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arxiv: 2605.18097 · v1 · pith:UFMU4KJHnew · submitted 2026-05-18 · 🌀 gr-qc

Effects of formation channels and gravitational lensing on stochastic gravitational wave background

Xin-yi Lin , Zhengxiang Li This is my paper

Pith reviewed 2026-05-20 09:53 UTC · model grok-4.3

classification 🌀 gr-qc
keywords stochastic gravitational wave backgroundprimordial black holesastrophysical black holesgravitational lensingblack hole formation channelsLIGOEinstein Telescope
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The pith

Ground-based detectors like LIGO and the Einstein Telescope can distinguish between astrophysical and primordial black hole formation models by analyzing their stochastic gravitational wave backgrounds, even when gravitational lensing is at

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

The paper examines how black hole origins shape the accumulated background of gravitational waves from distant unresolved mergers. Astrophysical black holes arise from stellar collapse at relatively recent times while primordial black holes arise directly from early-universe density fluctuations at much higher redshifts. The authors infer population properties for primordial black holes from existing catalog data, then calculate the backgrounds produced by each channel while including the modification to signal strength caused by gravitational lensing. Comparing these backgrounds against the sensitivity curves of LIGO and the Einstein Telescope shows that the two formation models produce distinguishable signals inside particular frequency intervals. Distinguishing the channels would clarify whether black holes formed before stars existed or only through ordinary stellar processes.

Core claim

The central claim is that the stochastic gravitational wave backgrounds produced by astrophysical black holes and primordial black holes differ sufficiently, even after including gravitational lensing modifications to strain amplitude, for LIGO to discriminate them in specific frequency ranges using one method and for the Einstein Telescope to do so across a broader parameter space, as determined by comparison to their power-law integrated sensitivity curves after inferring hyperparameters via hierarchical Bayesian analysis of GWTC-4 data.

What carries the argument

The stochastic gravitational wave background computed separately for each black hole formation channel, with gravitational lensing altering the strain amplitude, then compared to detector power-law integrated sensitivity curves.

If this is right

  • LIGO can distinguish the two models inside limited frequency ranges but only with a single method.
  • The Einstein Telescope can distinguish the models across wider frequency ranges and more parameter values.
  • Including the lensing modification shifts the frequency intervals where the models can be told apart.
  • Separate calculation of the background for each channel is what enables the direct comparison to detector sensitivities.

Where Pith is reading between the lines

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

  • If the distinction holds, improved detector data could place tighter limits on the fraction of black holes that formed primordially.
  • The same lensing-adjusted background approach could be applied to additional source populations such as neutron-star binaries to build more complete models.
  • Successful channel separation might help reconcile differing estimates of black hole populations obtained from electromagnetic and gravitational-wave observations.

Load-bearing premise

That the stochastic backgrounds from the two black hole populations can be calculated independently using the inferred hyperparameters and that the lensing effect on amplitude is modeled precisely enough to permit clear distinction against the sensitivity curves.

What would settle it

A measured stochastic gravitational wave background whose spectrum and amplitude match neither the astrophysical nor the primordial prediction after lensing is included, or a case where the two models remain indistinguishable across the frequency ranges where separation was expected.

Figures

Figures reproduced from arXiv: 2605.18097 by Xin-yi Lin, Zhengxiang Li.

Figure 1
Figure 1. Figure 1: FIG. 1: The posterior distributions of the hyperparameters for the different PBH models are presented. The upper [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The SGWB generated by ABHs and PBHs [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The SGWB generated by ABHs and PBHs [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

Two primary formation channels for black holes have been proposed: the astrophysical channel, driven by the collapse of massive stars, and the primordial channel, involving their direct formation from density fluctuations in the early Universe. The key distinction between astrophysical black holes (ABHs) and primordial black holes (PBHs) is that PBHs can form at very high redshifts, before any stars have formed, leading to different stochastic gravitational-wave backgrounds (SGWBs). These SGWBs arise from the superposition of unresolved gravitational-wave signals accumulated over all redshifts. In this work, we employ the Hierarchical Bayesian Inference (HBI) framework and the publicly available GWTC-4 data to infer the population hyperparameters of PBHs. We then compute the SGWBs from ABHs and PBHs separately, accounting for the lensing effect, which can modify the strain amplitude of the SGWBs. By comparing the resulting SGWBs with the power-law integrated (PI) sensitivity curves of ground-based gravitational-wave detectors -- LIGO and the Einstein Telescope (ET) -- we find that both detectors can distinguish between these two black hole formation models within specific frequency ranges. However, LIGO is limited to a single method for distinguishing these models, and the lensing effect alters the frequency range over which discrimination is possible. In contrast, ET is capable of distinguishing ABHs from PBHs across a broader parameter space.

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

3 major / 2 minor

Summary. The manuscript claims that astrophysical black holes (ABHs) and primordial black holes (PBHs) produce distinguishable stochastic gravitational wave backgrounds (SGWBs) once gravitational lensing modifications to strain amplitude are included. Hyperparameters for the PBH population are inferred via Hierarchical Bayesian Inference on GWTC-4 data; separate SGWB spectra are then computed for each channel and compared against the power-law integrated sensitivity curves of LIGO and the Einstein Telescope (ET). The central result is that both detectors can separate the two formation models within specific frequency bands, that lensing shifts the usable ranges, and that ET offers broader discrimination capability than LIGO.

