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I have understanding sphere eversion as #1 on my bucket list (if I ever get a round tuit) but understanding metric expansion seems to be a rapidly receding possibility :-)

Question: Suppose it takes me 100 million years to evert a sphere and I manage to live that long. If I watched extremely red-shifted galaxies near the edge of the observable universe for a very long time, how would their appearance change (aside from their own natural evolution)? Would their red shift z remain constant or increase/decrease? Would more of them appear?

If the radius observable universe is growing and by the rate and mechanisms described in the answer linked below, I'm thinking that at a given observing red-shifted wavelength new galaxies would appear farther away, and so the ones that were at the limit before must have somehow become less red-shifted.

This question was inspired by this answer to Does the mass of the observable universe ever change? but here I'd like to ask for supporting sources as that will allow me to read further. Thanks!

Related in Physics SE, my previous efforts to try to understand metric expansion:

Somewhat related:

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    $\begingroup$ The redshift of galaxies entering the horizon decreases from infinity to a minimum value, after which is increase to infinity again. I started writing an answer, but then I found this one and this one on physics.SE, which I think answer your question. $\endgroup$
    – pela
    Jul 4 at 11:16
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    $\begingroup$ Your question about observing at a given redshift is also answered there: If you follow the dashed lines (e.g. the one called "z = 50") in answer #2's spacetime diagram, you'll see that in the beginning more and more distant galaxies will have that redshift, but at some point in the future, galaxies with that redshift will be progressively nearby. $\endgroup$
    – pela
    Jul 4 at 11:18
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    $\begingroup$ @pela I checked briefly and I can see I will need to look at them again in the morning here, and I'm not confident I'll be able to understand them sufficiently to know the answer to my question here with any confidence. If those answers tell us just how the current horizon galaxies' appearance will be different in 100 million years, if their red shift z will go up or down and if we'll see more, and that can be summarized briefly here, that would be an ideal answer; a few sentences linking to those answers as authoritative sources. $\endgroup$
    – uhoh
    Jul 4 at 12:47
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    $\begingroup$ Okay, I tried to have a go at it. $\endgroup$
    – pela
    Jul 5 at 11:13
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    $\begingroup$ I made my own spacetime diagram, omitting some information that wasn't necessary. $\endgroup$
    – pela
    Jul 6 at 13:56
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tl;dr Their redshift would first decrease from $\infty$ to $\sim60$, then increase to $\infty$ again. And more eventually appear.

The answer to this question is somewhat non-trivial, and will depend on the cosmology of the universe you're considering. But in our Universe, in which dark energy supposedly accelerates expansion, what happens can be summarized as this:

Qualitative description

As time goes, light from more and more distant regions will have had the time to reach us, so new matter enters our (particle) horizon all the time. For an infinitesimally small period of time, the redshift $z$ of that matter is infinite, but will then decrease with time, as the matter moves further into our observable Universe. However, at some point later on, the accelerated expansion will speed up the matter, increasing its redshift.

Quantitative description

The behavior is best understood using comoving coordinates, i.e. the coordinates that expand along with the Universe. In these coordinates, galaxies and other matter stay fixed (except for a peculiar velocity which is of no importance to the principle). The relation between real, physical distances and comoving distances is $$ d_\mathrm{phys} = a\,d_\mathrm{com}, $$ where $a$ is the scale factor (the "size") of the Universe, normalized to be $a=1$ today.

Consider an observer at a time when the scale factor of the Universe was $a_1$, observing light emitted earlier when the scale factor was $a_2 < a_1$. The redshift $z_{12}$ observed by this observer is $$ z_{12} = \frac{a_1}{a_2} - 1. $$ Now this is simple enough; the non-trivial part enters when calculating how $a$ evolves with time, which is done by integrating the Friedmann equation. This can only be solved analytically in simple cases; in general it must be solved numerically.

Numerically solving the Friedmann equation can also yield the distance to an observed object. In this way you can convert a redshift, or a corresponding scale factor, to a distance. But you can also do the opposite; that is, convert a given, observed redshift at a time $t_1=t(a_1)$ of light emitted at time $t_2=t(a_2)$ to a (comoving) distance $d_{12}$.

