Is the redshift clustering of long-duration gamma-ray bursts significant?

The 26 long-duration gamma-ray bursts (GRBs) with known redshifts form a distinct cosmological set, selected differently than other cosmological probes such as quasars and galaxies. Since the progenitors are now believed to be connected with active star formation and since burst emission penetrates dust, one hope is that with a uniformly selected sample, the large-scale redshift distribution of GRBs can help constrain the star formation history of the universe. However, we show that strong observational biases in ground-based redshift discovery hamper a clean determination of the large-scale GRB rate and hence the connection of GRBs to the star formation history. We then focus on the properties of the small-scale (clustering) distribution of GRB redshifts. When corrected for heliocentric motion relative to the local Hubble flow, the observed redshifts appear to show a propensity for clustering: eight of 26 GRBs occurred within a recession velocity difference of 1000 km s(-1) of another GRB. That is, four pairs of GRBs occurred within 30 h(65)(-1) Myr in cosmic time, despite being causally separated on the sky. We investigate the significance of this clustering using a simulation that accounts for at least some of the strong observational and intrinsic biases in redshift discovery. Comparison of the numbers of close redshift pairs expected from the simulation with that observed shows no significant small-scale clustering excess in the present sample; however, the four close pairs occur in only about 20% of the simulated data sets (the precise significance of the clustering is dependent on the modeled biases). We conclude with some impetuses and suggestions for future precise GRB redshift measurements.