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Burdening (or not) gravitational waves in the presence of primordial black holes

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper argues that a PBH-reheating universe writes a four-part gravitational-wave fossil record, readable band by band, and that a memory-burden double peak would expose the back-reaction.

desk verdict A genuinely useful unburdened GW template, but the burdened double-peak diagnostic is algebraically wrong and needs a major fix. read the letter →

arxiv 2509.02701 v1 pith:ICX5APYQ submitted 2025-09-02 hep-ph astro-ph.COgr-qc

Mathieu Gross , Md Riajul Haque , Yann Mambrini This is my paper · ORCID
classification hep-phastro-ph.COgr-qc
keywords primordialblackholesgravitationalwavebackgroundPBHreheatingmemoryburdenHawkingevaporationscalar-inducedwavesinflatonscatteringstochastic
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 argues that in a universe where primordial black holes (PBHs) reheat the cosmos after inflation, the stochastic gravitational-wave background observed today is the sum of four physically distinct sources, each leaving a recognizable imprint in a different frequency band. It derives analytic shapes and characteristic frequencies for each component: inflationary tensor modes that re-enter during radiation, PBH domination, or inflaton oscillations; gravitational waves from inflaton scattering at the highest frequencies; scalar-induced waves from the Poisson density fluctuations of the PBH population; and gravitons emitted directly by Hawking evaporation. It then redoes the calculation under the memory-burden hypothesis, in which evaporation is slowed once the hole has radiated a significant fraction of its mass, and shows that the burdened spectrum is pushed toward lower frequencies and acquires a double evaporation peak. If the picture is right, a future broadband gravitational-wave measurement would read out the reheating temperature, the PBH mass and abundance, the inflaton equation of state, and possibly the memory-burden parameters $n$ and $q$.

What carries the argument

The load-bearing machinery is the PBH evaporation law. Semiclassically $dM_{\rm BH}/dt = -\epsilon M_P^4/M_{\rm BH}^2$; the memory-burden generalization divides this by $S^n$, where $S=\frac12 (M_{\rm BH}/M_P)^2$ is the black-hole entropy, introducing two free parameters: $q$, the mass fraction at which burdening begins, and $n$, the power of the suppression. This single ansatz fixes the reheating temperature and the duration of PBH domination, and through them every characteristic frequency in the spectrum. Together with transfer-function methods for tensor modes produced during inflation, graviton production by inflaton scattering, and scalar (Poisson) density fluctuations, it produces the region-by-region shapes A1 through D$_2$ in Fig. 1.

What would settle it

Search for the spectrum with a detector sensitive to $\Omega_{\rm GW}h^2\simeq10^{-18}$ across $10^{-5}$--$10^7$ Hz and check for the flat A1 plateau, the $f_0^{-2}$ slope in A2, the $f_0^{2/5}$ slope in A3, the B1/B2 peak at $f_{\rm UV}^0\simeq 4.0\times10^6\,(M_{\rm BH}/1\,{\rm g})^{-5/6}$ Hz, and the evaporation bump near $f_{{\rm ev}}^0\simeq 1.5\times10^{13}\,(M_{\rm BH}/1\,{\rm g})^{1/2}$ Hz. For the burden picture, look for a double evaporation peak with $f_{{\rm ev}2}^0/f_{{\rm ev}1}^0\simeq 1.2\, q^{1+4n/3}\,[2n(3+2n)]^{2/3}(M_{\rm in}/M_P)^{4n/3-1}$; a single peak would rule out the two-stage burdened evaporation, while a different ratio would fix $n$ at a value inconsistent with the power-law ansatz.

