Calibration of gamma-ray burst luminosity indicators

Abstract

Several gamma-ray burst (GRB) luminosity indicators have been proposed which can be generally written in the form of L̂ = c π x<inf>i</inf>^a<inf>i</inf>, where c is the coefficient, x<inf>i</inf> is the ith observable, and a<inf>i</inf> 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 cosmolog-ical brethren. Calibrating the standard candles using GRBs in a narrow redshift range (Az) near a fiducial redshift has been proposed recently. Here we elaborate such a possibility and propose to calibrate {a<inf>i</inf>} 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<inf>iso</inf> = cE^a<inf>1</inf> <inf>p</inf> t^a<inf>2</inf> <inf>b</inf>, as an example and test the validity of this approach through simulations, where Ei<inf>so</inf> is the isotropic energy of prompt gamma-rays, E<inf>p</inf> is the spectral break energy, and t<inf>b</inf> is the temporal break time of the optical afterglow light curve. We show that while c strongly depends on the cosmological parameters, neither a<inf>1</inf> nor a<inf>2</inf> does 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 {a<inf>i</inf>}, 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<inf>c</inf> ~ 1 or z<inf>c</inf> ~ 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<inf>iso</inf>, E<inf>p</inf> and t<inf>b</inf>, 25 GRBs within a redshift bin of δz ~ 0.30 would give fine calibration to the Liang-Zhang luminosity indicator. © 2006 The Authors. Journal compilation © 2006 RAS.