We consider two empirical relations using data only from the prompt emission of Gamma-Ray Bursts ( GRBs ) , the peak energy ( E _ { p } ) - peak luminosity ( L _ { p } ) relation ( so called Yonetoku relation ) and the E _ { p } -isotropic energy ( E _ { iso } ) relation ( so called Amati relation ) .
Both relations show high correlation degree , but they also have larger dispersion around the best fit function rather than the statistical uncertainty .
Then we first investigated the correlation between the residuals of L _ { p } and E _ { iso } from the best function , and found that a partial linear correlation degree is quite small of \rho _ { L _ { p } ~ { } E _ { iso } \cdot E _ { p } } = 0.379 .
This fact indicates that some kinds of independence may exist between Amati and Yonetoku relation even if they are characterized by the same physical quantity E _ { p } , and similar quantities L _ { p } and E _ { iso } which mean the brightness of the prompt emission .
Therefore we may have to recognize two relations as the independent distance indicators .
From this point of view , we compare constraints on cosmological parameters , \Omega _ { m } and \Omega _ { \Lambda } , using the Yonetoku and the Amati relation calibrated by low-redshift GRBs with z < 1.8 .
We found that they are different in 1- \sigma level , although they are still consistent in 2- \sigma level .
In this paper , we introduce a luminosity time T _ { L } defined by T _ { L } \equiv E _ { iso } / L _ { p } as a hidden parameter to correct the large dispersion of the Yonetoku relation .
A new relation is described as ( L _ { p } / { 10 ^ { 52 } ~ { } { erg~ { } s ^ { -1 } } } ) = 10 ^ { -3.87 \pm 0.19 } ( E _ { p } / { keV } ) ^ { 1. % 82 \pm 0.08 } ( T _ { L } / { s } ) ^ { -0.34 \pm 0.09 } .
We succeeded in reducing the systematic error about 40 % level , and might be regarded as ” Fundamental plane ” of GRBs .
We show a possible radiation model for this new relation .
Finally , applying the new relation to high-redshift GRBs with 1.8 < z < 5.6 , we obtain ( \Omega _ { m } , \Omega _ { \Lambda } ) = ( 0.17 ^ { +0.15 } _ { -0.08 } , 1.21 ^ { +0.07 } _ { -0.61 } ) , which is consistent with the concordance cosmological model .