Watch

Extended reading notes

Core claim

The paper's central claim is a decomposition. With a PBH population that dominates and reheats the Universe, the present-day gravitational-wave abundance $\Omega_{\rm GW}h^2$ as a function of present frequency $f_0$ is assembled from region A1 (a flat plateau from modes entering during radiation), A2 (falling as $f_0^{-2}$ from modes entering during PBH domination), A3 (rising as $f_0^{(6w_\phi-2)/(3w_\phi+1)}$ from modes entering during inflaton oscillations), a sudden cutoff C from modes that never left the horizon, the peak B1/B2 from scalar-induced waves sourced by PBH number-density fluctuations, and a high-frequency evaporation bump D1. The paper verifies this architecture for potentials $V(\phi)\propto \phi^k$ during reheating, with $k=6$ as the benchmark. Under memory burden it claims the same architecture survives but with reheating occurring later and at lower frequency: the gap ratio $f_{\rm UV}^0/f_{\rm IR}^0$ grows from about $4200$ to about $3.3\times 10^6$ for $n=1$, $q=1/2$, and the evaporation bump splits into two peaks D$'_1$ and D$'_2$ whose frequency ratio $f_{{\rm ev}2}^0/f_{{\rm ev}1}^0 \simeq 1.2\, q^{1+4n/3}\,[2n(3+2n)]^{2/3}\,(M_{\rm in}/M_P)^{4n/3-1}$ gives direct access to $n$.

Load-bearing premise

The entire burdened half of the paper rests on the ansatz that once a PBH has radiated a fraction $1-q$ of its mass, its evaporation rate is suppressed by an inverse power of its entropy, $S^{-n}$, with $q$ and $n$ free parameters and benchmark $q=1/2$, $n=1$; the paper itself states that there are no real microscopic motivated values for $q$.

Editorial extensions

If this is right

  • The flat A1 plateau amplitude, $\Omega_{\rm A1} h^2\simeq 2.2\times10^{-18}$, directly measures the inflationary Hubble scale $H_{\rm end}$, independent of the PBH parameters.
  • The break frequencies $f_{\rm RH}^0$ and $f_{\rm BH}^0$ and the A3 slope encode the sequence of eras and the inflaton equation of state: for $V(\phi)\propto\phi^6$, $\Omega_{\rm GW}\propto f_0^{2/5}$ in region A3.
  • The density-fluctuation peak B1/B2 scales as $f_{\rm UV}^0\simeq 4.0\times10^6\,(M_{\rm BH}/1\,{\rm g})^{-5/6}$ Hz and $f_{\rm IR}^0\simeq 947\,(M_{\rm BH}/1\,{\rm g})^{-3/2}$ Hz in the semiclassical case, so its position measures the PBH mass.
  • The evaporation bump D1 has an amplitude $\simeq 7.5\times10^{-7}$ that is independent of $M_{\rm BH}$, making it a clean marker of a PBH reheating phase even when other features are obscured.
  • With memory burden, the whole spectrum is enlarged toward lower frequencies, and the double-peak ratio $f_{{\rm ev}2}^0/f_{{\rm ev}1}^0$ provides a direct measurement of the burden exponent $n$.

Reading between the lines

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

  • The paper itself notes two simplifying choices: it assumes the burden effect begins during the PBH-dominated era and it neglects accretion. Either assumption would shift the predicted frequencies, so testing them is necessary before precise data comparisons can be drawn.
  • Beyond the paper: the double peak is the most falsifiable burden prediction, but its exact ratio depends on the power-law form $S^{-n}$; a two-peak signal with a different ratio would indicate a different back-reaction law rather than ruling out memory burden itself.
  • Beyond the paper: if the high-frequency evaporation peak is ever detected at the predicted amplitude, it could serve as a standard candle for the reheating epoch even if lower-frequency regions remain below detector sensitivity.
  • Beyond the paper: the same machinery could be applied to non-Poisson or strongly clustered PBH initial conditions, which would change the slope of the B1/B2 spectrum and could mimic or mask the memory-burden signature.
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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 computes the total present-day gravitational-wave background from a post-inflationary phase in which primordial black holes reheat the Universe while the inflaton oscillates in a V(phi) proportional to phi^k potential. It combines four contributions: inflationary tensor modes re-entering during radiation, PBH domination, and inflaton domination (regions A1-A3); gravitons from inflaton scattering (region C); scalar-induced gravitational waves from PBH number-density fluctuations (regions B1/B2); and direct Hawking graviton emission (region D1). A memory-burden extension is added in which evaporation is suppressed by a factor 1/S^n after the PBH mass drops below q M_in, producing the additional features D'_1 and D'_2. The advertised observational signatures are the low-frequency enlargement of the spectrum under memory burden, the frequency ratio f_UV0/f_IR0, and especially the double-peak ratio f_ev2,0/f_ev1,0, which is claimed to give 'a direct access to n'.

