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

 

[sardar]

I would like to clarify that the inference of a closed universe from the FLRW metric is based on the assumption that the universe is homogeneous and isotropic on large scales. This means that the universe looks the same in all directions and at all points in space.

The FLRW metric is a solution to the Einstein field equations, which describe the relationship between the curvature of spacetime and the matter and energy content of the universe. In this metric, the curvature is directly related to the matter density, with higher density leading to positive curvature and lower density leading to negative curvature.

Therefore, if the matter density is high enough, the curvature will be positive, and the universe will be closed. This is because positive curvature causes the expansion of the universe to eventually slow down and reverse, resulting in a closed, finite universe. On the other hand, if the matter density is too low, the curvature will be negative, and the universe will be open and infinite.

The inference of a closed universe from the FLRW metric is based on the understanding that the local curvature of spacetime is directly related to the overall global properties of the universe. This means that if the local curvature is positive, the overall universe must also be closed.

As for why an infinite universe of positive local curvature is ruled out, this is because an infinite universe would require an infinite amount of matter and energy, which is not supported by our observations. In a closed universe, the total amount of matter and energy is finite, allowing for a positive curvature without requiring an infinite amount of matter.

In summary, the inference of a closed universe from the FLRW metric is based on the assumption of homogeneity and isotropy, and the understanding that the local curvature is directly related to the overall global properties of the universe. An infinite universe of positive local curvature is ruled out due to the finite amount of matter and energy observed in our universe.