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The black hole - pair instability boundary for high stellar rotation

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Rotation at about 60% of critical speed can fully mix a massive star, capping black hole masses before pair instability near the critical CO core mass of roughly 35–36 solar masses.

desk verdict A solid follow-up that extends the Mcrit boundary to fast rotation, but the 35-36 M_sun cap sits on the uncalibrated CHE assumption and the LIGO bump alignment is partly built from the input cutoff. read the letter →

arxiv 2504.17009 v1 pith:2P6DDABE submitted 2025-04-23 astro-ph.SR astro-ph.GAastro-ph.HE

Ethan R. J. Winch , Gautham N. Sabhahit , Jorick S. Vink , Erin R. Higgins This is my paper · ORCID
classification astro-ph.SRastro-ph.GAastro-ph.HE
keywords blackholemassgappair-instabilitysupernovastellarrotationchemicallyhomogeneousevolutionmechanicallossgravitationalwavesourcesmassivestarlowmetallicitystars
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

This paper tries to establish that fast stellar rotation, at roughly 60% of critical speed, can make a massive star's entire interior chemically homogeneous, so that the star evolves like an already stripped star. In that state, the final black hole mass before a pair-instability supernova is set by the critical carbon-oxygen core mass, which the authors compute as 36.3 solar masses, with their high-rotation experiment giving 35.0 ± 2.3 solar masses. If true, the heaviest black holes formed from single rapidly rotating low-metallicity stars should pile up around that cap, naturally producing the bump seen near 35 solar masses in gravitational-wave black hole merger data. The paper also adds a mechanical mass-loss treatment for stars at critical rotation and derives fits for core and final masses across rotation rate and metallicity.

What carries the argument

The load-bearing object is the critical carbon-oxygen core mass, defined through a global stability integral of the first adiabatic index; the star becomes pair-unstable when this integrated index drops below the stability threshold. The new mechanism is the chemically homogeneous (stripped-star) limit reached at high rotation: once rotational mixing homogenizes the whole star, the core mass saturates at the final mass, and the pair-instability boundary becomes a hard cap on the final black hole mass. The implementation also uses a rotation-boosted radiative wind formalism combined with a mechanical mass-loss term that switches on when the surface speed crosses 98% of the critical value, preventing models from spinning up to supercritical rotation.

What would settle it

A survey of rotating low-metallicity massive stars that finds no chemically homogeneous objects at the rotation rates assumed, or a single-star black hole above about 36 solar masses whose progenitor was fully mixed, would break the cap.

Watch

Extended reading notes

Core claim

The central claim is that rotation at Ω/Ω_crit ≈ 0.6 triggers chemically homogeneous evolution, turning a massive low-metallicity star into a stripped star whose core mass equals its final mass. For these fully mixed stars, the helium core mass loses its independent meaning, and the pair-instability boundary is set by a fixed critical carbon-oxygen core mass, M_CO,crit = 36.3 M_sun (with the high-rotation extension giving 35.0 ± 2.3 M_sun). Consequently, the maximum black hole mass from this channel is capped by the critical core mass rather than by the initial stellar mass. Population synthesis using the new fits produces a strong pile-up of black hole progenitors near 27–36 M_sun with a steep drop just above 36 M_sun, matching the bump in the gravitational-wave black hole mass distribution.

Load-bearing premise

The load-bearing premise is that rotation near 60% of critical speed really does mix the entire star chemically, an assumption the paper acknowledges is still under debate and lacks direct empirical support.

Editorial extensions

If this is right

  • The maximum black hole mass from single, rapidly rotating, low-metallicity stars is set by the critical CO core mass of about 35–36 M_sun; heavier black holes in the pair-instability gap would require a different formation path.
  • Population synthesis with the new fits produces a black hole progenitor pile-up around 27–36 M_sun and a steep drop just above 36 M_sun, matching the observed bump in gravitational-wave remnant masses.
  • Below about 1/100th solar metallicity, mechanical mass loss dominates over radiative winds, often contributing 35–70% of the total mass lost, which further lowers the final black hole masses.
  • Stars rotating faster than roughly 0.4–0.5 of critical speed become chemically homogeneous and stripped, so final mass equals core mass and the helium core mass ceases to be a useful predictor for these objects.
  • The presence or absence of the internal magnetic-field dynamo changes final masses significantly only at higher metallicity, not at low metallicity, contrary to a previously proposed explanation for differences between stellar evolution codes.

