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The correlation between the peak photon energy of the internal spectrum  E  p, i  and isotropic equivalent radiated energy  E  iso  (the  Amati relation ) is explored in a scalar field model of dark energy. Using an updated data set of 109 high‐redshift gamma‐ray bursts (GRBs), we show that the correlation parameters only weakly depend on the cosmological model. Once the parameters of the Amati relation have been determined, we use this relation to construct a  fiducial  GRB Hubble diagram (HD) that extends up to redshifts ∼8. Moreover, we apply a local regression technique to estimate, in a model‐independent way, the distance modulus from the recently updated Union Type Ia supernova (SNIa) sample, containing 557 SNIa spanning the redshift range of 0.015 ≤ z ≤ 1.55. The derived calibration parameters are used to construct an updated GRB HD, which we call the  calibrated  GRB HD. We also compare the  fiducial  and  calibrated  GRB HDs, which turned out to be fully statistically consistent, thus indicating that they are not affected by any systematic bias induced by the different calibration procedures. This means that the high‐redshift GRBs can be used to test different models of dark energy settling the circularity problem. Furthermore, we investigate possible evolutionary effects that might have important influence on our results. Our analysis indicates that the presently available GRB data sets do not show  statistically unambiguous  evolutionary effect with the cosmological redshift. Finally, we propose another approach to calibrate the GRB relations, by using an approximate luminosity–distance relation, which holds in any cosmological model. We use this calibration of the Amati relation to construct an  empirical approximate  HD, which we compare with the  calibrated  GRB HD. We finally investigate the implications of this approach for the high‐redshift  cosmography .

Standardizing the gamma‐ray bursts with the Amati E p,i – E iso relation: the updated Hubble diagram and implications for cosmography