The Impact of Molecular Hydrogen Cooling on the Galaxy Formation Threshold

We study the impact of molecular (${\rm H_2}$) and atomic (HI) hydrogen cooling on the galaxy formation threshold. We calculate the fraction of dark matter (DM) halos that exceeds a critical mass required for star formation, $M_{\mathrm{crit}}(z)$, as a function of their peak mass. By convolving analytic halo mass accretion histories (MAHs) with models for $M_{\mathrm{crit}}(z)$, we predict that halos with peak virial masses below $\sim 10^8~M_{\mathrm{\odot}}$ can form stars before reionization through ${\rm H_2}$ cooling. These halos remain dark when only HI cooling and reionization are modeled. However, less than $\approx 10\%$ of halos with peak masses below $\sim 10^{7}~M_{\mathrm{\odot}}$ ever exceed $M_{\mathrm{crit}}(z)$, even when ${\rm H_2}$ cooling is included; this threshold is primarily set by relative streaming motion between DM and baryons imprinted at recombination. We obtain similar results using subhalo MAHs from an extremely high-resolution cosmological DM--only zoom-in simulation of a Milky Way (MW) analog (particle mass $6.3\times 10^3~M_{\mathrm{\odot}}$). Based on the abundance of MW satellites, these results imply that at least some known ultra-faint dwarf galaxies formed through ${\rm H_2}$ cooling. This work sharpens predictions for the galaxy formation threshold and demonstrates how its essential features emerge from the underlying distribution of halo growth histories.