Pith. sign in

REVIEW 4 major objections 7 minor 1 cited by

Neutrino phenomenology and Dark matter in a left-right asymmetric model with non-holomorphic modular $A_{4}$ group

T0 review · 4 major / 7 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read One non-supersymmetric modular model claims to reproduce neutrino data and dark matter.

desk verdict A coherent new modular-flavor construction, but the keV sterile-neutrino DM claim rests on couplings that are never given, and the neutrino 'predictions' are refits of the input ranges. read the letter →

arxiv 2509.01205 v1 pith:6AZ3DTAE submitted 2025-09-01 hep-ph

Bhabana Kumar , Mrinal Kumar Das This is my paper · ORCID
classification hep-ph
keywords neutrinooscillationsmodularsymmetryA4flavorTypeIIseesawsteriledarkmatterpolyharmonicMaaßformsleft-rightasymmetricmodelneutrinolessdoublebetadecay
topics Dark Matter
open problems Dark Matter
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

This paper tries to show that a single non-supersymmetric extension of the Standard Model, built on the Γ3 modular group (isomorphic to A4) with neutrino Yukawa couplings given by weight-zero polyharmonic Maaß forms, can explain both measured neutrino mixing and the observed dark matter abundance. Neutrino masses come from a Type II seesaw dominance mechanism within a left-right asymmetric gauge structure, with one sterile neutrino per generation. After scanning the modular parameter τ and fitting two dimensionless couplings to the three-sigma oscillation ranges, the model reproduces the mixing parameters only for normal mass ordering, predicts θ23 near 40°–45° (lower octant) and δCP in two narrow bands, and places the effective Majorana mass between 10⁻³ eV and 0.1 eV. It also yields a 10–30 keV sterile neutrino whose active–sterile mixing production gives the measured relic density, with a decay width of 10⁻³⁵ to 10⁻³⁴ s⁻¹. If these outputs hold up, the construction connects neutrino physics, neutrinoless double-beta decay, and dark matter in one minimal setup.

What carries the argument

The load-bearing object is the set of weight-zero, level-three polyharmonic Maaß forms of the modular group Γ3 (A4): Yukawa couplings that satisfy a Laplacian condition instead of holomorphicity, keeping explicit dependence on the modulus τ at modular weight zero. Three of them, Y_{3,1}, Y_{3,2}, Y_{3,3}, form an A4 triplet and enter, through the couplings b1–b10, every neutral-lepton mass matrix (ML, MD, MR, M). The VEV hierarchy MR > M > MD >> ML makes the 9×9 mass matrix block-diagonalize to mν = ML, MR = (v_R/v_L)ML, and MS = −M M_R⁻¹ Mᵀ; the first relation is Type II seesaw dominance, and the last fixes the sterile masses and active–sterile mixing that feed the relic-density and decay-w

What would settle it

Take the paper's Lagrangian, assign concrete values to b3–b10 (or scan them), and recompute MS = −M M_R⁻¹ Mᵀ; if no choice gives a 10–30 keV lightest sterile neutrino with active–sterile mixing whose production reaches Ωh² = 0.1187 while obeying X-ray bounds, the dark-matter claim is refuted. Alternatively, a future neutrinoless-double-beta experiment that pushes the effective Majorana mass below 0.001 eV would sit outside the predicted 0.001–0.1 eV band.

Watch

Extended reading notes

Core claim

Central claim: a non-supersymmetric left-right asymmetric model, with Γ3 (A4) modular flavor symmetry and weight-zero polyharmonic-Maaß-form Yukawa couplings, fits neutrino oscillation data and contains a sterile-neutrino dark-matter candidate. Type II seesaw dominance gives mν = ML, MR = (v_R/v_L)ML, and MS = −M M_R⁻¹ Mᵀ after block-diagonalizing the 9×9 mass matrix. With τ scanned and b1,b2 fitted to 3σ oscillation ranges, the model rejects inverted ordering via the cosmological summed-mass bound and, for normal ordering, predicts θ23 in [40°,45°], δCP in [0°,50°] and [280°,350°], and 10⁻³ eV < m_ee < 0.1 eV. The lightest sterile neutrino at 10–30 keV then gives Ωh² = 0.1187 via active–ste

Load-bearing premise

The dark-matter predictions require values for the couplings b3–b10 that control the Dirac, right-handed, and sterile-right-handed mass matrices, but the paper never states those values, so the sterile-neutrino relic-density curves cannot be independently reproduced without filling them in.

