Long-term Evolution of Tightly-Packed Stellar Black Holes in AGN Disks: Formation of Merging Black-Hole Binaries via Close Encounters

We study the long-term evolution of two or more stellar black holes (BHs) on initially separated but unstable circular orbits around a supermassive BH (SMBH). Such a close-packed orbital configuration can naturally arise from BH migrations in the AGN disk. Dynamical instability of the orbits leads to close encounters between two BHs, during which the BH separation r_p becomes less than the Hill radius R_H. In the very close encounters (with r_p several orders of magnitude less than R_H), tight merging BH binaries can form with the help of gravitational wave emission. We use N-body simulations to study the time evolution of close encounters of various degrees of “closeness” and the property of the resulting binary BH mergers. For a typical “SMBH + 2 BHs” system, the averaged cumulative number of close encounters (with r_p ≲ R_H) scales proportional to t^0.5. The minimum encounter separation r_p, which spans several orders of magnitude, follows a linear cumulative distribution P(<r_p) that is proportional to r_p. From these, we obtain a semi-analytical expression for the averaged rate of binary captures that lead to BH mergers. Our results suggest that close-packed BHs in AGN disks may take a long time (≳10⁶ orbits around the SMBH) to experience very close encounters and form a bound binary, although this time can be much shorter if the initial BH orbits are highly aligned. The BH binary mergers produced in this scenario always have high eccentricities when entering the LIGO band, and have a broad distribution of orbital inclinations relative to the original AGN disk. We also carry out a preliminary exploration on the effects of the gas disk and find that simple gas drags on the BHs do not lead to an enhanced BH binary capture rate.