### Measuring dark energy with the $E_{\rm iso}-E_{\rm p}$ correlation of GRBs using model-independent methods

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In this paper, we use two model-independent methods to standardize long gamma-ray bursts (GRBs) using the $E_{\rm iso}-E_{\rm p}$ correlation, where $E_{\rm iso}$ is the isotropic-equivalent gamma-ray energy and $E_{\rm p}$ is the spectral peak energy. We update 42 long GRBs and try to make constraint on cosmological parameters. The full sample contains 151 long GRBs with redshifts from 0.0331 to 8.2. The first method is the simultaneous fitting method. The extrinsic scatter $\sigma_{\rm ext}$ is taken into account and assigned to the parameter $E_{\rm iso}$. The best-fitting values are $a=49.15\pm0.26$, $b=1.42\pm0.11$, $\sigma_{\rm ext}=0.34\pm0.03$ and $\Omega_m=0.79$ in the flat $\Lambda$CDM model. The constraint on $\Omega_m$ is $0.55<\Omega_m<1$ at the 1$\sigma$ confidence level. If reduced $\chi^2$ method is used, the best-fit results are $a=48.96\pm0.18$, $b=1.52\pm0.08$ and $\Omega_m=0.50\pm0.12$. The second method is using type Ia supernovae (SNe Ia) to calibrate the $E_{\rm iso}-E_{\rm p}$ correlation. We calibrate 90 high-redshift GRBs in the redshift range from 1.44 to 8.1. The cosmological constraints from these 90 GRBs are $\Omega_m=0.23^{+0.06}_{-0.04}$ for flat $\Lambda$CDM, and $\Omega_m=0.18\pm0.11$ and $\Omega_{\Lambda}=0.46\pm0.51$ for non-flat $\Lambda$CDM. For the combination of GRB and SNe Ia sample, we obtain $\Omega_m=0.271\pm0.019$ and $h=0.701\pm0.002$ for the flat $\Lambda$CDM, and for the non-flat $\Lambda$CDM, the results are $\Omega_m=0.225\pm0.044$, $\Omega_{\Lambda}=0.640\pm0.082$ and $h=0.698\pm0.004$. These results from calibrated GRBs are consistent with that of SNe Ia. Meanwhile, the combined data can improve cosmological constraints significantly, comparing to SNe Ia alone. Our results show that the $E_{\rm iso}-E_{\rm p}$ correlation is promising to probe the high-redshift universe.