Relativistic accretion and burdened primordial black holes

Suvashis Maity

REVIEW 4 major objections 4 minor 1 cited by

Relativistic accretion and burdened primordial black holes

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Combining relativistic accretion with memory-burdened evaporation lowers the minimum initial mass for primordial black holes surviving today to roughly $10^3$ g, and shifts the resulting dark-matter and dark-radiation constraints.

desk verdict Useful PBH parameter map combining relativistic accretion and memory burden, but the printed accretion coefficient is off by ~2500 and every quantitative result inherits the typo. read the letter →

arxiv 2507.02821 v2 pith:DWKSCTCM submitted 2025-07-03 astro-ph.CO hep-th

Suvashis Maity This is my paper · ORCID

classification astro-ph.COhep-th

keywords primordialblackholesmemoryburdenholeevaporationrelativisticaccretiondarkmatterrelicabundanceradiationeffectivenumberofdegreesfreedomPBH

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 tries to establish that two effects usually studied separately—accretion, which grows primordial black holes, and memory-burdened evaporation, which slows their mass loss—combine to change the fate of low-mass primordial black holes. In the combined picture, a black hole first accretes until its mass reaches a saturation value $M_{\rm acc}$, then evaporates as if it had started at that larger mass, with evaporation further delayed once quantum memory-burden backreaction sets in. The result is an explicit survival threshold: with memory-burden parameters $\kappa=2$, $q=1/2$ and a radiation-dominated background, initial masses as low as $1400$ g (for relativistic accretion) can survive to the present, versus $5600$ g with no accretion and about $10^{15}$ g in the standard semiclassical case. A sympathetic reader should care because this reopens a low-mass window for primordial black holes as dark matter and moves the parameter space for dark matter and dark radiation emitted during evaporation.

What carries the argument

The load-bearing object is the piecewise mass-evolution law of Eq. (9): an accretion phase in which the mass follows the relativistic spherical accretion growth toward the saturation value $M_{\rm acc}$, followed by a semiclassical evaporation phase and then a memory-burdened phase with evaporation rate $-(\epsilon M_{\rm Pl}^4/M^2)S^{-\kappa}$. The memory burden is the suppression of evaporation once the black hole's entropy $S=M^2/(2M_{\rm Pl}^2)$ has dropped, with $q$ fixing the transition mass and $\kappa$ the suppression power. This machinery replaces a two-process dynamics by an effective single-process evolution starting from $M_{\rm acc}$, which is what makes the survival threshold and the subsequent abundance formulas analytically tractable.

What would settle it

Integrate the full system $\dot{M}=\lambda_c M^2 \rho_\infty/(16\pi M_{\rm Pl}^4)-\epsilon M_{\rm Pl}^4 S^{-\kappa}\Theta(t-t_q)/M^2$ numerically for representative values of $\kappa$, $q$, $M_{\rm in}$, and $w$ without the sequential split, and compare the resulting evaporation times and survival masses with the paper's piecewise formulas; a difference of an order of magnitude in the survival threshold would show the sequential approximation fails.

Watch

Extended reading notes

Core claim

The central claim is that relativistic accretion and memory-burdened evaporation are not independent corrections but combine through a simple mass rescaling. During the early dense phase, accretion dominates and the mass approaches $M_{\rm acc}=M_{\rm in}[1-\lambda_c\gamma/(2(1+w))]^{-1}$, which is about $4M_{\rm in}$ in a radiation-dominated universe for relativistic accretion; evaporation takes over afterward with this saturated mass as the effective initial condition, and once the black hole has radiated down to a fraction $q$ of that mass, the memory burden suppresses further emission by the entropy power $S^{-\kappa}$. The paper derives the survival threshold and finds, for $\kappa=2$, $q=1/2$, $w=1/3$, minimum present-day survival masses of $5600$ g (no accretion), $3200$ g (nonrelativistic accretion), and $1400$ g (relativistic accretion). It further shows that accretion lowers the maximum initial mass that evaporates before big bang nucleosynthesis, shrinks the allowed $(m_j, M_{\rm in})$ plane for dark matter produced by evaporation, shifts the $f_{\rm PBH}$ constraints in the new mass window, and alters the predicted $\Delta N_{\rm eff}$ from dark radiation.

