In a classic paper, Bohr and Rosenfeld (1933) showed that the quantization of matter implies the quantization of the electromagnetic field. In other words, it is logically inconsistent to have classical electromagnetic waves interacting with a quantum measuring apparatus. Crucially, the same is not true for gravity, due to the existence of a fundamental length scale, the Planck length. 

In fact, not only is the quantization of gravity not a logical necessity, it is also unobservable within Einstein gravity. Freeman Dyson first showed that it is impossible to detect a single graviton with high probability in any realistic experiment, and conjectured a censorship effect that precludes the observation of the quantization of the gravitational field in Einstein’s theory. Dyson's argument is based on the striking fact that the absorption cross section for gravitons is universal and equal to the planck area within numerical factors of order one. 

Dyson’s work was motivated by the hope that the failures in reconciling General Relativity with Quantum Mechanics were really due to the fact that the gravitational field is a purely classical entity. In the dichotomous world he envisioned, the geometric theory of gravity would peacefully coexist with the quantum realm, and the obstruction in observing individual gravitons would be attributable to their non existence.

In our own paper,  we propose a type of experiment that would be able to distinguish between classical and quantum gravity at the observational level. The experiment consists in looking for dark lines in gravitational spectra, due to gravitational absorption by quantum bound states. Gravitational absorption lines would probe quantum gravity in two distinct ways. First of all, since General Relativity forbids them, they would explore and constrain exotic physics close to the Planck scale. Secondly, and perhaps more importantly, they would confirm that the gravitational field is quantized at low energies, effectively proving the existence of gravitons.

Here you can find an essay I wrote for the Danish Development Research Network that was inspired by this piece of research.

Conventional wisdom says that you can't have gravitational atoms because gravity is weak: electric and nuclear forces always dominate at microscopic scales. This is not necessarily true. While it is true that gravity is weak for visible particles, it could be strong for a heavy particle that we haven't found yet. It could even be the dark matter particle, as in the PIDM scenario. In fact, in the minimal PIDM scenario, purely gravitational atoms are naturally produced by thermal scattering in the SM plasma, the same mechanism responsible for dark matter genesis. 

By their very nature, gravitational atoms are incredibly heavy. In our paper we derive a lower bound on their mass based on tidal disruption in galaxies. It turns out that the mass has to be at least 10-8  in natural units. That's approximately the mass of a bacterium! The heaviest atoms are not much bigger than the Planck length and they weigh almost as much as a grain of sand. These heaviest atoms are usually unstable and decay to gravitons in the early universe, sourcing a highly energetic, isotropic, and nearly monochromatic gravitational wave signal. In the paper we derive a lower bound on the frequency of the signal: it's 1013 Hz. We don't know of any other process in nature that would give such a strong monochromatic signal at such high frequencies.  

Most of the matter in the universe is invisible, meaning that it hardly interacts with ordinary (baryonic) matter. In particular, it doesn't emit photons, so we cannot see it directly. We can only infer its existence from the gravitational effects it has on galaxies and the universe as a whole. 

While the evidence for the existence of dark matter so far stems purely from its gravitational influence, the dark matter particle is usually assumed to have other-than-gravitational interactions. In other words, the physics beyond the Standard Model responsible for dark matter is typically assumed to emerge much below the Planck scale. In the WIMP scenario, for example, new particles and symmetries are expected to arise around the TeV scale. This also makes it reasonably feasible to see the dark matter particle in direct detection experiments.

However, new physics below the Planck scale is not a necessity. There is nothing fundamentally wrong in having an extended desert between the electroweak scale and the Planck scale, or possibly the GUT scale. New physics is bound to emerge close to the Planck scale to restore unitarity in gravitational scattering, but in principle no new fundamental scale is needed to make sense of our world. In a minimal approach to the dark matter problem, no new scale should be added to solve the problem if it’s not needed. Thus, the most minimal dark matter model is one in which the dark matter particle is connected to the new degrees of freedom that emerge at the Planck scale to unitarize gravity. In this new paradigm, the dark matter particle is intimately linked to the theory of quantum gravity; it has a natural mass close to the Planck scale, and has only gravitational (planck-suppressed) interactions. We call this scenario Planckian Interacting Dark Matter (PIDM). 

The PIDM scenario has a natural string theory realization as the Kaluza-Klein excitation of the graviton in orbifold compactifications of string theory, as well as in models of monodromy inflation and in Higgs inflation. In these setups, dark matter is, in a sense, both a particle and a modification of gravity.