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I have read this question:

What is the difference between gravitons and gravitational waves?

I have read this on wikipedia:

However, if gravitons are the quanta of gravitational waves, then the relation between wavelength and corresponding particle energy is fundamentally different for gravitons than for photons, since the Compton wavelength of the graviton is not equal to the gravitational-wave wavelength. Instead, the lower-bound graviton Compton wavelength is about 9×109 times greater than the gravitational wavelength for the GW170104 event, which was ~ 1,700 km. The report[16] did not elaborate on the source of this ratio. It is possible that gravitons are not the quanta of gravitational waves, or that the two phenomena are related in a different way.

So are gravitational waves made up of gravitons or are they just waves in the fabric of spacetime?

It is a contradiction, because spacetime itself should not be made of anything that we know as of today, we do not know what spacetime is made of, we do not even know if there is a fabric of spacetime or what it is.

Wikipedia says:

Gravitational waves are disturbances in the curvature (fabric) of spacetime

So GWs are disturbances in the fabric of spacetime, and gravitons are the quanta of GWs. This would mean that spacetime is made of gravitons, which has to be a contradiction.

Question:

  1. are GWs made up of gravitons, or not?

  2. are gravitons the quanta of GWs or not?

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Suppose you start with some spacetime geometry. This could be any geometry but when explaining gravitational waves it's usual to start with a flat spacetime so the metric would be the Minkowski metric $\eta$. Now introduce a small perturbation to the metric $h$ (we won't worry at the moment exactly how this perturbation could be created). What we find is that we can write a perturbation $h$ that propagates through spacetime like a wave, and this is what we call a gravitational wave.

Conceptually the gravitational wave is not that different from an electromagnetic wave, though it is a tensor wave not a vector wave like an EM wave. This gravitational wave carries energy just like an EM wave carries energy.

As mentioned in a comment we don't have a full quantum theory of gravity, but for low energies we can obtain an effective theory using the same quantum field theory ideas as for electromagnetism. If we do this we get a particle called a graviton that is described as a state of the metric just as a photon is described as a state of the electromagnetic field. A gravitational wave is built up from gravitons in the same fashion that an electromagnetic field is built up from photons. However in both cases the relationship between the particle and the classical wave is a complicated one. A light wave is not a hail of little photons and a gravitational wave is not a hail of little gravitons. In both cases the particle is best understood as the exchange of energy between the wave and whatever that wave is interacting with.

So the answer to your question (2) is that if you accept that the effective theory for the gravitational field is a valid description then yes the gravitons are the quanta of gravitational waves. The answer to your question (1) is that the relationship between gravitons and gravitational waves is complicated, but then that's equally true for EM waves.

The big question is whether the naive quantisation of the metric is physically meaningful. An analogy often used is that if we didn't know water was made up from molecules we might attempt to quantise water by quantising the Navier Stokes equations. Mathematically this is perfectly possible, but of course it is physically meaningless. What we don't know if how far this analogy applies to gravity.

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There is an accepted theory of gravitational waves, and they have been detected. There is no accepted fundamental theory of quantum gravity and thus no accepted fundamental theory of gravitons. (However, physicists do make theoretical calculations of processes involving gravitons, using an approximate theory that should be valid at energies below the Planck energy. You can calculate Feynman diagrams with gravitons in a similar way to calculating Feynman diagrams with photons.) Gravitons have not been detected, and the possibility of detecting them is not within our current technological capabilities, now or in the foreseeable future.

Your first Wikipedia quote is from a paragraph discussing the possibility of massive gravitons. If gravitons exist, they are more likely massless and thus have an infinite Compton wavelength, just like the photon.

Physicists do not talk about quanta of waves but rather quanta of quantum fields. Waves are classical phenomena and do not have quanta.

There is no reason to assume spacetime is made of something, any more than there is a reason to assume an electromagnetic field or a quark field is made of something. (No, these quantum fields are not made of photons and quarks, and, if gravitons exist, spacetime would not be made of them.)

In currently accepted theories, spacetime and quantum fields are ontologically fundamental things which are not made of other things. There is a widespread assumption that the metric field of spacetime must be a quantum field like all other fields. In this case it is expected that it would have quanta (“gravitons”) like all quantum fields do. However, technical problems with this idea, such as non-renormalizability, have not been overcome. This has led to radically different approaches, such as string theory and loop quantum gravity, which are not yet successful or accepted theories.

A more recent speculative idea is that spacetime is not ontologically fundamental but rather an emergent phenomena arising from quantum entanglement. I don’t understand what happens to gravitons in this scenario.

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