Load-bearing premise

The argument assumes that a black hole finishes growing by accretion and saturates before it starts losing much mass to evaporation, so the two processes can be treated one after the other; if they overlap substantially, the numerical thresholds shift.

Editorial extensions

If this is right

Primordial black holes with initial masses as low as roughly $10^3$ g can survive to today, so the $f_{\rm PBH}$ dark-matter window opens at masses far below the standard $10^{15}$ g threshold.
The maximum initial mass that evaporates completely before big bang nucleosynthesis drops to a few grams under relativistic accretion, so black holes that would have evaporated before BBN instead persist past it.
The allowed $(m_j, M_{\rm in})$ parameter space for dark matter emitted during evaporation shrinks as the memory-burden parameter $\kappa$ grows, and shrinks further when accretion, especially relativistic accretion, is included.
The critical initial abundance $\beta_{\rm cr}$ for primordial black holes to dominate the Universe before evaporation decreases by orders of magnitude with accretion, so a much smaller initial PBH fraction can alter cosmology.
The predicted $\Delta N_{\rm eff}$ contribution from dark radiation shifts with the accretion treatment and with the memory-burden parameters, and the shifts are large enough that future CMB measurements with sensitivity around $\Delta N_{\rm eff}\sim 0.06$ could distinguish the cases.

Reading between the lines

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

Beyond the paper: if the sequential split between accretion and evaporation is correct, the same 'replace $M_{\rm in}$ by $M_{\rm acc}$' rescaling should carry over to spinning black holes, where running existing Kerr evaporation codes with the rescaled initial mass would test whether the survival-window widening survives rotation.
Beyond the paper: the paper treats the transition into the memory-burdened phase as sudden; a gradual transition would smooth the mass evolution and could shift the survival threshold by an order of magnitude at fixed $\kappa$, a direct extension of the present calculation.
Beyond the paper: because accretion changes the black-hole lifetime, the redshift and amplitude of secondary gravitational waves from evaporation would change as well; inserting the rescaled masses into existing stochastic-gravitational-wave computations gives a testable signature of the accretion boost.

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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

Short version: the paper is a useful parameter-space study, but as written its central numbers rest on an internal contradiction. I checked Eqs. (3)-(5) by hand. Integrating dM/dt = lambda_c 16pi M^2 rho_inf / M_Pl^4 with rho_inf = 4 M_Pl^2 / [3(1+w)^2 t^2] and t_in = M_in / [6pi gamma (1+w) M_Pl^2] gives M(t) = M_in [1 - (128 pi^2 lambda_c gamma / (1+w)) (1 - t_in/t)]^{-1}, not the paper's B = lambda_c gamma / [2(1+w)]. The printed coefficient is too large by a factor of 256 pi^2. With lambda_c^R at w = 1/3, B is about 1.9e3, so the denominator crosses zero almost immediately and no finite M_acc ~ 4 M_in exists. If instead Eq. (3) should read 1/(16 pi), then every later formula--survival masses 5600/3200/1400 g, beta_cr, Delta N_eff--goes through as written. This looks like a typo, but it is load-bearing and has to be fixed before the numbers can be trusted.

What the paper does well: it cleanly combines two established effects into one mass evolution law, Eq. (9), and reduces to prior memory-burden results when lambda_c = 0 and to standard Hawking evaporation when kappa = 0, q = 1. The parameter maps for emitted DM, f_PBH, and Delta N_eff are laid out honestly, with kappa, q, beta, and w treated as free parameters rather than fitted. The author acknowledges the sudden-onset memory-burden and sequential accretion-then-evaporation idealizations. The citation pattern is appropriate, including the author's own prior derivations; self-citation here is not a problem.

Softer spots: the sequential approximation is asserted rather than quantitatively demonstrated, and the sudden memory-burden transition is a known simplification. Those are secondary compared with the coefficient issue.

Bottom line: this deserves a serious referee, but not acceptance in its current form. The referee should ask the author to correct Eq. (3), re-run the numerics, and check the accretion coefficient against ref. [98]. Once that is done, I would cite it as a useful map of the joint parameter space. As it stands, I would not.

