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A Possible Triple Formation Scenario of Binary Black Hole Merge With One In Pair-instability Supernova Mass Gap

T0 review · 4 major / 7 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Triple Stars Can Forge Black Holes in the 'Forbidden' Mass Gap

desk verdict A new isolated triple channel for PISN mass-gap BBH mergers — plausible physics, but the rate estimate rests on an unvalidated common-envelope prescription and a hand-tuned example. read the letter →

arxiv 2607.06067 v1 pith:BOBPIXW4 submitted 2026-07-07 astro-ph.HE astro-ph.SR

Tian Huang , Xizhen Lu , Guoliang Lü , Chunhua Zhu , Sufen Guo , Helei Liu , Zhuowen Li , Zhijun Wang
show 3 more authors
Lei Li Wei-Min Gu Nurzada Beissen
This is my paper · ORCID
classification astro-ph.HEastro-ph.SR
keywords evolutionmasstriplebinaryblackdrivesframeworkgw190706
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

Some LIGO gravitational-wave detections contain black holes weighing 45–130 solar masses — a range where pair-instability supernovae should destroy stars entirely, leaving no remnant. This paper proposes that isolated hierarchical triple stars (three stars orbiting each other in a stable configuration) can bypass this destruction. In the proposed channel, two stars in a very tight inner orbit undergo tidal synchronization that drives chemically homogeneous evolution (CHE), bypassing the giant-phase expansion that normally leads to problematic mass transfer. These stars collapse directly into black holes. When the third, outer star later expands into a giant, it engulfs the inner binary in a triple common envelope (TCE) phase, shrinking the inner orbit enough to force the two black holes to merge promptly. Because the inner black holes are equal-mass and spin-aligned, the merger produces no gravitational recoil kick, so the merged remnant stays bound to the third star's collapsing core. The result is a binary containing one massive second-generation black hole sitting squarely in the PISN mass gap, paired with a smaller first-generation black hole. The paper traces a specific evolutionary track reproducing GW190706 (primary ~72.8 solar masses, secondary ~39.3 solar masses, effective spin ~0.45) and computes a volumetric merger rate of ~0.011 Gpc⁻³ yr⁻¹ at z≈0.68, accounting for roughly 22% of the empirically inferred rate for this sub-population.

What carries the argument

Chemically homogeneous evolution (CHE) in the tidally locked inner binary; the triple common envelope (TCE) phase computed via the SCATTER angular-momentum-conservation formalism; the zero-recoil condition from equal-mass, spin-aligned inner mergers; and the direct-collapse (failed supernova) channel for forming first-generation black holes without natal kicks.

What would settle it

If future GW catalogs show that PISN mass-gap events with high positive χ_eff are rarer than ~22% of the sub-population rate, or if their spin-orbit misalignments are inconsistent with the strictly coplanar, spin-aligned topology this channel requires, the triple-CHE-TCE pathway would be constrained to a smaller contribution or ruled out as the dominant channel.

Watch

Extended reading notes

Core claim

The paper constructs a complete evolutionary chain from a coplanar hierarchical triple of ~109 solar-mass stars at low metallicity (Z=0.001) through CHE of the inner binary, direct collapse via failed supernovae, a TCE phase driven by the tertiary's giant expansion, and prompt inner-binary merger, arriving at a final BBH system whose masses (72.8 + 39.3 solar masses) and effective spin (χ_eff ≈ 0.45) match GW190706. The rate calculation, built from empirically motivated initial-parameter distributions and a triple-star fraction of 0.73, yields a birth probability P_sys ≈ 1.41×10⁻⁹ per massive star, translating to ~22% of the observed mass-gap merger rate.

Load-bearing premise

The entire channel depends on the SCATTER triple common-envelope formalism, an angular-momentum-conservation prescription with fitted parameters (η, A, B, δ) calibrated on a limited sample of post-CE binaries. If this formalism misestimates the post-CE orbital shrinkage by even a factor of a few, the inner binary either fails to merge promptly or the system is disrupted, invalidating both the rate prediction and the GW190706 evolutionary track.

