2012ApJ...745..168L

An apparent redshift dependence of the jet opening angles (θ_j) of gamma-ray bursts (GRBs) is observed from the current GRB sample. We investigate whether this dependence can be explained with instrumental selection effects and observational biases by a bootstrapping method. Assuming that (1) the GRB rate follows the star formation history and the cosmic metallicity history and (2) the intrinsic distributions of the jet-corrected luminosity (L_γ) and θ_j are a Gaussian or a power-law function, we generate a mock Swift/Burst Alert Telescope (BAT) sample by considering various instrumental selection effects, including the flux threshold and the trigger probability of BAT, the probabilities of a GRB jet pointing to the instrument solid angle, and the probability of redshift measurement. Our results reproduce the observed θ_j- z dependence well. We find that in the case of L_γ∝θ² _j good consistency between the mock and observed samples can be obtained, indicating that both L_γ and θ_jare degenerate for a flux-limited sample. The parameter set (L_γ, θ_j) = (4.9x10⁴⁹ erg/s, 0.054 rad) gives the best consistency for the current Swift GRB sample. Considering the beaming effect, the derived intrinsic local GRB rate is accordingly 2.85 x10²/Gpc3/yr, inferring that ∼0.59% of Type Ib/c supernovae may be accompanied by a GRB.