  We examine the relationship between a number of long gamma-ray burst (lGRB)
properties (isotropic emitted energy, luminosity, intrinsic duration, jet
opening angle) and redshift. We find that even when accounting for conservative
detector flux limits, there appears to be a significant correlation between
isotropic equivalent energy and redshift, suggesting cosmological evolution of
the lGRB progenitor. Analyzing a sub-sample of lGRBs with jet opening angle
estimates, we find the beaming-corrected lGRB emitted energy does not correlate
with redshift, but jet opening angle does. Additionally, we find a
statistically significant anti-correlation between the intrinsic prompt
duration and redshift, even when accounting for potential selection effects. We
also find that - for a given redshift - isotropic energy is positively
correlated with intrinsic prompt duration. None of these GRB properties appear
to be correlated with galactic offset. From our selection-effect-corrected
redshift distribution, we estimate a co-moving rate density for lGRBs, and
compare this to the global cosmic star formation rate (SFR). We find the lGRB
rate mildly exceeds the global star formation rate between a redshift of 3 and
5, and declines rapidly at redshifts above this (although we cannot constrain
the lGRB rate above a redshift of about 6 due to sample incompleteness). We
find the lGRB rate diverges significantly from the SFR at lower redshifts. We
discuss both the correlations and lGRB rate density in terms of various lGRB
progenitor models and their apparent preference for low-metallicity
environments.
