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

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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: ϕ(L) ∝ L⁻β (β> 0, Lmin ≤ L ≤ Lmax), data from the BATSE 4B catalog can be used to constrain slope index β and the dynamic range width Log Lmax Lmin. 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 10²<Lmax/Lmin < 10⁷.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 θmax < 1/γ however, the effects of relativistic beaming would not dominate.