Is the universe finite or infinite?

[twofish-quant]

Either one. Just not textbooks expressly designed to investigate speculative areas of astronomy.

It would help if you gave me some authors.

No doubt there are graduate textbooks on MOND, on loop quantum gravity and on microscopic black holes.

Not really. MOND and LQC are changing too quickly for there to be much in the way of textbooks, so you end up with review papers and paper collections. Microscopic black holes are very interested from a theory standpoint, but there isn't much to say about them.

Good luck with that, I'm sure they'll be thrilled to have your expertise weighing in.

Well yes.

They told you that eternal inflation is a mainstream consensus idea? I doubt that strongly.

No they told me that

1) inflation doesn't require zero curvature
2) the current model of cosmology doesn't assume flatness

I don't see any quotes from them in your argument.

Give me a few days. If I can get you a personal email from one of the three people confirming my points, will you concede the argument? Also, I want to define the question, because I don't want to get into a situation where I bug someone who is busy, get an e-mail, and then you argue that the e-mail doesn't refute your point.

Conversely if you concede those two points now, you save me the effort of writing an e-mail.

What are you claiming they said, and why don't you think it is making it to the WMAP website?

1) inflation doesn't require zero curvature
2) the current model of cosmology doesn't assume flatness

Because the WMAP website was intended for non-technical people, and they simplify a lot of stuff in ways that could be misleading. That's why I'd prefer a reference to something stronger. If you have a citation to a paper in ApJ or a graduate textbook that argues that inflation is inconsistent with non-zero curvature, that would be different than a public affairs website.

 

[Ken G]

Give me a few days. If I can get you a personal email from one of the three people confirming my points, will you concede the argument? Also, I want to define the question, because I don't want to get into a situation where I bug someone who is busy, get an e-mail, and then you argue that the e-mail doesn't refute your point.

Obviously, it is very important to detail the issue correctly. The way you have paraphrased my arguments makes me doubt your version would have much resemblance. For one thing, you insist that I'm claiming that inflation implies the universe is flat. Of course it does no such thing, inflation is a model, it does not constrain the universe, rather the universe, in concert with the goals and demonstrable benefits of science, constrains the model. What I'm actually saying is threefold:

1) For inflation to continue to be regarded as a good model of what really happened in our universe, with no need for anthropic thinking, then curvature must never be detected, and conversely, if curvature is detected, then inflationary models will be up to their ears in anthropic justifications, so much so that much of their original purpose (to escape anthropic thinking) will be lost.

2) The current best model of the universe is a flat, infinite model that obeys the cosmological principle. Occam's razor contributes significantly to making this our best model. Its success is by no means a claim that the universe is actually flat or infinite, for indeed no model can ever make such a claim, given that we cannot see far enough to check it, and never will.

3) The question "is the universe infinite" could never be answered "yes" by any scientific means imaginable. It could only be answered "no", and we already know it cannot be so answered, because we already know we cannot see the limit of the universe. Even if we detect some tiny positive curvature, it would only mean that our best model was now a closed finite model, and again by Occam's razor-- not by any testable claim on the actual geometry of the universe that we cannot see. The best model is never a claim on things that observations are moot about, such things are adopted in the model purely based on Occam's razor. We must stop pretending that science can determine truths even after we have discovered that the observations cannot.

These are the three points I have repeated over and over, and I have never said, or thought, anything else of importance to this discussion. Anyone whose opinion you'd like to solicit on those three points would be more than welcome, indeed quite informative. But the way you have characterized my points is completely inaccurate, and framing the issue as you put it above would have no value whatsoever.

 

[twofish-quant]

1) For inflation to continue to be regarded as a good model of what really happened in our universe, with no need for anthropic thinking, then curvature must never be detected, and conversely, if curvature is detected, then inflationary models will be up to their ears in anthropic justifications, so much so that much of their original purpose (to escape anthropic thinking) will be lost.

