Conformal-time quadratic action for an inflaton perturbation ($S^{(2)}=\frac{1}{2}\int d\tau d^{3}x\{(f^{\prime }{)}^2-(\nabla f{)}^2+(a^{\prime \prime }/a-a^2{V}^{\prime \prime })f^2\}$)

Neglecting metric perturbations, the rescaling $f=a\delta \varphi$ removes the expansion friction from the scalar perturbation action after a boundary term is discarded. Its canonical momentum is $f^{\prime }$, and each real Fourier oscillator has time-dependent squared frequency $k^2+a^2{V}^{\prime \prime }-a^{\prime \prime }/a$. The oscillator can become inverted on superhorizon scales; the canonical commutation relations still apply.