  The motion of a pseudo-scalar field $X$ during inflation naturally induces a
significant amplification of the gauge fields to which it is coupled. The
amplified gauge fields can source characteristic scalar and tensor primordial
perturbations. Several phenomenological implications have been discussed in the
cases in which (i) $X$ is the inflation, and (ii) $X$ is a field different from
the inflation, that experiences a temporary speed up during inflation. In this
second case, visible sourced gravitational waves (GW) can be produced at the
CMB scales without affecting the scalar perturbations, even if the scale of
inflation is several orders of magnitude below what is required to produce a
visible vacuum GW signal. Perturbativity considerations can be used to limit
the regime in which these results are under perturbative control. We revised
limits recently claimed for the case (i), and we extend these considerations to
the case (ii). We show that, in both cases, these limits are satisfied by the
applications that generate signals at CMB scales. Applications that generate
gravitational waves and primordial black holes at much smaller scales are at
the limit of the validity of this perturbativity analysis, so we expect those
results to be valid up to possibly order one corrections.
