How distant were the furthest currently-observable cosmic events when their currently-observed radiation was emitted?

Question

(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)?

Answer 1

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.

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