And I strongly disagree. If it turns out that there is curvature that imposes a natural scale to the inflation. At that point we can look at the details of the inflation mechanism to see what physics would cause inflation to stop that that scale. For example, if inflation stops when some energy level reaches Planck's constant, then whatever stops inflation could be some quantum tunneling effect.

In physics this is called "hand waving" but it's a useful technique.

2) The current best model of the universe is a flat, infinite model that obeys the cosmological principle.

Again this is incorrect. Cosmologists do not assume that the universe is flat.

Occam's razor contributes significantly to making this our best model.

I'd argue that it doesn't. Where we don't add a term, there are reasons why a term is avoided. Occam's razor tends to be overused as a justification, and it's not that really useful in complex systems.

3) The question "is the universe infinite" could never be answered "yes" by any scientific means imaginable.

I think that you are limiting yourself. I *might* agree to the statement, if you put "currently imaginable" in that statement. Also, the statement of "assuming the universe is isotropic, must it also be infinite?" is something that *can* be answered yes or no.

We must stop pretending that science can determine truths even after we have discovered that the observations cannot.

But this is "proof by lack of imagination." Unknown is not unknowable, and if you want to convince me that something is "unknownable" then you have to give some quasi-mathematical proof of it. Then you run into the issue of whether something is unknowable is itself unknowable.

 

[twofish-quant]

Also just to clarify. Is assertion 1) something you got from someone else or something you made up. It makes a difference between if it's something you got from someone else, then it's easier if you just put a link to where ever you got the idea.

Also one other point is that neither 1) or 2) is "mainstream cosmology."

 

[marcus]

Ken I like some of your posts on other threads e.g. the multiverse issue. thoughtful and cogent. In this case your point #3 is extremely well taken. We don't expect scientists to claim X is the absolute truth. We are happy if they offer the simplest best fit model that has been devised so far and the most reliable model so far for predicting future observations.

So, as you say in point #3 if some positive curvature is discovered (with 95% certainty say) then the simplest best fit model becomes spatially finite. But like any scientific finding that would be provisional and no one can predict the future discoveries. The model might be revised down the road a ways.

That said, you might want to relax your points #1 and #2. I've always understood inflation as having leveled things out enough to be consistent with what we see today. Inflation is consistent with some slight residual curvature.

The treatment of inflation in Loop cosmology does not require fine-tuning and makes an adequate inflation era highly probable. It is consistent with some curvature and if curvature were detected would not bring on the "anthropery" bogeyman. Whether you get threatened by anthopery is to some extent model dependent. Some recent Ashtekar papers about inflation. So point #1 is not terribly firm.

Point #2 is a rather one-sided invocation of Occam, I think. Some people would put Occam on the side of a spatially finite universe, other things equal. I find the finite volume case easier to imagine, simpler. The infinite case with its infinite amount of matter and energy is quite a stretch to imagine. Uniformly distributed too! Infinite energy homogeneously distributed throughout infinite volume!

What you think Occam tells you is to some extent a matter of taste and community consensus. One doesn't want to be too dogmatic about what Occam says is "best". I think anyway.

There's a lot of good in what you say, here and elsewhere, but I think you might relax slightly on points 1 and 2 here.

... What I'm actually saying is threefold:

1) For inflation to continue to be regarded as a good model of what really happened in our universe, with no need for anthropic thinking, then curvature must never be detected, and conversely, if curvature is detected, then inflationary models will be up to their ears in anthropic justifications, so much so that much of their original purpose (to escape anthropic thinking) will be lost.

2) The current best model of the universe is a flat, infinite model that obeys the cosmological principle. Occam's razor contributes significantly to making this our best model. Its success is by no means a claim that the universe is actually flat or infinite, for indeed no model can ever make such a claim, given that we cannot see far enough to check it, and never will.

