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REVIEW 3 major objections 6 minor 122 references

Gravitational wave signatures of primordial black hole accretion during early matter domination

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read PBH accretion during an early matter-dominated era produces a double-peaked gravitational-wave spectrum.

desk verdict Nice new twist—PBH accretion during EMD—but the EMD GW peak is computed from Pζ(kO) alone and the log-normal peak at kp is left unintegrated; the double-peak claim needs a full calculation before it carries the detectability. read the letter →

arxiv 2505.21885 v1 pith:SYYMPBDC submitted 2025-05-28 hep-ph astro-ph.COgr-qc

Rouzbeh Allahverdi , James B. Dent , Ngo Phuc Duc Loc , Tao Xu This is my paper
classification hep-phastro-ph.COgr-qc
keywords primordialblackholesearlymatterdominationBondi-Hoyleaccretionscalar-inducedgravitationalwavesasteroid-masswindowdarkstochasticwavebackgroundHawkingevaporation
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 argues that primordial black holes formed in the early radiation-dominated universe can grow substantially during a later phase of early matter domination, with masses increasing by up to two orders of magnitude. If true, this accretion rescues black holes that would otherwise have Hawking-evaporated by now, repositioning them inside the asteroid-mass window where they could account for all of dark matter. The same early-universe history generates a gravitational-wave background with two peaks: a high-frequency peak from the curvature perturbations that created the black holes, and a low-frequency peak from the sudden end of the matter-dominated phase. The paper maps the black-hole mass ranges in which LISA and BBO could see one or both peaks, and reports signal-to-noise ratios above unity across most of the asteroid-mass window.

What carries the argument

The central mechanism is Bondi-Hoyle accretion in the collisionless-fluid limit during the matter-dominated era, where the PBH mass scales as $M\propto a$ until a disk forms at the Eddington time $t_{\rm Edd}$ and shuts off efficient accretion; this yields the maximal-mass relation $M_{\rm max}\approx 2M_i\sigma_{H,O}^{-6/5}(T_O/T_i)^{8/5}$. The gravitational-wave machinery is scalar-induced gravitational waves: the enhanced curvature perturbation $P_\zeta$ that creates the PBHs sources tensor modes at second order during radiation domination, while the sudden EMD-to-RD transition amplifies a separate low-frequency peak. The relation $k_O/k_p=T_O/T_i$ connects the two peak scales to the PBH formation mass and the onset of matter domination, so a two-peak detection would constrain both the accretion history and the duration of the EMD epoch.

What would settle it

A LISA or BBO measurement of a stochastic background in the relevant band that shows only one peak, or two peaks whose frequency ratio disagrees with $k_O/k_p=T_O/T_i$ for the assumed $\sigma_{H,O}$, would rule out the scenario. A numerical simulation of accretion onto a PBH between $10^{14}$ and $10^{17}$ g in a collisionless matter-dominated background that gives far less than an order-of-magnitude mass growth would also falsify the central mass-growth claim.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that a PBH of initial mass $M_i$ formed during radiation domination can reach $M_{\rm max} \approx 2 M_i \sigma_{H,O}^{-6/5}(T_O/T_i)^{8/5}$ by Bondi-Hoyle accretion during an early matter-dominated era, corresponding to growth factors around 200 for $\sigma_{H,O}=10^{-5}$ and around 20 for $\sigma_{H,O}=10^{-3}$. The same scalar perturbations that form the PBHs induce tensor modes at second order, so the scenario predicts a stochastic gravitational-wave background with two peaks: the high-frequency peak traces PBH formation during radiation domination and is diluted by the later matter era, while the low-frequency peak comes from the sudden transition back to radiation domination. For PBHs comprising all of dark matter, the paper identifies detection windows for both peaks together: roughly $10^{15}\,{\rm g} \lesssim M_{\rm PBH} \lesssim 10^{21}\,{\rm g}$ for mass growth of order 20 and $2\times10^{15}\,{\rm g} \lesssim M_{\rm PBH} \lesssim 5\times10^{18}\,{\rm g}$ for mass growth of order 200.

Load-bearing premise

The load-bearing premise is that during early matter domination each PBH accretes as a collisionless fluid with mass growing as $M\propto a$ until disk formation halts accretion at the Eddington time; if angular momentum or disk physics makes accretion less efficient, the predicted growth factors and detection windows shrink.

