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I'm wondering how it is possible that light cannot reach us, even all the way out there where metric expansion is making the distance between our galaxy and their galaxy increase at greater than $c$.

Assuming the expansion is relatively linear (that's why Hubble's constant is "constant", I suppose; and even if it's not linear, or even discontinuous, this argument still works), it'll be exactly like the Ant on a Rubber Rope problem, in which the ant always gets to the end of the rope no longer how quickly it's expanding. Thus, the light will eventually reach us.

However, How Are Galaxies Receding Faster Than Light Visible To Observers? suggests that some light traveling in our direction never actually reaches us. Thus, there are some galaxies out there whose light will never, ever, ever (assuming infinite expansion, heat death, no interstellar extinction, etc.) reach us. But that doesn't make sense with the Ant on a Rope argument!

An extension to this problem would be (if I'm correct in the above), what would it actually look like? Would it appear that the galaxy is receding slower and slower, since the light beam is so awkwardly distorted? How massively would it be redshifted?

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A few points:

"the ant always gets to the end of the rope no longer how quickly it's expanding" is incorrect. The problem is that the rope cannot extend indefinitely, it will eventually slow down. Same with the universe, if it slows down, light that at the current expansion rate would not be able to reach us, will be able to do so at the slower rate.

"Thus, there truly is a limit to the volume of the universe that we will ever observe. But... that doesn't make sense!" It does make sense to many people, including me.

"Would it appear that the galaxy is receding slower and slower, since the light beam is so awkwardly distorted?", No remember that they will look as receding faster the farther they are. But their light will be changed to lower frequencies by the Doppler effect. So in practice the galaxies will become invisible to visible light well before they reach the edge of the "observable" universe. You will see faster and fainter galaxies until you will no longer see anything.

Update: in the commentary by nocieurghq, he links to a webpage were they show that the ant actually reaches any given point if the expansion rate is constant. Luckily for me, I will not have to burn my PhD certificate. The reason is that the assumptions of what is a constant rate are different. In the example of the rope, the rate of rope extension is actually decreasing with time, rather than constant. To see this, consider what a "constant rate" as defined in cosmology would look like. If the rope starts with a length $L$ at $t=t_0$, and then expands to a length $2L$ at $t=t_0+\Delta t$, then, if the rate is constant, it should expand to $4L$ at $t=t_0+2\Delta t$. That is, whatever the length, if the rate is constant, the length doubles after a time $2\Delta t$. That is not what happens in the rope example in which one end travels at constant speed. In such a case, at $t=t_0+2\Delta t$ the length will be $3L$ rather than $4L$. Thus the effective expansion rate actually decreases in that example.

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  • $\begingroup$ $\dot a=\text{const}$ is called a constant rate of expansion in cosmology because the relative velocities of galaxies are constant. Expansion slowing down means galaxies are slowing down relative to each other. It's a reasonable definition (and completely standard). $\endgroup$
    – benrg
    Jan 17 at 4:40
  • $\begingroup$ good point, thanks $\endgroup$ Jan 17 at 4:46

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