B Shape of Universe | Is It Same from Every Point?

[wolram]

This may be a silly question, but how do we give a shape to the universe if it is The same from every point we look?

 

[Arman777]

This may be a silly question, but how do we give a shape to the universe if it is The same from every point we look?

We don't give a shape.

We can make an argument that universe looks the same at larger scales (you can further search cosmological principle). This argument reduces the possible geomtries that universe can have.

1-Flat with zero curvature
2-Spherical geometry with positive curvature
3-Hyperbolic geometry with negative curvature

From observation we can conclude or derive the geometry of universe.

Recent data (Plank 2015 results or etc) shows that universe is flat and has zero curvature.

It may have another shape like a torus maybe but even in that case the size of torus must be huge cause obersvable universe seems flat over a large scale.

 

[PeroK]

This may be a silly question, but how do we give a shape to the universe if it is The same from every point we look?

You could ask the same about the surface of the Earth, assuming it were ideally spherical. Its curvature can be defined by its being a 2D surface embedded in 3D space: i.e. it's the surface of a 3D sphere.

Or, its shape can be defined using differential geometry to define the infinitesimal distance in any direction. Using spherical polar coordinates this is:

$dS^2 = R^2(d\theta^2 + \sin^2 \theta d\phi^2)$, where $0 \le \phi < 2\pi$ and $0 \le \theta \le \pi$, and $R$ is some parameter, which equates to the radius of the 3D sphere above.

This "line element", in fact, encapsulates the shape of the sphere's surface without directly appealing to an embedding in a higher dimension.

Note that this is the distance along the surface, not taking any shortcuts through the body of the Earth!

 

[wolram]

I get the analogy of the Earth, but as regards to the universe what does flat mean, surly due to gravity and dark matter the universe must be globular, but how can that be flat?

 

[PeroK]

I get the analogy of the Earth, but as regards to the universe what does flat mean, surly due to gravity and dark matter the universe must be globular, but how can that be flat?

It's flat on a large scale. But, locally, where there are galaxies, black holes or simply where there are stars or planets you have locally curved spacetime.

 

[Arman777]

surly due to gravity and dark matter the universe must be globular, but how can that be flat?

Why it would be globular ? And how can it be globular and flat at the same time ?

There's something you should mention, are you talking about the universe or observable universe.

We can't now the geometry of the universe, (since we can't travel or observe the whole universe etc), but we can claim that observable universe is flat. Becasue that is what we observe (by experiment and measurement)

The universe can have a spherical geomtery or it can be just flat as ours, but If its sphere the R (radius of the sphere) must be so huge that, we observe, the observable universe as flat.

but as regards to the universe what does flat mean

Mathematically it means that curvature is zero, In example a plane is a flat becasue, when you set a triangle, and you measure the angles it gives you $π$. But in spherical geometry it gives you more then $π$.

Of course in metric terms it would be more different

Try to think that you are a small ant, and you are traveling on a piece of paper. Wherever you walk or go it feels like you are on a flat surface and when you draw a triagle and measure the angles you get $π$. Observable universe is a piece of paper and you are an ant.

If we were bigger then ant, then we could have notice the curvature, but since we are not, we cannot know the real geometry of the universe.

(thats why our ancestors thought that Earth is flat in the first place casue it was hard to observe that Earth is spherical, in our case we cannot observe or don't have tools to see the real geometry of the universe, we are more like a bacteria respect to the universe)

 

[phinds]

The universe can have a spherical geomtery or it can be just flat as ours, but If its sphere the R (radius of the sphere) must be so huge that, we observe, the observable universe as flat.

But if the universe were a sphere, that would clearly imply a preferred direction and there is zero evidence of such a thing.

 

[PeterDonis]

as regards to the universe what does flat mean

It means spatially flat: a spacelike slice of constant time for a comoving observer is Euclidean 3-space.