Editorial extensions

If this is right

  • If the model is right, future neutrino data should confirm normal mass ordering and lower-octant θ23 in the 40°–45° range.
  • Neutrinoless double-beta decay searches should eventually find an effective Majorana mass between 10⁻³ eV and 0.1 eV, correlated with the lightest neutrino mass.
  • A monoenergetic X-ray line from sterile-neutrino decay should appear for a dark-matter mass between 10 and 30 keV, with a width of 10⁻³⁵ to 10⁻³⁴ s⁻¹.
  • The model's excluded CP-phase region, roughly 50°–280°, is a sharp target for long-baseline CP-violation experiments.
  • Weight-zero non-holomorphic modular forms offer a flavor-symmetry mechanism that works without supersymmetry or flavon fields.

Reading between the lines

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

  • Because b1 and b2 are fitted to the same 3σ oscillation ranges, the reproduction of those ranges is expected; the testable content is the correlation pattern among θ23, δCP, Σmν, and m_ee across the scanned τ, which could be mapped onto explicit confidence regions.
  • The dark-matter window depends on the unspecified couplings b3–b10 and on the chosen VEVs (vL = 0.01 eV, vR = v′ = 10 TeV); varying these values would reveal how robust the 10–30 keV preferred range really is.
  • The q-expansions are truncated at q⁶ with no error estimate; extending them or evaluating them near the boundary of the fundamental domain could shift the quoted parameter ranges.
  • Since µS and ⟨χL⟩ were set to zero, turning either on would alter active–sterile mixing and could be probed against the same X-ray and relic-density constraints.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 7 minor

Summary. The manuscript constructs a non-supersymmetric left-right asymmetric model with a Gamma_3 modular A4 symmetry, using weight-zero polyharmonic Maaß forms as Yukawa couplings and Type II seesaw dominance for neutrino masses. The authors scan the modulus tau in the fundamental domain, fix the charged-lepton Yukawa couplings from trace conditions, and fix the active-neutrino couplings b1,b2 using the NuFIT 3 sigma ranges of oscillation parameters. They then quote predictions for the mass ordering, theta23 octant, delta_CP, J_CP, the effective Majorana mass, and a sterile-neutrino dark matter candidate with mass 10-30 keV, active-sterile mixing, Dodelson-Widrow relic density, and decay width.

Significance. If the results were fully supported, the model would be a compact simultaneous explanation of neutrino oscillation data and dark matter, with the notable virtues of avoiding flavons and supersymmetry and using non-holomorphic modular symmetry. The paper contains several useful ingredients: explicit mass matrices from polyharmonic Maaß forms, an analysis of normal vs inverted hierarchy, and a discussion of 0nu beta beta bounds. However, the current manuscript does not provide enough information to verify the central claims. The neutrino-sector results are acceptance-rejection fits with no goodness-of-fit, and the dark-matter predictions depend on couplings that are never specified. These issues are load-bearing, not cosmetic.