Reading between the lines

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

  • Editorial inference: If the cap is real, the 35 M_sun pile-up in gravitational-wave data should be strongest for mergers whose progenitors formed at low metallicity, and it should weaken near solar metallicity where winds remove angular momentum before homogenization.
  • Editorial inference: The same mechanism would sharpen the upper edge of the second mass gap at about 36 M_sun rather than the wider 50–120 M_sun range, implying that any future single-star black hole above roughly 40 M_sun would require an alternative formation channel such as dynamical mergers or later accretion.
  • Editorial inference: The paper's fits predict that the black hole mass cap is nearly independent of initial mass for fast rotators above about 80 M_sun; a cluster of remnants all lying within 35 ± 3 M_sun would support the mechanism, while a wide scatter above the cap would weaken it.
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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 / 4 minor

Summary. This paper extends the pair-instability boundary study of Winch et al. (2024) to rapidly rotating, low-metallicity massive stars. The authors implement a mechanical mass-loss prescription in MESA for models that approach the OmegaGamma limit, run an M_crit experiment up to Omega/Omega_crit = 0.8, and find that the critical CO core mass for the BH/PPI boundary remains approximately constant (M_CO,crit,high = 35.0 +/- 2.3 Msun, within error of the earlier 36.3 Msun). They then run a grid of models, derive empirical fits for final and core masses, and use these in a simple population synthesis. The central claim is that fast rotators become chemically homogeneous, so that the final BH mass is capped by the CO critical core mass, producing a pile-up around 27-36 Msun that they associate with the LIGO/Virgo bump at about 35 Msun.

Significance. If the central claim holds, the paper provides a concrete mechanism - rotation-induced chemical homogeneity - that links stellar evolution to the observed bump in the black hole mass distribution near 35 Msun, a topic of current interest. The M_crit experiment at high rotation is a useful independent check of the critical core mass criterion, and the mechanical mass-loss implementation in MESA is a practical contribution that will be of value to the massive-star modeling community. The paper is clearly organized and the synthetic population is a straightforward illustration of the model consequences. However, the headline conclusion rests heavily on the realization of chemically homogeneous evolution at Omega/Omega_crit ~ 0.6, an assumption the authors themselves note is debated, and the population feature is in part imposed by applying the derived critical core mass cutoff to the fits. These points need to be addressed before the central claim can be considered robust.

