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Hawking radiation emitted from galaxy clusters
photons amassed in the recesses of intergalactic space ... due to the reflective capacity of the intervening material
radiation pressure caused by this buildup of photons between galaxy clusters
If Hawking radiation were to be anything, it would almost certainly be gamma-ray photons.
Ok, what would it be other than gamma-ray photons then?
and what wavelength would these photons be given the gravitational potential energy they're subject to near a black hole event horizon?
In any case, any and all photonic radiation (either incoming or outgoing) near an event horizon will be de facto gamma radiation due to the blueshifting near the horizon.
Einstein’s theory of general relativity predicts that the wavelength of electromagnetic radiation will lengthen as it climbs out of a gravitational well. Photons must expend energy to escape, but at the same time must always travel at the speed of light, so this energy must be lost through a change of frequency rather than a change in speed. If the energy of the photon decreases, the frequency also decreases. This corresponds to an increase in the wavelength of the photon, or a shift to the red end of the electromagnetic spectrum – hence the name: gravitational redshift. This effect was confirmed in laboratory experiments conducted in the 1960s.
The exact opposite happens in very strong gravitational fields.
You are choosing the wrong hill to die on.https://energyeducation.ca/encyclopedia/Radiant_heat#:~:text=Radiant%20heat%2C%20also%20known%20as,(unlike%20convection%20and%20conduction).
That you seem ignorant of the fact that gamma rays can impart heat to an object.