  In this review, we focus on whether a canonical quantization of general
relativity can produce testable predictions for cosmology. In particular, we
examine how this approach can be used to model the evolution of primordial
perturbations. This program of quantum geometrodynamics, first advocated by
John Wheeler and Bryce DeWitt, has a straightforward classical limit, and it
describes the quantum dynamics of all fields, gravitational and matter. In this
context, in which a classical background metric is absent, it is necessary to
discuss what constitutes an observation. We first address this issue in the
classical theory and then turn to the quantum theory. We argue that predictions
are relational, that is, relative to physical clocks and rods, and that they
can be straightforwardly obtained in a perturbative approach with respect to
Newton's constant, which serves as a coupling parameter. This weak-coupling
expansion leads to a perturbative Hilbert space for quantum cosmology, and to
corrections to the dynamics of quantum fields on a classical, fixed background
metric. These corrections imply modifications of primordial power spectra,
which may lead to signatures in the anisotropy spectrum of the Cosmic Microwave
Background (CMB) radiation, for which we discuss concrete results. We conclude
that the subject of quantum geometrodynamics, the oldest and most conservative
approach to canonical quantum gravity, not only illuminates conceptual issues
in quantum gravitation, but may also lead to observational prospects in
cosmology and elsewhere.
