If a heavy species and a light species are in thermal equilibrium in the early universe and decouple from everything else (because the interaction rate falls du

If a heavy species and a light species are in thermal equilibrium in the early universe and decouple from everything else (because the interaction rate falls due to expansion) at the same time (i.e., same temperature), the heavy species cools faster with the expansion of the universe than the light one does. This is because the temperature of a relativistic gas cools like ~1/a while the temperature of a non-relativistic gas cools like ~1/a^2, where a is the expansion factor. Thus, in that simplified case, the heavier one really is colder, not just moving slower. Now, it turns out there's an opposing effect where, depending on how the DM couples to normal matter, the heavier particle decouples earlier (and thus at a hotter temperature) because creation/annihilation have to stop once the temperature drops below the energy scale associated with the rest mass, and this effect can often dominate the previous one. The thing that actually matters, functionally, (and thus the thing that is used to classify DM types) is the thermal velocity relative to the speed of the Hubble flow during structure formation. It is of course true that, as you say, at fixed temperature a species of low mass will have higher velocity than a species of high mass, but CDM and HDM are not at the same temperature due to the opposing expansion and decoupling effects mentioned above (among other things). My original point, in response to andsoitis, was that heat is an abstract thermodynamic concept that does not depend on the particle species. The names "hot" and "cold" were not metaphorical, even if the boundaries between CDM and HDM are not literally drawn using a threshold temperature. For instance, per the symmetry article: > “Even though the universe was very hot at the time, axions would have been very cold at birth and would stay cold forever, which means that they are absolutely cold dark matter.” Even though axions are very light, Graham says, “because they exist at close to absolute zero, the temperature where all motion stops, they are essentially not moving. They’re kind of this ghostly fluid, and everything else moves through it.” This is why it's wrong for phys.org to say that "cold", "warm", and "hot" do not refer not to the temperatures of the matter itself. Likewise, even if "hard" and "soft" cheeses are formally defined by whether they are dried and aged (such that there could be a few "hard" cheeses that are softer than the hardest "soft" cheeses), it would be wrong to say "'hard' and 'soft' do not refer to the firmness of the cheese itself".