Mass functions as evidence for hierarchical black hole mergers ✦
Using a flexible mixture model of the population for 259 binary black hole mergers from GWTC-5, we show that the mass function of the high-spin subpopulation reproduces, peak by peak, the predicted mass distribution of merger remnants from the low-spin (stellar-origin) subpopulation up to ~80 M☉. The morphological match, quantified by the Bhattacharyya coefficient (up to ~0.95), is naturally explained if rapidly spinning black holes are themselves products of previous mergers; alternative scenarios require fine-tuning. This serves as decisive evidence for hierarchical mergers. Additionally, the sharp upper cutoff of the low-spin subpopulation at m_max,1 = 54.2^{+7.7}_{-7.2} M☉ yields an astrophysical S-factor for the 12C(α,γ)16O reaction: S_300 = 151^{+30}_{-26} keV·b (68% confidence), consistent with the theoretical reference value. The entire observed black hole population is explained by stellar collapse followed by dynamical hierarchical assembly, without invoking primordial black holes.
Context
How stellar-mass black holes grow after birth is one of astrophysics' central questions. According to Einstein's general relativity, accelerating masses emit gravitational waves, and Schwarzschild first found the exact solution for a non-rotating black hole. Observations with LIGO, Virgo, and KAGRA detectors (created with key contributions from Rainer Weiss) revealed a population of merging black holes with large masses and high spins. Two competing hypotheses for their origin exist: growth through repeated black hole mergers in dense star clusters (hierarchical mergers) or gas accretion onto isolated holes. Distinguishing them means understanding stellar populations and cluster dynamics, and tracing the chain from stellar evolution to the formation of massive compact objects.
Methods
Researchers applied hierarchical Bayesian analysis to 259 black hole merger events from the GWTC-5 gravitational-wave catalog. They built a flexible mixture model of two subpopulations describing mass, spin, and redshift distributions within the framework of Einstein's equations, which predict signal characteristics. Mass functions were modeled with splines, accounting for selection effects and detector noise. Separate distributions were recovered for the first subpopulation (low spins, interpreted as products of stellar evolution and core collapse) and the second (high spins — hierarchical remnant candidates). Gravitational wave emission during merger is described by the quadrupole formula, allowing source parameter extraction.
Results
The main result: the mass function shape of the second subpopulation is nearly identical to the final mass distribution after black hole mergers of the first. This similarity is shown in the figure: up to 80 solar masses, the curves practically coincide. The Bhattacharyya coefficient, measuring overlap of normalized distributions, reaches 0.95 in the [40,80] M⊙ interval, indicating a statistically significant match. Mass peaks of the first subpopulation at ~10 M⊙ and ~35 M⊙ reappear in the second at ~17 M⊙ and ~58 M⊙, consistent with expected mass buildup from mergers. Additionally, a sharp cutoff of the low-spin population at 54.2+7.7-7.2 M⊙ is interpreted as the lower edge of the mass gap caused by pair-instability supernovae. This allowed computation of the S-factor for the 12C(α,γ)16O reaction, crucial for nucleosynthesis in massive stars: S300 = 151+30-26 keV·barn, in excellent agreement with theoretical estimates.
Implications
The discovered morphological link is direct proof that high-spin massive black holes are products of earlier black hole mergers of the first subpopulation. An alternative scenario, such as growth via accretion from an accretion disk, would require fine-tuning to reproduce the same spectral features. Thus, hierarchical assembly dominates the formation of the observed binary black hole population, and primordial black holes are not needed to explain it. The derived upper limit on the maximum mass of low-spin holes also challenges single stellar evolution models predicting slowly rotating black holes up to ~93 M⊙.
Future development
As data accumulate from new observing runs of LIGO–Virgo–KAGRA and future detectors like Einstein Telescope and Cosmic Explorer, mass functions will be measured with even greater precision. This will not only strengthen the evidence for hierarchical black hole mergers but also allow detailed study of contributions from higher generations (third and beyond) and link them to the birth of intermediate-mass black holes. Modeling of cluster dynamics and accretion flows will also gain new observational constraints.
Impact
The results will impact compact object astrophysics, stellar evolution theory, nuclear astrophysics (via refined nucleosynthesis), and cosmology, refining the black hole merger rate and star formation history.
Next steps
Next steps include incorporating time delays between merger generations and analyzing contributions from different dynamical environments (globular clusters, active galactic nucleus disks) via detailed numerical simulations and further gravitational wave observations.
Key open problems
The work directly addresses open problems: the origin of the observed black hole mass function, the nature of the mass gap from pair-instability supernovae, the rate of the 12C(α,γ)16O nuclear reaction, and the formation mechanisms of binaries in dense environments.
🎯 The pair-instability supernova mass gap, where black holes are almost never born, was long considered a 'forbidden zone,' but now we see how hierarchical mergers fill it with remnants of previous generations, like a cosmic nesting doll.
🎬 The idea of hierarchical mergers echoes the 'nesting-doll universe' from Liu Cixin's 'The Three-Body Problem,' where civilizations embed worlds within each other, but here nature itself builds increasingly massive black holes through a cascade of collisions.
Key numbers
- Bhattacharyya coefficient: 0.95
- maximum mass of low-spin black holes: 54.2 M☉
- S-factor for ¹²C(α,γ)¹⁶O at 300 keV: 151 keV·barn
- number of BBH events in GWTC-5: 259
- fraction of hierarchical mergers: ~10%