Calibration of Gamma-Ray Burst Luminosity Indicators
  Several gamma-ray burst (GRB) luminosity indicators (LIs)have been proposed. Unlike SNe Ia, calibration of GRB LIs using a low-redshift sample is difficult. Based on the Bayesian theory, here we propose an approach to calibrate these LIs without introducing a low-redshift GRB sample. The essential points of our approach include, (1) calibrate the power-law indices in the LIs with a sample of GRBs in a narrow redshift range (Delta z); and (2) marginalize the coefficient of the LIs over a reasonable range of cosmological parameters. We take our newly discovered multivariable GRB LIs as an example and test the validity of our approach through simulations. We show that while the coefficient strongly depends on the cosmological parameters, the power-law indices do not 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, while a lower redshift is preferable for the marginalization of the coefficient. The best strategy would be to collect GRBs within a narrow redshift bin around a fiducial intermediate redshift (e.g. z~1 or z~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, spectral break energy, and the optical temporal break time, 25 GRBs within a redshift bin of Delta z ~ 0.30 would give fine calibration to our luminosity indicator(abridged). 
Enwei Liang and Bing Zhang (UNLV)