Path Integral for Inflationary Perturbations

TL;DR

This paper develops a path integral formulation for quantum cosmological perturbations during inflation, simplifying the calculation of interactions without solving energy and momentum constraints explicitly.

Contribution

It introduces a novel path integral approach that avoids explicit constraint solutions, streamlining the derivation of inflationary perturbation interactions.

Findings

Derives free propagators from gauge-invariant quadratic actions.

Reproduces known tree-level 3- and 4-point functions.

Framework facilitates future loop calculations.

Abstract

The quantum theory of cosmological perturbations in single field inflation is formulated in terms of a path integral. Starting from a canonical formulation, we show how the free propagators can be obtained from the well known gauge-invariant quadratic action for scalar and tensor perturbations, and determine the interactions to arbitrary order. This approach does not require the explicit solution of the energy and momentum constraints, a novel feature which simplifies the determination of the interaction vertices. The constraints and the necessary imposition of gauge conditions is reflected in the appearance of various commuting and anti-commuting auxiliary fields in the action. These auxiliary fields are not propagating physical degrees of freedom but need to be included in internal lines and loops in a diagrammatic expansion. To illustrate the formalism we discuss the tree-level 3-point and 4-point functions of the inflaton perturbations, reproducing the results already obtained by the methods used in the current literature. Loop calculations are left for future work.