3) The question "is the universe infinite" could never be answered "yes" by any scientific means imaginable. It could only be answered "no", and we already know it cannot be so answered, because we already know we cannot see the limit of the universe. Even if we detect some tiny positive curvature, it would only mean that our best model was now a closed finite model, and again by Occam's razor-- not by any testable claim on the actual geometry of the universe that we cannot see. The best model is never a claim on things that observations are moot about, such things are adopted in the model purely based on Occam's razor. We must stop pretending that science can determine truths even after we have discovered that the observations cannot.

These are the three points I have repeated over and over, and I have never said, or thought, anything else of importance to this discussion. Anyone whose opinion you'd like to solicit on those three points would be more than welcome, indeed quite informative. But the way you have characterized my points is completely inaccurate, and framing the issue as you put it above would have no value whatsoever.

 

[Ken G]

That said, you might want to relax your points #1 and #2. I've always understood inflation as having leveled things out enough to be consistent with what we see today. Inflation is consistent with some slight residual curvature.

Here's the problem. You have an inflation model, and it has some parameter in it, perhaps the shape of some scalar potential function. You then look at the curvature today, run your GR backward until inflation ends, and try to match up what you get. Certainly you can take any current curvature, no matter how close to flat it is, and you'll get an answer to this exercise. The issue is what is the "size of the target" you are trying to match. If current curvature is not detectable at the, say, .0001 level, then you have a vast range of possible curvatures at the end of inflation, you just map from .0001, to 0, all the way back, and what you get is a hugely wide range of possible curvatures at the end of inflation. Now you have some hope that a plausible inflation model, that is consistent with other established physics, will "hit the target."

Now imagine some observation was just done that detects spatial curvature, say it's in the range .0001 to .0002. Play the same game, map that backward to the end of inflation, and now you have only a factor of 2 in parameter space-- that's the size of the target you have to "hit" with your inflation model. twofish-quant is saying that he has the hope that a plausible inflation scenario that is based on some atomic scale will rather magically hit this target. I'm saying that's pure hope, but at least it's a plausible hope if you have orders of magnitude of possible curvatures that fit with the modern observations. But let's say that a miracle occurs and a natural-sounding inflation model with some built-in established subatomic scale hits the target with finite curvature today. That will certainly seem like a convincing case for that inflation model, a slam dunk even.

But look at the cost we've had to pay-- first of all, we seem to have gotten really lucky to have hit the target, but that's what we are using to justify faith in our model. What's worse is, we now have to wonder why that subatomic scale happens to be set just to hit that tiny range, out of all the orders of magnitude of possiblities for a subatomic scale, so as to just barely generate measurable curvature today! The inflation model seems correct, even undeniable, but it's lost its main purpose: to be able to see the universe as not special or finely tuned. We'd be right back to anthropic reasoning-- the subatomic scale must be coming out that way so as to create a universe with small but measurable curvature because we couldn't exist in all the other more generic universes where that was not the case.

The treatment of inflation in Loop cosmology does not require fine-tuning and makes an adequate inflation era highly probable. It is consistent with some curvature and if curvature were detected would not bring on the "anthropery" bogeyman.

But I think it would. Look at it this way. Take the model you have in mind, and partition its possible parameters into two sets-- the set that leads to unobservable curvature, and the set that leads to small but observable curvature. Of course throw out the set that we've already falsified because it leads to huge curvature. Really do this, it should be easy enough with whatever model you have in mind. Now ask a simple question-- what is the relative measure of those two sets? Is it not true that the unobservable curvature parameter set has vastly larger measure than the observable curvature set? So how is it not fine tuned if we detect curvature tomorrow? How do you answer the question: why that parameter set and not the other parameter set, if the other one was orders of magnitude larger?

Point #2 is a rather one-sided invocation of Occam, I think. Some people would put Occam on the side of a spatially finite universe, other things equal.