Editorial extensions

If this is right

  • If this channel is real, a subset of PISN mass-gap BBH mergers should show high positive effective spins (~0.4–0.7) from second-generation remnant spin, distinguishable from cluster-formed hierarchical mergers which would show more isotropic spin orientations.
  • The channel predicts a correlation between mass-gap primary mass and effective spin: the 2g remnant spin of ~0.686 is inherited from the inner merger, so systems near the center of the gap should cluster around χ_eff ≈ 0.45 when paired with a non-spinning 1g companion.
  • Future space-based detectors could potentially detect the tertiary companion's gravitational signature or residual eccentricity in pre-merger systems, providing a direct test of the triple-origin hypothesis.
  • The 22% rate contribution leaves room for complementary channels (cluster dynamics, AGN disks) to produce the remaining mass-gap events, making this a partial rather than exclusive explanation.

Reading between the lines

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

  • The channel's predictability hinges on the inner binary having a mass ratio of exactly unity after early over-contact equilibration. If real systems retain q < 1, the gravitational recoil from the inner merger would be nonzero, potentially unbinding the system and reducing the rate well below 22%.
  • The SCATTER formalism is calibrated on post-CE binaries, not on triple systems with the extreme mass ratios and configurations studied here. If its orbital-shrinkage predictions are off by even a modest factor, the inner BBH may not merge promptly (t_GW > 1 yr threshold), breaking the chain.
  • The assumption of zero natal spin for 1g black holes (from efficient Tayler-Spruit angular momentum transport) is load-bearing for the χ_eff prediction. If 1g BHs retain even modest spins, the effective-spin signature would shift, potentially weakening the match with GW190706.
  • The delay-time approximation (neglecting delays beyond ~10 Myr) may underestimate or overestimate the rate at z≈0.68 depending on the true cosmic metallicity distribution at that redshift.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. The manuscript proposes an isolated hierarchical triple stellar evolution channel to produce binary black hole (BBH) mergers with one component in the pair-instability supernova (PISN) mass gap. The channel combines tidally driven chemically homogeneous evolution (CHE) in a tight inner binary with a subsequent triple common envelope (TCE) phase computed via the SCATTER formalism. A specific evolutionary track is shown to reproduce the observed properties of GW190706 (m1≈72.8 M☉, m2≈39.3 M☉, χeff≈0.45). A volumetric merger rate of ~0.011 Gpc⁻³ yr⁻¹ is derived, claimed to account for ~22% of the empirical rate for this sub-population at z≈0.68.

Significance. The question of how black holes in the PISN mass gap form is actively debated, and an isolated triple channel is a legitimate and interesting contribution to this discussion. The paper provides a concrete, step-by-step evolutionary pathway and a falsifiable rate prediction. The combination of CHE (which naturally produces aligned-spin, equal-mass inner binaries with zero recoil) with a TCE-driven inspiral is a physically motivated mechanism. However, the significance of the rate claim is limited by the unvalidated nature of the SCATTER formalism for triples and the unjustified survival fraction, both of which are load-bearing for the quantitative prediction.

