Motivated by the Wilkinson Microwave Anisotropy Probe (WMAP) results indicating an early epoch of reionization, we consider alternative cosmic star formation models that are capable of reionizing the early intergalactic medium. We develop models that include an early burst of massive stars (with several possible mass ranges) combined with standard star formation. We compute the stellar ionizing flux of photons, and we track the nucleosynthetic yields for several elements: D, 4He, C, N, O, Si, S, Fe, and Zn. We compute the subsequent chemical evolution as a function of redshift, both in the intergalactic medium and in the interstellar medium of forming galaxies, starting with the primordial objects that are responsible for the reionization. We apply constraints from the observed abundances in the Ly? forest and in damped Ly? clouds in conjunction with the ability of the models to produce the required degree of reionization. We also consider possible constraints associated with the observations of the two extremely metal-poor stars HE 0107-5240 and CS 22949-037. We confirm that an early top-heavy stellar component is required, since a standard star formation model is unable to reionize the early universe and reproduce the abundances of the very metal-poor halo stars. A bimodal (or top-heavy) initial mass function (IMF; 40-100 M?) is our preferred scenario, compared with the extreme mass range (100 M?) often assumed to be responsible for the early stages of reionization. A mode of even more extreme stellar masses in the range ?270 M? has also been considered. All massive stars in this mode collapse entirely into black holes, and as a consequence, chemical evolution and reionization are decorrelated. The ionizing flux from these very massive stars can easily reionize the universe at z ~ 17. However, the chemical evolution in this case is exactly the same as in the standard star formation model, and the observed high-redshift abundances are not reproduced. We show that the initial top-heavy mode, which originally was introduced to reionize the early universe, produces rapid initial metal pollution. The existence of old, C-rich halo stars with high [O/Fe] and [C/Fe] ratios is predicted as a consequence of these massive stars. The recently observed abundances in the oldest halo stars could trace this very specific stellar population. The extreme mass range is disfavored, and there is no evidence, nor any need, for a hypothesized primordial population of very massive stars in order to account for the chemical abundances of extremely metal-poor halo stars or of the intergalactic medium. The combined population of early-forming normal (0.1-100 M?) and massive (40-100 M?) stars can simultaneously explain the cosmic chemical evolution and the observations of extremely metal-poor halo stars and also account for early cosmological reionization.
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