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(Edited for clarity. Thanks to James K and Connor Garcia.)

This question about the most distant, observable cosmic objects made me wonder if we know the distance that was between us and them at the time (13.4 billion years ago, in the example linked) of the initial emission from them of the light that we can now see.

(Answer: Thanks to Connor Garcia for pointing out that the answer of 2.66 billion light-years was in the notes of the wikipedia page of the galaxy (GN-z11, currently the oldest and most distant known galaxy in the observable universe) that I linked to in my own question. I can always count on stack exchange users to helpfully point out that the answer I was looking for was just a bit more effort away, lol. But seriously, thanks for the help Connor.)

I can understand how expansion has caused them to travel a current distance of 30+ billion light years from us (or I can at least understand how the distance is larger than light could travel in 14 billion years), but I haven't been able to find a statement about their distance from us when they originally emitted the light we're seeing today.

Can anyone give me insight into how close this 13.4 billion year-old galaxy was when it emitted the light we're seeing today?

Is it as simple as the galaxy being 13.4 billion light years away at the proposed time of 400 million years after the Big Bang? And if so, did space really expand so fast in just 400 million years that objects could be 13.4 billion light years away from each other (and some much further, I'll venture to assume)?

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    $\begingroup$ "Inflation" is used as a technical term by cosmologists, and isn't correct here. You should probably replace it with "expansion", to avoid complications. $\endgroup$
    – James K
    Aug 19 at 21:20
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    $\begingroup$ The answer is in the note in the wiki page you linked to (emphasis added by me): "the distance of 2.66 billion light-years between GN-z11 and the Milky Way at the time when the light was emitted increased by a factor of (z+1)=12.1 to a distance of 32.2 billion light-years during the 13.4 billion years it has taken the light to reach us" $\endgroup$
    – Connor Garcia
    Aug 19 at 21:24
  • $\begingroup$ However, I think an excellent answer is still possible which would explain how the "2.66 billion light year" figure can be obtained. $\endgroup$
    – Connor Garcia
    Aug 19 at 21:35
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    $\begingroup$ @ConnorGarcia thanks! I hadn't thought to check for notes. I would be interested in an explanation, but I was primarily interested in just ballpark figures, because I thought the distance to them at the time of their emission was maybe just the "light travel distance" of 13.4 billion years in the example of GN-z11. $\endgroup$
    – Glycoversi
    Aug 20 at 0:02
  • $\begingroup$ Thanks for the clarification @JamesK, I edited my question to reflect it. $\endgroup$
    – Glycoversi
    Aug 20 at 0:03
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tl;dr No, it's unfortunately not that simple.

Cosmological distances

The comoving distance to an object observed to have a redshift $z$ — i.e. the coordinates that expand along with the Universe — is calculated by integrating the Friedmann equation, assuming some values$^\dagger$ for the expansion rate $H_0$ and the density parameters $\{\Omega_r,\Omega_m,\Omega_k,\Omega_\Lambda\}$ (radiation, matter, curvature, and dark energy): $$ d_\mathrm{com}(z) = \frac{c}{H_0}\int_0^z \frac{dz}{\sqrt{ \Omega_r(1+z)^4 + \Omega_m(1+z)^3 + \Omega_k(1+z)^2 + \Omega_\Lambda }}, $$ where $c$ is the speed of light.

In general, this equation doesn't have an analytical solution, but must be solved numerically. I usually use Python's astropy module for this.

By definition, the physical distance $d_\mathrm{phys}$ equals the comoving distance today. Since redshift evolves linearly with the scale factor $a$, and since $a$ is defined to be $1$ today, observing an object to have a redshift $z$ means that the light we see was emitted when $a$ was equal to $1/(1+z)$. For instance, a galaxy that emitted some light when the Universe was a quarter if its current size (so that $a$ was $0.25$) would be seen to have a redshift $z=1/a-1=3$.

In other words, the distance to an object at redshift $z$ when it emitted the light we see, is a factor (1+z) smaller than it is today (disregarding the galaxies' relatively small peculiar velocities).

The most distant galaxy

The current galaxy redshift record holder is, as you say, GN-z11 with $z=11.1$ (Oesch et al. 2016). Solving the equation above yields a current distance of $$ d_\mathrm{phys,now}=32.2\,\mathrm{Glyr}, $$ (a Glyr is a billion lightyears), and hence the distance between GN-z11 and the Milky Way was, when it emitted its light some 400 Myr after the Big Bang $$ d_\mathrm{phys,then}=\frac{32.2\,\mathrm{Glyr}}{1+11.1}=2.66\,\mathrm{Glyr}. $$

As a small curiosity I can tell you, that GN-z11 recede from us with a velocity of roughly $4c$ when it emitted its light, while today it "only" recedes at $2.2c$. And yes, that's allowed.

The cosmic microwave background

There are perhaps some higher-redshift galaxies, but they haven't been confirmed spectroscopically. But depending on your definition of "event", you could argue that the cosmic microwave background with $z\simeq1100$ is the most distance. The current distance the the gas that emitted the observed CMB is $45.4\,\mathrm{Glyr}$ (very close to the edge of the observable Universe at $46.3\,\mathrm{Glyr}$). Hence, that gas was only $42\,\mathrm{Mlyr}$ (million lightyears) when it emitted its light, shortly after the Big Bang.


$^\dagger$Roughly, $H_0\simeq70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, and $\{\Omega_r,\Omega_m,\Omega_k,\Omega_\Lambda\}$ $\simeq$ $\{\lt 10^{-4},0.3,0,0.7\}$

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  • $\begingroup$ thank you. this clarified the 2.66 Glyr answer perfectly. The more I look slightly into these astronomical topics, the more respect I have for you physicists and mathematicians. The fact that we only started photographing stars 150 years ago and we've lept from speculating the Universe is maybe 100,000 light years across to just the observable Universe reaching 126 BILLION light years across (if I read the last paragraph of your linked post right), makes me hopeful about our future if we can teach our children more critical thinking, humility, and wonder about this existence. $\endgroup$
    – Glycoversi
    Aug 20 at 16:56
  • $\begingroup$ I just realized that the "small curiosity" you mentioned compels me to ask how GN-z11 could have already been moving 4c away from us so early on? Was the comoving distance still about the same as now and the space metric was just expanding at a larger rate back then? Or does it have to do with inflation still occurring from the theorized origin of the Universe? (Now that I'm typing this out, I'm also realizing that I'm very unclear about how inflation from the Big Bang is different from cosmological expansion, if you happen to have any straight-forward explanation). $\endgroup$
    – Glycoversi
    Aug 20 at 17:36
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    $\begingroup$ @Glycoversi I agree, the speed with which not only our technology, but also our consciousness, accelerates is fantastic. A few back-comments: The 126 Glyr you mention is a bit too much; the radius is ~46 Glyr, so ~92 Glyr across. The comoving distance between galaxies is the same at all time (except for a small "peculiar", or "real" motion); that is the definition of comoving coordinates. But yes, space was expanding faster in the past but decelerated due to mutual attraction of all matter, then later accelerated due to dark energy. Inflation is similar, but had (probably) a different… $\endgroup$
    – pela
    Aug 21 at 8:56
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    $\begingroup$ …origin, was much much faster, and lasted only until the Universe was some $10^{-33}$ seconds old. I don't have a straight-forward explanation though, since inflation is too far from my comfort zone, but normally the mechanism driving inflation is thought to be a different field from that of dark energy. $\endgroup$
    – pela
    Aug 21 at 9:01

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