Referee Report

4 major / 4 minor

Summary. The paper studies the joint evolution of primordial black holes (PBHs) under relativistic or nonrelativistic accretion and memory-burdened Hawking evaporation. It derives analytic expressions for the PBH mass history, the minimum initial mass for survival until today, the maximum initial mass for evaporation before BBN, the critical abundance for PBH domination, the relic abundance of dark matter emitted during evaporation, the stable-PBH dark-matter fraction f_PBH, and the contribution to ΔN_eff. The central quantitative illustrations take κ=2, q=1/2 and show that accretion, especially relativistic accretion, shifts the survival threshold from about 5600 g (no accretion) to 3200 g (nonrelativistic) and 1400 g (relativistic) for w=1/3, with analogous shifts in the DM and ΔN_eff parameter spaces.

Significance. If the quantitative results are correct, the paper extends the memory-burden PBH program by demonstrating that the same mechanism that allows sub-10^15 g PBHs to survive also gains importance from early-universe accretion, lowering the survival boundary by a factor of a few and modifying the DM and dark-radiation constraints. The analytic treatment covers two equations of state and provides falsifiable predictions, e.g., the ΔN_eff reach of CMB-S4/CMB-HD for given PBH masses. The paper is not a fit to data: β, κ, q, and w are scanned parameters, and the analysis transparently rests on prior memory-burden literature and on the Bondi-type accretion formalism. The main weaknesses are the internal inconsistency of the printed accretion equation, the unquantified timescale-separation assumption, and the breakdown of the stated κ=0 limit in several general formulas.

major comments (4)

[II.A, Eqs. (3)–(5)] Equation (3) as printed is inconsistent with Eq. (4). Substituting ρ∞ = 4M_Pl^2/[3(1+w)^2t^2] and t_in = M_in/[6πγ(1+w)M_Pl^2] into Eq. (3) gives dM/dt = [128π^2 λ_c γ/(1+w)] M^2/(M_in t^2), whose solution is Eq. (4) with B = 128π^2 λ_c γ/(1+w), not the printed B = λ_c γ/[2(1+w)]. Since 256π^2 ≈ 2.5×10^3, for λ_c^R at w=1/3 the denominator in Eq. (4) crosses zero almost immediately and no finite M_acc exists, whereas Eq. (5) and all subsequent results use M_acc/M_in ≈ 4. If the intended accretion law has the Bondi coefficient 1/(16π) rather than 16π, the authors must correct Eq. (3) and state this normalization explicitly, because every quantitative result in Eqs. (10)–(14), (17), (22), (26)–(35), and (43) is a monotone function of (1 − λ_cγ/[2(1+w)])^{-1}.
[II.C, Eqs. (8)–(14)] The sequential treatment of accretion and evaporation is asserted rather than established. Equation (9) introduces t_acc without giving an expression for it, and the text only states that the interval [t_in, t_acc] is “typically much shorter” than [t_acc, t_ev]. Since the survival thresholds in Eqs. (13)–(14) and the β_cr values are computed by replacing M_in with M_acc as the effective initial mass, the authors should derive t_acc and demonstrate t_acc ≪ t_q and t_acc ≪ t_κ^ev for the full ranges of M_in and κ shown in Figs. 1–6. If accretion and evaporation overlap significantly, the quoted 5600/3200/1400 g thresholds and the corresponding DM and ΔN_eff contours would change, even though the qualitative direction of the accretion effect is robust.
[II.B, Eqs. (7), (12)–(14); III, Eqs. (26)–(27)] The claimed limit κ=0, q=1 is not recovered by the general formulas. In Eq. (12), t_κ^ev ∝ [2κ(3+2κ)]^{-1} diverges as κ→0 instead of reducing to the Hawking time M_in^3/(3ϵM_Pl^4); Eq. (14) then gives M_in ≥ 0 at κ=0; and Eqs. (26)–(27) vanish at κ=0. Since κ=0 is shown as a reference case in Figs. 2–4 and 6, the authors should state that the memory-burdened formulas apply for κ>0 and that the κ=0 curves are computed from the separate semiclassical expressions.
[II.C, Eqs. (16)–(17)] The factor q in Eq. (16) accounts for the mass qM_acc at memory-burden onset, but the accretion factor M_acc/M_in appears to be missing. For λ_c≠0, the late-time PBH mass is qM_acc = qM_in(1 − λ_cγ/[2(1+w)])^{-1}, so Eq. (16) should read t_BH = t_in[qβ(1 − λ_cγ/[2(1+w)])^{-1}]^{-(1+w)/(2w)}. Consequently β_cr in Eq. (17) should contain an extra factor (1 − λ_cγ/[2(1+w)]) relative to the printed expression; for relativistic accretion at w=1/3 this changes the quoted β_cr by M_acc/M_in ≈ 4. Please justify the present form or correct it.

