Binary Black Hole Merger Rates in AGN Disks versus Nuclear Star Clusters: Loud beats Quiet

Galactic nuclei are promising sites for stellar origin black hole (BH) mergers, as part of merger hierarchies in deep potential wells. We show that binary black hole (BBH) merger rates in active galactic nuclei (AGN) should always exceed merger rates in quiescent galactic nuclei (nuclear star clusters, NSCs) around supermassive BHs (SMBHs) without accretion disks. This is primarily due to average binary lifetimes in AGN that are significantly shorter than in NSCs. The lifetime difference comes from rapid hardening of BBHs in AGN, such that their semi-major axes are smaller than the hard-soft boundary of their parent NSC; this contrasts with the large average lifetime to merger for BBHs in NSCs around SMBHs, due to binary ionization mechanisms. Secondarily, merger rates in AGNs are enhanced by gas-driven binary formation mechanisms. Formation of new BHs in AGN disks are a minor contributor to the rate differences. With the gravitational wave detection of several BBHs with at least one progenitor in the upper mass gap, and signatures of dynamical formation channels in the $\chi_{\rm eff}$ distribution, we argue that AGN could contribute $\sim 25\%-80\%$ of the LIGO-Virgo measured rate of $\sim 24 \rm{Gpc}^{-3} \rm{yr}^{-1}$.