It is certainly true that Occam's razor is never clear-cut, but if you just look purely at the model, with no extraneous baggage that says the model is supposed to be the actual reality, then it is clear enough that a model with a non-arbitrary value for the flatness (i.e., flat) is simpler than one with an arbitrarily chosen value of curvature (how do you even give a value to it?). Also, it is much easier to use for doing calculations, which is the key issue I would say.

I find the finite volume case easier to imagine, simpler.

Simplicity of imagining is a different flavor of Occam's razor, it's hard to say if a "best model" is the one that has the fewest arbitrarily chosen parameters (like flat versus some essentially randomly chosen curvature that is not refuted by observation), or the one that is "easiest to picture." My point is that once you are on board with point #3, you are relieved of any philosophical issues with an infinite universe, because you are not claiming the universe is infinite-- you are just fitting what we see to a flat model, like fitting a tangent plane to a manifold where you cannot measure any deviation between the two. The tangent plane is mathematically non-arbitrary, but it is hard to picture because it goes off to infinity in all directions. If we saw that as a bad thing, we could certainly do all of calculus to any precision with circles and spheres of small enough curvature, but we don't, we have lines and planes, because they are mathematically simpler, though harder to picture.

 

[Ken G]

Also just to clarify. Is assertion 1) something you got from someone else or something you made up.

Neither. It comes from me, but it stems from a logical argument. I summarized that argument again just above. So if you want to critique it, you do better finding an actual flaw in the logic.

Also one other point is that neither 1) or 2) is "mainstream cosmology."

You don't seem to even understand what I'm saying with 1) or 2), so I'm suspicious of your judgements of these points. For example, you insist on claiming that I have said that cosmologists assume the universe is flat. That is so completely different from anything I've said, or even thought, that I have no idea where you are even getting that from, but you can't be reading very carefully.

 

[twofish-quant]

Now imagine some observation was just done that detects spatial curvature, say it's in the range .0001 to .0002. Play the same game, map that backward to the end of inflation, and now you have only a factor of 2 in parameter space

If the curvature is positive then at some point in the life of the universe it will take all values from 1e-16 to infinity. We happen to catch it at 0.001, but wait a few billion years and it will be 0.002. Then 0.3, then 0.5, then 2, then 1000, then at some point dark energy takes over and it goes down again.

twofish-quant is saying that he has the hope that a plausible inflation scenario that is based on some atomic scale will rather magically hit this target.

If there is *any* curvature, no matter how small, then the universe at some point in it's life will take all values between that small curvature and infinity.

But look at the cost we've had to pay-- first of all, we seem to have gotten really lucky to have hit the target, but that's what we are using to justify faith in our model.

No we don't. It's not a matter of hitting a target. As long as the inflation ends with *any* positive curvature, then things will work. It doesn't matter whether the minimum curvature is 10^-32, 10^-50, or 10^-100. Once the universe starts expanding from *any* small curvature, it will take *all* values between that small number and some limit at which when dark energy takes over.

So it doesn't *matter* what the minimum curvature is. It could be *any* number below observation. If it is 10^-100, it will eventually blow up to be 0.001. If it is 10^-32, it will eventually blow up to be 0.001. The only "magic" is that we see it at 0.001 rather than 0.002 or 0.01 which is what we will see if we wait a few billion years.

But I think it would. Look at it this way. Take the model you have in mind, and partition its possible parameters into two sets-- the set that leads to unobservable curvature, and the set that leads to small but observable curvature. Of course throw out the set that we've already falsified because it leads to huge curvature. Really do this, it should be easy enough with whatever model you have in mind. Now ask a simple question-- what is the relative measure of those two sets? Is it not true that the unobservable curvature parameter set has vastly larger measure than the observable curvature set? So how is it not fine tuned if we detect curvature tomorrow?

You are making quasi-anthropic arguments which I dislike. And no. It doesn't work that way. Suppose you end inflation with a undetectable positive curvature. This positive curvature will take all values from 10^-whatever and some large number at which you have dark energy take over.