 

[PeterDonis]

if the universe were a sphere, that would clearly imply a preferred direction

No, it woudn't; there is no preferred direction in a 3-sphere. A 3-sphere is spatially isotropic.

You might be confusing this with the case of a spherically symmetric, but not homogeneous, spacetime such as Schwarzschild spacetime. In this case, yes, the radial direction is different from the tangential directions. But that's due to the fact that there is a mass at the center, but not anywhere else; i.e., the spacetime is not homogeneous. In a closed FRW universe, which has the spatial geometry of a 3-sphere, that's not the case: the density is the same everywhere, and all directions are the same

 

[wolram]

Thank you for your replies, I can see now that when we talk about the universe having shape we only mean the observable universe

 

[phinds]

Thank you for your replies, I can see now that when we talk about the universe having shape we only mean the observable universe

That's not very exciting since the OU is just a sphere centered on you.

 

[Valentin Kanev]

Using the balloon analogy, what's inside the balloon skin? Is it the Universe from a moment ago?

 

[phinds]

Using the balloon analogy, what's inside the balloon skin? Is it the Universe from a moment ago?

You are misunderstanding the analogy. I recommend the link in my signature.

 

[_PJ_]

Using the balloon analogy, what's inside the balloon skin? Is it the Universe from a moment ago?

That's one of the biggest flaws of such analogies.
Ther balloon does not represent the universe nor anything about the shape, topology or geometry of the universe. The SURFACE of the balloon 'skin', where drawings of galazy clusters might be made, represents the Relative distances between those clusters, that's about it.

 

[_PJ_]

That's not very exciting since the OU is just a sphere centered on you.

I find a universe centred on !e particularly exciting :)

- a joke only.

 

[AgentSmith]

I find a universe centred on !e particularly exciting :)

- a joke only.

But my research shows its centered on me. :)

 

[Tom Mcfarland]

A Play. Title: Unseen Shapes (the "balloon analogy" revisited)

Scene 1: You are in what appears locally to be an isotropic Flatland (E²). You look around in every direction. Far away objects
are rather dim, and you suspect that beyond your range of vision, there is more. Indeed, the gods of mathematics have
revealed that if you could see far enough (which you cannot), you would see the same small point-like object in every direction. If these gods
are correct, then you must live in S² (a 2-sphere, "the balloon").

Scene 2: You are now in what appears locally to be an isotropic Solid-land (E³). You look around in every direction. Far away objects
are again rather dim, and you suspect that beyond your range of vision, there is more. Indeed, the gods of mathematics have
revealed that if you could see far enough (which you cannot), you would see the same small point-like object in every direction. If these gods
are correct, then you must live in S³ (a 3-sphere).

End of Play. Do you see any problem with its logic?

I have read and probably understand this imbedded and related link

https://www.physicsforums.com/insights/balloon-analogy-good-bad-ugly/

Tom McFarland

 

[PeterDonis]

Do you see any problem with its logic?

Not as far as it goes, but in our real universe there are no "gods of mathematics" to tell us whether in fact we could see the same object in all directions if we could see far enough. And we have no evidence that we could. So we have no evidence that your play is describing our actual universe.

 

[Tom Mcfarland]

PeterDonis:

Sorry to post this question twice. I feared that I had posted incorrectly the first time.

On the issue...

Is it not true that the big bang thesis posits a point-like start to our universe which we are
currently researching, limited by our ability to "see" before the origin of CMB. Indeed,
Even the universe at "last scattering" (when the CMB was produced) would qualify as
sufficiently point-like and Omni-directional to force the current shape to be close to S³, via the above logic.

Thus, in scene 1 above, if an ideal point P is seen in all directions, you get S² for sure, but if P is merely
very small compared to all of space (a big point, like the "last scattering" surface), this still forces the topology to be S²,
like a balloon with your finger plugging its hole (P being your finger). The same applies to S³.

 

[Tom Mcfarland]

Peter Donis:

Here is another neat topological way to convince yourself that the universe must have the topology of S³.