I can add the full set of equations, if you like, or you can have a look at this and this answers on physics.SE. For now, I've just implemented them in Python and plotted the (hopefully) illuminating diagram below.

Example

Below is a spacetime diagram, showing the Universe at all times (along the $y$ axis) and, correspondingly, at all scale factors, as a function of comoving distances (along the $x$ axis). A given epoch is a horizontal line (e.g. "now"), and a given position — also called a worldline — is along a vertical line (e.g. "here").

Horizons in the spacetime diagram

Everything we observe lie on our part light cone (red) which, as time goes, converges toward our event horizon (orange); the part of the Universe we will ever see. The observable Universe is the part of the "Now" line inside the green lines marking the particle horizon.

Curves of constant redshift

The dashed, cyan lines show the curves of constant redshift, calculated as described above. That is, at any time (given by a horizontal slice of your choice), a redshift (given by one of the dashed lines of your choice) will be observed for objects lying at a comoving distance given by the crossing of those two chosen lines.

spacetime

As an example, let's consider the highest-redshift observed galaxy, GN-z11, which is currently observed at $z=11.1$.

The worldline of GN-z11 is show as a black, dashed line, and our view of GN-z11 today is marked by a star lying on our past light cone. GN-z11 was not inside our observable Universe at the time we see it today; it only entered our cosmic horizon (the green line) when the Universe was just over $t = 4\,\mathrm{Gyr}$ old. As time went, its redshift decreased from $\infty$, to $100$ at $t\sim6\,\mathrm{Gyr}$, to $30$ at $t\sim8\,\mathrm{Gyr}$, to $11.1$ today.

In the future, GN-z11's redshift will continue to decrease to around $t\sim20\,\mathrm{Gyr}$, at which time we will observe it to have $z\sim9$. After this, it will increase without bounds.

Example #2

Now you ask specifically to "extremely redshifted galaxies", i.e. galaxies appearing now (which we would see "at the Big Bang" and hence not yet as a galaxy). The comoving distance to such a (proto-)galaxy — let's call it "$\textsf{uhoh}$" — would be (almost) equal to the distance to the particle horizon. Its worldline is marked in the spacetime diagram, and as indicated, we will see its redshift decreasing from $\infty$ to $z\simeq60$, and back to $\infty$.

When we observe $z_\mathsf{uhoh}\simeq60$, the age of the Universe will be just over $t=30\,\mathrm{Gyr}$, but we will see $\textsf{uhoh}$ as it was when the Universe was $t\simeq\mathrm{200}\,\mathrm{Myr}$ which, coincidentally, was just around the time when the first galaxies formed.

This can be seen from noticing that the worldline of $\textsf{uhoh}$ just grazes the $z_\mathrm{obs}=60$ line at $t\sim30\,\mathrm{Gyr}$, then following the $45\circ$ line from the point $\{t,d\}=\{30,0\}$ (our past light cone in the future) back to $\textsf{uhoh}$'s worldline, seeing that these lines cross at $t\simeq200\,\mathrm{Gyr}$.

In the far future, as $z_\mathsf{uhoh}\rightarrow\infty$, we will see $\textsf{uhoh}$ converge toward an age of $t\simeq500\,\mathrm{Myr}$, because this is where $\textsf{uhoh}$ crosses our event horizon.

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  • $\begingroup$ @uhoh What "dz/dt = 0" means is that, if at a given time you observe a bunch of galaxies with different redshifts, then all those with a redshift z_obs that is smaller than the value, z', that satisfies dz/dt = 0 will — if you wait a little — acquire a larger redshift. Conversely, all those with z_obs > z' will after a little while have a smaller z_obs. This "threshold redshift" slowly increases with time, and is currently z_obs ≈ 2. I'm not sure there's anything meaningful about this particular value, though. Anyway, thanks a lot, I'll consider writing it up somewhere :) $\endgroup$
    – pela
    Jul 9 at 9:41
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    $\begingroup$ @uhoh Okay, I’ll do the same then :) $\endgroup$
    – pela
    Jul 13 at 7:29
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    $\begingroup$ Too late :D I’ll leave the first one, which elaborates on dz/dt. $\endgroup$
    – pela
    Jul 13 at 7:32

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