Neutrino mass constraints in the Schwarzschild-de Sitter black-hole dark energy model with ACT DR6 and DESI DR2 data

Guo-Hong Du; Jing-Fei Zhang; Sheng-Han Zhou; Tian-Nuo Li; Xin Zhang; Yi-Min Zhang; Zhao-Yu Li

REVIEW 2 major objections 5 minor 1 cited by

Neutrino mass constraints in the Schwarzschild-de Sitter black-hole dark energy model with ACT DR6 and DESI DR2 data

T0 review · 2 major / 5 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read A black-hole dark-energy model drives a positive neutrino-mass preference, but fits the data worse than ΛCDM.

desk verdict Solid MCMC paper: SdSDE + latest ACT/DESI/SN data yields a clear positive ∑mν preference that the authors themselves flag as likely compensation, not a better fit. read the letter →

arxiv 2607.03183 v1 pith:HGCKN4UI submitted 2026-07-03 astro-ph.CO astro-ph.GAgr-qchep-ph

Sheng-Han Zhou , Tian-Nuo Li , Guo-Hong Du , Yi-Min Zhang , Zhao-Yu Li , Jing-Fei Zhang , Xin Zhang This is my paper · ORCID

classification astro-ph.COastro-ph.GAgr-qchep-ph

keywords neutrinomassNeffSchwarzschild-deSitterdarkenergyDESIDR2ACTDR6dynamicalparametercompensation

topics Dark Energy

open problems Dark Energy

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

DESI has tightened neutrino-mass bounds inside standard cosmology until they nearly collide with the floor set by oscillation experiments, and has also favored dynamical dark energy. This paper asks what happens to the sum of neutrino masses when the late-time expansion is instead fixed by a Schwarzschild–de Sitter black-hole dark-energy template whose equation of state is locked to the redshift evolution of the cosmic black-hole mass density. Using Planck and ACT CMB, DESI BAO, and two supernova samples, the authors find that the model systematically prefers a positive total neutrino mass whenever that mass is free, at roughly the 3–4σ level depending on whether the effective number of relativistic species is also free. The preference is traced to a phantom-like expansion history that is partially offset by a larger neutrino mass and by a lower Neff. At the same time, the absolute goodness of fit is markedly worse than the corresponding ΛCDM extension, especially in the BAO and supernova distances. The authors therefore caution that the positive-mass signal may be parameter compensation rather than a genuine improvement, and that the result must be re-tested with future high-precision data.

What carries the argument

The SdSDE effective dark-energy density ρDE(z) = ρDE,0 exp[F(z)], where F(z) is a fixed cubic polynomial taken from an astrophysical fit to the cosmic black-hole mass density; the resulting equation of state wDE(z) is completely determined and phantom-like at late times.

What would settle it

A future data combination (or a re-analysis that frees the cubic coefficients or allows phantom-divide crossing) that either removes the positive ∑m u preference inside SdSDE or reverses the χ² ranking so that SdSDE fits better than the corresponding ΛCDM extension.

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Extended reading notes

Core claim

Within the fixed SdSDE background, every data combination that lets the sum of neutrino masses vary returns a positive central value (for example ∑m u = 0.207+0.047-0.052 eV with CMB+DESI+DES-Dovekie, falling to 0.162+0.055-0.056 eV when Neff is also free). The same model systematically pulls Neff below its standard value of 3.044. Best-fit χ² comparisons nevertheless show that ΛCDM with the same neutrino extensions is strongly preferred, so the positive-mass preference is interpreted as possible parameter compensation rather than an improved global description of the data.

Load-bearing premise

The entire dark-energy history is locked to three numerical coefficients taken from a prior black-hole mass-density fit and is never re-calibrated against the cosmological data used here.

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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

The useful takeaway is simple: inside the fixed SdSDE template, free ∑mν prefers a positive mass (0.207^{+0.047}_{-0.052} eV with CMB+DESI+DES-Dovekie, ~4σ from zero; still ~0.16 eV when Neff is free), while the same data strongly prefer ΛCDM+∑mν+Neff by Δχ^{2} of 18–66. The authors say this out loud and treat the mass shift as geometric compensation from the phantom-like wDE(z). That honesty is the paper’s real contribution.

