Constraints from Conformal Symmetry on the Three Point Scalar Correlator in Inflation

TL;DR

This work investigates how approximate conformal symmetry during inflation constrains the scalar bispectrum by deriving Ward identities that relate ⟨OOO⟩ to ⟨OOOO⟩ via the wave function of the universe. Using a two-gauge, reparametrization-invariant framework, the authors show that in the slow-roll regime the three-point function is generically suppressed, with magnitude set by ${\dot{\bar{φ}}/H}$ and typically yielding $f_{NL} \sim O(( {\dot{\bar{φ}}/H})^2)$. The Ward identities fix ⟨OOO⟩ in terms of ⟨OOOO⟩ up to a homogeneous piece S_h corresponding to a dim-3 primary, whose normalization is expected to be small in slow-roll. Consequently, conformal symmetry provides only a weak constraint on the three-point function, and a significant observational deviation would challenge models with approximate conformal invariance during inflation, potentially hinting at additional operators or higher-spin fields. The paper also outlines a formal prescription to obtain ⟨OOO⟩ from ⟨OOOO⟩ and discusses the implications for interpreting non-Gaussianity in the cosmic microwave background.

Abstract

Using symmetry considerations, we derive Ward identities which relate the three point function of scalar perturbations produced during inflation to the scalar four point function, in a particular limit. The derivation assumes approximate conformal invariance, and the conditions for the slow roll approximation, but is otherwise model independent. The Ward identities allow us to deduce that the three point function must be suppressed in general, being of the same order of magnitude as in the slow roll model. They also fix the three point function in terms of the four point function, upto one constant which we argue is generically suppressed. Our approach is based on analyzing the wave function of the universe, and the Ward identities arise by imposing the requirements of spatial and time reparametrization invariance on it.