Quantum Cosmological Backreactions I: Cosmological Space Adiabatic Perturbation Theory

The search for quantum gravity fingerprints in currently available cosmological data that have their origin from the Planck era is of growing interest due to major recent progress both in the theoretical modelling as well as the observational precision. Unsurprisingly, the theoretical predictions are very sensitive to the quantum effects that occur close to the classical big bang singularity. It is therefore of substantial interest to describe these effects as precisely as possible. This is the first in a series of papers that aim at improving on the treatment of quantum effects that arise due to backreactions between matter and geometry. The technique we employ is space adiabatic perturbation theory (SAPT) in the form developed in seminal papers by Panati, Spohn and Teufel. SAPT is a generalisation of the more familiar Born Oppenheimer Approximation (BOA) that applies well in systems that allow a split of the degrees of freedom into two sets that propagate on rather different time scales such as the homogeneous and inhomogeneous field modes in cosmology. We will show that this leads to presently neglected correction terms in the quantum Friedman equations. In the present paper we adapt and generalise SAPT to the hybrid approach to quantum cosmology developed by Mena Marugan et al. that allows for a systematic quantum separation of the (in)homogeneous modes. Since SAPT was developed for quantum mechanics rather than quantum field theory, several challenges have to be met.