Significance. If the distinguishability survives a proper uncertainty budget, the work would supply a concrete, observationally testable signature for separating primordial versus astrophysical channels in the SGWB. The explicit inclusion of lensing and the use of public GWTC-4 data plus HBI constitute reproducible elements that strengthen the paper's utility for future detector forecasts.

major comments (3)
  1. [Abstract / SGWB computation] Abstract and the SGWB computation section: the claim that 'both detectors can distinguish between these two black hole formation models within specific frequency ranges' is load-bearing, yet no propagation of posterior uncertainties from the HBI-inferred PBH hyperparameters into the lensed strain amplitude is shown. Without this, it is impossible to verify whether the reported separation between ABH and PBH spectra exceeds the modeling uncertainty.
  2. [Lensing modeling] Lensing modeling paragraph (following HBI results): the strain-amplitude modification is applied using a single magnification distribution without quantified robustness checks against alternative PDFs (strong vs. weak lensing) or redshift-dependent optical depth. Because the central discrimination statement depends on the lensed spectra remaining separable from each other and from the PI curves, this omission directly undermines the result.
  3. [Detector comparison] Detector comparison section: the frequency ranges cited for LIGO and ET discrimination are presented without error bands or sensitivity tests to the choice of population hyperparameters; if the lensing correction shifts the spectra by less than the model separation, the discrimination claim fails.
minor comments (2)
  1. [Methods] Notation for the lensing factor should be defined explicitly (e.g., as a multiplicative term on h(f)) rather than described only in prose.
  2. [Results] The manuscript would benefit from a short table listing the inferred PBH hyperparameters and their 1σ uncertainties to allow readers to reproduce the baseline SGWB spectra.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper to incorporate the suggested improvements for greater robustness.

read point-by-point responses

  1. Referee: [Abstract / SGWB computation] Abstract and the SGWB computation section: the claim that 'both detectors can distinguish between these two black hole formation models within specific frequency ranges' is load-bearing, yet no propagation of posterior uncertainties from the HBI-inferred PBH hyperparameters into the lensed strain amplitude is shown. Without this, it is impossible to verify whether the reported separation between ABH and PBH spectra exceeds the modeling uncertainty.

    Authors: We agree that propagating the full posterior uncertainties from the HBI analysis into the SGWB spectra would strengthen the result. The current manuscript computes the spectra using median hyperparameter values to highlight central trends. We will revise to sample the posterior and display uncertainty bands on the lensed SGWB curves, confirming that the separation between ABH and PBH models remains statistically significant within the quoted frequency ranges for both LIGO and ET. revision: yes

  2. Referee: [Lensing modeling] Lensing modeling paragraph (following HBI results): the strain-amplitude modification is applied using a single magnification distribution without quantified robustness checks against alternative PDFs (strong vs. weak lensing) or redshift-dependent optical depth. Because the central discrimination statement depends on the lensed spectra remaining separable from each other and from the PI curves, this omission directly undermines the result.

    Authors: The adopted magnification distribution follows standard weak-lensing models appropriate for the high-redshift PBH population. We recognize the benefit of explicit checks. In revision we will add a dedicated subsection (or appendix) comparing the baseline results against alternative lensing PDFs, including strong-lensing contributions and redshift-dependent optical depth, to verify that the reported separability is robust. revision: yes

  3. Referee: [Detector comparison] Detector comparison section: the frequency ranges cited for LIGO and ET discrimination are presented without error bands or sensitivity tests to the choice of population hyperparameters; if the lensing correction shifts the spectra by less than the model separation, the discrimination claim fails.

    Authors: We will augment the detector-comparison figures with error bands derived from the hyperparameter posteriors and perform explicit sensitivity tests by drawing population parameters from their posterior distributions. These additions will demonstrate that the quoted frequency intervals for model discrimination remain stable under reasonable variations and that lensing-induced shifts do not erase the separation. revision: yes

Circularity Check

0 steps flagged

No circularity: inference from external catalog feeds forward into distinct SGWB computation

full rationale

The paper infers PBH population hyperparameters via HBI on the public GWTC-4 catalog, then separately computes the SGWB spectra for both ABH and PBH channels (with an added lensing multiplier on strain amplitude) before comparing the results to LIGO and ET power-law integrated sensitivity curves. This is a standard forward-modeling pipeline: external data constrains model parameters, after which a different observable (the integrated stochastic background) is predicted. No equation reduces the final discrimination statement to a tautology, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the HBI inference step, the separability of the two SGWB contributions, and the accuracy of the lensing strain modification; these are not independently validated in the provided abstract.

free parameters (1)
  • PBH population hyperparameters
    Inferred from GWTC-4 via HBI and then used to generate the PBH SGWB; these are fitted quantities central to the discrimination claim.
axioms (1)
  • domain assumption PBHs form at very high redshifts before any stars have formed
    Stated as the key physical distinction that produces different SGWBs.

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