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.

Figure 1

The plot (in Planck units) shows the variation of energy density as a function of $x$ in the loop quantization of the noncompact Bianchi-II model when $\sin ({\overline{\mu }}_i{c}_i)=1$ is substituted in Eq. (3.4).

Figure 2

Behavior of energy density $\rho$ as a function of $v$ is shown (in Planck units).

Figure 3

Variation of energy density ${\rho }_q$ is shown versus $v$ in Planck units when the approach to classical singularities is isotropic. The minimum occurs at $-753.09{\rho }_{\mathrm{Pl}}$.

Figure 4

Variation of energy density ${\rho }_{\mathrm{q}}$, in Planck units, is shown versus $p_2$ and $p_3$ (with $p_1=p_2$). The energy density diverges as two of the triads tend to zero while the other approaches infinity. For other trajectories ${\rho }_{\mathrm{q}}$ is finite.

Figure 5

Variation of shear scalar ${\sigma }^2$ for Bianchi-IX is shown versus $p_2$ and $p_3$ (with $p_1=p_2$). For the isotropic approach to singularity there exists a local maximum.