Conformal Invariance and the Four Point Scalar Correlator in Slow-Roll Inflation

TL;DR

This work computes the four-point scalar trispectrum in the canonical slow-roll inflation model by exploiting the wave function of the universe and AdS/CFT-inspired techniques. By performing a careful gauge fixing and employing conformal Ward identities, it resolves subtleties unique to de Sitter space and demonstrates that the trispectrum is consistent with conformal invariance, the flat-space limit, and operator-product expansion expectations. The final result, which agrees with prior calculations (Seery 2008), decomposes into a conformal-invariance–determined ET term and a model-dependent CF term, with the ET piece fixed by $\langle O O T_{ij} \rangle$ and universally constrained by symmetry. The analysis showcases the power of holographic-style methods in cosmology and provides nontrivial tests of conformal symmetry in inflationary correlators, while highlighting the subtle gauge issues intrinsic to de Sitter space.

Abstract

We calculate the four point correlation function for scalar perturbations in the canonical model of slow-roll inflation. We work in the leading slow-roll approximation where the calculation can be done in de Sitter space. Our calculation uses techniques drawn from the AdS/CFT correspondence to find the wave function at late times and then calculate the four point function from it. The answer we get agrees with an earlier result in the literature, obtained using different methods. Our analysis reveals a subtlety with regard to the Ward identities for conformal invariance, which arises in de Sitter space and has no analogue in AdS space. This subtlety arises because in de Sitter space the metric at late times is a genuine degree of freedom, and hence to calculate correlation functions from the wave function of the Universe at late times, one must fix gauge completely. The resulting correlators are then invariant under a conformal transformation accompanied by a compensating coordinate transformation which restores the gauge.

Figures (2)

• Figure 1: Three different contribution corresponding to S,T and U-channel to the scalar four point correlator are shown in the three figures. The brown solid vertical line represents the 3-dimensional boundary of $AdS_4$ at $z=0$, the black solid lines are boundary to bulk scalar propagators whereas the green wavy lines are graviton propagators in the bulk.
• Figure 2: S-channel Feynman diagram for scattering of minimally coupled scalars