The Formation Rates of Population III Stars and Chemical Enrichment of Halos during the Reionization Era

[abridged] The First Stars in the Universe form out of pristine primordial gas clouds that have been radiatively cooled to a few hundreds of degrees Kelvin either via molecular or atomic (Lyman-Alpha) hydrogen lines. This primordial mode of star formation is eventually quenched once radiative and/or chemical (metal enrichment) feedbacks mark the transition to Population II stars. In this paper we present a model for the formation rate of Population III stars based on Press-Schechter modeling coupled with analytical recipes for gas cooling and radiative feedback. Our model also includes a novel treatment for metal pollution based on self-enrichment due to a previous episode of Population III star formation in progenitor halos. With this model we derive the star formation history of Population III stars, their contribution to the re-ionization of the Universe and the time of the transition from Population III star formation in minihalos to that in more massive halos where atomic hydrogen cooling is also possible. We consider a grid of models highlighting the impact of varying the values for the free parameters used, such as star formation and feedback efficiency. The most critical factor is the assumption that only one Population III star is formed in a halo. In this scenario, metal free stars contribute only to a minor fraction of the total number of photons required to re-ionize the universe. In addition, metal free star formation is primarily located in minihalos and chemically enriched halos become the dominant locus of star formation very early in the life of the Universe, at redshift z~25. If instead multiple metal free stars are allowed to form out of a single halo, then there is an overall boost of Population III star formation, with a consequent significant contribution to the re-ionizing radiation budget.