Selection Effects on the Observed Redshift Dependence of GRB Jet Opening Angles

Apparent redshift dependence of the jet opening angles ($\theta_{\rm j}$) of gamma-ray bursts (GRBs) is observed from 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 ($L_{\rm \gamma}$) and $\theta_{\rm j}$ are a Gaussian or a power-law function, we generate a mock {\em Swift}/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 well reproduce the observed $\theta_{\rm j}-z$ dependence. We find that in case of $L_{\gamma}\propto \theta_{\rm j}^2$ good consistency between the mock and observed samples can be obtained, indicating that both $L_{\rm \gamma}$ and $\theta_{\rm j}$ are degenerate for a flux-limited sample. The parameter set $(L_{\rm \gamma}, \theta_{\rm j})=(4.9\times 10^{49} \rm {erg\ s}^{-1},\ 0.054 {rad})$ gives the best consistency for the current {\em Swift} GRB sample. Considering the beaming effect, the derived intrinsic local GRB rate accordingly is $2.85\times 10^2$ Gpc$^{-3}$ yr$^{-1}$, inferring that $\sim 0.59$ of Type Ib/c SNe may be accompanied by a GRB.