  Gamma Ray Bursts (GRBs) and galaxies at high redshift represent complementary
probes of the star formation history of the Universe. In fact, both the GRB
rate and the galaxy luminosity density are connected to the underlying star
formation. Here, we combine a star formation model for the evolution of the
galaxy luminosity function from z=0 to z=10 with a metallicity-dependent
efficiency for GRB formation to simultaneously predict the comoving GRB rate.
Our model sheds light on the physical origin of the empirical relation often
assumed between GRB rate and luminosity density-derived star formation rate:
Rgrb(z) = \epsilon(z)*SFR_{obs}(z), with \epsilon(z) (1+z)^{1.2}. At z<4,
\epsilon(z) is dominated by the effects of metallicity evolution in the GRB
efficiency. Our best-fitting model only requires a moderate preference for
low-metallicity, that is a GRB rate per unit stellar mass about four times
higher for log(Z/Zsun)<-3 compared to log(Z/Zsun)>0. Models with total
suppression of GRB formation at log(Z/Zsun)>0 are disfavored. At z>4, most of
the star formation happens in low-metallicity hosts with nearly saturated
efficiency of GRB production per unit stellar mass. However at the same epoch,
galaxy surveys miss an increasing fraction of the predicted luminosity density
because of flux limits, driving an accelerated evolution of \epsilon(z)
compared to the empirical power-law fit from lower z. Our findings are
consistent with the non-detections of GRB hosts in ultradeep imaging at z>5,
and point toward current galaxy surveys at z>8 only observing the top 15-20 %
of the total luminosity density.
