MAGICS III. Seeds sink swiftly: nuclear star clusters dramatically accelerate seed black hole mergers

Merger rate predictions of Massive Black Hole (MBH) seeds from large-scale cosmological simulations differ widely, with recent studies highlighting the challenge of low-mass MBH seeds failing to reach the galactic center, a phenomenon known as the seed sinking problem. In this work, we tackle this issue by integrating cosmological simulations and galaxy merger simulations from the MAGICS-I and MAGICS-II resimulation suites with high-resolution $N$-body simulations. Building on the findings of MAGICS-II, which showed that only MBH seeds embedded in stellar systems are able to sink to the center, we extend the investigation by incorporating nuclear star clusters (NSCs) into our models. Utilizing $N$-body resimulations with up to $10^7$ particles, we demonstrate that interactions between NSCs and their surrounding galactic environment, particularly tidal forces triggered by cluster interactions, significantly accelerate the sinking of MBHs to the galactic center. This process leads to the formation of a hard binary in $\lesssim 500$ Myr after the onset of a galaxy merger. Our results show that in 8 out of 12 models, the high stellar density of the surrounding NSCs enhances MBH hardening, facilitating gravitational wave (GW) mergers by redshift $z = 4$. We conclude that at $z > 4$, dense NSCs serve as the dominant channel for MBH seed mergers, producing a merger rate of $0.3$--$0.6\, \mathrm{yr}^{-1}$ at $z = 4$, which is approximately 300--600 times higher than in non-NSC environments. In contrast, in environments without NSCs, surrounding dark matter plays a more significant role in loss-cone scattering.