The correlation between the peak photon energy of the internal spectrum Ep,i
and isotropic equivalent radiated energy Eiso (the Amati relation) is explored
in a scalar field model of dark energy. Using an updated data set of 109 high
redshift GRBs, we show that the correlation parameters only weakly depend on
the cosmological model. Once the parameters of Amati relation have been
determined we use this relation to construct a fiducial GRBs Hubble diagram
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 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 GRBs Hubble diagram, which we call the calibrated GRBs
HD. We also compare the fiducial and calibrated GRBs 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 GRBs datasets 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 GRBs HD. We finally
investigate the implications of this approach for the high redshift
cosmography.Comment: 18 figures and 1 table, accepted for publication on MNRA
