Calibration of Gamma-Ray Burst Luminosity Indicators

Several gamma-ray burst (GRB) luminosity indicators have been proposed, which can be generally written in the form of L̂ = c ∏ x ai i , where c is the coefficient, xi is the i-th observable, and ai is its corresponding power-law index. Unlike Type-Ia supernovae, calibration of GRB luminosity indicators using a low-redshift sample is difficult. This is because the GRB rate drops rapidly at low redshifts, and more importantly, some nearby GRBs appear to be different from their cosmological brethren. Based on the Bayesian theory, here we propose an approach to calibrate the GRB luminosity indicators without introducing a low-redshift GRB sample. The essential points of our approach include, (1) calibrate {ai} with a sample of GRBs in a narrow redshift range (∆z); and (2) marginalize the c value over a reasonable range of cosmological parameters. We take our newly discovered multi-variable GRB luminosity indicator, Eiso = cE a1 p t a2 b , as an example and test the validity of our approach through simulations. We show that while c strongly depends on the cosmological parameters, neither a1 nor a2 does as long as ∆z is small enough. The selection of ∆z for a particular GRB sample could be judged according to the size and the observational uncertainty of the sample. There is no preferable redshift to perform the calibration of the indices {ai}, while a lower redshift is preferable for c-marginalization. The best strategy would be to collect GRBs within a narrow redshift bin around a fiducial intermediate redshift (e.g. zc ∼ 1 or zc ∼ 2), since the observed GRB redshift distribution is found to peak around z = (1 − 2.5). Our simulation suggests that with the current observational precision of measuring GRB isotropic energy (Eiso), spectral break energy (Ep), and the optical temporal break time (tb), 25 GRBs within a redshift bin of ∆z ∼ 0.30 would give fine calibration to the Liang-Zhang luminosity indicator.