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

Apparent redshift dependence of the jet opening angles ($θ_{\rm j}$) 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 ($L_{\rm γ}$) and $θ_{\rm j}$ 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 $θ_{\rm j}-z$ dependence. We find that in case of $L_γ\propto θ_{\rm j}^2$ good consistency between the mock and observed samples can be obtained, indicating that both $L_{\rm γ}$ and $θ_{\rm j}$ are degenerate for a flux-limited sample. The parameter set $(L_{\rm γ}, θ_{\rm j})=(4.9\times 10^{49} \rm {erg\ s}^{-1},\ 0.054 {rad})$ 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.