Isotropic Loop Quantum Cosmology with Matter II: The Lorentzian Constraint

TL;DR

This paper solves the Lorentzian Hamiltonian constraint in isotropic loop quantum cosmology with a massless scalar field, showing quantum geometry removes classical singularities but introduces new conditions for smooth evolution.

Contribution

It extends previous Euclidean analyses by solving the Lorentzian constraint, revealing how quantum geometry affects singularity resolution and the conditions for smooth cosmological evolution.

Findings

Quantum geometry removes classical singularities.

Smooth behavior only occurs for positive or negative times.

A minimal initial energy of the order of the Planck energy is required.

Abstract

The Lorentzian Hamiltonian constraint is solved for isotropic loop quantum cosmology coupled to a massless scalar field. As in the Euclidean case, the discreteness of quantum geometry removes the classical singularity from the quantum Friedmann models. In spite of the absence of the classical singularity, a modified DeWitt initial condition is incompatible with a late-time smooth behavior. Further, the smooth behavior is recovered only for positive or negatives times but not both. An important feature, which is shared with the Euclidean case, is a minimal initial energy of the order of the Planck energy required for the system to evolve dynamically. By forming wave packets of the matter field an explicit evolution in terms of an internal time is obtained.