major comments (4)
  1. [Sec. 4, Eqs. (10)-(11), Fig. 6] The dark-matter analysis is unreproducible because the couplings b3-b10 are never specified. The sterile mass matrix M_S = -M M_R^{-1} M^T and the active-sterile mixing U_alpha S in Eq. (10)-(11) depend on M_D, M_R, and M, i.e., on b3-b5, b6-b7, and b8-b10 in Eqs. (23)-(25), but the paper gives no values, no scan ranges, and no benchmark point. With the stated VEVs v_R = v' = 10 TeV and O(1) couplings, the naive expectation is m_S ~ v'^2/v_R ~ 10 TeV and mixing of order M_D/M, far from the claimed 10-30 keV and sin^2 2theta ~ 10^-10. Reproducing Fig. 6 requires tuned small couplings whose sizes and ranges are not stated, so the dark-matter curves are not a genuine prediction of the model as presented.
  2. [Eq. (10) vs Eq. (24)] There is a direct contradiction in the right-handed neutrino mass matrix normalization. Eq. (10) states M_N = M_R = (v_R/v_L) M_L, while Eq. (24) defines M_R = (v_L/v_R) times a matrix of b6,b7. These differ by roughly 30 orders of magnitude with the quoted VEVs (v_L = 0.01 eV, v_R = 10 TeV). The ratio M_R/M enters M_S in Eq. (10) and the condition M_R >> M used for block diagonalization is never verified. The manuscript must state which normalization is correct and show the corresponding benchmark parameters.
  3. [Sec. 4, Figs. 3-4] The claimed 'predictions' for theta23, delta_CP, and J_CP are circular as presented. The text says b1 and b2 are determined using the 3 sigma values of the neutrino oscillation parameters, and UPMNS is generated from those same parameters. The scan envelopes in Figs. 3-4 are therefore consequences of the input ranges, not independent model predictions. To support the claims of lower-octant preference and specific delta_CP intervals, the authors should provide a goodness-of-fit or likelihood comparison, or at least show the input distribution and state which parameters are free versus fixed. Without this, the central neutrino result is merely a parameter scan that accepts all points within the NuFIT ranges.
  4. [Sec. 4, Fig. 2 and IH conclusion] The conclusion that inverted hierarchy is 'strongly disfavoured' because 'no data points were found' is not robust as stated. The scan ranges of b1,b2 for IH are not given, and Fig. 2(b) presumably corresponds to some unspecified scan. It is possible that the IH scan simply did not explore the relevant parameter region. The authors should specify the scan ranges and the number of points for both NH and IH, and ideally quantify the fraction of accepted points, before claiming a preference for normal ordering.
minor comments (7)
  1. [Eq. (1)] The formula for the effective Majorana mass has notation 'Uνν2 ei' which is garbled; it should presumably be sum_i |U_ei|^2 m_i.
  2. [Sec. 2, after Eq. (7)] The text says 'the first term in the equation (8)' but the displayed equation is (7).
  3. [Sec. 5, text after Fig. 6] The sentence 'using Eqs. (21) and (4)' for relic abundance and decay width is wrong: the relic abundance formula is Eq. (3), not Eq. (21) (which is the Yukawa Lagrangian).
  4. [Eq. (23)] There are apparent typos in the Dirac mass matrix: the (3,2) entry contains '-b1 + b4' and the (3,3) entry contains '2b2', which are inconsistent with the b3/b4 structure of the rest of the matrix and with the symmetric form expected from the Lagrangian. Please check all entries.
  5. [Sec. 4 heading] The section heading reads 'Numarical Analysis'; this should be 'Numerical Analysis'.
  6. [Figs. 3 and 5] Several figures lack clear axis labels or legends in the text. For example, Fig. 3 should state explicitly which quantity is on each axis, and Fig. 5 should label the horizontal bound lines. Please improve figure readability.
  7. [Appendix A, Eqs. (A.7)-(A.9)] The q-expansions are truncated at q^6 with no error estimate. For Im(tau) in the fundamental domain, |q| <= e^{-pi sqrt(3)} ~ 0.004, so the truncation is plausibly safe, but this should be stated explicitly.

Circularity Check

2 steps flagged · score 6.0 of 10

Neutrino 'predictions' for θ23, δCP, JCP and m_ee are inherited from the NuFIT inputs used to fit b1 and b2; the DM sector is under-specified rather than circular.

  1. fitted input called prediction [Section 4 (Numerical Analysis), after Eq. (26); Figs. 3–4]
    "The UPMNS matrix is generated using the neutrino oscillation parameters given in Table 2, while Ul is constructed by calculating the eigenvectors of the charged lepton mass matrix. ... Finally, the unknown free parameters b1 and b2 are determined using the 3σ values of neutrino oscillation parameters together with the Yukawa couplings (Y(0)3,1, Y(0)3,2, Y(0)3,3) obtained by randomly varying τ in the upper half of the complex plane. ... Figure 3 illustrates the parameter space of θ23 and δCP as obtained from the model."

