The Amati relation in the fireshell model

Context. The cosmological origin of gamma-ray bursts (GRBs) has been firmly established, with redshifts up to z = 6.29. They are possible candidates for use as distance indicators for testing cosmological models in a redshift range hardly achievable by other cosmological probes. Asserting the validity of the empirical relations among GRB observables is now crucial for their calibration. Aims. Motivated by the relation proposed by Amati and collaborators, we look within the fireshell model for a relation between the peak energy E-p of the vF(v). total time-integrated spectrum of the afterglow and the total energy of the afterglow E-aft, which in our model encompasses and extends the prompt emission. Methods. The fit within the fireshell model, as for the canonical GRB 050315, uses the complete arrival time coverage given by the Swift satellite. It is performed simultaneously, self-consistently, and recursively in the four BAT energy bands (15-25 keV, 25-50 keV, 50-100 keV, and 100-150 keV), as well as in the XRT one (0.2-10 keV). It uniquely determines the two free parameters characterizing the GRB source, the total energy E-tot(e +/-) of the e(+/-) plasma and its baryon loading B, as well as the effective CircumBurst Medium (CBM) distribution. We can then build two sets of gedanken GRBs varying the total energy of the electron-positron plasma E-tot(e +/-) and keeping the same baryon loading B of GRB 050315. The first set assumes the one obtained in the fit of GRB 050315 for the effective CBM density. The second set assumes instead a constant CBM density equal to the average value of the GRB 050315 prompt phase. Results. For the first set of gedanken GRBs we find a relation E-p alpha(E-aft)(a), with a = 0.45 +/- 0.01, whose slope strictly agrees with the Amati one. Such a relation, in the limit B -> 10(-2), coincides with the Amati one. Instead, no correlation is found in the second set of gedanken GRBs. Conclusions. Our analysis excludes the proper GRB (P-GRB) from the prompt emission, extends all the way to the latest afterglow phases, and is independent of the assumed cosmological model, since all gedanken GRBs are at the same redshift. The Amati relation, on the other hand, includes the P-GRB, focuses only on the prompt emission, being therefore influenced by the instrumental threshold that fixes the end of the prompt emission, and depends on the assumed cosmology. This might explain the intrinsic scatter observed in the Amati relation.