George F. R. Ellis writes in section 2.4.2 of his outstanding survey paper, Issues in the Philosophy of Cosmology:
Let’s start by defining a few of the terms that Ellis uses above.
particle horizon – the distance beyond which light has not yet had time to reach us in all the time since the Big Bang
decoupling – the time after the Big Bang when the Universe had expanded and cooled enough that it was no longer a completely ionized opaque plasma; atoms could form and photons began traveling great distances without being absorbed
worldlines – the path of a photon (or any particle or object) in 4-dimensional spacetime: its location at each and every moment in time
LSS – last scattering surface
COBE – Cosmic Background Explorer
(And, Planck should be added now, too)
Now the question. Do we live in a small, big, or really big universe? The best answer we can give now (or, perhaps, even in the future) is that we live in a really big universe, though it is unlikely to be infinite. Ellis himself provides a cogent argument in section 9.3.2 of the paper referenced here that infinity, while a very useful mathematical tool, does not ever exist in physical reality. We shall investigate this topic in a future posting.
Even though general relativity shows us how matter defines the geometry of our observable universe, it tells us nothing about the topology of our universe, in other words, its global properties. Do we live in a wrap-around universe where if we set off in one direction and traveled long enough, we’d eventually return to the same point in spacetime? Is the topology of our universe finite or infinite? At the moment it appears that we are not able to observe enough of the universe to discern its topology. If that is true, we may never be able to understand what type of universe we live in. But observational cosmologists will continue to search for the imprint of topology on our visible universe at the largest scales.
Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.
Liddle, A.R. 2015, An Introduction to Modern Cosmology, 3rd ed., Wiley, ISBN: 978-1-118-50214-3.