How can they rule out a closed universe?

[RagingHadron]

If my understanding is correct, they use shapes of things great distances apart and they compare certain properties measured to what is calculated for a closed, curved or flat universe. But my questions is if a 2-manifold is topologically homeomorphic to any 2-sphere and the same is true of 3-manifolds and 3-spheres, how could they rule out a closed universe using any 2 or 3-manifolds?

 

[marcus]

A spatially finite universe has not been ruled out.

If my understanding is correct, they use shapes of things great distances apart and they compare certain properties measured to what is calculated for a closed, curved or flat universe. But my questions is if a 2-manifold is topologically homeomorphic to any 2-sphere and the same is true of 3-manifolds and 3-spheres, how could they rule out a closed universe using any 2 or 3-manifolds?

There was a NASA report that came out with the 5-year data from the WMAP mission which concluded that, while we don't know spatial finite or infinite IF spatially finite then with 95% confidence the circumference is AT LEAST 600-some billion LY.

That is, with 95% confidence space is either infinite or very big.

Obviously that is not the same as saying infinite.

Another term they use is "nearly flat". That is, the overall average curvature of the portion we can see is, with 95% confidence, zero or at least very small.
===========================

FYI a compact 2D manifold is not necessarily homeomorphic to a 2-sphere. It might be topologically equiv. to a torus---surface of donut. Or some other stuff. To a first approximation it is not worth worrying about more sophisticated topology issues, in cosmology. In my humble opinion.

It's good for your mind. But in practical---observation based---cosmology they mostly don't worry about donuts. They just say things like "nearly flat" and "very large if not actually infinite". Humans don't know a lot, we are just getting started appreciating the universe. It's wonderful to have inferred even the very little we have already. Wonderful to have instruments like WMAP mapping the ancient light---the microwave background sky.

 

[skydivephil]

I don't believe a closed universe is ruled out. As I understand it , the universe is measured to be flat. So if there is any curvature its way beyond our horizon, that's all. If anyone thinks that's wrong please let me know.

 

[RagingHadron]

I didn't mean to imply that I thought that it had been ruled out, I was just wondering how their methods could rule it out

 

[marcus]

Oh! OK have a look at
http://arxiv.org/abs/0803.0547

Look at table 2. they have a 95% confidence interval on the overall curvature.

All they can do is narrow that down.

The closer to zero, the more likely it's infinite.

What you could be asking is how do they estimate the overall curvature. That is an absolutely fascinating process. they undoubtably are going to narrow the interval down, I would guess next year or 2014. Let's talk about how they constrain curvature.

 

[RagingHadron]

I'm more asking whether it's valid. Maybe my understanding of homeomorphism is wrong...

 

[marcus]

First learn how to read the results. I assume you are looking at table 2 of
http://arxiv.org/pdf/0803.0547v2.pdf

Look in the WMAP+BAO+SN column which is where they combined ALL the data relevant to curvature. Not just the WMAP mission data.

Look in the row that says Positive curvature case. that would be the 3-sphere case.

Look where it gives the estimate for the RADIUS OF CURVATURE. That would be like the radius of the 3-sphere if you pictured it embedded in a higher dimensionality which we merely imagine, do not know exists.

The table tells you that with considerable confidence the radius R in that case has to be
> 22 h^-1 Gpc

OK you have to learn how to interpret that. G means billion. pc means parsec (i.e. 3.26 lightyear) so multiply 22 times 3.26 and call the answer billion lightyears.

And h is a little number cosmologists use which encodes what you believe the Hubble parameter is. A common value to assign to it is h = 0.71 because you think the present value of the Hubble parameter is 71 km/s per Mpc.
But you might also think h is 0.706 if you and your co-authors are using 70.6 km/s per Mpc in your work. Having this little order-of-magnitude unity (approx = 1) number in the formulas is helpful because it gives everybody the freedom to adjust, as estimates of the Hubble parameter are refined.

So multiply 22 times 3.26 and divide by 0.71 and call the lowerbound answer for the RADIUS that many billion lightyears.

Then if you want a lowerbound for the circumference you can multiply by 2 pi.

Space has to be AT LEAST that big. So if you could freeze expansion and head off straight in some direction at the speed of light it would take at least 600-some billion years before you completed circumnavigation and got back home. At least. It is only a 95% confidence lowerbound estimate.

Now the fascinating thing is how they arrive at the confidence interval for the curvature radius, or equivalently for the curvature parameter Omega-sub-k

Look at footnote g of table 2 to find out how you calculate the radius R from the parameter Ω_k.

One way might be to count the number of galaxies in balls of different radius. If we live in overall zero curvature then the number should increase with the cube of the radius. (after you adjust for expansion, and rates of galaxy formation). The idea is that if it weren't for distances expanding then in a nice flat (zero curvature) universe the volume would increase as the CUBE of how far out you look. And counting galaxies is a way of estimating the volume. that's oversimple but it gives an idea of how one might get a handle on curvature.

Another way is to study the SIZES of the TEMPERATURE FLUCTUATIONS in the ancient light.
this is a snapshot of the universe taken in year 380,000. All that light was released right about that time. Our region of space was 1000 times smaller and full of hot gas that had just then become transparent. We understand something about the PHYSICS of that hot gas. How its pressure and density would have been throbbing randomly in patches of lower/higher pressure. So astrophysicists can estimate the statistics of the sizes of the patches of lower/higher temperature. what the actual sizes were in year 380,000.

And then they can compare what the sizes were then with how big they look now. Admittedly it gets somewhat sophisticated and involved. But it gives another way to get a handle on the overall curvature. Several other people here (say e.g. B. Powell) can explain this more clearly.

