Quantum Fate of Singularities in Anisotropic Cosmological Models

This thesis studies the possibility of the quantum avoidance of gravitational singularities in anisotropic cosmological models. For that purpose, we review the fundamentals of spatially homogeneous cosmological models and quantum cosmology based on the Wheeler-DeWitt equation in minisuperspace. Furthermore, we introduce a generalized dynamical system which is designed to emulate some of the main features of the cosmological models. After studying its geometric properties, we start to investigate how one can approach the canonical quantization of such a system. The main focus of our analysis is on the factor ordering problem in the Wheeler-DeWitt equation. The considerations motivate us to formulate criteria for singularity avoidance, that respect the conformal geometry of the con�guration space of the spatially homogeneous models. We then go on by studying some speci�c models with and without matter. In particular we examine classical and quantum properties of the Bianchi type I, II and IX and the Kantowski-Sachs universe. The criteria we developed previously are applied to see under which circumstances singularities can be avoided. If the potential terms are negligible when compared against the velocity terms in the gravitational action, the approach towards the singularity is called asymptotically velocity term dominated. We �nd that such singularities can be resolved, if the dimension of the minisuperspace is suffciently large. The underlying mechanism is a spreading of wave packets in minisuperspace. We also consider the non-diagonal Bianchi IX model with tilted dust. This model is relevant in the context of the BKL scenario. We pay particular attention to the asymptotic regime close to the singularity and the temporal behavior of curvature invariants in this regime

On the dynamics of the general Bianchi IX spacetime near the singularity

We show that the complex dynamics of the general Bianchi IX universe in the vicinity of the spacelike singularity can be approximated by a simplified system of equations. Our analysis is mainly based on numerical simulations. The properties of the solution space can be studied by using this simplified dynamics. Our results will be useful for the quantization of the general Bianchi IX model.Comment: 20 pages, 5 figures, minor change

Comparing the dynamics of diagonal and general Bianchi IX spacetime

We make comparison of the dynamics of the diagonal and nondiagonal Bianchi IX models in the evolution towards the cosmological singularity. Apart from the original variables, we use the Hubble normalized ones commonly applied in the examination of the dynamics of homogeneous models. Applying the dynamical systems method leads to the result that in both cases the continuous space of critical points is higher dimensional and they are of the nonhyperbolic type. This is a generic feature of the dynamics of both cases and seems to be independent on the choice of phase space variables. The topologies of the corresponding critical spaces are quite different. We conjecture that the nondiagonal case may carry a new type of chaos different from the one specific to the usually examined diagonal one.Comment: 25 pages, 2 figures, version including numerical simulations of dynamic

Classical and quantum cosmology of Born-Infeld type models

We discuss Born-Infeld type fields (tachyon fields) in classical and quantum cosmology. We first partly review and partly extend the discussion of the classical solutions and focus in particular on the occurrence of singularities. For quantization, we employ geometrodynamics. In the case of constant potential, we discuss both Wheeler-DeWitt quantization and reduced quantization. We are able to give various solutions and discuss their asymptotics. For the case of general potential, we transform the Wheeler-DeWitt equation to a form where it leads to a difference equation. Such a difference equation was previously found in the quantization of black holes. We give explicit results for the cases of constant potential and inverse squared potential and point out special features possessed by solutions of the difference equation.Comment: 14 pages, 8 figures, published versio

Curvature invariants for the Bianchi IX spacetime filled with tilted dust

We present an analysis of the Kretschmann and Weyl squared scalars for the general Bianchi IX model filled with tilted dust. Particular attention is given to the asymptotic regime close to the singularity for which we provide heuristic considerations supported by numerical simulations. The present paper is an extension of our earlier publication (Kiefer et al., in Eur Phys J C 78:691, 2018)

Singularity avoidance in anisotropic quantum cosmology

We discuss the fate of the classical singularities in quantum cosmological models. We state our criteria of singularity avoidance and apply them to Friedmann-Lemaˆıtre models models as well as, in more detail, to the anisotropic case of a Bianchi I universe. We find that the classical singularities are generally avoided in these cases

Singularity avoidance in Bianchi I quantum cosmology

We extend recent discussions of singularity avoidance in quantum gravity from isotropic to anisotropic cosmological models. The investigation is done in the framework of quantum geometrodynamics (Wheeler-DeWitt equation). We formulate criteria of singularity avoidance for general Bianchi class A models and give explicit and detailed results for Bianchi I models with and without matter. We find that the classical singularities can generally be avoided in these models.Comment: 15 pages, 8 figures, final versio

Hamiltonian formulation of dust cloud collapse

We consider the gravitational collapse of a self-gravitating spherical dust cloud in the Hamiltonian formalism. We address both homogeneous and inhomogeneous cases. Our novel derivation of the Hamiltonian of the system is based on an improved variational principle. It differs from usual treatments due to the presence of an extra boundary term added to the Hilbert action. As expected, the standard equations of motion are retrieved. However, differently from other treatments, the total Hamiltonian obtained with our procedure in the Schwarzschild time gauge is identical to the total mass of the system as measured from infinity, as it would be expected. Implications for the quantization of the system arc suggested