  We prove existence and uniqueness of solutions of the semiclassical Einstein
equation in flat cosmological spacetimes driven by a quantum massive scalar
field with arbitrary coupling to the scalar curvature. In the semiclassical
approximation, the backreaction of matter to curvature is taken into account by
equating the Einstein tensor to the expectation values of the stress-energy
tensor in a suitable state. We impose initial conditions for the scale factor
at finite time and we show that a regular state for the quantum matter
compatible with these initial conditions can be chosen. Contributions with
derivative of the coefficient of the metric higher than the second are present
in the expectation values of the stress-energy tensor and the term with the
highest derivative appears in a non-local form. This fact forbids a direct
analysis of the semiclassical equation, and in particular, standard recursive
approaches to approximate the solution fail to converge. In this paper we show
that, after partial integration of the semiclassical Einstein equation in
cosmology, the non-local highest derivative appears in the expectation values
of the stress-energy tensor through the application of a linear unbounded
operator which does not depend on the details of the chosen state. We prove
that an inversion formula for this operator can be found, furthermore, the
inverse happens to be more regular than the direct operator and it has the form
of a retarded product, hence causality is respected. The found inversion
formula applied to the traced Einstein equation has thus the form of a fixed
point equation. The proof of local existence and uniqueness of the solution of
the semiclassical Einstein equation is then obtained applying the Banach fixed
point theorem.