What is new is the first published posterior on ∑mν and Neff inside this particular black-hole-inspired DE model with ACT DR6 + DESI DR2 + the two SN samples. The pipeline is standard and clean (CAMB + Cobaya, R−1 < 0.02, public likelihoods). Contours, 1-D posteriors, H0/S8 shifts, the ∑mν–Neff correlation, and the χ^{2} breakdown by probe are all reported carefully. They also show that the fixed SdSDE EoS does not track the w0wa reconstruction preferred by DESI+SN, which explains why BAO and SN drive the worse fit.

The soft spot is exactly the one the authors inherit: the cubic F(z) coefficients are frozen from an external astrophysical black-hole mass-density fit and never varied, plus an ad-hoc cutoff at z~10 that forces the model back to ΛCDM. That choice sets the late-time expansion and therefore the trade-off against neutrino mass. It is not circular in the sense of forcing ∑mν by construction, but it does mean the positive-mass preference is a property of this particular rigid template rather than a robust DE-independent result. No other internal inconsistency or data-handling problem shows up.

This is for people already tracking how neutrino-mass bounds move under dynamical DE after DESI. It is a clean cautionary data point, not a resolution of the tension and not a new physical model of DE. Methods are reproducible in principle; a serious referee should see it. I would accept it for peer review and would cite the numerical intervals if I am writing on model dependence of ∑mν.

Referee Report

2 major / 5 minor

Summary. The paper constrains ∑mν and Neff in the Schwarzschild–de Sitter black-hole dark energy (SdSDE) model of Hayashi, using Planck+ACT DR6 CMB, DESI DR2 BAO, and DES-Dovekie/PantheonPlus SN data. With the DE density fixed by a cubic F(z) taken from an external black-hole mass-function fit (Eqs. 1–6), SdSDE+ ∑mν yields a positive mass preference, e.g. ∑mν = 0.207^{+0.047}_{-0.052} eV (~4σ) for CMB+DESI+DES-Dovekie, reduced to 0.162^{+0.055}_{-0.056} eV when Neff is free (Tables II–III, Figs. 1–2). The authors attribute this to compensation with a phantom-like fixed wDE(z) and a systematic pull of Neff below 3.044, and they show that ΛCDM+∑mν+Neff is strongly preferred by best-fit χ^{2} (Δχ^{2} ≈ 18–66; Table IV). The central claim is therefore carefully hedged: the positive-mass preference is real within SdSDE but may be parameter compensation rather than an improved global fit.

Significance. The work is a timely, carefully executed test of model dependence of DESI-era neutrino-mass bounds in a concrete black-hole-inspired DE template. Strengths include a standard CAMB+Cobaya pipeline with public likelihoods, Gelman–Rubin R−1<0.02, transparent multi-dataset posteriors, and an honest χ^{2} decomposition that undercuts any over-claim of model preference. The explicit comparison of the fixed SdSDE EoS to w0waCDM (Fig. 5) and the documentation of the ∑mν–Neff–H0–S8 compensation direction (Figs. 3–4) make the result useful for the broader literature on DE–neutrino degeneracies. The result is incremental rather than transformative, but it is a clean, falsifiable data point for black-hole DE scenarios.

major comments (2)