Instead of taking multiple universes, let's just take one.

Now let's take a random point in the life of a universe with a positive curvature.

You have inflation and it reduces the curvature to some random small number. Now let's evolve the universe. It turns out that for most of the life of the universe, you will have a detectable curvature. If inflation ends with *any* positive curvature no matter how small, then at some point in the life of the universe, that curvature will take every positive value. So someone at some point will wonder why they observe 0.001, someone else will wonder why they observe 0.3, someone else will wonder they observe 1.0, and there is nothing to explain. You observe X, because you happen to be at the stage of the universe where you see X.

My point is that once you are on board with point #3, you are relieved of any philosophical issues with an infinite universe, because you are not claiming the universe is infinite-- you are just fitting what we see to a flat model, like fitting a tangent plane to a manifold where you cannot measure any deviation between the two.

Philosophy would be easy if we didn't have to worry about observations.

 

[twofish-quant]

Neither. It comes from me, but it stems from a logical argument. I summarized that argument again just above. So if you want to critique it, you do better finding an actual flaw in the logic.

If it comes from someone else, then it's likely that it's already been critiqued and I can pop up google and replay the conversation.

For example, you insist on claiming that I have said that cosmologists assume the universe is flat.

You are saying that the best model used by cosmologists assumes a flat universe, and I'm saying that's not the case.

That is so completely different from anything I've said, or even thought, that I have no idea where you are even getting that from, but you can't be reading very carefully.

Communication is difficult, and if you aren't making that assertion, then what assertion you are making. You have said that LCDM assumes a flat universe, and that's false.

 

[Ken G]

If the curvature is positive then at some point in the life of the universe it will take all values from 1e-16 to infinity. We happen to catch it at 0.001, but wait a few billion years and it will be 0.002. Then 0.3, then 0.5, then 2, then 1000, then at some point dark energy takes over and it goes down again.

But that's just it, the dark energy has already taken over. So we are pretty much at the curvature "peak" right now. That's the problem with a peak curvature that just happens to be what we can barely measure, why on Earth would life come along at just the time when it can barely measure the curvature? That's the "fine tuning problem" that you would be staring at if curvature is detected, and that's what would steal most of the wind from inflation's sails.

If there is *any* curvature, no matter how small, then the universe at some point in it's life will take all values between that small curvature and infinity.

No, not with dark energy.

Once the universe starts expanding from *any* small curvature, it will take *all* values between that small number and some limit at which when dark energy takes over.

Exactly, and if curvature is detected, then we will have the fine tuning problem that dark energy is taking over at exactly the point when the curvature is barely detectable by intelligent life. That's just the fine tuning that Weinberg argued is evidence for a multiverse, in relation to the amount of dark energy-- you would be in the exact same boat, but now in regard to curvature instead. You would need an anthropic argument to escape the appearance of fine tuning, and it would have to magically be consistent with the same anthropic argument that is supposed to be what let's dark energy be 10¹⁰⁰ time weaker than it "ought to" be. If we reject this is an escape hatch for the inflation theory, then we have no explanation for why the universe has a sense of humor that it will just let us glimpse the curvature before dark energy washes it away.

Instead of taking multiple universes, let's just take one.

I agree completely, I don't think resorting to multiple universes is a fair way to make a theory seem palatable or plausible. That's exactly why I claim any inflation proponent should be hoping we never detect curvature, and indeed, should probably even be confident we never will. There's just no reason for the parameters of a working inflationary model to be so well perched at that arbitrary tipping point that would suddenly seem very special indeed.

You have inflation and it reduces the curvature to some random small number. Now let's evolve the universe. It turns out that for most of the life of the universe, you will have a detectable curvature.

