Cosmological perturbations in Hybrid Loop Quantum Cosmology: Mukhanov-Sasaki variables

TL;DR

This paper investigates cosmological perturbations within Loop Quantum Cosmology using Mukhanov-Sasaki variables, clarifying gauge independence, deriving effective equations, and comparing different quantization approaches.

Contribution

It introduces a hybrid quantization framework with a Born-Oppenheimer ansatz to analyze inhomogeneous perturbations and compares its results with existing dressed metric approaches.

Findings

Derived an approximate Schrödinger equation for perturbations.

Proved the similarity of dynamics with the dressed metric approach under certain conditions.

Obtained effective equations for Mukhanov-Sasaki variables with various factor orderings.

Abstract

We study cosmological perturbations in the framework of Loop Quantum Cosmology, using a hybrid quantization approach and Mukhanov-Sasaki variables. The formulation in terms of these gauge invariants allows one to clarify the independence of the results on choices of gauge and facilitates the comparison with other approaches proposed to deal with cosmological perturbations in the context of Loop Quantum Theory. A kind of Born-Oppenheimer ansatz is employed to extract the dynamics of the inhomogeneous perturbations, separating them from the degrees of freedom of the Friedmann-Robertson-Walker geometry. With this ansatz, we derive an approximate Schr\"odinger equation for the cosmological perturbations and study its range of validity. We also prove that, with an alternate factor ordering, the dynamics deduced for the perturbations is similar to the one found in the so-called "dressed metric approach", apart from a possible scaling of the matter field in order to preserve its unitary evolution in the regime of Quantum Field Theory in a curved background and some quantization prescription issues. Finally, we obtain the effective equations that are naturally associated with the Mukhanov-Sasaki variables, both with and without introducing the Born-Oppenheimer ansatz, and with the different factor orderings that we have studied.