Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 310 / 4 / c

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.