An analytic model of angular momentum transport by gravitational torques: from galaxies to massive black holes

We present analytic calculations of angular momentum transport and gas inflow in galaxies, from scales of similar to kpc to deep inside the potential of a central massive black hole (BH). We compare these analytic calculations to numerical simulations and use them to develop a sub-grid model of BH growth that can be incorporated into semi-analytic calculations or cosmological simulations. Motivated by both analytic calculations and simulations of gas inflow in galactic nuclei, we argue that the strongest torque on gas arises when non-axisymmetric perturbations to the stellar gravitational potential produce orbit crossings and shocks in the gas. This is true both at large radii similar to 0.01-1 kpc, where bar-like stellar modes dominate the non-axisymmetric potential, and at smaller radii less than or similar to 10 pc, where a lopsided/eccentric stellar disc dominates. The traditional orbit-crossing criterion is not always adequate to predict the locations of, and inflow due to, shocks in gas+stellar discs with finite sound speeds. We derive a modified criterion that predicts the presence of shocks in stellar-dominated systems even absent formal orbit crossing. We then derive analytic expressions for the loss of angular momentum and the resulting gas inflow rates in the presence of such shocks. We test our analytic predictions using hydrodynamic simulations at a range of galactic scales, and show that they successfully predict the mass inflow rates and quasi-steady gas surface densities with a small scatter similar or equal to 0.3 dex. We use our analytic results to construct a new estimate of the BH accretion rate given galaxy properties at larger radii, for use in galaxy and cosmological simulations and semi-analytic models. While highly simplified, this accretion rate predictor captures the key scalings in the numerical simulations. By contrast, alternate estimates such as the local viscous accretion rate or the spherical Bondi rate fail systematically to reproduce the simulations and have significantly larger scatter.