    The two parameters b1 and b2 that determine the entire light-neutrino mass matrix ML are solved from the same NuFIT 3σ oscillation observables (mixing angles, δCP, mass-squared differences) that define the target UPMNS. The predicted θ23, δCP and JCP are therefore the fitted input passed through the model matrices; agreement with the oscillation data is built into the fit, not an independent test. Figures 3–4 are thus outputs of a fitting procedure, not genuine predictions.

  2. fitted input called prediction [Section 4–5, Eq. (1), Fig. 5]
    "We have also studied neutrinoless double beta decay (0νββ) and calculated the effective Majorana mass arising from the standard contribution. ... Furthermore, the effective Majorana mass is predicted to vary from 0.1 eV down to 10−3 eV as the lightest neutrino mass decreases from 0.12 eV to 10−4 eV."

    The effective Majorana mass m_ee is computed from Eq. (1) using the same Uν and mνi that were used to fix b1 and b2. Because the model's light-neutrino sector is fitted to reproduce exactly those UPMNS parameters and mass eigenvalues, the quoted 10^-3–0.1 eV range is simply the standard 0νββ formula evaluated on the fitted input ranges. No modular-model constraint independent of the fit enters this observable, so presenting it as a model prediction is a renaming of the input rather than a new result.

full rationale

The central circularity is in the neutrino sector: the paper explicitly determines b1 and b2 using the 3σ values of the neutrino oscillation parameters, and the UPMNS matrix is generated from those same parameters. The subsequent plots of θ23, δCP and JCP are therefore outputs of the fit, not independent predictions. The m_ee range is likewise inherited from the standard formula evaluated on the fitted quantities. This warrants a score of 6: one or more 'predictions' reduce by construction, while some model content (the NH/IH comparison and the Planck-bound check) remains structural rather than definitional. I do not score the dark-matter sector as circular. The couplings b3–b10 and their scan ranges are never stated, so the curves in Fig. 6 cannot be independently reproduced; however, this is a completeness/reproducibility failure, not a demonstrated reduction to input. No equation or statement shows that the 10–30 keV range is forced by the fit itself. The self-citations to Kumar & Das 2025a,b are contextual reviews of the non-holomorphic modular framework and are not load-bearing for the present derivation. Thus the overall circularity is partial, not total.

Assumptions & free parameters 6 free parameters · 4 assumptions · 1 invented entities

The central construction rests on modular symmetry and polyharmonic Maaß form literature, on the chosen Type II seesaw dominance, and on many scanned or hand-chosen parameters. The neutrino fit is essentially a scan over tau, the lightest neutrino mass, and b1,b2; the DM analysis additionally needs b3-b10, which are not reported. This ledger shows the paper is not self-contained and its predictions are partly refits of NuFIT input.

