Effective Hamiltonian for Isotropic Loop Quantum Cosmology

TL;DR

This paper derives an effective Hamiltonian for isotropic loop quantum cosmology, providing a non-singular, continuum approximation that captures quantum effects at small volumes while aligning with classical dynamics at large scales.

Contribution

It introduces a differential equation approximation for the fundamental difference equation, leading to an effective Hamiltonian that models quantum cosmological dynamics reliably across all volumes.

Findings

The effective Hamiltonian is non-singular and extends through zero volume.

The effective dynamics match classical behavior at large volumes.

The approach offers a reliable model for quantum cosmological phenomenology.

Abstract

For a class of solutions of the fundamental difference equation of isotropic loop quantum cosmology, the difference equation can be replaced by a differential equation valid for {\em all} values of the triad variable. The differential equation admits a `unique' non-singular continuation through vanishing triad. A WKB approximation for the solutions leads to an effective continuum Hamiltonian. The effective dynamics is also non-singular (no big bang singularity) and approximates the classical dynamics for large volumes. The effective evolution is thus a more reliable model for further phenomenological implications of the small volume effects.