# Can we consider the Universe to be some kind of 3-sphere?

This is probably a naive question. I'm learning a bit of cosmology and I've recently covered the so called angular size-redshift relation, which states that in an expanding Universe the angular size of an object decreases with distance to the observer (as in Euclidean geometry) but at some point it starts to grow in angular size with distance (which departs from Euclidean predictions). If I understand this correctly, this is because at some distance the observed object is so far away that we are looking to the deep past, which means the object lies in the young Universe, and thus a Universe that was way smaller (due to the expansion of space). This means that the object would look bigger (in terms of angular size), since it occupied a larger portion of the whole Universe back then (at that distance).

Another way to see this is by understanding that, in principle, the farther away we see the larger the spherical carcases surrounding us we are probing, but at the same time, since the Universe expands this means we are looking also to spherical carcases that have smaller and smaller volumes (as we look into the past). Thus the angular size of objects must appear larger and larger at some point with increasing distance.

At the same time I remember that in a 2-sphere (the surface of a 3D sphere) if you (an observer) where located at the north pole (for example) and had a stick measuring 1 meter, you could say that the angular size of it decreases with distance (as it goes farther away you need less longitudinal lines to cover the angular size of the stick). But at some point (the equator) this behaviour would start to change. As we move the 1 meter stick closer to the south pole, as viewed from the north pole we would start to see it as spanning more longitudinal lines, so we would see it as an increase in angular size with distance.

So, this mathematical description of what we would see in a 2-sphere reminds me a lot of the angular size-redshift relation in the cosmology of an expaning Universe.

My question is, in which way we can conceive this universe (if we can) as a 3-sphere? I know that we currently don't know if the curvature of space is flat, hyperbolic or spherical (because the observations of the curvature in the CMB are compatible with all these options for now). But that is about the curvature of space and I feel that in the case of my question I'm tackling a different set of properties that could be represented by a 3-sphere topology, what are those? Space-time geometry against space geometry?

Sorry if this is all a bit confusing, and thanks in advance everyone.

• This may be of interest: Topology of the Universe: Theory and Observations Feb 4, 2022 at 22:37
• I highly recommend popular book The Poincare Conjecture: In Search of the Shape of the Universe by Donal O'Shea. Feb 5, 2022 at 13:55

In traditional big-bang cosmology, although the cross-sections of the light cones are spheres whose radii approach $$0$$ as $$t\to 0$$, the big bang is a singularity and not a point on the manifold, so the light cone doesn't close into a topological 3-sphere. It doesn't even make sense to complete the light cone with a point representing the big bang, because points nearby aren't in causal contact (the horizon problem). It's better to think of the $$t\to 0$$ limit of the spheres as a sphere of indeterminate size, rather than a point.