Loop Quantum Cosmology with Complex Ashtekar Variables

TL;DR

This paper develops a version of Loop Quantum Cosmology with a complex Barbero-Immirzi parameter, demonstrating that it still resolves the Big Bang singularity with a quantum bounce, and analyzing the resulting modified cosmological dynamics.

Contribution

It introduces complex Ashtekar variables into LQC, extending the theory via an analytic continuation of the Hamiltonian constraint and exploring the implications for cosmological singularity resolution.

Findings

Complex LQC removes the classical singularity with a quantum bounce.

The area spectrum becomes continuous in complex LQC.

The pre- and post-bounce phases exhibit asymmetric dynamics.

Abstract

We construct and study Loop Quantum Cosmology (LQC) when the Barbero-Immirzi parameter takes the complex value $\gamma=\pm i$. We refer to this new quantum cosmology as complex Loop Quantum Cosmology. We proceed in making an analytic continuation of the Hamiltonian constraint (with no inverse volume corrections) from real $\gamma$ to $\gamma=\pm i$ in the simple case of a flat FLRW Universe coupled to a massless scalar field with no cosmological constant. For that purpose, we first compute the non-local curvature operator (defined by the trace of the holonomy of the connection around a fundamental plaquette) evaluated in any spin $j$ representation and we find a new close formula for it. This allows to define explicitly a one parameter family of regularizations of the Hamiltonian constraint in LQC, parametrized by the spin $j$. It is immediate to see that any spin $j$ regularization leads to a bounce scenario. Then, motivated particularly by previous results on black hole thermodynamics, we perform the analytic continuation of the Hamiltonian constraint defined by $\gamma=\pm i$ and $j=-1/2+is$ where $s$ is real. Even if the area spectrum is now continuous, we show that the so-defined complex LQC removes also the original singularity which is replaced by a quantum bounce. In addition, the maximal density and the minimal volume of the Universe are obviously independent of $\gamma$. Furthermore, the dynamics before and after the bounce are no more symmetric, which makes a clear distinction between these two phases of the evolution of the Universe.