One of the possible ends to the universe is the Big Crunch, although I understand the current understanding is a flat universe*. If the universe were in a collapsing period, would we be able to tell? What would we observe? How long after the start of the collapse would we be able to tell?
We could tell through the Doppler shift of the light from other galaxies, just as we can tell the universe is currently expanding–except that we’d see galaxies hurtling towards each other instead of hurtling away from each other.
You are correct in saying that the observed spatial geometry is flat, or at least very near to flat. In older (FLRW) cosmological models a flat Universe corresponds to a Universe that expands for ever, but whose expansion rate slows down so that it approaches zero at an infinite time in the future. However observations lead us to believe that there is a mysterious negative pressure energy in the Universe called dark energy that causes the rate of expansion to accelerate. So, for the most part, the big crunch has been ruled out as the ultimate fate of the Universe.
Of course it is still an interesting and valid question to ask if the Universe were to go into a contracting phase what would we observe?
The evidence for expansion of the Universe is the redshift (Hubble shift) of distant stars, so unsurprisingly the most direct evidence for contraction would be blueshift. However the red shift/blue shift of the Universe depends on the net expansion/contraction of the Universe between when the light was emitted and when it was received.
In most models which end in a big crunch precede as below:
big bang —> expanding phase —> contracting phase —> big crunch
As an expanding phase directly precedes the contracting phase, this means that early in the contracting phase we would still observe all but the very nearest objects to be redshifted. This is because for such objects the amount of expansion the Universe would’ve undergone since the light we observe from them was emitted is greater than the amount of contraction. In theory we could observe blueshift in nearby objects, but in practice the amount of blueshift would be too small to observe. It would only be once we were a significant amount of time into the contracting phase that blue shift would start to occur in further away objects and would become observable.
Beyond directly observing the blueshift, we could still possibly work out the Universe is contracting by observing the redshift and notice that the expansion of the Universe in earlier times was slowing down quick enough to currently put us into a contracting phase.
Some have speculated that the arrow of time would suddenly reverse when the Universe entered into a contracting phase, but this is highly contentious and difficult to justify.
Hubble shift is not Doppler shift, or at least is not commonly understood in that way. As explained above, even though a galaxy is contracting towards us it can still be redshifted.
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I’m not sure I understand the bolded phrase. Are you saying that some people theorize that time would run backwards? How would that work?
So at what point would space start becoming crowded- galaxies colliding and so forth. And which effect would predominate- more and more of space being filled with black holes, or the cosmic background radiation becoming too hot for survival? IOW, what are things like for any surviving observers close to the end?
Yes: the way I understand it, is that proponents of this idea believe that entropic arrow of time would be reversed, meaning the usual order of irreversible processes would be reversed. This would be difficult to justify as otherwise the turnaround from expanding to contracting is pretty much undetectable. Some believe the reversal might happen sometime into the contracting phase, but that is still problematic.
The thing is, saying that the entropic arrow of time would reverse when the Universe started contracting is exactly equivalent to saying that the Universe would just suddenly end for no discernible reason right at the moment of maximum expansion.
The evolution of the Universe would be symmetric, so assuming the early phases of such a Universe was approximately like ours, the evolution of the temperature in late times would be the reverse. This would mean that about 17 million years before the big crunch the ambient temperature of the Universe would become such that it may be possible for life to survive without the energy input from the star, however about 10 million years before the big crunch the Universe would become hotter than the boiling point of water and a few million after that so hot even iron would melt.
It would be sometime after the temperature became too high for life to exist anywhere in the Universe that collisions between stars become common and coalescing black holes would pose a hazard to observers.