Closing the Mass Window for Stupendously Large Black Holes

Antonio J. Iovino; Christopher Gerlach; Nicholas Leister; Yann Gouttenoire

REVIEW 2 major objections 3 minor 1 cited by

Closing the Mass Window for Stupendously Large Black Holes

T0 review · 2 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Stupendously large black holes are excluded as a significant dark matter component because their isocurvature perturbations exceed CMB limits.

desk verdict Robust exclusion of SLABs as dark matter via Poisson isocurvature CMB bounds; residual technical quirks affect boundary details, not the headline. read the letter →

arxiv 2508.08238 v1 pith:PX53RJ7F submitted 2025-08-11 astro-ph.CO gr-qchep-ph

Christopher Gerlach , Yann Gouttenoire , Antonio J. Iovino , Nicholas Leister This is my paper · ORCID

classification astro-ph.COgr-qchep-ph PACS 98.80.-k95.35.+d

keywords primordialblackholesstupendouslylargeSLABisocurvatureperturbationscosmicmicrowavebackgrounddarkmatterPoissonshotnoisebaryonacousticoscillations

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

Primordial black holes with masses $M \gtrsim 10^{11}\,M_\odot$ — the 'stupendously large' range — are too massive to reside within galaxies and so cannot make up all of the dark matter, yet earlier work suggested they could still be a significant fraction. This paper shows that such black holes cannot be significant: their rareness and discreteness generate an isocurvature component in the dark-matter density field whose power spectrum, scaled by $f_{\mathrm{PBH}}^2$, exceeds the 95% CMB+BAO upper limits across $M_{\mathrm{PBH}} \in [10^{8}, 10^{19}]\,M_\odot$ unless $f_{\mathrm{PBH}}$ is very small. The resulting exclusion curve, shown in pink in Fig. 2, closes the mass window that earlier work had left open. A sympathetic reader would take the paper's central claim to be that the CMB has already settled this part of the dark-matter question.

What carries the argument

The carrying object is the Poisson (shot-noise) isocurvature power spectrum of the discrete PBH population. For randomly placed black holes, the formation-epoch density contrast obeys $\langle(\delta_{\mathrm{iso}}^f)^2\rangle = \frac{2}{3\pi}\left(\frac{k}{k_{\mathrm{PBH}}}\right)^3 \frac{3j_1(k/k_f)}{k/k_f}$, where $k_{\mathrm{PBH}}$ is the inverse comoving interspacing of the black holes and the spherical Bessel factor $j_1$ imposes a cutoff at the comoving PBH radius at formation. This spectrum, multiplied by $f_{\mathrm{PBH}}^2$ and evolved with the standard isocurvature transfer function $T_{\mathrm{iso}}(a)$, is the quantity bounded by the CMB and BAO data.

What would settle it

Recompute the excluded region using $\langle I_{\mathrm{DM}}^2\rangle(a_{\mathrm{rec}}) = T_{\mathrm{iso}}^2(a_{\mathrm{rec}}) f_{\mathrm{PBH}}^2 \langle(\delta_{\mathrm{iso}}^f)^2\rangle$ as the quantity compared with the 95% CL limits from Ref. [26]; if the curve for $M_{\mathrm{PBH}} = 10^{11}\,M_\odot$ and $f_{\mathrm{PBH}} = 10^{-2}$ falls below the limit, the central exclusion claim would be falsified at that normalization.

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Extended reading notes

Core claim

The paper's central claim is that the Poisson shot noise of a monochromatic, non-spinning PBH population with $M_{\mathrm{PBH}} \in [10^{8}, 10^{19}]\,M_\odot$ produces a dark-matter isocurvature perturbation whose primordial power spectrum, $f_{\mathrm{PBH}}^2 \langle(\delta_{\mathrm{iso}}^f)^2\rangle$, exceeds the 95% CL CMB+BAO constraints on the isocurvature power spectrum derived in Ref. [26]. The full perturbation is $\langle I_{\mathrm{DM}}^2\rangle(a) \simeq T_{\mathrm{iso}}^2(a) f_{\mathrm{PBH}}^2 \langle(\delta_{\mathrm{iso}}^f)^2\rangle$, with the linear transfer function $T_{\mathrm{iso}}(a) = (2+3y)/(2+3y_f)$, and the spectrum that is compared with the constraints is Eq. (18) evaluated without that transfer function. Under both ultraviolet cut-off prescriptions considered, the exclusion region in the $(M_{\mathrm{PBH}}, f_{\mathrm{PBH}})$ plane rules out SLABs as a significant dark matter component; using the three-point $\beta_{\mathrm{iso}}$ limits from the Planck satellite gives a slightly weaker but qualitatively identical bound.

