Evolutions and Calibrations of Long Gamma-Ray Bursts Luminosity
  Correlations Revisited
  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 $\Omega_{\rm m}$ and coefficients into account.
Neglecting these correlation information can break the degeneracies and obtain
the biased constraint on $\Omega_{\rm m}$ 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 $\Omega_{\rm m}$ 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 $\Omega_{\rm m}$ using the low-redshift calibration
method, we need a large number of GRB samples with high precisions.
