The "in-in" Formalism and Cosmological Perturbations

TL;DR

This paper introduces an efficient operator formalism for calculating higher order cosmological perturbations using the in-in formalism, simplifying complex diagram evaluations in primordial spectrum analyses.

Contribution

It presents a novel, transparent method for evaluating higher order diagrams in the in-in formalism, including loop corrections to primordial spectra.

Findings

Simplified evaluation of 2-vertex, 1-loop gravitational wave corrections.

Application to generalized N-point, 2-vertex diagrams.

Demonstrated efficiency in handling complex perturbation diagrams.

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.