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

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.