Effective Hamiltonian of the k=+1 FRW model from the K-Quantization method in LQC

TL;DR

This paper constructs an effective Hamiltonian for the closed FRW model in loop quantum cosmology using the K-Quantization method, confirming the holonomy correction's form and analyzing the model's quantum bounce behavior.

Contribution

It introduces a new Hamiltonian constraint operator for the k=+1 FRW model in LQC using the K-Quantization and 55-scheme, and derives the effective dynamics including the modified Friedmann equation.

Findings

Effective Hamiltonian matches flat space holonomy correction.

Numerical simulations show a quantum bounce with minimum volume.

Infrared behavior and 55-parameter constraints are consistent with results.

Abstract

In the effective theory of loop quantum cosmology LQC, the influence of the holonomy correction (with $\overline{\mu}$-scheme) on the homogeneous and the inhomogeneous cosmological models have been extensively studied in the case of flat space. In this paper,using the K-Quantization method and the $\overline{\mu}$-scheme,in the same framework of LQC,we construct the Hamiltonian constraint operator of the closed FRW model (k=+1), we show that the effective semi-classical limit gives the same holonomy correction expression that was already used in the case of a flat space. We also derive the modified Friedman equation. The numerical investigation is performed to plot the effective evolution of the massless scalar field in term of the volume,which indicate a minimum volume at the bounce and also a suitable infrared behavior.The condition that we apply to constrain the value of $\overline{\mu}$ in the semi-classical limit is consistent with the numerical results.