  We investigate the cosmological implications of modified gravities induced by
the quantum fluctuations of the gravitational metric. If the metric can be
decomposed as the sum of the classical and of a fluctuating part, of quantum
origin, then the corresponding Einstein quantum gravity generates at the
classical level modified gravity models with a nonminimal coupling between
geometry and matter. As a first step in our study, after assuming that the
expectation value of the quantum correction can be generally expressed in terms
of an arbitrary second order tensor constructed from the metric and from the
thermodynamic quantities characterizing the matter content of the Universe, we
derive the (classical) gravitational field equations in their general form. We
analyze in detail the cosmological models obtained by assuming that the quantum
correction tensor is given by the coupling of a scalar field and of a scalar
function to the metric tensor, and by a term proportional to the matter
energy-momentum tensor. For each considered model we obtain the gravitational
field equations, and the generalized Friedmann equations for the case of a flat
homogeneous and isotropic geometry. In some of these models the divergence of
the matter energy-momentum tensor is non-zero, indicating a process of matter
creation, which corresponds to an irreversible energy flow from the
gravitational field to the matter fluid, and which is direct consequence of the
nonminimal curvature-matter coupling. The cosmological evolution equations of
these modified gravity models induced by the quantum fluctuations of the metric
are investigated in detail by using both analytical and numerical methods, and
it is shown that a large variety of cosmological models can be constructed,
which, depending on the numerical values of the model parameters, can exhibit
both accelerating and decelerating behaviors.
