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Pair-Instability Gap Black Holes in Population III Star Clusters: Pathways, Dynamics, and Gravitational Wave Implications

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Pith's one-line read Population III star clusters form pair-instability gap black holes mainly through binary black hole mergers, and the survivors of gravitational-wave recoil merge again at rates (0.005–0.017 per year per cubic gigaparsec) that make these…

desk verdict Solid extension of an established simulation program: the low-mass cluster runs and spin-dependent recoil post-processing are new, but the headline rates depend on an approximate GW-kick treatment and one printed formula needs fixing. read the letter →

arxiv 2504.20392 v1 pith:OQ2JGS6V submitted 2025-04-29 astro-ph.HE astro-ph.GA

Weiwei Wu , Long Wang , Shuai Liu , Yining Sun , Ataru Tanikawa , Michiko Fujii This is my paper · ORCID
classification astro-ph.HEastro-ph.GA
keywords PopulationIIIstarsN-bodysimulationspair-instabilitymassgapbinaryblackholemergersgravitationalwaverecoilGW190521starclustersastronomy
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

Black holes in the pair-instability mass gap—between about 65 and 120 solar masses—are not expected to form from ordinary single-star evolution, because the star is instead blown apart by a pair-instability supernova. The gravitational-wave event GW190521 appears to contain a black hole in exactly this gap. This paper argues that dense clusters of Population III stars, the first stars in the universe, can fill the gap: in its N-body simulations, binary black hole mergers produce roughly 90% of these gap black holes, while direct stellar collisions produce the rest. Most of the merger-made black holes survive the gravitational-wave recoil of the merger that created them, especially in massive clusters, and can merge again with another black hole to make GW190521-like signals. Assuming a top-heavy initial mass function (many massive stars), the authors estimate average rates of 0.005–0.017 per year per cubic gigaparsec for a gap black hole merging with another black hole, with upper limits of 0.030–0.106, and predict that roughly 43–98% of these events would be detectable by space-borne gravitational-wave observatories.

What carries the argument

The central objects are pair-instability gap black holes (PIBHs), defined here as black holes with masses 65–120 solar masses, objects that ordinary single-star evolution cannot produce. The mechanism that carries the argument is the hierarchical merger chain inside a dense cluster: a top-heavy initial mass function seeds many massive black holes, primordial binaries or dynamical encounters form black hole binaries, those binaries merge by gravitational radiation into a gap black hole, and the remnant can pair up with another black hole and merge again. The filter on that chain is gravitational-wave recoil: spins are assigned to merging components from isolated-binary evolution tables, a numerical-relativity recoil formula gives each merged remnant a kick, and remnants whose kick exceeds the cluster escape velocity are removed from the accounting. The retained population sets the predicted merger rates and the gravitational-wave strain signals that the authors compare with detector sensitivity curves.

What would settle it

Re-run the same N-body simulations with gravitational-wave recoil applied self-consistently at every binary black hole merger and count the resulting secondary PIBH-BH mergers; if that count falls below the post-processing estimates by more than the simulation scatter, the central rate claim fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the pair-instability gap in dense Population III star clusters is filled chiefly by repeated binary black hole mergers, not by stellar collisions. In a 100,000-solar-mass cluster with primordial binaries, the simulations produce about 49 gap black holes per cluster through black hole mergers and only 1.9 through stellar collisions; without primordial binaries, the collision channel produces none and the merger channel weakens sharply. Gravitational-wave recoil ejects roughly 10–50% of the merger-made gap black holes, depending on spins and cluster escape speed, and the retained fraction goes on to form secondary mergers. The resulting volumetric merger rates are 0.005–0.017 per year per cubic gigaparsec on average and up to 0.106, which the authors compare with rates from other cluster channels for GW190521-like events. Because these clusters form at redshift 20 and above, the model also implies a population of high-redshift sources for space-borne detectors.

Load-bearing premise

The rate numbers assume that a gravitational-wave kick only removes the merged black hole from the cluster when the kick exceeds the escape speed, and does not otherwise alter the cluster or the sequence of later mergers.

