Most of gamma-ray bursts ( GRBs ) models have predicted that the intrinsic isotropic energy is limited to below \sim 10 ^ { 53 - 54 } ergs .
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 ( T _ { 90 } > 2.5 sec ) and 149 short bursts ( T _ { 90 } < 2.5 sec ) , and compare the statistics calculated in each subgroup .
The \langle V / V _ { max } \rangle of the long bursts is 0.2901 \pm 0.0113 , and that of the short bursts is 0.4178 \pm 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 \sim 14 ( \alpha = 1.0 ) and \sim 6 ( \alpha = 2.0 ) for the long bursts and \sim 3 ( \alpha = 1.0 ) and \sim 1.6 ( \alpha = 2.0 ) for the short bursts where \alpha is the photon index .
The large z _ { max } value for the short bursts is rather surprising in that the \langle V / V _ { max } \rangle 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 ^ { \prime } } , i.e . f _ { > z ^ { \prime } } .
When we take { z ^ { \prime } } = 3.42 and apply the luminosity function derived for the cylindrical-beam , the expected f _ { > z ^ { \prime } } is \sim 75 ~ { } \%~ { } ( \alpha = 1.0 ) and \sim 50 ~ { } \%~ { } ( \alpha = 2.0 ) for long bursts .
When we increase the opening angle of the conic beam to \Delta \theta = 3 ^ { \circ } .0 , f _ { > z ^ { \prime } } decreases to \sim 20 ~ { } \%~ { } ( \alpha = 1.0 ) at { z ^ { \prime } = 3.42 } .
If we assume \alpha = 2.0 , the conic-beam with \Delta \theta = 3 ^ { \circ } .0 can not 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 \langle V / V _ { max } \rangle might be explained by the standard model .