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arxiv: 2607.01818 · v1 · pith:L3K2YTPDnew · submitted 2026-07-02 · 🌌 astro-ph.CO · gr-qc· hep-ph

Gravitational Waves from Primordial Black Holes: Connecting Low-Frequency Scalar-Induced Signatures to High-Frequency Binary Mergers

Ashu Kushwaha This is my paper

Pith reviewed 2026-07-03 06:58 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-ph
keywords primordial black holesscalar-induced gravitational wavesbinary mergersISCO frequencycurvature perturbationsstochastic gravitational wave backgroundgravitational wave spectrum
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The pith

For monochromatic primordial black holes, scalar-induced gravitational waves connect directly to binary merger signals through the ISCO frequency, with the merger peak at 1.79 times that frequency independent of mass.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the same enhanced curvature perturbations producing primordial black holes generate both a low-frequency stochastic background of scalar-induced gravitational waves and high-frequency waves from subsequent PBH binary mergers. It derives a model-independent link between the SIGW frequency scale and the innermost stable circular orbit frequency of the binaries for a monochromatic mass function. The peak frequency of the full merger spectrum is fixed at 1.79 times the ISCO frequency because emission is strongest near that orbit. Ellipsoidal collapse produces a stronger SIGW signal than spherical collapse when PBH abundance constraints are applied to the curvature spectrum. This framework allows the same early-universe fluctuations to be tested in widely separated frequency bands.

Core claim

For a monochromatic PBH mass function, the SIGW frequency corresponds to the ISCO frequency of the binaries, and the peak of the merger GW spectrum satisfies f_peak = 1.79 f_ISCO independent of binary masses, unifying the low-frequency scalar-induced background with the high-frequency merger signal from the same primordial fluctuations.

What carries the argument

The mass-independent relation f_peak = 1.79 f_ISCO arising from maximal GW emission near the ISCO, which maps the SIGW frequency scale onto the binary merger spectrum.

If this is right

  • Ellipsoidal collapse yields a substantially stronger SIGW background than spherical collapse under the same PBH abundance constraints.
  • PBH abundance limits on the primordial curvature power spectrum translate directly into predictions for both the low-frequency SIGW amplitude and the high-frequency merger rate.
  • The same set of primordial fluctuations can be constrained simultaneously in low-frequency and high-frequency GW bands.
  • The ISCO-to-peak relation holds regardless of the specific binary masses as long as the mass function remains monochromatic.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Multi-band GW observations could test whether a single early-universe fluctuation spectrum accounts for both signals without additional assumptions about PBH clustering.
  • Relaxing the monochromatic assumption would likely decorrelate the two frequency bands, offering a way to bound the width of the PBH mass function from joint non-detections.
  • The framework suggests that SIGW upper limits could already restrict the expected merger rate in the high-frequency band even before direct merger detections.

Load-bearing premise

The PBH mass function must be monochromatic to keep the SIGW frequency tied to the binary ISCO frequency in a consistent, model-independent way.

What would settle it

A measured merger spectrum whose peak deviates from 1.79 times the ISCO frequency inferred from an observed SIGW background, or signals from the two channels that cannot be produced by any single monochromatic PBH population consistent with abundance bounds.

Figures

Figures reproduced from arXiv: 2607.01818 by Ashu Kushwaha.

Figure 1
Figure 1. Figure 1: FIG. 1. Present-day SIGW energy density obtained using the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Stochastic GW background produced by mergers [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

