Light emitted from galaxies receding faster than $c$

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?

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.
• $\dot a=\text{const}$ 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). Jan 17 at 4:40