Questions about Isotropy in Our Universe and Implications

[Buzz Bloom]

TL;DR Summary

Do the average observable charateristics, within a determined angular area of the CBR, demonstrate approximate isotropy of our universe?
What does this imply about possible topological shapes for our universe?i

That is, is there a angular circle size which is approximately in all directions statistically the same with respect to its CBR temperature variation contents (after an adjustment for the veocity of the Earth in some specific direction)?

If the answer to the above question is "Yes", is this result sufficient evidence surporting the assumption that our universe is isotropic?

If this is correct, is it adequate evidence that our universe cannot have a toroidal shape? The argument that comes to mind is that a hyper-torus can be represented as a cube with opposite faces identified. It has a continuous translation symmetry, but no continuous rotation symmetry. It is not isotropic becasue it is possible to have straight lines from a chosen origin that returns to itself with different angles relative the six parallel surfaces, AND such different paths will NOT be the same length.

 

[PeterDonis]

angular circle size

What do you mean by "angular circle size"?

 

[Buzz Bloom]

I apologize for an inappropriate vocabulary. I visualize a circle in space perpendicular to my line of sight. By "angle size" I meant the "solid angle" corresponding to a circle on the surface of a sphere with its size described in terms of square degrees (SD) or square minutes (SM) or square seconds (SS). 1 SD = 3600 SM. 1 SM = 3600 SS. Another unit is the Steradian based on square radians. There are 4π Steradians on the entire sphere.

 

[PeterDonis]

I visualize a circle in space perpendicular to my line of sight.

It would be a 2-sphere, not a circle, as your further comments show. So basically, you are asking whether, after making an appropriate adjustment for the motion of the Earth, the temperature of the CMB is isotropic? The answer to that is yes, to about one part in 100,000.

 

[Buzz Bloom]

It would be a 2-sphere, not a circle

Hi Peter:

I am confused by your use of "2-sphere". As I understand the term, a 2-sphere is the 2D surface of a 3D sphere. What I was trying to describe is the solid angle of the 2D area of a circle on a 2-sphere in terms of the perspective of an observer at the center of the 2-sphere which corresponds to the total source of the CBR we on Earth observe.

Thank you very much for your "yes" answer to my first and second questions. I am still hoping someone will answer my question about whether or not this conclusion logically implies that our universe cannot have a torus topology.

Regards,
Buzz

 

[PeterDonis]

As I understand the term, a 2-sphere is the 2D surface of a 3D sphere.

That is one possible realization of a 2-sphere, but not the only one. Another is all possible directions that you can look from a given point in space. The set of all those directions also forms a 2-sphere.

What I was trying to describe is the solid angle of the 2D area of a circle on a 2-sphere in terms of the perspective of an observer at the center of the 2-sphere which corresponds to the total source of the CBR we on Earth observe.

We observe CBR coming to us from all directions, so the "total source" is the distribution of observed properties of the CBR over a 2-sphere. Which is, as I said, isotropic to about one part in 100,000.

Thank you very much for your "yes" answer to my first and second questions.

I only answered "yes" to the first question. The answer to your second question is "no". The CBR is not the only thing in the universe that we can see. In order to have sufficient evidence for isotropy we need to look at everything we can see, not just the CBR. It turns out that, as far as we can tell, everything else we can see is also isotropic, but not necessarily to the same accuracy as the CBR. But there's no way to know that just by looking at the CBR.

I am still hoping someone will answer my question about whether or not this conclusion logically implies that our universe cannot have a torus topology.

The answer to this question is also "no". The kind of anisotropy you describe for a 3-torus topology is not necessarily inconsistent with everything we can observe up to now being isotropic. For the 3-torus anisotropy to show up in our observations, there would have to have been sufficient time for light to travel all the way around the universe in at least some directions. If the 3-torus is large enough, that would not be the case.

 

[kimbyd]

There's no evidence of large-scale anisotropy that differs from basic inflationary expectations. Therefore there are no apparent observable effects for any sort of global topology, indicating that any nontrivial global topology must have observational features much larger than our observable universe.

E.g., if our universe is a 3-sphere, the radius of curvature of said sphere must be much greater than the radius of the observable universe.