Hybrid Quantization: From Bianchi I to the Gowdy Model

TL;DR

This paper advances the loop quantum cosmology of Bianchi I and Gowdy T^3 models by establishing a well-defined evolution framework based on discrete volume, enabling complete quantization of these inhomogeneous cosmologies.

Contribution

It introduces a hybrid quantization approach combining loop and Fock methods for Gowdy models and demonstrates a well-posed initial value problem using a discrete volume evolution variable.

Findings

The Hamiltonian constraint acts as a discrete evolution equation.

Physical solutions are fully determined by initial data on a constant volume slice.

The quantization scheme is successfully completed for both models.

Abstract

The Gowdy cosmologies are vacuum solutions to the Einstein equations which possess two space-like Killing vectors and whose spatial sections are compact. We consider the simplest of these cosmological models: the case where the spatial topology is that of a three-torus and the gravitational waves are linearly polarized. The subset of homogeneous solutions to this Gowdy model are vacuum Bianchi I spacetimes with a three-torus topology. We deepen the analysis of the loop quantization of these Bianchi I universes adopting the improved dynamics scheme put forward recently by Ashtekar and Wilson-Ewing. Then, we revisit the hybrid quantization of the Gowdy $T^3$ cosmologies by combining this loop quantum cosmology description with a Fock quantization of the inhomogeneities over the homogeneous Bianchi I background. We show that, in vacuo, the Hamiltonian constraint of both the Bianchi I and the Gowdy models can be regarded as an evolution equation with respect to the volume of the Bianchi I universe. This evolution variable turns out to be discrete, with a strictly positive minimum. Furthermore, we argue that this evolution is well-defined inasmuch as the associated initial value problem is well posed: physical solutions are completely determined by the data on an initial section of constant Bianchi I volume. This fact allows us to carry out to completion the quantization of these two cosmological models.