Evolutions and Calibrations of Long Gamma-Ray Bursts Luminosity Correlations Revisited

Abstract

Luminosity correlations of long Gamma-ray bursts (GRB) are extensively proposed as an effective complementarity to trace the Hubble diagram of Universe at high redshifts, which is of great importance to explore properties of dark energy. Recently, several empirical luminosity correlations have been statistically proposed from GRB observations. However, to treat GRB as the distance indicator, there are two key issues: the redshift evolution of luminosity correlations and their calibrations. In this paper, we choose the Amati relation, the correlation between the peak spectra energy and the equivalent isotropic energy of GRBs ($E_{\rm p}-E_{\rm iso}$), as an example, and find that the current GRB dataset implies that there could be a evolution of the luminosity correlation with respect to the redshift. Therefore, we propose an extended Amati relation with two extra redshift-dependent terms to correct the redshift evolution of GRB relation. Secondly, we carefully check the reliability of the calibration method using the low-redshift GRB data. Importantly, we find that the low-redshift calibration method does not take whole correlations between $Ω_{\rm m}$ and coefficients into account. Neglecting these correlation information can break the degeneracies and obtain the biased constraint on $Ω_{\rm m}$ which is very sensitive to values of parameters for the calibration. A small shift of parameters of "calibrated" relation could significantly change the final constraint on $Ω_{\rm m}$ in the low-redshift calibration method. Finally, we simulate several GRB samples with different statistical errors and find that, in order to correctly recover the fiducial value of $Ω_{\rm m}$ using the low-redshift calibration method, we need a large number of GRB samples with high precisions.