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Constraints on the primordial curvature power spectrum at small scales between $3\times 10^{18}$ and $4.5\times 10^{21}~\rm Mpc^{-1}$

T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Using memory-burdened primordial black holes, the paper derives new upper limits on the primordial curvature power spectrum between $4.5\times10^{18}$ and $1.8\times10^{21}~\mathrm{Mpc}^{-1}$, with the strongest reaching…

desk verdict Routine Press-Schechter conversion of new memory-burdened PBH constraints gives genuinely new P_R limits, but the headline bounds rest on a speculative mechanism and the derivation is under-disclosed. read the letter →

arxiv 2411.18887 v5 pith:WGXKFBIB submitted 2024-11-28 astro-ph.CO gr-qchep-ph

Yupeng Yang This is my paper · ORCID
classification astro-ph.COgr-qchep-ph
keywords primordialcurvaturepowerspectrumblackholesmemoryburdenHawkingradiationbigbangnucleosynthesishigh-energyneutrinosgamma-raybackgroundsmall-scaleperturbations
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 tries to close a gap in our picture of the very early universe: the size of density ripples on comoving scales $10^{18}$ to $10^{21}~\mathrm{Mpc}^{-1}$, far too small to appear in the cosmic microwave background, has been almost unconstrained. It argues that recent advances in black-hole physics, especially the 'memory burden' effect that lets light primordial black holes survive to the present day, turn existing gamma-ray, high-energy-neutrino, and Big Bang nucleosynthesis bounds into new upper limits on the primordial curvature power spectrum. The tightest new bound reaches $\mathcal{P}_{\mathcal{R}}\lesssim 10^{-1.7}$ over a wide swath of this range. If the assumptions hold, these are the strongest constraints yet in a regime previous literature had barely probed.

What carries the argument

The load-bearing object is the Press--Schechter-type mapping from the PBH initial mass fraction $\beta(M_{\mathrm{PBH}})$ to the curvature power spectrum via the smoothed density-contrast variance: $\beta\simeq \mathrm{erfc}(\delta_c/\sqrt{2\pi}\,\sigma(R))$, with $P_\delta = \frac{4(1+w)^2}{(5+3w)^2}\left(k/aH\right)^4 \mathcal{P}_{\mathcal{R}}$. This conversion is fed by three external inputs: deuterium-abundance BBN limits on $\beta$, gamma-ray and neutrino bounds on the present abundance of memory-burdened PBHs, and neutrino bounds on PBH mergers. The memory-burden effect itself---the strong slowing of Hawking mass loss once a black hole has radiated away about half its mass---is what makes light PBHs ($M_{\mathrm{PBH}}\lesssim 10^{15}~\mathrm{g}$) survive to today, so that their radiation and mergers can be observed at all.

What would settle it

A first-principles computation of black-hole evaporation that shows the memory-burden suppression of the mass-loss rate does not persist after half the mass is gone would invalidate the gamma-ray and neutrino constraints and with them the new $\mathcal{P}_{\mathcal{R}}$ limits. Observationally, a future high-energy neutrino telescope that detects a flux matching the merged-memory-burdened-PBH prediction would confirm the mechanism, while a null detection at projected sensitivity would push all the limits lower.

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Extended reading notes

Core claim

On the paper's own terms, the central result is a set of new upper limits on the primordial curvature perturbation power spectrum $\mathcal{P}_{\mathcal{R}}$ for comoving wavenumbers $4.5\times10^{18}\lesssim k\lesssim 1.8\times10^{21}~\mathrm{Mpc}^{-1}$, a range that earlier PBH-based studies had left almost untouched. These follow from recasting existing limits on the initial or present mass fraction of light PBHs with masses $10^4\lesssim M_{\mathrm{PBH}}\lesssim 10^{10}~\mathrm{g}$ through the standard PBH-formation mapping. The strongest limits, $\mathcal{P}_{\mathcal{R}}\lesssim 10^{-1.7}$ on $7\times10^{18}\lesssim k\lesssim 3\times10^{20}~\mathrm{Mpc}^{-1}$, come from high-energy neutrinos emitted after mergers of memory-burdened PBHs; non-mergered memory-burdened PBHs give comparable limits from gamma rays and neutrinos on the largest scales, while deuterium-abundance bounds from Big Bang nucleosynthesis anchor the smallest scales.

Load-bearing premise

The whole chain of new limits depends on the memory-burden modification of Hawking radiation: once a PBH loses half its initial mass, its evaporation is effectively halted, so light PBHs survive to today and radiate the photons and neutrinos whose non-observation sets the constraints; if that modification is wrong, the derived $\mathcal{P}_{\mathcal{R}}$ limits do not follow.

