A simple physical model for long-duration gamma ray bursts ( GRBs ) is used to fit the redshift ( z ) and the jet opening-angle distributions measured with earlier GRB missions and with Swift .
The effect of different sensitivities for GRB triggering is sufficient to explain the difference in the z distributions of the pre- Swift and Swift samples , with mean redshifts of \langle z \rangle \cong 1.5 and \langle z \rangle \cong 2.7 , respectively .
Assuming that the emission properties of GRBs do not change with time , we find that the data can only be fitted if the comoving rate-density of GRB sources exhibits positive evolution to z \gtrsim 3 – 5 .
The mean intrinsic beaming factor of GRBs is found to range from \approx 34 – 42 , with the Swift average opening half-angle \langle \theta _ { j } \rangle \sim 10 ^ { \circ } , compared to the pre- Swift average of \langle \theta _ { j } \rangle \sim 7 ^ { \circ } .
Within the uniform jet model , the GRB luminosity function is \propto L ^ { -3.25 } _ { * } , as inferred from our best fit to the opening angle distribution .
Because of the unlikely detection of several GRBs with z \lesssim 0.25 , our analysis indicates that low redshift GRBs represent a different population of GRBs than those detected at higher redshifts .
Neglecting possible metallicity effects on GRB host galaxies , we find that \approx 1 GRB occurs every 600,000 yrs in a local L _ { * } spiral galaxy like the Milky Way .
The fraction of high-redshift GRBs is estimated at 8 – 12 % and 2.5 – 6 % at z \geq 5 and z \geq 7 , respectively , assuming continued positive evolution of the GRB rate density to high redshifts .