Significance. If the central formulas were correct, this would be a useful unified template for gravitational-wave searches in PBH-reheating scenarios. The paper has genuine strengths: the PGW matching between eras is internally consistent (for example, Eq. (41) and Eq. (45) meet at f_BH0); the authors explicitly flag the non-linear cutoff ambiguity in Appendix D.4 and display its impact in Fig. 9; and they are transparent about the absence of a microscopic derivation for gamma and for the burden-onset parameter q. The burdened UV/IR relation Eq. (79) follows from Eqs. (76) and (46). However, the diagnostic that the paper emphasizes most, the two-peak ratio of Eq. (92), is not supported by the displayed algebra, so the central memory-burden conclusion is not yet established.

major comments (3)
  1. [Sec. VI.B, Eq. (92)] The ratio f_ev2,0/f_ev1,0 is announced as the direct probe of n, but Eq. (92) does not follow from Eqs. (85) and (90). Dividing Eq. (90) by Eq. (85) as printed gives f_ev2,0/f_ev1,0 = 3^(-5/6) 2^(-7n/6) (3+2n)^(-2/3) q^(2n/3) (M_in/M_P)^(4n/3), whereas Eq. (92) states 1.2 q^(1+4n/3) [2n(3+2n)]^(2/3) (M_in/M_P)^(4n/3-1). The two expressions disagree in the q exponent, the mass exponent, and the n=0 prefactor. In the semiclassical limit n=0, q=1, both Eqs. (85) and (90) are claimed to reduce to Eq. (82), so the ratio must tend to 1; Eq. (92) instead gives 0 through the factor (2n)^(2/3). Since Sec. VI and the abstract use this ratio as the observational signature of memory burden, this is a load-bearing error that must be repaired.
  2. [Sec. VI.B, Eqs. (85) and (90)] The two defining equations are inconsistent in the limit they are both stated to reproduce. Substituting n=0, q=1 into Eq. (85) gives a prefactor proportional to 3^(7/12), while Eq. (90) gives a prefactor proportional to 3^(-1/4) for the same physical frequency; the two expressions therefore differ by a factor 3^(5/6) and cannot both equal Eq. (82). The origin appears to lie in the orientation of the scale-factor ratio in Eq. (86) and in the q- and 3-dependences of Eq. (85). The double-peak calculation should be re-derived from the definitions of a_ev1, a_RH, and T_RH before the benchmark frequencies in Eqs. (88) and (91) are quoted.
  3. [Sec. VI.B and Sec. II.C] The claim that observing the double peak gives 'a direct access to n' should be qualified by the explicit model dependence of Eq. (12). Since q and n are free parameters and the paper itself states that there are no microscopic motivated values for q, the peak positions and their ratio are predictions of the power-law entropy-suppression ansatz, not of memory burden in general. The paper should present at least a concrete cross-check, for example the behavior of f_ev2,0/f_ev1,0 as n tends to 0 at fixed q, and should state what would change for a non-power-law suppression of the evaporation rate.
minor comments (5)
  1. [Sec. IV.B] The sentence 'on needs to solve the Boltzmann equation' should read 'one needs to solve the Boltzmann equation'.
  2. [Sec. V.A] The word 'wich' should be 'which' in the sentence describing the scalar fluctuations.
  3. [Appendix B.2] The word 'sumarized' should be 'summarized', and the sentence 'The usual workaround for PBH evaporation is to get the generic instantaneous emission before integrating it with respect to the energy in order to get the proper mass evolution' is grammatically incomplete.
  4. [Abstract] The phrase 'For the first time' is a strong claim; given the overlap with Refs. [30, 80, 81], the phrasing should be softened or the precise new element should be stated explicitly.
  5. [Eq. (84)] The parenthetical 'with g_RH = 106.75' is not closed; please close the parenthesis.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the combined GW spectrum is derived from explicit ansätze and re-derived inputs, with the reported Eq. (92) inconsistency being a non-circular algebraic error.

