Assuming that dark matter is not made of WIMPs (weakly interacting massive particles), but interacts only gravitationally, what would be the possible mechanism giving mass to dark matter particles? If they don't interact weakly, they couldn't get mass from interacting with the Higgs field. The energy of gravitational interactions alone does not seem to be sufficient to account for a large particle mass. Would this imply that dark matter consists of a very large number of particles with a very small mass, perhaps much smaller than of neutrinos? Or do we need quantum gravity to explain the origin of mass of dark matter?
Dark Matter – What Gives Mass to Dark Matter Particles?
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There are some standing anomalies that could be explained by non-gravitational dark matter interactions. For example, Fermi-LAT is an indirect detection experiment (i.e. an experiment that looks for the debris of a dark matter decay that occurred far from Earth), and it currently reports an excess of gamma rays. There are occasional claims that nontrivial dark matter interactions make galaxy simulations work better. Excess cooling of astrophysical objects, such as white dwarfs, also "hints" at dark matter which could carry extra energy away. Finally, there are even some direct detection experiments like DAMA which claims to see dark matter interactions.
However, these anomalies generally don't point in any coherent direction, and often fade away. Most of the time, all they reflect is an incomplete understanding of astrophysics. So you're completely right: the default hypothesis, if you're an astrophysicist, is that dark matter has no non-gravitational interactions.
But particle physicists are investigating a different problem: they're trying to explain what dark matter is made of and how it got here. This always takes place within specific models. No direct detection experiment just looks for "dark matter" per se; instead they look for specific dark matter candidates.
The longest-running direct detection experiments look for particles whose mass is about a hundred times the proton's, and whose interactions are primarily electroweak. This specific guess is because such dark matter candidates exist in many supersymmetric extensions of the Standard Model, which were favored for other reasons. In such models you automatically get dark matter interactions; you can't get rid of them unless you purposefully tune the interaction to zero. Moreover, when you run the numbers you find that such models automatically produce the right amount of dark matter in the early universe, a result known as the WIMP miracle. (Production is another reason you want interactions; you could just postulate you get the right amount by tweaking the initial conditions, but it would be nicer to explain why.)
These days, a wider range of dark matter candidates are getting attention, leading to a great variety of cheap direct detection experiments, which target each candidate's particular interactions. Everybody working in this field (which comprises less than 1% of physics at large) is perfectly aware that all these candidates could be nonexistent, and that the whole enterprise might have been completely doomed from the start. But that is true for any scientific endeavour; that risk is what makes it exciting in the first place!
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I think this question contains a misconception unfortunately caused by popular science descriptions of the Standard Model.
The question seems to assume there needs to be some concrete source that particles "get" mass from, as if mass is a resource like money and the Higgs field is giving it out. But that's not right. In a generic field theory there is no issue adding a new field $\psi$ whose particles have mass. The only thing you have to do is make sure the Lagrangian has a term proportional to $\psi^2$.
You might protest that this violates the conservation of energy because the mass has to "come from" somewhere, but that's not right. Mass is the energy price for creating a particle. I don't create money by changing the pricetag of an item in a store.
The reason science popularizers say that mass must come from the Higgs mechanism is because of a peculiarity of the Standard Model (SM). The symmetries of the SM forbid a term such as $\psi^2$ for any field $\psi$ in the SM, so we need a trick to get a mass term. In brief, the Higgs field $\phi$ allows us to write terms like $\phi \psi^2$ which do respect the symmetry. This is an interaction term, but we can set up the Lagrangian so the Higgs field $\phi$ acquires a constant part, yielding the $\psi^2$ mass term we wanted.
However, once you start speculating about dark matter models, especially dark matter that does not interact with the electroweak force at all, these constraints don't apply and generically there is nothing forbidding a $\psi^2$ term. There's no need for any special mechanism for "giving" mass. You just treat mass exactly like you did in high school, intro mechanics and quantum mechanics: write it down, call it $m$ and call it a day.