The Inverse Scale Factor in Isotropic Quantum Geometry

TL;DR

This paper quantizes the inverse scale factor in isotropic loop quantum cosmology, showing it remains finite at the classical singularity, indicating a resolution of the Big Bang singularity.

Contribution

It introduces a new quantization of the inverse scale factor that remains bounded and explicitly computes its eigenvalues, demonstrating improved behavior near the singularity.

Findings

Eigenvalues match classical expectations at large scales

Eigenvalues remain finite near the Planck scale

Classical singularity is resolved in loop quantum cosmology

Abstract

The inverse scale factor, which in classical cosmological models diverges at the singularity, is quantized in isotropic models of loop quantum cosmology by using techniques which have been developed in quantum geometry for a quantization of general relativity. This procedure results in a bounded operator which is diagonalizable simultaneously with the volume operator and whose eigenvalues are determined explicitly. For large scale factors (in fact, up to a scale factor slightly above the Planck length) the eigenvalues are close to the classical expectation, whereas below the Planck length there are large deviations leading to a non-diverging behavior of the inverse scale factor even though the scale factor has vanishing eigenvalues. This is a first indication that the classical singularity is better behaved in loop quantum cosmology.