Formation of primordial black holes (PBHs) requires a significant enhancement of curvature perturbations. This mechanism leaves a twofold gravitational-wave (GW) signature: a \emph{low-frequency} stochastic background of scalar-induced GWs (SIGWs) and a distinct \emph{high-frequency} signal from subsequent PBH binary mergers. We leverage this shared origin to establish a consistent, \emph{model-independent} connection between these two observables for a monochromatic PBH mass function. Using PBH abundance constraints on the primordial curvature power spectrum, we evaluate the stochastic SIGW background for spherical and ellipsoidal collapse models, demonstrating that the ellipsoidal scenario yields a significantly stronger signal. Furthermore, we analyze the GW signal from PBH binary mergers and find a direct correspondence between the SIGW frequency and the innermost stable circular orbit (ISCO) frequency of the binaries. Because GW emission is nearly maximal near the ISCO, we additionally show that the peak of the full merger GW spectrum relates to the ISCO frequency via $f_{\text{peak}} = 1.79 \, f_{\text{ISCO}}$, a relation that is independent of the binary masses. Remarkably, this unified framework connects these distinct GW channels, enabling the same primordial fluctuations to be probed across widely separated frequency bands.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that primordial curvature perturbations enhanced enough to form PBHs produce both a low-frequency stochastic SIGW background and a high-frequency GW signal from subsequent PBH binary mergers. For a monochromatic PBH mass function, it establishes a direct frequency correspondence between the SIGW peak (set by the comoving scale k) and the ISCO frequency of the binaries, evaluates the SIGW amplitude for spherical versus ellipsoidal collapse using PBH abundance constraints (finding the ellipsoidal case stronger), and derives that the peak of the full merger spectrum satisfies f_peak = 1.79 f_ISCO independent of binary mass because emission is maximal near ISCO.

Significance. If the frequency mapping and numerical relation hold, the work supplies a concrete multi-band probe of the same primordial fluctuations, linking SIGW constraints on the curvature power spectrum to observable high-frequency merger signals. The explicit comparison of collapse models and the mass-independent ratio are concrete strengths that could guide future observational strategies.

major comments (1)
  1. [Abstract / derivation of Eq. for f_peak] Abstract and §3 (or equivalent derivation section): the factor 1.79 in f_peak = 1.79 f_ISCO is presented as following from maximal emission near ISCO plus dimensional scaling, but the precise numerical coefficient requires an explicit integral or fitting procedure over the merger waveform; without the intermediate steps shown, it is impossible to verify whether the value is robust or sensitive to waveform modeling choices.
minor comments (2)
  1. [Abstract] The repeated use of 'model-independent' should be qualified in the abstract and introduction to emphasize that the frequency correspondence is derived under the monochromatic mass-function assumption stated later in the text.
  2. [Figures] Figure captions and axis labels for the SIGW spectra should explicitly state the PBH mass (or k-scale) chosen for each curve so that the frequency correspondence to the ISCO can be read off directly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract / derivation of Eq. for f_peak] Abstract and §3 (or equivalent derivation section): the factor 1.79 in f_peak = 1.79 f_ISCO is presented as following from maximal emission near ISCO plus dimensional scaling, but the precise numerical coefficient requires an explicit integral or fitting procedure over the merger waveform; without the intermediate steps shown, it is impossible to verify whether the value is robust or sensitive to waveform modeling choices.

    Authors: We agree that the explicit steps leading to the coefficient 1.79 were not shown. The value follows from the peak of the GW energy spectrum for circular binaries, obtained by integrating the quadrupole formula (or standard phenomenological waveforms) over the inspiral-to-merger transition, where emission peaks near the ISCO. In the revised manuscript we will add the intermediate calculation (or the fitting formula from the Phenom/EOBNR family) in §3, together with a brief statement on robustness to waveform modeling within current uncertainties. This will make the result verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central result is a frequency correspondence between SIGW and PBH binary signals that follows directly from both observables tracing the same primordial curvature scale k under the explicitly stated monochromatic mass-function assumption; this is a model feature rather than a hidden self-definition or fitted input renamed as prediction. The additional mass-independent ratio f_peak = 1.79 f_ISCO is obtained from standard scaling (both frequencies ∝ 1/M) plus the known near-ISCO peak of GW emission, without any reduction to prior self-citations, ansatzes, or uniqueness theorems imported from the authors. No load-bearing steps in the provided abstract or description reduce by construction to the inputs; the derivation remains self-contained once the monochromatic assumption is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the paper relies on standard domain assumptions in PBH cosmology rather than introducing new free parameters or entities.

axioms (1)
  • domain assumption PBH abundance constraints on the primordial curvature power spectrum are used to normalize the SIGW amplitude
    Invoked to evaluate the stochastic background for both collapse models.

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discussion (0)

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