I don't agree, I think that for the vast majority of ways to set up that universe, the curvature will remain way too small to detect, because the one-two punch of inflation and dark energy will insure that. That holds whether you imagine a cosmological constant or a quintessence-type continuous inflation. You have to really fine tune the combination of inflation and dark energy to both have a universe that inflates enough to be anything like what we see (and, dare I say it, to support life), but still leave a window for detectable curvature for a few billion years out of that vastly aging universe-- exactly when life comes along. That's the problem I've been talking about, this bizarre "glimpse of curvature" phenomenon, which has no "natural" explanation at all, and would sorely tax the whole spirit of using inflation to recover a "natural" feel.

 

[Ken G]

You are saying that the best model used by cosmologists assumes a flat universe, and I'm saying that's not the case.

No, a thousand times no. Not only did I never say that, I bent over backward many times over to stress that is exactly not what I am saying.

Communication is difficult, and if you aren't making that assertion, then what assertion you are making.

Yes, communication is the hardest thing, so let me repeat again what I have been saying. Cosmologists make models, and they make the models only as complicated as necessary to fit the data. The current models that do that are flat, and use the cosmological principle, so they are infinite models, like a derivative is an infinite model of a function even if the function becomes uncertain outside of some compact region. As I've repeated many times now, no such flat and infinite model can say anything at all about the infiniteness of the universe, indeed I stressed several times (there are at least three threads we are debating, so it's hard to keep track of where!) that the question "is the universe infinite" is fundamentally a question that science could never answer in the positive. But science can certainly, and does, make infinite models, if it has no evidence for making them finite. No such model is a claim on what we cannot observe, nor would there ever be any scientific validity in making any claims on such a region.

You have said that LCDM assumes a flat universe, and that's false.

Never said anything like that, nope. I said LCDM is a flat model, which is a totally different claim. It just says the simplest model that fits what we see is a flat infinite universe model, that is not at all "assuming the universe is flat." We don't make assumptions about the universe, we embed assumptions into models, in order to make good models, not in order to use models to make claims on the universe. The logic goes the other way-- observations of the universe inform our models, our models do not inform the universe that is outside the observations we used to make the model.

 

[marcus]

No, a thousand times no. Not only did I never say that, I bent over backward many times over to stress that is exactly not what I am saying. Yes, communication is the hardest thing, so let me repeat again what I have been saying. Cosmologists make models, and they make the models only as complicated as necessary to fit the data. The current models that do that are flat, and use the cosmological principle, so they are infinite models, like a derivative is an infinite model of a function even if the function becomes uncertain outside of some compact region. As I've repeated many times now, no such flat and infinite model can say anything at all about the infiniteness of the universe, indeed I stressed several times (there are at least three threads we are debating, so it's hard to keep track of where!) that the question "is the universe infinite" is fundamentally a question that science could never answer in the positive. But science can certainly, and does, make infinite models, if it has no evidence for making them finite. No such model is a claim on what we cannot observe, nor would there ever be any scientific validity in making any claims on such a region.
Never said anything like that, nope. I said LCDM is a flat model, which is a totally different claim. It just says the simplest model that fits what we see is a flat infinite universe model, that is not at all "assuming the universe is flat." We don't make assumptions about the universe, we embed assumptions into models, in order to make good models, not in order to use models to make claims on the universe. The logic goes the other way-- observations of the universe inform our models, our models do not inform the universe that is outside the observations we used to make the model.

Things are getting clearer. You are not saying that mainstream cosmologists believe the universe is spatially flat, or infinite.

You are claiming that the predominant model in use, the LCDM, comes in only one version and that version is spatially infinite with zero curvature.

If you have never seen a cosmologist use a version of LCDM which has overall slightly positive curvature, then this claim is certainly understandable! It would square with your experience for you to insist that there is only the one version in use, with infinite space and matter.

However my experience is different from yours. I have seen top level cosmologists use different versions of LCDM, and for example, calculate a lower bound for the radius of curvature for the spatially finite positive curved version of LCDM.

