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Soft limits in holographic cosmology

Paul McFadden

TL;DR

The paper provides a non-perturbative holographic derivation of the inflationary consistency relations by deriving soft theorems for stress-tensor correlators in the dual QFT from the diffeomorphism Ward identities and then mapping these results to cosmological correlators via holographic formulae. By carefully defining the soft limit and projecting onto a helicity basis, it obtains leading and subleading squeezed-limit behaviour for all 3-point functions of curvature perturbations and gravitons, valid for both weakly and strongly coupled dual theories. The authors verify the predictions in slow-roll inflation and in strongly coupled holographic cosmologies, reinforcing the universality of the consistency relations and demonstrating the method's generality. The approach opens avenues to extend these results to higher-point functions and more exotic cosmologies, while highlighting the crucial role of semilocal contact terms in obtaining correct soft behaviour.

Abstract

We study the soft limits of cosmological correlators from a holographic perspective, showing how the inflationary consistency relations arise from the diffeomorphism invariance of the dual quantum field theory. Starting from the corresponding Ward identity, by taking moments we derive the leading and subleading behaviour of the stress tensor 3-point function in the limit as one momentum vanishes. These results are non-perturbative and valid in quantum field theories of a very general nature. Exploiting the known mapping of correlators in the dual quantum field theory to those of the cosmology, we then obtain the leading and subleading soft behaviour of all cosmological 3-point correlators of curvature perturbations and gravitons. Our results thus provide a holographic derivation of all leading and subleading consistency relations for cosmological 3-point functions, and our method is easily generalised. We verify our results explicitly for slow-roll inflation and for strongly coupled holographic cosmologies with a perturbative dual description.

Soft limits in holographic cosmology

TL;DR

The paper provides a non-perturbative holographic derivation of the inflationary consistency relations by deriving soft theorems for stress-tensor correlators in the dual QFT from the diffeomorphism Ward identities and then mapping these results to cosmological correlators via holographic formulae. By carefully defining the soft limit and projecting onto a helicity basis, it obtains leading and subleading squeezed-limit behaviour for all 3-point functions of curvature perturbations and gravitons, valid for both weakly and strongly coupled dual theories. The authors verify the predictions in slow-roll inflation and in strongly coupled holographic cosmologies, reinforcing the universality of the consistency relations and demonstrating the method's generality. The approach opens avenues to extend these results to higher-point functions and more exotic cosmologies, while highlighting the crucial role of semilocal contact terms in obtaining correct soft behaviour.

Abstract

We study the soft limits of cosmological correlators from a holographic perspective, showing how the inflationary consistency relations arise from the diffeomorphism invariance of the dual quantum field theory. Starting from the corresponding Ward identity, by taking moments we derive the leading and subleading behaviour of the stress tensor 3-point function in the limit as one momentum vanishes. These results are non-perturbative and valid in quantum field theories of a very general nature. Exploiting the known mapping of correlators in the dual quantum field theory to those of the cosmology, we then obtain the leading and subleading soft behaviour of all cosmological 3-point correlators of curvature perturbations and gravitons. Our results thus provide a holographic derivation of all leading and subleading consistency relations for cosmological 3-point functions, and our method is easily generalised. We verify our results explicitly for slow-roll inflation and for strongly coupled holographic cosmologies with a perturbative dual description.

Paper Structure

This paper contains 22 sections, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Soft theorems for stress tensor correlators
  3. Defining the soft limit
  4. The diffeomorphism Ward identity
  5. Moments of the diffeomorphism Ward identity
  6. First moment
  7. Second moment
  8. Converting to a helicity basis
  9. Soft limits in holographic cosmology
  10. Holographic formulae
  11. Cosmological consistency relations
  12. Explicit tests
  13. Slow-roll inflation
  14. Strongly coupled holographic cosmologies
  15. Discussion
  16. ...and 7 more sections

Figures (2)

  • Figure 1: Due to momentum conservation the momenta in a 3-point function form a triangle. In the squeezed limit we send the magnitude of $\vec{q}_1$ to zero while holding its direction fixed. The remaining vectors $\vec{q}_2$ and $\vec{q}_3$ are then given in terms of $\vec{q}_1$ and the fixed vector $\vec{q}$ as shown.
  • Figure 2: Given arbitrary vectors $\vec{q}$ and $\delta\vec{q}$ we can orient our coordinates such that \ref{['orientation']} holds.