Noncommutative effective loop quantum cosmology: Inclusion of a potential term

Abstract

We construct and study a simple noncommutative scheme (theta-deformation) for the effective Loop Quantum Cosmology of the flat Friedmann-Lema\^itre-Robertson-Walker model in the presence of a homogeneous scalar field $\phi$ with a potential $\mathcal{V}(\phi)=\frac{1}{2}m^2\phi^2$. We first conduct a simple analysis from the corresponding Hamilton equations of motion considering a generic term $\mathcal V(\phi)$. It is observed that the characteristic Big Bounce of Loop Quantum Cosmology is preserved under such noncommutative extension. When specializing to the quadratic case, numerical solutions to the corresponding Hamilton equations exhibiting an early inflationary epoch with a sufficiently large number of e-foldings are found. It is concluded that, in this noncommutative setup, solutions exist which are in the overall compatible with the early universe predicted by standard (effective) Loop Quantum Cosmology (i.e. a bouncing and inflationary early universe). The issue of the genericness of a sufficiently long inflationary period on the space of solutions in this noncommutative construct remains to be addressed.