Pair-Instability Gap in Stellar Black Holes

• The pair-instability gap is a theoretical mass interval (~45–130 M☉) where electron–positron pair production in massive stars softens the core’s equation of state and suppresses black hole formation.
• Research shows that factors like nuclear reaction rates, stellar rotation, and envelope retention significantly modify the gap boundaries and influence the resulting black hole masses.
• Observational and simulation studies indicate that dynamical channels such as stellar mergers and collisions can repopulate the gap, challenging classical models of first-generation stellar collapse.

The pair-instability gap is the predicted dearth of stellar-origin black holes produced when very massive stellar cores enter the electron–positron pair-production regime, soften their equation of state, and undergo either pulsational pair-instability mass ejection or complete pair-instability supernova disruption. In the gravitational-wave literature, this “upper black-hole mass gap” is often quoted near $45$–$130\,M_\odot$, but the adopted boundaries vary across studies, and several recent works emphasize that the lower edge depends sensitively on core structure, angular-momentum transport, envelope retention, and the $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$ rate (Farmer et al., 2019, Marchant et al., 2020, Woosley et al., 2021).

1. Physical origin of the gap

The physical mechanism is pair creation in radiation-pressure-supported cores. After core-helium burning, sufficiently massive stars develop large carbon–oxygen cores; at high temperature, photons create $e^\pm$ pairs, removing pressure-supporting radiation and softening the equation of state. One study summarizes the relevant thermodynamic regime as densities between $\approx 10^2$ and $10^6\ {\rm g\ cm^{-3}}$ and temperatures above $6\times 10^8$ K (Ballone et al., 2022). In the same picture, the effective adiabatic index is driven toward or below $4/3$, the core contracts, and explosive carbon or oxygen burning follows (Farmer et al., 2019).

Two regimes are distinguished. In pulsational pair instability, the star survives one or more thermonuclear pulses but ejects substantial mass before final collapse; in full pair-instability supernovae, the explosion unbinds the whole star and leaves no compact remnant (Farmer et al., 2019). This is why the gap is fundamentally a consequence of pre-collapse thermonuclear evolution rather than of black-hole formation physics at core collapse. The most massive black holes below the gap are produced by stars that enter the pulsational regime but are not fully disrupted, whereas stars above the pulsational regime are completely destroyed until collapse resumes on the far side of the gap (Farmer et al., 2019, Franciolini et al., 2024).

2. Boundaries, definitions, and theoretical uncertainty

The literature does not use a single pair-instability interval. Several population and detector studies adopt a gap around $45$–$130\,M_\odot$ (Son et al., 2020, Marchant et al., 2020, Franciolini et al., 2024), while collision and young-cluster studies often work with $130\,M_\odot$0–$130\,M_\odot$1 (Ballone et al., 2022, Costa et al., 2022, Mangiagli et al., 2019). Other analyses phrase the lower boundary more broadly as $130\,M_\odot$2–$130\,M_\odot$3 and the upper boundary as $130\,M_\odot$4 (O'Brien et al., 2021). This dispersion reflects different progenitor assumptions: stripped helium stars, H-rich stars retaining envelopes, rotating stars, or post-merger products.

For bare helium-star progenitors, one influential MESA study found the lower edge of the mass gap—equivalently the maximum first-generation black-hole mass below the gap—to be approximately $130\,M_\odot$5, with only small shifts from metallicity, mixing, winds, neutrino physics, and resolution, but a much larger shift from the uncertain $130\,M_\odot$6 rate. In that calculation, the edge moves from roughly $130\,M_\odot$7 to $130\,M_\odot$8–$130\,M_\odot$9 across the quoted $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$0 nuclear uncertainty (Farmer et al., 2019). Woosley and Heger similarly argued that a value $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$1–$^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$2 is reasonable for the lower boundary and that current uncertainties can raise the upper boundary $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$3 as large as $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$4 (Woosley et al., 2021).

Rotation modifies the lower edge further. For stripped helium stars at $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$5, the inferred lower edge is $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$6 for non-rotating models, $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$7 for rapidly rotating models with efficient Spruit–Tayler angular-momentum transport, and $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$8 without Spruit–Tayler transport (Marchant et al., 2020). This places the uncertainty not only in initial rotation but also in angular-momentum coupling.

