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REVIEW 3 major objections 5 minor 62 references

Ultradense Dark Matter Halos with Poisson Noise from Stellar-Mass Primordial Black Holes

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper argues that the discrete, Poisson-distributed positions of stellar-mass primordial black holes add an isocurvature term to the matter power spectrum that strongly boosts the formation of ultradense dark matter halos, with the…

desk verdict A clean parameter scan confirming known Poisson-noise effects on UDMHs, but the multi-component dark matter claim rests on comparing UDMH mass fractions to PBH abundance bounds. read the letter →

arxiv 2502.00914 v2 pith:MASZBGGQ submitted 2025-02-02 astro-ph.CO gr-qc

Saeed Fakhry , Javad T. Firouzjaee This is my paper · ORCID
classification astro-ph.COgr-qc
keywords primordialblackholesPoissonnoiseultradensedarkmatterhalospowerspectrumhalomassfunctionsmall-scalestructureisocurvatureperturbations
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 establish that the discrete, random spatial distribution of stellar-mass primordial black holes—their Poisson noise—leaves a measurable imprint on the formation of ultradense dark matter halos in the radiation-dominated era. It constructs a modified matter power spectrum that adds an isocurvature term from PBH shot noise to the usual adiabatic spectrum, and shows that this term strongly enhances small-scale density fluctuations, especially for more massive PBHs and higher PBH fractions. The authors find that the resulting UDMH mass function shifts toward higher masses as the PBH mass grows, and that the steepness of the small-scale cutoff, parameterized by $n$, controls whether the predictions favor multi-component dark matter (low $n$) or a single-component stellar-mass-PBH dark matter (high $n$). A sympathetic reader would care because these ultradense structures could be observable probes of PBH dark matter that are distinct from the usual continuous-fluid approximation.

What carries the argument

The central object is the modified linear matter power spectrum $P(k)=P_{\mathrm{ad}}(k)+P_{\mathrm{iso}}(k)$ of Eq. (23), where $P_{\mathrm{ad}}$ is the nearly scale-invariant adiabatic spectrum and $P_{\mathrm{iso}}$ is the PBH Poisson-noise (isocurvature) contribution with amplitude proportional to $f_{\mathrm{PBH}}M_{\mathrm{PBH}}$ and an exponential cutoff controlled by the parameter $n$. This spectrum converts into halo abundances through the excursion-set peak-height variable $\nu=\delta_c/\sigma(M)$ and the multiplicity functions PS, ST, DP1, DP2; the DP-family barriers include angular momentum and dynamical friction, which raise the predicted UDMH abundance.

What would settle it

A measurement of the small-scale matter power spectrum at $k\sim10^2$–$10^4\,\mathrm{Mpc}^{-1}$ (for example from Lyman-$\alpha$ forest or 21-cm observations) that shows no isocurvature bump of the predicted height, or a high-resolution N-body simulation of stellar-mass PBH shot noise whose UDMH mass function does not shift toward higher masses with increasing $M_{\mathrm{PBH}}$, would settle against the central claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that the shot noise from a discrete population of stellar-mass PBHs cannot be ignored: it contributes an isocurvature term $P_{\mathrm{iso}}(k) = Q(k)\exp[-(k-k')^n/(2\sigma^n)]$ to the matter power spectrum, with amplitude $A_{\mathrm{iso}} = 3.2\times10^{-12} f_{\mathrm{PBH}}(M_{\mathrm{PBH}}/30M_\odot)$ at the pivot scale. This Poisson-noise term enhances small-scale power by many orders of magnitude, and when fed through excursion-set halo mass functions it shifts the differential mass function of UDMHs toward higher masses as $M_{\mathrm{PBH}}$ increases. The steepness index $n$ of the exponential cutoff controls how much of this enhancement survives: low $n$ leaves a broad small-scale boost that favors multi-component dark matter, while high $n$ suppresses low-mass UDMHs and brings predictions into line with a single-component PBH dark matter scenario. The paper also claims that mass functions including angular momentum and dynamical friction (DP1 and DP2) yield more UDMHs than the Press-Schechter form.

Load-bearing premise

The load-bearing premise is that the Poisson-noise contribution from PBHs is accurately described by an exponential cutoff with the chosen location, width, and steepness ($k'\simeq10^2\,\mathrm{Mpc}^{-1}$, $\sigma\simeq4.5\times10^2\,\mathrm{Mpc}^{-1}$, free index $n$); if the true small-scale damping or PBH clustering differs, the predicted UDMH abundances and the favored dark matter composition would change.

