Propagation of quantum gravity-modified gravitational waves on a classical FLRW spacetime

The linearized Einstein field equations provide a low-energy wave equation for the propagation of gravitational fields which may originate from a high energy source. Motivated by loop quantum gravity, we propose the polymer quantization scheme to derive the effective propagation of such waves on a classical Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime. To overcome the challenge of polymer quantizing a time-dependent Hamiltonian, we rewrite such a Hamiltonian in a time-independent manner in the extended phase space, polymerize it, and then transform it back to the usual phase space. In this way we obtain a time-dependent polymer Hamiltonian for the gravitational waves. We then derive the effective equations of motion and show that (i) the form of the waves is modified, (ii) the speed of the waves depends on their frequencies, and (iii) quantum effects become more apparent as waves traverse longer distances.