Conformal invariance of scalar perturbations in inflation

TL;DR

This paper demonstrates that scalar perturbations in certain inflationary models exhibit full conformal invariance, independent of dynamics, with specific fixed forms for correlation functions, providing a new way to test inflation models.

Contribution

It establishes the conformal invariance of scalar perturbations in inflationary models with decoupled scalars and derives explicit forms of correlation functions based on this invariance.

Findings

Scalar perturbations are fully conformally invariant in certain inflation models.

3-point function is fixed by two constants due to conformal symmetry.

4-point function depends on two parameters, fewer than in non-conformal cases.

Abstract

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.