Shape of Universe | Is It Same from Every Point?

[PeterDonis]

Your hypothetical example seems to violate conservation of energy. That is, a single flash contains a finite amount of energy E

Not if the universe is spatially infinite, which it is in my example (and our current best-fit model of our actual universe).

I appreciate the parallel with the CMB, but I had assumed that the 2-sphere slice (S) of the surface of last scattering which we see today is not the same slice which we see tomorrow.

That's correct. And it's also true in the simplified example I gave. Both scenarios are the same in this respect, so I don't see what your point is with this comment.

If and when these slices are receding from us at speed c, we will no longer see them ??

The slices we see are not the slices "now"; they are the slices 13 and a fraction billion years ago, when the CMB was emitted. We can't "no longer see them"; they don't go "out of our view". Think carefully about what it means that the CMB was emitted everywhere in the universe at a single instant of time 13 and a fraction billion years ago. (Technically it wasn't really a single instant, but that simplification will do for this discussion.)

Also, "receding at speed c" doesn't mean what you think it means. This is another reason why you need to spend time studying a cosmology textbook. Many of the intuitive ideas you have are wrong when applied to our current best-fit model of the universe.

I am not sure which side of the slice S you are calling "interior" and which the "exterior". That is, which side are "we" on?

"We" are not in the slice at all, because it was 13 and a fraction billion years ago. But if you extended the Earth's worldline back that far, the spatial point at which it intersected the 3-surface of last scattering would be in the interior of the 2-sphere S representing the "slice" of the CMB that an idealized observer who followed the Earth's extended worldline would see at any instant after the time of CMB emission.

Looking around, we seem to be on the interior of some spherical slice S??

This is vague, because there are a number of different 2-spheres that we could say we "seem to be" in the interior of. A more precise version that's relevant to this discussion is what I said above; the point on the Earth's worldline representing us now is one of the points on the Earth's worldline to which my statement above applies.

So you are claiming that the side of S containing the early
universe has unknown topology?

I don't understand what you mean. S is a 2-sphere lying within a spacelike 3-surface of constant time--i.e., S is a 2-sphere in "space" as it was 13 and a fraction billion years ago, a few hundred thousand years after the Big Bang. How would this space "contain the early universe"?

 

[Bandersnatch]

@Tom Mcfarland the example of CMBR emission in static space-time described by Peter in post #29 can be represented on a space-time diagram with two spatial dimensions suppressed:

The triangles represent paths of light (null worldlines), which in the case with one spatial dimension suppresed (2D+time) become cones. Hence 'light cones'.
The dashed line is a spatial slice of constant time representing the universe undergoing transition into trasparent state; it is the time when CMBR was emitted. It is a line in 1+1, a surface in 2+1 (hence the 'surface of last scattering'), and a volume in 3+1.
The intersection points in 2+1 dimensions become a circle, and in 3+1 dimensions - a sphere. That's the currently observable snapshot of the surface of last scattering.
The portion of the surface of last scattering enclosed by the observable 2-sphere at intersection with light cones is a 3-ball (marked as red dashed line). The exterior is the entire 3D volume with the 3-ball cut off.
The observable portion of the surface of last scattering grows as the apex of the light cone travels into the future.

It should be noted that the above space-time diagram is applicable both to the static and expanding cases. All we have to do is use appropriate scaling for the axes. In the static case, the scaling is linear. For the expanding universe, using comoving distance vs conformal time recovers the above shape, as can be seen on the following diagram:

(source for the second diagram: Lineweaver & Davis, 2003; fig. 1)
... and the same relationships apply.

None of this is strictly speaking relevant to the topology of the universe, I think.