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REVIEW 4 major objections 3 minor 3 cited by

Primordial black hole induced gravitational waves in $f(R)$ gravity

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper argues that in R^(1+ε) f(R) gravity, scalar perturbations from ultra-light primordial black holes grow exponentially during the early matter-dominated era, producing an induced gravitational-wave signal with a distinctive low-fre

desk verdict A new frequency scaling from R^(1+ε) gravity that deserves a close look, but the exponential growth claim is unverified in the abstract. read the letter →

arxiv 2508.03939 v1 pith:LXHTAM2U submitted 2025-08-05 astro-ph.CO astro-ph.HEgr-qchep-th

Theodoros Papanikolaou , Salvatore Capozziello This is my paper
classification astro-ph.COastro-ph.HEgr-qchep-th
keywords primordialblackholesinducedgravitationalwavesf(R)gravityR^(1+ε)earlymatter-dominatederascalarperturbationswavedetectors
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 show that a minimal modification of general relativity, the R^(1+ε) f(R) model, can dramatically amplify the gravitational waves emitted by tiny primordial black holes. In standard relativity, such black holes create an early matter-dominated phase and then evaporate, leaving behind a second-order gravitational-wave background from their density fluctuations. The paper claims that in R^(1+ε) gravity these fluctuations grow exponentially instead of the slower growth of GR, so even a very small black hole abundance produces a strong signal. The predicted spectrum rises linearly with frequency at low frequencies, Ω_GW ∝ f, and features a sharp spike near the nonlinear cutoff, placing it within the sensitivity of LISA, ET, BBO and SKA.

What carries the argument

The central mechanism is the growth of scalar perturbations in f(R) gravity during a matter-dominated universe. The R^(1+ε) model introduces an effective instability for the extra scalar degree of freedom carried by f(R) gravity, making the density contrast grow exponentially on subhorizon scales rather than with the power-law growth of general relativity. This amplified scalar source feeds the second-order tensor modes, producing the enhanced and spectrally shaped induced gravitational-wave background.

What would settle it

A second-order or nonlinear calculation of scalar perturbations in R^(1+ε) gravity during the matter-dominated era showing that the exponential growth is cut off by backreaction well before the claimed cutoff scale; for instance, if the one-loop correction to the effective equation reduces the growth rate substantially, the predicted gravitational-wave spectrum would not reach the amplitudes or the linear tail quoted in the paper.

Watch

Extended reading notes

Core claim

The central claim is that in the R^(1+ε) f(R) gravity model with ε≪1, scalar perturbations sourced by primordial black hole isocurvature fluctuations grow exponentially during the early matter-dominated era before Big Bang Nucleosynthesis. Because induced gravitational waves at second order are generated by products of scalar perturbations, this exponential growth enhances the gravitational-wave spectrum even for very small initial PBH abundances. The resulting Ω_GW(k) has a characteristic shape: a linear frequency dependence on infrared scales away from the peak, Ω_GW ∝ f for f ≪ f_peak, and a spiky feature near the nonlinear cutoff scale where perturbation theory breaks down. The paper fur

Load-bearing premise

The assumption that scalar perturbations in R^(1+ε) gravity keep growing exponentially as described by linear perturbation theory up to the nonlinear cutoff scale; if nonlinearities or backreaction quench the growth earlier, the predicted enhancement and the Ω_GW ∝ f low-frequency scaling would be suppressed or altered.

Editorial extensions

If this is right

  • If the claim holds, induced gravitational waves from ultra-light primordial black holes become a sensitive probe of R^(1+ε) gravity, even at black hole abundances too small for other tests.
  • The predicted Ω_GW ∝ f low-frequency tail is a distinctive observational fingerprint that separates R^(1+ε) gravity from general-relativistic predictions.
  • The spiky peak near the nonlinear cutoff marks the edge of perturbation theory, so the shape and height of the peak carry information about the nonlinear scale and the viability of the model.
  • A signal in the LISA, ET, BBO or SKA bands would allow direct constraints on the parameter ε and on the primordial black hole abundance during the early matter-dominated era.
  • The paper's contrast with R^2 gravity suggests that only certain curvature corrections to GR leave an observable imprint in this channel; R^2 models remain effectively indistinguishable from GR here.

