Nonminimally coupled scalar fields in homogeneous universes

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.