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Multiple Soft Limits of Cosmological Correlation Functions

Austin Joyce, Justin Khoury, Marko Simonović

TL;DR

The paper addresses how cosmological correlation functions behave when more than one external momentum is taken to be soft. It develops two complementary methods—the background-wave argument and fixed-time 1PI Ward identities—to derive double-soft and multi-soft consistency relations, revealing new ${\cal O}(q^2)$-level constraints beyond the standard single-soft limit. The authors verify these identities in resonant non-Gaussianity and small sound speed models, extend the framework to arbitrarily many soft legs, and discuss implications for large-scale structure. The results offer a robust, model-independent set of null tests for single-field inflation and lay groundwork for incorporating tensor modes and LSS into the same symmetry-based formalism.

Abstract

We derive novel identities satisfied by inflationary correlation functions in the limit where two external momenta are taken to be small. We derive these statements in two ways: using background-wave arguments and as Ward identities following from the fixed-time path integral. Interestingly, these identities allow us to constrain some of the O(q^2) components of the soft limit, in contrast to their single-soft analogues. We provide several nontrivial checks of our identities both in the context of resonant non-Gaussianities and in small sound speed models. Additionally, we extend the relation at lowest order in external momenta to arbitrarily many soft legs, and comment on the many-soft extension at higher orders in the soft momentum. Finally, we consider how higher soft limits lead to identities satisfied by correlation functions in large-scale structure.

Multiple Soft Limits of Cosmological Correlation Functions

TL;DR

The paper addresses how cosmological correlation functions behave when more than one external momentum is taken to be soft. It develops two complementary methods—the background-wave argument and fixed-time 1PI Ward identities—to derive double-soft and multi-soft consistency relations, revealing new -level constraints beyond the standard single-soft limit. The authors verify these identities in resonant non-Gaussianity and small sound speed models, extend the framework to arbitrarily many soft legs, and discuss implications for large-scale structure. The results offer a robust, model-independent set of null tests for single-field inflation and lay groundwork for incorporating tensor modes and LSS into the same symmetry-based formalism.

Abstract

We derive novel identities satisfied by inflationary correlation functions in the limit where two external momenta are taken to be small. We derive these statements in two ways: using background-wave arguments and as Ward identities following from the fixed-time path integral. Interestingly, these identities allow us to constrain some of the O(q^2) components of the soft limit, in contrast to their single-soft analogues. We provide several nontrivial checks of our identities both in the context of resonant non-Gaussianities and in small sound speed models. Additionally, we extend the relation at lowest order in external momenta to arbitrarily many soft legs, and comment on the many-soft extension at higher orders in the soft momentum. Finally, we consider how higher soft limits lead to identities satisfied by correlation functions in large-scale structure.

Paper Structure

This paper contains 18 sections, 122 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conformal symmetries and adiabatic modes
  3. Background-wave derivation
  4. Single-soft relation
  5. Double-soft dilation result
  6. First order in derivatives
  7. Full coordinate transformation at ${\cal O}(q^2)$
  8. Ward identity derivation
  9. A single soft leg
  10. Double dilation
  11. Dilation & SCT
  12. Double SCT
  13. Checks of double-soft identities
  14. Resonant non-Gaussianity
  15. Small speed of sound
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Contributions to the double squeezed limit \ref{['bwave2dil']}. Left: Two soft modes combine and are connected to the hard modes via a soft internal line. Right: Both soft lines come from the same vertex as the hard modes, resulting in a double-dilation on the remaining hard modes.
  • Figure 2: Contributions to the $N$-squeezed limit \ref{['Ndilations']}. Left: Contribution where the all soft modes to combine to factorize the diagram via a soft internal line. Middle: Contribution all but one of the soft modes combine to factorize the diagram, while the last mode comes from the same vertex as the hard mode, causing the remaining hard modes to feel two dilations. Right: Diagram where all soft modes come from the same vertex as the hard lines, causing the correlation function to feel $N$ dilations.