  We express the in-in functional determinant giving the one-loop effective
potential for a scalar field propagating in a cosmological spacetime in terms
of the mode functions specifying the vacuum of the theory and then apply
adiabatic regularization to make this bare potential finite. In this setup, the
adiabatic regularization offers a particular renormalization prescription that
isolates the effects of the cosmic expansion. We apply our findings to
determine the radiative corrections to the classical inflaton potentials in
scalar field inflationary models and also we derive an effective potential for
the superhorizon curvature perturbation \zeta\ encoding its scatterings with
the subhorizon modes. Although the resulting modifications to the cosmological
observables like nongaussianity turn out to be small, they distinctively appear
after horizon crossing.
