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ABSTRACT A long-standing problem in galactic simulations is to resolve the dynamical friction (DF) force acting on massive black hole particles when their masses are comparable to or less than the background simulation particles. Many sub-grid models based on the traditional Chandrasekhar DF formula have been proposed, yet they suffer from fundamental ambiguities in the definition of some terms in Chandrasekhar’s formula when applied to real galaxies, as well as difficulty in evaluating continuous quantities from (spatially) discrete simulation data. In this work, we present a new sub-grid DF estimator based on the discrete nature of N-body simulations, which also avoids the ambiguously defined quantities in Chandrasekhar’s formula. We test our estimator in the gizmo code and find that it agrees well with high-resolution simulations where DF is fully captured, with negligible additional computational cost. We also compare it with a Chandrasekhar estimator and discuss its applications in real galactic simulations. 

ABSTRACT Possible formation scenarios of supermassive black holes (BHs) in the early universe include rapid growth from less massive seed BHs via super-Eddington accretion or runaway mergers, yet both of these scenarios would require seed BHs to efficiently sink to and be trapped in the Galactic Centre via dynamical friction. This may not be true for their complicated dynamics in clumpy high-z galaxies. In this work, we study this ‘sinking problem’ with state-of-the-art high-resolution cosmological simulations, combined with both direct N-body integration of seed BH trajectories and post-processing of randomly generated test particles with a newly developed dynamical friction estimator. We find that seed BHs less massive than $$10^8\, \mathrm{M}_\odot$$ (i.e. all but the already-supermassive seeds) cannot efficiently sink in typical high-z galaxies. We also discuss two possible solutions: dramatically increasing the number of seeds such that one seed can end up trapped in the Galactic Centre by chance, or seed BHs being embedded in dense structures (e.g. star clusters) with effective masses above the mass threshold. We discuss the limitations of both solutions.