  We predict the redshift distribution of long Gamma-Ray Bursts (GRBs) with
Monte Carlo simulations. Our improved analysis constrains free parameters with
three kinds of observation: (i) the log(N)-log(P) diagram of BATSE bursts; (ii)
the peak energy distribution of bright BATSE bursts; (iii) the HETE2 fraction
of X-ray rich GRBs and X-ray flashes. The statistical analysis of the Monte
Carlo simulation results allow us to carefully study the impact of the
uncertainties in the GRB intrinsic properties on the redshift distribution. The
comparison with SWIFT data then leads to the following conclusions. The Amati
relation should be intrinsic, if observationally confirmed by SWIFT. The
progenitor and/or the GRB properties have to evolve to reproduce the high mean
redshift of SWIFT bursts. Our results favor an evolution of the efficiency of
GRB production by massive stars, that would be ~6-7 times higher at z~7 than at
z~2. We finally predict around 10 GRBs detected by SWIFT at redshift z>6 for a
three year mission. These may be sufficient to open a new observational window
over the high redshift Universe.
