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

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