Model-independent distance calibration of high-redshift gamma-ray bursts and constrain on the $\Lambda$CDM model

Gamma-ray bursts (GRBs) are luminous enough to be detectable up to redshift $z\sim 10$. They are often proposed as complementary tools to type-Ia supernovae (SNe Ia) in tracing the Hubble diagram of the Universe. The distance calibrations of GRBs usually make use one or some of the empirical luminosity correlations, such as $\tau_{\rm lag}-L$, $V-L$, $E_p-L$, $E_p-E_{\gamma}$, $\tau_{\rm RT}-L$ and $E_p-E_{\rm iso}$ relations. These calibrating methods are based on the underling assumption that the empirical luminosity correlations are universal over all redshift range. In this paper, we test the possible redshift dependence of six luminosity correlations by dividing GRBs into low-$z$ and high-$z$ classes according to their redshift smaller or larger than 1.4. It is shown that the $E_p-E_{\gamma}$ relation for low-$z$ GRBs is consistent with that for high-$z$ GRBs within $1\sigma$ uncertainty. The intrinsic scatter of $V-L$ relation is too larger to make a convincing conclusion. For the rest four correlations, however, low-$z$ GRBs differ from high-$z$ GRBs at more than $3\sigma$ confidence level. As such, we calibrate GRBs using the $E_p-E_{\gamma}$ relation in a model-independent way. The constraint of high-$z$ GRBs on the $\Lambda$CDM model gives $\Omega_M=0.302\pm 0.142(1\sigma)$, well consistent with the Planck 2015 results.