Editorial extensions

If this is right

  • PBHs in the asteroid-mass window could be the entirety of dark matter and still be observable through a stochastic gravitational-wave background, a population that is otherwise extremely hard to probe.
  • Measuring the ratio of the two peak frequencies would recover the ratio $T_O/T_i$, effectively measuring when the early matter era began.
  • Accretion means the final PBH mass is not the formation mass; searches for Hawking evaporation or microlensing must target the heavier, post-accretion masses rather than the initial spectrum.
  • The scenario requires the early matter era to end above roughly 1 MeV, and the paper finds the detectable mass ranges are compatible with this nucleosynthesis bound.
  • The larger the curvature perturbation at the onset of matter domination (up to the linear-regime limit), the wider the detectable mass range and the higher the signal-to-noise ratio at BBO.

Reading between the lines

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

  • A joint detection of both peaks would test the accretion law itself: the peak-frequency ratio is fixed by the model parameters, so an observed ratio that disagrees with the Bondi-Hoyle expectation would point to a different accretion efficiency or a non-sudden transition.
  • The same mass growth should shift the PBH spin distribution, since angular momentum determines when the Eddington cutoff acts; future spin measurements from mergers or superradiance could corroborate or exclude the scenario.
  • Because accretion is claimed to be significant regardless of initial mass, smaller PBHs that would have evaporated could be pushed into the observable window; this implies evaporation searches should see a depleted low-mass population relative to formation predictions.
  • Relaxing the linear-regime cutoff would likely strengthen the low-frequency peak, so the quoted detection windows are conservative rather than optimistic.
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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 / 6 minor

Summary. The paper proposes a scenario in which primordial black holes (PBHs) form during an initial radiation-dominated (RD) phase and subsequently accrete during an early matter-dominated (EMD) epoch, growing in mass by up to two orders of magnitude. It then computes the stochastic gravitational wave (GW) background sourced by scalar perturbations in two regimes: the high-frequency peak from PBH formation during the initial RD phase, and a low-frequency peak from the sudden EMD-to-RD transition. For a log-normal curvature power spectrum on top of a flat CMB-scale spectrum, and for f_PBH=1, the paper identifies double-peaked GW spectra, gives LISA and BBO detectability windows on the f_PBH vs final-mass plane, and computes signal-to-noise ratios for the asteroid-mass PBH dark matter window.

Significance. If the double-peak prediction is robust, it would provide a distinctive observational signature connecting asteroid-mass PBH dark matter to a non-standard early matter epoch, and the mass-growth mechanism is an interesting application of existing Bondi-Hoyle accretion results. The paper is clearly written and correctly recovers the no-EMD limits: Eq. (2.8) reduces to the standard PBH abundance relation of Ref. [16], and Eq. (3.8) recovers the peak wavenumber relation of Ref. [111]. The explicit SNR curves and detectability maps for LISA and BBO are useful for future searches. However, the central low-frequency peak calculation is based on an unjustified reduction of the EMD-induced GW signal to P_zeta(k_O) alone, and the chosen linear-regime cutoff is not consistent with the large log-normal peak of Eq. (4.1). These issues affect the amplitude, position, and detectability of the low-frequency peak, so the main double-peak claim is not yet established as presented.

