CMB from CFT

TL;DR

The authors leverage the conformal symmetry of de Sitter space to derive a model-independent, highly constrained form for the three-point function involving two scalar and one tensor perturbation during single-field inflation. By working with the late-time wavefunction and employing both direct momentum-space and spinor-helicity techniques, they derive a set of differential constraints whose unique solution fixes the momentum dependence and normalization of ⟨ζ ζ γ⟩ in terms of the two-point functions. The result provides a sharp, symmetry-based prediction for the bispectrum that can test whether special conformal transformations were preserved during inflation, with corrections only arising from slow-roll breaking or other symmetry-violating effects. This connects cosmological perturbations to 3d CFT correlators and offers a concrete, testable imprint of conformal invariance on the CMB and large-scale structure.

Abstract

During inflation, spacetime is approximately described by de Sitter space which is conformally invariant with the symmetry group SO(1,4). This symmetry can significantly constrain the quantum perturbations which arise in the inflationary epoch. We consider a general situation of single field inflation and show that the three point function involving two scalar modes and one tensor mode is uniquely determined, up to small corrections, by the conformal symmetries. Special conformal transformations play an important role in our analysis. Our result applies only to models where the inflaton sector also approximately preserves the full conformal group and shows that this three point function is a good way to test if special conformal invariance was preserved during inflation.