Homogeneous cosmological models with non-vanishing intrinsic curvature require a special treatment when they are quantized with loop quantum cosmological methods. Guidance from the full theory which is lost in this context can be replaced by two criteria for an acceptable quantization, admissibility of a continuum approximation and local stability. A quantization of the corresponding Hamiltonian constraints is presented and shown to lead to a locally stable, non-singular evolution compatible with almost classical behavior at large volume. As an application, the Bianchi IX model and its modified behavior close to its classical singularity is explored.

We set up a canonical Hamiltonian formulation for a theory of gravity based on a Lagrangian density made up of the Hilbert-Palatini term and, instead of the Holst term, the Nieh-Yan topological density. The resulting set of constraints in the time gauge are shown to lead to a theory in terms of a real SU (2) connection which is exactly the same as that of Barbero and Immirzi with the coefficient of the Nieh-Yan term identified as the inverse of Barbero-Immirzi parameter. This provides a topological interpretation for this parameter. Matter coupling can then be introduced in the usual manner, without changing the universal topological Nieh-Yan term.

In classical general relativity, the generic approach to the initial singularity is very complicated as exemplified by the chaos of the Bianchi IX model which displays the generic local evolution close to a singularity. Quantum gravity effects can potentially change the behavior and lead to a simpler initial state. This is verified here in the context of loop quantum gravity, using methods of loop quantum cosmology: the chaotic behavior stops once quantum effects become important. This is consistent with the discrete structure of space predicted by loop quantum gravity.PACS numbers: 04.60. Pp,98.80.Jk,98.80.Bp According to the celebrated singularity theorems of classical general relativity, the backward evolution of an expanding universe leads to a singular state where the classical theory ceases to apply. An extensive analysis of the approach to the singularity, in the general context of inhomogeneous cosmologies, has culminated in the BKLscenario [1]. According to this scenario, as the singularity is approached, the spatial geometry can be viewed as a collection of small patches each of which evolves essentially independently as a homogeneous model, most generally the Bianchi IX model. This is justified by the observation that interactions between the patches are negligible because time derivatives dominate over space derivatives close to a singularity.The approach to the singularity of a Bianchi IX model is described by a particle moving in a potential with exponential walls (corresponding to the increasing curvature) bounding a triangle (Fig. 1). During its evolution the particle is reflected at the walls resulting in an infinite number of oscillations (of the scale factors) when the singularity is approached. This classical behavior can be shown to lead to a chaotic evolution by using an analogy with a billiard valid in the asymptotic limit close to the singularity [2].To appreciate implications of the chaotic approach to the Bianchi IX singularity, observe that at any given time the spatial slice can be decomposed into a collection of almost homogeneous patches, the size of the patches being controlled by the magnitude of space derivatives in the equations of motion. During subsequent evolution when the curvatures grow, these patches have to be subdivided to maintain the homogeneous approximation. This subdivision is also controlled by evolution of the individual patches. Since the patches are homogeneous only to a certain approximation, a subdivision of a given patch at a certain time leads to new patches with slightly different initial conditions. The chaotic approach to the Bianchi IX singularity then implies that their geometries will depart rapidly from each other, and the patches have to be fragmented more and more the closer one comes to the singularity. This rapid fragmentation suggests a very complicated and presumably fractal structure of the spatial geometry at the classical singularity. Note that both of these features, the breaking up of a spatial slice into approximately homogeneous patche...

For a class of solutions of the fundamental difference equation of isotropic loop quantum cosmology, the difference equation can be replaced by a differential equation valid for all values of the triad variable. The differential equation admits a 'unique' non-singular continuation through vanishing triad. A WKB approximation for the solutions leads to an effective continuum Hamiltonian. The effective dynamics is also non-singular (no big bang singularity) and approximates the classical dynamics for large volumes. The effective evolution is thus a more reliable model for further phenomenological implications of the small volume effects.

The Bianchi IX model has been used often to investigate the structure close to singularities of general relativity. Its classical chaos is expected to have, via the BKL scenario, implications even for the approach to general inhomogeneous singularities. Thus, it is a popular model to test consequences of modifications to general relativity suggested by quantum theories of gravity. This paper presents a detailed proof that modifications coming from loop quantum gravity lead to a non-chaotic effective behavior. The way this is realized, independently of quantization ambiguities, suggests a new look at initial and final singularities.