major comments (3)
  1. [Sec. 2.2 and Sec. 3.2 (cf. Sec. 4.1, Fig. 13)] The paper's main conclusion - that rapid rotation caps BH masses near 35-36 Msun - depends on models becoming chemically homogeneous at Omega/Omega_crit ~ 0.6. The authors explicitly acknowledge in Sec. 2.2 that the physics of rotationally induced CHE is still under debate and that there is no empirical evidence that low-metallicity WR stars rotate faster than their high-metallicity counterparts. The D_ES = 0 test in Sec. 4.1 (Fig. 13) changes the HRD track and final mass, but the authors do not report whether that model still becomes chemically homogeneous or retains an H envelope. This is load-bearing: if CHE does not occur at the adopted mixing efficiency, the maximum BH mass reverts to the H-rich boundary (~93 Msun) and the LIGO/Virgo bump interpretation disappears. Please run a set of models with reduced rotational mixing efficiencies (e.g., D_ES = 0, reduced secular shear, or a factor of a few lower diffusion coefficients) and report the resulting M_final, M_CO, and whether the star is still fully mixed at core He exhaustion. This directly tests the robustness of the 36 Msun cap.
  2. [Sec. 3.3 and Sec. 3.4 (Eqs. 14-15, Fig. 12)] The population synthesis in Sec. 3.4 applies the critical CO core mass criterion from Sec. 3.2 as a hard cutoff to the fitted final masses, and the resulting pile-up below ~36 Msun is therefore largely a construction of that imposed criterion rather than an independent prediction. The large error on the CO core mass fit (M_CO,fit = +/- 9.09 Msun) is not propagated into the population histogram, so the sharp drop at 36.3 Msun and the claimed alignment with the LIGO/Virgo bump at ~35 Msun are not robust. Please propagate the fit uncertainties into the synthetic population (e.g., via Monte Carlo over the fit coefficients) and show the resulting distribution with error bands. In addition, show the histogram produced from the final-mass fit alone, without the M_CO,crit cutoff, so the reader can see what fraction of the pile-up is physical rather than imposed. If the bump disappears or shifts substantially, the wording should be softened from an alignment claim to a consequence of the adopted criterion.
  3. [Sec. 3.2 (Fig. 7) and Sec. 3.4] The high-rotation M_crit experiment is performed at a single fiducial metallicity (Z = 1/10 Z_sun, as stated in the Fig. 7 caption), while the population synthesis spans Z/Z_sun from 1/5 to 1/1000. The occurrence of CHE and the effectiveness of mechanical mass loss are metallicity-dependent, so the applicability of the 35-36 Msun cap across the full metallicity range is not directly demonstrated. Please clarify whether the constancy of M_CO,crit at high rotation has been checked at other metallicities, or state explicitly that the metallicity dependence is assumed to be negligible and justify that assumption with references or a small number of test models.
minor comments (4)
  1. [Sec. 3.3 (Eqs. 14-16)] The typeset equations for the fits are garbled: the branch conditions and coefficients run together, making it difficult to reproduce the fits. Please reformat these equations with explicit piecewise definitions and list all coefficients, ideally also in a machine-readable table.
  2. [Table 2] Table 2 lists the grid parameters but the layout is confusing; it is not obvious which combinations of M_ZAMS, Z/Z_sun, and Omega/Omega_crit are actually run. A full factorial grid or an explicit statement of which combinations were omitted would help.
  3. [Sec. 3.4 and Table 4] The quantity called M_BH in Table 4 is actually the final stellar mass before collapse, not a directly predicted BH mass, since no core-collapse simulation is performed. The caption should make this explicit, even though the text notes it.
  4. [Sec. 4.1 (Fig. 13)] The comparison with Sibony et al. (2024) shows a ~20% final-mass difference for similar initial conditions. The authors attribute this to rotational mixing and mass-loss implementation, but the discussion is brief; a sentence summarizing the main reason for the difference (e.g., GENEC vs. MESA mixing treatment) would strengthen the comparison.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the critical core mass is re-derived from the Stothers stability integral, and the population cutoff is an explicit input, not a fitted prediction.

full rationale

The derivation chain is not circular. The critical CO core mass is obtained in this paper's M_crit experiment by integrating the first adiabatic index with the Stothers (1999) global stability criterion (Eqs. 11-12), running models to the end of core Oxygen burning; it is not fitted to LIGO/Virgo data. The high-rotation value M_CO,crit,highOmega/Omegacrit = 35.0 +/- 2.3 Msun is newly computed in Section 3.2, not merely imported from Winch et al. (2024). The population synthesis in Section 3.4 does impose that criterion: 'We apply our CO critical core mass criterion from Section 3.2 into our fit from Equation 14, and final mass fit from Equation 15,' and the paper transparently states that the steep drop above ~36 Msun 'is due to the aforementioned critical core mass criterion.' Thus the sharp upper edge of the predicted BH mass distribution is an input, but it is a physically derived input, and the paper does not disguise it as an independent population-synthesis prediction. The comparison to the LIGO/Virgo bump is therefore a test of an independently derived physical boundary, not a self-fulfilling fit. The only caveat is that the realization of chemically homogeneous evolution at Omega/Omegacrit ~0.6 is model-dependent and the authors themselves state that 'The physics of rotationally-induced chemically homogenous evolution is still under debate'; this is a scientific risk, not a circularity. The self-citation to Winch et al. (2024) for the original 36.3 Msun value is not load-bearing because the value is re-derived and verified here within errors.

