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.
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