We study the luminosity function (LF), the comoving rate and the detection rate of Long Gamma-Ray Burst (LGRBs) to high redshift, using galaxy catalogues constructed by combining high-resolution N-body simulations with semi-analytic models of galaxy formation. We assume the collapsar model and different metallicity thresholds, and conclude that LGRBs are not good tracers of the star formation history in the universe. Then using the log N-log P diagram for BATSE bursts, we determine the LF (with and without evolution with redshift) and the formation rate of LGRBs, obtaining constraints on the slope of the power-law. We check the resulting redshift distribution with SWIFT data updated to 2009 August, finding that models where LGRBs have as progenitors stars with Z<0.3Z_sun and without evolution of the LF are in agreement with the data. We also predict that there are about ~1% of GRBs at redshift z>6.
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