On the Evolution of Disk-embedded Binaries: Framing Local Models in Global Context

The disks of active galactic nuclei (AGNs) have in recent years been recognized as possible sites for gravitational-wave sources, leading to a series of numerical studies on the evolution of disk-embedded black hole binaries. The majority of these works have been carried out so far using the shearing box, a local Cartesian domain corotating with the binary center-of-mass around the supermassive black hole. The local nature of this framework allows for focusing computational power close to the binary at the expense of detaching the gas flow around the binary from the global dynamics. In this paper, we provide a framework to assess the applicability of the shearing box for studying the long-term evolution of the orbital elements of the embedded binary in viscous hydrodynamic disks. We accomplish this by identifying the conditions under which relevant global timescales are longer than the gas-induced evolution timescale of the embedded binary across various AGN disk models. For black hole masses of interest, we report the existence of radii beyond which the global influence of the disk may be reasonably neglected, supporting the use of the shearing box. More generally, we introduce a systematic approach to link local simulations with the global problem they aim to approximate while providing a way to gauge their accuracy. This will prove to be essential, as we seek to add additional physics, such as magnetic fields and radiative transport, to develop more realistic models for black hole binary mergers and their potential electromagnetic signatures in AGN disks.