Quantum cosmological backreactions. I. Cosmological space adiabatic perturbation theory

Quantum cosmology, including quantum cosmological inhomogeneities, is a promising framework for describing the very early universe in which all degrees of freedom are being considered as dynamical and quantum. However, many previous discussions on this subject consider the quantum cosmological perturbations as test fields on a curved spacetime with effective quantum modifications and thus neglect certain interactions of the subsystems, namely the backreaction of the inhomogeneous quantum fields on the cosmological background. In a series of four papers, of which this is the first, we aim at improving on the treatment of quantum effects that arise due to backreactions between matter and geometry as well as between the cosmological perturbations and the homogeneous degrees of freedom. We employ the technique of space adiabatic perturbation theory in the form developed by Panati, Spohn, and Teufel which relies on the familiar Born-Oppenheimer approximation. We extend the scheme to quantum field theory in the cosmological perturbative setting and show that this leads to presently neglected correction terms in the quantum Friedmann equations. In this first article, we provide a detailed introduction to the iterative scheme and examine the generic challenges encountered in its application to perturbative quantum cosmology. Our results will allow for a direct comparison to cosmological observations.