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Diagrammatic approach to non-Gaussianity from inflation

Christian T. Byrnes, Kazuya Koyama, Misao Sasaki, David Wands

TL;DR

This work develops a comprehensive diagrammatic approach to compute primordial non-Gaussianity from inflation using the δN formalism. It provides explicit real- and Fourier-space rules for drawing and evaluating Feynman-type diagrams at tree and loop levels, and shows how vertex renormalisation absorbs dressed-vertex contributions to yield a simplified, renormalised expansion. The authors apply the method to derive the power spectrum up to two loops and give leading loop corrections to the bispectrum and trispectrum, including when initial field perturbations are Gaussian and when they are non-Gaussian. The framework clarifies the role of large-scale cutoffs and non-Gaussian initial conditions, offering a systematic, scalable toolkit for confronting inflationary models with higher-order observational statistics.

Abstract

We present Feynman type diagrams for calculating the n-point function of the primordial curvature perturbation in terms of scalar field perturbations during inflation. The diagrams can be used to evaluate the corresponding terms in the n-point function at tree level or any required loop level. Rules are presented for drawing the diagrams and writing down the corresponding terms in real space and Fourier space. We show that vertices can be renormalised to automatically account for diagrams with dressed vertices. We apply these rules to calculate the primordial power spectrum up to two loops, the bispectrum including loop corrections, and the trispectrum.

Diagrammatic approach to non-Gaussianity from inflation

TL;DR

This work develops a comprehensive diagrammatic approach to compute primordial non-Gaussianity from inflation using the δN formalism. It provides explicit real- and Fourier-space rules for drawing and evaluating Feynman-type diagrams at tree and loop levels, and shows how vertex renormalisation absorbs dressed-vertex contributions to yield a simplified, renormalised expansion. The authors apply the method to derive the power spectrum up to two loops and give leading loop corrections to the bispectrum and trispectrum, including when initial field perturbations are Gaussian and when they are non-Gaussian. The framework clarifies the role of large-scale cutoffs and non-Gaussian initial conditions, offering a systematic, scalable toolkit for confronting inflationary models with higher-order observational statistics.

Abstract

We present Feynman type diagrams for calculating the n-point function of the primordial curvature perturbation in terms of scalar field perturbations during inflation. The diagrams can be used to evaluate the corresponding terms in the n-point function at tree level or any required loop level. Rules are presented for drawing the diagrams and writing down the corresponding terms in real space and Fourier space. We show that vertices can be renormalised to automatically account for diagrams with dressed vertices. We apply these rules to calculate the primordial power spectrum up to two loops, the bispectrum including loop corrections, and the trispectrum.

Paper Structure

This paper contains 22 sections, 48 equations, 13 figures.

Table of Contents

  1. Introduction
  2. The primordial curvature perturbation
  3. Diagrams for Gaussian field perturbations
  4. Real space diagrams
  5. Disconnected diagrams
  6. Fourier space diagrams
  7. Comments on the diagrams and their rules
  8. Power spectrum
  9. Evaluation of the 1 loop diagrams
  10. Renormalisation
  11. Renormalised real space diagrams
  12. Renormalised Fourier space diagrams
  13. Power spectrum
  14. Bispectrum
  15. Extension to non-Gaussian fields in Fourier space
  16. ...and 7 more sections

Figures (13)

  • Figure 1: The 2-point function at tree (a) and one loop level (b), and the 3- and 4-point functions at tree level, (c) and (d) respectively. All figures in this paper were drawn using JaxoDraw Binosi:2003yf.
  • Figure 2: The terms that are associated with every vertex and propagator.
  • Figure 3: The tree level (a) and 1 loop correction (b) for the power spectrum.
  • Figure 4: The two loop terms for the power spectrum with a Gaussian initial field.
  • Figure 5: Graphical representation of $\langle\tilde{N}_A\rangle$.
  • ...and 8 more figures