There are several handles one can get on curvature. They are all "model-dependent". You have to set up a mathematical model of the geometry of the universe and make observational measurements that you put into the model and adjust the model to, and get curvature out. Where it says "WMAP+BAO+SN" that means using 3 different types of observation to get 3 different handles.

the model is called Friedman equation and it is derived from the 1915 Einstein equation, which is basically a law of geometry, how geometry (in conjunction with matter) evolves.
I wish I could do a better job of intuitively explaining how one infers estimates of curvature from various observations.

I think the confidence interval for Ω_k is one of the more beautiful achievements of cosmology. BTW there is a conventional sign reversal and negative Ω_k signifies positive curvature---i.e. finiteness.

As they continue to shrink the interval, it might continue to straddle zero (which would suggest flatness and possibly infiniteness) OR it might get a little off to one side, say the negative side and stop straddling zero. And that would signify spatial finiteness. To shrink the interval requires improved instruments. Currently the successor to WMAP is taking data. It is called Planck and it orbits the sun about a million miles farther out from the sun than the Earth is. The Planck data will be reported in 2013, I think. Not this year anyway.

 

[RagingHadron]

Ohh ok thank you soo much. It makes more sense now

 

[bapowell]

With regards to the mention of homeomorphism, these transformations preserve certain topological characteristics of the space. But intrinsic curvature is a geometric -- not a topological -- property, and it is not a homeomorphism invariant. We can therefore tell by measuring the curvature of the universe whether it has spherical (curvature K=1), hyperbolic (K=-1), or Euclidean (K=0) geometry. These are the only three choices up to isometries.

 

[RagingHadron]

Bapowell, that's the answer I was looking for haha. That's what confused me

 

[Chalnoth]

If my understanding is correct, they use shapes of things great distances apart and they compare certain properties measured to what is calculated for a closed, curved or flat universe. But my questions is if a 2-manifold is topologically homeomorphic to any 2-sphere and the same is true of 3-manifolds and 3-spheres, how could they rule out a closed universe using any 2 or 3-manifolds?

If our observable universe is extremely flat (which it appears to be), then we won't ever be able to rule out a closed universe. We'll just have tighter and tighter bounds on how flat our universe is.

 

[ImaLooser]

If our observable universe is extremely flat (which it appears to be), then we won't ever be able to rule out a closed universe. We'll just have tighter and tighter bounds on how flat our universe is.

Quite so. Nevertheless the flat Universe has been declared the "standard model." The reasoning is that the curvature tends to become magnified over time. So for it to be so close to flat now seems like too much of a coincidence.

 

[Chalnoth]

Quite so. Nevertheless the flat Universe has been declared the "standard model." The reasoning is that the curvature tends to become magnified over time. So for it to be so close to flat now seems like too much of a coincidence.

Stating that the universe is absolutely flat is not part of the standard model. Just very, very flat. And inflation offers a good explanation for this.

 

[bapowell]

Quite so. Nevertheless the flat Universe has been declared the "standard model." The reasoning is that the curvature tends to become magnified over time. So for it to be so close to flat now seems like too much of a coincidence.

That's the flatness problem, which is resolved by inflation as Chalnoth says. Inflation gives us an observable universe that is very close to flat, but by no means needs to be exactly flat.

 

[twofish-quant]

To describe why inflation creates a nearly flat universe. Imagine that you are in any shape say a balloon. Now imagine that shape expands a huge amount. Whatever the shape was before, after you inflate it, it looks flat. Witness the earth. It's round, but when you look at a small enough part of it, it looks flat.

Measuring the exact amount of flatness is good because that will tell us something about how the universe inflated.

One other thing was that before 1995 or so it was often assumed that if a number about the universe was close to zero that it would be zero. The idea was that the universe had some fundamental symmetries, so if it looked like something would be close to zero, then symmetry would make that number zero. This no longer is a strong argument, because there turn out to be plenty of numbers that end up close to zero that aren't zero, and the numbers that are are getting don't see to be "magic" (i.e. we get things like 0.7 rather than 1.0 or 0.0).

That means that when we look at the observations that say that the universe is close to flat, people no longer assume that it's going to be flat.

 

[twofish-quant]

Quite so. Nevertheless the flat Universe has been declared the "standard model."

It actually hasn't. Someone needs to change the Lambda-CDM wikipedia page since it states that a flat universe is part of the model, when it isn't.

The reasoning is that the curvature tends to become magnified over time. So for it to be so close to flat now seems like too much of a coincidence.

It's something of an "anti-coincidence". There are some constants that do get magnified over time (namely the fraction of dark energy), but flatness isn't one of them. Flatness comes out naturally from inflation.

Also, just out of curious where did you get the idea that flatness was a coincidence? I worry that there may be some popular author that misspoke and I'd like to "nip this in the bud".

(And i just changed the wikipedia article)

 

[bapowell]

Also, just out of curious where did you get the idea that flatness was a coincidence? I worry that there may be some popular author that misspoke and I'd like to "nip this in the bud".

He's referring I believe to the flatness problem; in the non-inflationary universe flatness is unstable and the universe evolves away from flatness. This "coincidence" is a reference to the extreme fine tuning of the curvature that must be done at the start of the expansion.

 

[Farahday]

If the cosmos is indeed a spherical spacecage then for every point in it there exists another point within a finite distance at which travel in any direction would not increase the distance between the two points. A traveler who could instantaneously traverse sufficient distance would begin to return to point of origin. If space is isotrophic and homogeneous - ALL points within the spacecage are points where progress becomes regress...and transportation would suffer greatly.