major comments (4)
  1. §II.B, Eqs. (4)–(9): The SCATTER formalism is the load-bearing physical ingredient of the entire channel. The post-TCE inner-orbit shrinkage (Eq. 9) determines whether t_GW ≤ 1 yr (Eq. 18), which is the criterion for a prompt inner-binary merger. SCATTER was calibrated on post-CE binaries (Di Stefano et al. 2023, ref [77]), not on triple systems. The extension to triples assumes that the envelope mass partitioning between inner and outer orbits follows the Roche-lobe approximation in Eq. (8), but no 3D hydrodynamic validation of this assumption is cited. If the post-TCE inner orbit is larger than predicted by a factor of ~2–3, t_GW will exceed the tertiary helium-star lifetime (~0.1–1 Myr), the inner binary will not merge promptly, and no 2g+1g system forms. The manuscript should explicitly acknowledge this uncertainty and, at minimum, provide a sensitivity analysis showing how the rate和
  2. §III.C, Eq. (21): The survival fraction f_surv = 0.5 is introduced without justification ('Adopting a fiducial survival fraction of f_surv = 0.5 to account for dynamical disruption'). This parameter enters linearly into the final rate. More critically, it is unclear what f_surv is meant to capture: dynamical disruption, the fraction of systems satisfying t_GW ≤ 1 yr, or both? If it includes the latter, the rate calculation becomes partially circular, because P_sys is already computed from narrow parameter windows (§III.B) chosen because they produce the desired outcome, and f_surv then implicitly absorbs the unknown SCATTER success fraction. The paper should either compute this fraction from the simulations or provide a clear physical justification for the adopted value, with an exploration of how the rate scales with it.
  3. §III.C, Eq. (21): The quantity ⟨M⟩ in Eq. (21) is never defined. It presumably represents the mean stellar mass for IMF normalization, but its value is not stated. Since the rate is linear in 1/⟨M⟩, this omission makes the rate calculation unreproducible. Please define ⟨M⟩ and state its value.
  4. §III, §III.C: There is an internal inconsistency in the event count. §III states that after excluding events with negative χ_eff, 'the remaining sources, notably GW230824 and GW190706' are well-explained (implying N≈2). Yet in §III.C, Eq. (22), N_obs = 6 is used to compute the empirical rate R_obs ≈ 0.050 Gpc⁻³ yr⁻¹. Additionally, GW230824 does not appear in the initial sample of six events listed in §III (GW230107, GW230928, GW230820, GW190706, GW190620, GW170729). Please clarify which events are used for the rate baseline and reconcile the filtering described in §III with the N_obs = 6 used in Eq. (22).
minor comments (7)
  1. Figure 1 is difficult to parse: the orbital separation and mass labels overlap, and the time axis is non-linear with unmarked jumps (e.g., from 3.644 to 3.772 to 24.609 Myr). A cleaner version with a logarithmic time axis or separate panels would help.
  2. §III.A: The statement that ω_spin/ω_crit ≈ 1.21 triggers CHE should cite the specific threshold criterion used (e.g., from Li et al. 2025, ref [71]) and note its uncertainty.
  3. §III.A: The paper states the tertiary helium core 'also experiences violent episodic mass loss via PPISNe before undergoing a FSNe.' The pre-SN mass of the tertiary core and the resulting BH mass (39.3 M☉) should be traced explicitly, as is done for the inner binary components.
  4. §III.B: The outer period range ΔP_out ∈ [400, 1200] days is stated, but the GW190706 fiducial system has a_out = 3600 R☉, which for ~109 M☉ total mass corresponds to P_out ≈ 1400 days — outside this range. Please reconcile.
  5. Abstract and §IV: 'GW230107, GW230820, and GW230928' are cited as GWTC-4 events but no reference is given for these beyond the GWTC-4 catalog (ref [12]). If these are preliminary, this should be noted.
  6. §II.E, Eq. (14): The SFRD parameters a=0.015, b=2.7, c=2.9, d=5.6 are attributed to Madau & Dickinson (2014) and Madau & Fragos (2017) but the values differ between those references. Please clarify which parameter set is used.
  7. Several typographical issues: 'Merge' in the title should be 'Mergers'; 'A V AILABLE' in the Data Available section; inconsistent spacing in equations.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee raises four major points: (1) the SCATTER formalism lacks hydrodynamic validation for triples and a sensitivity analysis is needed; (2) the survival fraction f_surv=0.5 is unjustified and its physical meaning unclear; (3) ⟨M⟩ in Eq. (21) is undefined; (4) an internal inconsistency exists in the event count and event list between §III and §III.C. We agree that points (3) and (4) require correction, and that points (1) and (2) warrant additional discussion and sensitivity analysis. We do not claim that the current rate prediction is robust to order-unity uncertainties in the SCATTER formalism, and we will revise the manuscript to make this explicit.

read point-by-point responses

  1. Referee: SCATTER formalism is load-bearing, calibrated on post-CE binaries not triples, no 3D hydrodynamic validation cited. If post-TCE inner orbit is larger by factor 2-3, t_GW exceeds tertiary helium-star lifetime and no 2g+1g system forms. Should acknowledge uncertainty and provide sensitivity analysis.

