Simulation results for the the GW recoil bottleneck. Shown is the threshold escape velocity V_esc that is required for SMBH growth in at least one third of the cases as a function of the slope β of the BH mass-function ψ​ ∝ ​m^−β. The blue lower curves are for cold disks, with spin-orbit misalignments θ​ = ​0 − 10°, while the red upper curves correspond to hot disks, with θ​ = ​0 − 30°. The solid and dashed curves refer to cases where the mass of the initial primary BH m₀ is either drawn from the seed BH mass function in (10² − 10⁴ M_⊙) or selected to be 10⁴ M_⊙, respectively. The fiducial escape velocity for an FFB disk at z_ffb​ = ​9 is V_esc​ ≃ ​170 km s⁻¹ when ignoring compaction events, and it could rise to several hundred  km s⁻¹ after compaction (horizontal dotted lines). Naturally, the required escape velocity for BH growth is lower for a steeper BH mass function. For no BH spins, or fully aligned spin and orbit, there is always growth for V_esc​ ≥ ​100 km s⁻¹ (not shown). For a cold disk, The fiducial V_esc​ ∼ ​200 km s⁻¹ of FFB is sufficient for BH growth if β​ ≥ ​1.5 (or β​ ≥ ​1) for a random m₀ (or m₀​ = ​10⁴ M_⊙). The bottleneck becomes more severe for a hot disk, where a non-negligible growing fraction can be obtained either with a V_esc that has been boosted by compaction events to the level of a few hundred  km s⁻¹, or with the original FFB V_esc​ ∼ ​200 km s⁻¹ if m₀​ > ​3​ × ​10⁴ M_⊙ (not shown).