  This is the third paper in a series of four in which we use space adiabatic
methods in order to incorporate backreactions among the homogeneous and between
the homogeneous and inhomogeneous degrees of freedom in quantum cosmological
perturbation theory.
  In this paper we consider a particular kind of cosmological perturbation
theory which starts from a gauge fixed version of General Relativity. The gauge
fixing is performed using a material reference system called Gaussian dust. The
resulting system has no constraints any more but possesses a physical
Hamiltonian that drives the dynamics of both geometry and matter. As observable
matter content we restrict to a scalar field (inflaton). We then explore the
sector of that theory which is purely homogeneous and isotropic with respect to
the geometry degrees of freedom but contains inhomogeneous perturbations up to
second order of the scalar field.
  The purpose of this paper is to explore the quantum field theoretical
challenges of the space adiabatic framework in a cosmological model of
inflation which is technically still relatively simple. We compute the quantum
backreaction effects from every energy band of the inhomogeneous matter modes
on the evolution of the homogeneous geometry up to second order in the
adiabatic parameter. These contributions turn out to be significant due to the
infinite number of degrees of freedom and are very sensitive to the choice of
Fock representation chosen for the inhomogeneous matter modes.
