  We develop a comprehensive framework in which the existence of solutions to
the semiclassical Einstein equation (SCE) in cosmological spacetimes is shown.
Different from previous work on this subject, we do not restrict to the
conformally coupled scalar field and we admit the full renormalization freedom.
Based on a regularization procedure, which utilizes homogeneous distributions
and is equivalent to Hadamard point-splitting, we obtain a reformulation of the
evolution of the quantum state as an infinite-dimensional dynamical system with
mathematical features that are distinct from the standard theory of
infinite-dimensional dynamical systems (e.g., unbounded evolution operators).
Nevertheless, applying methods closely related to Ovsyannikov's method, we show
existence of maximal/global solutions to the SCE for vacuum-like states, and of
local solutions for thermal-like states. Our equations do not show the
instability of the Minkowski solution described by other authors.
