The tight relation of star formation with molecular gas indicated by observations and assumed in recent models implies that the efficiency with which galaxies convert their gas into stars depends on gas metallicity. This is because the abundance of molecular hydrogen is sensitive to the abundance of dust, which catalyzes the formation of H_2 and helps to shield it from dissociating radiation. In this study we point out that in the absence of significant pre-enrichment by Population III stars forming out of zero metallicity gas, such H_2-based star formation is expected to leave an imprint in the form of bi-modality in the metallicity distribution among dwarf galaxies and in the metallicity distribution of stars within individual galaxies. The bi-modality arises because when gas metallicity (and dust abundance) is low, formation of molecular gas is inefficient, the gas consumption time scale is long, and star formation and metal enrichment proceed slowly. When metallicity reaches a critical threshold value star formation and enrichment accelerate, which leads to rapid increase in both stellar mass and metallicity of galaxies. We demonstrate this process both using a simple analytical model and full cosmological simulations. In contrast, observed metallicity distributions of dwarf galaxies or stars within them are not bi-modal. We argue that this discrepancy points to substantial early stochastic pre-enrichment by population III stars to levels Z ~ 0.01 Z_sun in dense, star forming regions of early galaxies.
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