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Conformal consistency relations for single-field inflation

Paolo Creminelli, Jorge Noreña, Marko Simonović

TL;DR

This work extends single-field inflationary consistency relations by incorporating a subleading $1/q^2$ term in the squeezed limit, tying it to the conformal variation under special conformal transformations and placing the curvature perturbation correlators on an $SO(4,1)$-invariant footing with non-linear realizations. It provides a comprehensive derivation in exact de Sitter, extends to gravity and slow-roll, and generalizes to models with non-trivial speed of sound and resonant non-Gaussianities, including tensor-mode extensions and soft internal lines. The results constrain the structure of all $n$-point functions in single-field models, showing that the leading and subleading squeezed-limit behaviors are governed by conformal symmetry, with the ${1/q^2}$ term typically suppressed relative to the ${1/q^3}$ term. These conformal consistency relations offer a robust, symmetry-based framework for connecting squeezed-limit observables across a broad class of inflationary models and have potential implications for interpreting non-Gaussianity measurements in CMB and large-scale structure.

Abstract

We generalize the single-field consistency relations to capture not only the leading term in the squeezed limit---going as 1/q^3, where q is the small wavevector---but also the subleading one, going as 1/q^2. This term, for an (n+1)-point function, is fixed in terms of the variation of the n-point function under a special conformal transformation; this parallels the fact that the 1/q^3 term is related with the scale dependence of the n-point function. For the squeezed limit of the 3-point function, this conformal consistency relation implies that there are no terms going as 1/q^2. We verify that the squeezed limit of the 4-point function is related to the conformal variation of the 3-point function both in the case of canonical slow-roll inflation and in models with reduced speed of sound. In the second case the conformal consistency conditions capture, at the level of observables, the relation among operators induced by the non-linear realization of Lorentz invariance in the Lagrangian. These results mean that, in any single-field model, primordial correlation functions of ζare endowed with an SO(4,1) symmetry, with dilations and special conformal transformations non-linearly realized by ζ. We also verify the conformal consistency relations for any n-point function in models with a modulation of the inflaton potential, where the scale dependence is not negligible. Finally, we generalize (some of) the consistency relations involving tensors and soft internal momenta.

Conformal consistency relations for single-field inflation

TL;DR

This work extends single-field inflationary consistency relations by incorporating a subleading term in the squeezed limit, tying it to the conformal variation under special conformal transformations and placing the curvature perturbation correlators on an -invariant footing with non-linear realizations. It provides a comprehensive derivation in exact de Sitter, extends to gravity and slow-roll, and generalizes to models with non-trivial speed of sound and resonant non-Gaussianities, including tensor-mode extensions and soft internal lines. The results constrain the structure of all -point functions in single-field models, showing that the leading and subleading squeezed-limit behaviors are governed by conformal symmetry, with the term typically suppressed relative to the term. These conformal consistency relations offer a robust, symmetry-based framework for connecting squeezed-limit observables across a broad class of inflationary models and have potential implications for interpreting non-Gaussianity measurements in CMB and large-scale structure.

Abstract

We generalize the single-field consistency relations to capture not only the leading term in the squeezed limit---going as 1/q^3, where q is the small wavevector---but also the subleading one, going as 1/q^2. This term, for an (n+1)-point function, is fixed in terms of the variation of the n-point function under a special conformal transformation; this parallels the fact that the 1/q^3 term is related with the scale dependence of the n-point function. For the squeezed limit of the 3-point function, this conformal consistency relation implies that there are no terms going as 1/q^2. We verify that the squeezed limit of the 4-point function is related to the conformal variation of the 3-point function both in the case of canonical slow-roll inflation and in models with reduced speed of sound. In the second case the conformal consistency conditions capture, at the level of observables, the relation among operators induced by the non-linear realization of Lorentz invariance in the Lagrangian. These results mean that, in any single-field model, primordial correlation functions of ζare endowed with an SO(4,1) symmetry, with dilations and special conformal transformations non-linearly realized by ζ. We also verify the conformal consistency relations for any n-point function in models with a modulation of the inflaton potential, where the scale dependence is not negligible. Finally, we generalize (some of) the consistency relations involving tensors and soft internal momenta.

Paper Structure

This paper contains 16 sections, 108 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. The limit of exact de Sitter
  3. Derivation of the consistency relations in exact de Sitter
  4. Small speed of sound
  5. Including gravity
  6. Adiabatic modes at order $k$
  7. Derivation of the conformal consistency relations including gravity
  8. Conformal consistency relation for slow-roll inflation
  9. Including the deviation from scale invariance
  10. Resonant non-Gaussianities
  11. Consistency relations for tensor modes
  12. Soft internal lines
  13. Conclusions and outlook
  14. Non-linear metric in matter dominance
  15. Consistency relations at the level of the action
  16. ...and 1 more sections

Figures (1)

  • Figure 1: A constant background mode can be removed by a dilation. A background with a constant gradient can be removed by a special conformal transformation.