full rationale

The derivation chain is self-contained rather than circular. The burdened spectrum follows from the stated memory-burden ansatz, Eq. (12), whose parameters q and n are explicitly acknowledged as unconstrained: the paper says 'There are no real microscopic motivated values for q' (Sec. II.C). Choosing q=1/2, n=1 as benchmarks is a model-parameter choice, not a fit to the predicted GW frequencies, so the burdened predictions are conditional consequences, not inputs renamed as outputs. The PGW, inflaton-scattering, scalar-fluctuation, and PBH-evaporation components are derived in the main text and appendices (e.g., critical beta in Appendix B, inflaton scattering in Appendix C, evaporation spectra in Appendix B.4, and scalar-induced GW in Appendix D), so the self-citations to [16,28,30,69-73] are not load-bearing in a circular sense: the cited results are external publications and are largely re-derived here. The central claim that a double peak would give 'a direct access to n' is a model prediction, not a definitional identity. One important caveat is non-circular: the algebra behind the headline ratio Eq. (92) appears internally inconsistent, since substituting n=0, q=1 does not reproduce the semiclassical limit and direct division of Eq. (90) by Eq. (85) yields different prefactors and exponents; that is a correctness/mathematical-error concern, not a circularity, and under the stated rules it does not warrant a circularity score above zero.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The central claim rests on a small number of scenario parameters (q, n, Min, beta, w_phi, Hend, gamma) and on several domain assumptions inherited from the PBH and GW literature. The paper is transparent about the weakest ones: the memory burden ansatz has no microscopic derivation and gamma is a conventional estimate. No new particles or forces are introduced.

