We use two model-independent methods to standardize long gamma-ray bursts ( GRBs ) using the E _ { iso } - E _ { p } correlation ( \log E _ { iso } = a + b \log E _ { p } ) , where E _ { iso } is the isotropic-equivalent gamma-ray energy and E _ { p } is the spectral peak energy .
We update 42 long GRBs and attempt to constrain the 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 .
We take the extrinsic scatter \sigma _ { ext } into account and assign it to the parameter E _ { iso } .
The best-fitting values are a = 49.15 \pm 0.26 , b = 1.42 \pm 0.11 , \sigma _ { ext } = 0.34 \pm 0.03 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 \pm 0.18 , b = 1.52 \pm 0.08 , and \Omega _ { m } = 0.50 \pm 0.12 .
The second method uses type Ia supernovae ( SNe Ia ) to calibrate the E _ { iso } - E _ { 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 \pm 0.11 and \Omega _ { \Lambda } = 0.46 \pm 0.51 for non-flat \Lambda CDM .
For the combination of GRB and SNe Ia sample , we obtain \Omega _ { m } = 0.271 \pm 0.019 and h = 0.701 \pm 0.002 for the flat \Lambda CDM and the non-flat \Lambda CDM , and the results are \Omega _ { m } = 0.225 \pm 0.044 , \Omega _ { \Lambda } = 0.640 \pm 0.082 , and h = 0.698 \pm 0.004 .
These results from calibrated GRBs are consistent with that of SNe Ia .
Meanwhile , the combined data can improve cosmological constraints significantly , compared to SNe Ia alone .
Our results show that the E _ { iso } - E _ { p } correlation is promising to probe the high-redshift universe .