Consistency condition for inflation from (broken) conformal symmetry

TL;DR

The paper develops a holographic, symmetry-based framework to derive inflationary constraints on the bispectrum by mapping late-time correlators to a weakly broken 2+1D Euclidean CFT via Ward identities. It shows that while conformal perturbation theory informs the structure, it must be supplemented by the inflationary evolution (Callan-Symanzik dynamics) to reproduce the standard squeezed-limit consistency relation for the three-point function. The key result is that the inflationary bispectrum in the squeezed limit emerges from a combination of two- and three-point conformal correlators, with the evolution equation ensuring the proper cancellation of contributions and yielding the familiar Maldacena-type consistency condition. The work highlights subtle distinctions between conformal perturbation theory and slow-roll, and outlines a path toward deriving the full bispectrum and higher-point functions in this symmetry-guided, holographic language.

Abstract

We investigate the symmetry constraints on the bispectrum, i.e. the three-point correlation function of primordial density fluctuations, in slow-roll inflation. It follows from the defining property of slow-roll inflation that primordial correlation functions inherit most of their structure from weakly broken de Sitter symmetries. Using holographic techniques borrowed from the AdS/CFT correspondence, the symmetry constraints on the bispectrum can be mapped to a set of stress-tensor Ward identities in a weakly broken 2+1-dimensional Euclidean CFT. We construct the consistency condition from these Ward identities using conformal perturbation theory. This requires a second order Ward identity and the use of the evolution equation. Our result also illustrates a subtle difference between conformal perturbation theory and the slow roll expansion.

Figures (2)

• Figure 1: The cosmic evolution can be seen as a reversed renormalization group flow, from the IR fixed point of the dual theory to the UV fixed point of the dual theory. Inflation occurs at a certain intermediate stage during the renormalization group flow. As is usual for inflation, a pivot point along the flow is chosen around which the slow-roll expansion can be studied. We observe the effects of inflation at late times, corresponding to the UV fixed point of the renormalization group flow.
• Figure 2: The inflationary phase at an intermediate point in the renormalization group flow may be approximated by a conformal fixed point. The dashed line indicates a marginal renormalization group flow from one UV theory to another, for an operator with exactly marginal dimension $\Delta=3$. The validity of the slow-roll approximation suggests that expressions in the quasi-fixed point can be approximated by a $(\Delta-3)$-expansion.