    Authors: We agree that the SCATTER formalism has not been validated against 3D hydrodynamic simulations of triple common envelopes, and that this is a genuine uncertainty affecting the rate prediction. We will add an explicit caveat in §II.B acknowledging this limitation. Regarding the sensitivity analysis: the referee's concern is well-taken. The critical question is whether the post-TCE inner orbit a_in,f is small enough that t_GW ≤ 1 yr (Eq. 18). In our fiducial track, a_in,f ≈ 13.5 R☉ yields t_GW ≈ 0.05 yr, which is well below the 1 yr threshold. If a_in,f were larger by a factor of 2-3, t_GW would scale as a^4 (Eq. 18), giving t_GW ≈ 0.8-6.4 yr. At the lower end (factor ~2), the system would still satisfy t_GW ≤ 1 yr; at factor ~3, it would not. We will include a sensitivity analysis showing how the rate scales with a multiplicative factor on a_in,f, demonstrating that the channel survives for moderate (~2x) overestimates of the post-TCE inner orbit but fails for larger ones. We will also note that 3D hydrodynamic simulations of TCE phases are needed to resolve this uncertainty definitively. revision: yes

  2. Referee: f_surv = 0.5 introduced without justification. Unclear what it captures: dynamical disruption, fraction satisfying t_GW ≤ 1 yr, or both? If it includes the latter, rate calculation becomes partially circular since P_sys is already computed from narrow parameter windows. Should compute from simulations or provide physical justification with exploration of rate scaling.

    Authors: The referee is correct that f_surv = 0.5 is insufficiently justified and that its physical scope is ambiguous. To clarify: f_surv is intended to account for dynamical disruption of the triple system during the TCE phase and subsequent evolution — i.e., systems that are disrupted by natal kicks, dynamical instabilities, or envelope ejection that unbinds the inner binary. It is not intended to absorb the fraction of systems satisfying t_GW ≤ 1 yr, which is already implicitly encoded in the narrow parameter windows of §III.B. However, we acknowledge that this distinction is not clearly stated in the manuscript, and the referee is right that the current presentation risks circularity. We will revise the text to explicitly define f_surv as the fraction of systems surviving dynamical disruption (excluding the t_GW criterion), and we will add a rate scaling showing R_triple ∝ f_surv for f_surv ∈ [0.1, 1.0]. We note honestly that we cannot compute f_surv from first principles without a full population synthesis of TCE outcomes, which is beyond the scope of this paper. We will adopt f_surv = 0.5 as a fiducial value but present the rate as scaling linearly with this parameter. revision: yes

  3. Referee: ⟨M⟩ in Eq. (21) is never defined. Presumably mean stellar mass for IMF normalization, but value not stated. Rate is linear in 1/⟨M⟩, making calculation unreproducible. Please define and state its value.

    Authors: The referee is correct. ⟨M⟩ represents the mean stellar mass obtained by integrating the Kroupa IMF over the standard mass range [0.1, 150] M☉. Using the IMF P(M) ∝ M^{-2.3} for M > 0.5 M☉ (with the full Kroupa piecewise form below that), we compute ⟨M⟩ ≈ 0.35 M☉. We will add this definition and value to the manuscript, and we will show the explicit substitution so that the rate calculation is reproducible. We thank the referee for catching this omission. revision: yes

  4. Referee: Internal inconsistency in event count. §III says after excluding negative χ_eff events, 'remaining sources, notably GW230824 and GW190706' are well-explained (implying N≈2). But §III.C Eq. (22) uses N_obs=6. Also GW230824 does not appear in the initial sample of six events listed in §III. Clarify which events are used for rate baseline and reconcile filtering with N_obs=6.

