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The structure of correlation functions in single field inflation

Sarah Shandera

TL;DR

The paper investigates when single-field inflation with higher-derivative interactions yields perturbative, Gaussian-like fluctuations with a well-defined hierarchy of correlation functions. Using a general $P(X,\phi)$ framework and an EFT viewpoint, it shows that a perturbative regime characterized by $X/M^4\lesssim1$ and a small sound speed $c_s$ leads to a scalable, hierarchical structure of $n$-point functions, with higher-order cumulants suppressed by powers of $\mathcal{P}_\zeta^{1/2}/c_s^2$. The analysis extends to six-derivative terms, establishing that unless perturbativity fails, the dominant contributions follow from a finite set of operators, and that the DBI action provides a concrete, symmetry-protected example of this behavior. The findings clarify how derivative-driven non-Gaussianity can be large yet still amenable to perturbative, Gaussian-based expansions, and they outline the conditions under which this hierarchical picture holds in realistic models. This has implications for interpreting higher-order statistics in CMB and LSS data and for connecting observed non-Gaussianity to the UV structure of inflation.

Abstract

Many statistics available to constrain non-Gaussianity from inflation are simplest to use under the assumption that the curvature correlation functions are hierarchical. That is, if the n-point function is proportional to the (n-1) power of the two-point function amplitude and the fluctuations are small, the probability distribution can be approximated by expanding around a Gaussian in moments. However, single-field inflation with higher derivative interactions has a second small number, the sound speed, that appears in the problem when non-Gaussianity is significant and changes the scaling of correlation functions. Here we examine the structure of correlation functions in the most general single scalar field action with higher derivatives, formalizing the conditions under which the fluctuations can be expanded around a Gaussian distribution. We comment about the special case of the Dirac-Born-Infeld action.

The structure of correlation functions in single field inflation

TL;DR

The paper investigates when single-field inflation with higher-derivative interactions yields perturbative, Gaussian-like fluctuations with a well-defined hierarchy of correlation functions. Using a general framework and an EFT viewpoint, it shows that a perturbative regime characterized by and a small sound speed leads to a scalable, hierarchical structure of -point functions, with higher-order cumulants suppressed by powers of . The analysis extends to six-derivative terms, establishing that unless perturbativity fails, the dominant contributions follow from a finite set of operators, and that the DBI action provides a concrete, symmetry-protected example of this behavior. The findings clarify how derivative-driven non-Gaussianity can be large yet still amenable to perturbative, Gaussian-based expansions, and they outline the conditions under which this hierarchical picture holds in realistic models. This has implications for interpreting higher-order statistics in CMB and LSS data and for connecting observed non-Gaussianity to the UV structure of inflation.

Abstract

Many statistics available to constrain non-Gaussianity from inflation are simplest to use under the assumption that the curvature correlation functions are hierarchical. That is, if the n-point function is proportional to the (n-1) power of the two-point function amplitude and the fluctuations are small, the probability distribution can be approximated by expanding around a Gaussian in moments. However, single-field inflation with higher derivative interactions has a second small number, the sound speed, that appears in the problem when non-Gaussianity is significant and changes the scaling of correlation functions. Here we examine the structure of correlation functions in the most general single scalar field action with higher derivatives, formalizing the conditions under which the fluctuations can be expanded around a Gaussian distribution. We comment about the special case of the Dirac-Born-Infeld action.

Paper Structure

This paper contains 17 sections, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of inflation from $P(X,\phi)$
  3. Inflationary Fluctuations
  4. Some notation for cumulants
  5. Hierarchical structure and the local model
  6. Fluctuations and effective field theory
  7. Structure of tree-level correlation functions for a perfect fluid
  8. Expanding the most general action with higher derivatives
  9. Expansion at six derivatives
  10. Estimating the higher derivative contributions
  11. Example: the DBI action
  12. The lowest order brane action in an AdS background
  13. Higher order corrections
  14. Conclusions
  15. Calculating correlation functions
  16. ...and 2 more sections

Figures (1)

  • Figure 1: For a brane moving in an AdS background geometry, the sound speed continually decreases until the geometry changes near $r_{min}$. If $c_s$ gets too small ($c^{-2}_s>\mathcal{P}^{-1/2}_{\zeta}\approx10^{9/2}$), the perturbative calculation of correlation functions based on the lowest order brane action is no longer valid.