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 (L) over cap = c Pi x(i)(ai), where c is the coefficient, x(i) is the ith observable, and a(i) is its corresponding power-law index. Unlike in 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 some nearby GRBs may be different from their cosmological brethren. Calibrating the standard candles using GRBs in a narrow redshift range (Delta z) near a fiducial redshift has been proposed recently. Here we elaborate such a possibility and propose to calibrate {a(i)} based on the Bayesian theory and to marginalize the c value over a reasonable range of cosmological parameters. We take our newly discovered multivariable GRB luminosity indicator, E-iso = cE(p)(a1)t(b)(a2), as an example and test the validity of this approach through simulations, where E-iso is the isotropic energy of prompt gamma-rays, E-p is the spectral break energy, and t(b) is the temporal break time of the optical afterglow light curve. We show that while c strongly depends on the cosmological parameters, neither a(1) nor a(2) does as long as Delta z is small enough. The selection of Delta 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 {a(i)}, 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. z(c) similar to 1 or z(c) similar to 2), as the observed GRB redshift distribution is found to peak around this range. Our simulation suggests that with the current observational precisions of measuring Eiso, Ep and tb, 25 GRBs within a redshift bin of Delta z similar to 0.30 would give fine calibration to the Liang-Zhang luminosity indicator.