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Constraint on Neutrino Statistics from Cosmological Data

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Cosmological data can now exclude purely bosonic neutrinos at 95% confidence.

desk verdict A solid, honest update of the de Salas et al. neutrino-statistics analysis with newer data, but the headline exclusion of bosonic neutrinos rests on Planck's H0 and evaporates if H0 shifts upward by ~1 km/s/Mpc. read the letter →

arxiv 2501.12264 v2 pith:ARMNINOO submitted 2025-01-21 astro-ph.CO hep-ph

YiCheng Dai , Wei Liao This is my paper · ORCID
classification astro-ph.COhep-ph
keywords neutrinostatisticsBose-EinsteinFermi-DiraccosmologicalconstraintscosmicmicrowavebackgroundbaryonacousticoscillationsmassHubbleconstant
topics The Hubble Tension
open problems The Hubble Tension
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

The paper asks whether cosmology can tell us not just how heavy neutrinos are but what quantum statistics they obey. It extends the standard cosmological model with a single parameter $\kappa_\nu$ that interpolates between Fermi-Dirac statistics ($\kappa_\nu=1$) and Bose-Einstein statistics ($\kappa_\nu=-1$), and fits the resulting model to CMB and baryon acoustic oscillation (BAO) data. The central claim is that purely bosonic neutrinos are excluded at 95% confidence, while purely fermionic and mixed-statistics neutrinos remain viable. The exclusion works through a degeneracy among $\kappa_\nu$, the sum of neutrino masses, and the Hubble constant, so the CMB's tight constraint on the Hubble constant is what pushes the low-statistics branch out. If the claim holds, cosmology becomes a direct probe of the spin-statistics connection for neutrinos.

What carries the argument

The load-bearing object is the statistical parameter $\kappa_\nu$ entering the neutrino phase-space distribution $f = 1/(e^{E/T} + \kappa_\nu)$, which controls the neutrino energy density at every epoch: Bose-Einstein neutrinos carry $8/7$ times the photon-scaled energy density of Fermi-Dirac neutrinos when relativistic and $4/3$ times when non-relativistic. The argument runs through the angular scale of the sound horizon at recombination, $\theta_s = r_s/D_A$, because increasing $\kappa_\nu$, increasing the neutrino mass sum, and increasing the Hubble constant all move $\theta_s$ in the same direction. That near-degeneracy is the machinery: the CMB's sharp measurement of $\theta_s$ and of $H_0$ converts into a lower bound on $\kappa_\nu$, especially once small masses and BAO shrink the other degeneracy directions.

What would settle it

Fix the Hubble constant at values 2, 5, and 8 km s$^{-1}$ Mpc$^{-1}$ above the CMB-inferred one and rerun the same fit; if the 95% lower bound on $\kappa_\nu$ crosses $-1$ for any of these shifts, the exclusion claim fails as stated. The authors' CMB+BAO+SNe fit, which effectively includes such a shift, already produces a bound of $-1$, so this comparison is directly checkable with published chains.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that current CMB data alone, and CMB combined with BAO distances, rule out the possibility that neutrinos obey Bose-Einstein rather than Fermi-Dirac statistics. In the authors' fit, the 95% lower bounds on the statistical parameter are $\kappa_\nu > -0.317$ for CMB only and $\kappa_\nu > -0.489$ for CMB+BAO, both excluding the purely bosonic value $\kappa_\nu=-1$. Adding supernova and local-$H_0$ information reverses this exclusion, but the paper treats that combination as statistically compromised by the Hubble tension and reports it only for comparison. The fits also favor small neutrino masses, and allowing mixed statistics raises the BAO-based neutrino-mass upper limit slightly, from 0.072 eV to 0.078 eV, which modestly eases the tension with oscillation lower bounds. The second conclusion is that the data prefer fermionic or nearly fermionic statistics, with mixed statistics still open below the 95% boundary.

Load-bearing premise

The load-bearing premise is that the Hubble constant derived from CMB data is accurate; if the true value is instead closer to the higher number favored by local distance measurements, the degeneracy could let purely bosonic neutrinos survive.