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

The central claims rest on standard but unverified stellar physics assumptions (mixing, mass loss, stability criterion) and on the authors' previously derived critical core mass. No new particles or forces are introduced. The free parameters are mostly calibration choices from the 54-model grid and the mass-loss implementation.

free parameters (7)
  • Overshooting parameter alpha_ov = 0.1
    Chosen to match the fiducial model of Winch et al. (2024) and to produce the largest black holes; not derived from data.
  • Semiconvective efficiency alpha_sc = 0.1
    Same as previous fiducial; controls mixing above the convective core.
  • Initial rotation rates Omega/Omega_crit = 0.4, 0.6, 0.8
    Discrete grid values; the critical threshold for chemical homogeneity at ~0.6 is inferred from these points.
  • Mechanical mass loss trigger v/v_crit,MM = 0.98
    Ad hoc numerical choice to avoid strict unity; no sensitivity study is presented.
  • CAK force multiplier alpha = 0.52
    Fixed for high temperatures following Lamers et al. (1995); affects rotation-enhanced mass-loss boost magnitude.
  • Polynomial fit coefficients in Eqs 14-16 = As reported in Equations 14-16
    Fitted to the 54-model grid; quoted errors are +-9.09 solar masses (M_CO), +-2.86 solar masses (M_final), +-3.04 solar masses (M_He).
  • Spruit-Tayler dynamo efficiency D_ST = 0 for main grid; 1 in test models
    Turned off for the main grid after tests showed small effect on final mass at low Z; a modeling choice.
assumptions (7)
  • domain assumption Stothers (1999) global stability criterion with <Gamma_1> determines the onset of pair-instability.
    Used as the M_crit test in Section 2.4; it is an integral criterion for spherical stars and is assumed to remain valid for rotating, distorted models.
  • domain assumption Ledoux criterion with MLT efficiency alpha_MLT = 1.82 and step overshooting alpha_ov = 0.1 accurately describes convection and boundary mixing.
    Adopted in Section 2.2 from Choi et al. (2016); directly sets core size and hence core mass.
  • domain assumption Diffusive rotational mixing (Heger et al. 2000), including Solberg-Hoiland, secular shear, Eddington-Sweet and GSF instabilities, drives chemical homogeneity at Omega/Omega_crit ~ 0.6.
    The entire stripped-star channel rests on this mixing being efficient enough to homogenize the star; the paper itself notes this physics is debated and lacks empirical support (Section 2.2).
  • domain assumption The Maeder & Meynet (2000) Omega-Gamma limit and rotation-boost factor describe mass loss near break-up.
    Equations 2 to 7; the paper relaxes one approximation but keeps the overall formalism, including the numerical search for Gamma_Omega = 1.
  • ad hoc to paper The critical CO core mass M_CO,crit = 36.3 solar masses derived in Winch et al. (2024) is a universal boundary that can be applied as a hard cutoff to the fitted final masses.
    Used to generate the population pile-up in Section 3.4; this paper tests the value at high rotation and finds 35.0 +- 2.3, so the constancy is partially self-fulfilling in the population synthesis.
  • domain assumption The adopted radiation-driven wind recipes (Vink et al. 2001/2011, de Jager et al. 1988, Sander & Vink 2020, Vink 2017 floor) are applicable at low metallicity.
    Section 2.3.1; these recipes set the baseline mass loss that rotation enhances, so they shape all final masses.
  • domain assumption 1D MESA models with the implemented mechanical mass loss capture the mass and angular momentum loss of near-critical rotators.
    Section 2.3.3; the mechanical loss is a parameterized switch at v/v_crit = 0.98 with no comparison to multi-D simulations or observed eruptive mass loss.

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Pith. "Pith review of The black hole - pair instability boundary for high stellar rotation." pith.science (2026). https://pith.science/paper/2P6DDABE

@misc{pith2026250417009,
  author       = {Pith},
  title        = {Pith review of: The black hole - pair instability boundary for high stellar rotation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2P6DDABE}},
  note         = {Machine review of arXiv:2504.17009}
}
abstract

The Pair Instability (PI) boundary is crucial for understanding heavy merging Black Holes (BHs) and the second mass gap's role in galactic chemical evolution. So far, no works have critically and systematically examined how rotation and mass loss affect the PI boundary or BH masses below it. Rapid rotation significantly alters stellar structure and mass loss, which is expected to have significant effects on the evolution of stellar models. We have previously derived a critical core mass independent of stellar evolution parameters, finding the BH (Pulsational) PI boundary at $M_{ CO, crit} = 36.3 M_\odot$ for a carbon-oxygen (CO) core. Using MESA, we model massive stars around the PI boundary for varying rotation rates and metallicities. We implement mechanical mass loss in MESA, studying its effects on massive stars in low-metallicity environments. Below $1/100$th $Z_\odot$, mechanical mass loss dominates over radiative winds. We check the BH-PI boundary for rapid rotators to confirm our critical core mass criterion and derive model fits describing rotation's impact on core and final masses. Fast rotators reach a point (typically $\Omega / \Omega_{crit} \approx 0.6$) where the entire star becomes chemically homogeneous, evolving like a stripped star. This lowers the maximum BH mass before PI to its critical core mass of $M_{CO, crit} = 36.3 M_\odot$, aligning with the bump feature in the BH mass distribution observed by LIGO/VIRGO.

