Double black hole mergers in nuclear star clusters: eccentricities, spins, masses, and the growth of massive seeds

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 which 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 IMBHs with masses of $O({10}^2)$-$O({10}^4){\mathrm{M}}_{\odot }$ can be formed solely through hierarchical mergers in timescales of a few $100$ Myrs to a few Gyrs. Clusters with escape velocities $\gtrsim 400$ km s${}^{-1}$ inevitably form high-mass IMBHs. The spin distribution of IMBHs with masses $\gtrsim {10}^3$ M${}_{\odot }$ is strongly clustered at $\chi \sim 0.15$; while for lower masses, it peaks at $\chi \sim 0.7$. Eccentric mergers are more frequent for equal-mass binaries containing first-and/or second-generation BHs. Metal-rich, young, dense clusters can produce up to $20\mathrm{\%}$ of their DBH mergers with eccentricity $\ge 0.1$ at $10\mathrm{H}\mathrm{z}$, and $\sim 2$-$9\mathrm{\%}$ of all in-cluster mergers can 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 ($\sim {10}^8$ M${}_{\odot }$) and density ($\sim {10}^8$ M${}_{\odot }$pc${}^{-3}$) can have about half of all of their DBH mergers with primary masses $\ge 100$ M${}_{\odot }$. The fraction of in-cluster mergers increases rapidly with increasing cluster escape velocity, being nearly unity for $v_{\mathrm{e}\mathrm{s}\mathrm{c}}\gtrsim 200$ km s${}^{-1}$. Cosmological merger rate of DBHs from nuclear clusters varies $⪅0.01-1$ Gpc${}^{-3}$yr${}^{-1}$.