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Are there any theoretical predictions or "educated speculation" by cosmologists on how matter will be distributed at very, very distant times in the future?

For example, will matter be:

  • agglutinated into a big cold ball?
  • diluted across the universe in tiny bits?
  • large clusters of matter here and there?
  • something else?
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  • $\begingroup$ I adjusted the wording a bit to make sure this was asking for answers based on science and would not require opinion-based answers. I did that because this ended up in the review queue as potentially opinion-based, even though there are no close votes yet. I'm pretty sure the adjustment does not have any significant impact on the fit of the answers, and I wouldn't want to see this closed! $\endgroup$ – uhoh Dec 31 '19 at 14:19
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The basic answer to your question is "yes to all of the above, in different eras".

If we use the concordance $\Lambda$CDM model, with the cosmological constant remaining constant, then the universe will keep on expanding at an accelerating rate forever. The first big effect of this is that gravitationally bound clusters of matter such as galaxy clusters will become separated by ever larger distances (but they themselves are not changed noticeably by this). The current cosmic web of galaxies and voids disintegrates into little separated droplets over the next few hundred billion years.

Inside these "island universes" local gravitational interactions tend to make galaxies merge with each other, eventually ending up as a single big elliptic galaxy.

Over long time (around $10^{20}$ years) close-encounters between stars lead to some stars being flung out of the galaxy into the outer darkness, and the others bunching up closer to the central black hole. Eventually this reduces the galaxies to big black holes and scattered stars. The scattered stars cool down to low temperatures.

If proton decay happens, then somewhere beyond $10^{33}$ years or so the scattered stars quietly and slowly evaporate into elementary particles. The black holes are also expected to evaporate, and after $\sim 10^{99}$ years or so the only thing left will be separate stable elementary particles - electrons, positrons, neutrinos, photons, maybe some stable dark matter. Due to the expansion each particle will be alone in its visible universe.

A good overview is Adams and Laughlin's "The Five Ages of the Universe" (or their paper, for more detail).

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The current possibilities are:

  • The Big Freeze
  • The Big Rip
  • The Big Crunch
  • The Big Bounce
  • The Big Slurp

However, with our current incomplete understanding of dark energy, we can't say for certain.

Based on TV documentaries about cosmology, it seems that a combination of the Big Freeze and Big Rip are the current favorites -- at least among science communicators.

The Big Freeze is: Everything will get cold, the black holes will lose energy from Hawking radiation, and eventually there will be total entropy/no information.

The Big Rip is: Everything will be ripped apart by dark energy. No more protons or neutrons (I'm not sure about electrons and quarks, though).

The Big Crunch is: Everything becomes one black hole.

The Big Bounce is: After becoming one black hole, there's another Big Bang.

The Big Slurp: The universe ends "early" with vacuum decay exerting a force that causes all matter to just stop being matter, by changing the universal constants. Such an end would propagate at the speed of light, so we wouldn't see it coming (since no information can travel faster than the speed of light).

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