First, consider the universe starting with us now, extending out (and backward in time) to the surface S of last scattering (at which CMB was produced). We see this surface isotropically in all directions in microwave light. This is a 3-ball (the interior of a big 2-sphere plus boundary S). Hold this in your mind.

Secondly, consider the postulated universe from the big bang out to the same surface of last scattering S. Currently we cannot see this region, but it would also be a 3-ball with the same boundary S as above, if the gods of mathematics are correct. Hold this in your mind.

Now, topologically merge these two 3-balls along their common boundary S. This union is a topological 3-sphere, the universe from big bang to now.

This merger is analogous to merging two 2-dimensional discs (like the upper and lower crusts of a pie) along their circular boundaries, to obtain a 2-sphere.

 

[PeterDonis]

Is it not true that the big bang thesis posits a point-like start to our universe

No, it is not true. The "point-like start" is an artifact of a particular class of idealized models. It is not a feature of our actual current best-fit model of the universe.

However, let's put that aside and consider the idealized models I just referred to. Even in those idealized models, it is not true that the spatial topology of the universe must be $S^3$. It is perfectly possible to have such an idealized model, with a "point-like start", with spatial topology $R^3$ (i.e., infinite). The reason is that the "point-like start" is not a point in space; it is a moment of time. Your arguments for the spatial topology having to be $S^3$ are only valid if the "point" that is visible in all directions is a point in space. (The technical way of putting it is that, for your argument to be valid, the "point" must be a timelike curve--it's a curve when you include the time dimension. But the "point-like start" in the idealized models I referred to is spacelike, not timelike.)

Here is another neat topological way to convince yourself that the universe must have the topology of S³.

Nope. This argument also fails for the reason I gave above. Even if the "point" is actually a small region, it would have to be a timelike "world tube" (a small bundle of timelike worldlines) for your argument to be valid; but the "tube" occupied by, for example, the region of last scattering is spacelike, not timelike.

 

[PeterDonis]

First, consider the universe starting with us now, extending out (and backward in time) to the surface S of last scattering (at which CMB was produced). We see this surface isotropically in all directions in microwave light. This is a 3-ball (the interior of a big 2-sphere plus boundary S).

No, it isn't. You're leaving out the time dimension. Our observable universe back to the surface of last scattering is a 4-ball, not a 3-ball. More precisely, it's the past light cone of the Earth at its present point in spacetime, which is a 4-volume bounded by a spacelike surface in the past (the surface of last scattering) and a null cone whose apex is the Earth's present event.

However, this only applies to the observable universe. It does not in any way rule out the possibility that the full spacelike surface of last scattering is spatially infinite. We can't rule that out because we can only see a portion of that surface, and that surface, as I have said, is spacelike, not timelike, so we can't think of it as a "point-like region visible in all directions" in the sense your arguments require.

 

[Tom Mcfarland]

Thank you for your patience...I am not a trained cosmologist. I have replied to you point-for-point, but the time dimension is a struggle for me.

You write:
>>The "point-like start" ... is not a feature of our actual current best-fit model

Accepted. The logic in favor of S³ does require a particular starting moment, or a point-like start, which was used to simplify the argument.

You write:
>>argument (in favor of S³ topology) also fails ... Even if the "point" is actually a small region

Again, the "small point-like region" device was a rhetorical convenience, but not a logical requirement in support of S³ topology

You write:
>>...Your arguments for the spatial topology having to be S³ are only valid if the "point" that is visible in all directions is a point in space

This is not true. The argument is valid as long as the same object is visible in all directions. The object needn't be a point. This object
then becomes the "point at infinity" and ties off the balloon, making it a sphere. However, the second argument (merging balls) avoids this issue

You write:
>>You're leaving out the time dimension. Our observable universe back to the surface of last scattering is a 4-ball, not a 3-ball.

Accepted, but when we view the CMB, we are looking at a very narrow interval on the time axis, so I have treated this surface of last scattering (S) as a purely space-like surface, albeit at a particular time. You have also agreed to this in (*) below.