Load-bearing premise

The exclusion boundaries assume the formation-epoch spectrum $f_{\mathrm{PBH}}^2 \langle(\delta_{\mathrm{iso}}^f)^2\rangle$ is the quantity to compare with the CMB/BAO limits; if the transfer function $T_{\mathrm{iso}}^2(a_{\mathrm{rec}})$ should be included because the CMB constrains the later-time amplitude, the boundaries shift by up to an order of magnitude.

Editorial extensions

If this is right

SLABs in the mass range $[10^8, 10^{19}]\,M_\odot$ cannot make up a significant part of dark matter, closing the window that earlier SLAB studies had proposed.
The nanohertz gravitational-wave background detected by pulsar timing arrays is very unlikely to be dominated by SLAB mergers, since a population large enough to explain it would violate the CMB isocurvature bounds.
Proposals that PBH-induced density fluctuations seed the early massive galaxies seen by JWST are in tension with these limits.
Future CMB experiments, e.g., LiteBIRD and the Simons Observatory, should tighten the same bounds, pushing the allowed $f_{\mathrm{PBH}}$ to even smaller values.
If isocurvature and adiabatic modes were fully anticorrelated, the constraint on the isocurvature fraction would strengthen to $\mathcal{O}(10^{-3})$, tightening the abundance limits further.

Reading between the lines

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

Editorial extension: the same Poisson-shot-noise argument is not specific to SLABs; applied to other discrete dark-matter candidates, such as intermediate-mass black holes or MACHOs in unconstrained mass windows, it would yield analogous CMB isocurvature bounds that could be tested against current data.
Editorial extension: if the transfer-function normalization ambiguity goes the other way, the exact $f_{\mathrm{PBH}}$ upper limits in Fig. 2 could move by up to an order of magnitude, but the qualitative exclusion of SLABs as a dominant component would likely survive because the excluded region spans many decades.
Editorial extension: generalizing to an extended PBH mass function would make the isocurvature spectrum scale-dependent; future CMB lensing or 21-cm observations could discriminate between monochromatic and extended cases.

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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

Colleague,

This letter does one useful thing: applies the known Poisson shot-noise isocurvature calculation (Inman & Ali-Haïmoud) to the SLAB mass range and compares against the recent broad-k CMB+BAO bounds from Buckley et al. The result — SLABs cannot be a significant DM component — is solid. The calculation is standard but correctly executed, the UV cut-off choices are conservative and clearly stated, and the comparison with the external constraints is the right thing to do. The paper also connects to the PTA and JWST debates, which is fair context but not load-bearing.

The soft spots are real but minor. Eq. (15) writes the isocurvature spectrum with a single j1 factor, and that expression turns negative above k/kf ~ 4.5, which is unphysical. What they presumably want is the squared window or the appropriate small-k limit; the exclusion, however, rides on low-k superhorizon modes, so the high-k sign flip does not affect the headline. The text says that the comparison uses Eq. (18) "without the transfer function," which is the correct primordial normalization; the reader's worry about an ambiguous mapping is a presentation issue, not a physics error, but the sentence could be clearer. No code or data are included, which is a minor annoyance given the simplicity of the calculation.

The citation practice is fine. They use external constraints from Ref. [26] and Planck, note the concurrent Lyman-alpha closure of part of the range, and do not lean on self-citation except where the result is theirs.

Bottom line: the qualitative conclusion is robust and the paper is worth a serious referee. I would accept it for review; it is a clean, useful addition to the PBH constraints literature, and the boundary-level technical issues are fixable with a clarifying note and a corrected form factor. Not groundbreaking, but not sloppy.

