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REVIEW 4 major objections 5 minor 6 cited by

The Hierarchical Merger Scenario for GW231123

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper argues that GW231123 is best explained as a 2G+2G hierarchical merger — both observed black holes being remnants of earlier mergers — not as a 2G+1G or 3G+2G event.

desk verdict The 2G+2G claim for GW231123 reverses under the paper's own alternative waveform; read this as a useful sensitivity study, not a settled origin. read the letter →

arxiv 2509.08298 v1 pith:7CARB4CO submitted 2025-09-10 astro-ph.HE astro-ph.GAastro-ph.SRgr-qc

Guo-Peng Li , Xi-Long Fan This is my paper · ORCID
classification astro-ph.HEastro-ph.GAastro-ph.SRgr-qc
keywords GW231123hierarchicalmergersblackholegravitationalwaves2G+2Gspinpriorpair-instabilitymassgapwaveformmodel
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

GW231123 is a detected black-hole merger with component masses near 137 and 103 solar masses and high spins, properties that ordinary stellar collapse struggles to produce. The paper asks whether it formed hierarchically, with one or both observed black holes being remnants of earlier mergers. Using Bayesian model comparison, it argues that the most likely channel is 2G+2G: both components are second-generation black holes. The implied progenitors are four first-generation black holes of roughly 75, 66, 64, and 59 solar masses, and the observed high spins match the ~0.7 spin expected of merger remnants. The paper also reports that this conclusion is waveform-dependent, since an alternative waveform model reverses the preference.

What carries the argument

The central tool is the Bayesian odds ratio O_i^j = (p(d|H_i)/p(d|H_j))×(p(H_i)/p(H_j)), comparing the three hypotheses 2G+1G, 2G+2G, and 3G+2G. The evidence terms are built from generation-specific priors synthesized by a parametric population model that starts from the inferred 1G mass/spin distribution and adds a retention condition: a merger remnant participates in the next merger only if its kick velocity stays below the host escape speed. The prior odds are set proportional to the retention probabilities of the required earlier mergers. The GW231123 posterior from NRSur7dq4 provides the data term, and the mass-dependent overlap between that posterior and the mass distributions of 1G, 2

What would settle it

A re-analysis of GW231123 with a waveform model validated for asymmetric, precessing, high-mass binaries, or with an additional detector's data, would settle the central claim by fixing the secondary mass: if the secondary is near 111 solar masses the 2G+2G scenario wins, while a secondary near 55 solar masses makes 2G+1G win (odds 0.03).

Watch

Extended reading notes

Core claim

On the paper's own terms, assuming GW231123 is hierarchical, the 2G+2G hypothesis is strongly preferred over 2G+1G and 3G+2G. With the NRSur7dq4 posterior, escape speed 100 km/s, and a 1G maximum mass of 85 solar masses, the odds ratios are 1.28×10^4 versus 2G+1G and 2.81×10^8 versus 3G+2G. The primary is then a remnant of ~75+66 solar-mass progenitors, and the secondary of ~64+59 solar-mass progenitors, with remnant spins peaking near 0.69. The paper stresses that this depends on an informed 0.7-peaked spin prior and on the waveform: with IMRPhenomXO4a the 2G+2G versus 2G+1G odds fall to 0.03, favoring 2G+1G.

Load-bearing premise

The 2G+2G conclusion rests on the assumption that the NRSur7dq4 posterior for GW231123 is the correct one; the paper's own comparison shows that with the IMRPhenomXO4a waveform the 2G+2G hypothesis becomes only 3% as probable as 2G+1G.

Editorial extensions

If this is right

  • If GW231123 is a 2G+2G event, ordinary hierarchical assembly in a low-escape-speed environment can place black holes in the pair-instability gap without invoking unusual stellar collapse physics.
  • The four inferred first-generation progenitors have masses consistent with standard stellar-collapse black holes, so an apparent upper-mass-gap event does not by itself require new formation physics.
  • The post-merger spins cluster near 0.69, which means parameter estimation for repeated-merger candidates should use an informed spin prior peaked near 0.7 instead of a uniform prior.
  • Waveform choice is decisive: replacing NRSur7dq4 with IMRPhenomXO4a changes the odds ratio for 2G+2G versus 2G+1G by six orders of magnitude and reverses the favored channel.
  • Higher escape speeds (300 km/s) weaken the preference for 2G+2G over 2G+1G but do not overturn it, while a lower 1G maximum mass strengthens the 2G+2G case.

