Numerics of Bianchi type II and type IX spacetimes in effective loop quantum cosmology

Abstract: We numerically solve the effective loop quantum cosmology dynamics for the vacuum Bianchi type II and type IX spacetimes, in particular studying how the Kasner exponents evolve across the loop quantum cosmology bounce. We find that when the spatial curvature is negligible at the bounce then the Kasner exponents transform according to the same simple equation as for a Bianchi type I spacetime in effective loop quantum cosmology, while there are departures from this transformation rule in cases where the spatial curvature is significant during the bounce. We also use high-precision numerics to compute the evolution of a Bianchi type IX spacetime through multiple bounces and recollapses, and find indications of chaotic behaviour. Interestingly, the numerics indicate that it is during the classical recollapse, and not the loop quantum cosmology bounce, that nearby solutions diverge most strongly.