How do we infer a closed universe from FLRW metric?

[pellman]

The Friedmann–Lemaître–Robertson–Walker metric is a solution of the field equations of GR. It tells us the local behavior of spacetime, that is, g(x) at a given spacetime point x

If the matter density is high enough, the curvature is positive. It is said then that the universe is closed. How is this global property of the manifold inferred from the local metric? Why is an infinite universe of positive local curvature ruled out?

 

[marcus]

you measure curvature on the largest scale you can
if it kept coming up positive, you might think "closed" was the most plausible explanation
though you could never be entirely sure of course.

so far there is no clear indication. the large scale curvature when measured comes out near zero but the 95% confidence interval straddles zero. both pos and neg are possible

while you are waiting to hear more conclusive indications you might like to think about how largescale curvature is measured. we could talk about that in this thread. it's fascinating. real ingenuity is involved.

 

[George Jones]

The Friedmann–Lemaître–Robertson–Walker metric is a solution of the field equations of GR. It tells us the local behavior of spacetime, that is, g(x) at a given spacetime point x

If the matter density is high enough, the curvature is positive. It is said then that the universe is closed. How is this global property of the manifold inferred from the local metric? Why is an infinite universe of positive local curvature ruled out?

I believe the cosmological principle, i.e., spatial homogeneity and isotropy, requires space (not spacetime) to be a maximally symmetric Riemannian manifold. The maximally symmetric 3-dimensional Riemannian manifold with positive intrinsic curvature is S^3 with the standard spatial metric.

 

[marcus]

just as a thoughtexperiment, suppose distances were not expanding, imagine a static universe---how then might you estimate largescale curvature?

Well you could see if the volume of a ball increases as the cube of the radius, or whether at very large radius it begins to grow more slowly than the cube.

How would you estimate volume? In a static situation you might simply count galaxies, assuming a constant number per unit volume