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Constraining primordial black holes and primordial curvature power spectrum with extragalactic muon neutrino

T0 review · 2 major / 5 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Neutrino flux from WIMP annihilation around primordial black holes caps their dark-matter fraction at a few times 10^{-5} and the small-scale curvature power at about 10^{-1.65}.

desk verdict Clean multi-channel extension of UCMH-neutrino limits that also maps to P_R; useful complementary numbers, not a paradigm shift. read the letter →

arxiv 2607.10524 v1 pith:UVYKXAB6 submitted 2026-07-12 astro-ph.CO

Yupeng Yang This is my paper · ORCID
classification astro-ph.CO
keywords primordialblackholesWIMPsultracompactminihalosextragalacticneutrinosIceCubecurvaturepowerspectrummixeddarkmatter
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 asks how much of the dark matter can be primordial black holes if the rest is ordinary WIMPs that pile up around them. After each black hole forms, WIMPs accrete into dense ultracompact minihalos whose annihilation rate is far higher than in ordinary galactic halos. The resulting extragalactic muon-neutrino flux is calculated for three annihilation channels and two IceCube event classes (contained and upward-going). Requiring that flux not exceed the measured atmospheric neutrino background for a one-year exposure yields upper limits on the PBH fraction. The tightest bound is f_PBH ~ 4 imes10^{-5} for 1 TeV WIMPs annihilating directly to muon neutrinos around 10^3 solar-mass black holes. That abundance limit is then converted, via the standard Press-Schechter mapping, into an upper bound on the primordial curvature power spectrum of order 10^{-1.65} at wave-numbers around 3 imes10^{12} Mpc^{-1}. The result supplies an independent neutrino-based check on small-scale primordial fluctuations that is complementary to gamma-ray and CMB constraints.

What carries the argument

The piecewise WIMP density profile inside each ultracompact minihalo (inner r^{-3/4}, intermediate r^{-3/2}, outer r^{-9/4}, capped by the annihilation density ho_max). The square of this profile enters the neutrino luminosity, so the entire f_PBH and P_R limits scale directly with the assumed cusp strength.

What would settle it

A one-year IceCube analysis that isolates the high-energy muon sample and finds an excess (or a tighter null result) above atmospheric background in the energy window set by a 1 TeV WIMP annihilating to u_ u¯ would directly confirm or rule out the quoted f_PBH ~ 4 imes10^{-5} bound.

Watch

Extended reading notes

Core claim

For a mixed WIMP-plus-PBH dark-matter cosmology, the extragalactic muon-neutrino flux produced by WIMP annihilation inside the ultracompact minihalos that form around PBHs cannot exceed the atmospheric neutrino background. The strongest one-year IceCube limit obtained from this requirement is f_PBH ~ 4 imes10^{-5} (upward events, u_ u¯ channel, m_ u = 10^3 GeV, M_PBH = 10^3 M_ u). Mapping that abundance limit through the Press-Schechter formalism yields P_R ~ 10^{-1.65} at k ~ 3 imes10^{12} Mpc^{-1}.

Load-bearing premise

The calculation treats the multi-power-law density profile of WIMPs around every primordial black hole, fixed by kinetic-decoupling temperature and a hard annihilation cap, as exact; any softening of that cusp would weaken the neutrino signal and the derived limits.

Editorial extensions

If this is right

  • For PBH masses above roughly 10^{-11} M_ u the neutrino limits become nearly mass-independent, so the same f_PBH ceiling applies across a wide intermediate-mass window.
  • The corresponding P_R bound is stronger than pure-PBH limits over 10^7 ≲ k ≲ 10^{13} Mpc^{-1}, tightening the allowed amplitude of small-scale primordial fluctuations.
  • Direct annihilation to muon neutrinos yields the strongest constraint; other channels (μ^+μ^-, τ^+τ^-) give limits weaker by factors of a few to ten.
  • Upward-going events generally out-perform contained events once the WIMP mass is high enough for long muon tracks, reversing the hierarchy seen at lower mass.

Reading between the lines

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

  • If future multi-year IceCube or KM3NeT exposures improve the high-energy atmospheric background subtraction by even a factor of a few, the same minihalo calculation would push f_PBH into the 10^{-6} range and P_R correspondingly lower.
  • Because the neutrino limits are still four orders of magnitude weaker than isotropic gamma-ray bounds on the same minihalos, a joint gamma-plus-neutrino analysis could test whether the density-profile assumptions are consistent across messengers.
  • The same UCMH luminosity that produces the neutrino flux also sources high-energy electrons and positrons; a parallel AMS-02 or future space-based positron bound would provide an independent cross-check of the annihilation rate 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.