Reading between the lines

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

  • Inference: A decisive next step would be to rerun this comparison with a waveform model that is validated for asymmetric, precessing, high-mass systems; the discriminating observable is the secondary-mass posterior, since it moves from ~111 to ~55 solar masses between the two waveform models.
  • Inference: If high-generation black holes really spin near 0.7, a population-wide reanalysis of massive high-spin events with the informed prior could reclassify some previously reported events as hierarchical, a consequence the paper does not work out.
  • Inference: The reversal under IMRPhenomXO4a could also mean the true arrangement is not among the three channels tested; a 3G+1G or 4G+2G scenario might accommodate the asymmetric mass posterior that makes 2G+1G look better.
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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 / 5 minor

Summary. The paper analyzes the very massive, high-spin gravitational-wave event GW231123 under the hypothesis of hierarchical black-hole mergers. It compares three formation scenarios—2G+1G, 2G+2G, and 3G+2G—by computing prior odds from retention probabilities and Bayes factors from population-level mass/spin priors, using the published LVK posterior distributions obtained with the NRSur7dq4 and IMRPhenomXO4a waveform models. Under NRSur7dq4 with V_esc = 100 km/s and M_max = 85 M_sun, the authors report odds of ~1.28e4 and ~2.8e8 favoring 2G+2G over 2G+1G and 3G+2G, and they infer a progenitor scenario with four 1G BHs. They also report that using IMRPhenomXO4a reverses the 2G+2G versus 2G+1G comparison (odds 0.03) and that the post-merger spin values are dominated by the assumed spin prior. The paper is transparent about these caveats, but the abstract and conclusions present the NRSur7dq4-based odds as the primary result.

Significance. If the central claim were robust, the paper would provide a concrete, testable progenitor scenario for GW231123 and strengthen the case that hierarchical mergers are observable in current LIGO-Virgo-KAGRA data. The methodology is clearly formulated, the paper makes use of publicly available LVK posterior samples, and the authors are commendably explicit about the waveform-model reversal and the prior-dominated nature of the inferred post-merger spins. However, the headline odds ratio of >O(10^3) is conditional on a single waveform model, and the paper does not supply a physical or statistical reason to prefer NRSur7dq4 over IMRPhenomXO4a for this unusually massive event. The evidentiary integral in Eq. (2) is also not connected to a concrete numerical scheme. These issues affect the load-bearing claim and therefore need to be addressed before the result can be considered robust.

major comments (4)
  1. [Abstract; §III A, Table I] The abstract's claim that 'odds ratios of >O(10^3)' favor the 2G+2G scenario is contradicted by the paper's own Table I: for IMRPhenomXO4a, log B_{2G+1G}^{2G+2G} = -1.48 and O_{2G+1G}^{2G+2G} = 0.03, i.e., the 2G+1G scenario is preferred. The text notes this reversal but does not justify why NRSur7dq4 should be trusted over IMRPhenomXO4a for this event. Because the conclusion reverses under an alternative waveform model used in the LVK analysis, the paper must either (i) provide a quantitative justification for preferring NRSur7dq4 (e.g., a waveform-model Bayes factor, goodness-of-fit comparison, or evidence that IMRPhenomXO4a is inadequate for such high masses) or (ii) reframe the abstract and conclusions as a conditional statement. A caveat in the text is not sufficient while the unqualified odds statement remains in the abstract.
  2. [§II B, Eq. (2)] The evidence p(d|H_i) is defined as an eight-dimensional integral over masses and spins, but the manuscript never states how this integral is evaluated numerically. The Bayes factors in Table I are the central quantitative result, and they depend on both the prior volume and the normalization of the likelihood over the integration domain. The authors should specify the estimator used (e.g., importance sampling, nested sampling, harmonic mean), how the LVK posterior samples are reweighted to compute p(d|H_i), the integration bounds, and convergence checks. Without this, the numbers in Table I are not reproducible and the model comparison cannot be independently verified.
  3. [§III B, Fig. 3] The paper states that the post-merger spin distributions of ~0.69 are 'dominated by the prior' and 'reflect the effect of the spin prior that clusters around 0.7.' This is an explicit admission that the quoted spin values are not data-driven measurements. Presenting these values as results in the text and in Fig. 3 without a clear separation between 'prior input' and 'posterior output' is misleading, especially because the abstract uses high spins as motivation. The authors should clearly mark these as prior-driven predictions and, ideally, quantify the information gain from the data or remove them from the set of inferred quantities.
  4. [§II B, after Eq. (3)] The dismissal of the 'primary=1G, secondary=2G' case is argued by saying that when the mass difference is 'relatively large' the probability can be ignored. This is not a valid criterion in general. The actual reason is that the primary-mass posterior from both waveform models lies above the assumed maximum 1G BH mass of 85 M_sun, so a 1G primary is excluded by construction. The paper should replace the hand-waving statement with this explicit argument and note that the exclusion depends on the chosen M_max value.
minor comments (5)
  1. [§IV, item (1)] The conclusion quotes log Bayes factor 8.35 for 2G+2G versus 3G+2G, but Table I gives 8.38. Please correct the inconsistency.
  2. [Eq. (1) and surrounding text] The dataset is denoted 'd^1' in one place and 'd' elsewhere. Use a single symbol, e.g., 'd', and remove the superscript.
  3. [Footnote 1] The Zenodo link is given without a description. Please state whether it contains the LVK posterior samples, the analysis code, or both, and provide a version/DOI reference.
  4. [Table I, IMRPhenomXO4a row] The '/' for the 3G+2G comparison is explained as computational cost, but the paper should state whether the 3G+2G evidence was not computed or whether an upper/lower bound was used. This affects the interpretation of the row.
  5. [Fig. 1 caption] The caption mentions a maximum 1G BH mass of ~85 M_sun, but Section III A also tests M_max = 65 M_sun. Consider adding the 65 M_sun truncation to the figure or explaining why only 85 M_sun is shown.

