RELATIVISTIC BEAMING AND GAMMA-RAY BURSTS

Gamma-ray bursts are likely to be relativistically beamed and nonspherical. We study the statistics of gamma-ray bursts by assuming gamma-ray radiation is produced in a cone with a large Lorentz factor. Such a geometry naturally introduces a luminosity function due to different orientations of the cone. The width and shape of the luminosity function depend on the opening angle and the Lorentz factor. The luminosity function generally increases the redshift of the most distant bursts. Large Lorentz factors and opening angles provide sharply peaked luminosity functions that are almost indistinguishable statistically from standard candles. We find it impossible to constrain the opening angle of the cone and in general the width of a wide range of luminosity functions. The opening angle is, however, constrained by the observed rate of gamma-ray bursts and the total rate of their progenitors. In the cosmological context, the opening angle must satisfy DELTAtheta greater-than-or-similar-to 10(-3)/R(i)1/2, where R(i) is the rate of progenitors per year per galaxy. This requires the emission to be quasispherical for the neutron star merger model and DELTAtheta greater-than-or-similar-to 5-degrees for the failed supernovae model as proposed by Woosley.