    Authors: The referee has identified a genuine inconsistency in our manuscript, and we thank them for catching it. The issue is twofold. First, GW230824 was erroneously mentioned in §III as a well-explained event; it does not appear in our initial sample of six events (GW230107, GW230928, GW230820, GW190706, GW190620, GW170729) and should not have been referenced. This is an error we will correct. Second, there is a logical inconsistency between the filtering described in §III (which excludes negative-χ_eff events, leaving fewer than six) and the use of N_obs = 6 in Eq. (22). The intent was to use all six events as the observational baseline for the empirical rate, while noting that our model specifically explains the subset with positive χ_eff (such as GW190706). However, this logic is not clearly presented. We will revise §III and §III.C to either (a) use N_obs = 6 as the total population baseline and explicitly state that our channel accounts for a subset, or (b) recompute R_obs using only the positive-χ_eff subset. We will adopt option (a) for consistency with the 22% comparison, but will clarify the distinction between the full baseline and the subset our model targets. revision: yes

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity found; the rate calculation and GW190706 match are fitting/assumption-laden but not self-reducing

full rationale

The paper's derivation chain does not exhibit structural circularity. (1) The GW190706 match: initial conditions (M1=M2=109.2 M☉, M3=108.7 M☉, specific orbital separations) are tuned to reproduce GW190706, but the output masses (~72.8 and 39.3 M☉) and spin (χeff≈0.45) emerge from stellar evolution physics (winds, PPISN, failed SNe, NR fitting formulas), not from the initial conditions by definition. This is fitting, not circularity. (2) The rate calculation (Eq. 21): P_sys is computed from parameter windows (M1∈[107,140] M☉, Pin∈[1.2,2.0] days, etc.) that are defined by the physical requirement of producing gap BHs, and the probabilities within those windows come from externally observed distributions (IMF from Kroupa, period/mass-ratio distributions from Moe & Di Stefano 2017). The rate is then compared to an independently derived empirical baseline (R_obs from GWTC-4 catalog events). This is standard population synthesis methodology, not circular: the windows are defined by physics, the probabilities by observation, and the comparison is to external data. (3) The SCATTER formalism (Eqs. 4–9) is cited from Di Stefano et al. 2023/2026 (refs [76,77]), who are not co-authors of this paper — this is an external prescription, not a self-citation. The η parameter (Eq. 6) is fitted to post-CE binaries and extrapolated to triples, which is an unvalidated extrapolation (a correctness risk), but not circularity since the fit target (binary CE outcomes) differs from the prediction target (triple CE outcomes). (4) The self-citation to Li et al. [71] for CHE parameters is load-bearing, but that work has independent content (its own stellar evolution models) and does not define its inputs in terms of the present paper's results. The f_surv=0.5 is a free assumption, not a fitted-then-predicted quantity. Overall, the central claims have independent content and the derivation does not reduce to its inputs by construction.

Assumptions & free parameters 8 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles, forces, or physical entities. It combines existing physical prescriptions (CHE, TCE/SCATTER, MOBSE) within the TSE code. The main burden is carried by free parameters (8 listed) and domain assumptions (5 listed), particularly the unvalidated SCATTER triple CE formalism and the hand-tuned initial conditions for the GW190706 track.

