Quantum cosmological gravitational waves?

General relativity and its cosmological solution predicts the existence of tensor modes of perturbations evolving on top of our Friedman-Lema\^itre-Robertson-Walker expanding Universe. Being gauge invariant and not necessarily coupled to other quantum sources, they can be seen as representing pure gravity. Unambiguously showing they are indeed to be quantised would thus provide an unquestionable proof of the quantum nature of gravitation. This review will present a summary of the various theoretical issues that could lead to this conclusion.
Comment: Invited chapter for the Section "Perturbative Quantum Gravity" of the "Handbook of Quantum Gravity" (Eds. C. Bambi, L. Modesto and I.L. Shapiro, Springer Singapore, expected in 2023). New version matches the accepted version