We investigate the well-measured spectral and energetic properties of 20 gamma-ray bursts ( GRBs ) in their cosmological rest frames .
We find a tight relation between the isotropic-equivalent \gamma -ray energy E _ { iso } , the local peak energy E ^ { \prime } _ { p } of the \nu F _ { \nu } spectrum , and the local break time t ^ { \prime } _ { b } of the GRB afterglow light curve , which reads E _ { iso } t ^ { \prime } _ { b } \propto E ^ { \prime 1.95 \pm 0.08 } _ { p } ( \chi ^ { 2 } _ { \nu } = 1.40 ; \Omega _ { M } = 0.27 , \Omega _ { \Lambda } = 0.73 ) .
Such a power-law relation can be understood via the high-energy radiation processes for the GRB prompt emission accompanying the beaming effects .
We then consider this relation as an intrinsic one for the observed GRB sample , and obtain a constraint on the mass density \Omega _ { M } = 0.24 ^ { +0.16 } _ { -0.12 } ( 1 \sigma ) for a flat \Lambda CDM universe , and a \chi ^ { 2 } _ { dof } = 1.33 for \Omega _ { M } \approx 0.3 and \Omega _ { \Lambda } \approx 0.7 .
Ongoing GRB observations in the Swift era are expected to confirm this relation and make its cosmological utility progress much .