Bridging Scales in Black Hole Accretion and Feedback: Subgrid Prescription from First Principles

TL;DR

This work builds a horizon-to-galaxy bridge for black hole accretion and feedback using long-duration, scale-spanning GRMHD simulations across spins $|a_*|=0$--0.9 and Bondi radii $R_B/r_g=4\times10^2$--$2\times10^6$, delivering first-principles, spin-dependent subgrid prescriptions for low-Eddington hot flows. It provides two complementary prescriptions: (i) time-averaged relations for $\overline{\dot M}$ and $\overline{\dot E_{\rm fb}}$ as simple power-laws in $R_B$ with linear spin dependence, and (ii) lognormal distributions for $\dot M$ and $\eta$ to capture strong intrinsic variability; both depend only on $R_B/r_g$ and $a_*$ and are robust to initial conditions. A key result is the spin evolution $\overline{s}(a_*)\approx -3.7a_*$, yielding a spindown timescale $t_s\simeq 12\left(\frac{10^{-3}}{f_{\rm Edd}}\right)\,\text{Gyr}$, implying spins are effectively frozen during quiescent accretion and evolve mainly during high-$f_{\rm Edd}$ episodes or mergers. The study cautions against directly extrapolating small-domain GRMHD results to galaxies and outlines practical steps to implement these prescriptions in cosmological simulations and semi-analytic models, while noting limitations such as the restricted spin range, neglect of radiative cooling, and assumed magnetic geometries.

Abstract

Understanding how supermassive black holes (BHs) couple to their host galaxies across a vast spatial and temporal dynamic range remains a central challenge in galaxy evolution. Using the multizone framework -- designed to capture bidirectional inflow--outflow from the event horizon to the Bondi scale -- we present a suite of long-duration GRMHD simulations spanning BH spins $|a_\ast|=0$--0.9 and Bondi radii $R_B/r_g=4\times10^2$--$2\times10^6$. From these simulations we derive spin-dependent subgrid prescriptions from first principles, applicable to hot accretion flows with low-Eddington ratios ($f_{\rm Edd}\lesssim10^{-3}$), for adoption in cosmological simulations and semi-analytic models. We provide compact analytic fits for the time-averaged accretion rate $\dot M(R_B,a_\ast)$ and feedback power $\dot E_{\rm fb}(R_B,a_\ast)$ with respect to the Bondi rate $\dot{M}_B$, which are largely insensitive to the initial gas configuration and magnetic field strength. To capture intrinsic time-variability, we also quantify the full distributions of $\dot M$ and feedback efficiency $η$, both well described by lognormal statistics, with widths that increase toward larger $R_B$. We further measure self-consistent spin evolution in the hot accretion mode, finding that the spin-up parameter varies as $s(a_\ast)\simeq -3.7\,a_\ast$, which implies a very long spindown timescale $t_s\simeq 12(10^{-3}/f_{\rm Edd})\,{\rm Gyr}$. Thus, BH spins are effectively frozen during phases of quiescent accretion. Compared to conventional small-domain GRMHD calculations, our simulations, which reach dynamical equilibrium across horizon-to-galaxy scales, yield systematically different long-term accretion, feedback, and spin properties, cautioning against direct extrapolation from small-scale GRMHD simulations when constructing galactic-scale subgrid models.

Figures (6)

• Figure 1: Time averaged accretion rate $\overline{\dot{M}}$ (top), feedback power $\overline{\dot{E}_{\rm fb}}$ (middle), and feedback efficiency $\overline{\eta}$ (bottom), for simulations with varying Bondi radii $R_B$ and BH spins $a_*$. The scale at the bottom shows the Bondi radius $R_B$ in units of the gravitational radius $r_g$, and the scale at the top shows the corresponding asymptotic temperature $T_\infty$. Colors represent different BH spins. The approximate formulae for $\overline{\dot{M}}$ (\ref{['eq:Mdot_powerlaw']}) and $\overline{\dot{E}_{\rm fb}}$ (\ref{['eq:Efb_powerlaw']}) are shown as straight lines in the top and middle panels.
• Figure 2: Radial profiles of the time-averaged feedback efficiency $\overline{\eta(r)}$. Each panel corresponds to a single BH spin $a_*$ and colors represent different Bondi radii $R_B$. The scale at the bottom shows the radius in units of $r_g$, and the scale at the top shows the physical radius in pc for M87's BH mass. For the largest Bondi radius, $R_B = 2\times10^6\,r_g$, the kinetic $\overline{\eta_{\rm kinetic}(r)}$ (dashed) and thermal $\overline{\eta_{\rm thermal}(r)}$ (dotted) components are also overplotted, showing the conversion of kinetic to thermal energy as the jet outflow crosses $R_B$ (vertical lines).
• Figure 3: Distributions of $\ln(\dot{M}/\dot{M}_B)$ (top row) and $\ln(\eta)$ (bottom row). Each column is for a given BH spin $a_*$ and the blue and red curves are for the smallest and largest Bondi radii, $R_B\approx 400\,r_g$ and $\approx 2\times 10^6\,r_g$, respectively. Lognormal curves corresponding to each run are overplotted for comparison. The means of the lognormal variable $X$, calculated using \ref{['eq:mean_lognormal']} from log-space quantities $\mu_{\ln X}$ and $\sigma_{\ln X}$, are shown as vertical lines.
• Figure 4: The spinup parameter $s$ (\ref{['eq:s_def']}) as a function of BH spin $a_*$. Fiducial runs without initial gas angular momentum are shown in grayscale. Blue and red open circles represent runs with initially co- and counter-rotating gas, respectively, and agree well with the equivalent fiducial runs without rotation. Narayan2022's small-scale GRMHD simulations are shown as dotted lines for prograde (blue) and retrograde (red). Multizone simulations limited to much shorter durations from Cho2025, shown as stars, agree better with the Narayan2022 curves.
• Figure 5: Time-averaged accretion rate $\overline{\dot{M}}$ and feedback power $\overline{\dot{E}_{\rm fb}}$ for the $a_*=0.9$ runs with different ICs: fiducial non-rotating (black), co-rotating (blue), counter-rotating (red), and weakly magnetized (green). For reference, the fitted relations $\overline{\dot{M}}(R_B,a_*=0.9)$ and $\overline{\dot{E}_{\rm fb}}(R_B,a_*=0.9)$ (Equations \ref{['eq:Mdot_powerlaw']} and \ref{['eq:Efb_powerlaw']}) are shown as black lines.