Reading between the lines

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

  • If exponential growth of scalar perturbations is generic in some class of modified gravity theories, then strong induced gravitational-wave backgrounds during early matter domination may be a common feature, not unique to R^(1+ε); the linear low-frequency scaling could serve as a template for such models.
  • The exact height of the spiky peak depends on the choice of the nonlinear cutoff scale, which is not computed from first principles in the paper; a precise prediction requires modeling backreaction and nonlinear saturation.
  • One could test the mechanism by numerically solving the full nonlinear scalar perturbation equations in R^(1+ε) gravity during matter domination: if backreaction quenches the exponential growth before the claimed cutoff, the predicted amplitude and slope would be suppressed.
  • The same second-order tensor source mechanism might amplify gravitational waves from other isocurvature sources, such as domain walls or cosmic strings, if those sources are modulated by an exponential-growth scalar mode in modified gravity.
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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

4 major / 3 minor

Summary. The manuscript studies second-order gravitational waves induced by scalar perturbations sourced by primordial black hole (PBH) density fluctuations during an early matter-dominated era, in the context of f(R) modified gravity. The abstract reports that for the R^2 model deviations from general relativity are small, while for R^(1+ε) with ε≪1 scalar perturbations grow exponentially during matter domination. This growth is claimed to enhance the induced GW signal even for very small PBH abundances, yielding a distinctive infrared scaling Ω_GW(k) ∝ f for f ≪ f_peak and a spiky feature near the non-linear cutoff, with detectability by LISA, ET, BBO and SKA.

Significance. If the claimed exponential growth and the resulting GW phenomenology are correct, the paper would identify a new observational signature of f(R) gravity that is qualitatively different from standard PBH-induced GW predictions. The predicted linear IR slope and spiky peak would be falsifiable by future detectors, which gives the work potential significance. However, the current manuscript—at least as represented by the abstract—does not provide the derivations, parameter choices, or consistency checks needed to establish these claims.

major comments (4)
  1. [Abstract] The central claim that 'R^(1+ε) gravity features ... an exponential growth of scalar perturbations during a matter-dominated era' is stated without any supporting equation or derivation. This exponential growth is the linchpin of the entire paper: it is what converts a small initial PBH abundance into an enhanced GW signal. The manuscript must provide the linearized f(R) field equations, the scalaron effective mass, the evolution equation for the curvature perturbation during the eMD era, and the explicit solution exhibiting exponential growth. It must also specify the gauge and initial conditions. Without this, the claimed enhancement is not established.
  2. [Abstract] The abstract itself acknowledges a 'non-linear cut-off scale below which perturbation theory breaks down,' and the spiky behaviour is claimed to occur 'close to' that scale. This is exactly the regime where the exponential growth is supposed to produce the enhanced signal. The manuscript must show that the growth is not an artifact of linear perturbation theory and that non-linearities or backreaction do not quench it before the scales relevant to the GW peak. If the growth is truncated earlier, the predicted Ω_GW ∝ f scaling and the peak shape would not hold. This is a load-bearing issue that needs to be addressed with a controlled non-linear or backreaction estimate.
  3. [Abstract] The detectability claim for LISA, ET, BBO and SKA is not quantitatively substantiated. The amplitude and peak frequency of the induced GW spectrum depend on at least the f(R) parameter ε, the PBH mass, the initial PBH abundance, and the reheating temperature. The abstract does not report the values used or the resulting peak amplitudes relative to the noise curves. The manuscript needs a parameter-space scan showing for which ranges of ε and PBH parameters the signal falls above the sensitivity curves, and it must demonstrate that those parameters are consistent with BBN, CMB, and other constraints. Otherwise the claim 'well within the sensitivity curves' is unsupported.
  4. [Abstract] The claimed infrared scaling Ω_GW(k) ∝ f for f ≪ f_peak is a distinctive quantitative prediction and needs to be derived explicitly. In standard PBH-induced GW scenarios the IR tail typically has a different power, so this linear scaling must follow from the scalar source spectrum, the transfer functions, and the f(R) modification. The manuscript should show the calculation leading to this scaling and explain why the non-linear cutoff does not contaminate the IR regime. As written, the claim is an assertion rather than a demonstrated result.
minor comments (3)
  1. [Abstract] The phrase 'induce abundantly gravitational waves' is grammatically awkward; suggest 'induce gravitational waves abundantly' or 'produce a copious gravitational-wave background.'
  2. [Abstract] Quantities such as f_peak, ε, and 'very small initial PBH abundances' should be defined quantitatively in the abstract or introduction. Without numerical values, the reader cannot gauge the strength of the claimed enhancement.
  3. [Abstract] The phrase 'the initial isocurvature in nature PBH energy density fluctuations' is unclear. Presumably the authors mean that PBH density fluctuations are initially isocurvature perturbations, but this should be stated precisely.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found; the f(R) gravity effects are derived consequences rather than fitted inputs.

