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 Originally Posted by OngBonga
Distant galaxies don't exist anymore.
This kinda depends on what you mean by a galaxy's existence.
Galaxies don't "burn out" like stars do. Galaxies are self-sustaining in a sense. Exploding stars send out incomprehensibly massive shock waves through the galaxy. They spread out fused and more importantly, not fused atoms into a nebula.
Exploding stars create a zone with a "high concentration" (on galactic scales - its still more sparse than most man-made vacuum chambers) of materials that could form into a star, as well as other shizwaz that could form planets and other stuffs. They also send out a shock wave that presses other nebulae from other supernovae into each other, and cause new star formation. In this sense, galaxies are efficient at using up some fuel, separating out unfusable materials and re-mixing the fusable materials into new stars.
The process of fusing Hydrogen into heavier elements is actually very slow compared to the amount of Hydrogen fuel available in the Galaxy. For example, the Milky Way is ~13.4 billion years old (in "proper time" i.e. an observer in the Milky Way the whole time would say that's how long they've been here - time is wonky). The age of the universe in proper time is only 13.7 billion years. So you can see that the age of the universe hasn't been very long, yet, on the time scale of galactic evolution.
Andromeda and the Milky Way will collide in ~ 4 billion years or so (IIRC). Would you say that both galaxies cease to exist and a new galaxy or collection of galaxies is born? Serious question. If you would say that, then it is fair to say that some distant galaxies no longer exist, due to mergers with other galaxies. If you would not say that, then even the most distant galaxies probably still exist.
But this is playing fast and loose with time. What we see of those distant galaxies, here and now, is what they are, here and now. There's no way to relate "here and now" to "there and now" without describing how you plan to traverse that spacetime. To the photons we observe of those distant galaxies, no time has passed between leaving that galaxy and being observed on Earth, due to photons experiencing infinite time dilation in their (non-inertial) rest frames.
So the manner in which you cross spacetime makes a difference in how you relate the time between 2 locations in spacetime. The time between events is relative. The order of events is relative - unless they are causally linked.
If you and I coordinated our clocks, and flashed a bright light at each other at the same time from some distance apart, we'd see our own light happen before we saw the other's light happen. So the order of events is relative. But in this case, the turning on of each of our lights was not caused by the other person turning on their light, so there is no causal link between these events.
If we change our experiment such that the moment I see your light, I turn on my light, then the time between these events will vary based on who's observing (I see your light an my light simultaneously, you see a distinct gap in time between them), but no observer would ever see my light turning on before your light, as now there is a causal link in the order of events. The time between your light turning on and my light turning on will vary, but the order will not.
 Originally Posted by OngBonga
When we look very far away, we're looking very far back in time. We're not observing the universe expanding now, we're observing it expanding in the past. And it seems to be expanding more in the distant past than in the recent past. Which implies the expansion isn't accelerating, on the contrary it's slowing down (technically still acceleration but you get the gist).
There's more than 1 expansion in this story (or more than 1 phase in a single expansion).
There's the big bang. First expansion.
A short time later (to us / on a human scale - it was the entire age of the universe back then) the universe went through a phase of exponential expansion called cosmic inflation.
After that, the expansion rate slowed down dramatically over a relatively short period.
The rate of expansion dropped dramatically after inflation, but it didn't stop. It has continued ever since.
When we look out in space, we look back in time. This is correlated; closer objects in space are closer in time. Further in space means further in time. So we are probing the history of the universe when we look at galaxies of varying distances. As we look at different distances, we look at different times. As we explore the red shift of different distances, we explore the expansion rate of the time between us/here and there/then. So we can build a map in time that shows the rate of expansion as it changes over time.
The part where you said, "it seems to be expanding more in the distant past than in the recent past," is correct and incorrect. It was expanding more during the period of cosmic inflation, but then the rate dropped (inflation ended) to an apparent minimum. As it has been increasing since then, as well, albeit at a far lower rate. That rate has still been increasing since then, just at a much slower rate than what lead to cosmic inflation.
 Originally Posted by OngBonga
Of course, shortly after the big bang, the universe experienced a rapid period of insane expansion. Is that not what we're observing when we see red shift of distant galaxies?
No, it is not. The period of cosmic inflation happened prior to the formation of atoms, let alone stars and galaxies.
The formation of stars and galaxies doesn't happen for another 200 - 500 million years.
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