Generalized tests for eight GRB luminosity relations

Long-duration gamma-ray bursts (GRBs) have eight luminosity relations where observable burst properties can yield the burst luminosity and hence distance. This turns GRBs into useful tools of cosmology. Recently, two tests have been proposed (by Nakar & Piran and by Li) for which one of the eight relations is claimed to have significant problems. In this Letter, we generalize these tests and apply them to all eight GRB luminosity relations. (1) All eight relations pass the Nakar & Piran test after accounting for the uncertainties on the data and the dispersions of the correlations. (2) All eight relations are good when the GRB redshifts are known, for example, for calibration of the relations and for GRB Hubble diagram purposes. (3) We confirm the earlier results that the E-gamma,E-iso-E-peak Amati relation must produce very large error bars whenever an unknown redshift being sought is greater than or similar to 1.4. (4) The E-gamma-E-peak relation of Ghirlanda et al. must produce very large error bars whenever an unknown redshift being sought is greater than or similar to 3.4. (55) The other six relations have no problem at all from the ambiguity test of Li.