free parameters (7)
  • q = 1/2
    Fraction of initial mass at which memory burden begins; benchmark from refs [17,19], no microscopic derivation (Sec. II.C).
  • n = 1
    Power of the entropy suppression 1/S^n in Eq. (12); benchmark chosen for figures, also n=1/2 shown in appendix.
  • Min = 1 g
    Initial PBH mass benchmark; scans include 1e2 and 1e4 g; constrained range 1 g to 1e9 g (Sec. II.A).
  • beta = 1e-5 (unburdened), 1e-8 (burdened)
    Initial PBH fraction; chosen above beta_c to ensure PBH domination before evaporation; controls f_BH0 and the A3 amplitude.
  • w_phi = 1/2
    Inflaton equation of state during reheating, corresponding to V ~ phi^6; benchmark for Fig. 1; varied to 1/3 and 1 in Figs. 5-6.
  • Hend = 2.5e-6 M_P
    Inflationary Hubble scale benchmark, affecting the overall PGW amplitude; chosen within tensor-to-scalar constraints.
  • gamma = w^{3/2}
    Carr-Hawking collapse efficiency in Min = 4 pi gamma M_P^2 / H_in; paper notes accurate determination is lacking (Appendix B.1).
assumptions (7)
  • ad hoc to paper Memory burden evaporation ansatz, dMBH/dt = (1/S^n) dMBH/dt|semiclassical (Eq. 12), with onset at MBH = q Min.
    Introduced without microscopic derivation; the paper states there are no motivated values for q and models the suppression by analogy with information theory.
  • domain assumption Carr-Hawking collapse efficiency, Min = 4 pi gamma M_P^2 / H_in with gamma = w^{3/2} (Eq. B6).
    Conventional estimate; appendix B.1 admits a complete analytical treatment in the post-inflationary regime is lacking.
  • domain assumption PBHs are uncorrelated, giving Poisson density fluctuations with P_Phi(k) ~ (k/k_UV)^3 (Eq. D7).
    The B1/B2 spectra rely on this Poisson isocurvature power spectrum from [76].
  • domain assumption Greybody factor fixed at the geometrical-optics value 27/4 in the mass-loss equation (Eq. 1).
    Standard approximation; the paper notes more complex greybody factors exist in the literature.
  • domain assumption Accretion onto PBHs is neglected.
    Stated in Sec. II.A; cited works [45,46,86,87] indicate accretion can enhance mass growth by an order of magnitude, with phenomenological implications.
  • domain assumption Inflaton scattering computed with the equivalence between Feynman diagrams and Bogoliubov modes for k >> k_end (Sec. IV).
    Used for region C; relies on [62,68]; the paper also uses the exact mode sum in its numerics.
  • domain assumption Instantaneous evaporation approximation for peak-frequency estimates in Secs. V and VI.
    Used to derive fIR, fUV and the D1 peak; the paper acknowledges the approximation is valid near the peak where Gamma ~ H.

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Cite this review

Pith. "Pith review of Burdening (or not) gravitational waves in the presence of primordial black holes." pith.science (2026). https://pith.science/paper/ICX5APYQ

@misc{pith2026250902701,
  author       = {Pith},
  title        = {Pith review of: Burdening (or not) gravitational waves in the presence of primordial black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ICX5APYQ}},
  note         = {Machine review of arXiv:2509.02701}
}
abstract

We present the spectrum of primordial gravitational wave (GW) expected from the presence of primordial black holes (PBH) and inflaton in the early Universe. For the first time, we combine the waves produced by the PBH decay, with their density fluctuation counterpart, as well as their effects on the GW produced by the inflaton {\it after} (high frequency modes) and {\it before} (low frequency modes) the end of inflation. We generalize our study for a potential $V(\phi)\propto \phi^k$ during reheating. We also extend our study, taking into account a possible memory burden effect to see how it can affect the shape of the spectrum.

Figures

Figures reproduced from arXiv: 2509.02701 by the authors.

Figure 1
Figure 1. Minimal Gravitational wave spectra coming from a [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Example of a gravitational process contributing to [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. GW spectra in the unburden case with wϕ = 1/2 for M = 1, 102 and 104 g respectivelly for black, red and green with β = 10−5 , 10−6 and 10−7 10−5 10−1 103 107 1011 1015 1019 1023 f0(Hz) 10−24 10−21 10−18 10−15 10−12 10−9 10−6 10−3 100 ΩGW h 2 LISA ET BBO Planck 18 COrE/Euclid CVL Res. Cavities GB β > βcrit [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: GW spectra in the burden case with wϕ = 1/2 for M = 1, 102 and 104 g respectivelly for black, red and green with β = 10−8 , 10−13 and 10−17 10−5 10−1 103 107 1011 1015 1019 1023 f0(Hz) 10−24 10−21 10−18 10−15 10−12 10−9 10−6 10−3 100 ΩGW h 2 LISA ET BBO Planck 18 COrE/…
Figure 5
Figure 5. Figure 5: GW spectra in the unburdened case with wϕ = 1/3, 1/2 and 1 respectivelly for black, red and green with β = 10−4 , 10−5 and 10−9 . The PBH mass is fixed to M = 1g [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: GW spectra in the burden case with wϕ = 1/3, 1/2 and 1 respectivelly for black, red and green with β = 10−7 , 10−8 and 10−13. The PBH mass is fixed to M = 1g the possibility of memory burden effects in the process of PBH evaporation. We then distinguished typical signa…
Figure 7
Figure 7. Figure 7: The restriction on the reheating temperature for [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: The reheating temperature as a function of [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Fig [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]

