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Why do some physicists think that dark matter interacts via the weak interaction? Is there evidence of that, or is it theoretically likely, or is it just the easy option to test?
This is due to the (at this point mostly historical) 'WIMP miracle'. I'll try to summarize.
The initial conditions of the universe were (probably) set by the decay of a scalar field known as the inflaton into standard model particles. No one knows its preferred decay channel (whether it predominantly decays into photons, fermions, neutrinos, the Higgs, etc). It turns out that for the standard model particles, this doesn't matter: you have a very hot and dense early universe, and particle interactions happen all over the place, and very quickly the universe reaches a thermal distribution of the types of particles--this sets the ratios of all the fundamental particles to what they should be.
If the dark matter particle didn't interact at all, then there's no way for it to be produced in this spate of temperature-inspired particle interactions. Since it's unlikely that the inflaton would decay in just the right amount to give the density of dark matter we see today, you want it to interact just a little bit, so that in the early universe, it can be thermally produced. Too much though, and we'd have noticed it in the sky. The weak force provides such an interaction term, with particles at the characteristic electroweak scale (~100 GeV) giving cross sections (interactions with the standard model) of ⟨σ(χχ→S.M.)v⟩≃3×10−26m3s⟨σ(χχ→S.M.)v⟩≃3×10−26cm3s, which is low enough that WIMP to standard model interactions should be vanishingly improbable today, but high enough for thermal production in the early universe. The WIMP miracle is that a particle around that mass with around the predicted cross section gives around the right proportions for the amount of dark matter relative to normal matter (1000% or so).
The problem is we've looked at particles in that mass range pretty hard (with both direct detection experiments like LUX and particle accelerators like the LHC) and haven't found anything. So a 'standard' WIMP is all but excluded, and 'nonstandard' WIMPs are decidedly less miraculous.