Does a positive curvature necessarily indicate the finiteness of the universe?

Question

Imagine the following situation: more and more accurate measurements of the average density of the Universe reveal that it is greater than the critical one, which corresponds to the model of a closed Universe with positive curvature. Will this necessarily mean that the entire large Universe is closed and has a positive curvature? Could it be that the curvature will simply be positive in our region of the large Cosmos, but at very large distances it will change, or is this impossible?

Answer 1

If the universe had a positive average curvature then it would be finite. The converse is not correct - a universe with zero or positive average curvature could be either finite or infinite, depending on its overall topology.

Note that even a finite universe could still be so large that the limited size of the observable universe might mean we can only observe a very small part of it.

Answer 2

Yes, it's possible that the uniformity that we observe is only local.

The Lemaître–Tolman–Bondi family of exact solutions to GR generalizes the FLRW and Schwarzschild geometries, and can describe, for example, a huge uniform FLRW region with a slight positive curvature (small enough to be consistent with current data) surrounded by an infinite Schwarzschild vacuum. I think similar exact solutions can be constructed with a nonzero cosmological constant.

If you want the FLRW region to be irregular in shape instead of precisely spherical, or you want to include radiation or any matter with a peculiar velocity, then there's no hope of finding an exact solution. But that doesn't mean such configurations don't make sense. There's no special cosmological magic to the FLRW geometry; it just happens to be the gravitational field of a matter distribution that's simple enough that its gravitational field can be calculated exactly. The lack of more realistic solutions is due to the difficulty of solving coupled second-order partial differential equations and not any deeper property of GR.

In some inflationary models, inflation starts in a small region of a larger non-FLRW universe when the conditions for it are randomly met in that region. The rest of the universe is still there after the end of inflation. As far as I know, the almost-uniform almost-flat region produced in these models has a finite volume.