Editorial extensions

If this is right

  • On scales $7\times10^{18}\lesssim k\lesssim 3\times10^{20}~\mathrm{Mpc}^{-1}$ the primordial curvature power spectrum is capped at $\mathcal{P}_{\mathcal{R}}\lesssim 10^{-1.7}$, a tighter bound than any previously available in that window.
  • Light PBHs with initial masses $3.3\times10^3\lesssim M_{\mathrm{PBH}}\lesssim 8.5\times10^9$ g cannot be a large dark-matter component, because their surviving radiation or merger products would exceed observed gamma-ray and neutrino fluxes.
  • The deuterium-based BBN bound extends the constraints to the smallest wavenumbers, $4.5\times10^{18}\lesssim k\lesssim 7\times10^{18}~\mathrm{Mpc}^{-1}$, where no other probe reaches.
  • Future high-energy neutrino detectors can strengthen all of these limits, since the current constraints sit just above their projected sensitivities.

Reading between the lines

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

  • If the memory-burden activation point or the exponent $k$ in the slowed mass-loss rate $dM_{\mathrm{PBH}}'/dt=(4\pi G M_{\mathrm{PBH}}^2)^{-k}\,dM_{\mathrm{PBH}}/dt$ differs from the values assumed here, the derived $\mathcal{P}_{\mathcal{R}}$ floor would shift; mapping the limits as a function of these parameters would quantify the model dependence.
  • The same machinery could be pushed to wavenumbers above $10^{21}~\mathrm{Mpc}^{-1}$ if PBH constraints extend below $M_{\mathrm{PBH}}\sim 10^3$ g, potentially connecting to the existing LSP and Planck-mass-relic bounds.
  • The conversion from $\beta$ to $\mathcal{P}_{\mathcal{R}}$ assumes Gaussian density perturbations; detectable non-Gaussianity at PBH scales would change the mapping and require a separate analysis.
  • A confirmed detection of the neutrino signal from merging memory-burdened PBHs would not only tighten the curvature bounds but would independently corroborate the memory-burden picture of black-hole evaporation.
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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

5 major / 5 minor

Summary. The paper compiles recent constraints on the abundance of light primordial black holes (PBHs) — a BBN deuterium bound from Boccia, Iocco and Visinelli, gamma-ray and neutrino bounds on memory-burdened PBHs from Thoss et al., Chianese et al., and Zantedeschi and Visinelli — and converts them into upper limits on the primordial curvature power spectrum P_R at comoving scales 4.5×10^18 ≲ k ≲ 1.8×10^21 Mpc^{-1}. The conversion uses a Press–Schechter-type Gaussian integral relating the PBH mass fraction β to the density-variance σ, together with the standard relation β ≃ 1.6×10^{-25} f_PBH (M_PBH/g)^{1/2}. The resulting bounds reach P_R ≲ 10^{-1.7} over part of the range and are claimed to be stronger than previous LSP and Planck-mass-relic limits. The paper is short and explicitly states in the abstract that the memory-burden assumptions enter the derivation; the main body, however, does not quantify the sensitivity of the final limits to those assumptions.

Significance. If the conversion steps are correct, the paper provides a useful map of existing PBH-abundance constraints onto P_R in a scale window that has received less attention than the CMB/Lyman-α scales and the very small PBH scales. The derivation is not circular: the input constraints come from external analyses of BBN, gamma-ray, and neutrino observations rather than from P_R itself. The paper's reliance on independent groups' calculations [21,26-28] is a strength, as is the explicit disclosure in the abstract that the memory-burden model is an assumption. However, the central claim is conditional on current model-dependent PBH evaporation physics, and the manuscript does not provide several load-bearing derivations needed to reproduce Fig. 2. The paper does not include machine-checked proofs or public code, so reproducibility rests on the clarity of the analytic steps, which is currently insufficient.