Given the above paragraph, I think we are agreed that whatever the initial conditions of the universe (point or smear), the surface S is a space-like 2-sphere which contains a 4-ball universe from time t=0 up to the time of last scattering , right?

Now you write:
>>we can only see a portion of that surface (S)

Has this claim actually been verified, or just assumed. If you are correct here, my argument is not falsified, but it is no longer complete, and I give up.
However, I have 2 reasons to think you are incorrect here. Firstly , since your claim would require mapping a portion of S continuously onto the surface
which we see today as a spherical map of the CMB, then such a mapping would necessarily display discontinuities which we do not see. Thus,
if you take Massachusetts alone, and try to stretch and paste it to cover the entire globe...you will see discontinuities. Secondly, even if some part
of S were not visible to us today, how would we know this? The needed data is inaccessible ! Indeed, you seem to recognize this as you write...

>>(*) Our observable universe back to the surface of last scattering is a 4-ball, not a 3-ball. ... bounded by a spacelike surface in the past (the surface of last scattering S) and a null cone whose apex is the Earth's present eventAt this point, we can now return to my 2nd argument (merging balls) modified to accommodate your protests.

We have a 4-ball bounded by a space-like S containing the (invisible) early universe. and another 4-ball containing the visible universe, also bounded by S. As before, we merge these two objects by identifying their common boundaries S. We now have a universe with one time dimension (t = 0 to now), and whose spatial context is S³ (positive curvature, not flat).

 

[PeterDonis]

when we view the CMB, we are looking at a very narrow interval on the time axis, so I have treated this surface of last scattering (S) as a purely space-like surface, albeit at a particular time.

You're not addressing my argument. My argument is not that you need to look at a larger "time slice" around the surface of last scattering. My argument is that your argument requires that the "point" or "region" (you are correct that it doesn't have to be a single spatial point) being looked at is timelike, not spacelike. Since the surface of last scattering is spacelike (as you agree), your argument does not apply to our observations of it.

Has this claim actually been verified

The surface of last scattering was a finite time ago, and light travels at a finite speed. That means we are only seeing a finite intersection of our past light cone with the surface of last scattering.

If you're asking how we know that finite intersection does not comprise the entirety of the surface of last scattering, we know that if the spatial topology of our universe were in fact $S^3$, its radius of curvature would have to be much, much larger than the size of our observable universe. That ensures that the portion we see of the surface of last scattering is only a portion; it can't be the whole surface. If it were, we would see strong evidence of positive spatial curvature, and we don't.

We have a 4-ball bounded by a space-like S containing the (invisible) early universe. and another 4-ball containing the visible universe, also bounded by S.

In other words, S is the surface of last scattering. If that is the case, then we have two 4-balls, separated by a spacelike 3-surface, yes. But you have not shown that that spacelike 3-surface is an $S^3$. See below.

As before, we merge these two objects by identifying their common boundaries S. We now have a universe with one time dimension (t = 0 to now)

Yes. But what topology does this 4-manifold have? You are claiming that it has topology $S^3 \times R$; but you haven't shown that. See below.

and whose spatial context is S³ (positive curvature, not flat).

Nope. The construction of an $S^3$ by merging two 3-balls works like this: you have two 3-balls, each with a 2-sphere as a boundary. You identify the two boundaries and "glue" the two 3-balls together at their common boundary; the resulting manifold is then an $S^3$.

But in the case you are describing, you have two 4-balls, each bounded by the same 3-surface. But you can't assume that that 3-surface (the surface of last scattering) is an $S^3$; that's what you are trying to prove. So you can't show that the 4-manifold you get when you "glue" the two 4-balls together at their common boundary has topology $S^3 \times R$, which is what you are claiming.

 

[Tom Mcfarland]

Peter:

I am grateful for your feedback, but I feel I should take a couple of cosmology courses here at UW-Madison before continuing this thread. I am currently taking a related physics course. However, here are some parting comments.

