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Take a primordial black hole and measure the Hawking radiation over a large amount of time by gamma-ray detectors, as well as a Large neutrino detector. Using theoretical calculations about the composition of Hawking radiation combined with the detected neutrino flux you can determine the neutrino luminosity. Combine that with the luminosity in the form of gamma rays and whatever particles the Black hole is hot enough to produce, and then compare that with the loss in mass over time. If you detect a discrepancy, it's likely the missing flux is in the form of gravitons.

Would this experiment work? I did also consider using a primordial black hole cold enough to not emit neutrinos but the Hawking radiation calculator I used gave a black hole lifetime greater than most estimates of the lifetime of a proton.

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In principle, I see no reason for why that would not work. However, notice that even the existence of primordial black holes can be thought of as questionable and their detection even more. Furthermore, as mentioned in the comments, any particle should be taken into account, including particles we don't yet know to exist.

It should also be added that gravitons, just as photons, are massless particles. Hence, you don't really need a small black hole to get graviton emission. It seems to be that it would be more fruitful to use cold black holes as a test for gravitons, as that could allow to search for massless particles.

It might interest you to know that using Hawking radiation to probe new particles is indeed a recent proposal, although the case I know of is as a search for dark matter. See arXiv: 2107.00013 [hep-ph] if this interests you.

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  • $\begingroup$ Using cold black holes won’t work because your equipment will disappear to proton decay undergo proton decay before the black hole loses a significant portion of its mass. $\endgroup$
    – blademan9999
    Commented Jul 31, 2023 at 16:31
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A graviton is a quantum of gravitational radiation, and when people talk about demonstrating the existence of gravitons, they mean demonstrating that gravitational radiation is quantized. Your experiment doesn't do that. At best, if you solved all of the other problems, it would only show that gravitational radiation exists, which we already know.

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    $\begingroup$ Hawking radiation obeys Plank spectrum, so gravitational Hawking radiation does imply quantization of gravity. $\endgroup$
    – A.V.S.
    Commented Jul 31, 2023 at 17:59
  • $\begingroup$ @A.V.S. ...maybe? It doesn't seem to make any sense for gravity to be unquantized in the first place. If, somehow, the universe makes that weird combination work, I don't see why it wouldn't include gravitational radiation from an evaporating black hole in an amount that's consistent with quantization of gravity, because... that amount is required for consistency, somehow. It's implausible, but no more implausible than the premise. $\endgroup$
    – benrg
    Commented Jul 31, 2023 at 18:53
  • $\begingroup$ Before planks law there was the UV-catastrophe. Given how hawking radiation works, you should have gravitational radiation emitted unless gravitons exist. $\endgroup$
    – blademan9999
    Commented Jul 31, 2023 at 19:19

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