major comments (5)
  1. [II.C and Fig. 2] The mapping from PBH mass to comoving wavenumber k is never given. Equations (1)-(5) connect β and f_PBH to σ and hence to P_R(k), but no formula connects M_PBH to the scale R or k. The x-axis of Fig. 2 is therefore not reproducible from the text. Please state explicitly the relation k(M_PBH) used, including the assumed values of γ and g_{*i}, and clarify whether the mass appearing in the constraints of Refs. [26-28] is the initial PBH mass or the present memory-burdened mass.
  2. [Eq. (3)] The printed approximation β ≃ erfc(δ_c/(√(2π)σ(R))) is not the result of the preceding Gaussian integral. For the probability distribution in Eq. (1), the Press–Schechter integral evaluates to erfc(δ_c/(√2 σ(R))). If the numerical conversion used the printed factor √(2π), the inferred σ is smaller by a factor √π, which shifts P_R by a factor π (about 0.5 dex). Please correct the formula and verify that the curves in Fig. 2 use the standard result.
  3. [II.B and II.C] The conversion of f_PBH constraints from memory-burdened PBHs into the initial mass fraction β via Eq. (5) assumes that the PBH mass has not changed between formation and the present. In the memory-burden scenario adopted here, a PBH loses roughly half its mass before the burden becomes active, so the present abundance and the initial abundance are related by an additional factor involving the initial-to-present mass ratio, and the mass used for the k-mapping should be the formation mass. Please state whether the external constraints are on f_PBH or β, and derive the conversion consistently; as written, the limits could be biased at the O(1) level in β.
  4. [II.A] The choice δ_c = 0.42 is the most aggressive value in the quoted simulation range 0.42 ≲ δ_c ≲ 0.66 and produces the strongest (lowest) P_R limits. Because β is exponentially sensitive to δ_c/σ, the headline bound P_R ≲ 10^{-1.7} depends strongly on this choice. Please either justify δ_c = 0.42 for the formation scenarios considered or present the resulting limits for representative values such as δ_c = 0.5 and 0.66.
  5. [Abstract and II.B] The abstract states that the memory-burden effect 'halts further evaporation', while Section II.B gives dM'_PBH/dt = (4πGM_PBH^2)^{-k} dM_PBH/dt with k>0, which suppresses but does not stop the mass loss. This distinction matters for whether the lightest PBHs survive to the present and for the spectra used in Refs. [26-28]. Please align the model description and, given that the strongest limits at k ≳ 2×10^19 Mpc^{-1} rest entirely on memory-burdened PBH constraints, add a quantitative sensitivity statement (for example, the dependence of Fig. 2 on the suppression exponent k and on the activation threshold).
minor comments (5)
  1. [Title and text] The title in the provided full text uses '4.5×10^18 and 1.8×10^21 Mpc^{-1}', while the abstract and the arXiv-title version use different limits ('3×10^18' and '4.5×10^21'). Please unify the quoted range.
  2. [Fig. 2 caption] The caption states the plotted range as 1.5×10^18 to 2.5×10^21 Mpc^{-1}, but the text claims constraints over 4.5×10^18 to 1.8×10^21 Mpc^{-1}; please explain the extra range or restrict the caption to the claimed interval.
  3. [Throughout] There are repeated spelling and grammar errors, including 'constrains' for 'constraints', 'baryion' for 'baryon', 'curvatue' for 'curvature', 'mergered' for 'merged', 'investigatons' for 'investigations', 'throry' for 'theory', and 'EGERT' for 'EGRET'.
  4. [Section II.A] In Eq. (3), the integration variable is written as dσ(R) in the intermediate step; it should be dδ(R). The final erfc form should also be checked against the corrected prefactor.
  5. [Summary] The phrase 'we have derived new constraints' could be misleading because the observational constraints are imported from Refs. [21,26-28]; phrases such as 'compiled and converted' would more accurately describe the paper's contribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: P_R limits are converted from independent PBH-abundance constraints via standard formulas.

full rationale

The paper's derivation chain takes external limits on the PBH abundance (β or f_PBH) from Refs. [21,26,27,28] and converts them to upper limits on the primordial curvature power spectrum P_R using the standard Press-Schechter-type relations (Eqs. 3-5). The input constraints come from BBN, gamma-ray, and neutrino observations of memory-burdened PBHs; these do not presuppose any specific value of P_R. The conversion is a one-way mapping from the observed abundance bound to P_R, with no fitted parameter that is later renamed as a prediction. The author's own cited works (e.g., Refs. [6,9,16,30,41]) are used only as background references for previous PBH constraints and are not load-bearing for the central step. The memory-burden model dependence identified in the skeptical note is an assumption about black-hole physics, not a circular reuse of the target quantity, and the paper explicitly discloses that different memory-burden parameters would yield modified limits. Therefore no step reduces to its own input, and no circularity is present.