[1] I had assumed that the surface of last scattering (S) was a 2-sphere, not a 3-sphere, like the colorful maps of the CMB we have seen.

[2] I had intended to glue the two 3-balls exactly as you did. Indeed, I describe this in my original post, to which you agreed.

[3] I don't think I understand your use of the terms "time-like" and "space-like", in spite of my attempting to also use them.

[4] I still feel that the spatial topology of our universe will be S³, that is, we live within the surface of a 4-ball with a very large diameter. I have other reasons for this "feeling", but currently these reasons are fanciful, without evidence, so I won't waste your time with them. However, if the topology were indeed S³, I can envision several big currently unsolved problems acquiring elegant solutions, such as "what is dark matter?" and "Why is space expanding?" When the James Webb wakes up, I expect a big change in the standard model.

Cheers, Tom

 

[PeterDonis]

I had assumed that the surface of last scattering (S) was a 2-sphere, not a 3-sphere, like the colorful maps of the CMB we have seen.

I have no idea why you would assume this. The surface of last scattering is a moment in time, i.e,. a spacelike surface of constant time. That is a 3-surface, not a 2-surface (and, as I have already noted, we do not know that its topology is that of a 3-sphere, so you can't assume that; you would have to prove it, and you haven't).

The maps of the CMB that you refer to are the appearance of the CMB on our sky, which is a 2-sphere; but the reason our sky is a 2-sphere is that it is a projection of our past light cone, which is a null 3-surface composed of 2-sphere "slices", each of which is the intersection of our past light cone with a 3-surface of constant time. The maps of the CMB that you see are maps of one particular 2-sphere "slice" of our past light cone, the one that intersects the 3-surface of last scattering. But obviously this is not an image of the complete surface of last scattering; it's just an image of one particular 2-sphere in it, the 2-sphere corresponding to events that were at just the right spatial position at the time of CMB emission for the light they emitted in our direction to be just reaching us now.

I had intended to glue the two 3-balls exactly as you did.

You've lost me; which 3-balls are you referring to? There aren't any 3-balls we've discussed whose gluing together supports your argument about the surface of last scattering.

(Btw, it helps a lot to use the PF quote feature, as I have been doing, to make it clear exactly which parts of my posts you are responding to.)

I don't think I understand your use of the terms "time-like" and "space-like"

Then you definitely need to spend some time studying GR. Sean Carroll's online lecture notes on GR give a good introduction, and include a chapter on basic cosmology. I would strongly advise you to work through them. "Timelike" and "spacelike" are extremely basic concepts in GR and you should definitely understand how they work; and that's not something I think we can fix in this discussion.

 

[Tom Mcfarland]

Peter

You write:
>>Btw, it helps a lot to use the PF quote feature

Good idea, but I do not know how to use this feature. Hence I have improvised this substitute device.

You write (in response to my assumption that the surface of last scattering was a 2-sphere):
>>I have no idea why you would assume this.

I see now why we were not successfully communicating. I had treated the surface of last scattering as identical to the particular "slice" of our 4-D space-time universe from which the currently visible CMB originated, which you acknowledge is a 2-sphere. I then gave this particular slice the name "S", and ignoring any time contribution, used S as the boundary between two disjoint regions of space, each of which is a 3-ball. Gluing these two 3-balls yields a 3-sphere.

I appreciate your prodding me to account for the passage of time, and hope to re-think the above spatial idea to justify positive curvature.
Somehow, this should follow from the currently visible S² slice of the surface of last scattering.

You write:
>>you definitely need to spend some time studying GR

I plan to make "time" (ahem) to do this...at this link, right?

https://arxiv.org/abs/gr-qc/9712019

Previously, I had been viewing Leonard Susskind videos and Wikipedia tutorials

This will probably be my last post to this thread.

Cheers, Tom McFarland

 

[PeterDonis]

I do not know how to use this feature.

