Constraints on Jets and Luminosity Function of Gamma-ray Bursts Associated with Supernovae

If Gamma-ray Bursts (GRBs) are generally associated with supernovae like 1998bw, a relatively wide intrinsic luminosity function is implied, which indicates the existence of a large amount of undetected dim bursts, and a much higher event-rate than is often assumed. If it is assumed that the intrinsic luminosity function of GRBs is a power-law: ${\phi (L)} \propto {L^{-\beta}}$ ($\beta > 0$, $L_{min} \leq L \leq L_{max}$), data from the BATSE 4B catalog can be used to constrain slope index $\beta$ and the dynamic range width Log ${L_{max}\over L_{min}}$. Using a K-S test comparison with the observational Log $N$ - Log $P$, we find constraints on the GRB fireball model, GRB jets, and the possible GRB contribution to cosmic gamma-ray background. We find the acceptable dynamic range for $10^2<L_{max}/L_{min} < 10^7$.Our results show that jet model is more likely to be related more highly energetic explosion than fireball model.Our studies also show that the luminosity function provided by a purely special relativistic effect on a jet is outside of the K-S test acceptable range. Were intrinsic beaming to confine the jet to $\theta_{max} < 1/\gamma$ however, the effects of relativistic beaming would not dominate.