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

Ashish Shukla, Sandip P. Trivedi, V. Vishal

TL;DR

This work analyzes how conformal Ward identities constrain inflationary correlators in momentum space, proving their validity for generalized single-field models in the probe limit. It then investigates α-vacua, showing α-vacua preserve conformal invariance for a probe scalar but lead to back-reaction issues in full inflation, where the Maldacena consistency condition can fail. A bulk, AdS/CFT-inspired calculation in Euclidean AdS recovers the standard BD three-point function for scalar perturbations, offering a complementary perspective on the dS/CFT-like structure of inflationary correlators. The results underscore the role of conformal symmetry as a model-independent constraint on inflationary physics and highlight subtle back-reaction effects when non-Bunch-Davies initial states are considered.

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 $α$-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 $α$-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.

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

TL;DR

This work analyzes how conformal Ward identities constrain inflationary correlators in momentum space, proving their validity for generalized single-field models in the probe limit. It then investigates α-vacua, showing α-vacua preserve conformal invariance for a probe scalar but lead to back-reaction issues in full inflation, where the Maldacena consistency condition can fail. A bulk, AdS/CFT-inspired calculation in Euclidean AdS recovers the standard BD three-point function for scalar perturbations, offering a complementary perspective on the dS/CFT-like structure of inflationary correlators. The results underscore the role of conformal symmetry as a model-independent constraint on inflationary physics and highlight subtle back-reaction effects when non-Bunch-Davies initial states are considered.

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 -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 -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.

Paper Structure

This paper contains 28 sections, 230 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basic Properties of de Sitter Space
  3. Conformal Invariance and General Single Field Models of Inflation
  4. The Probe Approximation: A Massless Interacting Scalar Field in de Sitter Space
  5. Defining the $\alpha$-vacua
  6. The Three Point Function in the $|\alpha\rangle_{\mathrm{I}}$ Vacua
  7. Computing the Three Point Function
  8. Checking for Conformal Invariance
  9. Correlation Functions with $\alpha$-vacua in Inflation
  10. Basic Setup
  11. The Two Point Function
  12. The Three Point Function
  13. Squeezed Limit and the Consistency Condition
  14. Bulk Calculation of the Scalar Three Point Function in Inflation
  15. Computing the Coefficient Function $\langle OOO \rangle$
  16. ...and 13 more sections

Figures (3)

  • Figure 1: The three point function in de Sitter space, with one leg being the time-derivative of the background $\bar{\phi}$.
  • Figure 2: The qualitative three point bulk interaction vertex between two scalars and a graviton.
  • Figure 3: Feynman-Witten diagrams for the s, t, and u channel processes contributing to the calculation of the EAdS partition function. $z=0$ is the $\text{EAdS}$ boundary. The wavy line denotes the bulk-to-bulk graviton propagator, the solid lines represent the bulk-to-boundary propagators for the scalar field $\delta\phi$, and the dashed line represents the background. The exchanged graviton carries momentum $\boldsymbol{k}$.