Cosmological perturbations meet Wheeler DeWitt

TL;DR

This paper explores approximate solutions to the Wheeler DeWitt equation in cosmology, comparing them with standard perturbation theory, and clarifies the probabilistic interpretation and gauge choices involved.

Contribution

It introduces a WKB-based approach to connect quantum cosmological wavefunctions with classical backgrounds and standard perturbation results, including gauge analysis and probabilistic interpretation.

Findings

Verified relation between WKB solutions and perturbation wavefunctions in specific models

Analyzed gauge dependence and probabilistic interpretation of wavefunctions

Discussed deviations from classical trajectories and higher derivative terms

Abstract

We study approximate solutions of the Wheeler DeWitt (WdW) equation and compare them with the standard results of cosmological perturbation theory. In mini-superspace, we introduce a dimensionless gravitational coupling $\alpha$ that is typically very small and functions like $\hbar$ in a WKB expansion. We seek solutions of the form $\Psi = e^{iS/\alpha} \psi$ that are the closest quantum analog of a given classical background spacetime. The function $S$ satisfies the Hamilton-Jacobi equation, while $\psi$ obeys a Schr\"odinger-like equation and can be given a probabilistic interpretation. The semiclassical limit suggests a specific relation between $\psi$ and the standard perturbation-theory wavefunction $\psi_P$. We verify this relation in two main examples: a scalar field with a purely exponential potential, of which simple scaling solutions are known and a slow-roll scenario expanded in the vicinity of the origin in field space. Each example is worked out in two different gauges, that are the minisuperspace equivalent of unitary gauge and spatially flat gauge. We discuss possible deviations from the classical background trajectory as well as the higher ``time" derivative terms that are present in the WdW equation but not in the perturbative approach. We clarify the \emph{conditional probability} content of the wavefunctions and how this is related with the standard gauge fixing procedure in perturbation theory.