Symmetry Constraints in Inflation, $\alpha$-vacua, and the Three Point Function

TL;DR

This paper investigates conformal symmetry constraints on inflation models, examines $ ext{alpha}$-vacua and their impact on three-point functions, and discusses the consistency of Ward identities and non-Gaussian features in different vacua.

Contribution

It extends Ward identity analysis to generalized single field inflation models and explores the behavior of three-point functions in $ ext{alpha}$-vacua and Bunch-Davies vacuum.

Findings

Ward identities hold for generalized models in momentum space.

Ward identities are satisfied in $ ext{alpha}$-vacua up to contact terms.

Ward identities are violated in $ ext{alpha}$-vacua for scalar perturbations, likely due to back-reaction issues.

Abstract

The Ward identities for conformal symmetries in single field models of inflation are studied in more detail in momentum space. For a class of generalized single field models, where the inflaton action contains arbitrary powers of the scalar and its first derivative, we find that the Ward identities are valid. We also study a one-parameter family of vacua, called $\alpha$-vacua, which preserve conformal invariance in de Sitter space. We find that the Ward identities, upto contact terms, are met for the three point function of a scalar field in the probe approximation in these vacua. Interestingly, the corresponding non-Gaussian term in the wave function does not satisfy the operator product expansion. For scalar perturbations in inflation, in the $\alpha$-vacua, we find that the Ward identities are not satisfied. We argue that this is because the back-reaction on the metric of the full quantum stress tensor has not been self-consistently incorporated. We also present a calculation, drawing on techniques from the AdS/CFT correspondence, for the three point function of scalar perturbations in inflation in the Bunch-Davies vacuum.