Scalar and Vector Perturbations in Quantum Cosmological Backgrounds

TL;DR

This paper simplifies the formulation of scalar and vector perturbations in quantum cosmological backgrounds using canonical transformations, enabling easier analysis even when the background metric is quantized.

Contribution

It introduces a method to derive simple lagrangians and hamiltonians for cosmological perturbations without relying on classical background equations.

Findings

Simplified form of perturbation lagrangians and hamiltonians.

Applicable to quantized background metrics.

Independent of classical Einstein equations for background.

Abstract

Generalizing a previous work concerning cosmological linear tensor perturbations, we show that the lagrangians and hamiltonians of cosmological linear scalar and vector perturbations can be put in simple form through the implementation of canonical transformations and redefinitions of the lapse function, without ever using the background classical equations of motion. In particular, if the matter content of the Universe is a perfect fluid, the hamiltonian of scalar perturbations can be reduced, as usual, to a hamiltonian of a scalar field with variable mass depending on background functions, independently of the fact that these functions satisfy the background Einstein's classical equations. These simple lagrangians and hamiltonians can then be used in situations where the background metric is also quantized, hence providing a substantial simplification over the direct approach originally developed by Halliwell and Hawking.