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

The central claim rests on the adopted values of δc, γ, g_*i, the Press-Schechter formalism, the memory-burden model, and the external PBH constraints. No new entities are introduced. The most fragile inputs are the memory-burden model and the choice of δc.

free parameters (3)
  • δc (critical collapse threshold) = 0.42
    Chosen from the lower end of the simulation range 0.42-0.66 (Ref. [29]). Since the P_R limit scales roughly as δc^2, this choice gives the tightest, most aggressive bounds without a conservative discussion.
  • γ (collapse fraction of horizon mass) = 0.2
    Adopted in Eq. (5) to convert f_PBH to β_PBH; a standard Press-Schechter choice.
  • g_*i (relativistic degrees of freedom at formation) = ≈100
    Adopted in Eq. (5); a standard value for PBH formation in the early universe.
assumptions (5)
  • domain assumption Primordial curvature perturbations are Gaussian and PBHs form via Press-Schechter collapse with a Gaussian window function (Eqs. 1-3).
    The conversion from β to σ assumes Gaussian statistics and the Press-Schechter form; non-Gaussianity would alter the relation.
  • domain assumption The mapping β = 1.6×10^-25 f_PBH (M/g)^1/2 (Eq. 5) is valid with the adopted γ and g_*i.
    Used to convert present-day PBH abundance limits to the initial mass fraction.
  • domain assumption The memory-burden effect activates after a PBH loses half its mass and effectively halts further evaporation, so light PBHs survive to today.
    Central to the validity of the gamma-ray and neutrino constraints (Refs. [26-28]) used to bound f_PBH; the paper states different assumptions would modify the limits.
  • domain assumption The external constraints from Refs. [21,26,27,28] are correct and correctly interpreted as upper bounds on β or f_PBH.
    The paper replots these constraints (Fig. 1) without re-deriving them.
  • domain assumption A standard relation converts PBH mass M to comoving scale k, and σ(R) is effectively proportional to P_R at the scale R, with the integral in Eq. (2) dominated by one scale.
    The paper does not display this conversion, so the reader must assume the standard mass-to-scale mapping and a single-scale approximation.

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Cite this review

Pith. "Pith review of Constraints on the primordial curvature power spectrum at small scales between $3\times 10^{18}$ and $4.5\times 10^{21}~\rm Mpc^{-1}$." pith.science (2026). https://pith.science/paper/WGXKFBIB

@misc{pith2026241118887,
  author       = {Pith},
  title        = {Pith review of: Constraints on the primordial curvature power spectrum at small scales between $3\times 10^18$ and $4.5\times 10^21~\rm Mpc^-1$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WGXKFBIB}},
  note         = {Machine review of arXiv:2411.18887}
}
abstract

The primordial curvature power spectrum $\mathcal{P}_\mathcal{R}$ has been measured with high precision on large scales $10^{-4}\lesssim k\lesssim 3~\rm Mpc^{-1}$ based on observations of the cosmic microwave background, Lyman-$\alpha$ forest and large scale structure. On small scales $3\lesssim k \lesssim 10^{23}~\rm Mpc^{-1}$, constraints are primarily derived from studies on primordial black holes (PBHs). In particular, for very small scales $10^{17}\lesssim k\lesssim 10^{23}~{\rm Mpc^{-1}}$, current limits come exclusively from investigations of the lightest supersymmetric particles produced by PBH radiation and the stable Planck-mass relics after their evaporation. Recent findings also indicate that the evaporation of light PBHs ($M_{\rm PBH}\lesssim 10^{9}~\rm g$) can modify the expansion rate of the Universe and the baryon-to-photon ratio, thereby affecting the primordial abundance of light nuclei. Moreover, it has been proposed that the ``memory burden'' effect can slow down the mass loss rate of black holes, allowing light PBHs to survive until today. Based on recent theoretical advancements in black hole physics and existing constraints on the initial mass fraction of light PBHs with masses $10^{3}\lesssim M_{\rm PBH}\lesssim 2\times 10^{9}~\rm g$, and especially the recent constraints on memory-burdened PBHs, we derive new and tighter upper limits on $\mathcal{P}_\mathcal{R}$ on small scales $3\times 10^{18}\lesssim k\lesssim 4.5\times 10^{21}~\rm Mpc^{-1}$, a regime that has been underexplored in previous literature. These constraints are derived under the specific assumption that the memory burden effect activates after the PBH loses half of its initial mass and subsequently halts further evaporation, and different assumptions on the memory burden parameters would lead to modified limits.

Figures

Figures reproduced from arXiv: 2411.18887 by the authors.

Figure 1
Figure 1. FIG. 1. New constraints on the (initial) mass fraction of light [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Constraints on the power spectrum of primordial cur [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.