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

摘要

Most gamma-ray burst (GRB) models have predicted that the intrinsic isotropic energy is limited to below ~1053-1054 ergs. Recently claimed high redshifts, correlation with supernovae, and connections to 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 s) and 149 short bursts (T90 2.5 s), 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 Δθ = 30, f z' decreases to ~20% (α = 1.0) at z' = 3.42. If we assume α = 2.0, the conic beam with Δθ = 30 cannot explain the redshift distribution of the observed GRBs. We conclude that the beaming-induced luminosity functions are compatible with the redshift distribution of the observed GRBs, although the apparent "Euclidean" value of V/Vmax might be explained by the standard model.