Effects of Luminosity Functions Induced by Relativistic Beaming on Statistics of Cosmological Gamma-Ray Bursts

Most of gamma-ray bursts (GRBs) models have predicted that the intrinsic isotropic energy is limited to below ∼ 10ergs. Recently claimed high redshifts, correlation with supernovae, and connection to the cosmic star formation activity point to a different energy requirement and source evolution history possibly with strong beaming. We study the effects of the beaming-induced luminosity function on statistics of observed GRBs, assuming the cosmological scenario. We select and divide the BATSE 4B data into 588 long bursts (T90 > 2.5 sec) and 149 short bursts (T90 < 2.5 sec), and compare the statistics calculated in each subgroup. The 〈V/Vmax〉 of the long bursts is 0.2901 ± 0.0113, and that of the short bursts is 0.4178 ± 0.0239. For luminosity function models, we consider a cylindrical-beam and a conic-beam. We take into account the spatial distribution of GRB sources as well. A broad luminosity function is naturally produced when one introduces beaming of GRBs. We calculate the maximum detectable redshift of GRBs, zmax. The estimated zmax for the cylindrical-beam case is as high as ∼ 14 (α = 1.0) and ∼ 6 (α = 2.0) for the long bursts and ∼ 3 (α = 1.0) and ∼ 1.6 (α = 2.0) for the short bursts where α is the photon index. The large zmax value for the short bursts is rather surprising in that the 〈V/Vmax〉 for this subgroup is close to the so-called Euclidean value, 0.5. We calculate the fraction of bursts whose redshifts are larger than a certain redshift z, i.e. f>z′ . When we take z = 3.42 and apply the luminosity function derived for the cylindrical-beam, the expected f>z′ is ∼ 75 % (α = 1.0) and ∼ 50 % (α = 2.0) for long bursts. When we increase the opening angle of the conic beam to ∆θ = 3.0, f>z′ decreases to ∼ 20 % (α = 1.0) at z = 3.42. If we assume α = 2.0, the conic-beam with ∆θ = 3.0 can not explain the redshift distribution of the observed GRBs. We conclude that the beaming-induced luminosity functions are compatible with the