Circularity Check

1 steps flagged · score 4.0 of 10

One secondary result—post-merger spins near 0.7—is explicitly prior-dominated; the central 2G+2G vs 2G+1G comparison is an independent Bayesian calculation, not a circular reduction.

  1. self definitional [Section III B (Progenitors of GW231123), after Figure 3]
    "The two progenitor mergers of GW231123 both show post-merger spin distributions sharply peaked near χ∼0.7, with 0.69+0.04−0.05 and 0.69+0.04−0.04, reflecting the effect of the spin prior that clusters around 0.7 [10]."

    The 2G population prior used in the analysis (Appendix A and Figure 1) is constructed so that high-generation BH spin distributions peak near χ~0.7–0.9. The posterior for the post-merger (2G) remnant spin is then dominated by this prior because the GW data constrain only the final binary spins of GW231123, not the spins of the hypothetical 2G remnants. The paper's own conclusion (3) states these spins are 'dominated by the prior.' Thus the claimed peak at 0.69 is a direct restatement of the assumed spin prior, not an inference from the data.

full rationale

The only clear circular step is the spin result in Section III B: the post-merger spins of ~0.7 are explicitly attributed to the spin prior that clusters at that value, making that particular 'prediction' equivalent to an input assumption by the paper's own admission. This is a supporting result, not the central claim. The central claim—that GW231123 is a 2G+2G merger with odds >10^3 over 2G+1G—is obtained by a hierarchical Bayesian evidence computation: the likelihood from the NRSur7dq4 posterior is integrated against hypothesis-specific priors synthesized from the GWTC-3 1G population and a kick/remnant model. That is an independent calculation, not a reduction to a fitted parameter or a self-citation theorem. The authors do cite their own earlier population models (refs. [37,38]) for the 2G/3G priors, but those are physical models with stated assumptions, not an unverified uniqueness claim, and the odds are not a pure restatement of the priors. The dramatic reversal under IMRPhenomXO4a is a robustness/correctness issue—the paper itself reports it—but it is not circularity, because the waveform choice is an external input, not something derived from the hypotheses being compared. On balance, the paper has one self-definitional supporting result while the central model comparison retains independent content, so the circularity score is 4.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The paper adds no new entities. Its burden is carried by assumed population models, a hand-chosen escape speed and 1G mass cutoff, and a spin prior that predetermines the reported remnant-spin result. The prior odds themselves rest on a retention-probability proportionality that is not derived from rates.

