound. But it is an expectation, not a demonstration. If CMB lensing sharpens the Mtot likelihood or adds hierarchy-dependent shape, the quantitative comparison could shift. I would not call this a fatal flaw, but it is the load-bearing unverified step. A second, minor caveat: the profile likelihoods are parabolic fits, so near the boundary there is some numerical uncertainty; they acknowledge this.

Who it's for: anyone interpreting DESI-era neutrino mass bounds, especially people who want to know whether the choice of hierarchy matters. The answer — it doesn't, as long as you impose the right lower bound — is useful and should be in the literature. I'd send it to peer review; it deserves a serious referee. I'd cite it in my own work on neutrino mass constraints, with the lensing caveat noted.

Referee Report

1 major / 4 minor

Summary. This paper tests whether the common assumption of three degenerate-mass (DM) neutrinos remains a good approximation for the physically motivated normal (NH) and inverted (IH) hierarchies when inferring the sum of neutrino masses from Planck and DESI data. The authors run both Bayesian MCMC analyses and frequentist profile-likelihood analyses for DM, one-massive/two-massless (1M), NH, and IH, using Planck PR3 CMB spectra and lensing plus DESI BAO. They find that, for Planck alone and for Planck+DESI BAO, the tightest Bayesian bounds come from the DM (and, in combination with DESI, 1M) approximation, while NH and IH produce looser bounds because of the lower mass floors M_tot > 0.06 eV and > 0.1 eV imposed by oscillation experiments. The frequentist profile likelihoods show that all hierarchies fit the data similarly well, and the authors demonstrate in Fig. 5 that imposing the NH/IH lower bounds on the DM approximation reproduces the full NH/IH posteriors. The central conclusion is that the DM approximation is adequate for the Planck+DESI BAO dataset, provided the corresponding lower neutrino mass bounds are enforced.

Significance. If the central claim holds, the paper provides a useful validation of the DM approximation used in DESI-era neutrino mass analyses, while also clarifying that the apparent loosening of constraints under NH/IH is driven by the oscillation-motivated lower bounds rather than by differences in the cosmological predictions. The frequentist cross-check with Feldman-Cousins intervals is a valuable addition, as it separates prior effects from model differences. The paper ships publicly available likelihood code and plotting notebooks, which strengthens reproducibility. The main limitation is that the analysis excludes ACT DR6 and Planck PR4 lensing, which are part of the DESI 2024 baseline; the authors state that they expect this not to change the conclusions, but that expectation is not tested here.

major comments (1)

[Section III and Sec. V/Abstract] The central claim that the DM approximation remains good "also for Planck+DESI data" is established only for Planck PR3 CMB spectra and lensing plus DESI BAO, not for the full DESI 2024 baseline, which includes ACT DR6 and Planck PR4 lensing. Since CMB lensing is the observable most sensitive to the different free-streaming scales of DM, NH, and IH, the conclusion requires either an explicit test with those datasets or a restriction of the headline claim to the dataset actually analyzed. The current wording of the abstract and conclusions overstates the scope of the test, despite the acknowledgment in Sec. III.

minor comments (4)

[Table II] The frequentist upper limits for NH and IH without lower bounds (0.07 eV and 0.06 eV) lie below the oscillation lower bounds for those hierarchies; the text notes this, but the table would be clearer if these entries were explicitly flagged as unphysical.
[Sec. IV.B] The statement that the four profile likelihoods agree "at the level of the numerical noise in the minimizations" is not quantified; reporting the minimizer tolerance or a small repeated-minimization study would make this claim more concrete.
[Sec. IV.B] The phrase "more than 2 sigma away from 0.06 eV" is ambiguous because no sigma-interpretation is defined for the extrapolated parabolic minimum; a short clarification would help.
[Sec. III] The sentence "we expect that it will not alter the main conclusions" is a reasonable caveat, but it should be repeated in the abstract or conclusions so that readers do not conflate the tested dataset with the full DESI baseline.

Circularity Check

0 steps flagged · score 1.0 of 10

No meaningful circularity: the hierarchy comparison is an empirical likelihood/prior comparison; only a minor methodological self-citation and an acknowledged lensing-data omission.

