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I've heard this referred to as a "dark sector", or "hidden sector":

https://en.wikipedia.org/wiki/Hidden_sector

The scary thought is that, were it true that there is voluminous richly-interacting dark matter, we would be the actual dark sector.

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"Imagine a universe, like ours, overlapping ours, except where some of its matter spontaneously rips itself apart, and other matter can be mashed together if you squeeze it hard enough." "That's nightmare fuel! Fortunately it's probably impossible, so far as we can tell."

I can imagine other scary thoughts, though. Ever read any Warhammer lore?

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It's possible that there could be an entirely different charge mechanism that works exactly like ours but they don't interact at all; but the whole premise of dark matter is that it doesn't seem to have any self-interaction outside of gravity.
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The premise of dark matter is that it is something with gravity that does not otherwise interact with any of our detectors (e.g. "normal" matter); or that any such interactions are weak enough that it is plausible we have not noticed.

By itself, that does not exclude the possibility of dark matter having other interactions which do not interfere with our detectors.

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Our observations are much more consistent with a form of dark matter which does not self-interact, or which does so incredibly weakly.

Don’t ask me how or why, but this is essentially the universal (pun intended) consensus amongst cosmologists.

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That doesn't rule out "dark matter having other interactions which do not interfere with our detectors" Dark sector theories which include other dark particles or new particle interactions are not controversial amongst cosmologists in this space. For example there's a whole area of study around "dark photons" which would mix with our photons and interact with dark matter.
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I guess so. If you could sort of measure the dark matter distribution, there might be a structure under it, but strictly speaking isn’t it limited to saying it’s not at least likely to emit light but does have mass.
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It's very hard to explain the gravitational halo around the galaxies if your dark matter can interact with itself. If it interacted like normal matter, it would have a distribution similar to the gases, and not spread way into intergalactic space.
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But it does bound it: if dark matter has self-interactions then the apparent distribution would be different. Regular matter forms stars and galaxies and all the structures we see because it can self-interact.

Since where we see dark matter mass shadows we don't see structure formation, what self interaction it may have must be very, very limited.

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You might like this old paper then:

https://arxiv.org/abs/0903.0660v1

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