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

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 $\mathrm{\mid }{a}_{\ast }\mathrm{\mid }=0$–0.9 and Bondi radii $R_B\mathrm{/}{r}_g=4\times {10}^2$–$2\times {10}^6$. From these simulations we derive spin-dependent subgrid prescriptions from first principles, applicable to hot accretion flows with low-Eddington ratios ($f_{\mathrm{E}\mathrm{d}\mathrm{d}}\lesssim {10}^{-3}$), 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 ${\dot{E}}_{\mathrm{f}\mathrm{b}}(R_B,a_{\ast })$ 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 $\eta$, 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 $s(a_{\ast })\simeq -3.7a_{\ast }$, which implies a very long spindown timescale $t_s\simeq 12({10}^{-3}\mathrm{/}{f}_{\mathrm{E}\mathrm{d}\mathrm{d}})\mathrm{G}\mathrm{y}\mathrm{r}$. 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.