major comments (3)
  1. [Sec. 4, Eq. (4.2)] As written, Eq. (4.2) does not yield the values of T_O/T_i quoted in Tables 1 and 2: for sigma=1, A~3x10^-2 and sigma_H,O=10^-5, the argument of the logarithm is (5/2)(2*pi)^(1/4)/(A^(1/2) sigma_H,O) ~ 2x10^6, so the expression inside the square root is negative and k_O/k_p is imaginary. The tables are reproduced if the ratio is inverted, i.e., if the numerator contains sigma_H,O and the denominator contains A^(1/2). This central relation therefore needs to be corrected; otherwise the derivation of T_O/T_i and all subsequent mass-growth and detectability results cannot be verified from the printed equations.
  2. [Sec. 3.2 and the paragraph after Eq. (4.1)] The EMD contribution to Omega_GW is computed using only P_zeta(k_O), based on the assertion that the EMD-sourced GW spectrum 'narrowly peaks at the wavenumber corresponding to the mode entering horizon at the onset of EMD.' This assertion is not established by the cited sudden-transition calculation of Ref. [90], whose decomposition in Eq. (3.9) includes a resonant peak at the reheating scale k_R and, in general, receives contributions from the full convolution of P_zeta(k_1)P_zeta(k_2). For the log-normal spectrum of Eq. (4.1) with sigma=1 and A~3x10^-2, P_zeta(k_p) is about 10^-2, while P_zeta(k_O) is 10^-9 to 10^-5, a difference of several to many orders of magnitude. No bound is provided on the contribution of the log-normal peak to Omega_GW,EMD, nor on the contribution from modes with k>k_O that have become nonlinear by the end of the EMD. Without an explicit evaluation of the full spectrum through the sudden-transition kernel, the position and amplitude of the low-frequency peak in Figs. 2 and 3, and hence the LISA/BBO detectability windows, are not robust.
  3. [Sec. 3.2, Eqs. (3.10)-(3.12)] The paper states that it restricts to the linear perturbation regime, but the chosen EMD duration does not keep the modes of Eq. (4.1) linear. Setting delta(t_R)=1 for the mode k_O implies a_R/a_O ~ (g_*,O/g_*,R)^(1/3) (T_O/T_R)^(4/3). For sigma_H,O=10^-5 and T_O/T_i ~ 3x10^-3, this is a_R/a_O ~ 10^5; for sigma_H,O=10^-3 it is ~200. The modes near the log-normal peak have sigma_H(k_p) ~ (4/9) sqrt(P_zeta(k_p)) ~ 5x10^-2, so their density contrast at reheating is delta(k_p,t_R) ~ 5000 (for sigma_H,O=10^-5) or ~10 (for sigma_H,O=10^-3). These modes are therefore strongly nonlinear during the EMD and cannot be described by the linear second-order calculation used for the EMD signal. Either the contribution of these modes to the reheating-transition GW must be shown to be negligible, or the nonlinear regime must be treated, before the double-peak prediction can be considered reliable.
minor comments (6)
  1. [Sec. 3.2, after Eq. (3.9)] The statement that the EMD-sourced GW spectrum 'narrowly peaks at the wavenumber corresponding to the mode entering horizon at the onset of EMD' should be reconciled with the three-component decomposition taken from Ref. [90] in Eq. (3.9), in particular with the resonant component at the reheating scale; please clarify which component and which wavenumber is being referred to.
  2. [Eq. (3.7)] The notation 'g,0' in Eq. (3.7) should read 'g_*,0' for consistency with the rest of the paper.
  3. [Abstract and Sec. 1] The abstract and introduction quote the detectable mass ranges '10^15 g < M_PBH < 10^21 g (2x10^15 g < M_PBH < 5x10^18 g)' without the explicit qualifiers that these are for f_PBH=1 and sigma_H,O=10^-3 (10^-5); adding these qualifiers would avoid overstatement.
  4. [Fig. 1 caption] The caption 'Onset of EMD and used to determine nonlinear cutoff' appears to be missing a word or phrase; please rephrase.
  5. [Sec. 4, bullet list after Fig. 3] The text says that both the RD and the EMD detectable regions are 'vertical' on the f_PBH-M_PBH plane; this is confusing because the two contributions have different parametric dependences and the reader cannot see the distinction in Fig. 3. Please clarify what is meant.
  6. [Appendix B] The conclusion that no significant PBH population is produced during EMD relies on the spin-suppressed formation rate of Eq. (B.1); please state explicitly that this is an assumption imported from Ref. [102] and note the sensitivity of the sigma_H,O range to that formula.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is self-contained, with external grounding for accretion and EMD gravitational-wave calculations.

full rationale

The paper's central claims are conditional predictions from stated inputs, not reductions to those inputs. PBH formation abundance (Eq. 2.1) uses the standard Gaussian collapse formula; accretion growth (Eqs. 2.4, A.2, A.7) is taken from the external Bondi-Hoyle treatment of De Luca et al. [105]; the EMD-sourced gravitational-wave spectrum (Eq. 3.9) follows the independent sudden-transition formalism of Inomata et al. [90]; and detector sensitivities are external (Schmitz [119]). The amplitude A of the log-normal curvature spectrum is fixed by the explicit benchmark f_PBH = 1 through Eqs. (2.1), (2.8) and (3.13), not by fitting to gravitational-wave data, so the GW amplitudes and SNR are consequences of that benchmark rather than fitted quantities renamed as predictions. The EMD duration is transparently chosen in Eq. (3.12) to keep the perturbation at the onset scale linear while maximizing the signal, an explicit modeling choice. The mass-growth factor is a derived consequence of the assumed log-normal shape, the chosen sigma_H,O, and the resulting T_O/T_i via Eq. (4.2); it is not an input disguised as an output. Self-citations (e.g., [21, 76, 77, 109, 111, 43]) appear in contextual or standard-formula roles and are not load-bearing; the key EMD and accretion results are external. The skeptic's concern that the EMD phase also amplifies modes near the PBH-formation peak kp is a potential issue of omitted contributions or robustness, but it is not a circularity: the paper's own assertion that the EMD GW 'narrowly peaks' at the onset scale is an approximation to be checked, not a definitional equivalence. Therefore no circular step can be exhibited from the paper's equations, and the appropriate finding is no significant circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The model depends on a small number of tuned parameters and a set of domain assumptions about the early-universe expansion history and accretion physics. The log-normal peak amplitude A is normalized to f_PBH = 1, so the GW amplitude is not a free prediction.

