I always had kind of a difficult time reconciling the idea that the Universe may be infinite beyond what we can ever observe, with the fact that the confines of the observable universe go all the way back to the Big Bang. And I recognize this is the limitations of my own capacity to visualize how time and space fit together, but I think having an answer to this question will help me:

When we look at the CMB, ~13 billion light years away plus expansion, we look at objects that were exactly how far away from “us” at the time that they were actually this way? How big was the sphere containing all the things we can account for at the time the oldest light we’re receiving was emitted? How big was the entire universe compared to that sphere at that time? And if we know these things then how can we be at a loss as to what size the entire universe must be today?


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First, "how far away" something in a distant part of the universe is has to be handled carefully since the universe is expanding in a nonlinear fashion. The most useful measure for this kind of question is co-moving distance. Co-moving spatial coordinates essentially follows the matter of the universe (which is usually nearly at rest as seen by the comoving frame), so it makes sense to use them to discuss how far away an object was in the distant past - today the object is still likely close to the same comoving coordinate. The other kind of distance, proper distance, changes with the expansion of the universe and is what we would measure if we at the current cosmic epoch put a lot of rulers between us and the object. By convention we set these distances equal at the present, so the object at co-moving distance $d$ has the same proper distance.

The reframed first part of the question is: how far (in co-moving coordinates) is the edge of our past light cone when it reaches the surface of last scattering? This happens at redshift $z~1100$ [1]. Using the calculator [2] I get a distance of 13.9547 Gpc or 45.514 Gly. (At that time the scale factor was $a(t)=1/(1+z)\approx 0.0009082$, so the proper distance was 41 million lightyears.)

Note that it might even make sense to ask how far away our light cone was at the Big Bang: at least for standard cosmological models you get a finite answer of about 46 Gly (see figure 1 in [1]).

This leads to the second part of the question: why doesn't this information tell us anything about the size of the universe? (I assume the size here denotes size in co-moving coordinates)

Obviously, if the universe had been smaller than 46 Gly, we would have seen something (leaving explicit boundaries aside, it would have had to be some bounded manifold like a 3-torus or a Poincare dodecahedron; see [3,4]) - so far there is no evidence for any pattern in the CMB compatible with this. So in a sense we have a bit of (negative) information about compatible sizes from the CMB.

But if you look at figure 1 of [1], you could extend the x-axis arbitrarily far: there is no reason to assume it has to be finite, and even if it is, there is no reason for it to be close to our horizon distance.

[1] Davis, T. M., & Lineweaver, C. H. (2004). Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe. Publications of the Astronomical Society of Australia, 21(1), 97-109. https://arxiv.org/abs/astro-ph/0310808

[2] http://www.astro.ucla.edu/%7Ewright/CosmoCalc.html

[3] https://arxiv.org/abs/gr-qc/9605010v2

[4] http://www.scholarpedia.org/article/Cosmic_Topology


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