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At leading order in $|k\tau |$, the inflationary Fourier mode and its derivative areThus the leading parts of $\hat{f}$ and $\hat{\pi }$ contain the same quadrature of the creation and annihilation operators and satisfyTheir commutator consequently vanishes in this leading approximation. The exact nonzero result in part b is carried by the subleading decaying mode. Relative to the rapidly growing anticommutator, its effect is suppressed by powers of $|k\tau |$, which is the squeezing captured by the classicality parameter of a cosmological perturbation. The perturbation can therefore be treated as an effectively classical stochastic field on superhorizon scales even though its exact quantum commutator is unchanged.

Expand the quantized field using the normalized modeThe canonical commutation relation fixes the Wronskian $M_{\mathrm{Pl}}^2{z}^2(u_k{u}_k^{\ast \prime }-u_k^{\ast }{u}_k^{\prime })=i$. Writing $x=k\eta$, the primed correlators areEach unprimed expression includes $(2\pi {)}^3{\delta }^{(3)}(\mathbf{k}+{\mathbf{k}}^{\prime })$. The imaginary part of the mixed correlator givesConsequently the classicality parameter of a cosmological perturbation isIt tends to zero on superhorizon scales, so the perturbation becomes effectively classical even while its amplitude grows as $a^3$. In ordinary slow-roll inflation, $\mathcal{R}$ instead freezes to a constant; its analogous normalized commutator also vanishes, but behaves as $|k\eta |$ for the leading de Sitter mode rather than $|k\eta {|}^3$.