Redshifts and luminosities for 112 gamma-ray bursts

Two different luminosity indicators have recently been proposed for gamma-ray bursts that use gamma-ray observations alone. They relate the burst luminosity (L) with the time lag between peaks in hard and soft energies ( lag) and the spikiness or variability of the burst's light curve (V). These relations are currently justified and calibrated with only six or seven bursts with known redshifts. We have examined BATSE data for tau (lag) and V for 112 bursts. (1) A strong correlation between tlag and V exists, and it is exactly as predicted from the two proposed relations. This is proof that both luminosity indicators are reliable. (2) GRB 830801 is the all-time brightest burst, yet with a small V and a large tlag, and hence it is likely the closest known event, being perhaps as close as 3.2 Mpc. (3) We have combined the luminosities as derived from both indicators as a means to improve the statistical and systematic accuracy when compared with the accuracy from either method alone. The result is a list of 112 bursts with good luminosities and hence redshifts. (4) The burst-averaged hardness ratio rises strongly with the luminosity of the burst. (5) The burst luminosity function is a broken power law, with the break at L = 2 x 10(52) ergs. The numbers in logarithmic bins scale as above the break and as L-1.7 +/-0.1 below the break. (6) The comoving number density of GRBs varies with redshift roughly as (1 + z)(3.5 +/-0.3) between 0.2 < z < 5. This demonstrates that the burst rate follows the star formation rate at low redshifts, as expected since long bursts are generated by very massive stars. Excitingly, this result also provides a measure of the star formation rate out to z similar to 5 with no effects from reddening, and the rate is rising uniformly for redshifts above 2.