Pair-Instability Mass Gap: Theory & Observations

• Pair-instability mass gap is defined as the BH mass range (~45–120 M☉) absent due to complete or pulsational supernova disruptions in massive stars.
• Stellar evolution models (e.g., MESA, SEVN) consistently place the lower gap edge near 45 M☉, with sensitivity to nuclear reaction rates and envelope retention.
• Alternative channels such as stellar mergers and hierarchical BH growth can partly fill the gap, offering distinct gravitational-wave signatures for astrophysical tests.

The pair-instability mass gap is a theoretically predicted region in the black hole (BH) mass spectrum, traditionally spanning approximately $45$–$120~M_\odot$, where standard stellar evolutionary mechanisms preclude the formation of BHs. This gap arises due to the onset of the pair-instability supernova (PISN) phenomenon and its less disruptive variant, pulsational pair-instability supernova (PPISN), in the late evolutionary stages of massive stars, resulting in either complete disruption or significant mass loss that limits the remnant BH mass. The precise understanding of the pair-instability mass gap is integral to interpreting gravitational-wave detections of BH mergers, constraining models of massive star evolution, probing nuclear astrophysics (notably the $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$ reaction rate), and tracing the origin of outlier, high-mass merging BHs (Mangiagli et al., 2019, Farmer et al., 2019, O'Brien et al., 2021).

1. Physical Origin and Theoretical Definition

The pair-instability mass gap is produced by the interplay of electron–positron pair creation and thermonuclear burning in massive stellar cores. As a massive star evolves past core helium burning, core temperatures can become sufficiently high ($kT \gtrsim m_e c^2$) for energetic photons to produce $e^+e^−$ pairs. The resulting pressure deficit—quantified by a drop in the adiabatic index $\Gamma_1$ below $4/3$—triggers dynamic contraction and, depending on the core mass, ignites explosive burning:

• PPISN (pulsational pair-instability supernova): Episodic, mass-losing thermonuclear pulses lower the mass of the star, typically capping the BH remnant mass at $\sim 45~M_\odot$ for helium star progenitors (Farmer et al., 2019, Renzo et al., 2020).
• PISN (pair-instability supernova): For larger core masses, the explosive oxygen burning completely unbinds the entire star, leaving no remnant.

Traditionally, BHs below $\sim 45~M_\odot$ form via direct core collapse, while those in the range $\sim 45$–$120~M_\odot$ are absent, thus defining the "mass gap." For $M_{\rm rem}\gtrsim 120~M_\odot$, very massive, low-metallicity stars (especially $M_{\rm ZAMS}\gtrsim 260~M_\odot$) can directly collapse to form BHs above the gap if they avoid prior disruption, and metal-poor environments are key for retaining enough pre-collapse mass (Mangiagli et al., 2019, Franciolini et al., 23 Jan 2024).

2. Stellar Evolutionary Modeling and Gap Robustness

Advanced stellar evolution modeling (using codes such as MESA, SEVN, or MOBSE) demonstrates the following:

• Maximum BH mass below the gap: Calculations for hydrogen/helium-star progenitors consistently locate the lower edge near $45~M_\odot$ (Farmer et al., 2019), with uncertainty primarily driven by the $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$ reaction rate. Metallicity introduces only a modest ($\sim 7\%$) variation; mixing and wind prescription have minor effects ($\sim 1$–$4~M_\odot$) (Farmer et al., 2019, Renzo et al., 2020).
• Core mass thresholds: The final BH mass as a function of core mass and metallicity is captured by analytic fits (e.g., $$M_{BH}=a_1 M_{CO}^2 + a_2 M_{CO} + a_3 \log_{10}Z + a_4$$ for $38\le M_{CO}\le60~M_\odot$) (Farmer et al., 2019).
• Sensitivity to input physics: The location of the mass gap lower edge is robust against environmental changes and the majority of microphysics, but can be shifted (e.g., up to $40$–$56~M_\odot$) by uncertainties in the $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$ rate (Farmer et al., 2019, Woosley et al., 2021), by up to $\sim5~M_\odot$ for different treatments of convection (Renzo et al., 2020), and further influenced by rotational physics (Marchant et al., 2020).
• Hydrogen envelope retention: Models that retain a massive hydrogen envelope at collapse (especially at low metallicity) can produce black holes below the classic pair-instability gap but with masses approaching $93.3~M_\odot$ (Winch et al., 29 Jan 2024) and, under certain circumstances (if the envelope collapses), shift the lower gap edge up to $70~M_\odot$ or even higher (Costa et al., 2020).

