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In cosmology, we speak of galaxies racing away from us due to the expansion of the universe. The light from these galaxies appears red-shifted. If the galaxies are far enough, then the space between us actually expands faster than light, and light from these galaxies never reaches us.

My question is what would happen in a Big Crunch universe - where space contracted faster than light. How would a galaxy getting closer faster than the speed of light appear to us? Would its light be extremely blue-shifted? Would we see photons emitted later arriving first, so it would appear as though it is going back in time, and getting farther away?

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In an ideal FLRW recollapsing universe, the collapse is just the time reversal of the expansion. The Doppler shift factor at all times is $a(t_\text{emission})/a(t_\text{detection})$, and during the collapse, that's a blueshift instead of a redshift.

Realistically, the big crunch would be much more chaotic than the big bang, because entropy would continue to increase, and so it wouldn't be well described by FLRW cosmology.

Regardless of the details, though, there is never a "superluminal blueshift" where light arrives backwards.

Recession speeds in cosmology are defined in such a way that the speed $c$ has no special significance. See this answer for more information. Galaxies with a recession speed higher than $c$ aren't traveling faster than light (i.e., outside of the light cone), and it isn't true as a rule that light from them never reaches us. (In ΛCDM, there is a cosmological horizon beyond which we can't see light, but it doesn't coincide with the point at which the recession speed is $c$.)

In a collapsing universe, galaxies can have an approach speed (defined in the same way) that is larger than $c$, but light from those galaxies would still reach us before the galaxy did, and would reach us with a large positive blueshift, not a negative Doppler shift (backwards).

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    $\begingroup$ I'm confused. There are whole articles explaining that due to the expansion rate of the universe, some galaxies' light will never reach us. en.m.wikipedia.org/wiki/Observable_universe $\endgroup$
    – Alon Navon
    Commented Sep 22, 2021 at 6:42
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    $\begingroup$ "This fact can be used to define a type of cosmic event horizon whose distance from the Earth changes over time. For example, the current distance to this horizon is about 16 billion light-years, meaning that a signal from an event happening at present can eventually reach the Earth in the future if the event is less than 16 billion light-years away, but the signal will never reach the Earth if the event is more than 16 billion light-years away." $\endgroup$
    – Alon Navon
    Commented Sep 22, 2021 at 6:43
  • $\begingroup$ "Light that is emitted today from galaxies beyond the more-distant cosmological event horizon, about 5 gigaparsecs or 16 billion light-years, will never reach us, although we can still see the light that these galaxies emitted in the past." en.m.wikipedia.org/wiki/Expansion_of_the_universe $\endgroup$
    – Alon Navon
    Commented Sep 22, 2021 at 6:49
  • $\begingroup$ @AlonNavon I just meant that there isn't a rule that when the recession speed passes $c$, the object becomes invisible. The recession speed is $c$ at $c/H_0\approx 14\text{ Gly}$, which is strictly less than the $16\text{ Gly}$ distance to the cosmological horizon. The matter that emitted the CMBR had a recession speed at that time of around $70c$, I think. In open matter-dominated models, recession speeds can be arbitrarily large, but there's no horizon and all light eventually reaches everywhere in the universe. Etc. $\endgroup$
    – benrg
    Commented Sep 22, 2021 at 7:28
  • $\begingroup$ From Alon's link to the Observable Universe on Wikipedia: Assuming dark energy remains constant (an unchanging cosmological constant), so that the expansion rate of the universe continues to accelerate, there is a "future visibility limit" beyond which objects will never enter our observable universe at any time in the infinite future, because light emitted by objects outside that limit could never reach the Earth.- can you reconcile that with your statements, @benrg? Assuming you and the article are both correct, how do you arrive at different conclusions? $\endgroup$
    – AnoE
    Commented Sep 22, 2021 at 8:10

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