Sec. II.A, Eqs. (1)–(6) and the coefficients a=0.00658, b=−0.104, c=−0.348: the entire DE evolution is locked to an external cubic fit and never varied or re-calibrated against the cosmological data. Because the positive ∑mν preference is driven by this fixed phantom-like wDE(z), the central claim is only as robust as that template. At minimum the authors should (i) quantify sensitivity by varying a,b,c within the uncertainties of the Sicilia et al. mass-function fit, or (ii) present a one-parameter rescaling of F(z) and show how the ∑mν posterior moves. Without such a test, the reported ~3–4σ mass preference remains tied to an untested external prior.
Sec. II.A: the smooth cutoff at z_cut ≃ 10 that forces SdSDE → ΛCDM at high redshift is introduced by hand and is not varied. Early-universe quantities that enter the CMB likelihood (sound horizon, Neff inference) therefore depend on an arbitrary transition. A short robustness check with alternative cutoffs (e.g. z_cut = 5 and 20) or a continuous matching function would confirm that the Neff pull and the residual ∑mν preference are not artifacts of this choice.

minor comments (5)

Table IV: the χ^{2} decomposition is valuable; adding the effective number of data points or reduced χ^{2} for BAO and SN would make the degradation easier to interpret across datasets.
Fig. 5: the w0waCDM bands are shown only for two data combinations; including the PantheonPlus combination for completeness would match the rest of the paper.
Sec. III: the ~4.0σ / ~2.9σ “descriptive significance” statements for ∑mν should be clarified as distance-from-zero under the prior (not a model-comparison significance), to avoid misreading.
Notation: the manuscript mixes ∑mν, P mν and ∑m_ν; a single consistent symbol would improve readability.
References: a brief pointer to other cosmologically coupled black-hole DE implementations (beyond Hayashi) would help place SdSDE in the wider literature.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: fixed external DE template yields data-driven neutrino posteriors; mild self-citation is contextual only.

full rationale

The paper's central results are MCMC posterior constraints on ∑mν and Neff under the SdSDE background (Tables II–III, Figs. 1–4). The DE density is prescribed by a cubic F(z) whose coefficients are taken unchanged from an external astrophysical fit (Hayashi 2026 / Sicilia et al. 2022) and are never varied against the CMB+BAO+SN likelihoods used here (Sec. II.A, Eqs. 1–6); a high-z cutoff is likewise imposed by hand. Neutrino parameters are free, sampled with standard priors, and constrained by independent public data (Planck/ACT, DESI DR2, DES-Dovekie/PantheonPlus). The reported positive-mass preference is therefore a model-dependent inference, not forced by construction or by a fitted DE parameter renamed as a prediction. The authors themselves show via best-fit χ² (Table IV) that the corresponding ΛCDM extensions are strongly preferred (Δχ² ≈ 18–66), and they interpret the mass shift as parameter compensation—an honest, non-circular reading of the same chains. Self-citations to prior Zhang-group papers on DE–neutrino degeneracies supply only background context and are not load-bearing for the new numerical results. No uniqueness theorem, self-definitional identity, or ansatz-smuggled prediction appears. Score 1 reflects only the ordinary presence of author self-citations that do not close the derivation loop.

Assumptions & free parameters 4 free parameters · 4 assumptions · 1 invented entities

The central claim rests on a fixed phenomenological DE density taken from an external BH-mass-density fit, standard neutrino free-streaming physics, and public cosmological likelihoods. No new free DE parameters are introduced; the only free neutrino parameters are the usual ∑mν and Neff. The model is therefore almost entirely inherited, with the new content being the MCMC constraints themselves.

free parameters (4)

∑mν = 0.207^{+0.047}_{-0.052} eV (SdSDE+∑mν); 0.162^{+0.055}_{-0.056} eV (with Neff)
Varied with flat prior U[0,5] eV; the positive central value is the main reported result.
Neff = 2.72±0.18 (CMB+DESI+DES-Dovekie)
Varied with flat prior U[0,7]; systematically pulled below 3.044 and positively correlated with ∑mν.
baseline ΛCDM parameters (Ωb h², Ωc h², H0, τ, As, ns)
Standard six parameters varied under the usual flat priors listed in Table I; they absorb geometric and amplitude shifts induced by the fixed SdSDE background.
a, b, c coefficients of F(z) = a=0.00658, b=−0.104, c=−0.348
Fixed by hand to the cubic fit of the cosmic BH mass density (a=0.00658, b=−0.104, c=−0.348); never re-fit to the cosmological data used here.

