Formation of first star clusters under the supersonic gas flow -- II. Critical halo mass and core mass function

The formation and mass distribution of the first stars depend on various environmental factors in the early universe. We compare 120 cosmological hydrodynamical simulations to explore how the baryonic streaming velocity (SV) relative to dark matter affects the formation of the first stars. We vary SV from zero to three times its cosmic root-mean-square value, $v_{\rm SV}/\sigma_{\rm SV}=0-3$, and identify 20 representative halos from cosmological simulations. For each model, we follow the evolution of a primordial star-forming cloud from the first appearance of a dense core (with gas density > $10^{6}\,{\rm cm^{-3}}$) until 2 Myr later. In each model, higher SV systematically delays the formation of primordial clouds, formed inside more massive halos ($10^{5}-10^{7}\,{\rm M}_\odot$), and promotes cloud-scale fragmentation and multiple-core formation. The number and total mass of dense cores increase with increasing SV. More than half of models with $v_{\rm SV}/\sigma_{\rm SV} \ge 1.5$ form three or more dense cores in a single halo. In extreme cases, up to 25 cores form at once, which leaves a massive first star cluster. On the other hand, models with $v_{\rm SV}/\sigma_{\rm SV} \leq 1$ form only one or two cores in a halo. In addition, HD-cooling is often enabled in models with low SV, especially in low-z, where HD-cooling is enabled in more than 50% of models. This leads to the formation of the low-mass first star. SV shapes the resulting initial mass function of the first stars and plays a critical role in setting the star-forming environment of the first galaxies.