Approximation methods in Loop Quantum Cosmology: From Gowdy cosmologies to inhomogeneous models in Friedmann-Robertson-Walker geometries

TL;DR

This paper introduces approximation techniques in Loop Quantum Cosmology to simplify the complex Hamiltonian of Gowdy models with inhomogeneities, enabling analysis of inhomogeneous cosmologies as effective FRW models.

Contribution

It develops justified approximation methods to reduce the complexity of the Hamiltonian constraint in hybrid quantized Gowdy models, connecting inhomogeneous models to effective FRW descriptions.

Findings

Approximate the homogeneous Hamiltonian as an FRW model with anisotropies.

Valid in high-energy regimes and for smooth anisotropy profiles.

Simplify the inhomogeneous Hamiltonian to an FRW model with manageable matter content.

Abstract

We develop approximation methods in the hybrid quantization of the Gowdy model with linear polarization and a massless scalar field, for the case of three-torus spatial topology. The loop quantization of the homogeneous gravitational sector of the Gowdy model (according to the improved dynamics prescription) and the presence of inhomogeneities lead to a very complicated Hamiltonian constraint. Therefore, the extraction of physical results calls for the introduction of well justified approximations. We first show how to approximate the homogeneous part of the Hamiltonian constraint, corresponding to Bianchi I geometries, as if it described a Friedmann-Robertson-Walker (FRW) model corrected with anisotropies. This approximation is valid in the high-energy sector of the FRW geometry (concerning its contribution to the constraint) and for anisotropy profiles that are sufficiently smooth. In addition, for certain families of states associated to regimes of physical interest, with negligible effects of the anisotropies and small inhomogeneities, one can approximate the Hamiltonian constraint of the inhomogeneous system by that of an FRW geometry with a relatively simple matter content, and then obtain its solutions.