Different approaches to the second order Klein-Gordon equation

TL;DR

This paper derives the second order Klein-Gordon equation for a scalar field coupled to gravity using two different methods, revealing an unexpected result in the treatment of the field equations.

Contribution

It presents a novel derivation of the second order Klein-Gordon equation in perturbation theory via Einstein equations and action methods, highlighting an unexpected finding.

Findings

Derivation of the second order Klein-Gordon equation using Einstein equations

Derivation from the action after integrating out constraints

Identification of an unexpected result in field equation treatment

Abstract

We derive the Klein--Gordon equation for a single scalar field coupled to gravity at second order in perturbation theory and leading order in slow-roll. This is done in two ways: we derive the Klein--Gordon equation first using the Einstein field equations, and then directly from the action after integrating out the constraint equations. We also point out an unexpected result regarding the treatment of the field equations.