Editorial extensions

If this is right

  • Purely bosonic neutrinos are excluded at the 95% confidence level by both CMB-only and CMB+BAO fits, so any particle-physics model that predicts $\kappa_\nu=-1$ is now in tension with cosmology.
  • Mixed-statistics neutrinos with $\kappa_\nu \gtrsim -0.5$ at 2$\sigma$ remain allowed, meaning the observationally viable distributions are closer to fermionic than bosonic.
  • Allowing $\kappa_\nu$ to vary relaxes the BAO-based upper bound on the neutrino mass sum from 0.072 eV to 0.078 eV, making the cosmology-oscillation mass tension slightly smaller.
  • Tighter future measurements of the Hubble constant and of BAO distances should sharpen the bound on $\kappa_\nu$ and could eventually distinguish fermionic from mixed statistics.

Reading between the lines

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

  • As an editorial extension, the same degeneracy implies that if the Hubble tension is eventually resolved toward the higher local value, the $\kappa_\nu$ bound would probably cross $-1$ on its own; the paper's own CMB+BAO+SNe fit is a preview of that regime and is why the authors discount it.
  • Because the degeneracy runs through the sound-horizon scale, any early-universe mechanism that changes the sound horizon (extra relativistic species, early dark energy) could mimic part of the neutrino-statistics signal, so the bound is model-dependent rather than pure spin-statistics evidence.
  • A direct quantitative test: refit with a Gaussian prior on $H_0$ whose central value is varied from 67 to 73 km s$^{-1}$ Mpc$^{-1}$ and map the 95% lower bound on $\kappa_\nu$; the curve would identify exactly which future $H_0$ measurement would flip the exclusion.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper extends ΛCDM by a neutrino statistics parameter κν, interpolating between Fermi-Dirac (κν=1) and Bose-Einstein (κν=-1) occupation, and fits the model to CMB-only (Planck PR3 + ACT/PR4 lensing) and CMB+BAO (DESI DR1/DR2) datasets. The headline finding, stated in the abstract and Table II, is that purely bosonic neutrinos are excluded at 95% confidence for the CMB and CMB+BAO analyses, with lower bounds κν > -0.317 and κν > -0.489, respectively. The CMB+BAO+SNe combination, which includes a local H0 constraint via the supernova absolute magnitude, shifts the lower bound to -1 and no longer excludes bosonic neutrinos; the paper attributes this reversal to the Hubble tension and cautions that this combination is not statistically well-founded. A secondary conclusion is that CMB+BAO data prefer fermionic statistics, while mixed statistics remain possible.

Significance. If the central claim holds, the paper would provide a novel cosmological constraint on a fundamental property of neutrinos, going beyond earlier bounds that allowed bosonic statistics at 95% CL. The paper is transparent about the CMB+BAO+SNe reversal, gives a clear physical discussion of the κν–Σmν–H0 degeneracy, and includes a post-submission check with DESI DR2. These are genuine strengths. However, as argued below, the headline exclusion is conditional on the assumed H0 scale and on an unverified implementation of the modified statistics in the Boltzmann code, so the significance is currently provisional.