full rationale

The paper's central claim is that the degenerate-mass (DM) approximation remains adequate for the normal/inverted hierarchies for Planck+DESI BAO data once the oscillation-motivated lower bounds are imposed. This is a data-driven comparison, not a derivation from an assumed conclusion. The NH and IH models are defined by external neutrino-oscillation mass splittings (Sec. I, Eq. 1), and the DM+prior variants are constructed in Appendix A; the agreement between DM with NH/IH priors and the full NH/IH analyses is an empirical result that could have failed if the likelihood were sensitive to the mass ordering. The frequentist statement that 'imposing the lower limit dominates over the physical modeling differences' is supported by the profile likelihoods in Fig. 4, which are computed from the data rather than imposed. The self-citation of [62] for pinc and the Feldman-Cousins construction is methodological and is explicitly qualified: 'we do not explicitly test these assumptions and acknowledge that frequentist coverage might only be approximately given,' so it is not load-bearing. The one notable scope limitation is stated in Sec. III: 'we exclude ACT and Planck PR4 data from our analysis but expect that it will not alter the main conclusions'; this narrows the support for the 'Planck+DESI' headline but is a caveat, not a circular step. No fitted parameter is renamed as a prediction, and no uniqueness theorem or ansatz is imported from the authors' prior work. Score 1 reflects the minor methodological self-citation and the acknowledged exclusion of lensing data, not any self-referential derivation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a data-analysis robustness check and introduces no new free parameters, ad hoc constants, or invented entities. It relies on the standard cosmological model, external oscillation inputs, and established statistical methods. The only modeling choices relevant to the central claim are the oscillation lower bounds and the dataset selection, both explicitly discussed.

assumptions (4)

domain assumption Flat ΛCDM cosmology with Neff = 3.046 fixed is assumed throughout.
Section III, paragraph 3: 'We use the Boltzmann solver CLASS for predictions of the linear spectra assuming the ΛCDM model.' The analysis does not explore extended models such as dynamical dark energy, which could change the neutrino mass constraints.
domain assumption Neutrino oscillation mass splittings from [11] are used to compute the lower bounds for NH and IH.
Section I: delta m21^2 = 7.5e-5 eV^2 and |delta m3l^2| = 2.45e-3 eV^2 are taken from Esteban et al. These values set the hard lower bounds (Mtot > 0.06 eV NH, > 0.1 eV IH) that drive the main result.
domain assumption The Feldman-Cousins construction and parabolic extrapolation of profile likelihoods are assumed valid for NH/IH without explicit testing.
Section III, final paragraph: 'Here, we do not explicitly test these assumptions and acknowledge that frequentist coverage might only be approximately given.' The frequentist limits in Table II depend on this assumption.
domain assumption halofit is used for non-linear corrections to the matter power spectrum.
Section III, paragraph 3: '...we fix the number of relativistic degrees of freedom, Neff = 3.046' and use halofit. Non-linear modeling affects the lensing and BAO predictions, and the authors note precision settings matter for small scales.

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Cite this review

Pith. "Pith review of Revisiting the impact of neutrino mass hierarchies on neutrino mass constraints in light of recent DESI data." pith.science (2026). https://pith.science/paper/MB6HAOFE

@misc{pith2026241203546,
  author       = {Pith},
  title        = {Pith review of: Revisiting the impact of neutrino mass hierarchies on neutrino mass constraints in light of recent DESI data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MB6HAOFE}},
  note         = {Machine review of arXiv:2412.03546}
}

abstract

Recent results from DESI combined with cosmic microwave background data give the tightest constraints on the sum of neutrino masses to date. However, these analyses approximate the neutrino mass hierarchy by three degenerate-mass (DM) neutrinos, instead of the normal (NH) and inverted hierarchies (IH) informed by terrestrial neutrino oscillation experiments. Given the stringency of the upper limits from DESI data, we test explicitly whether the inferred neutrino constraints are robust to the choice of neutrino mass ordering using both Bayesian and frequentist methods. For Planck data alone, we find that the DM hierarchy presents a good approximation to the physically motivated hierarchies while showing a strong dependence on the assumed lower bound of the prior, confirming previous studies. For the combined Planck and DESI baryon acoustic oscillation data, we find that assuming NH ($M_\mathrm{tot} < 0.13\,\mathrm{eV}$) or IH ($M_\mathrm{tot} < 0.16\,\mathrm{eV}$) loosens the Bayesian upper limits compared to the DM approximation ($M_\mathrm{tot} < 0.086\,\mathrm{eV}$). The frequentist analysis shows that the different neutrino models fit the data equally well and the loosening of the constraints can thus be attributed to the lower bounds induced by NH and IH. Overall, we find that the DM hierarchy presents a good approximation to the physically motivated hierarchies also for Planck+DESI data as long as the corresponding lower neutrino mass bounds are imposed.

Figures

Figures reproduced from arXiv: 2412.03546 by the authors.

**Figure 2.** ). For a given fixed value of Mtot < 0.1 eV, DM fits Planck data slightly better than the 1M approximation, while at Mtot ≥ 0.1 eV, the 1M approximation starts to fit the data better than the other mass hierarchies. This leads to a flatter profile likelihood for the 1M approximation and thus looser constraints. The profile likelihoods of DM, NH and IH, on the other hand, show ex0.00 0.05 0.10 0.15 0.20 0.25 0.30 … view at source ↗

**Figure 3.** FIG. 3. Posterior corner plot under [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Profile likelihoods under [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5. Posterior corner plot under [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 7 Pith papers

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