Minimum mass (top) and minimum x_H2 fraction (bottom) of halos hosting clumps in our simulation as a function of redshift. The blue solid lines show the evolution of these minimums and the dashed lines show the subsequent mass or x_H2 evolution of the relevant ’minimal’ halo, usually increasing in both mass and H₂ fraction. In the top panel, the horizontal lines show the critical mass found by Yoshida et al. (2003) and Machacek et al. (2003).