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

TL;DR

This work develops a rigorous framework to evaluate when local shearing-box simulations can faithfully represent the long-term evolution of black-hole binaries embedded in AGN disks. By defining four key dimensionless ratios and multiple global-timescale comparisons (libration, viscous, migration) against the inspiral timescale, the authors quantify the regimes where global disk dynamics significantly feed back on the binary. Applying the framework to standard α-disk models and global SG03/TQM05 disk models, they map regions of parameter space where local simulations are robust and identify when global effects—especially the horseshoe flow and, in some models, radial flows—must be included. The study provides practical guidance and quantitative criteria to connect local simulations to the global context, with implications for modeling BBH mergers and their electromagnetic signatures in AGN disks. This framework lays the groundwork for incorporating additional physics (MHD, radiative transfer) while retaining a clear criterion for the validity of local models.

Abstract

The disks of Active Galactic Nuclei (AGN) 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 co-rotating 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.

Figures (7)

• Figure 1: The dimensionless measure of the inspiral timescale in Equation \ref{["eq. inspiral'"]} measured in previous shearing box simulation, as a function of the chosen binary mass to thermal mass ratio $q/h^3$ in Equation \ref{['eq. R_H/H_or_q/h']}.
• Figure 2: Representative results from local and global hydrodynamical simulations of a disk-embedded binary. Left column: Average gas density within the Hill sphere (top) and rolling averages of binary accretion rate (middle) and gravitational torque (bottom) for local (red) and global (blue) simulations. Deviations appear after half a libration time as indicated by the dashed line. Right column: Density snapshot from local simulation before half a libration time (top), and from global simulations before (middle) and after (bottom). The dashed orange circle indicates the binary Hill sphere.
• Figure 3: Radial dependence of $H/R$, surface density $\Sigma$ and radial velocity $v_r$ for a standard $\alpha$-disk (left) and models of SQ03 Sirko_2003 (middle) and TQM05 Thompson_2005 (right). The models are shown for three different SMBH masses. For the $\alpha$-disk and SG03 model (both with $\alpha=0.01$) the shaded regions span $l_e\in [0.1,1]$ while for the TQM05 model they span $m\in[0.1,0.3]$.
• Figure 4: Evaluation of timescale ratios in Equations \ref{['eq. tau_libation_ratio']}, \ref{['eq. tau_viscous_ratio']}, and \ref{['eq. tau_migration_ratio']} for a standard $\alpha$-disk with SMBH mass $M=10^8M_{\odot}$, Eddington fraction $l_e=0.1$, and $\alpha=0.01$, assuming $\tau'_{\rm inspiral}=1$. The left- and right- tilted hashed regions have $m_b<5M_{\odot}$ and $m_b>500M_{\odot}$, respectively, while $Q\geq 1$ outwards of the bold dotted line. Left panel: Coloured regions indicate parts of the $(R,q/h^3)$ parameter space for which the respective timescales are shorter than the inspiral timescale. In the yellow region the Hill mass is larger than the binary mass, see Equation (\ref{['eq. disk_mass_limit']}). Right panel: Contours of $\tau_{\rm libration}/(2\tau_{\rm inspiral})$. The squares and circles show the minimum radii at which the simulations presented in Figure \ref{['fig: sim_data_points']} accurately capture the long-term binary evolution. Above the green line, the binary is expected to form a gap, as evaluated from Equation \ref{['eq. gap_formation']}.
• Figure 5: Similar to Figure \ref{['fig: alpha_single']} but for the SQ03 disk model with $l_e=0.1$, $\alpha=0.01$ (top) and the TQM05 disk model with $m=0.2$ (bottom), both with $M=10^8M_{\odot}$. Beyond the vertical dotted line the disk is marginally stable with Toomre's $Q\sim1$.