Dynamical Friction Models for Black-Hole Binary Formation in AGN Disks

Stellar-mass black holes (sBH) embedded in gaseous disks of active galactic nuclei (AGN) can be important sources of detectable gravitational radiation for LIGO/Virgo when they form binaries and coalesce due to orbital decay. In this paper, we study the effect of gas dynamical friction (DF) on the formation of BH binaries in AGN disks using $N$-body simulations. We employ two simplified models of DF, with the force on the BH depending on $\Delta {\bf v}$, the velocity of the sBH relative to the background Keplerian gas. We integrate the motion of two sBH initially on circular orbits around the central supermassive black hole (SMBH), and evaluate the probability of binary formation under various conditions. We find that both models of DF (with different dependence of the frictional coefficient on $|\Delta{\bf v}|$) can foster the formation of binaries when the effective friction timescale $\tau$ satisfies $\Omega_{\rm K}\tau\lesssim 20-30$ (where $\Omega_{\rm K}$ is the Keplerian frequency around the SMBH): prograde binaries are formed when the DF is stronger (smaller $\tau$), while retrograde binaries dominate when the DF is weaker (larger $\tau$). We determine the distribution of both prograde and retrograde binaries as a function of initial orbital separation and the DF strength. Using our models of DF, we show that for a given sBH number density in the AGN disk, the formation rate of sBH binaries increases with decreasing $\tau$ and can reach a moderate value with a sufficiently strong DF.