Gamma-Ray Bursts in $1.8 < z < 5.6$ Suggest that the Time Variation of the Dark Energy is Small

We calibrated the peak energy-peak luminosity relation of GRBs (so called Yonetoku relation) using 33 events with the redshift $z < 1.62$ without assuming any cosmological models. The luminosity distances to GRBs are estimated from those of large amount of Type Ia supernovae with $z<1.755$. This calibrated Yonetoku relation can be used as a new cosmic distance ladder toward higher redshifts. We determined the luminosity distances of 30 GRBs in $1.8 < z < 5.6$ using the calibrated relation and plotted the likelihood contour in $(Omega_m,Omega_Lambda)$ plane. We obtained $(Omega_m, Omega_{Lambda})= (0.37^{+0.14}_{-0.11}, 0.63^{+0.11}_{-0.14})$ for a flat universe. Since our method is free from the circularity problem, we can say that our universe in $1.8 < z < 5.6$ is compatible with the so called concordance cosmological model derived for $z < 1.8$. This suggests that the time variation of the dark energy is small or zero up to $zsim 6$.