Editorial extensions

If this is right

  • For fixed $f_{\mathrm{PBH}}$, increasing $M_{\mathrm{PBH}}$ from $1\,M_\odot$ to $100\,M_\odot$ shifts the differential UDMH mass function to higher masses because fewer, heavier PBHs produce stronger shot noise.
  • Lower values of the suppression index $n$ leave more small-scale power and boost the UDMH abundance, a regime in which the predicted halo counts exceed PBH observational bounds and favor multi-component dark matter.
  • Higher $n$ values damp low-mass UDMH formation and bring the predicted distributions into closer agreement with OGLE, LVK, and ultra-faint-dwarf constraints, supporting a single-component PBH dark matter scenario for lighter PBHs.
  • The DP1 and DP2 mass functions, which include angular momentum and dynamical friction, consistently produce higher UDMH abundances than the Press-Schechter form, especially in the high-mass tail.
  • The modified power spectrum's small-scale amplitude is enhanced by at least seven orders of magnitude relative to large scales, making the Poisson-noise effect potentially accessible to small-scale structure probes.

Reading between the lines

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

  • Because the exponential cutoff parameters ($k'$, $\sigma$, $n$) are phenomenological, the same analytic machinery could be re-run with a physically motivated damping spectrum to see whether the multi-component versus single-component preference survives.
  • The Poisson-noise contribution is generic to any discrete compact-object dark matter component, so the predicted UDMH boost should apply to other massive compact halo objects, not only to PBHs.
  • If UDMHs are as abundant as claimed at low $n$, their annihilation or lensing signatures could be searched for in gamma-ray and microlensing data even when the PBH fraction is too small to be seen directly.
  • The degeneracy among $n$, $M_{\mathrm{PBH}}$, and $f_{\mathrm{PBH}}$ means that observational upper limits on UDMH abundance translate into joint constraints, not independent bounds on the PBH fraction.
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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

3 major / 5 minor

Summary. The paper studies the effect of Poisson noise from stellar-mass primordial black holes (PBHs) on the formation of ultradense dark matter halos (UDMHs) during the radiation-dominated era. It constructs a modified power spectrum that adds a PBH-induced isocurvature term with an exponential cutoff to the adiabatic CDM power spectrum, and then uses excursion-set theory with several mass-function prescriptions (PS, ST, DP1, DP2) to compute differential mass fractions of UDMHs. The authors vary the PBH mass, the PBH fraction, and a suppression parameter n, and compare the resulting mass functions with observational upper limits on PBH abundance from OGLE, LVK, and ultrafaint dwarf disruption. The central claims are that heavier PBHs shift UDMH mass functions to higher masses, lower n boosts UDMH abundance and favors multi-component dark matter, higher n aligns with single-component dark matter, and the DP1/DP2 mass functions predict higher abundances than Press-Schechter.

Significance. If the calculation were robust, the paper would be a useful exploration of an interesting and understudied effect: the discreteness of PBHs can strongly affect small-scale density fluctuations and thus the formation of compact dark matter structures. The paper is clearly written in its main equations and covers a reasonable parameter space, and the use of several halo mass functions is a helpful comparison. However, the advertised observational conclusions are currently not supported by the comparison made in Figs. 2-4, and the quantitative results are governed by an empirically unspecified cutoff in the isocurvature power spectrum. The paper would be strengthened by re-expressing the comparison in terms of a single constrained quantity and by treating the suppression parameter as a free knob in a conditional model exploration rather than as the basis for claims about which dark matter scenario is favored.

