  We used a sample of GRBs detected by Fermi and Swift to reanalyze the
correlation discovered by Amati et al. (2002) between Epi, the peak energy of
the prompt GRB emission, and Eiso, the energy released by the GRB assuming
isotropic emission. This correlation has been disputed by various authors, and
our aim is to assess whether it is an intrinsic GRB property or the consequence
of selection effects. We constructed a sample of Fermi GRBs with homogeneous
selection criteria, and we studied their distribution in the Epi-Eiso plane.
Our sample is made of 43 GRBs with a redshift and 243 GRBs without a redshift.
We show that GRBs with a redshift follow a broad Epi-Eiso relation, while GRBs
without a redshift show several outliers. We use these samples to discuss the
impact of selection effects associated with GRB detection and with redshift
measurement. We find that the Epi-Eiso relation is partly due to intrinsic GRB
properties and partly due to selection effects. The lower right boundary of the
Epi-Eiso relation stems from a true lack of luminous GRBs with low Epi. In
contrast, the upper left boundary is attributed to selection effects acting
against the detection GRBs with low Eiso and large Epi that appear to have a
lower signal-to-noise ratio. In addition, we demonstrate that GRBs with and
without a redshift follow different distributions in the Epi-Eiso plane. GRBs
with a redshift are concentrated near the lower right boundary of the Epi-Eiso
relation. This suggests that it is easier to measure the redshift of GRBs close
to the lower Epi-Eiso boundary. In this context, we attribute the controversy
about the reality of the Amati relation to the complex nature of this relation
resulting from the combination of a true physical boundary and biases favoring
the detection and the measurement of the redshift of GRBs located close to this
boundary.
