Exploring the parameter space of hierarchical triple black hole systems

Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51b, CH-1290 Versoix, Switzerland

^⋆ Corresponding author: mara.attia@unige.ch

Received: 15 April 2025
Accepted: 17 June 2025

Abstract

We present a comprehensive exploration of hierarchical triple black hole (BH) systems to address the long-standing “initial separation” problem in gravitational wave (GW) astrophysics. This problem arises because isolated BH binaries must have extremely small initial separations to merge within a Hubble time via GW emission alone, separations at which their stellar progenitors would have merged prematurely. Using a modified version of the JADE secular code that incorporates GW energy loss, we systematically investigate a seven-dimensional parameter space consisting of the masses of the three BHs (5 − 100 M_⊙ for the inner binary components, 1 − 200 M_⊙ for the tertiary), inner and outer semimajor axes (1 − 200 AU and 100 − 10 000 AU, respectively), outer orbit eccentricity (0 − 0.9), and mutual inclination between orbits (40° −80°). We employed an innovative adaptive Markov chain Monte Carlo approach that preferentially samples the transition boundary between merging and nonmerging configurations, allowing us to efficiently map the merger probability landscape with nearly 15 million distinct simulations. Our results reveal that the parameter space regions most conducive to mergers correspond to systems with asymmetric inner binary masses, large inner separations where the von Zeipel–Lidov–Kozai (ZLK) mechanism can operate effectively without being suppressed by relativistic precession, small outer separations providing stronger perturbations, and large outer eccentricities that bring the tertiary closer at pericenter. Merger probability is also globally positively correlated with mutual inclination, with some irregular features departing from monotonicity. For nonmerging systems, we developed a classification scheme based on the presence of GW emission and ZLK oscillations, identifying distinct regions in parameter space for each category and providing a better understanding of the hierarchical triple channel. Additionally, we trained a neural network that predicts merger outcomes with an area under the receiver operating characteristic (ROC) curve score of 99% and an overall accuracy of 95%, increasing to 99.7% accuracy for the ∼80% of predictions made with high confidence. This model enables rapid population synthesis studies without requiring computationally expensive dynamical simulations. We validated our secular approach through comparison with direct N-body integrations for select systems, finding good qualitative agreement in merger outcomes for 87% of test cases, confirming that our methodology effectively captures the essential dynamics governing triple BH evolution, while enabling exploration at an unprecedented scale. Our results provide crucial insights into which configurations of hierarchical triples can resolve the initial separation problem and can serve as viable progenitors for the growing catalog of GW detections.