Absolute Rate, Evolving Luminosity function, and Evolving Jet Opening Angle Distribution for Long Gamma-Ray Bursts

Using the luminosity function and apparent GRB rate inferred from the spectral peak energy-luminosity relation, we discuss the absolute GRB formation rate taking into account both the jet-luminosity evolution and jet opening angle evolution effects. In the case of no jet opening angle evolution, the jet-corrected luminosity of high-redshift GRB is much larger than the typical value of 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 of low-z GRBs. Therefore, it is preferable that both evolution effects are taken into account. We also estimate the local GRB event rate in a galaxy of 10^{-7}-10^{-5}/yr