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It is often said that inflation with a single slow rolling field minimally coupled to gravity produces, on super Hubble scales, a spectrum of density perturbations which is (almost) Gaussian. I don't quite understand the meaning of this, I don't see a Gaussian function (something of the form $e^{-\alpha x^2}$) appearing anywhere. For instance, here https://ned.ipac.caltech.edu/level5/Sept02/Kinney/Kinney4_7.html they explicitly calculate the power spectrum and obtain a constant value (it doesn't even depend on the wavenumber $k$). Where does the Gaussian comes from?

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