A calibrated model for N-body dynamical friction acting on supermassive black holes

Black holes are believed to be crucial in regulating star formation in massive galaxies, which makes it essential to faithfully represent the physics of these objects in cosmological hydrodynamics simulations. Limited spatial and mass resolution and the associated discreteness noise make following the dynamics of black holes especially challenging. In particular, dynamical friction, which is responsible for driving massive black holes towards the centres of galaxies, cannot be accurately modelled with softened N-body interactions. A number of subgrid models have been proposed to mimic dynamical friction or directly include its full effects in simulations. Each of these methods has its individual benefits and shortcomings, while all suffer from a common issue of being unable to represent black holes with masses below a few times the simulated dark matter particle mass. In this paper, we propose a correction for unresolved dynamical friction, which has been calibrated on simulations run with the code ketju, in which gravitational interactions of black holes are not softened. We demonstrate that our correction is able to sink black holes with masses greater than the dark matter particle mass at the correct rate. We show that the impact of stochasticity is significant for low-mass black holes (M-BH <= 5 M-DM) and propose a correction for stochastic heating. Combined, this approach is applicable to next generation cosmological hydrodynamics simulations that jointly track galaxy and black hole growth with realistic black hole orbits.