Contradictory Cosmological Values: Crisis for Cosmology?

[Buzz Bloom]

TL;DR Summary

A 5 Nov 2019 article gives as the calculated value of the radius of curvature of a closed universe to be 10 Gpc. Wikipedia gives the size of the observable universe as "about 14.3 billion parsecs".

These two independent sources have cosmological values that seriously contradict each other.

The article "Planck evidence for a closed Universe and a possible crisis for cosmology"

https://www.nature.com/articles/s41550-019-0906-9​

arXiv:1911.02087v1 [astro-ph.CO] 5 Nov 2019​

gives as the calculated value of the radius of curvature of a closed universe to be 10 Gpc. This value is based on a calculated 3.4 standard deviations range of values for the ΛCDM energy density parameter Ω_k:

-0.0007 < Ω_k < -0.095 with a 99% confidence level.​

Paraphrasing Wikipedia: the radius of the observable universe is about 14.3 Gpc.

https://en.wikipedia.org/wiki/Observable_universe​

For this value Wikipedia cited a Scientific American Article for which the following has its abstract.

https://dialnet.unirioja.es/servlet/articulo?codigo=6596344​

I am hoping someone here at the PFs can explain to me how the currently best value for the radius of a finite universe can be significantly smaller than the size of the observable universe. I understand that the four parameters of the two different ΛCDM models involved will be different, but I do not have access to sources for the values of these variables for these two models, so I cannot analyze how these variable values could produce this strange comparison.

ADDED
I just made an observation regarding the two results.

The value of the finite universe volume is 2π²R_F³, and the volume of the observable universe is (4/3)πR_OU³.
Thus the ratio of volumes is

V_F/V_OU = (3π/2) (10/14.3)³ = 1.6​

That is, the Finite universe is 1.6 times larger in volume than the Observable Universe. This relatively small ratio would have an effect on observing the CMB boundary, The finite universe CMB boundary would be about 1/2 the area that it would be in a flat universe.

​

ANOTHER ADDED
I unintentionally cited an abstract rather than the whole article above. The link to a PDF file of the whole article is in my post #3.

AND STILL ANOTHER ADDED

My post #15 explains (briefly) my discovery of my conceptual errors regarding an aspect of this thread's curvature topic related to CMB watts per square meter detection.

 

[mathman]

My guess: radius calculation is wrong or misunderstood. Link gives only abstract not paper.

 

[Buzz Bloom]

Link gives only abstract not paper.

Hi mathman:

My bad. The following is a link to a PDF file. The radius value is on page 2.

https://arxiv.org/pdf/1911.02087v1.pdf​

Regards,
Buzz

 

[mathman]

Paper is over my head. It would need an astrophysicist to comment intelligently.

 

[Keith_McClary]

Paper is over my head. It would need an astrophysicist to comment intelligently.

There are some here:
https://astronomy.stackexchange.com/search?q=radius+universe

 

[Bandersnatch]

I am hoping someone here at the PFs can explain to me how the currently best value for the radius of a finite universe can be significantly smaller than the size of the observable universe.

What's the issue here? A ~10 Gpc radius of curvature (assuming a hypersphere) still translates to something like ~200 Gly 'circumference'* - a good couple times larger than the diameter of the observable universe. And even if it were smaller, what problem do you see there? It would only mean that you could observe the same structures repeatedly, at different epochs.

This relatively small ratio would have an effect on observing the CMB boundary, The finite universe CMB boundary would be about 1/2 the area that it would be in a flat universe.

Can you explain why that should be the case? And what do you mean by smaller area, observationally?

*I'm just hoping I'm generalising the notion of the circumference correctly.

 

[PeterDonis]

*I'm just hoping I'm generalising the notion of the circumference correctly.

You don't have to "generalize" it at all; it's still just $2 \pi R$, where $R$ is the radius.

 

[PAllen]

What's the issue here? A ~10 Gpc radius of curvature (assuming a hypersphere) still translates to something like ~200 Gly 'circumference'* - a good couple times larger than the diameter of the observable universe. And even if it were smaller, what problem do you see there? It would only mean that you could observe the same structures repeatedly, at different epochs.Can you explain why that should be the case? And what do you mean by smaller area, observationally?

*I'm just hoping I'm generalising the notion of the circumference correctly.

The circumference doesn't change with dimension. So, about 62.8 Gpc would be expected, still no contradiction. The volume of the 3-sphere (equiv. area of surface of 4-ball) would be $2\pi^2R^3$, or about 20,000 cubic Gpc.

 

[Buzz Bloom]

So, about 62.8 Gpc would be expected, still no contradiction.

Hi Paul:

Thank you for your post.

I may be misinterpreting the geometry, but it seems to me (ignoring the universe expansion for this part of the discussion) that in a non-expanding finite universe of radius R_FU = 10 Gpc, the maximum distance between two points is

D = πR_FU = 31.4 Gpc.​

What I would like to calculate, but I am having difficulty in doing this, is the comparison of two ΛCDM universe models of the current universe both as they are at the current time:

(1) a flat universe, and​

(2) a finite universe.​

To do this I need for each model is the current best values for the four model parameters, and the best value for

a = scale factor​

at the time of recombination, i.e., the beginning time for the CMB.

