Loop quantum cosmology with complex Ashtekar variables

We construct and study loop quantum cosmology (LQC) when the Barbero-Immirzi parameter takes the complex value gamma = +/- i. We refer to this new approach to quantum cosmology as complex LQC. This formulation is obtained via an analytic continuation of the Hamiltonian constraint (with no inverse volume corrections) from real gamma to gamma = +/- i, in the simple case of a flat FLRW Universe coupled to a massless scalar field with no cosmological constant. For this, we first compute the non-local curvature operator (defined by the trace of the holonomy of the connection around a fundamental plaquette) evaluated in an arbitrary spin j representation, and find a new close formula for its expression. This allows us 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 bouncing scenario. Then, motivated in particular by previous results on black hole thermodynamics, we perform the analytic continuation of the Hamiltonian constraint to values of the Barbero-Immirzi parameter given by gamma = +/- i and to spins j = 1/2(-1 + is) where s is real. Even if the area spectrum then becomes continuous, we show that the complex LQC defined in this way does also replace the initial big-bang singularity by a big-bounce. In addition to this, the maximal density and the minimal volume of the Universe are obviously independent of gamma. Furthermore, the dynamics before and after the bounce is not symmetrical anymore, which makes a clear distinction between these two phases of the evolution of the Universe.