  (Abridged) The contemporary discoveries of galaxies and gamma ray bursts
(GRBs) at high redshift have supplied the first direct information on star
formation when the universe was only a few hundred million years old. The
probable origin of long duration GRBs in the deaths of massive stars would link
the universal GRB rate to the redshift-dependent star formation rate density,
although exactly how is currently unknown. As the most distant GRBs and
star-forming galaxies probe the reionization epoch, the potential rewards of
understanding the redshift-dependent ratio Psi(z) of the GRB rate to star
formation rate are significant and include addressing fundamental questions
such as incompleteness in rest-frame UV surveys for determining the star
formation rate at high redshift and time variations in the stellar initial mass
function. Using an extensive sample of 112 GRBs above a fixed luminosity limit
drawn from the Second Swift Burst Alert Telescope catalog, we compare the
cumulative redshift distribution N(< z) of GRBs with the star formation density
rho_sfr(z) measured from UV-selected galaxies over 0 < z < 4. Strong evolution
(e.g., Psi(z) \propto (1+z)^{1.5}) is disfavored, while more modest evolution
(e.g., Psi(z) \propto (1+z)^{0.5}) is consistent with the data. If such trends
continue beyond z ~ 4, we find the discovery rate of distant GRBs implies a
star formation rate density much higher than that inferred from UV-selected
galaxies. We show that such a star formation history would over-predict the
observed stellar mass density at z > 4 measured from rest-frame optical
surveys. The resolution of this important disagreement is currently unclear,
and the GRB production rate at early times is likely more complex than a simple
function of star formation rate and progenitor metallicity.
