  In this work, we investigate the physical aspects of the inflection-point
inflation scenario and assess its observational viability in light of current
Cosmic Microwave Background (CMB) data. The model we consider encapsulates the
inflaton with a pseudo-scalar (the dark matter candidate) in a complex neutral
scalar singlet. The cosmological constraints on the parameters of inflation
derived at a high energy scale are translated to a low energy scale by running
these parameters. Ensuring the entire Lagrangian to be invariant under a $Z_3$
symmetry with the adequate transformation of the fields, the imaginary part of
the singlet decouples from the other scalars of the model. We then investigate
if the observational viability of inflation is also compatible with this
pseudo-scalar being the dark matter component.We show that the CMB constraints
on the inflationary parameters assure that the pseudo-scalar is stable and
provides the correct relic dark matter abundance only when the pseudo-scalar is
thermally produced.
