Constraining the cosmological parameters and transition redshift with gamma-ray bursts and supernovae

A new method of measuring cosmology with gamma-ray bursts (GRBs) has been proposed by Liang and Zhang recently. In this method, only observable quantities including the rest-frame peak energy of the nu F-nu(E'(p)), the isotropic energy of GRB (E-gamma,E-iso) and the break time of the optical afterglow light curves in the rest frame (t'(b)) are used. By considering this method, we constrain the cosmological parameters and the redshift at which the Universe expanded from the deceleration to acceleration phase. We add five recently detected GRBs to the sample and derive E-gamma,E-iso/10(52) erg = (0.93 +/- 0.25) (E'(p)/100 keV)(1.91 +/- 0.32)(t'(b)/1 d)(-0.93 +/- 0.38) for a flat universe with Omega(M)= 0.28 and H-0=71.0 km s(-1) Mpc(-1). This relation is independent of the medium density around bursts and the efficiency of conversion of the explosion energy to gamma-ray energy. We consider the E-gamma,E- iso(E'(p), t'(b)) relationship as a standard candle and find >0.05 < Omega(M) < 0.48 Omega(Lambda) < 1.15 (at the 1 sigma confidence level). In a flat universe with the cosmological constant, we obtain 0.25 < Omega(M) < 0.46 and 0.54 < Omega(Lambda) < 0.78 at the 1 sigma confidence level. The transition redshift is z(T)= 0.69(-0.12)(+0.11). Combining 20 GRBs with 157 Type Ia supernovae (SNe Ia), we find Omega(M)= 0.29 +/- 0.03 for a flat universe and the transition redshift is z(T)= 0.61(-0.05)(+0.06), which is slightly larger than the value found by considering SNe Ia alone. In particular, we also discuss several dark-energy models in which the equation of state w(z) is parametrized, and investigate constraints on the cosmological parameters in detail.