Book Description

The present thesis addresses the cosmological backreaction problem, i.e., the question of whether and to which extent cosmological inhomogeneities affect the global evolution of the Universe. We will thereby focus on, but not restrict to, backreaction in a purely quantum theoretical framework which is adapted to describe situations during the earliest phases of the Universe. Our approach to evaluating backreaction uses a perturbative and constructive mathematical formalism, denoted space adiabatic perturbation theory, that is inspired by but which extends the well--known Born--Oppenheimer approximation to molecular systems.The underlying idea of this scheme is to separate the system into an adiabatically slow and a fast part, similar to the separation of nuclear and electronic subsystems in a molecular setting. Such a distinction is reasonable if a corresponding perturbation parameter can be identified. In case of molecular systems, such a parameter arises as the ratio of the light electron and heavy nuclear masses. In the case of the here considered cosmological systems, we identify the ratio of the gravitational and the matter coupling constants as a suitable perturbative parameter. In a first step, we apply the space adiabatic formalism to a toy model and compute the backreation of a homogeneous scalar field on a homogeneous and isotropic geometry. We restrict the computations to second order in the adiabatic perturbations and obtain an effective Hamilton operator for the geometry.In the sequel, we apply space adiabatic perturbation theory to an inhomogeneous cosmology and calculate backreaction effects of the inhomogeneous quantum cosmological fields on the global quantum degrees of freedom. Therefore, it is necessary to first extend the scheme adequately for an application to infinite dimensional field theories. In fact, the violation of the Hilbert--Schmidt condition for quantum field theories prevents a direct application of the scheme. A solution is obtained by a transformation of variables which is canonical up to second order in the cosmological perturbations. This allows us to compute an effective Hamilton operator for a cosmological field theory previously deparametrized by a timelike dust field, as well as the identification of an effective Hamilton constraint for a system with gauge--invariant cosmological perturbations. Both objects act on the global degrees of freedom and include the backreaction of the inhomogeneities up to second order in the adiabatic perturbation theory.We conclude that it is a priori inadmissible to neglect cosmological backreaction. However, due to the general difficulties associated with finding solutions for coupled gravitational systems, the concrete evaluation of the operators found here must remain the subject of future research. One obstacle is the occurrence of indefinite mass squares associated with the perturbation fields which are the result of the previous transformations (which however, already appear in independent problems, for example in the use of Mukhanov--Sasaki variables) . A further complication in the final quantization and search for appropriate solutions arises from the non--polynomial dependence on the global degrees of freedom. We discuss these obstacles in detail and point to possible solutions.