Closed FRW model in Loop Quantum Cosmology

TL;DR

This paper develops a new loop quantum cosmology model for a closed, homogeneous, isotropic universe, overcoming previous technical challenges and analyzing the quantum geometry and spectral properties of the gravitational Hamiltonian.

Contribution

It constructs the first detailed LQC model for the closed ($k=1$) universe, distinguishing local properties of quantum geometry between SU(2) and SO(3) topologies.

Findings

Quantum Hamiltonian is a difference operator based on 3-volume quantization.

The operator is essentially self-adjoint and bounded above by zero.

Eigenvectors form a basis and spectral projections are estimated.

Abstract

The basic idea of the LQC applies to every spatially homogeneous cosmological model, however only the spatially flat (so called $k=0$) case has been understood in detail in the literature thus far. In the closed (so called: k=1) case certain technical difficulties have been the obstacle that stopped the development. In this work the difficulties are overcome, and a new LQC model of the spatially closed, homogeneous, isotropic universe is constructed. The topology of the spacelike section of the universe is assumed to be that of SU(2) or SO(3). Surprisingly, according to the results achieved in this work, the two cases can be distinguished from each other just by the local properties of the quantum geometry of the universe. The quantum hamiltonian operator of the gravitational field takes the form of a difference operator, where the elementary step is the quantum of the 3-volume derived in the flat case by Ashtekar, Pawlowski and Singh. The mathematical properties of the operator are studied: it is essentially self-adjoint, bounded from above by 0, the 0 itself is not an eigenvalue, the eigenvectors form a basis. An estimate on the dimension of the spectral projection on any finite interval is provided.