Particle physics came to the rescue, sort of. Quite independently of the dark matter question, particle theorists had hypothesized about "Supersymmetry" to account for some oddities in their older theories. Among other things, Supersymmetry ("SUSY") predicts that the Big Bang should have churned out a huge number of "neutralinos" - massive, stable, neutral particles that a) do not emit or absorb light, and b) do not clump together into stars and planets, but rather drift diffusely around galaxies. So, in fact, it is entirely reasonable that the missing "dark matter" consists of a sea of SUSY neutralinos bathing every galaxy in invisible mass.
Neutralinos don't interact with light, but they do interact with each other. Every once in a while, two of them will collide and emit a little spray of particles. Among the spray will be some high-energy positrons. This is the exciting part: there is nothing else in the Universe that is expected to emit little bursts of high-energy positrons. Other processes can generate positrons, true, but they are generated at all energies, spread out all over the spectrum. These neutralino collisions emit positrons at high energy and not at low energy! Since this is happening all over the galaxy, the positrons will just be a part of the general background cosmic ray spectrum. If AMS sees extra positrons at high energy, that's a good indication that there are neutralinos colliding out there!
Aside from the neutralino idea, some other hypothetical dark-matter particles (Kaluza-Klein bosons, for example) would produce similar positron results.
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