  The subject of this thesis is the coupling of quantum fields to a classical
gravitational background in a semiclassical fashion. It contains a thorough
introduction into quantum field theory on curved spacetime with a focus on the
stress-energy tensor and the semiclassical Einstein equation. Basic notions of
differential geometry, topology, functional and microlocal analysis, causality
and general relativity will be summarised, and the algebraic approach to QFT on
curved spacetime will be reviewed. Apart from these foundations, the original
research of the author and his collaborators will be presented: Together with
Fewster, the author studied the up and down structure of permutations using
their decomposition into so-called atomic permutations. The relevance of these
results to this thesis is their application in the calculation of the moments
of quadratic quantum fields. In a work with Pinamonti, the author showed the
local and global existence of solutions to the semiclassical Einstein equation
in flat cosmological spacetimes coupled to a scalar field by solving
simultaneously for the quantum state and the Hubble function in an
integral-functional equation. The theorem is proved with a fixed-point theorem
using the continuous functional differentiability and boundedness of the
integral kernel of the integral-functional equation. In another work with
Pinamonti the author proposed an extension of the semiclassical Einstein
equations which couples the moments of a stochastic Einstein tensor to the
moments of the quantum stress-energy tensor. In a toy model of a Newtonianly
perturbed exponentially expanding spacetime it is shown that the quantum
fluctuations of the stress-energy tensor induce an almost scale-invariant power
spectrum for the perturbation potential and that non-Gaussianties arise
naturally.
