  We revisit the calculation of quantum-gravitational corrections to the power
spectra of scalar and tensor perturbations in the Born-Oppenheimer approach to
quantum gravity. We focus on the issue of the definition of the inner product
of the theory and the unitarity of the corrections to the dynamics of the
cosmological perturbations. We argue that the correction terms are unitary,
provided the inner product is defined in a suitable way, which can be related
to a notion of gauge-fixing the time variable and the use of conditional
probabilities in quantum cosmology. We compare the corrections obtained within
this framework to earlier results in the literature and we conclude with some
remarks on the physical interpretation of the correction terms.
