Effects of Luminosity Functions Induced by Relativistic Beaming on   Statistics of Cosmological Gamma-Ray Bursts
  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 (T$_{90}>2.5$ sec) and 149 short bursts (T$_{90}<2.5$ sec), and compare the statistics calculated in each subgroup. The $<V/V_{\rm max}>$ of the long bursts is $ 0.2901\pm 0.0113$, and that of the short bursts is $0.4178\pm 0.0239$, which is a Euclidean value. 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, $z_{\rm max}$. The estimated $z_{\rm max}$ for the cylindrical-beam case is as high as $\sim 14$ for the long bursts and $\sim 3$ for the short bursts. The large $z_{\rm max}$ value for the short bursts is rather surprising in that the $<V/V_{\rm max}>$ 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. $f_{\rm > z'}$. When we take $z'=3.42$ and apply the luminosity function derived for the cylindrical-beam, the expected $f_{\rm > z'}$ is $\sim 75 %$ for long bursts. When we increase the opening angle of the conic beam to $\Delta \theta =3^\circ.0$, $f_{\rm > z'}$ decreases to $\sim 20 %$ at $ {\rm z'=3.42}$. We conclude that the beaming-induced luminosity functions are compatible with the redshift distribution of observed GRBs and that the apparent Euclidean value of $<V/V_{\rm max}>$ may not be due to the Euclidean space distribution but to the luminosity distribution. 
Chunglee Kim, Heon-Young Chang, and Insu Yi (Korea Institute for   Advanced Study)