  Inflation is often described through the dynamics of a scalar field,
slow-rolling in a suitable potential. Ultimately, this inflaton must be
identified as the expectation value of a quantum field, evolving in a quantum
effective potential. The shape of this potential is determined by the
underlying tree-level potential, dressed by quantum corrections from the scalar
field itself and the metric perturbations. Following [1], we compute the
effective scalar field equations and the corrected Friedmann equations to
quadratic order in both scalar field, scalar metric and tensor perturbations.
We identify the quantum corrections from different sources at leading order in
slow-roll, and estimate their magnitude in benchmark models of inflation. We
comment on the implications of non-minimal coupling to gravity in this context.
