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The "in-in" Formalism and Cosmological Perturbations

Peter Adshead, Richard Easther, Eugene A. Lim

TL;DR

This paper clarifies how to implement the in-in formalism efficiently for cosmological perturbations, emphasizing explicit initial-state (BD vacuum) specification and careful contour handling to avoid spurious divergences in multi-vertex diagrams. It demonstrates, via a one-loop scalar-loop correction to the tensor spectrum and a general two-vertex N-point topology, that proper vacuum treatment yields finite results and substantially simplifies the algebra by producing factorized integral structures. The results confirm that scalar-loop corrections to gravitational waves are minuscule (scaling as $(H/M_p)^4$ with possible logarithmic modulations), while providing a robust methodological framework for higher-order in-in calculations relevant to upcoming precision cosmology. Overall, the paper provides practical prescriptions to extend in-in computations to complex diagrams with confidence in their physical validity.

Abstract

We describe an efficient scheme for evaluating higher order contributions to primordial cosmological perturbations using the "in-in" formalism, which is the basis of modern calculations of non-Gaussian and higher order contributions to the primordial spectrum. We show that diagrams with two or more vertices require careful handling. We present an implementation of the operator formalism in which these diagrams can be evaluated in a simple and transparent fashion. We illustrate our methodology by evaluating the correction to the primordial gravitational wave spectrum generated by scalar loops, a 2-vertex, 1-loop interaction. We then look at a generalized $N$-point, 2-vertex diagram.

The "in-in" Formalism and Cosmological Perturbations

TL;DR

This paper clarifies how to implement the in-in formalism efficiently for cosmological perturbations, emphasizing explicit initial-state (BD vacuum) specification and careful contour handling to avoid spurious divergences in multi-vertex diagrams. It demonstrates, via a one-loop scalar-loop correction to the tensor spectrum and a general two-vertex N-point topology, that proper vacuum treatment yields finite results and substantially simplifies the algebra by producing factorized integral structures. The results confirm that scalar-loop corrections to gravitational waves are minuscule (scaling as with possible logarithmic modulations), while providing a robust methodological framework for higher-order in-in calculations relevant to upcoming precision cosmology. Overall, the paper provides practical prescriptions to extend in-in computations to complex diagrams with confidence in their physical validity.

Abstract

We describe an efficient scheme for evaluating higher order contributions to primordial cosmological perturbations using the "in-in" formalism, which is the basis of modern calculations of non-Gaussian and higher order contributions to the primordial spectrum. We show that diagrams with two or more vertices require careful handling. We present an implementation of the operator formalism in which these diagrams can be evaluated in a simple and transparent fashion. We illustrate our methodology by evaluating the correction to the primordial gravitational wave spectrum generated by scalar loops, a 2-vertex, 1-loop interaction. We then look at a generalized -point, 2-vertex diagram.

Paper Structure

This paper contains 6 sections, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Specifying Initial Conditions in "in-in"
  3. Gravitational Waves from Loops of Scalars
  4. The N-point 2-vertex Interaction
  5. Conclusions
  6. Acknowledgments

Figures (2)

  • Figure 1: The Feynman diagram corresponding to 2-point graviton correlation with a scalar loop. As we explain below, the diagram may be evaluated off-shell, so the momenta on the external legs have distinct labels, to avoid unphysical divergences.
  • Figure 2: A sample diagram with $N=8$ legs, two $m=8$ point vertices and $p=4$ internal lines.