Effective Dynamics in Bianchi Type II Loop Quantum Cosmology

TL;DR

This paper numerically studies the effective equations of Bianchi II Loop Quantum Cosmology, revealing that classical singularities are replaced by bounces, with complex anisotropic behaviors and implications for understanding space-like singularities.

Contribution

It provides a detailed numerical analysis of Bianchi II LQC solutions, highlighting new behaviors of anisotropies and singularity resolution mechanisms.

Findings

Big-bang singularity replaced by a bounce

Multiple directional and global bounces observed

Solutions can have densities exceeding isotropic bounds

Abstract

We numerically investigate the solutions to the effective equations of the Bianchi II model within the "improved" Loop Quantum Cosmology (LQC) dynamics. The matter source is a massless scalar field. We perform a systematic study of the space of solutions, and focus on the behavior of several geometrical observables. We show that the big-bang singularity is replaced by a bounce and the point-like singularities do not saturate the energy density bound. There are up to three directional bounces in the scale factors, one global bounce in the expansion, the shear presents up to four local maxima and can be zero at the bounce. This allows for solutions with density larger than the maximal density for the isotropic and Bianchi I cases. The asymptotic behavior is shown to behave like that of a Bianchi I model, and the effective solutions connect anisotropic solutions even when the shear is zero at the bounce. All known facts of Bianchi I are reproduced. In the "vacuum limit", solutions are such that almost all the dynamics is due to the anisotropies. Since Bianchi II plays an important role in the Bianchi IX model and the Belinskii, Khalatnikov, Lifshitz (BKL) conjecture, our results can provide an intuitive understanding of the behavior in the vicinity of general space-like singularities, when loop-geometric corrections are present.