We describe cosmological galaxy formation simulations with the adaptive mesh refinement code Enzo that incorporate a star formation prescription regulated by the local abundance of molecular hydrogen. We show that this H2-regulated prescription leads to a suppression of star formation in low mass halos (M_h < ~10^10 M_sun) at z>4, alleviating some of the dwarf galaxy problems faced by theoretical galaxy formation models. H2 regulation modifies the efficiency of star formation of cold gas directly, rather than indirectly reducing the cold gas content with supernova feedback. We determine the local H2 abundance in our most refined grid cells (76 proper parsec in size at z=4) by applying the model of Krumholz, McKee, & Tumlinson, which is based on idealized 1D radiative transfer calculations of H2 formation-dissociation balance in ~100 pc atomic--molecular complexes. Our H2-regulated simulations are able to reproduce the empirical (albeit lower z) Kennicutt-Schmidt relation, including the low Sigma_gas cutoff due to the transition from atomic to molecular phase and the metallicity dependence thereof, without the use of an explicit density threshold in our star formation prescription. We compare the evolution of the luminosity function, stellar mass density, and star formation rate density from our simulations to recent observational determinations of the same at z=4-8 and find reasonable agreement between the two.
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