A different branch of work emphasizes hydrogen-envelope retention. Detailed low-$^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$9 MESA grids that make no prior assumption about envelope stripping find that, if the H envelope falls into the black hole, the maximum black-hole mass below pair instability can reach $e^\pm$0 (Winch et al., 2024). Related PARSEC models with H-rich stars at $e^\pm$1 find a lower edge of about $e^\pm$2 for the standard $e^\pm$3 rate if the residual H-rich envelope collapses, a gap of $e^\pm$4–$e^\pm$5 for the $e^\pm$6 rate, a gap of $e^\pm$7–$e^\pm$8 for the $e^\pm$9 rate, and complete removal of the gap in the $\approx 10^2$0 case when dredge-up operates; the authors stress that this dredge-up is particularly sensitive to convection and mixing assumptions (Costa et al., 2020).

3. Channels that populate or circumvent the gap

The pair-instability gap does not imply that black holes in the interval cannot exist; it implies that standard first-generation stellar collapse is suppressed there. Several channels can repopulate the interval.

One channel is stellar collision or stellar merger in dense clusters. Hydrodynamical simulations of a head-on collision between a $\approx 10^2$1 core-helium-burning primary and a $\approx 10^2$2 main-sequence secondary show that the collision ejects about $\approx 10^2$3 of the initial mass, leaving a bound remnant of $\approx 10^2$4, with the main-sequence star depositing most of its mass around the primary core and producing a helium-enriched envelope with $\approx 10^2$5 (Ballone et al., 2022). Follow-up stellar-evolution calculations based on that hydrodynamical remnant find that the merger product avoids pair instability and collapses as a blue supergiant to a black hole of $\approx 10^2$6 (Costa et al., 2022).

A related dynamical young-cluster study found that up to $\approx 10^2$7 of all simulated black holes can lie in the pair-instability gap, depending on metallicity; $\approx 10^2$8, $\approx 10^2$9, and $10^6\ {\rm g\ cm^{-3}}$0 of all BBHs have at least one component in the gap at $10^6\ {\rm g\ cm^{-3}}$1, $10^6\ {\rm g\ cm^{-3}}$2, and $10^6\ {\rm g\ cm^{-3}}$3, respectively; and $10^6\ {\rm g\ cm^{-3}}$4 of detectable BBH mergers at Advanced LIGO/Virgo design sensitivity have at least one component in the gap if all stars form in young star clusters (Carlo et al., 2019). Population III cluster simulations likewise produce pair-instability-gap black holes through stellar collisions and BBH mergers, with BBH mergers accounting for about $10^6\ {\rm g\ cm^{-3}}$5 of the contributions in that study (Wu et al., 29 Apr 2025).

By contrast, classical isolated binary evolution appears much less effective. Even under deliberately extreme super-Eddington prescriptions, one population-synthesis study finds that at most $10^6\ {\rm g\ cm^{-3}}$6 of merging BBHs contain a black hole with $10^6\ {\rm g\ cm^{-3}}$7, only $10^6\ {\rm g\ cm^{-3}}$8 have $10^6\ {\rm g\ cm^{-3}}$9, and no merging systems exceed $6\times 10^8$0 in total mass (Son et al., 2020). Another recent analysis argues that efficient first Roche-lobe overflow followed by highly non-conservative second mass transfer can mimic a cutoff in the lighter black-hole component near $6\times 10^8$1, even when the true pair-instability limit for the heavier component is much higher; in that model, the fraction of systems in which the less massive black hole exceeds $6\times 10^8$2 is negligible (Olejak, 20 Apr 2026). This suggests that not every observational cutoff near $6\times 10^8$3 uniquely diagnoses pair instability.

4. Observational status in gravitational-wave catalogs

The modern observational discussion was catalyzed by GW190521, whose primary mass was reported as $6\times 10^8$4, inside the traditionally disfavored interval (Ballone et al., 2022). A detection-space outlier analysis of the LIGO–Virgo BBH sample concluded that, if the pair-instability gap begins at or below $6\times 10^8$5, GW190521 is firmly an outlier to the stellar-mass BBH population, with $6\times 10^8$6; for a lower boundary of $6\times 10^8$7–$6\times 10^8$8, the remaining population also becomes difficult to reconcile with a purely stellar-mass model (O'Brien et al., 2021).