Referee Report

2 major / 5 minor

Summary. The paper studies a mixed WIMP+PBH dark-matter scenario in which WIMPs accrete onto PBHs to form UCMHs with the piecewise density profile of Eq. (1) (capped by the annihilation density of Eq. (5)). It computes the extragalactic muon-neutrino flux from WIMP annihilation (Eq. (7)) for the channels μ⁺μ⁻, τ⁺τ⁻ and νμν̄μ, converts the flux into contained and upward muon events via the standard formulae (8)–(11), and obtains 2σ upper limits on f_PBH by requiring that the signal not exceed the atmospheric-neutrino background for a one-year IceCube exposure (Eq. (14)). The strongest limits are f_PBH ∼ 10^{-4} (4 imes10^{-5}) for contained (upward) events at m_χ = 10^3 GeV and M_PBH = 10^3 M_⊙ in the νμν̄μ channel. These bounds are then mapped, via the Gaussian Press–Schechter relation, onto upper limits on the primordial curvature power spectrum, reaching P_R ∼ 10^{-1.65} at k ∼ 3 imes10^{12} Mpc^{-1}.

Significance. If the adopted UCMH density profile is accepted, the work supplies a clean, complementary neutrino-based bound on mixed WIMP+PBH dark matter that extends previous μ⁺μ⁻-only analyses to three channels and a wider PBH mass range, and that improves existing P_R limits over 10^7 ≲ k ≲ 10^{13} Mpc^{-1}. The calculation is fully standard once the profile is fixed, the comparison with an independently measured atmospheric background is free of circularity, and the results are directly falsifiable with IceCube data. The explicit conversion of the f_PBH limits into P_R constraints further increases the paper’s utility for early-universe cosmology.

major comments (2)
  1. The entire set of f_PBH and P_R limits rests on the piecewise density profile of Eq. (1) (with transition radii fixed by the kinetic-decoupling parameters of Eqs. (2)–(4) and the hard annihilation cap of Eq. (5)). While this profile is taken from the literature, no quantitative assessment is given of how the annihilation luminosity (and therefore the quoted limits) changes if the inner cusp is softened, if T_KD is varied within its theoretical uncertainty, or if a different ho_max prescription is used. A short sensitivity study or an explicit statement of the scaling of the luminosity with these parameters is needed before the strongest numerical claims can be regarded as robust.
  2. Section 3.2 adopts energy-independent effective volume V_eff = 0.04 km^{3} and area A_eff = 1 km^{2} for IceCube. Because the muon spectrum hardens with m_χ and the atmospheric background falls steeply, an energy-dependent acceptance would shift the relative strength of the contained versus upward limits and could alter the quoted best-case numbers by a non-negligible factor. Either a justification that the constant approximation is adequate at the energies of interest or a recalculation with published IceCube effective areas is required.
minor comments (5)
  1. The abstract and the final paragraph of Sec. 3.2 both state that the strongest limits come from the νμν̄μ channel, yet Fig. 2 shows that for m_χ = 10^{2} GeV the contained-event limits from μ⁺μ⁻ are competitive; a brief clarifying sentence would avoid confusion.
  2. The assumption of a monochromatic PBH mass function is never stated explicitly; a short remark in Sec. 3 or 4 would make the scope of the P_R bounds clearer.
  3. Several typographical issues appear (missing spaces after commas, inconsistent use of “WIMPs” vs “WIMP”, and the repeated reference “[17, 17, 18]”). A careful proof-reading pass is recommended.
  4. Figure 1 caption should specify that the curves assume f_PBH = 1; the body text does so, but the caption does not.
  5. The 1:1:1 flavor ratio after oscillation is adopted without citation or discussion of possible deviations for the direct νμν̄μ channel; a one-sentence reference would suffice.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: f_PBH limits are standard flux-vs-external-ATM upper bounds; P_R follows from literature Press-Schechter conversion.

full rationale

The derivation chain is self-contained and non-circular. The neutrino flux (Eq. 7) is linear in f_PBH; the muon rates (Eqs. 8, 10) and event counts (Eq. 13) are computed from that flux using fixed literature inputs (DarkSUSY spectra, canonical ⟨σv⟩, energy-independent V_eff/A_eff, 1:1:1 flavor ratio). Upper limits on f_PBH are then obtained by the ordinary statistical requirement that the predicted signal not exceed the independently measured atmospheric background (Eq. 12) via the ζ statistic (Eq. 14). The subsequent conversion to P_R uses the standard Gaussian Press-Schechter formulae (Eqs. 15–19) with literature values of δ_c; no parameter is fitted to the same data that is later “predicted.” Self-citations (e.g., to the author’s earlier muon-neutrino paper) supply only the prior context that is being extended; they are not load-bearing uniqueness theorems or ansatzes that force the present numerical results. The piecewise UCMH density profile (Eq. 1) is an external modeling assumption taken from the literature, not a circular definition. Consequently the claimed bounds do not reduce to their inputs by construction.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central numerical claims rest on a chain of standard cosmological and particle-physics assumptions plus a handful of fixed experimental and theoretical parameters taken from the literature; no new free functions are fitted to data, but several conventional choices (〈σv〉, δ_c, effective volume) directly set the quoted numbers.

