Shingo Hirano, Yusuke Sakai, Hideyuki Umeda

The first generations of stars ionised and enriched their host galaxies and seeded the growth of massive black holes. Models often assume that Pop III stellar masses in different minihaloes are stochastic realisations of a single universal initial mass function (IMF). We use 138 cosmological zoom-in hydrodynamics simulations to test this assumption and to map the first-star IMF across redshift, halo mass, and baryon-dark matter streaming velocity (SV). We construct a dense-cloud merger tree and assign first-star masses by mapping the radial gas accretion-rate profile to stellar mass, yielding per-halo stellar mass functions without imposing any a priori IMF. The high-mass tail and multiplicity increase systematically with redshift, halo mass, and SV. Low-mass, low-SV haloes form only one or a few first stars, whereas massive, high-SV haloes host rich first star clusters and commonly produce very massive ($\gtrsim10^3$-$10^4\,{\rm M}_\odot$) first stars. Even in a fixed environment, halo-to-halo scatter remains substantial. Nevertheless, group-averaged IMFs converge to well-defined forms, ruling out a single universal IMF at the halo level across the range of environments probed here. Mapping our seeds onto the redshift-mass plane, we show that high-SV and massive haloes preferentially populate the heavy-seed regime relevant to luminous high-redshift sources. At the same time, low-SV environments are consistent with single/few-event enrichment signatures in metal-poor stars. Our results deliver a practical, physically motivated prescription for per-halo IMF.