Seeding Cores: A Pathway for Nuclear Star Clusters from Bound Star Clusters in the First Billion Years

We model the formation of star clusters in a dwarf galaxy progenitor during the first 700 Myr of cosmic history using a cosmological radiation-hydrodynamic simulation with a sub-grid star formation efficiency (SFE) model calibrated from AU-scale radiation-MHD simulations of molecular clouds with varying mass, density, and metallicity. In comparison to a constant SFE model, our model yields more bursty star formation, a more abundant massive star cluster population, and overall a higher stellar mass. Clouds reach SFEs up to 80\%, forming bound star clusters (densities $\sim10^{2-4} ~{\rm M_\odot\:pc^{-2}}$, radii $\lesssim 3~{\rm pc}$) resembling those observed by the James Webb Space Telescope (JWST) in strongly lensed galaxies. Star clusters follow a flat power-law mass function ${\rm d}N/{\rm d}\log M \propto M^\Gamma$ with slope $\Gamma \sim -0.4$. The most massive star clusters ($10^{4-5} ~{\rm M_\odot}$) grow through mergers and have metallicity spreads of $0.05 - 0.1$ dex that roughly scale with mass. The second burst of star formation produce loosely bound star clusters with higher metallicities: $-1.95 < \log(Z/{\rm Z_\odot}) < -1.50$ at lower SFEs (2 - 20\%). At $z \sim 8.7$, a nuclear star cluster (NSC) is seeded, growing 83\% of its mass ($2.4 \times 10^5 ~{\rm M_\odot}$, $20\%$ of the galaxy's stellar mass) through mergers with pre-existing clusters and the rest through in-situ star formation. The early formation of NSCs has interesting implications for seeding supermassive black holes and the population of _little red dots_ recently discovered by JWST at $z \gtrsim 5$.