free parameters (6)
  • b1, b2 (active neutrino Yukawa couplings) = |b1|, |b2| about 1-8 for NH
    Fitted to NuFIT 3 sigma oscillation parameters in Sec. 4; they set the light neutrino mass matrix M_L.
  • a1, a2, a3 (charged lepton Yukawa couplings) = Not reported
    Fixed by charged lepton masses via Eqs. (18)-(20).
  • b3-b10 (Dirac, RH neutrino and sterile Yukawa couplings) = Not reported
    Enter M_D, M_R and M; required for sterile neutrino mass and mixing in the DM analysis, but values are never given.
  • tau (modulus) = Random scan over fundamental domain, no best-fit value
    Scan point controls all Yukawa values via q-expansions; scatter plots hide the sampling details.
  • lightest neutrino mass m1 (NH) = Scanned 1e-5 to 0.1 eV
    Used to build m_diag; no best fit quoted.
  • VEVs vL, vR, v' = 0.01 eV, 10 TeV, 10 TeV
    Chosen by hand to enforce Type II dominance and keV sterile masses; not derived.
assumptions (4)
  • domain assumption The A4 charge assignments and zero modular weights in Table 2 define the allowed Yukawa contractions.
    Invoked in Secs. 3.1 and 3.2; no derivation of why this assignment is natural.
  • domain assumption Polyharmonic Maaß forms of weight zero and level 3 have the q-expansions (A.7)-(A.9), truncated at q^6.
    Taken from the Qu-Ding literature; truncation error is not quantified.
  • domain assumption Type II seesaw dominance, m_nu = M_L, with hierarchy M_R > M > M_D >> M_L and vL = 0.01 eV.
    Used in Eq. (10); no check that loop corrections or other terms are negligible.
  • domain assumption Sterile neutrino dark matter is produced by the Dodelson-Widrow mechanism, with relic abundance Eq. (3) and decay width Eq. (4), and is subject to Lyman-alpha and X-ray constraints.
    Standard formulas from the cited literature; assumed to apply to this model's sterile neutrinos.
invented entities (1)
  • Three sterile neutrinos S_i, one per generation independent evidence
    purpose: The lightest one is the keV warm dark matter candidate and mixes with active neutrinos.
    The predicted mass 10-30 keV and sin^2 2theta around 1e-11 to 1e-9 are testable with X-ray telescopes; the paper already uses existing X-ray and Lyman-alpha bounds in Fig. 6.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Neutrino phenomenology and Dark matter in a left-right asymmetric model with non-holomorphic modular $A_{4}$ group." pith.science (2026). https://pith.science/paper/6AZ3DTAE

@misc{pith2026250901205,
  author       = {Pith},
  title        = {Pith review of: Neutrino phenomenology and Dark matter in a left-right asymmetric model with non-holomorphic modular $A_4$ group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6AZ3DTAE}},
  note         = {Machine review of arXiv:2509.01205}
}
abstract

We present a model constructed within a non-supersymmetric framework capable of explaining both current neutrino oscillation data and the observed dark matter relic abundance. In this study, the Yukawa couplings are expressed as polyharmonic Maa\ss{} forms, and the non-supersymmetric left-right symmetric model is realized through the $\Gamma_{3}$ modular group, with neutrino masses generated via the Type II seesaw dominance mechanism. The analysis focuses on determining the neutrino oscillation parameters, the effective Majorana mass arising from the standard contribution, and the dark matter relic density. Our results indicate that the model strongly favours the normal mass hierarchy over the inverted one and prefers the lower octant for the mixing angle $\theta_{23}$. Furthermore, the effective Majorana mass is predicted to lie in the range $10^{-3}\,\text{eV}$ to $0.1\,\text{eV}$. In addition, the lightest sterile neutrino present in the model is considered a viable dark matter candidate. A sterile neutrino mass in the range $10~\text{keV}$ to $30~\text{keV}$ is found to yield consistent results for both the relic density and active-sterile mixing angles.

Figures

Figures reproduced from arXiv: 2509.01205 by the authors.

Figure 1
Figure 1. Parameter space of absolute value of b1 and b2 for NH. After determining the free parameters, we calculated the sum of the neutrino masses P mν, the mixing angle θ23, and the CP￾violating phase ∆CP from the model. We observed that the cal￾culated value of P mν lies within the Planck bound (i.e., below 0.12 eV) only for the NH case. For the IH scenario, no data points were found to satisfy the Planck bound [PITH_FUL… view at source ↗
Figure 3
Figure 3. Parameter space of of θ23 and δCP. 0 50 100 150 200 250 300 350 -0.02 -0.01 0.00 0.01 0.02 δCP JCP [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Variation of the Jarlskog invariant parameter with the Dirac CP phase. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: Variation of effective mass with the lightest neutrino mass. We have also explored the dark matter sector of the model, considering only the NH scenario. The model contains sterile neutrinos, which can serve as potential dark matter candidates. As discussed in Drewes e…
Figure 6
Figure 6. Figure 6: Variation of the active–sterile mixing angle, relic density, and decay [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Neutrino mass and leptogenesis in the non-SUSY modular $A^\prime_5$ inverse seesaw model

    hep-ph 2026-03 conditional novelty 6.0 of 10

    Three non-SUSY A5-prime modular inverse-seesaw models fit neutrino oscillation data and can generate the observed baryon asymmetry via TeV-scale resonant leptogenesis.

