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Double Soft Limits of Cosmological Correlations

Mehrdad Mirbabayi, Matias Zaldarriaga

TL;DR

This work derives and tests double-soft identities for cosmological correlators, linking two long-wavelength modes to lower-order correlators through background-field and wavefunction-symmetry methods. It extends Weinberg's adiabatic-mode construction to second order, demonstrating uniqueness and constructing explicit second-order transformations, with an infinite set of single-soft identities and a finite set of double-soft identities. The paper provides concrete examples (two uniform modes, uniform+gradient, two gradients) and multiple consistency checks, and discusses observational implications for single-field inflation via squeezed and double-squeezed limits. Overall, it illuminates how spontaneously broken spacetime symmetries constrain cosmological correlations and offers tools for testing inflationary models through precise soft-limit relations.

Abstract

Correlation functions of two long-wavelength modes with several short-wavelength modes are shown to be related to lower order correlation functions, using the background wave method, and independently, by exploiting symmetries of the wavefunction of the Universe. These soft identities follow from the non-linear extension of the adiabatic modes of Weinberg, and their generalization by Hinterbichler et. al. The extension is shown to be unique. A few checks of the identities are presented.

Double Soft Limits of Cosmological Correlations

TL;DR

This work derives and tests double-soft identities for cosmological correlators, linking two long-wavelength modes to lower-order correlators through background-field and wavefunction-symmetry methods. It extends Weinberg's adiabatic-mode construction to second order, demonstrating uniqueness and constructing explicit second-order transformations, with an infinite set of single-soft identities and a finite set of double-soft identities. The paper provides concrete examples (two uniform modes, uniform+gradient, two gradients) and multiple consistency checks, and discusses observational implications for single-field inflation via squeezed and double-squeezed limits. Overall, it illuminates how spontaneously broken spacetime symmetries constrain cosmological correlations and offers tools for testing inflationary models through precise soft-limit relations.

Abstract

Correlation functions of two long-wavelength modes with several short-wavelength modes are shown to be related to lower order correlation functions, using the background wave method, and independently, by exploiting symmetries of the wavefunction of the Universe. These soft identities follow from the non-linear extension of the adiabatic modes of Weinberg, and their generalization by Hinterbichler et. al. The extension is shown to be unique. A few checks of the identities are presented.

Paper Structure

This paper contains 21 sections, 127 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Adiabatic modes at linear order
  3. Classification of linear adiabatic modes
  4. Single soft consistency conditions
  5. Cauchy formulation of inflationary correlation functions
  6. Single soft limit
  7. The infinite set of single soft consistency conditions
  8. A comment on counting adiabatic modes
  9. Double soft limit
  10. Adiabatic modes at second order
  11. The first three identities
  12. Checks of the double soft identities
  13. Conclusions
  14. The time-dependence of adiabatic modes
  15. Including tensor modes
  16. ...and 6 more sections

Figures (6)

  • Figure 1: A tree diagram.
  • Figure 2: Single soft diagrams with the soft mode (a) evolving freely, and (b) arising from a cubic interaction. Except for the one soft line entering the gray blob in (a), it consists entirely of hard modes at tree-level.
  • Figure 3: The double soft diagram corresponding to the evolution of the hard modes in the background of two soft modes.
  • Figure 4: Double soft diagrams with a cubic interactions among the soft modes. The hard modes evolve in the background of a single soft mode $q_3 = |{\boldsymbol q}_1+{\boldsymbol q}_2|$.
  • Figure 5: There are two ways of adding a single soft line ${\boldsymbol q}$: to vertices, and to propagators.
  • ...and 1 more figures