I occasionally wonder if it could actually be smaller. We might expect to see spatial correlation in the CMB, but that might be swamped buy the other sources of inhomogeneities.
I just happened to be catching up on PBS Space Time, and they recently posted 2 videos related to this topic:
How much of the universe we can ever be able to see:
https://m.youtube.com/watch?v=eVoh27gJgME
How far into the universe we can ever get to:
https://m.youtube.com/watch?v=R-eIUxwcQo4
I had trouble visualising this, because eventually every other star in the universe would be outside the observable universe, so how could we still see the Cosmic Microwave Background Radiation from the big bang? But then I remembered that the Big Bang happened everywhere, so we’d still see the radiation from the space that existed between the stars, although it would be very redshifted and very sparse.
And this video confirms it, although Matt O’Dowd points out that eventually the CMBR would be so redshifted that we’d need an antenna larger than the Hubble Radius to see it, making even that evidence finally disappear.
That’s interesting, as a number of the popular science channels (e.g. PBS space time, Fermilab) seem to confidently assert that the universe is at least the size I mentioned.
I’m not saying you’re wrong, I am just saying that it is interesting that most physicists only really talk about the three options of ball-finite, open-infinite, saddle-infinite and not give much thought to other topologies.
I’m not 100% certain of this, but I expect by the time the CMBR is so redshifted that it requires an antenna too large to be built, the last stars will have long since ceased to radiate enough to maintain life on their planets.
You’re right to not be 100% certain.
CMBR will redshift beyond detectability within a few trillion years.
Star formation will cease by 100 trillion years.
Isn’t that because none of the other possibilities fit the equations?
The cases of spacetime of constant curvature are de Sitter space (positive), Minkowski space (zero), and anti-de Sitter space (negative). As such, they are exact solutions of the Einstein field equations for an empty universe with a positive, zero, or negative cosmological constant, respectively.
Or are the geometries not mappable onto these?
Chronos will know better than I, but I guess it is that there are topologies that are not constant curvature.
Eg you could have an “asteroids” style universe with zero curvature except at the edges.
(Maybe cosmologists don’t like such options as they would imply privileged reference frames…edges imply a centre?)
Two questions to answer.
One is whether such a topology simply doesn’t work in the field equations.
The other is whether we’re talking about the same thing. The three curvatures I talked about map to geometries (elliptic, Euclidean, and hyperbolic). There seem to be a variety of both constant and non-constant curvatures in topology. Whether those are relevant is my other question.
It’s been checked. It took a few weeks of supercomputer time, but no, there aren’t any of the correlations in the CMB that one would expect from this.
It’s presumed that the curvature is uniform, but that’s still possible with various nontrivial topologies. For flat space, there are something like a dozen different possible topologies (personally, I’m fond of the one where the unit cell is a rhombic dodecahedron). Positively or negatively curved spaces have fewer possibilities, but there are a few (one of the positively-curved ones is based on Platonic dodecahedrons).
Is it not true that if that is the case, the universe may even be much smaller than the observable universe since due to the finite speed of light we might not notice that it wraps around?
An old thread of mine:
A spherical Universe? - Factual Questions - Straight Dope Message Board
I don’t think so. AIUI, the main way we measure the curvature of the universe is by looking for lensing effects in the CMB; something about the size of the “clumps” of the CMB gives us information on how curved a path the photons have taken…I totally could give a doctoral thesis-level description of the mathematics, but I’ll give someone else the opportunity first 
But it isn’t (primarily) by looking for photons that have completed more than one lap.
I was referring to the point made by @Chronos about a zero curve kind of repeating like in an asteroid game
Can you stack them? Wow - yes you can.
I love the idea that we are trapped inside a honeycomb made of rhombuses.
Me too.
The point is, wrapping involves some form of curvature however you split it. “Flat + teleport” isn’t a topology.
And it’s either an even curve or not. If it’s an even curve, then we’re talking something akin to a ball, and the ball must be at least 500 times the size of the observable universe to look at flat as it does to us.
If it’s an uneven / discontinuous curve, flat over the volume of the observable universe then all bets are off. But we know the total is at least as big as the observable universe.
AIUI
I’m not sure it does. Space can, in theory, be ‘multiply connected’ by wormholes that connects two distant regions of space with no intervening space-like interval. If the universe is a (single) rhomboid dodecahedron with rhomboid wormhole mouths on each face, then passing out of one of the ‘top’ faces would take you to one of the bottom faces. The question remains - is the dodecahedron larger or smaller than the observable universe? If it is much larger, then we could never know the truth.
If it is much smaller, then we could try looking for repeating structures, but that would be a very tricky task, since we are not only looking outwards in space but backwards in time, making pattern-matching practically impossible. As well as the ‘Hubble horizon’ and the ‘particle horizon’, I expect that there is a ‘predictability horizon’, which limits the accuracy of our conceptions about conditions in the distant volumes of our universe. We see distant galaxies as they appeared 13 billion years ago; it seems utterly impossible that we could predict accurately what they look like ‘today’.
No need for any wormholes: a torus has the same local geometry everywhere as Euclidean space.
The curvature point Mijin made applies to a torus just much as it does to a ball. So either it’s a wormhole/teleportation or it’s 500 times the size of the observable universe.
[I think] you are misunderstanding that @Chronos was talking about a curved torus; what is meant is a flat torus with the same exact (flat) Robertson–Walker metric as flat Euclidean space. Curvature is zero everywhere. Globally there is a non-isotropy but there is no wormhole or non-flat region anywhere.
“A view from inside a 3-torus”:
