Several gamma-ray burst ( GRB ) luminosity indicators have been proposed , which can be generally written in the form of \hat { L } = c \prod x _ { i } ^ { a _ { i } } , where c is the coefficient , x _ { i } is the i -th observable , and a _ { i } 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 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 multi-variable GRB luminosity indicator , E _ { iso } = cE _ { p } ^ { a _ { 1 } } t _ { b } ^ { a _ { 2 } } , as an example and test the validity of this approach through simulations .
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 } \sim 1 or z _ { c } \sim 2 ) , since the observed GRB redshift distribution is found to peak around this range .
Our simulation suggests that with the current observational precision of measuring GRB isotropic energy ( E _ { iso } ) , spectral break energy ( E _ { p } ) , and the optical temporal break time ( t _ { b } ) , 25 GRBs within a redshift bin of \Delta z \sim 0.30 would give fine calibration to the Liang-Zhang luminosity indicator .