  We study the solutions of the semiclassical Einstein equation in flat
cosmological spacetimes driven by a massive conformally coupled scalar field.
In particular, we show that it is possible to give initial conditions at finite
time to get a state for the quantum field which gives finite expectation values
for the stress-energy tensor. Furthermore, it is possible to control this
expectation value by means of a global estimate on regular cosmological
spacetimes. The obtained estimates permit to write a theorem about the
existence and uniqueness of the local solutions encompassing both the spacetime
metric and the matter field simultaneously. Finally, we show that one can
always extend local solutions up to a point where the scale factor becomes
singular or the Hubble function reaches a critical value $H_c = 180\pi/G$,
which both correspond to a divergence of the scalar curvature, namely a
spacetime singularity.
