freyar
Extradimensional Explorer
I'm presuming this is obtained from the measurements of the energy density, which makes for a flat universe.
How do we get from that to the conclusion that there aren't joins? Does the lack of uniformity in directions (diagonals are longer than the perpindiculars) make that physically impossible? Or is it considered too strange?
(From a topologists point of view, a standard construction is to take a unit square and to identify the opposing sides to make a new quotient space of the original unit square. That makes for a flat, unbounded, yet finite, space. The space is not uniform in all respects.)
The energy density of the universe is (within measurement error) the amount that makes the universe spatially flat, yes. Due to the cosmological constant, the expansion is accelerating, again as mentioned earlier. [I am hesitant to use the phrase "open universe" since that had implications up until the discovery of the cosmological constant that no longer apply.]
However, even a flat universe can have a nontrivial topology. Your example of gluing the edges of a square together is a good one --- that's the mathematical definition of a 2D torus. So (the spatial part of) our universe could be a flat 3D torus or one of several other alternatives. (That's assuming it's actually exactly flat; there are other alternatives if it's positively or negatively curved.) How do we know if the universe is infinite or a finite torus/other example? We actually have to go measure. There are several groups that have looked at the cosmic microwave background (the oldest visible light); if the universe is finite and small enough, we'd be able to see repeated patterns due to seeing the same spot in the early universe from several directions (think about different ways you can shoot the same area on the screen in the old Asteroids game). So far, we don't see any repeated patterns like that, which means that the universe is either infinite or else finite but so big we can't see all the way around it.
There could be some local joins or curves (wormholes and black holes and such). But, on a broad scale, there is a point to be made: Mass/energy causes spacetime to curve. As far as we know, mass is the *only* thing that makes spacetime curve. Real physical spacetime cannot have arbitrary topological configurations *without* mass to make it so. This is why we say that the low energy density means spacetime is flat, at least within the observable universe.
This isn't really correct. There are finite, _flat_ spaces with nontrivial topology that solve Einstein's equations with _zero_ mass or energy density. A torus is an example. And our universe could have one of those spatial topologies. Black holes and wormholes do have curvature, though, so they require mass/energy.