What does the "size of the universe" mean if the Universe is infinite?

Question

There are questions that may seem to be similar to this one, but I've yet to find an answer.

I have come to understand that a flat universe, that is to say a curverature of $k=0$ which means that $S_k(r) = r$. The FLRW metric polar coordinates: $$ds^2 = -dt^2 + a^2(t) \left[ \frac{dr^2}{1 - kr^2} + r^2d\Omega^2 \right]$$ Now, since only $r$ is being altered, $dt = 0$ and $d\Omega$ = $0$. $$ds^2 = a^2(t) \frac{dr^2}{1 - kr^2}$$ This can be integrated to: $$s(r) = \frac{\sin^{-1}(\sqrt{k}r)}{\sqrt{k}}$$ So, by definition, the maximum value is when $\sin^{-1} = 90^\circ = \frac{\pi}{2}$ which occurs when $\sqrt kr=1$ To find the highest value, we replace $\sin^{-1}(\sqrt kr)$ with $\pi \over 2$ and get: $$s(r)_{\text max} = \frac{\pi}{2\sqrt k}$$ Therefore, as $k \to 0,\space s(r)\to \infty$. Given $k=0$, there is an infinite possible distance.

Now that we have that out of the way, when physicists talk about the size of the universe, by which I mean "when the universe was the size of a grapefruit" r a similar comparisson, space must have still been infinite, so what is this a description of?

Answer 1

All statements like "when the universe was the size of a grapefruit" refer to the currently observable universe. As the universe has a finite age and light travels at a finite speed (and there is nothing infinite going on with expansion), the observable universe is a finite patch.

I discussed some of the different notions of horizons in answering another question. The "observable universe" is taken to extend out to the particle horizon. That is, it includes precisely the points in our current time slice whose past worldlines (assuming they simply go with the expansion of space and have no peculiar velocity with respect to our reference frame) intersect the interior of our past light cone.

If you think of galaxies as marking these points, these are precisely the galaxies that we can see assuming arbitrarily good telescopes, since the light reaching us today was emitted as the galaxy crossed our past light cone.

Galaxies that started out too far away from us in an infinite universe haven't been able to get their photons to us. And indeed expansion will prevent most of them from ever getting to us.

The scale factor $a$ when the universe was the size of a grapefruit is simply the radius of a grapefruit divided by the radius of the current observable universe (about $46\ \mathrm{Gly}$), or something like $10^{-28}$ (corresponding to a redshift of about $z = 10^{28}$). The idea is that the galaxies (or rather their precursor quantum fluctuations) inside this grapefruit-sized volume are exactly the galaxies inside our observable universe today. In comoving coordinates the grapefruit is the same $46\ \mathrm{Gly}$ in radius then as our observable universe is now.