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The cosmic star formation histories are evaluated for different minimummasses of the initial halo structures, with allowance for realistic gasoutflows. With a minimum halo mass of 10^{7} - 10^{8} M_odot and a moderateoutflow efficiency, we reproduce both the current baryon fraction and the earlychemical enrichment of the IGM. The intensity of the formation rate of``normal'' stars is also well constrained by the observations: it has to bedominated by star formation in elliptical galaxies, except perhaps at very lowredshift. The fraction of baryons in stars is predicted as are also the type Iaand II supernova event rates. Comparison with SN observations in the redshiftrange z=0-2 allows us to set strong constraints on the time delay of type Iasupernovae (a total delay of \sim 4 Gyr is required to fit the data), the lowerend of the mass range of the progenitors (2 - 8 M_odot) and the fraction ofwhite dwarfs that reproduce the type Ia supernova (about 1 per cent). Theintensity in the initial starburst of zero metallicity stars below 270 M_\odotmust be limited in order to avoid premature overenrichment of the IGM. Onlyabout 10 - 20 % of the metals present in the IGM at z = 0 have been produced bypopulation III stars at very high z. The remaining 80 - 90 % are ejected laterby galaxies forming normal stars, with a maximum outflow efficiency occurringat a redshift of about 5. We conclude that 10^{-3} of the mass in baryons mustlie in first massive stars in order to produce enough ionizing photons to allowearly reionization of the IGM by z \sim 15.Comment: 51 pages, 23 eps figure

Hierarchical Growth and Cosmic Star Formation: Enrichment, Outflows, and Supernova Rates