free parameters (5)
  • 〈σv〉 = 3×10^{-26} cm^{3} s^{-1}
    Canonical thermal value 3×10^{-26} cm^{3} s^{-1} fixes ρ_max and therefore the annihilation luminosity of every UCMH.
  • V_eff / A_eff = 0.04 km^{3} / 1 km^{2}
    Energy-independent IceCube effective volume 0.04 km^{3} and area 1 km^{2} convert differential fluxes into event counts.
  • E_th^μ = 50 GeV
    Muon energy threshold of 50 GeV enters the range integral R(E_μ) for upward events.
  • δ_c = 0.42 (fiducial)
    Critical density contrast for PBH formation (0.42 / 0.66 / 1/3) controls the conversion from f_PBH to P_R.
  • γ, g_*i = 0.2, ≈100
    Collapse fraction γ=0.2 and relativistic degrees of freedom g_*i≈100 fix the β–f_PBH relation (Eq. 19).
assumptions (4)
  • domain assumption WIMP density profile inside a UCMH follows the three-segment power law of Eq. (1) with transitions fixed by kinetic decoupling.
    Taken from Eroshenko, Boucenna et al. and subsequent works; any deviation in the inner cusp directly rescales the annihilation rate.
  • domain assumption Primordial density perturbations are Gaussian and the Press-Schechter formalism with a top-hat window applies.
    Used without modification to convert β(M) into P_R (Eqs. 15–18).
  • domain assumption Neutrino flavor ratio at Earth is exactly 1:1:1 and atmospheric neutrinos are the sole background.
    Simplifies the conversion from source spectrum to detected muon flux; stated in Sec. 3.1.
  • standard math Standard ΛCDM expansion history and radiation-matter equality density ρ_eq.
    Enters the redshift integral of the extragalactic flux (Eq. 7).

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Pith. "Pith review of Constraining primordial black holes and primordial curvature power spectrum with extragalactic muon neutrino." pith.science (2026). https://pith.science/paper/UVYKXAB6

@misc{pith2026260710524,
  author       = {Pith},
  title        = {Pith review of: Constraining primordial black holes and primordial curvature power spectrum with extragalactic muon neutrino},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UVYKXAB6}},
  note         = {Machine review of arXiv:2607.10524}
}
abstract

We investigate a mixed dark matter scenario comprising weakly interacting massive particles (WIMPs) and primordial black holes (PBHs). After PBH formation, WIMPs can accrete onto them, forming ultracompact minihalos (UCMHs). The resulting WIMP number density within UCMHs is significantly enhanced compared to classical dark matter halo models, leading to a higher WIMP annihilation rate. Previous studies have focused mainly on the associated gamma-ray flux, we investigate the extragalactic neutrino flux from such annihilation. Considering the annihilation channels $\mu^{+}\mu^{-}$, $\tau^{+}\tau^{-}$, and $\nu_{\mu}\bar{\nu}_{\mu}$, we analyze two classes of neutrino events: upward and contained events. By requiring the neutrino flux from WIMP annihilation around PBHs does not exceed the atmospheric neutrino background, we derive upper limits on the fraction of dark matter in PBHs ($f_{\rm PBH}$) for a one-year exposure of the IceCube experiment. These limits depend on the annihilation channel, the masses of the WIMP and PBH, and the neutrino event type. The strongest constraints come from the $\nu_{\mu}\bar{\nu}_{\mu}$ channel, yielding $f_{\rm PBH} \sim 10^{-4}$ ($4\times 10^{-5}$) for contained (upward) events with $m_{\chi}=10^{3}$ GeV and $M_{\rm PBH}=10^{3} M_{\odot}$. Based on these bounds on PBHs, we further derive upper limits on the primordial curvature power spectrum $\mathcal{P}_{\mathcal{R}}$. From our strongest constraint, we obtain $\mathcal{P}_{\mathcal{R}} \sim 10^{-1.65}$ at the scale $k\sim 3\times 10^{12}~\mathrm{Mpc^{-1}}$.

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Forward citations

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