Referee Report

2 major / 3 minor

Summary. The paper derives the isocurvature power spectrum sourced by a population of Poisson-distributed, monochromatic, non-spinning primordial black holes (PBHs) with masses in the stupendously large black hole (SLAB) range, M >~ 1e11 solar masses. The central calculation models the PBH fluid as discrete objects with finite physical radius, obtains a shot-noise isocurvature spectrum, and compares its amplitude, after applying one of two UV cutoffs, with CMB+BAO upper bounds on the primordial isocurvature power spectrum taken from Ref. [26] and with the Planck isocurvature fractions of Eq. (30). The result is an exclusion region in the (M, f_PBH) plane that rules out SLABs as a significant dark matter component. The paper also places the new bounds in the context of existing PBH constraints and discusses implications for PTA gravitational-wave backgrounds and JWST early galaxy candidates.

Significance. If the result holds, this is a valuable model-independent exclusion: CMB isocurvature constraints are robust and do not rely on uncertain accretion or dynamical modeling, unlike several existing SLAB bounds. The qualitative conclusion is solid because the low-k, superhorizon-at-formation part of the Poisson spectrum is standard and the comparison with external CMB/BAO constraints is natural. The paper is also careful in presenting two UV cutoff prescriptions and in using the more conservative one as the primary bound. The main technical weakness is in the finite-size form factor, which enters the written spectrum with the wrong (unsquared) functional form; this affects the high-k tail and some boundary details but does not rescue a significant SLAB dark-matter fraction.

major comments (2)

[Eq. (15) and surrounding derivation] Equation (15) as written is not a valid power spectrum. The same-PBH contribution in Eq. (11) should involve the self-convolution of the PBH density profile, not a top-hat step, and the final Fourier-space expression should contain |W(k)|^2 = [3j1(k/kf)/(k/kf)]^2 rather than a single factor 3j1(k/kf)/(k/kf). As written, the spectrum becomes negative for k/kf above about 4.49, which is unphysical and contradicts the statement that the Bessel factor 'implements a cutoff'. The low-k limit is unaffected, so the central exclusion is robust, but the high-k tail and the quantitative boundaries in Fig. 2 for the largest masses, where k/kf at CMB scales can exceed unity, should be recomputed with the squared window.
[Eq. (18) and 'New Isocurvature Constraints'] The normalization epoch of the compared spectrum is ambiguous. Equation (18) defines <I_DM^2>(a) with an explicit T_iso^2(a) factor, but the text says the comparison to Ref. [26] uses '[Eq. (18)] without the transfer function'. The authors should define explicitly the primordial spectrum P_II(k) = f_PBH^2 <(delta_iso^f)^2> Theta(k-k_UV) with T_iso = 1, and state that the Aiso constraints of Ref. [26] refer to the primordial isocurvature amplitude. As it stands, a reader cannot tell whether Fig. 1 and Fig. 2 include a growth factor; if T_iso were accidentally included, the bounds would shift by a factor T_iso^2(a), which is order ten at recombination for the masses considered.

minor comments (3)

[Eqs. (17) and (29)] The effective degrees of freedom are written as g_star(k) and g_star,s(k), but the arguments are temperatures; the notation should be g_star(T_f) and g_star,s(T_f).
[Note added and Introduction] The note added states that Ref. [61] already closed the range 1e14-1e16 solar masses with Lyman-alpha data; this should be integrated into the main text and figure discussion rather than left as a post-submission note, because it changes the novelty statement for part of the claimed range.
[Figure 1 caption] The caption for Fig. 1 does not state whether the black curves include the transfer function T_iso(a) or represent the primordial spectrum with T_iso set to unity; this should be stated explicitly, especially given the ambiguity in Eq. (18).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PBH isocurvature spectrum is derived from Poisson statistics and compared to independent external CMB/BAO bounds; no fitted parameter is renamed as a prediction.

