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: 8 of 26 GRBs occurred within a recession velocity difference of 1000 km/s of another GRB. That is, 4 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. 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 only in about twenty percent of the simulated datasets (the precise significance of the clustering is dependent upon the modeled biases). We conclude with some impetuses and suggestions for future precise GRB redshift measurements.