Discreteness Corrections to the Effective Hamiltonian of Isotropic Loop Quantum Cosmology

TL;DR

This paper derives a new effective Hamiltonian for isotropic Loop Quantum Cosmology that explicitly incorporates discreteness corrections without relying on the continuum approximation, revealing significant deviations in certain cosmological scenarios.

Contribution

It introduces an alternative derivation of the effective Hamiltonian that directly includes discreteness effects, expressed as a closed form in the Barbero-Immirzi parameter.

Findings

Effective Hamiltonian includes discreteness corrections explicitly.

Corrections impose bounds on the universe's volume change rate.

Significant deviations occur in cosmological constant dominated universes.

Abstract

One of the qualitatively distinct and robust implication of Loop Quantum Gravity (LQG) is the underlying discrete structure. In the cosmological context elucidated by Loop Quantum Cosmology (LQC), this is manifested by the Hamiltonian constraint equation being a (partial) difference equation. One obtains an effective Hamiltonian framework by making the continuum approximation followed by a WKB approximation. In the large volume regime, these lead to the usual classical Einstein equation which is independent of both the Barbero-Immirzi parameter $\gamma$ as well as $\hbar$. In this work we present an alternative derivation of the effective Hamiltonian by-passing the continuum approximation step. As a result, the effective Hamiltonian is obtained as a close form expression in $\gamma$. These corrections to the Einstein equation can be thought of as corrections due to the underlying discrete (spatial) geometry with $\gamma$ controlling the size of these corrections. These corrections imply a bound on the rate of change of the volume of the isotropic universe. In most cases these are perturbative in nature but for cosmological constant dominated isotropic universe, there are significant deviations.