  We quantise and solve the dynamics of gravitational waves in a quantum
Friedmann-Lemaitre-Robertson-Walker spacetime filled with perfect fluid. The
classical model is formulated canonically. The Hamiltonian constraint is
de-parametrised by setting a fluid variable as the internal clock. The obtained
reduced (i.e. physical) phase space is then quantised. Our quantisation
procedure is implemented in accordance with two different phase space
symmetries, namely, the Weyl-Heisenberg symmetry for the perturbation
variables, and the affine symmetry for the background variables. As an
appealing outcome, the initial singularity is removed and replaced with a
quantum bounce. The quantum model depends on a free parameter that is naturally
induced from quantisation and determines the scale of the bounce. We study the
dynamics of the quantised gravitational waves across the bounce through three
different methods ("thin-horizon", analytical and numerical) which give
consistent results and we determine the primordial power spectrum for the case
of radiation-dominated universe. Next, we use the instantaneous
radiation-matter transition transfer function to make approximate predictions
for late universe and constrain our model with LIGO and Planck data. We also
give an estimate of the quantum uncertainties in the present-day universe.
