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Large galaxies appear to be dominated by dark matter, as do dwarf spheroidal galaxies with masses of perhaps $10^6 {\rm M}_{\odot}$. But (as far as I'm aware) there is no evidence for dark matter in galactic star clusters at these masses and below.

What are the smallest structures in the Universe that are dominated by dark matter and what sets any lower limit?

Or is it just that there are smaller dark matter structures, but they are incapable of attracting enough baryonic matter to make stars and hence reveal themselves?

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In the cold dark matter (WIMP) picture the mass limit for structures is quite low. I'm not sure if anyone knows how low, because in detail it depends on the mass of the actual dark matter particle and its free streaming length in the early Universe, but people seem to agree that forming Earth mass halos is plausible. For warm dark matter (e.g. sterile neutrinos) the cutoff is much higher, again depending on the particle properties, but something like $10^7\,{\rm M}_\odot$.

No dark matter structures with masses below about $10^9\,{\rm M}_\odot$ have yet been detected. As you suggest in your question, the idea is that these are too small to accumulate much gas and may never form even a single star. They may be expected to contain a bit of hydrogen and show up in surveys of the $21\,{\rm cm}$ hyperfine transition, but the line widths are so narrow that it is very challenging to tell such a signal apart from noise in a single channel of a radio detector. There is also an ongoing effort to find low mass dark matter structures via their gravitational properties.

Globular clusters appear to be a different thing altogether and I've never seen a claim for one with a measured dark matter component. It is precisely because they are dark matter free that they are classified as star clusters instead of galaxies. There are a couple of edge cases, though: a dwarf galaxy stripped down to leave only the core can look quite a bit like a GC. But at that point it would have lost almost all of its DM content as well.

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