3. Alternative Gap-filling Mechanisms and Their Diagnostics

Several astrophysical pathways can populate the pair-instability gap with BHs:

• Stellar mergers in star clusters: Dynamical interactions in dense, young clusters can lead to mergers of massive stars (or successive mergers), producing objects with small cores and oversized hydrogen envelopes (Carlo et al., 2019, Renzo et al., 2020, Costa et al., 2022, Ballone et al., 2022). Provided the core remains below the PPISN threshold while the envelope contributes most of the mass, these objects collapse directly into BHs occupying the mass gap.
• Hierarchical black hole mergers: In clusters, repeated mergers can assemble BHs above the mass gap, often producing remnants with high spin ($\chi\sim0.7$) (Baibhav et al., 2020, Gerosa et al., 2021). However, the combination of high mass and low spin ($m\gtrsim50~M_\odot$, $\chi\lesssim0.2$) is extremely difficult to achieve through hierarchical mergers, suggesting that observed low-spin, high-mass BHs in the gap require alternative astrophysical mechanisms (Gerosa et al., 2021).
• Super-Eddington accretion: In isolated binaries, phases of super-Eddington accretion can grow BHs into the gap, but significant orbital widening typically prevents such systems from merging within a Hubble time. This process contributes negligibly ($\lesssim2\%$) to the population of merging BBHs in the gap and does not yield total masses above $100~M_\odot$ (Son et al., 2020).
• Stellar evolutionary nuances: Variations in $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$, convective mixing, rotational support, envelope dredge-up, and metallicity may all shift the effective gap boundaries, sometimes allowing single stars to produce BHs in the gap under specific parameter choices (Costa et al., 2020, Woosley et al., 2021, Winch et al., 29 Jan 2024, Umeda et al., 2020).
• Modified gravity or exotic dark sectors: The presence of additional “fifth forces” or a dissipative dark sector can shift the PI mass gap edges downwards or enable “dark” BHs to populate otherwise forbidden ranges, providing a novel test for fundamental physics (Straight et al., 2020, Fernandez et al., 2022).

4. Demographics, Population Synthesis, and Event Rates

Population models, combining the initial mass function (IMF), star formation rate (SFR), and metallicity evolution across cosmic history, predict the cosmological merger rates for above-gap BH binaries:

Detector	Above-gap merger rate (per year)	Conditions
LIGO/Virgo (design)	$0.4$–$7~\text{yr}^{-1}$	Most pessimistic/optimistic
3G Detectors (ET)	$10$–$460~\text{yr}^{-1}$	Pessimistic/optimistic models

The predicted event rates for binaries with both components above the gap are highly sensitive to the assumed cosmic metallicity evolution. Models with rapid high-redshift metallicity decline (sSFR-sZ) yield an order-of-magnitude more above-gap events compared to moderate models (mSFR-mZ) (Mangiagli et al., 2019). The background of unresolved above-gap binaries in the LISA band may result in a gravitational-wave background with signal-to-noise ratios up to $\sim80$ (Mangiagli et al., 2019).

Merging BBHs with gap components produced by dynamical processes in clusters are expected to contribute a non-negligible fraction (up to $\sim5\%$) of all LIGO/Virgo detectable BBH mergers, especially at low metallicity (Carlo et al., 2019). For extremely massive systems (e.g., $M_{\rm rem}\gtrsim130~M_\odot$), next-generation detectors will localize the upper mass boundary with $\sim4\%$ precision for $N_{\rm det}=100$ above-gap mergers (Franciolini et al., 23 Jan 2024).