Editorial extensions

If this is right

  • If Population III clusters form with primordial binaries and a top-heavy initial mass function, their contribution to PIBH-BH mergers averages 0.005–0.017 per year per cubic gigaparsec and can reach 0.106, matching the inferred rate of GW190521-like events.
  • Because PIBH production scales nearly linearly with cluster mass, the numerous low-mass clusters of 1,000–10,000 solar masses collectively keep the volumetric rate comparable to that of the rarer 100,000-solar-mass clusters.
  • Primordial binaries are the controlling input: without them the average merger rate drops to 0–0.008 per year per cubic gigaparsec, so observed PIBH-BH rates would indirectly constrain the binary fraction of the first stars.
  • Most PIBH-BH mergers from this channel should appear at high redshift, and next-generation detectors—LISA, Taiji, TianQin, DECIGO, ET, CE—should catch a large fraction, allowing their redshifts and eccentricities to be measured.

Reading between the lines

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

  • A direct extension beyond the paper would be to re-run the models with gravitational-wave recoil applied self-consistently at each merger; if the retained fraction of gap black holes changes by more than the current post-processing estimate, the quoted rates would need revision.
  • The bimodal predicted spins of PIBH-BH mergers—concentrated near zero and near unity—could serve as a channel fingerprint: a future GW190521-like event with intermediate component spins would weigh against this Pop III cluster origin.
  • If the rates hold, the Pop III cluster channel predicts a distinguishably redshift-dependent mass spectrum in the gap, with more massive remnants at high redshift, which space-borne detectors could separate from lower-redshift globular or nuclear cluster channels.
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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

3 major / 5 minor

Summary. The paper uses N-body simulations of Population III star clusters with initial masses 10^3, 10^4, and 10^5 M_sun and primordial binary fractions of 0 and 1 to study the formation of black holes in the pair-instability mass gap (PIBHs, defined here as 65-120 M_sun). The clusters are evolved with petar and the bseemp stellar evolution prescription, including the L model, and are embedded in a mini NFW dark-matter halo. The authors find that PIBHs form predominantly through BBH mergers rather than stellar collisions (about 90% of PIBHs), that GW recoil ejects roughly 10-50% of these systems, and that the remaining PIBHs go on to form secondary PIBH-BH mergers. They report average merger rates of 0.005-0.017 yr^-1 Gpc^-3, upper limits up to 0.106 yr^-1 Gpc^-3, and estimate that 43.4% (LISA), 97.8% (Taiji), and 66.4% (Tianqin) of the simulated PIBH-BH events would be detectable by future space-borne detectors. The central claim is that Pop III star clusters are a significant formation site for GW190521-like events.

Significance. If the reported rates hold, the paper provides a quantitatively useful new channel for pair-instability-gap BH mergers and a concrete prediction for GW190521-like event rates from Pop III clusters. The strengths of the paper include a large simulation campaign (about 300 realizations per model), a clear decomposition of the stellar-collision and BBH-merger channels, and a transparent statement of the main approximation (the absence of GW kicks in the N-body evolution). The quoted rates are emergent simulation outputs rather than quantities fit to GW190521, which is a genuine strength. However, the post-processing treatment of GW recoil is load-bearing for the headline rates, and the printed f_peak formula is malformed, so the quantitative claims are provisional. The paper is a reasonable contribution to the discussion of GW190521-like events, but it needs revision before the central rates can be accepted.

