Path Integral for Inflationary Perturbations

TL;DR

This paper develops a canonical path-integral formulation for quantum inflationary perturbations in single-field inflation, deriving the perturbation action to all orders and showing how propagators for the gauge-invariant scalar mode $w$ and tensor modes $h^{TT}_{ij}$ emerge while constraints are handled without explicit solving. It reveals the necessary presence of commuting auxiliary fields (from constraints) and anticommuting ghosts (from gauge-fixing) that contribute to internal lines and loops, and it provides a complete in-in generating functional with diagrammatic rules to compute expectation values. The authors explicitly compute tree-level 3-point and 4-point functions of inflaton perturbations in the tensor gauge, reproducing Maldacena’s results and clarifying how auxiliary-field exchanges yield leading effective 4-point interactions. The framework is gauge-consistent (up to non-linear field redefinitions), generalizable to multi-field theories, and sets the stage for loop calculations and deeper analyses of non-Gaussianity, backreaction, and stochastic inflation.

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.