major comments (2)
  1. [Sec. 4, Fig. 5, Tables II and III] The paper's central claim that purely bosonic neutrinos are ruled out at 95% CL is not robust to the assumed H0 scale. The text itself states that the exclusion 'can be attributed to the stringent constraint on H0 by the CMB data,' and Table II shows that the CMB+BAO+SNe fit, which adds the local H0 information, shifts the 95% lower bound from -0.489 to -1, i.e., bosonic neutrinos become allowed. The paper dismisses this dataset as 'not statistically well-founded' because of the Hubble tension, but no calculation is given to show how the κν lower bound varies with the H0 prior. Because the Hubble tension is an unresolved systematic, the exclusion claim is conditional on the Planck H0 scale. The authors should quantify the prior dependence, for example by re-running the fits with a wide H0 prior or with a Gaussian prior centered on the local value, and should report the Δχ2 between κν=-1 and the best-fit for the CMB and CMB+BAO analyses. Without such a test, the abstract's unconditional statement is not supported.
  2. [Sec. 3 and Sec. 2, Eqs. (3)–(8)] The analysis uses a modified version of CLASS, but the paper neither provides the code nor describes how κν enters the perturbation equations. The CMB constraints come from the full angular power spectra, which depend on the neutrino perturbation hierarchy (density, velocity, anisotropic stress, free-streaming), not only on the background energy density computed in Eqs. (3)–(8). If the modification changes only the background cosmology, the resulting CMB spectra may be inconsistent with the assumed statistics. The authors should release the code or provide a complete derivation of the perturbation equations for the modified distribution, and validate the implementation, for instance by reproducing standard results for κν=1 and showing the response of the CMB spectra to κν in a test case. This is load-bearing because the exclusion claim relies on the CMB power spectra.
minor comments (4)
  1. [Throughout] There are several typos and notation errors: 'appoaches' in Sec. 2, 'unlablled' in Fig. 3 captions, 'adpoted' in Table I, and 'kν' in the Fig. 3 captions where κν is meant. These should be corrected.
  2. [References] Reference [35] is given the same arXiv number (2211.04492) as reference [20]; the Herold & Kamionkowski entry appears to have the wrong identifier.
  3. [Sec. 3 and Fig. 5] The label CMB+BAO+SNe is used for a dataset that includes a Gaussian constraint on the supernova absolute magnitude from [14], which effectively incorporates the local H0 measurement. The caption of Fig. 5 calls this 'local H0 measurement'; the terminology should be unified so readers can clearly see that the local H0 prior is included in the CMB+BAO+SNe combination.
  4. [Table III] Table III reports χ2 values 'obtained from the MCMC chains'; for best-fit comparisons, a proper χ2 minimization would be more reliable than values read off from MCMC chains.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the κν constraint is a posterior bound from external datasets, not a prediction defined by its inputs.

full rationale

The paper's central claim is that Planck CMB and CMB+BAO data exclude purely bosonic neutrinos (κν = -1) at 95% CL. This is a parameter-estimation result: κν is introduced as a free parameter in Eq. (3), the Boltzmann code CLASS is modified to include it, and the posterior is obtained by MCMC fits to external likelihoods (Planck PR3, ACT/Planck lensing, DESI BAO, Pantheon SNe). The quoted exclusion bound, κν > -0.317 (CMB) and κν > -0.489 (CMB+BAO) in Table II, is a data-driven constraint, not a quantity that is defined in terms of the fitted value. The paper's own analysis of the κν–H0 degeneracy is a physical explanation of the correlation, not a circular reduction: the degeneracy is computed from the model's effect on the sound horizon and angular diameter distance, and the exclusion is read off the posterior. The CMB+BAO+SNe result, which allows κν = -1 at 95% CL, is explicitly flagged as statistically ill-founded because of the Hubble tension, and the paper does not hide this qualification. There is no self-citation chain carrying the central premise: the comparison with de Salas et al. 2018 [8] concerns previous weaker bounds and is not load-bearing. The H0-dependence concern raised in the skeptic note is a robustness or prior-dependence issue, not circularity: the constraint could shift with different H0 assumptions, but that does not make the derivation equivalent to its inputs by construction. No equation is fitted and then renamed as a prediction, and no known result is repackaged under new coordinates. The analysis is self-contained against external benchmarks, so the appropriate finding is no significant circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a phenomenological interpolation of neutrino statistics (κν), the assumption that the neutrino temperature ratio (4/11)^{1/3} is unchanged, degenerate neutrino masses, and a modified CLASS code that is not described. The degeneracy among κν, Σmν, and H0 means the result inherits the systematics of the Planck H0 constraint.

free parameters (3)
  • κν (neutrino statistics parameter) = 0.60 (CMB), 0.93 (CMB+BAO), -0.04 (CMB+BAO+SNe) best-fit; lower bound -0.317 (CMB), -0.489 (CMB+BAO) at 95%
    The parameter of interest; interpolates between Fermi-Dirac (1) and Bose-Einstein (-1). Fitted to CMB and BAO data. The central claim is the exclusion of -1.
  • Σmν (sum of degenerate neutrino masses) = 0.082 eV (CMB), 0.010 eV (CMB+BAO), 0.034 eV (CMB+BAO+SNe) best-fit; upper bound 0.2114 eV (CMB), 0.0784 eV (CMB+BAO)…
    Neutrino mass sum, degenerate masses assumed. It is degenerate with κν and H0, affecting the constraint on κν.
  • H0 (present Hubble expansion rate) = 67.92 (CMB), 68.36 (CMB+BAO), 69.40 (CMB+BAO+SNe) km/s/Mpc best-fit
    Not sampled directly but derived from the angular scale θs and densities. The paper argues the Planck H0 constraint is what drives the κν exclusion.
assumptions (5)
  • domain assumption Flat ΛCDM universe
    The fitting model assumes spatial flatness; curvature is not varied (Table I).
  • ad hoc to paper Neutrino distribution f = 1/(e^{E/T}+κν) interpolating between FD and BE
    Phenomenological interpolation without microphysical derivation (Eq. 3). Prior work [8] used the same form.
  • domain assumption Tν/T = (4/11)^{1/3} remains valid for κν ≠ 1
    Stated in Sec. 2; relies on neutrinos decoupling before e+e- annihilation with entropy conservation.
  • domain assumption Degenerate neutrino masses
    Assumed for the main fits; a non-degenerate test in Sec. 4 gives similar results.
  • domain assumption The modified CLASS code correctly propagates κν into perturbation equations
    The paper does not detail the modification; it assumes the Boltzmann solver handles the altered distribution.

