  Recent observations in the total luminosity density have led to significant
progress in establishing the star formation rate (SFR) at high redshift.
Concurrently observed gamma-ray burst rates have also been used to extract the
SFR at high redshift. The SFR in turn can be used to make a host of predictions
concerning the ionization history of the Universe, the chemical abundances, and
supernova rates. We compare the predictions made using a hierarchical model of
cosmic chemical evolution based on three recently proposed SFRs: two based on
extracting the SFR from the observed gamma-ray burst rate at high redshift, and
one based on the observed galaxy luminosity function at high redshift. Using
the WMAP/Planck data on the optical depth and epoch of reionization, we find
that only the SFR inferred from gamma-ray burst data at high redshift suffices
to allow a single mode (in the initial mass function) of star formation which
extends from z = 0 to redshifts > 10. For the case of the more conservative SFR
based on the observed galaxy luminosity function, the reionization history of
the Universe requires a bimodal IMF which includes at least a coeval high (or
intermediate) mass mode of star formation at high redshift (z> 10). Therefore,
we also consider here a more general bimodal case which includes an
early-forming high mass mode as a fourth model to test the chemical history of
the Universe. We compute the abundances of several trace elements, as well as
the expected supernova rates, the stellar mass density and the specific SFR,
sSFR, as a function of redshift
  for each of the four models considered. We conclude that observational
constraints on the global metallicity and optical depth at high redshift favor
unseen faint but active star forming galaxies as pointed out in many recent
studies.
