Constraints onthe bulk Lorentz factor of gamma-ray burstswith the detection rate by Fermi LAT

The bulk Lorentz factor $(Γ)$ of the outflow is an essential parameter for understanding the physics of gamma-ray burst (GRB). Informations about the Lorentz factors of some individual GRBs have been obtained from the spectral features of the high-energy gamma-ray emissions (>100 MeV), assuming that the spectral breaks or cutoffs are due to the pair-production attenuation (i.e. $γγ → e^+ e^−$). In this paper, we attempt to interpret the dependence of the Large Area Telescope (LAT) detection rate of GRBs on the number of high-energy gamma-rays, taking into account the attenuation effect. We first simulate a long-GRB sample with Monte Carlo method using the luminosity function, rate distribution with redshift, and properties of the GRB spectrum. To characterize the distribution of the Lorentz factors, we assume that the Lorentz factors follow the relation $Γ= Γ_0 E^{k}_{iso,52⁠}$, where $E_{iso,52}$ is the isotropic photon energy in unit of 1052 erg. After taking into account the attenuation effect related with the above Lorentz factor distribution, we are able to reproduce the LAT-detected rate of GRBs as the function of the number of gamma-rays for suitable choice of the values of Γ0 and k. The result suggests that the distribution of the bulk Lorentz factor for the majority of GRBs is in the range of 50−250