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Conformal invariance of scalar perturbations in inflation

In inflationary models where the source of scalar perturbations is not the inflaton, but one or more scalars with negligible coupling with the inflaton, the resulting perturbations are not only scale invariant, but fully conformally invariant with conformal dimension close to zero. This is closely related to the fact that correlation functions can only depend on the de Sitter invariant distances. These properties follow from the isometries of the inflationary de Sitter space and are thus completely independent of the dynamics. The 3-point function is fixed in terms of two constants, while the 4-point function is a function of two parameters (instead of five as in the absence of conformal invariance). The conformal invariance of correlators can be directly checked in Fourier space, as we show in an explicit example. A detection of a non-conformal correlation function, for example an equilateral 3-point function, would imply that the source of perturbations is not decoupled from the inflaton.