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

Abstract

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, 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, 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 above physical quantities are generically unbounded, leading to the break down of geodesic equations. In contrast to the isotropic and Bianchi-I models, we find the role of energy conditions for Bianchi-II model and the inverse triad modifications for Bianchi-IX to be significant to obtain above bounds.These results bring out subtle physical distinctions between the quantization using holonomies over closed loops performed for isotropic and Bianchi-I models, and 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.

R+αR2loop quantum cosmology

“…It is important to stress that the replacement (3.2) is only valid for spatially flat models, which is our case. When the spatial curvature does not vanish, holonomy corrections have to be introduced in a nontrivial way (see, for instance, [20]). …”

A quantum gravitational inflationary scenario in Bianchi-I spacetime

“…It is found that GR is an excellent approximation to quantum dynamics at curvature scales small compared to the Planck value, however departures between the two theories become significant in the Planck regime which is described by the effective dynamics to an excellent accuracy. Using effective Hamiltonian method, bounds on expansion and shear scalars have been derived [52,53], and generic results on the absence of strong singularities in isotropic [54,55] and Bianchi-I spacetime [56] have been obtained.…”

Loop Quantum Cosmology

“…Another important implication results in the form of bounds on physical quantities which appear in geodesic and Raychaudhuri equations. It has been shown that energy density and expansion and shear scalars are bounded for isotropic [45] and anisotropic models [50,47] which has important implications to understand general resolution of singularities. This turns out to be true for isotropic models where it has been demonstrated that there is indeed a generic resolution of strong singularities [45,48].…”

“…The effective dynamics captures the underlying quantum evolution to an excellent accuracy for various models in LQC [11,12,33,28,30,43,44], and has been used to study detailed physical implications of loop quantization of cosmological spacetimes. As an example, using effective spacetime description, singularity resolution in LQC can be understood occurring due to bounds on the growth of spacetime curvature, which plays a conjugate role to geometry [45,46,47], and insights on the generic resolution of strong singularities and geodesic extendability in flat isotropic [45,48] and Bianchi-I models [46] have been obtained. Effective equations have also been used to gain insights on constraining quantization ambiguities [49,50], and have been used to explore the physics of Gowdy models [51].…”

“…Effective dynamics, which captures the key features of underlying quantum geometry, provides an excellent approximation of the quantum evolution and has been extensively used to extract novel physical predictions, such as, on bounds on energy density, expansion and shear scalars [45,47,50], resolution of strong singularities [45,46] and probability for inflation [72,73,74]. These have been also used to probe the physics beyond homogeneity approximation such as in Gowdy models [51] and imprint of quantum geometry on cosmological perturbations [52], which have also been derived using detailed properties of quantum spacetime [54].…”