Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 55 / 5 / b / Solution

The inflaton has quantum fluctuations while their wavelengths are shorter than the Hubble radius. Choosing the Bunch-Davies vacuum fixes the short-wavelength positive-frequency modes. During near-de Sitter slow-roll inflation, expansion carries each mode through $k=aH$, leaving a field fluctuation of amplitude approximately $H/(2\pi )$ on spatially flat slices.

The inflaton clock-shift origin of curvature perturbations converts this field displacement into a local time shift of magnitude $|\delta \Phi /\dot{\Phi }|$, and hence into a comoving curvature perturbation of magnitude $H|\delta \Phi /\dot{\Phi }|$. For the single-field adiabatic growing mode on an attractor, the superhorizon conservation of single-field comoving curvature makes this perturbation approximately constant at $k\ll aH$. These vacuum-generated perturbations seed the later density fluctuations.