Computer simulations have produced black-hole-binary mergers through two versions of a stellar-triple pathway: one in which the three-body system remains dynamically stable, and another in which it becomes unstable after the inner pair has become black holes. Direct n-body calculations found 50 stable-channel mergers and 135 unstable-channel mergers. Relative to the starting massive-triple population, those represented about 0.03% and 0.008%, respectively.
The study followed wide triples from stellar birth to black-hole-binary merger, including systems destabilized during stellar evolution and tracking both orbital motion and black-hole spins. Researchers combined population synthesis and orbit-averaged triple-dynamics equations with direct n-body evolution, adding post-Newtonian terms up to order 2.5 and coupled spin-precession equations.
A narrow route through stellar evolution
The starting model contained 52,464 massive triples. The population was split between 28,629 systems at Z = 0.0005 and 23,835 at Z = 0.005, with primary-star mass limits of 16 to 100 solar masses. Systems that were initially unstable or underwent Roche-lobe overflow, when a star's outer layers spill toward a companion, were discarded before the later evolution.
Most of the simulated triples did not reach the black-hole stage: they either disintegrated or experienced Roche-lobe overflow before the inner components became black holes. Around 0.5% became unstable after inner black-hole-binary formation, while approximately 1% formed stable inner black-hole binaries. At Z = 0.0005 and Z = 0.005, the unstable counts were 562 and 335, while the stable counts were 1,077 and 850. The orbit-averaged TRES calculation produced no inner-binary merger.
The direct n-body stage followed 1,927 stable black-hole-binary systems and 897 unstable ones. It identified 50 stable-channel inner-binary mergers and 135 unstable-channel mergers.
Orbital clues from the merger systems
At black-hole-binary formation, the angle between the inner and outer orbital planes in merging systems was more strongly concentrated near 90 degrees. An orbital quantity called the octupole parameter, a measure of the three-body orbital geometry, averaged near 10^-2 at that stage, compared with approximately 10^-3 at the zero-age main sequence, the model's starting point. The analysis labels supernova mass loss as the dominant contributor to the higher value.
At 10 Hz, 94% of stable-channel mergers were in the 10^-4 to 10^-2 eccentricity range. Eccentricity describes how stretched an orbit is. Unstable-channel mergers had a similar main range, but about 4.5% had e10Hz at or above 10^-1.5. The most extreme reported values were 0.02 for stable systems and 0.84 for unstable systems.
The spin results covered a wide range. The final effective spin, a single measure of the binary's combined spin, ran from -1 to 1 with a slight tendency toward zero. After strong three-body evolution and the inner binary's shrinkage and decoupling, the effective spin was effectively frozen at the value associated with the highest inner eccentricity.
For the spin-inclusive reruns, the black-hole spins began either aligned or randomly oriented, with a dimensionless spin parameter of 0.9 in both populations. But only about 8% of spin-inclusive systems, 29 in total, reached the merger point with spins evolved. In the initially random-spin subset, 27%, or 104 systems, reached that point. Some spin outcomes may therefore be missing from the reported distribution.
Rates remain model estimates
Using the modeled outcomes, the authors estimated formation efficiencies of 6 x 10^-7 for the stable route and 1.6 x 10^-7 for the unstable route. The corresponding merger rates were 4.99 and 1.42 per cubic gigaparsec per year, respectively.
The rate calculation assumed a triple fraction of 0.73, a single-star fraction of 0.06 and a binary-star fraction of 0.21. It did not account for higher-order multiples, so the reported rates are estimates tied to that assumed population mix.
The work remains a preprint
The document is an arXiv version 1 preprint dated 26 August 2026. It states that the data and code needed to reproduce its figures are publicly available on Zenodo.
Paper data and sources
Original title: Triple-induced mergers of black hole binaries: A comprehensive look at the role of stellar evolution, dynamical stability, and spin evolution
Authors: C. W. Bruenech, S. Toonen, T. Boekholt, A. Dorozsmai
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: 10.1051/0004-6361/202660214
Original paper · Full text