assumptions (4)

ad hoc to paper Effective DE density is exactly ρDE(z)=ρDE,0 exp[F(z)] with F a cubic polynomial whose coefficients are taken from a prior astrophysical BH mass-function fit and held fixed.
Sec. II.A, Eqs. (1)–(3); the entire late-time expansion history is locked by this choice.
ad hoc to paper A smooth cutoff at z_cut≃10 forces the model to ΛCDM at higher redshifts so that early-universe physics remains standard.
Explicitly introduced in Sec. II.A because the BH-population description is not meant to apply at arbitrarily high z.
domain assumption Three active neutrinos with degenerate masses, standard thermal history, and free-streaming suppression ΔP/P≈−8 Ων/Ωm on small scales.
Sec. II.B; conventional treatment used throughout the MCMC.
domain assumption Flat FLRW cosmology with the usual continuity equation for DE, yielding wDE(z)=−1+(1+z)/3 dF/dz.
Eqs. (4)–(6); standard background assumption.

invented entities (1)

SdSDE effective dark-energy density tied to cosmic black-hole mass density
purpose: Provides a fixed, phantom-like w(z) template motivated by non-singular black holes, used here as the background against which neutrino parameters are constrained.
Introduced by Hayashi (2026) and adopted without modification; no independent cosmological evidence for the specific cubic coefficients is supplied in the present work.

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Pith. "Pith review of Neutrino mass constraints in the Schwarzschild-de Sitter black-hole dark energy model with ACT DR6 and DESI DR2 data." pith.science (2026). https://pith.science/paper/HGCKN4UI

@misc{pith2026260703183,
  author       = {Pith},
  title        = {Pith review of: Neutrino mass constraints in the Schwarzschild-de Sitter black-hole dark energy model with ACT DR6 and DESI DR2 data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HGCKN4UI}},
  note         = {Machine review of arXiv:2607.03183}
}

abstract

Recent DESI observations have posed new challenges to $\Lambda$CDM, showing a preference for dynamical dark energy and yielding neutrino mass constraints within $\Lambda$CDM that approach the lower bound allowed by neutrino oscillation experiments. In this work, we investigate cosmological constraints on the key neutrino parameters, $\sum m_\nu$ and $N_{\rm eff}$, within the Schwarzschild-de Sitter black-hole dark energy (SdSDE) framework. We use cosmic microwave background (CMB) data from Planck and ACT DR6, baryon acoustic oscillation data from DESI DR2, and type Ia supernova data from DES-Dovekie and PantheonPlus. We find that SdSDE scenarios prefer a positive neutrino mass whenever $\sum m_\nu$ is allowed to vary. Using CMB+DESI+DES-Dovekie data, we obtain $\sum m_\nu=0.207^{+0.047}_{-0.052}~{\rm eV}$ for SdSDE+$\sum m_\nu$, reduced to $\sum m_\nu=0.162^{+0.055}_{-0.056}~{\rm eV}$ when $N_{\rm eff}$ is also varied. This arises from the positive correlation between $N_{\rm eff}$ and $\sum m_\nu$, together with the systematic preference of SdSDE for values of $N_{\rm eff}$ below the standard value. Furthermore, the best-fit $\chi^2$ comparison shows that $\Lambda$CDM with extended neutrino parameters is strongly preferred over the corresponding SdSDE extension. Overall, the positive neutrino mass preference induced by SdSDE may reflect parameter compensation rather than an improved global fit, a possibility that should be further tested with future high-precision observational data.

Figures

Figures reproduced from arXiv: 2607.03183 by the authors.

**Figure 2.** FIG. 2. One-dimensional marginalized posterior distributions of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Constraints on the cosmological parameters using the CMB+DESI, CMB+DESI+DES-Dovekie, and [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. One-dimensional marginalized posterior distributions of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. Comparison of the DE EoS evolution [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

Discussion (0). Continue with ORCID to comment.

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Reviewed July 12, 2026 · model on record in the stance chip above.

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