minor comments (4)

[Eq. (43)] The exponent in γ^{2w/(1+e)} should be 2w/(1+w); the “e” appears to be a typographical error.
[Sec. III, Eq. (21) and surrounding text] The notation “m_j < t_acc^BH” and “m_j < T_acc^BH” mixes dimensions; t_acc should be T_BH^acc, the black-hole temperature at the start of the memory-burdened phase.
[II.B, Eq. (12)] The statement that “setting κ=0 and q=1 recovers the standard case of Hawking evaporation” should be qualified, because Eq. (12) is singular at κ=0 and the first branch of Eq. (7) is the relevant one in that limit.
[I and IV] There are several typographical errors, e.g., “organisedd” in the Introduction and “lattitude” in the description of Eq. (45); these should be corrected in a proofreading pass.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: survival masses, beta_cr, DM-abundance, and Delta N_eff outputs are closed-form algebraic consequences of the stated input laws with scanned parameters, and the paper's self-citations are corroborative, not load-bearing; the dominant flagged issue is an internal arithmetic inconsistency (Eq. 3 vs Eq.

full rationale

I find no circular step. The paper's outputs (Eqs. 13, 14 survival and BBN bounds; Eq. 17 beta_cr; Eqs. 22-35 DM relic abundances; Eqs. 40-43 Omega_PBH; Eqs. 50-52 Delta N_eff) are closed-form algebraic consequences of two independently stated inputs: the accretion saturation factor Macc/M_in = [1 - lambda_c gamma/(2(1+w))]^{-1} (Eq. 5) and the memory-burdened evaporation law (Eqs. 6-7). The parameters kappa, q, beta, and w are scanned, not fitted to the outputs, and no observational data are used to infer the central factor; the formulas reduce to the known Hawking limit (kappa=0, q=1) and the no-accretion limit (lambda_c=0), so the derivation is self-contained against external benchmarks. The self-citations ([53], [29]) supply standard memory-burden and particle-emission machinery, but the one reused property (independence of the total number of emitted particles from the emission rate, Eq. 22) is re-derived in the text and is corroborated by independent groups ([51], [52]), so no load-bearing self-citation chain holds. A separate, serious internal-consistency problem should be weighed as correctness risk, not circularity: Eq. (3) prints dM/dt = lambda_c (16 pi) M^2 rho_infty / M_Pl^4, but direct integration with rho_infty = 4 M_Pl^2/[3(1+w)^2 t^2] and Eq. (1) yields M(t) = M_in [1 - 128 pi^2 lambda_c gamma/(1+w) (1 - t_in/t)]^{-1}, a coefficient 256 pi^2 ~ 2.5x10^3 times the paper's claimed solution Eq. (4); Eqs. (4)-(5) correspond instead to a coefficient 1/(16 pi). With the printed 16 pi, the w=1/3 relativistic growth factor would be ~1.9x10^3, so the denominator would cross zero almost immediately and no finite Macc would exist, meaning the printed Eq. (3) cannot yield the Macc ~ 4 M_in used in every later result. Every headline number (5600/3200/1400 g survival masses, beta_cr, DM and Delta N_eff contours) is a monotone function of the same factor (1 - lambda_c gamma/[2(1+w)])^{-1}, which makes this coefficient error the paper's dominant correctness risk; but because the outputs are not equivalent to the inputs by construction, this is an arithmetic issue, not circularity.

