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.
