Building on Nakar & Piran ’ s analysis of the Amati relation relating gamma-ray burst peak energies E _ { p } and isotropic energies E _ { iso } , we test the consistency of a large sample of BATSE bursts with the Amati and Ghirlanda ( which relates peak energies and actual gamma-ray energies E _ { \gamma } ) relations .
Each of these relations can be expressed as a ratio of the different energies that is a function of redshift ( for both the Amati and Ghirlanda relations ) and beaming fraction f _ { B } ( for the Ghirlanda relation ) .
The most rigorous test , which allows bursts to be at any redshift , corroborates Nakar & Piran ’ s result—88 % of the BATSE bursts are inconsistent with the Amati relation—while only 1.6 % of the bursts are inconsistent with the Ghirlanda relation if f _ { B } = 1 .
Even when we allow for a real dispersion in the Amati relation we find an inconsistency .
Modelling the redshift distribution results in an energy ratio distribution for the Amati relation that is shifted by an order of magnitude relative to the observed distribution ; any sub-population satisfying the Amati relation can comprise at most \sim 18 % of our burst sample .
A similar analysis of the Ghirlanda relation depends sensitively on the beaming fraction distribution for small values of f _ { B } ; for reasonable estimates of this distribution about a third of the burst sample is inconsistent with the Ghirlanda relation .
Our results indicate that these relations are an artifact of the selection effects of the burst sample in which they were found ; these selection effects may favor sub-populations for which these relations are valid .