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Is it possible that there are newly formed galaxies which are moving so fast that we will can't see them, even though they exist "inside" our visible universe?

I've been wondering if this could be a thing, or if I'm misinterpreting one of the following ideas.

  1. The universe is expanding and thought to be accelerating.
  2. Galaxies past about 4,200 parsecs are expanding away from us faster than the speed of light.
  3. We are moving away from those galaxies at the same speed relative to them.

Is it possible for us to "outrun" the light from a distant galaxy, to the point where we either can't or will never see it? Something about this seems wrong, but I can't quite figure out what.

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You're exactly right. What you've discovered, is the cosmological horizon, the distance beyond which we can no longer have any causal contact with distant galaxies. The fact that galaxies that are out of causal contact with each other in fact appear to be so uniformly distributed in density and temperature is considered a major problem in cosmology, the homogenity problem. An attempt to solve this is one of the reasons physicists have generally gone down the road to inflationary cosmology.