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

The paper introduces no new particles or forces. The central outputs depend on free parameters kappa, q, beta, w and on inherited models (memory burden, relativistic accretion) whose validity is not established within the paper. The quantitative claims are therefore conditional on these external inputs.

free parameters (4)

Memory burden exponent kappa = scanned over 0, 1, 2
Phenomenological parameter controlling the suppression of evaporation; currently undetermined (Sec. II.B).
Memory burden activation fraction q = 0.5
Fraction of initial mass at which memory burden switches on; adopted representative value, with 0<q<1 allowed (Sec. II.B).
Initial PBH abundance beta = scanned (e.g., 10^-19, 10^-21, 10^-37)
Initial energy density fraction of PBHs; treated as a free parameter (Sec. II).
Background equation of state w = 1/3 and 2/3
The paper presents these as representative toy models; the results depend strongly on w, especially through the accretion factor (Sec. II.A).

assumptions (4)

standard math Standard Hawking evaporation rate for Schwarzschild black holes
The baseline evaporation term dM/dt = -epsilon M_Pl^4/M^2 is assumed (Eq. 6), with graybody factor epsilon approximated as (27/4) g(T)/480.
domain assumption Memory burden model with rate suppression proportional to S^{-kappa} and sudden switch at q M_in
The core quantum effect is taken from Refs. [47-56]; the paper does not derive it from first principles and relies on its validity (Eqs. 6, 7).
domain assumption Steady-state relativistic accretion formula (Eq. 3) from [97,98] applies to PBHs in the early universe
The accretion rate with constant lambda_c and homogeneous background density is assumed; the paper notes the w toward 1 limit breaks down but otherwise adopts the formula wholesale.
ad hoc to paper Timescale separation: accretion saturates before evaporation dominates
Used to construct the piecewise mass evolution in Eq. (9); the paper asserts this separation but does not quantitatively validate it across the scanned parameter range.

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Pith. "Pith review of Relativistic accretion and burdened primordial black holes." pith.science (2026). https://pith.science/paper/DWKSCTCM

@misc{pith2026250702821,
  author       = {Pith},
  title        = {Pith review of: Relativistic accretion and burdened primordial black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DWKSCTCM}},
  note         = {Machine review of arXiv:2507.02821}
}

abstract

We examine the joint effects of relativistic accretion and memory-burdened evaporation on the evolution of primordial black holes (PBHs). The memory burden effect, which delays the evaporation by inducing a backreaction and making the evaporation rate scale as an inverse power law of the PBH entropy, opens up a new window that allows PBHs with $M \lesssim 10^{15}~\mathrm{g}$ to survive until the present epoch. Meanwhile, accretion increases the mass of PBHs, thereby enhancing their chances of survival for a given initial mass. We consider two main scenarios: one where PBHs evaporate completely before big bang nucleosynthesis, and another where PBHs persist until today. In the case of evaporation, we analyze the emission of dark matter (DM) and dark radiation (DR) during the process of evaporation. Conversely, in the other case, the surviving PBHs themselves can contribute as DM. We further investigate how relativistic and nonrelativistic accretion, together with memory-burdened evaporation, impact the parameter space of the emitted DM, the abundance of stable PBHs as DM, and the contribution of DR to the effective number of relativistic degrees of freedom, $\Delta N_{\mathrm{eff}}$.

Figures

Figures reproduced from arXiv: 2507.02821 by the authors.

**Figure 1.** FIG. 1. The evolution of the mass of PBHs under the combined effects of accretion and burdened evaporation is shown for two [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** FIG. 2. The allowed range of the initial PBH mass [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. The mass of DM produced from the evaporation of PBHs is shown as a function of the initial mass of PBHs for the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. The mass of DM produced from the evaporation of PBHs is shown as a function of the initial mass of PBHs for the case [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: FIG. 5. The constraints on the fraction of today’s DM energy density sourced by PBHs, denoted by [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: , we have shown the contribution to ∆Neff from the spin-0 bosonic components for the case of PBH domination. In the case of ∆Neff, the detailed behavior of g ev s with Tev is required, where for the case of PBH domination, the temperature at evaporation reads Tev = …

Discussion (0). Continue with ORCID to comment.

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