Abstract:
We develop the "triangulated" version of loop quantum cosmology, recently introduced in the literature. We focus on the "dipole" cosmology, where space is a three-sphere and the triangulation is formed by two tetrahedra. We show that the discrete fiducial connection has a simple and appealing geometrical interpretation and we correct the ansatz on the relation between the model variables and the Friedmann-Robertson-Walker scale factor. The modified ansatz leads to the convergence of the Hamiltonian constraint to the continuum one. We then ask which degrees of freedom are captured by this model. We show that the model is rich enough to describe the (anisotropic) Bianchi IX Universe, and give the explicit relation between the Bianchi IX variables and the variables of the model. We discuss the possibility of using this path in order to define the quantization of the Bianchi IX Universe. The model contains more degrees of freedom than Bianchi IX, and therefore captures some inhomogeneous degrees of freedom as well. Inhomogeneous degrees of freedom can be expanded in representations of the SU(2) Bianchi IX isometry group, and the dipole model captures the lowest integer representation of these, connected to hyper-spherical harmonic of angular momentum j=1.

Abstract:
A cosmological description of the universe is proposed in the context of Hamiltonian formulation of a Bianchi IX cosmology minimally coupled to a massless scalar field. The classical and quantum results are studied with special attention to the case of closed Friedmann-Robertson-Walker model.

Abstract:
The loop quantum cosmology "improved dynamics" of the Bianchi type IX model are studied. The action of the Hamiltonian constraint operator is obtained via techniques developed for the Bianchi type I and type II models, no new input is required. It is shown that the big bang and big crunch singularities are resolved by quantum gravity effects. We also present the effective equations which provide modifications to the classical equations of motion due to quantum geometry effects.

Abstract:
We consider the asymptotic behaviour of spatially homogeneous spacetimes of Bianchi type IX close to the singularity (we also consider some of the other Bianchi types, e. g. Bianchi VIII in the stiff fluid case). The matter content is assumed to be an orthogonal perfect fluid with linear equation of state and zero cosmological constant. In terms of the variables of Wainwright and Hsu, we have the following results. In the stiff fluid case, the solution converges to a point for all the Bianchi class A types. For the other matter models we consider, the Bianchi IX solutions generically converge to an attractor consisting of the closure of the vacuum type II orbits. Furthermore, we observe that for all the Bianchi class A spacetimes, except those of vacuum Taub type, a curvature invariant is unbounded in the incomplete directions of inextendible causal geodesics.

Abstract:
The Bianchi IX cosmological model (through Bianchi I and II) is analyzed in the framework of a generalized uncertainty principle. In particular, the anisotropies of the Universe are described by a deformed Heisenberg algebra. Three main results are in order. (i) The Universe can not isotropize because of the deformed Kasner dynamics. (ii) The triangular allowed domain is asymptotically stationary with respect to the particle (Universe) and its bounces against the walls are not interrupted by the deformed effects. (iii) No reflection law can be in obtained since the Bianchi II model is no longer analytically integrable.

Abstract:
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.

Abstract:
The influence of Inflation on initial (i.e. at Planck's epoch) large anisotropy of the Universe is studied, considering a more general metric than the isotropic one: the locally rotationally symmetric (L.R.S.) Bianchi IX metric. We find, then, a large set of initial conditions of intrinsic curvature and shear allowing an inflationary epoch that make the anisotropy negligible. These are not trivial because of the non-linearity of the Einstein's equations.

Abstract:
Within the context of finding the initial conditions of the universe we consider gravitational instantons falling into the Bianchi IX classification. That is, a Euclidean four-manifold with a metric that satisfies Einstein's equations with an induced metric on S^3 submanifolds that is homogeneous but anisotropic. As well as finding regular solutions to the field equations with a tunnelling scalar field, we also look at the case of singular instantons with a view to applying the results to generic potentials. The study is in agreement with the prejudice that instantons with higher symmetry have a lower Euclidean action, even when we consider the singular class of solutions. It is also found that the Euclidean action can diverge for simple potentials, showing that the Hawking Turok instanton had finite action owing to its symmetry.

Abstract:
The goal of this paper is to provide a new analysis of the classical dynamics of Bianchi type I, II and IX models by applying conventional Hamiltonian methods in the language of Ashtekhar variables. We show that Bianchi type II models can be seen as a perturbation of Bianchi I ones, and integrated. Bianchi IX models can be seen, in turn, as a perturbation of Bianchi IIs, but here the integration algorithm breaks down. This is an ''interesting failure'', bringing light onto the chaotic nature of Bianchi type IX dynamics.As a by product of our analysis we filled some gaps in the literature, such us recovering the BKL map in this context.

Abstract:
We investigate the stability of the Einstein static universe as a non-LRS Bianchi type IX solution of the Einstein equations in the presence of both non-tilted and tilted fluids. We find that the static universe is unstable to homogeneous perturbations of Bianchi type IX to the future and the past.