We present the redshift results from a Very Large Telescope (VLT) program aimed at
optimizing the legacy value of the Swift mission: to characterize a
homogeneous, X-ray selected, sample of 69 GRB host galaxies. Fifteen new redshifts have
been secured, resulting in a 77% (53/69) redshift completion, making the survey the most
comprehensive in terms of redshift completeness of any sample to the full Swift
depth, available to date. We present the cumulative redshift distribution and
derive a conservative, yet small, associated uncertainty. We constrain the fraction of
Swift GRBs at high redshift to a maximum of 14% (5%) for
z > 6 (z > 7). The mean redshift of the host
sample is assessed to be . Using this more complete sample, we confirm previous findings
that the GRB rate at high redshift () appears to be in excess of predictions based on
assumptions that it should follow conventional determinations of the star formation
history of the universe, combined with an estimate of its likely metallicity dependence.
This suggests that either star formation at high redshifts has been significantly
underestimated, for example due to a dominant contribution from faint, undetected
galaxies, or that GRB production is enhanced in the conditions of early star formation,
beyond those usually ascribed to lower metallicity.