full rationale

The central derivation is self-contained. Equation (15) follows from Poisson statistics for discrete PBHs via Eqs. (9)-(14), and Eq. (18) supplies the DM isocurvature power spectrum with f_PBH^2 and two explicit UV cut-off prescriptions, Eqs. (24) and (28). The paper compares this predicted primordial spectrum, identified as 'Eq. (18) without the transfer function', to external 95% CMB+BAO upper limits from Ref. [26] and to the Planck three-point beta_iso constraints in Eq. (30). No parameter is fitted to the constrained data: f_PBH and M_PBH are the scanned parameter axes, gamma_H = 1 is a stated conservative choice, and the two UV cut-offs are physically motivated rather than tuned to the CMB bounds. The cited prior works for the transfer function and k_PBH relations, Refs. [14] and [18], are external and not authored by the current authors. The only self-citations, Refs. [60] and [69], appear in contextual remarks about PTA interpretations and future ringdown searches, and they are not load-bearing for the central exclusion. The possible normalization ambiguity between Eq. (18)'s transfer-function-included amplitude and the 'primordial' spectrum used for comparison is a presentation issue, not a circular one, because the text explicitly specifies the quantity compared. Uncertainty in the window-function or transfer-function treatment can shift boundary details of the excluded region, but the low-k amplitude already exceeds the bounds, so no defect rescues a significant SLAB dark matter fraction. The paper's derivation does not reduce to its own inputs.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles, forces, or conserved quantities. The central calculation uses standard Poisson shot-noise statistics plus external CMB constraints. The only hand-set parameter is gamma_H=1, and the two UV cut-off prescriptions bracket the nonlinear regime. No constants are fitted to the data being constrained.

free parameters (2)

gamma_H (PBH mass fraction of horizon mass at formation) = 1 (assumed)
Set to 1 for definiteness in Eqs. (17) and (29). Smaller values would shift formation later and strengthen the derived bounds, so this is a conservative choice rather than a fit to data.
UV cutoff prescription k_UV = case A: k_PBH^NL; case B: k_DM^NL
Two hand-chosen prescriptions for truncating the isocurvature spectrum at nonlinear scales, defined in Eq. (19). They bracket the uncertainty but are not fitted to the CMB data.

assumptions (5)

domain assumption PBHs are Poisson-distributed and initially unclustered
Used in Eqs. (9)-(13) to write the two-point function of the PBH density field; footnote [17] notes that initial clustering would modify the relation.
domain assumption PDM perturbations are purely adiabatic at PBH formation
Eq. (2) following Ref. [14]; the entire isocurvature signal comes from the PBH discreteness term.
domain assumption Linear perturbation theory with the Meszaros transfer function is valid on CMB scales
Eq. (27) sets the isocurvature transfer function; the paper excises nonlinear scales with UV cutoffs but assumes linearity below those cutoffs.
domain assumption Baryons are neglected in the isocurvature transfer function
Stated before Eq. (27); a standard approximation that introduces O(1) corrections.
standard math CMB/BAO constraints from Ref. [26] are appropriate external benchmarks
The 95 percent CL bounds on delta-function isocurvature spectra from CLASS/MontePython fits are used without modification.

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Cite this review

Pith. "Pith review of Closing the Mass Window for Stupendously Large Black Holes." pith.science (2026). https://pith.science/paper/PX53RJ7F

@misc{pith2026250808238,
  author       = {Pith},
  title        = {Pith review of: Closing the Mass Window for Stupendously Large Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PX53RJ7F}},
  note         = {Machine review of arXiv:2508.08238}
}

abstract

We show that primordial black holes (PBHs) in the $\textit{Stupendously Large Black Hole}$ mass range ($M \gtrsim 10^{11}\,M_\odot$) produce isocurvature perturbations exceeding current $\textit{Planck}$ Cosmic Microwave Background limits, thereby excluding them as a significant dark matter component.

Figures

Figures reproduced from arXiv: 2508.08238 by the authors.

**Figure 1.** CMB+BAO 95% CL constraints on primordial DM isocurvature power spectrum taken from Ref. [26]. For comparison, the three black dots show the constraints derived by the Planck collaboration [16]. We show with black lines two representative cases for the PBH-induced isocurvature power spectrum defined in Eq.(18) with the case A cut-off (solid), Eq.(24), and case B cut-off (dashed), Eq.(28), for MPBH = 1010M⊙ with fPBH … view at source ↗

**Figure 2.** Current constraints on the abundance of a monochromatic, non-spinning PBH population. The new bounds from PBH-induced isocurvature perturbations derived in this work are shown in pink (CMB+BAO). The thicker curves use the broad-k constraints of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

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