Contrasting features of anisotropic loop quantum cosmologies: The role of spatial curvature

A characteristic feature of loop quantization of the isotropic and Bianchi-I spacetimes is the existence of universal bounds on the energy density and the expansion and shear scalars, independent of the matter content. We investigate the properties of these physical quantities in Bianchi-II and Bianchi-IX spacetimes, which have been recently loop quantized using the connection operator approach. Using the effective Hamiltonian approach, we show that for Bianchi-II spacetime the energy density and the expansion and shear scalars turn out to be bounded, albeit not by universal values. In Bianchi-IX spacetime, when the approach to the classical singularity is isotropic, the above physical quantities are bounded. In addition, for all other cases, where the approach to singularities is not isotropic and effective dynamics can be trusted, these quantities turn out to be finite. These results stand in sharp distinction to general relativity, where the above physical quantities are generically unbounded, leading to the breakdown of geodesic equations. In contrast to the isotropic and Bianchi-I models, we find the role of energy conditions for the Bianchi-II model and the inverse triad modifications for Bianchi-IX to be significant to obtain the above bounds. These results bring out subtle physical distinctions between (i) the quantization using holonomies over closed loops performed for isotropic and Bianchi-I models and (ii) the connection operator approach. We find that qualitative differences in physics exist for these quantization methods even for the isotropic models in the presence of spatial curvature. We highlight these important differences in the behavior of the expansion scalar in the holonomy based quantization and connection operator approach for isotropic spatially closed and open models.