= 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.
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