Abstract

We investigate the formation of intermediate-mass black holes (IMBHs) through hierarchical mergers of stellar-origin black holes (BHs), as well as BH mergers formed dynamically in nuclear star clusters. Using a semi-analytical approach that incorporates probabilistic, mass-function–dependent double-BH (DBH) pairing, binary–single encounters, and a mass-ratio–dependent prescription for energy dissipation in hardening binaries, we find that IMB Hs with masses of order 10²–10⁴ M⊙ can be formed solely through hierarchical mergers on timescales of a few hundred Myr to a few Gyr. Clusters with escape velocities ≳ 400 km s⁻¹ inevitably form high-mass IMB Hs. The spin distribution of IMB Hs with masses ≳ 10³ M⊙ is strongly clustered at χ ≈ 0.15, while for lower masses it peaks at χ ≈ 0.7. Eccentric mergers are more frequent for equal-mass binaries containing first- and second-generation BHs. Metal-rich, young, dense clusters can produce up to 20 of their DBH mergers with eccentricity ≥ 0.1 at 10 Hz, and ~ 2–9 of all in-cluster mergers form at > 10 Hz. Nuclear star clusters are therefore promising environments for the formation of highly eccentric DBH mergers, detectable with current gravitational-wave detectors. Clusters of extreme mass (∼ 10⁸ M⊙) and density (∼ 10⁸ M⊙ pc⁻³) can have about half of their DBH mergers with primary masses ≥ 100 M⊙. The fraction of in-cluster mergers increases rapidly with increasing escape velocity, approaching unity for Vesc ≳ 200 km s⁻¹. The cosmological DBH merger rate from nuclear clusters varies from ≲ 0.01 to 1 Gpc⁻³ yr⁻¹, where the large uncertainties stem from cluster initial conditions, number-density distributions, and the redshift evolution of nucleated galaxies.