major comments (3)
  1. [Section 3, Figs. 2-4] The central observational inference is based on comparing the differential mass fraction of UDMHs, df/dlogM, with shaded upper limits on the PBH abundance f_PBH from OGLE, LVK, and UFD disruption. These are different quantities: the shaded regions constrain the fraction of dark matter in stellar-mass black holes, while the curves describe the mass fraction in extended ultradense halos that can be much more massive than the seeding PBH (the plotted curves extend to ~10^6 M_sun for M_PBH = 1-100 M_sun). A single PBH can therefore produce a UDMH whose mass fraction lies above a PBH exclusion region while f_PBH itself is far below the bound. Consequently, the statement in Section 3 that the curves 'clearly violate' regions excluded by PBH constraints and thereby 'lend compelling support to a multi-component dark matter scenario' is unsupported as written. The same issue underlies the conclusion that low n favors multi-component and high n favors single-component dark matter. The comparison should be recast in terms of one constrained quantity, for example by converting the predicted UDMH abundances into the relevant observational observable or by explicitly stating that the curves are model predictions not directly bounded by PBH abundance limits.
  2. [Section 2.2, Eqs. (25)-(27)] The quantitative results are governed by the assumed exponential cutoff P_iso(k) = Q(k) exp(-[(k-k')/sigma]^n / 2) with k' ~ 10^2 Mpc^-1, sigma ~ 4.5 x 10^2 Mpc^-1, and free index n, together with the adopted amplitude A_iso = 3.2 x 10^-12 f_PBH (M_PBH/30 M_sun). The paper itself acknowledges that the microphysical origin of this suppression is model-dependent. Because P_iso is the dominant small-scale contribution, the conclusions that lower n boosts UDMH formation and that n = 2 or 3 brings the results into agreement with single-component dark matter are direct consequences of the chosen shape and parameters rather than robust predictions. The authors should either derive these parameters from a microphysical model, constrain them with independent data, show a sensitivity analysis over k' and sigma, or clearly frame the results as a conditional model exploration. Without such an addition, statements in the abstract and conclusions about favoring multi- versus single-component dark matter are overreaching.
  3. [Section 2.2 and Fig. 1] There is a notational inconsistency that affects reproducibility: Eq. (23) defines P(k) as the power spectrum of primordial curvature perturbations, while Fig. 1 is labeled as the modified matter power spectrum and Eq. (7) requires the matter power spectrum P(k,a). The connection is made through Eq. (9), but the text and figure should use consistent notation so that a reader can reproduce sigma^2(M,a) without guessing whether Eq. (23) is P_zeta(k) or P_matter(k).
minor comments (5)
  1. [Fig. 1 caption] The caption lists f_PBH = 1, 0.1, 0.01, 0.0001, and 0.0001; the last value is presumably 10^-5 and should be corrected.
  2. [Fig. 4] The lower-left panel appears to have a duplicated axis label 'MPBH=10M'; one of the two labels should be removed.
  3. [Figs. 2-4 captions and text] The shading colors are described inconsistently: the main text refers to blue, red, and green shaded regions for OGLE, LVK, and UFD, while the captions list orange, wheat, and cyan with different hatching.
  4. [Figs. 2-4 captions] The LVK citation in the figures is malformed as '?R. e. a. Abbott et al. 2022' and should be fixed to a standard author-year reference.
  5. [Eq. (12)] Equation (12) appears to have a typographical error in its prefactor: as printed it reads as a product of terms rather than a normalized first-crossing distribution with a denominator.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is a parameter-sensitivity study whose main dependencies are explicit consequences of the assumed isocurvature power spectrum; self-citations are contextual, not load-bearing.

full rationale

The paper's central calculation is a forward model: it assumes a modified matter power spectrum P(k)=Pad(k)+Piso(k) with a phenomenological isocurvature component (Eqs. 23-27), then evaluates excursion-set halo mass functions (PS, ST, DP1, DP2) to obtain differential UDMH mass fractions. The dependencies highlighted in the abstract and conclusions—stronger small-scale power for larger MPBH or fPBH, and stronger suppression for larger n—are direct numerical consequences of the assumed Piso(k) and Aiso scaling. This is model exploration, not a derivation of those scalings from independent principles, and the paper explicitly acknowledges the cutoff is phenomenological and model-dependent: 'While the exact microphysical origin of this suppression remains model-dependent, a phenomenological exponential cutoff has been widely adopted in the literature.' No fitted parameter is relabeled as a prediction, and no equation is defined in terms of the quantity it is said to predict. The DP1/DP2 versus PS ordering comes from adopting known mass-function fitting formulas from Del Popolo (2006) and Del Popolo et al. (2017), which are external to this paper and not justified by self-citation; the comparison is a numerical evaluation of those formulas, not a circular claim. The many citations to the authors' prior work appear in the introduction as contextual references to existing PBH/U DMH studies and are not load-bearing for the derivation. A separate, non-circular concern is that Figs. 2-4 compare the UDMH differential mass fraction df/dlogM with observational upper limits on the PBH abundance fPBH; these are different quantities, so the inference that exceeding the shaded regions implies a multi-component dark matter scenario is logically unsupported. That is a correctness/interpretation issue, not a circularity, and therefore does not raise the circularity score. Overall, the derivation chain is self-contained as a model study against external inputs, with no circular reduction identified.