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Cite this review

Pith. "Pith review of Constraint on Neutrino Statistics from Cosmological Data." pith.science (2026). https://pith.science/paper/ARMNINOO

@misc{pith2026250112264,
  author       = {Pith},
  title        = {Pith review of: Constraint on Neutrino Statistics from Cosmological Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ARMNINOO}},
  note         = {Machine review of arXiv:2501.12264}
}
read the original abstract

We investigate the impact of neutrino statistical property on cosmology and the constraints imposed by cosmological data on neutrino statistics. Cosmological data from probes such as Cosmic Microwave Background(CMB) radiation and Baryon Acoustic Oscillation(BAO) are used to constrain the statistical parameter of neutrino. This constraint is closely related to the degeneracy effects among neutrino statistical property, the sum of neutrino masses, and the Hubble constant. Our results show that purely bosonic neutrinos can be ruled out at 95\% confidence level and purely fermionic neutrinos are preferred.

Figures

Figures reproduced from arXiv: 2501.12264 by the authors.

Figure 1
Figure 1. Evolution of Hubble expansion rate H(a) with the scale factor a for different statistical parameter κν. Each curve is normalized to the case of κν = 1 and P P mν = 0.0eV. For (a), mν = 0 eV and H0 = 67 km s−1Mpc−1 are held fixed; for (b), Pmν = 0.06 eV and H0 = 67 km s−1Mpc−1 ; for (c), Pmν = 0.3 eV and H0 = 67 km s−1Mpc−1 ; for(d), Pmν = 0.9 eV and H0 = 67 km s−1Mpc−1 . The CMB temperature at present TCMB = 2.73 K,… view at source ↗
Figure 2
Figure 2. Evolution of Hubble expansion rate H(a) with the scale factor a for different Pmν. All curves are normalized to the case of κν = 1 and Pmν = 0.0 eV. For (a), κν = 1 and H0 = 67 km s−1Mpc−1 are held fixed; for (b), κν = −1 and H0 = 67 km s−1Mpc−1 . The vertical dashed line in each subplot indicates the scale factor at recombination, arec, for the case of Pmν = 0 eV. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The sound horizon rs(η), angular diameter DA(η) and peak scale parameter θs(η) at recombination. The variation with respect to κν are illustrated by the solid labeled (colored) lines in the subplots, where each label (color) corresponds to a different Pmν. The variations with respect to Pmν are depicted by solid unlabeled (black) lines. The dashed line describes the variations caused by H0. All the numerical results… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Evolution of Hubble expansion rate H(a) with the scale factor a for varying the present Hubble expansion rate H0. All curves are normalized to the corresponding one with H0 = 67 km s−1 Mpc−1 , which falls within the 1σ range of the Planck best-fit estimate, H0 = (67.4 …
Figure 5
Figure 5. Figure 5: The cosmological constraints for variable-neutrino-statistics model. The grey lines and [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: The cosmological constraints for variable-neutrino-statistics model. The left panel con [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Comparison of the results for CMB+BAO using the DESI DR1 data and the newly released DESI DR2 data. Dark-shaded areas denote the 1σ confidence intervals, while light-shaded areas indicate the 2σ confidence intervals. The red (blue) dashed line in the upper-left panel i…
Figure 8
Figure 8. Figure 8: The cosmological constraints for variable-neutrino-statistics model. This figure is iden [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

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