More recent GWTC-4-era analyses are not fully uniform in interpretation. One flexible hierarchical-Bayesian study of 153 BBHs finds no evidence for a sharp drop-off in component masses near $6\times 10^8$9–$4/3$0, favoring instead a smooth decline and placing the lower bound on a possible hidden cutoff at $4/3$1 (Ray et al., 21 Oct 2025). In contrast, two other GWTC-4 analyses report evidence for the gap in the secondary-mass distribution rather than the primary distribution. One infers a lower boundary of $4/3$2 at $4/3$3 credibility and interprets the absence of a corresponding feature in primary masses as evidence that hierarchical mergers partially fill the gap in $4/3$4 while leaving a cleaner imprint in $4/3$5 (Tong et al., 4 Sep 2025). Another finds a lower edge at $4/3$6 and a transition from a low-spin population below that scale to a high-spin, isotropic population above it, again consistent with hierarchical mergers occupying the gap (Antonini et al., 4 Sep 2025). A phase-space reconstruction analysis similarly reports a sharp truncation of the first-generation population at approximately $4/3$7 (Afroz et al., 11 Sep 2025).

Taken together, these observational studies suggest that the empirical signature of the pair-instability gap depends on the assumed population model, on whether the primary or secondary mass distribution is examined, and on how strongly hierarchical mergers are allowed to contaminate the high-mass regime.

5. Far side of the gap and future measurements

The upper boundary of the gap is conceptually distinct from the lower one. Theory often places the far-side onset near $4/3$8, corresponding to the minimum black-hole mass above the gap once stars become massive enough to collapse again rather than being fully disrupted (Franciolini et al., 2024). Very low-metallicity environments, especially Population III channels, are frequently invoked as the most plausible first-generation route to these far-side black holes (Mangiagli et al., 2019, Franciolini et al., 2024).

Assuming such far-side black holes exist, future detectors should measure the upper edge as a population feature. One forecasting study finds that next-generation ground-based detectors can constrain the minimum mass beyond the gap, $4/3$9, with relative precision $45$0 at $45$1 confidence, where $45$2 is the number of detected mergers with both components beyond the gap (Franciolini et al., 2024). Earlier rate forecasts for above-gap binaries similarly predict $45$3–$45$4 detections at LIGO/Virgo design sensitivity and $45$5–$45$6 for third-generation detectors, under assumptions that extremely metal-poor stars can form black holes above the gap (Mangiagli et al., 2019). These projections treat the far side of the gap as an observationally accessible probe rather than a purely theoretical boundary.

6. Implications for stellar physics and remaining controversies

The pair-instability gap has become a diagnostic of both stellar evolution and compact-object assembly. On the stellar-physics side, the lower edge is especially sensitive to the $45$7 rate. Farmer and collaborators argue that the lower edge near $45$8 is otherwise unusually robust for bare helium stars, with the nuclear rate as the dominant exception (Farmer et al., 2019). Several GW population studies already translate inferred gap locations into constraints on the astrophysical $45$9-factor at 300 keV, but the resulting values differ substantially—for example $130\,M_\odot$0 in one analysis and $130\,M_\odot$1 in another—showing that the inference remains model-dependent (Tong et al., 4 Sep 2025, Antonini et al., 4 Sep 2025).

On the compact-object side, the main controversy is whether current catalogs reveal a genuine pair-instability feature or an observationally similar structure generated by binary evolution and channel mixing. Current work supports all of the following statements: the first-generation stripped-star lower edge can sit near $130\,M_\odot$2; rotation can move it into the low $130\,M_\odot$3 range; H-envelope retention can raise the heaviest black holes below the onset of pair instability to $130\,M_\odot$4; stellar collisions can form $130\,M_\odot$5 black holes while still avoiding pair instability; and binary mass transfer can mimic a sharp cutoff in the lighter component distribution near $130\,M_\odot$6 (Marchant et al., 2020, Winch et al., 2024, Costa et al., 2022, Olejak, 20 Apr 2026).

A plausible implication is that “the pair-instability gap” is best treated not as a single immutable interval but as a model-dependent structure in the black-hole birth function, modified observationally by envelope retention, stellar mergers, hierarchical mergers, and population-selection effects. The central empirical task is therefore no longer only to locate a single lower edge, but to disentangle which parts of the high-mass black-hole population are first-generation stellar-collapse remnants and which are products of subsequent dynamical or binary processing.