Reference graph

Works this paper leans on

15 extracted references · 12 canonical work pages · cited by 1 Pith paper

  1. [1]

    P. A. R. Ade et al. Planck 2015 results. XIII. Cosmological parameters.Astron. Astrophys., 594:A13,

    work page 2015

  2. [9]

    doi: 10.1007/JHEP12(2024)189. A. Merle. keV sterile neutrino Dark Matter. PoS, NOW2016:082,

    doi:10.1007/jhep12(2024)189 2024

  3. [10]

    doi: 10.22323/1.283.0082. T. Nomura and H. Okada. Type-II seesaw of a non-holomorphic modular A4 symmetry. Phys. Lett. B , 868:139763, 2025a. doi: 10.1016/j.physletb.2025.139763. T. Nomura and H. Okada. Zee model in a non-holomorphic mod- ular A4 symmetry. Phys. Lett. B , 867:139618, 2025b. doi: 10.1016/j.physletb.2025.139618. T. Nomura, H. Okada, and O. ...

    work page arXiv 2025

  4. [11]

    doi: 10.1016/j.physletb.2024.139171. H. Okada and Y . Orikasa. A radiative seesaw in a non-holomorphic modular S 3 flavor symmetry. 1

    work page arXiv 2024

  5. [12]

    doi: 10.1016 /j.physletb.2012.12.036. P. Pritimita, N. Dash, and S. Patra. Neutrinoless Double Beta Decay in LRSM with Natural Type-II seesaw Dominance. JHEP, 10:147,

    work page 2012

  6. [14]

    doi: 10.1007 /JHEP08(2024)136. B.-Y . Qu, J.-N. Lu, and G.-J. Ding. Non-holomorphic modular flavor symmetry and odd weight polyharmonic Maaß form. 6

    work page 2024

  7. [2012]

    doi: 10.1016/j.nuclphysb.2012.01.017. F. J. de Anda and S. F. King. Modular flavour symmetry and orbifolds. JHEP, 06:122,

    doi:10.1016/j.nuclphysb.2012.01.017 2012

  8. [2013]

    doi: 10.1007 /JHEP08(2013)122. F. Bezrukov, H. Hettmansperger, and M. Lindner. keV sterile neutrino Dark Matter in gauge extensions of the Standard Model.Phys. Rev. D, 81:085032,

    work page 2013

Show all 15 references
  1. [2016]

    doi: 10.1007/JHEP10(2016)147. B.-Y . Qu and G.-J. Ding. Non-holomorphic modular flavor symmetry. JHEP, 08:136,

    2016 doi

  2. [2017]

    doi: 10.1088 /1475-7516/2017/01/025. I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, J. P. Pin- heiro, and T. Schwetz. NuFit-6.0: updated global analysis of three-flavor neutrino oscillations. JHEP, 12:216,

    2017

  3. [2020]

    doi: 10.1016 /j.nuclphysb.2020.115000. 9

    2020

  4. [2021]

    doi: 10.1007 /JHEP01(2021)037. G.-J. Ding, J.-N. Lu, S. T. Petcov, and B.-Y . Qu. Non-holomorphic modular S 4 lepton flavour models. JHEP, 01:191,

    2021

  5. [2023]

    doi: 10.1007 /JHEP06(2023)122. G.-J. Ding, F. Feruglio, and X.-G. Liu. Automorphic Forms and Fermion Masses. JHEP, 01:037,

    2023

  6. [2024]

    doi: 10.1007/JHEP12(2024)216. 8 S. Ferrara, . D. Lust, and S. Theisen. Target Space Modular Invariance and Low-Energy Couplings in Orbifold Compactifications. Phys. Lett. B , 233: 147–152,

    2024 doi

  7. [2025]

    doi: 10.1007/JHEP01(2025)191. S. Dodelson and L. M. Widrow. Sterile-neutrinos as dark matter. Phys. Rev. Lett., 72:17–20,

    2025 doi

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.