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 $|a_\ast|=0$--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 $\eta$, 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 $s(a_\ast)\simeq -3.7\,a_\ast$, 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.