If you highlight any portion of a post, "Quote" and "Reply" buttons will pop up. The "Reply" button is the one I use; it inserts the highlighted text directly into the edit area for posting a reply, enclosed in quote tags.

I had treated the surface of last scattering as identical to the particular "slice" of our 4-D space-time universe from which the currently visible CMB originated, which you acknowledge is a 2-sphere.

Yes, but that 2-sphere--i.e., the particular 2-sphere within the surface of last scattering, which is a spacelike 3-surface, from which the portion of the CMB that is just now visible on our sky originated--does not separate two 3-balls within the surface of last scattering. It encloses one 3-ball--the portion of the surface of last scattering that is the interior of the 2-sphere. But the other portion--the exterior of the 2-sphere--is not a 3-ball, or at least we don't know that it is, and you haven't shown that it is. It's a spacelike 3-surface of unknown topology (but our best current model says it has topology $R^3$) with a 3-ball (the interior of the 2-sphere) cut out.

I appreciate your prodding me to account for the passage of time, and hope to re-think the above spatial idea to justify positive curvature.

It won't.

at this link, right?

Yes, that's the one.

 

[PeterDonis]

It won't.

Perhaps it might help if I elaborate on this somewhat. Let's first consider a simpler scenario that illustrates the key point: a flat spacetime, empty of any significant matter and energy, but containing an observer (whose mass is negligible), in which a flash of light occurs at every point in space at one particular instant of time in the observer's rest frame. (The intensity of the flash of light is low enough that it also contains negligible energy, but it is enough to be perceptible to the observer.)

Now, what will that observer's sky look like? The answer is that it will be a continuous glow in all directions, from the time of the flash of light onwards. Why? Because at each instant of the observer's time, he is receiving a 2-sphere's worth of light rays from a slightly further distance away from him. For example, one year after the flash of light, he is receiving a 2-sphere's worth of light rays from points in space one light-year away from him. And one year and one second after the flash of light, he is receiving a 2-sphere's worth of light rays from points in space one light-year plus one light-second away from him. And so on.

Hopefully the point is now clear. The CMB looks to us exactly the same as the above description: a continuous glow in all directions. The only difference is that, because our universe has expanded by a factor of about 1100 since the CMB was emitted (i.e., since the instant of time at which the CMB light flash was emitted at every point in space), the CMB "glow" is redshifted by a factor of about 1100, so that instead of being a glow of visible light (at a temperature of about 3000 K, the roughly temperature at which the CMB emission occurred), it's a "glow" of microwaves at a temperature of a little under 3 K. But the fact that it appears from all directions is still due to the same cause as in the idealized scenario above: that at each successive instant we are seeing CMB radiation from all directions that was emitted a little further away from us than the instant before.

In other words, what I described above is how you take into account the passage of time when looking at the CMB. And, as is evident, doing so does not change at all the fact that you can't deduce an $S^3$ spatial topology for the universe from our observations of the CMB, any more than you could in the idealized flat spacetime scenario above. In fact, that idealized scenario should help by showing that there is a simple case(in which the spatial topology is obviously not $S^3$, yet you can still see radiation of the same temperature and intensity coming in continuously from all directions.

 

[Tom Mcfarland]

OK...I guess I am hooked on this.

Your hypothetical example seems to violate conservation of energy. That is, a single flash contains a finite amount of energy E,
but if a viewer sees even a diminished glow forever, eventually the integrated energy entering his eye will exceed E.

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. If and when these slices are receding from us at speed c, we will no longer see them ??
Waiting eagerly for the James Webb.

The CMB reminds me of a relativistic version of Olbers paradox .

Separately , you write:
>>(a slice of the) surface of last scattering (S)...encloses one 3-ball--... But the other portion--the exterior of the 2-sphere--is not a 3-ball

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?
Looking around, we seem to be on the interior of some spherical slice S?? So you are claiming that the side of S containing the early
universe has unknown topology?

 

[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.