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Reference graph

Works this paper leans on

139 extracted references · 3 canonical work pages · cited by 1 Pith paper

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    The aim of this appendix is to give to the reader the necessary tools to understand section III

    The spectrum For the reader interested by a detailed analysis of the PGW spectrum, we refer to [61, 88–94]. The aim of this appendix is to give to the reader the necessary tools to understand section III. Here, primordial gravitational waves refer to those generated as tensor perturbations due to vacuum fluctuations during inflation, in the ab- sence of a...

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    We now analyze how small TRH can be while remaining consistent with bounds on ∆ Neff, as- suming fixed values of the equation of state wϕ and the 16 inflationary energy scale

    Constraints on reheating For a stiff post-inflationary equation of state, the re- heating temperature TRH can be constrained by the ad- ditional relativistic degrees of freedom sourced by grav- itational waves. We now analyze how small TRH can be while remaining consistent with bounds on ∆ Neff, as- suming fixed values of the equation of state wϕ and the ...

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    However, if the initial PBH fraction β exceeds a critical threshold βc, the constraint on the reheating temperature TRH given in Eq

    PBH domination Up to this point, we have considered a scenario where PBHs do not dominate the energy density before evap- orating. However, if the initial PBH fraction β exceeds a critical threshold βc, the constraint on the reheating temperature TRH given in Eq. (A24) can be relaxed. This is because in a PBH-dominated phase, typically a matter-dominated ...

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    In this framework, the mass dissipation rate is suppressed by the black hole entropy as ∼ 1/Sn (for details, see the next section)

    Including the memory burden effect Now, let us modify the scenario by considering a phaseofmemoryburden that sets in after the black hole has lost a fraction q of its initial mass, instead of using the standard semiclassical approximation (i.e., Hawking evaporation). In this framework, the mass dissipation rate is suppressed by the black hole entropy as ∼...

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    PBH reheating After inflation ends, the inflaton field typically under- goes coherent oscillations around the minimum of its po- tential. The effective background equation of state, wϕ, depends on the shape of the potential, while the reheating duration is controlled by how rapidly the inflaton decays into Standard Model particles. In the standard picture...

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    Hawking evaporation The complete mass evolution of the PBH is described by a first phase of usual Hawking evaporation up to a time tq after which it will follow the burdened evolution described by Eq. (14). overall the mass evolution can be sumarized in the following form MBH(t) =Min 1− Γ(0) BH(t−tin) 1 3 θ(tq−t) +qMin h 1− Γ(n) BH(t−tq) i 1 3+2n θ(t−tq)....

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    The whole work is to properly integrate the contribution of the PBH popula- tion using the instantaneous emission relation given in Eq

    The spectrum Let’s now have a look at the shape of the PBH evap- oration spectra in more details. The whole work is to properly integrate the contribution of the PBH popula- tion using the instantaneous emission relation given in Eq. (B14). The GW abundance can be reconstructed using dρBH GW dtdk =nBH(t)k dNi dtdk . (B15) Since the memory burden affects o...

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    This peak has therefore a fixed frequency which depends only on the mass whileD′ 1 feels on top, the redshift aq/aRH which implies a dependence on the burden parameters

    We can at this level grasp a fundamental difference between D′ 1 and D′ 2 since D′ 2 is tied to the reheating temperature which is fixed by the last stage of PBH evaporation. This peak has therefore a fixed frequency which depends only on the mass whileD′ 1 feels on top, the redshift aq/aRH which implies a dependence on the burden parameters

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    (B16), we can compute the main peak value for the PBH evaporation ( D1 and D′ 2)

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