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In inflationary models, the approximate flatness of the power spectrum of primordial scalar perturbations is a consequence of the approximate de~Sitter symmetry.
We discuss an alternative possibility that the relevant symmetry is (approximate) conformal invariance. We illustrate the conformal scenario by toy model examples, and then discuss its features arising at non-linear level.
These are specific to the whole class of conformal models and give rise to potentially observable properties of the scalar perturbations, such as statistical anisotropy and particular form of non-Gaussianity.