Can Gamma-Ray Bursts Be Used to Measure Cosmology? A Further Analysis
  Three different methods of measuring cosmology with gamma-ray bursts (GRBs) have been proposed since a relation between the $\gamma$-ray energy $E_{\gamma} $ of a GRB jet and the peak energy $E_p$ of the $\nu F_{\nu}$ spectrum in the burst frame was reported by Ghirlanda and coauthors. In Method I, to calculate the probability for a favored cosmology, only the contribution of the $E_\gamma-E_p$ relation that is already best fitted for this cosmology is considered. We apply this method to a sample of 17 GRBs, and obtain the mass density $\Omega_M=0.15^{+0.45}_{-0.13}$ ($1\sigma$) for a flat $\Lambda$CDM universe. In Method II, to calculate the probability for some certain cosmology, contributions of all the possible $E_\gamma-E_p$ relations that are best fitted for their corresponding cosmologies are taken into account. With this method, we find a constraint on the mass density $0.14<\Omega_M<0.69$ ($1 \sigma$) for a flat universe. In Method III, to obtain the probability for some cosmology, contributions of all the possible $E_\gamma-E_p$ relations associated with their unequal weights are considered. With this method, we obtain an inspiring constraint on the mass density $0.16<\Omega_M<0.45$ ($1 \sigma$) for a flat universe, and a $\chi^2_{dof}=19.08/15=1.27$ for the concordance model of $\Omega_M=0.27$. (abridged) 
D. Xu, Z. G. Dai, E. W. Liang (NJU)