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

The calculation rests almost entirely on assumed inputs: a Poisson-noise amplitude from prior literature, a hand-chosen exponential cutoff, and mass functions calibrated elsewhere. The central prediction that lower n boosts UDMH formation is an algebraic consequence of the assumed cutoff, not an independent result.

free parameters (4)
  • Suppression parameter n = 1, 2, 3
    Controls the steepness of the exponential cutoff in Piso(k), Eq. (25); no physical derivation is given, and the paper's main conclusions about UDMH abundance and dark matter composition depend on it.
  • Cutoff wavenumber k' = ~10^2 Mpc^-1
    Position of the exponential suppression in Eq. (25); chosen by hand and sets the mass scale near 10^6 solar masses where the UDMH abundance peaks.
  • Cutoff width sigma = ~4.5 x 10^2 Mpc^-1
    Characteristic width of the suppression in Eq. (25); chosen by hand and directly shapes the small-scale enhancement.
  • Isocurvature amplitude Aiso = 3.2e-12 fPBH (M_PBH/30 M_sun)
    Sets the overall strength of the Poisson noise term in Eq. (27); taken from Gong & Kitajima 2017, not derived in this paper, and directly controls the small-scale boost.
assumptions (5)
  • domain assumption PBHs are randomly distributed with no correlations on scales larger than the Hubble horizon, giving pure Poisson shot noise with power spectrum n_PBH^-1.
    Invoked in Section 2.2, Eq. (22), citing Afshordi et al. 2003; if PBHs cluster, the isocurvature component is not Poisson.
  • domain assumption The excursion set formalism with a moving barrier B(S)=3(1+sqrt(S/5)) accurately predicts the abundance of UDMHs from linear density peaks.
    Used in Section 2.1, Eqs. (6)-(13), following Bond et al. 1991 and Delos & Silk 2023; this is a modeling assumption, not a first-principles derivation.
  • ad hoc to paper The exponential suppression of the isocurvature power spectrum, Eq. (25), with parameters k', sigma, and n is a valid description of small-scale damping.
    Introduced in Section 2.2 as a phenomenological cutoff; no microphysical derivation is given, and the paper's central conclusions are sensitive to it.
  • domain assumption Collapse along the smallest principal axis at density contrast delta_c = 3/(1-3e+p), with typical ellipticity e ~ sigma/sqrt(5 delta) and prolateness 0, determines halo formation during the radiation era.
    Used in Section 2, Eqs. (3)-(5), based on the Zeldovich approximation and Sheth et al. 2001; the mass functions inherit this ellipsoidal collapse threshold.
  • domain assumption PBHs do not accrete significantly after formation, so their mass is fixed at M_PBH.
    Stated in Section 2.2 before Eq. (21); accretion or mass growth would alter the number density and hence the shot-noise amplitude.

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Pith. "Pith review of Ultradense Dark Matter Halos with Poisson Noise from Stellar-Mass Primordial Black Holes." pith.science (2026). https://pith.science/paper/MASZBGGQ

@misc{pith2026250200914,
  author       = {Pith},
  title        = {Pith review of: Ultradense Dark Matter Halos with Poisson Noise from Stellar-Mass Primordial Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MASZBGGQ}},
  note         = {Machine review of arXiv:2502.00914}
}
abstract

In this work, we investigate the impact of Poisson noise from stellar-mass primordial black holes (PBHs) on the formation of ultradense dark matter halos (UDMHs). Our findings reveal that the discrete spatial distribution of PBHs significantly enhances small-scale density fluctuations, particularly for massive stellar-mass PBHs. Our results indicate that the modified power spectrum, incorporating both adiabatic and isocurvature contributions from PBH-induced Poisson noise, strongly depends on PBH mass and fraction. Specifically, increasing PBH mass shifts the differential mass function of UDMHs toward higher masses, while variations in the suppression parameter $n$ modulate the efficiency of UDMH formation at small scales. For lower values of $n$, our findings show a significant boost in UDMH abundance, favoring multi-component dark matter scenarios. Conversely, at higher values of $n$, the predicted UDMH distributions align more closely with single-component models dominated by stellar-mass PBHs. Furthermore, our analysis demonstrates that more realistic halo mass functions, which account for angular momentum and dynamical friction, consistently predict higher UDMH abundances compared to traditional Press-Schechter formalism.

Figures

Figures reproduced from arXiv: 2502.00914 by the authors.

Figure 1
Figure 1. Modified matter power spectrum as a function of wavenumber k, described by Eq. (23), while incorporating the impact of Poisson noise from stellar-mass PBHs in the mass range of MPBH = 1, 10 and 100M⊙, for various values of the fraction fPBH. Also, the suppression parameter is considered to be n = 1, 2, and 3. The shaded gray region indicates the range where the amplitude of the power spectrum on small scales is enha… view at source ↗
Figure 2
Figure 2. The differential mass fraction of dark matter in UDMHs is expressed as a function of mass M, incorporating the modified power spectrum derived in Eq. (23). This analysis accounts for Poisson noise contributions from PBHs with masses MPBH = 1, 10, and 100M⊙, while considering multiple values of the PBH fraction fPBH, and assumes a spectral index n = 1. The results for different mass functions are illustrated using va… view at source ↗
Figure 3
Figure 3. Similar to [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Similar to [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

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