major comments (3)
  1. [§3.4.1, Sec. 3.4.2, Table 2] The GW-recoil post-processing is the decisive uncertainty in the per-cluster merger counts and is not validated. The text states that the petar version used lacks the GW kick effect, and the authors respond by deleting PIBHs whose sampled kick exceeds vesc. Retained PIBHs are not given the impulsive recoil velocity at the merger, so their subsequent orbits, binary-formation probabilities, and later merger counts are computed exactly as if the kick had never happened. In m100000-bf1, 59% of PIBH-BH mergers are excluded, leaving 0.548 mergers per cluster; that number is therefore a rescaled count of un-kicked trajectories, not a dynamical prediction. The authors need either to rerun a subset of models with kicks implemented, or to provide a quantitative argument that the neglected recoil impulse does not bias the retained population and its subsequent mergers.
  2. [§3.4.1, Eqs. (3)-(5), Table 2] The total escape velocity is computed as vesc = vesc,s + vesc,h, but for two independent potential components the correct escape speed is sqrt(vesc,s^2 + vesc,h^2). With vesc,h approximately 53 km/s and vesc,s of order 15-20 km/s at BBH-merger times (Figure 10), the printed threshold of about 65 km/s for m100000-bf1 is roughly 15-20% too high relative to the correct value near 56 km/s. Because the ejected fraction sits near the retention boundary (59% of secondary mergers are excluded), this overestimate can translate directly into an overestimate of the retained PIBH-BH merger counts and of the rates in Table 3. The escape-velocity combination should be corrected and the sensitivity of the ejected fraction fe to the threshold should be quantified.
  3. [§3.5, Eq. (6)] The peak-frequency formula is malformed as printed. The bracket contains the expression '1.01678×10^5 − 5.57372×10^2 − 4.9271×10^3 + 1.68506×10^4' with no dependence on eccentricity and no apparent powers of (1-e^2); as written it is dimensionally inconsistent and cannot reproduce the standard Hamers (2021) form. Since f_peak determines where the characteristic-strain curves sit relative to detector sensitivity curves, the reported detection fractions (43.4% for LISA, 97.8% for Taiji, 66.4% for Tianqin) and Figure 13 depend on this formula. The correct expression must be provided and used; if the printed version is only a typographical corruption, this still needs fixing because the affected quantitative claims are load-bearing.
minor comments (5)
  1. [§2.2] W0 is described as the ratio of the core radius and the tidal radius; W0 is more precisely the dimensionless central potential of the King model, and the sentence should be corrected.
  2. [§2.4, Eq. (2)] The spin variables in Eq. (2) are not fully defined: the text should state explicitly that chi_tilde_parallel and chi_tilde_perpendicular are dimensionless spin components relative to the orbital angular momentum, and how the angle cos(Theta) is sampled in the post-processing.
  3. [§3.3] The statement that primordial binaries do not significantly change the merger-time distribution is based on a visual comparison of normalized cumulative curves; a quantitative comparison (e.g., a Kolmogorov-Smirnov test) would make the claim testable.
  4. [Table 3] The rows labeled 'Per mass [M_sun^-1]' appear to be merger counts per unit cluster mass rather than rates, and this should be stated explicitly in the table caption or text.
  5. [Throughout] There are several typographical errors: 'gravitaitonal' in §3.4.1, 'simulaiton' in §2.4, 'dependeonce' in §3.2, 'luminocity' in §2.3, 'prongeitor' in §2.4, 'younmg' in §5, and 'breifly' in §2.3. These should be corrected in a final pass.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: PIBH formation channels and merger rates are emergent N-body outputs, not inputs or fitted parameters.

full rationale

The paper's central quantities—PIBH formation counts (e.g., 49.09 per cluster via BBH mergers vs. 1.89 via collisions in m100000-bf1, Table 2), PIBH-BH merger counts (0.548 per cluster), and volumetric rates (0.005–0.017 yr^-1 Gpc^-3, Table 3)—are outputs of the petar N-body integrations and subsequent rate normalization, not quantities encoded in the initial conditions or in the bseemp stellar-evolution model. No parameter is fitted to GW190521; the comparison is made after the simulation results are produced. The spin/kick post-processing applies the external Gerosa & Kesden (2016) recoil formula to simulated masses using spins matched from bseemp isolated-binary tables; this is a model-based assignment, not a reverse-engineered fit to the target rate. Self-citations to Wang et al. (2022) and Liu et al. (2024b) establish the numerical setup and supply reused massive-cluster simulation data, but the load-bearing BBH-merger-dominance result is reproduced within the present runs and the low-mass cluster runs are new. The manuscript explicitly notes a genuine approximation: 'The petar code version (master branch, similar to v1.0) used in our simulations lacks the GW kick effect,' so ejected PIBHs are removed in post-processing. That is a modeling limitation affecting accuracy (as is the linear vesc = vesc,s + vesc,h addition), but it is not circular: the kick cutoff is an externally motivated physical condition, and the retained-merger counts are not defined to match any input. No uniqueness theorem, ansatz-via-citation, or definitional equivalence is load-bearing in the derivation chain.

Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

Central rates rest on a stack of unmeasured inputs: top-heavy IMF, binary fraction and properties, cluster mass, concentration, halo parameters, redshift, metallicity, and the L model of Pop III stellar evolution. The paper does not invent new entities, but it does rely on the post-hoc GW kick treatment and on assuming all Pop III stars form in clusters of one mass when converting counts to volume rates. Each of these choices is either from the same group's earlier papers or from hydrodynamical simulations with large uncertainties.

free parameters (8)
  • Pop III IMF power-law slope = -1
    Top-heavy IMF with slope -1 and mass limits 1-150 M_sun controls the number of massive BH progenitors; adopted from hydro simulations, not measured.
  • Primordial binary fraction = 0 or 1
    Binary fraction is set to either 0 or 100 percent with period/mass-ratio/eccentricity distributions from Sana et al. (2012), a metal-rich calibration applied to Pop III with no direct constraint.
  • Initial cluster mass = 10^3, 10^4, 10^5 M_sun
    Cluster mass is a scanned input; the rate for the real universe depends on the unknown mass function of Pop III clusters.
  • Initial half-mass radius = 1 pc
    Adopted as typical for present-day star clusters and kept fixed across models.
  • King concentration W0 = 9
    High central concentration is chosen because Wang et al. (2022) found it leads to IMBH formation; this concentrates dynamical activity.
  • NFW dark matter halo virial mass = 4e7 M_sun
    Taken from model A of Sakurai et al. (2017); sets the escape velocity that retains kicked PIBHs.
  • NFW halo concentration C = 15.3
    Sets the dark matter contribution to the central escape velocity, about 53 km/s.
  • PIBH mass gap boundaries = 65-120 M_sun
    The lower limit is chosen above the single-star remnant ceiling in the L model, shaping which remnants are counted as PIBHs.
assumptions (7)
  • domain assumption Pop III stellar evolution follows the bseemp L model at Z=2e-10, yielding single-star BH remnants of 40-60 M_sun and a PISN mass gap at 65-120 M_sun.
    The PIBH definition and all remnant masses rest on these fitting formulas; the authors note in Section 2.3 that the alternative M model changes isolated binary outcomes, so the L model is a load-bearing choice.
  • domain assumption petar and bseemp correctly model cluster dynamics, stellar/binary evolution, and external tides.
    All per-cluster merger counts are outputs of these codes; no cross-validation against a second N-body code or direct observation is provided.
  • ad hoc to paper GW recoil kicks can be accounted for in post-processing by deleting PIBHs whose sampled kick exceeds the instantaneous escape velocity, without changing the rest of the cluster evolution.
    Section 3.4.1 states the petar version used lacks GW kick; the validity of removing remnants after the fact rather than during the simulation is not tested.
  • standard math Peters (1964) orbital decay and the Gerosa & Kesden (2016) recoil formula govern merger timescales and kick velocities.
    Used to decide which BBHs merge and which PIBHs are ejected; coefficients are taken from numerical relativity literature and are not derived in this paper.
  • domain assumption The cosmic Pop III stellar mass density is 3.2e4 to 2e5 M_sun Mpc^-3 over z=10-20 and all Pop III stars form in clusters of a single simulated mass.
    Section 3.6 scales per-cluster merger counts by this density and explicitly calls the single-mass assumption unrealistic; the real cluster mass function is not modeled.
  • domain assumption Primordial binary orbital properties follow Sana et al. (2012) metal-rich star distributions.
    No direct constraints on Pop III binaries exist; the binary fraction (0 or 1) and period/mass-ratio/eccentricity distributions set the dominant PIBH formation channel.
  • domain assumption Pop III clusters form at z=20 with r_h=1 pc, W0=9, inside an NFW halo of Mvir=4e7 M_sun and concentration 15.3.
    These initial conditions set the escape velocities and dynamical densities; they are adopted from Sakurai et al. (2017) and Wang et al. (2022), not from a resolved Pop III cluster population model.

