2006MNRAS.369L..37L

Several gamma-ray burst (GRB) luminosity indicators have been proposed which can be generally written in the form of, 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 (Δz) near a fiducial redshift has been proposed recently. Here we elaborate such a possibility and propose to calibrate {lcub}a_i{rcub} 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^a1_pt^a₂^_b, as an example and test the validity of this approach through simulations, where E_isois the isotropic energy of prompt gamma-rays, E_pis the spectral break energy, and t_bis the temporal break time of the optical afterglow light curve. We show that while c strongly depends on the cosmological parameters, neither a₁nor a₂does as long as Δz is small enough. The selection of Δz 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 {lcub}a_i{rcub}, 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∼ 1 or z_c∼ 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 E_iso, E_pand t_b, 25 GRBs within a redshift bin of Δz∼ 0.30 would give fine calibration to the Liang-Zhang luminosity indicator.