Absolute Rate, Evolving Luminosity function, and Evolving Jet Opening   Angle Distribution for Long Gamma-Ray Bursts
  Using the luminosity function and the apparent GRB rate inferred from the spectral peak energy-luminosity relation, we investigate the absolute GRB formation rate, taking into account the effects of both the jet-luminosity evolution and the jet opening angle evolution. In the case that there is no jet opening angle evolution, the jet-corrected luminosity for high-redshift GRBs is much larger than typical values for low-z (z>~1) bursts. On the other hand, if the jet-luminosity does not evolve with time, the jet opening angle for high-z bursts should be much smaller than that for low-z GRBs. Therefore, it is preferable to take into account both evolution effects. We also estimate the local GRB event rate in a galaxy of ~10^{-7}-10^{-5}/yr. 
Tatsushi Matsubayashi, Ryo Yamazaki, Daisuke Yonetoku, Toshio   Murakami, Toshikazu Ebisuzaki