  The equations governing the evolution of non-minimally coupled scalar matter
and the scale factor of a Robertson-Walker universe are derived from a
minisuperspace action. As for the minimally coupled case, it is shown that the
entire semiclassical dynamics can be retrieved from the Wheeler-DeWitt equation
via the Born-Oppenheimer reduction, which properly yields the (time-time
component of the) covariantly conserved energy-momentum tensor of the scalar
field as the source term for gravity. However, for a generic coupling, the
expectation value of the operator which evolves the matter state in time is not
equal to the source term in the semiclassical Einstein equation for the scale
factor of the universe and the difference between these two quantities is
related to the squeezing and quantum fluctuations of the matter state. We also
argue that matter quantum fluctuations become relevant in an intermediate
regime between quantum gravity and semiclassical gravity and study several
cases in detail.