free parameters (8)
  • Metallicity Z=0.001 = 0.001
    Fixed to a single low value to minimize wind mass loss; no metallicity distribution is sampled despite the cosmic integration framework.
  • Inner binary mass ratio q_in=1 = 1.0
    Assumed exactly unity 'to facilitate modeling' (§III.A); the system is initialized post-mass-transfer equilibration.
  • SCATTER η parameters A, B = A=0.95, B=0.6
    Angular momentum transfer efficiency parameters fitted to post-CE binaries [77]; not independently derived.
  • SCATTER δ = 3
    Envelope mass distribution parameter, adopted as typical without sensitivity analysis in this paper.
  • Survival fraction fsurv = 0.5
    Adopted to account for dynamical disruption with no derivation or justification (§III.C).
  • Initial conditions for GW190706 track = M1=M2=109.2 M☉, M3=108.7 M☉, ain=36.1 R☉, aout=3600 R☉
    Hand-set to reproduce GW190706; no parameter space exploration shown.
  • Radiative mass loss fraction = 0.05
    5% of total mass lost in BBH merger, adopted from [114].
  • SFRD parameters a,b,c,d = 0.015, 2.7, 2.9, 5.6
    Cosmic SFR fitting parameters from Madau & Dickinson, used as inputs.
assumptions (5)
  • domain assumption Tidal synchronization in binaries with Pin<2 days drives CHE that fully bypasses giant expansion
    §II.A; adopted from Marchant et al. 2016 and Li et al. 2025. The threshold ωspin/ωcrit>1 is checked but the CHE outcome is assumed.
  • domain assumption 1g black holes from isolated stellar evolution have zero natal spin (χ1=χ2=0)
    §II.D; adopted from Fuller & Ma 2019 via Tayler-Spruit dynamo angular momentum transport.
  • domain assumption The SCATTER formalism correctly models triple common envelope orbital decay
    §II.B; the SCATTER mechanism [76,77] is a recently proposed prescription, not validated against 3D hydrodynamic simulations of triple CE.
  • domain assumption Stars with ZAMS mass ≳40 M☉ at Z=0.001 undergo direct collapse with ffb=1 (no natal kick)
    §II.C; adopted from Heger et al. 2003. Metallicity dependence of this threshold is not discussed.
  • standard math The Mardling & Aarseth stability criterion (Eq. 1) is sufficient for triple stability throughout evolution
    §II; empirical criterion widely used but derived for point-mass systems, not evolving stars with mass loss.

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Pith. "Pith review of A Possible Triple Formation Scenario of Binary Black Hole Merge With One In Pair-instability Supernova Mass Gap." pith.science (2026). https://pith.science/paper/BOBPIXW4

@misc{pith2026260706067,
  author       = {Pith},
  title        = {Pith review of: A Possible Triple Formation Scenario of Binary Black Hole Merge With One In Pair-instability Supernova Mass Gap},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BOBPIXW4}},
  note         = {Machine review of arXiv:2607.06067}
}
abstract

Observations of binary black hole (BBH) mergers detected by LIGO -- such as GW170729, GW190620, GW190706, GW230107, GW230820, and GW230928 -- feature high effective spins and primary black holes that fall squarely into the pair-instability supernova (PISN) mass gap ($\sim 45-130 \, M_{\odot}$). These events pose a significant challenge to standard stellar and binary evolution theories. To address this, we propose an isolated hierarchical triple stellar evolution channel. In this framework, tidal synchronization in tight inner binaries drives chemically homogeneous evolution (CHE), entirely bypassing giant expansion. A subsequent triple common envelope (TCE) evolution, triggered by the tertiary companion, rapidly drives the inner BBH to coalescence. Our model can provide a detailed evolutionary pathway that elegantly reproduces the properties of these GWs, such as GW190706. Assuming a low-metallicity environment ($Z = 0.001$), our framework predicts a volumetric merger rate of approximately $0.011 \, \mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$ at $z \approx 0.68$, accounting for $22\%$ of the empirical rate for this mass regime in the GWTC-4 catalog. This study demonstrates that primordial triple interactions are a highly efficient avenue for populating the PISN mass gap.

Figures

Figures reproduced from arXiv: 2607.06067 by the authors.

Figure 1
Figure 1. FIG. 1: The complete evolutionary pathway of GW190706, from the zero-age main sequence (ZAMS) to the final [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

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