Testing the Predictions of the Universal-Structured Jet Model of Gamma-Ray Bursts by Simulations

Based on the standard energy reservoir of gamma-ray burst (GRB) jets and the relationship between peak energy ($E_p$) of $\nu F_{\nu }$ spectrum and the equivalent-isotropic energy ($E_{\mathrm{i}\mathrm{s}\mathrm{o}}$), we test the GRB probability distributions as a function of viewing angle ($\theta$) and of viewing angle together with redshift $z$, $P(\theta )$ and $P(\theta ,z)$, predicted by the universal-structured jet (USJ) model with simulations on observational bases for a detection threshold of $S>4\times {10}^{-5}$ erg cm${}^{-2}$, where $S$ is the total fluence in an energy band $>20$ keV. We simulate a sample including ${10}^6$ GRBs, and then derive $P(\theta )$ and $P(\theta ,z)$. We show that both simulated and theoretical $P(\theta )$ and $P(\theta ,z)$ are consistent with each other. This consistency seems to be regarded as support for the USJ model. A small difference between the theoretical and simulated results might be due to the observed $E_p$ distribution used in our simulations, which perhaps suffers a little of biases for $E_p<100$ keV.