Measuring dark energy with the $E_{\rm iso}-E_{\rm p}$ correlation of
  gamma-ray bursts using model-independent methods
  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.