Figures

Figures reproduced from arXiv: 2504.17009 by the authors.

Figure 1
Figure 1. Sketch of where mechanical mass loss is relevant for models. Models in the blue region have a maximum Ω/Ωcrit < 1, meaning that these models do not require mechanical mass loss, while models in the red region have a maximum Ω/Ωcrit > 1. Adapted and extended from Winch et al. (2024). 2008; Vink 2022). Enormous amounts of mass can be lost via such line-driven winds, especially at high 𝑍 where the momentum transfer can… view at source ↗
Figure 2
Figure 2. Critical velocity evolution of a 100𝑀⊙ model with an initial rotation rate Ω/Ωcrit = 0.6 and a metallicity of 𝑍 = 1/100𝑍⊙. The critical velocities 𝑣crit,1 and 𝑣crit,2 defined in equations 2 and 4 are shown with the dashed blue, and dash-dot red lines respectively. The corresponding equatorial velocity of the model, 𝑣eq is shown by the black solid line. The dotted black vertical line shows the location where ΓEdd > 0… view at source ↗
Figure 3
Figure 3. Evolution of a 100𝑀⊙ star with an initial rotation of Ω/Ωcrit = 0.6 to the end of He burning. The mass-loss rate of the model is shown with the solid black line, while the evolution of rotation (Ω) is shown with the dotted blue line and the ΩΓ value of Equation 3 is shown with the red dashed line. The orange highlighted region is where our condition for enhanced mass loss is satisfied, and the model is undergoing me… view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: Hertsprung-Russel Diagrams of 100𝑀⊙ models from 1/10th 𝑍⊙ to 1/1000th 𝑍⊙ with an initial rotation rate of Ω = 0.6Ωcrit (as per models B1, B2, B3 in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 7
Figure 7. Figure 7: 𝑀crit experiment extension from Winch et al. (2024). Data points indicate the 𝑀crit values for that Ω/Ωcrit, and error bars are the uncertainties from the models used. Models above ∼ 0.5Ωcrit become fully mixed stripped stars and thus by the end of O burn, the star bec…
Figure 9
Figure 9. Figure 9: Combined plot of the datapoints for the main grid including fit lines for 𝑀CO. The solid maroon line denotes the 1:1 line for mass such that any deviation from this line represents mass loss. Blue, red and green data points are for models with metallicities of 1/1000th…
Figure 8
Figure 8. Figure 8: Profiles for two stars showing how the definition of the CO core can change the mass coordinate of the core. The top panel shows a H rich star with 𝑀ZAMS = 78𝑀⊙, Ω = 0.3Ωcrit, while the bottom panel shows a stripped star with 𝑀ZAMS = 120𝑀⊙, Ω = 0.8Ωcrit both at the end…
Figure 10
Figure 10. Figure 10: Combined plot of the datapoints for the main grid including fit lines for 𝑀He. The solid maroon line denotes the 1:1 line for mass such that any deviation from this line represents mass loss. Blue, red and green data points are for models with metallicities of 1/1000t…
Figure 11
Figure 11. Figure 11: Combined plot of the datapoints for the main grid including fit lines for 𝑀final. The solid maroon line denotes the 1:1 line for mass such that any deviation from this line represents mass loss. Blue, red and green data points are for models with metallicities of 1/10…
Figure 12
Figure 12. Figure 12: Graphical representation of the numbers plotted in [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]
Figure 13
Figure 13. Figure 13: Hertzprung-Russel Diagram comparing several 120𝑀⊙ models. The red line tracks the evolution of a model with Ω/Ωcrit = 0.4, while the black line tracks a model with no rotation. The blue track is for a model with rotation, but with the diffusion coefficient for the Edd…

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.