B What is the probability that the Universe is absolutely flat?

[Buzz Bloom]

So regardless the global curvature locally the universe can have small areas like our observable universe which are positively or negatively curved depending on local inhomogeneities?

Hi timmdeeg:

The assumption that the quote question is answered "yes" implies the corresponding universe is not (for large scales) isotropic and homogeneous. The means it is not compatible with a GR universe.

Regards,
Buzz

 

[PeterDonis]

The assumption that the quote question is answered "yes" implies the corresponding universe is not (for large scales) isotropic and homogeneous.

Not quite. It means that if the universe on large enough scales is isotropic and homogeneous, "large enough" must mean "larger than our observable universe". If the universe were spatially infinite, or spatially finite but much, much, much larger than our observable universe, this would not be a problem from a modeling standpoint; but it would mean that we would not be able to test the assumption of global isotropy and homogeneity directly by observations, since those are by definition restricted to our observable universe.

The means it is not compatible with a GR universe.

GR does not require that the universe is homogeneous and isotropic on any scale. The assumptions of global homogeneity and isotropy are made because they are simple (i.e., we know exact solutions for this case) and, at least so far, consistent with the data. But if for some reason global homogeneity and isotropy were ruled out, that would not in any way mean GR could no longer model the universe. It would just be a lot harder since we would not be able to use any exact solutions and would have to do the modeling numerically.

 

[metastable]

Are there any conceivable objects that could be large enough outside the observable universe to have any measurable effects within our observable universe?

 

[Jaime Rudas]

Do you know of any particular specified parameters that mathematically defines a theoretical 3-torus topologically shaped universe model which is finite, flat, isotropic, and homogeneous?

I think Friedman was referring to this when, in his 1924 paper On the Possibility of a World with Constant Negative Curvature of Space, he mentioned:

"In the present Notice it will be shown that it actually is possible to derive from the Einstein world equations a world with constant negative curvature of space.
[...]
At the end of this Notice we will touch upon the question of whether on the grounds of the curvature of space one is allowed at all to judge on its finiteness or infinitude.
[...]
We have convinced ourselves that the Einstein world equations possesses solutions that correspond to a world with constant negative curvature of space. This fact points out that the world equations taken alone are not suficient to decide the question of the finiteness of our world. Knowledge of the curvature of space gives us still no immediate hint on its finiteness or infinitude. To arrive at a definite conclusion on the finiteness of space, one needs some supplementary agreements."

 

[Buzz Bloom]

To arrive at a definite conclusion on the finiteness of space, one needs some supplementary agreements.

Hi Jaime:

Thank you for the Friedmann quotes.

I interpret this as saying that something that scientists currently do not have (after almost 100 years) is a reason to believe a flat finite universe is possible, except for the fact that such a model has not (yet) been proved to be impossible.

Regards
Buzz

 

[Jaime Rudas]

I interpret this as saying that something that scientists currently do not have (after almost 100 years) is a reason to believe a flat finite universe is possible, except for the fact that such a model has not (yet) been proved to be impossible.

Hi Buzz:

I interpret this as that general relativity plus the cosmological principle aren't a suitable instrument to determine if the universe is finite or infinite. That is, the fact that the universe is flat or negatively curved does not imply that is infinite, as it is frequently seen in popular science texts.

On the other hand, it seems logical to consider possible everything that we can't prove impossible.

Regards.

 

[Buzz Bloom]

On the other hand, it seems logical to consider possible everything that we can't prove impossible.

Hi Jaime:

I do not disagree with the quote. However, I do not think it is practical to give serious consideration to a possibility for which there is no evidence that the possibility is true except that there is no proof that it cannot be true.

In another thread there was a point that many scientists want a 4 sigma (99.99% confidence level) before taking a conclusion seriously. I find this to be a reasonable attitude. To take seriously a possibility just because it has not been proved impossible is like deciding to take a possibility seriously if it has 0.01% probability of being true.

Regards,
Buzz