Effects of luminosity functions induced by relativistic beaming on the statistics of cosmological gamma-ray bursts

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/V_max〉 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, z_max. The estimated z_max 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 z_max value for the short bursts is rather surprising, in that the 〈V/V_max〉 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 = 3^o.0, f_>z′ decreases to ∼20% (α = 1.0) at z′ = 3.42. If we assume α = 2.0, the conic beam with Δθ = 3^o.0 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/V_max〉 might be explained by the standard model.