Comparing hierarchical black hole mergers in star clusters and active galactic nuclei

Star clusters (SCs) and active galactic nuclei (AGNs) are promising sites for the occurrence of hierarchical black hole (BH) mergers. We use simple models to compare hierarchical BH mergers in two of the dynamical formation channels. We find that the primary mass distribution of hierarchical mergers in AGNs is higher than that in SCs, with the peaks of $\sim$$50M_{\odot }$ and $\sim$$13M_{\odot }$, respectively. The effective spin (${\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}$) distribution of hierarchical mergers in SCs is symmetrical around zero as expected and $\sim$$50\mathrm{\%}$ of the mergers have $\mathrm{\mid }{\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}\mathrm{\mid }>0.2$. The distribution of ${\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ in AGNs is narrow and prefers positive values with the peak of ${\chi }_{\mathrm{e}\mathrm{f}\mathrm{f}}\ge 0.3$ due to the assistance of AGN disks. BH hierarchical growth efficiency in AGNs, with at least $\sim$$30\mathrm{\%}$ of mergers being hierarchies, is much higher than the efficiency in SCs. Furthermore, there are obvious differences in the mass ratios and effective precession parameters of hierarchical mergers in SCs and AGNs. We argue that the majority of the hierarchical merger candidates detected by LIGO-Virgo may originate from the AGN channel as long as AGNs get half of the hierarchical merger rate.