In particular I want to compare the two current spherical areas of the CMB at recombination. The 1/2 number I came up with in post #1 was very rough and did not take into account the differences in the a(t) function for the two models. This function is needed to calculate the distance light travels from recombination to the observer (which is the radius of the visible universe for the flat model).

https://en.wikipedia.org/wiki/Observable_universe​

Sometimes astrophysicists distinguish between the visible universe, which includes only signals emitted since recombination (when hydrogen atoms were formed from protons and electrons and photons were emitted)​

The basic idea of such a ratio is that a sphere of radius R in a flat universe has a surface area
of

A = 4 π R²,​

while in a finite universe of radius R_FU

A = ((1/θ) sin θ) 4 π R²​

where

θ = πR/R_FU .​

Regards,
Buzz

 

[PAllen]

Hi Paul:

Thank you for your post.

I may be misinterpreting the geometry, but it seems to me (ignoring the universe expansion for this part of the discussion) that in a non-expanding finite universe of radius R_FU = 10 Gpc, the maximum distance between two points is

D = πR_FU = 31.4 Gpc.​

...

That's not the right number to compare to nominal observed universe size. From a given point, you could easily have a point 30 Gpc 'east' of you, and another 30 Gpc 'west' of your. That they are close together is not relevant to your observations.

 

[Buzz Bloom]

That's not the right number to compare to nominal observed universe size. From a given point, you could easily have a point 30 Gpc 'east' of you, and another 30 Gpc 'west' of your. That they are close together is not relevant to your observations.

Hi Paul:

I am hoping you can clear up my confusion.

By the way, I edited my previous post to correct the area formulas.

In the following discussion I am ignoring the fact that no message can be send from one place to another prior to recombination.

As I currently understand the meaning of the radius of the observable universe, it is the distance something could hypothetically travel at the speed of light from the big bang to an observer. Another way to say this is: it is the distance D at the present time between an observer and the place where the source of a signal is now from which a message was sent that is observed now by the observer.

As I understand it, in a stationary finite universe of radius R, a point at a distance R from an observer in a particular direction, and a second point at distance R from the observer in the exact opposite direction would be the exact same point in this finite universe.

Here is a thought experiment. I want to explore the implication of the CMB at a particular time, with respect to a volume between two concentric spheres of some fixed thickness, and with its center at the location of an observer. The particular time chosen is when this volume contains (mostly) hydrogen which is one-half ionized.

Is there any observable difference between such a volume in the flat universe and a finite universe with respect to the process of its completing its recombination?

It seems intuitively obvious to me (but of course possibly wrong) that if the flat observable universe has a radius close to twice the radius of the finite universe, there should be significant different behavior. The reason I think this is that the volume between the concetric spheres in the finite univerfse would be much much smaller that the volume in the flat universe.

Regards,
Buzz

 

[PAllen]

Radius of curvature is completely different from observable radius. Consider that the radius of curvature of a spatially flat universe is infinite. Given a 10 Gpc radius of curvature for closed universe, you can get light from over 30 Gpc away in every direction.

 

[Buzz Bloom]

Given a 10 Gpc radius of curvature for closed universe, you can get light from over 30 Gpc away in every direction.

Hi Paul:

Yes, I understand that. To be more precise, given a finite closed universe with a radius

R = 10 Gpc,​

an observer can hypothetically get (ignoring the opaqueness of matter prior to recombination) light from a distance D of a light source which at the time of observation is

D = πR = 31.4159... Gpc,​

provided that the expansion of the universe does not at the present time limit the size of the observable universe to be less than πR.

I would much appreciate a comment from you about the following which is modified from the last paragraph in my post #11.

Assume hypothethically that a flat universe model fitting the current cosmological database (CCDB) has values for the four ΛCDM variables and also

a = the scale factor,​

such that the radius of the corresponding observable universe is just a bit smaller than 31.4159 Gpc. Then I am guessing (since at the present time I cannot do a calculation) that the observable universe for a finite closed universe model with radius of 10 Gpc that also approximately matches the CCDB has a observable universe radius also a bit smaller than 31.4159 Gpc. If this guess is correct, then the surface area of the sphere corresponding to the CMB will be very much smaller for the finite universe model than for the flat universe. If this is correct, then would cosmologists likely have expectations that calculated observations of the CMB would be quite distinguishable between the two models?

If this is so, then my guess would be that there would also be some observational differences regarding the CMB between the two models descibed in post #1, although much smaller differences than the two models described in this post.

Regards,
Buzz

 

[PeterDonis]

Moderator's note: Some off topic posts have been deleted.

 

[Buzz Bloom]

I previously had what I now see to be an erroneous intuitive insight which falsely indicated that any curvature of the universe would result in different CMB power observations than that from a flat universe. I now understand that although the area producing the CMB photons in a finite universe would be smaller, each point on this surface would have a larger fraction of its radiation reaching the observer's apparatus. I now see that these two affects would cancel each other, so the CMB watts per meter squared detected would be exactly the same. This would also be the case for a hyperbolic universe.