  We conduct a comprehensive investigation of the detectability of the first
stars and their enrichment signatures in galaxy clusters. We show that the mean
metallicity of outflows from objects containing these Population III (PopIII)
stars is well above the critical transition metallicity (Z_cr \sim 10^-4) that
marks the formation of normal stars. Thus the fraction of PopIII objects formed
as a function of redshift is heavily dependent on the distribution of metals
and fairly independent of the precise value of Z_cr. Using an analytical model
of inhomogenous structure formation, we study the evolution of PopIII objects
as a function of the star formation efficiency, IMF, and efficiency of outflow
generation. For all models, PopIII objects tend to be in the 10^6.5-10^7.0
solar mass range, just large enough to cool within a Hubble time, but small
enough that they are not clustered near areas of previous star formation.
Although the mean metallicity exceeds Z_cr at a redshift of 15 in all models,
the peak of PopIII star formation occurs at z \sim 10, and such stars continue
to form well into the observable range. We discuss the observational properties
of these objects, some of which may have already been detected in ongoing
surveys of high-redshift Lyman-alpha emitters. Finally, we combine our PopIII
distributions with the yield models of Heger and Woosley (2002) to study their
impact on the intracluster medium (ICM) in galaxy clusters. We find that PopIII
stars can contribute no more than 20% of the iron observed in the ICM, although
their peculiar elemental yields help to reconcile theoretical models with the
observed Fe and Si/Fe abundances. However, these stars tend to overproduce S/Fe
and their associated SN heating falls far short of the observed level of 1 keV
per ICM gas particle.
