We estimate the distance modulus to long gamma-ray bursts ( LGRBs ) using the Type I Fundamental Plane , a correlation between the spectral peak energy E _ { p } , the peak luminosity L _ { p } , and the luminosity time T _ { L } ( \equiv E _ { iso } / L _ { p } where E _ { iso } is isotropic energy ) for small Absolute Deviation from Constant Luminosity ( ADCL ) .
The Type I Fundamental Plane of LGRBs is calibrated using 8 LGRBs with redshift z < 1.4 .
To avoid any assumption on the cosmological model , we use the distance modulus of 557 Type Ia supernovae ( SNeIa ) from the Union 2 sample .
This calibrated Type I Fundamental Plane is used to measure the distance moduli to 9 high-redshift LGRBs with the mean error \bar { \sigma } _ { \mu } = 0.31 , which is comparable with that of SNe Ia \bar { \sigma } _ { \mu } = 0.26 where \mu stands for the distance modulus .
The Type I Fundamental Plane is so tight that our distance moduli have very small uncertainties .
From those distance moduli , we obtained the constraint \Omega _ { M } = 0.22 \pm 0.04 for flat \Lambda CDM universe .
Adding 9 LGRBs distance moduli ( z > 1.4 ) to 557 SNeIa distance moduli ( z < 1.4 ) significantly improves the constraint for non-flat \Lambda CDM universe from ( \Omega _ { M } , \Omega _ { \Lambda } ) = ( 0.29 \pm 0.10 , 0.76 \pm 0.13 ) for SNeIa only to ( \Omega _ { M } , \Omega _ { \Lambda } ) = ( 0.23 \pm 0.06 , 0.68 \pm 0.08 ) for SNeIa and 9 LGRBs .