free parameters (5)
  • σ_H,O (curvature perturbation at onset of EMD) = 10^-5 and 10^-3
    Chosen as benchmark values; sets the onset scale k_O and the EMD duration via Eqs. (3.12) and (4.2). Not fitted to data but determines the mass-growth factor and GW amplitudes.
  • log-normal width σ = 1
    Chosen to produce an almost monochromatic PBH mass function (Sec. 4). Influences the relation k_O/k_p in Eq. (4.2).
  • A (log-normal peak amplitude) = ~0.03 (Table 1)
    Determined by requiring f_PBH = 1 for each initial mass M_i via Eqs. (2.1), (2.8), (3.13). This normalization fixes the high-frequency GW peak amplitude.
  • M_i (initial PBH mass) = 10^15 - 10^21 g scanned
    Input parameter scanned across the asteroid-mass window; final masses M_f and detection ranges are derived.
  • t_obs (SNR observation time) = 1 year
    Assumed in the SNR calculation, Eq. (4.3).
assumptions (6)
  • domain assumption Existence of an early matter-dominated epoch sourced by a bosonic field ϕ that behaves as cold matter and decays suddenly into radiation at reheating temperature T_R.
    Invoked in Secs. 2 and 3; the low-frequency GW peak relies on the sudden transition from EMD to RD.
  • domain assumption PBH accretion during EMD follows the Bondi-Hoyle regime with M ∝ a until the Eddington cutoff t_Edd (Eqs. A.2, A.7).
    This model yields the central mass-growth formula Eq. (2.4); it is adopted from [105] and its validity for the assumed collisionless fluid is discussed in Appendix A.
  • standard math The density contrast at horizon re-entry has Gaussian statistics with threshold δ_c ≈ 0.42 for PBH formation (Eq. 2.1).
    Standard PBH formation theory, cited from [4,88].
  • ad hoc to paper The enhanced curvature power spectrum is log-normal with width σ=1 on top of a flat CMB-scale spectrum (Eq. 4.1).
    Phenomenological choice motivated by ultra-slow-roll inflation; all quantitative results depend on this shape.
  • ad hoc to paper The mode entering at the start of EMD reaches δ=1 exactly at the end of EMD, i.e., the linear/nonlinear threshold sets the EMD duration (Eq. 3.12).
    Chosen to maximize the EMD GW peak while staying in the linear regime; if EMD is shorter, the low-frequency peak is weaker.
  • standard math Standard scalar-induced gravitational wave formulas for RD (Eq. 3.3) and for sudden-transition EMD (from [90]) apply.
    Used in Sec. 3 without re-derivation.

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Cite this review

Pith. "Pith review of Gravitational wave signatures of primordial black hole accretion during early matter domination." pith.science (2026). https://pith.science/paper/SYYMPBDC

@misc{pith2026250521885,
  author       = {Pith},
  title        = {Pith review of: Gravitational wave signatures of primordial black hole accretion during early matter domination},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SYYMPBDC}},
  note         = {Machine review of arXiv:2505.21885}
}
read the original abstract

We present a scenario in which primordial black holes (PBHs) form in a post-inflationary radiation-dominated (RD) phase and then experience significant accretion during a phase of early matter dominated (EMD). We show that PBH masses could grow by up to two orders of magnitude. Restricting to the linear perturbation regime, we compute the gravitational wave (GW) spectrum that features two peaks. The high-frequency peak is associated with the PBH formation in the RD phase, while the low-frequency peak is due to the sudden transition from EMD to the later, standard RD phase. We identify a PBH mass range where one or both peaks can be observed by a combination of different GW detectors. Finally, we show the signal-to-noise ratio of the total GW spectrum for PBHs in the asteroid mass window, where they could comprise the totality of dark matter.

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