  It is expected that the quantum nature of spacetime leaves its imprint in all
semiclassical gravitational systems, at least in certain regimes, including
gravitational waves. In this paper we investigate such imprints on
gravitational waves within a specific framework: space is assumed to be
discrete (in the form of a regular cubic lattice), and this discrete geometry
is quantised following Dirac's canonical quantisation scheme. The semiclassical
behavior is then extracted by promoting the expectation value of the
Hamiltonian operator on a semiclassical state to an effective Hamiltonian.
Considering a family of semiclassical states representing small tensor
perturbations to Minkowski background, we derive a quantum-corrected effective
wave equation. The deviations from the classical gravitational wave equation
are found to be encoded in a modified dispersion relation and controlled by the
discreteness parameter of the underlying lattice. For finite discretisations,
several interesting effects appear: we investigate the thermodynamical
properties of these modified gravitons and, under certain assumptions, derive
the tensor power spectrum of the cosmic microwave background. The latter is
found to deviate from the classical prediction, in that an amplification of UV
modes takes place. We discuss under what circumstances such effect can be in
agreement with observations.