free parameters (3)
  • escape speed V_esc = 100, 300 km/s (chosen)
    Host environment escape speed is not inferred; it sets the retention probability, prior odds, and the shape of higher-generation population priors.
  • maximum 1G BH mass M_max = 85, 65 M_sun (chosen)
    Truncation of the 1G mass distribution at the theorized pair-instability gap; varied by hand and strongly affects the odds ratios.
  • hierarchical remnant spin prior peak = chi ~ 0.7
    The 2G/3G spin distributions peak near 0.7 following Gerosa & Fishbach (2021); this input directly produces the claimed post-merger spin values.
assumptions (7)
  • standard math Bayes theorem and marginal likelihood computation
    Used in Eqs. 1 and 2 to define odds ratios and evidences.
  • domain assumption GW231123 is a hierarchical merger
    The introduction states the analysis is done 'under the assumption that the event originates from the hierarchical merger scenario.' The paper does not test this against non-hierarchical channels.
  • domain assumption 2G/3G population model from Li et al. (2023) and Li & Fan (2025) is faithful
    The priors p(m,chi|NG) and retention probabilities come from the authors' own parametric model; its accuracy is assumed, not independently validated on new data.
  • ad hoc to paper Prior probability of each hypothesis is proportional to retention probability
    Eq. 4 sets p(H) proportional to products of f_ret; this is a modeling choice, not derived from merger rates or delay-time distributions.
  • domain assumption Delay times between subsequent mergers are negligible
    Section II C: delay times are neglected with a brief justification that they are negligible in low escape-speed environments; this affects hierarchical merger rates and prior odds.
  • ad hoc to paper The 1G+2G case with primary=1G, secondary=2G can be ignored
    Section II B argues this is negligible when the mass difference is large, but the NRSur7dq4 masses are 129 and 111 solar masses, not a large difference.
  • domain assumption GWTC-3 1G population with Mmax truncation
    The 1G mass and spin priors follow GWTC-3 [36] with a sharp cutoff at Mmax; the choice of Mmax directly affects the 2G and 3G distributions.

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Cite this review

Pith. "Pith review of The Hierarchical Merger Scenario for GW231123." pith.science (2026). https://pith.science/paper/7CARB4CO

@misc{pith2026250908298,
  author       = {Pith},
  title        = {Pith review of: The Hierarchical Merger Scenario for GW231123},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CARB4CO}},
  note         = {Machine review of arXiv:2509.08298}
}
abstract

GW231123 exhibits exceedingly massive components and high spins, which challenges the formation of first-generation (1G) black holes from stellar collapse and implies that this event might originate from hierarchical mergers. Here we show that the \texttt{2G+2G} merger scenario for GW231123 is favored over a \texttt{2G+1G} (or \texttt{3G+2G}) merger, with odds ratios of $>$$\mathcal{O}(10^3)$. The primary (secondary) black hole is consistent with merging binary black holes with masses of $75^{+7}_{-7}\,M_\odot$ and $66^{+4}_{-10}\,M_\odot$ ($64^{+8}_{-10}\,M_\odot$ and $59^{+7}_{-9}\,M_\odot$; 90\% credible intervals), respectively. Our results reveal that the treatment of spin priors from the population level and waveform model choice are decisive in interpreting potential hierarchical gravitational-wave signals.

Figures

Figures reproduced from arXiv: 2509.08298 by the authors.

Figure 1
Figure 1. FIG. 1. Probability density distributions of masses ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Probability density distributions (left) of kick veloc [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Posterior probability density distributions of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Ensemble-Based Residual Tests of GW231123 across Waveform Models

    gr-qc 2026-08 conditional novelty 6.0 of 10

    GW231123's residuals are consistent with detector noise for all five waveform models, and the strain-level differences between models are too small to be detected.

  2. Across the Universe: GW231123 as a magnified and diffracted black hole merger

    astro-ph.GA 2025-12 conditional novelty 6.0 of 10

    GW231123's extreme mass and spins may be explained by a point-mass microlens embedded in a galaxy, reducing the inferred source mass to about 100-180 solar masses.

  3. A new group of low-spin $50-70M_\odot$ Black Holes and the high pair-instability mass cutoff

    astro-ph.HE 2025-10 conditional novelty 6.0 of 10

    GWTC-4.0 reveals low-spin black holes up to ~68.5 Msun, implying a higher pair-instability mass cutoff and a lower 12C(alpha,gamma)16O rate.

  4. Searching for Electromagnetic Counterpart Candidates to GW231123

    astro-ph.HE 2025-11 conditional novelty 5.0 of 10

    A crossmatch of GW231123's localization with ZTF AGN-flare catalogs yields six unconfirmed optical flare candidates, with association claims driven largely by the event's high primary mass.

  5. Exploring Hierarchical Merger Scenarios for GW241011 and GW241110

    astro-ph.HE 2026-07 conditional novelty 4.5 of 10

    GW241011 and GW241110 favor 2G+1G hierarchical-merger models over a fiducial 1G+1G population, with mild AGN-disk preference and one weak optical flare candidate.

  6. Considering lensing effect on gravitational wave signals from black holes in mass gap

    astro-ph.HE 2025-12 reject novelty 4.0 of 10

    Gravitational lensing could in principle explain apparent mass-gap black-hole mergers, but the required magnifications are μ≈12–39 for GW190521 and μ≈320–444 for GW231123 depending on assumptions—and the paper's abstr...

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