full rationale

Based on the material in scope (the abstract), the derivation chain runs from the f(R) action (R^2 and R^{1+ε} models) through the behaviour of scalar perturbations in a PBH-driven matter-dominated era to the second-order induced gravitational-wave spectrum. The model parameters (ε, PBH mass/abundance) are physical inputs; the claimed Ω_GW(k) ∝ f scaling and spiky peak are outputs of the perturbation calculation, not definitions of those inputs. No equation in the provided text equates the prediction with a fitted parameter or with an adopted ansatz by construction. The abstract explicitly flags the non-linear cut-off below which perturbation theory breaks down; that is a validity limitation, not a circular step, because it restricts the regime of the derived result rather than supplying the result itself. No load-bearing self-citation is visible in the abstract, and comparing the final spectrum to LISA/ET/BBO/SKA sensitivity curves is an external benchmark, which does not introduce circularity. Therefore no significant circularity is identified.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The model parameters (ε, PBH mass, PBH abundance) are inputs that the central prediction depends on. The axioms are background assumptions about the gravity theory and the cosmological scenario. No new entities are introduced beyond the known f(R) gravity and PBHs.

free parameters (3)
  • ε = ε << 1
    Model parameter in R^(1+ε) gravity; chosen to be small, but the abstract does not specify a numerical value.
  • PBH mass threshold = M_PBH < 5e8 g
    The abstract states this threshold for ultra-light PBHs to trigger an early matter-dominated era before BBN.
  • Initial PBH abundance = very small
    The paper claims an enhanced GW signal even for very small initial PBH abundances, but 'very small' is not quantified in the abstract.
assumptions (4)
  • domain assumption General relativity is extended to f(R) gravity with minimal models R^2 and R^(1+ε)
    The analysis assumes these f(R) models describe the gravitational interaction during the early universe.
  • domain assumption PBH evaporation drives an early matter-dominated era before BBN
    The abstract posits that ultra-light PBHs trigger an eMD era and reheat through evaporation.
  • domain assumption Initial PBH energy density fluctuations are isocurvature
    The abstract states that 'initial isocurvature in nature PBH energy density fluctuations' are the source of scalar perturbations.
  • standard math Standard second-order perturbation theory for induced GWs applies in f(R) gravity
    The computation of second-order induced tensor modes relies on perturbative methods extended from GR to f(R).

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Cite this review

Pith. "Pith review of Primordial black hole induced gravitational waves in $f(R)$ gravity." pith.science (2026). https://pith.science/paper/LXHTAM2U

@misc{pith2026250803939,
  author       = {Pith},
  title        = {Pith review of: Primordial black hole induced gravitational waves in $f(R)$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LXHTAM2U}},
  note         = {Machine review of arXiv:2508.03939}
}
abstract

Ultra-light primordial black holes (PBHs) with masses $M_\mathrm{PBH}<5\times 10^{8}\mathrm{g}$ can trigger an early matter-dominated (eMD) era before Big Bang Nucleosynthesis (BBN) and reheat the Universe through their evaporation. Notably, the initial isocurvature in nature PBH energy density fluctuations can induce abundantly gravitational waves (GWs) due to second-order gravitational effects. In this work, we study this induced GW signal within the context of $f(R)$ gravity theories investigating the effect of $f(R)$ gravity on the behaviour of scalar perturbations during the PBH-driven eMD era as well as on the source of the second-order induced tensor modes. In particular, we focus on two very minimal $f(R)$ models, that is $R^2$ and $R^{1+\epsilon}$, with $\epsilon\ll 1$, gravity theories as illustrative examples, finding at the end that $R^2$ gravity presents very small deviations from general relativity (GR) at the level of both the scalar and tensor perturbations. However, $R^{1+\epsilon}$ gravity features a different behaviour, exhibiting in particular an exponential growth of scalar perturbations during a matter-dominated era. This unique feature leads ultimately to an enhanced induced GW signal even for very small initial PBH abundances, being characterized by a linear frequency scaling on large infrared scales away from the peak frequency, i.e. $\Omega_\mathrm{GW}(k)\propto f$ for $f\ll f_\mathrm{peak}$, and a spiky behaviour close to the non-linear cut-off scale below which perturbation theory breaks down. Interestingly, one finds as well that the induced GW signal can be well within the sensitivity curves of GW detectors, namely that of LISA, ET, BBO and SKA.

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