I Shape of the universe, and a few other questions....

[Hypercube]

Hello everyone,

I don't normally come to Astro/Cosmo forums, but I stumbled upon a discussion between a mentor and a PF member here, which involved explanations on the geometry of the universe: difference between boundless vs unlimited, 2-torus vs 3-torus, why are tori boundless, etc. This got me curious about a few things, so I was wondering if someone could verify:

1. At present, do we know the shape of the universe?
2. Is the current assumption (or model) that the universe is flat, uniform and unlimited? We call that "3-plane"?
3. Speaking of these "exotic"-sounding shapes, like 3-torus and 3-sphere, can they also be a viable possibility?
4. If they are, that would suggest that they are so large that their curvature is practically undetectable? Hence, why we assume flat universe...?
5. In the discussion, someone mentioned Occam's razor; so, although we can keep coming up with differently closed surfaces of different shapes, can we say that the plane is the simplest one (and hence most probable)?
6. This is perhaps subjective, but does anyone else have a problem "accepting" the infinite universe with infinite mass, uniformly distributed? As an aspiring physicist I shouldn't go by what I "feel", but I cannot help that these infinities somehow go "against me" in some way. A 3-sphere would be boundless, but the mass would have to be finite, so I have no issues there...
7. Is Carroll An Introduction to Modern Astrophysics a good resource?
8. Do books on cosmology normally provide overview of GR? Or would you recommend to first read an introductory text on GR before I even dare open an astro/cosmology textbook?

Thank you.

My level: intermediate undergraduate, who has not taken any astro courses yet, and only conceptually knows what manifolds are. (This is probably wrong, but I think of it as some sort of generalisation of vector calculus.)

Edit: evidently, I can't count.

 

[kimbyd]

1. No. If there is a non-trivial topology (e.g. a torus or a sphere) that connects the universe at large scales, those scales are so much larger than the observable universe as to be unmeasurable.
2. The 3-plane is generally used by cosmologists. I don't think it should be assumed to be true, but it's accurate enough within the observable universe to make it useful.
3. I don't think there's any reason to exclude non-trivial topologies. In order to have an idea of what kinds of topologies are reasonable, you'd have to have a model for how that shape can form in the first place. Since we don't know how our observable universe originally formed, we can't really say anything about what shapes are possible or impossible.
4. Yes.
5. Occam's Razor isn't really useful without a physical model. As we don't have one for the topology, there's no good answer here.
6. Most definitely some people have difficulty accepting an infinite universe. That doesn't necessarily mean it's wrong, but it's worth being skeptical of such impulses.
7. I don't know that book, unfortunately.
8. You may get some limited overview of GR, but GR is complex enough that it very much justifies its own separate (and in-depth) study. Unfortunately, I'm terrible at book recommendations so I can't help further.

 

[andrewkirk]

3. Speaking of these "exotic"-sounding shapes, like 3-torus and 3-sphere, can they also be a viable possibility?

I suspect that a 3-torus would not be consistent with the Cosmological Principle, which includes the requirement that the universe be isotropic on the large scale. Unlike a 3-sphere, a 3-torus would have one or more preferred directions at any point, which would be the direction in which the length you have to go along a geodesic before it intersects itself is shortest. For a 3-sphere (more generally a n-sphere) that length is the same for all directions, but not for a 3-torus (more generally a n-torus).

A 3-sphere is not only a possibility but is one of the three candidates deemed possible under the Cosmological Principle: Flat, Hyperbolic or Elliptic. The 3-sphere is Elliptic.

 

[kimbyd]

I suspect that a 3-torus would not be consistent with the Cosmological Principle, which includes the requirement that the universe be isotropic on the large scale.

This is very true, but the cosmological principle is not necessarily true on scales much larger than the observable universe. It is certainly valid within the observable universe, but there's no reason to believe it extends infinitely. In fact, most proposals for how our universe began implicitly assume that the cosmological principle is violated at scales much larger than the observable universe.

For example, cosmic inflation was originally proposed to explain (among other things) the homogeneity of the observable universe. It does this by assuming a period of rapid expansion which has the impact of smoothing out inhomogeneities. This implies that if it were possible to observe the universe on much larger scales, those inhomogeneities would once again become apparent.

I've always seen the cosmological principle as the simplest set of assumptions that we could make, and there's never been any sort of justification or argument for why those assumptions must hold. They just happen to be a good approximation for our observable universe, and the reason why they are a good approximation is something that needs to be explained.

 

[berkeman]

I stumbled upon a discussion between a mentor and a PF member here,

Could you post a link to that thread? Thanks.

 

[Hypercube]

Could you post a link to that thread? Thanks.

Sure, this one.

Spoiler

Mentor had patience of a saint.