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Pith. "Pith review of Pair-Instability Gap Black Holes in Population III Star Clusters: Pathways, Dynamics, and Gravitational Wave Implications." pith.science (2026). https://pith.science/paper/OQ2JGS6V

@misc{pith2026250420392,
  author       = {Pith},
  title        = {Pith review of: Pair-Instability Gap Black Holes in Population III Star Clusters: Pathways, Dynamics, and Gravitational Wave Implications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OQ2JGS6V}},
  note         = {Machine review of arXiv:2504.20392}
}
abstract

The detection of the gravitational wave (GW) event GW190521 raises questions about the formation of black holes within the pair-instability mass gap (PIBHs). We propose that Population III (Pop III) star clusters significantly contribute to events similar to GW190521. We perform $N$-body simulations and find that PIBHs can form from stellar collisions or binary black hole (BBH) mergers, with the latter accounting for 90\% of the contributions. Due to GW recoil during BBH mergers, approximately 10-50% of PIBHs formed via BBH mergers escape from clusters, depending on black hole spins and cluster escape velocities. The remaining PIBHs can participate in secondary and multiple BBH formation events, contributing to GW events. Assuming Pop III stars form in massive clusters (initially 100,000 $M_\odot$) with a top-heavy initial mass function, the average merger rates for GW events involving PIBHs with 0% and 100% primordial binaries are $0.005$ and $0.017$ $\text{yr}^{-1} \text{Gpc}^{-3}$, respectively, with maximum values of $0.030$ and $0.106$ $\text{yr}^{-1} \text{Gpc}^{-3}$. If Pop III stars form in low-mass clusters (initial mass of $1000M_\odot$ and $10000 M_\odot$), the merger rate is comparable with a 100% primordial binary fraction but significantly lower without primordial binaries. We also calculate the characteristic strains of the GW events in our simulations and find that about 43.4% (LISA) 97.8% (Taiji) and 66.4% (Tianqin) of these events could potentially be detected by space-borne detectors, including LISA, Taiji, and TianQin. The next-generation GW detectors such as DECIGO, ET, and CE can nearly cover all these signals.

Figures

Figures reproduced from arXiv: 2504.20392 by the authors.

Figure 1
Figure 1. The evolution histories of three PIBH formation channels are illustrated, with example events taken from the m100000-bf1 models. In the left panel, a PIBH forms via a stellar collision, evolving into a PIBH without experiencing PISN. The middle panel shows a BBH formed through binary stellar evolution that merges into a PIBH via GW radiation. In the right panel, a dynamically formed BBH merges into a PIBH, with pote… view at source ↗
Figure 2
Figure 2. Mass spectrum of PIBHs from simulations with three different M and with primordial binaries. The upper and lowe panels show PIBHs formed through stellar collision and BBH merger channels, respectively. by binary stellar evolution. With the Sana et al. (2012) period distribution, some primordial binaries can evolve into BBH mergers through pure binary stellar evolu￾tion, without requiring dynamical encounters to shri… view at source ↗
Figure 4
Figure 4. The normalized cumulative distribution of the formation time of PIBHs through two channels [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figures from the paper (7 more)
Figure 5
Figure 5. Figure 5: Upper panel: m1 versus m2 for each BBH merger from all simulations with primordial binaries. Points of dif￾ferent colors indicate the generation of the merger. Lower panel: histogram of the mass ratio (q). tively. ζ is the angle between vm and v⊥. The formulas of the t…
Figure 6
Figure 6. Figure 6: Similar to [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 8
Figure 8. Figure 8: The spin distribution of BH components in bi￾nary systems within Pop III star clusters (red) and isolated binary evolution reference models (blue). Solid and dashed lines represent the spin distributions of the primary and sec￾ondary components, respectively. Additiona…
Figure 9
Figure 9. Figure 9: Considering with and without the spins of the BBHs, the kick velocity of the BH produced during the BBH merger for the m100000-bf1 model. 0 2000 4000 6000 8000 10000 12000 t [Myr] 0 10 20 30 40 50 vesc, s [k m/s] m100000-bf0 m100000-bf1 m10000-bf0 m10000-bf1 m1000-bf0 …
Figure 10
Figure 10. Figure 10: Evolution of the escape velocity only from stars of the star cluster over time. Using these spin values, we randomly sample the spin orbit angle, recalculate vk, and show its histogram in [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: The time distribution of PIBH-BH mergers for m100000 models. 3.4.2. Properties of BBH mergers involving PIBHs [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 13
Figure 13. Figure 13: illustrates the characteristic strain of GW signals as a function of their peak frequency, specifi￾cally focusing on mergers involving PIBHs. The analy￾sis includes information about the redshift (z) and mass ratio of each merger event, which are represented in the co…

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