5. Observational Diagnostics and Gravitational-Wave Signatures

The pair-instability mass gap provides several distinct empirical signatures, making it a key observable marker in gravitational-wave astrophysics:

• Mass cutoff and "pile-up": A sharp cutoff or "pile-up" just below the gap is a robust prediction for single-star evolution, with theoretical anchoring in the quasi-universal value of the lower edge ($\sim45~M_\odot$ for He stars, though as high as $\sim93~M_\odot$ for stars with large H envelopes) (Farmer et al., 2019, Winch et al., 29 Jan 2024).
• Component mass and spin: Detection of a binary merger with either component in the gap and low dimensionless spin ($\chi\lesssim0.2$) is highly inconsistent with hierarchical mergers (Gerosa et al., 2021).
• Multi-band GW astronomy: LISA and ground-based detectors together can identify above-gap binaries years before merger. The predicted number of multiband events merging within four years (allowing sequential LISA and LIGO/ET detection) ranges from 1–20, with future GW observatories (ET) sensitive to nearly all such sources (Mangiagli et al., 2019).
• Standard siren application: The near-universality of the lower-gap cutoff for first-generation BHs enables these events to serve as mass-based standard sirens, constraining cosmological parameters (Farmer et al., 2019, Son et al., 2020).
• Gap occupancy and astrophysical channel discrimination: Polluting the gap requires one or more of: dynamical interactions, exotic accretion, or unconventional physics (e.g. enhanced mixing, rotation, or new forces). The observed distribution of high-mass, low-spin BHs will separate formation channels (Gerosa et al., 2021, Carlo et al., 2019, Mangiagli et al., 2019).

6. Astrophysical and Fundamental Physics Significance

The paper of the pair-instability mass gap constrains both stellar and fundamental physics:

• Stellar and nuclear astrophysics: The location and occupancy of the gap test models of stellar evolution, stellar winds, internal mixing, and critically, the $^{12}$C$(\alpha,\gamma)^{16}$O reaction rate. A measured gap location acts as a sensitive probe of these processes.
• Pop III and cosmic history: Pop III stars in low-metallicity environments are key progenitors of gap and above-gap BHs, as their weak winds and top-heavy IMF facilitate the formation of both massive merger components and sizeable event rates for next-generation GW observatories (Wu et al., 29 Apr 2025, Mangiagli et al., 2019, Franciolini et al., 23 Jan 2024).
• Fundamental tests: The gap can constrain violations of the strong equivalence principle and modified gravity scenarios (Straight et al., 2020), and discriminates the possible role of a dark sector in early Universe structure formation (Fernandez et al., 2022).

7. Outstanding Issues and Future Directions

Several open questions and prospects remain:

• Uncertainties in input physics: Reducing uncertainties in convective overshooting, rotational mixing, and nuclear reaction cross-sections is imperative for robust mass gap predictions (Renzo et al., 2020, Woosley et al., 2021).
• Complex channels and outliers: The existence of GW events such as GW190521, with primary masses $85^{+21}_{-14}~M_\odot$, requires ongoing refinement of models involving stellar mergers, core-envelope configurations, or alternative physics (Costa et al., 2022, Costa et al., 2020).
• Next-generation observational constraints: Upcoming observatories (ET, CE, LISA) will localize the upper and lower mass gap edges, enable multi-band observations, and close in on the rate and nature of mergers populating the gap, with potential for $\lesssim4\%$ precision on the upper boundary (Franciolini et al., 23 Jan 2024).
• Simulations and multi-dimensional modeling: Further hydrodynamical simulations of stellar collisions, as well as improvements to cluster dynamics and binary population synthesis, will yield more realistic forecasts of gap-occupying BH frequency and GW signatures (Carlo et al., 2019, Ballone et al., 2022, Wu et al., 29 Apr 2025).

The pair-instability mass gap thus serves as an incisive astrophysical diagnostic for the late stages of massive star evolution, the nuclear physics of stellar interiors, the demographics of cosmological BH mergers, and potentially, the search for new physics beyond standard stellar and gravitational theory.