Loop Quantum Cosmology of Bianchi I Model in $\bar{\mu}$ and $\bar{\mu}'$ Schemes with Higher Order Holonomy Corrections

TL;DR

This paper extends loop quantum cosmology with higher order holonomy corrections to anisotropic Bianchi I models, showing that quantum bounces replace singularities and analyzing anisotropy behavior across different schemes.

Contribution

It develops effective dynamics for Bianchi I models with higher order holonomy corrections in both $ar{}$ and $ar{}'$ schemes, including matter effects and anisotropy analysis.

Findings

Singularity is avoided in the $ar{}'$ scheme with a generic quantum bounce.

Differences are observed in the bounce phase between the two schemes.

Anisotropy behavior varies before and after the bounce in both schemes.

Abstract

The detailed formulation of loop quantum cosmology with higher order holonomy corrections has been constructed recently in the homogeneous and isotropic spacetime, yet it is important to extend the higher order holonomy corrections to include the effects of anisotropy which typically grow during the collapsing phase. In this paper we investigate the Bianchi I model in $\bar{\mu}'$ scheme which truly captures the regularization of the Hamiltonian constraint. To compare with the earlier works and provide a comparison with the $\bar{\mu}'$ scheme, we also investigate the $\bar{\mu}$ scheme although it has many disadvantages. First we construct the effective dynamics with higher order holonomy corrections in a massless scalar field, then we extend it to the inclusion of arbitrary matter. Besides that, we also analyze the behavior of the anisotropy during the evolution of the universe. We find that in the $\bar{\mu}'$ scheme, the singularity is never approached and the quantum bounce is generic as in the isotropic case, regardless of the order of the holonomy corrections. Some differences in the bouncing phase of the two schemes are also found out. It is also shown that in the two schemes the behavior of the anisotropy is not the same before and after the bounce.