Luminosity and variability of collimated gamma-ray bursts

Within the framework of the internal shock model, we study the luminosity and variability in gamma-ray bursts from collimated fireballs. In particular, we pay attention to the role of the e^± pair photosphere produced by the internal shock synchrotron photons. It is shown that the observed Cepheid-like relationship between the luminosity and variability can be interpreted as a correlation between the opening angle of the fireball jet and the mass included at the explosion with a standard energy output. We also show that such a correlation can be a natural consequence of the collapsar model. Narrow jets, in which the typical Lorentz factors are higher than in wide jets, can produce more variable temporal profiles because of smaller angular spreading timescales at the photosphere radius. Using a multiple-shell model, we numerically calculate the temporal profiles of gamma-ray bursts and show that our simulations reproduce the observed correlation.