BH sinking timescale as a function of the total number of particles in the halo, for a pure DM NFW halo (left) and for a DM halo with a central Hernquist bulge (right). Diamonds indicate simulations without DF correction, circles indicate simulations with DF correction. The horizontal red dashed line shows the analytical sinking timescale from OTIS. In the left panel, teal-edged points, connected by teal dashed lines, adopt softening from Eq. (30), black-edged points connected by black dashed lines are for the softening from Eq. (29). In the right panel, teal edges and teal dashed lines correspond to ϵ_BH = ϵ_DM, while black edges with black dashed lines correspond to ϵ_BH = ϵ_∗. The color of each point indicates the BH orbital eccentricity, as highlighted in the colorbar.