You might recall this from the WMAP5 report by Komatsu et al (2010)

In other words, in my experience cosmologists do not jump to premature conclusions, do not gloss over different cases, and instead take the Omega confidence interval very seriously. Since the confidence interval has a substantial range above 1 that necessarily requires a spatial finite (but "nearly" flat) version of LCDM.

I think (if I understand you) we are getting closer to agreement because you are saying that cosmologists do not assume the U is spatially flat and infinite. I agree with you there.
If I understand correctly, you are merely saying that the LCDM model they use (but of course don't assume to be right) has only one version, which is exactly flat and infinite, spatially. And I disagree that there is only one sole model, not a confidence interval of different cases to which the model can be applied.

 

[ViewsofMars]

See the WMAP website at http://map.gsfc.nasa.gov/universe/uni_shape.html , where we find quotes like: [. . .]

and

The WMAP website oversimplifies things. I'll e-mail the maintainers of the site to get it changed.
1) from the graduate courses that I took in cosmology when I got my Ph.D. in astrophysics

2) from talking with cosmologists and supernova people, include one of the lead co-authors of the WMAP paper, one person that was a co-author on the supernova Ia investigation papers, and one person that has a Nobel prize in physics.

Brief comment, twofish-quant, the WMAP website is also used by the Smoot Group - Astrophysics and Cosmology. Might be worth reviewing their website:http://aether.lbl.gov/education.html

Dr. Smoot is also the Director of the Berkeley Center for Cosmological Physics and a winner of a Nobel Prize. I hope that helps with the ongoing discussion.

I have to go digging in my archives for further information I've stored to present to the discussions you are having with Ken on this topic and a few other topics.

 

[Ken G]

Things are getting clearer. You are not saying that mainstream cosmologists believe the universe is spatially flat, or infinite.

Right, most likely they believe almost as many different things as their are cosmologists, and indeed they are welcome to hold any personal beliefs they wish, but believing it wouldn't make it science.

You are claiming that the predominant model in use, the LCDM, comes in only one version and that version is spatially infinite with zero curvature.

There are always many multiple models in use, for a host of reasons, largely around the "buckshot" principle of doing science. But there is also a clear consensus on what is currently regarded as the best model, the model that is often heard in a sentence with "precision cosmology", and it is a model with no reason to include any curvature, so it doesn't. There's always the interplay between consensus and contrariness in science, and nowhere did I ever say that there is only one cosmological model that ever gets looked at-- I said there is one widely regarded best model, and Nobel prizes have been awarded.

If you have never seen a cosmologist use a version of LCDM which has overall slightly positive curvature, then this claim is certainly understandable!

Actually I have seen curved models invoked many times, my point is that none of those models ever gave us the value, the bang for the buck, that the flat model does. Indeed those models can now be seen to be largely a source of unnecessary complication. Almost all cosmology textbooks, for example, start out with the three possible geometries, and go to great lengths describing their differences, only to throw it all away when they come to describing the currently favored model! It's so much wasted overhead. I've no doubt that electromagnetism textbooks after Maxwell went to great lengths describing all the different ways light might operate in different frames if the speed of light was relative to an aether frame, but at some point, they realized that all that overhead was missing a key simplification that drastically simplified the mathematics of doing calculations-- Einstein's postulate. I'm saying flat models in cosmology are another example of just such a drastically beneficial mathematical simplification, to the point that it is becoming more and more apparent that we should embrace that simplification instead of fighting it every step of the way.

However my experience is different from yours. I have seen top level cosmologists use different versions of LCDM, and for example, calculate a lower bound for the radius of curvature for the spatially finite positive curved version of LCDM.

Certainly. And many experiments in the era of Michelson-Morely were aimed at placing an upper bound on how much the speed of light could deviate from c in various frames. But at some point, the mathematical simplicity of a basic unifying postulate overwhelms all that careful overhead, and you just embrace what has been jumping up and down waving its arms at you all the while.

In other words, in my experience cosmologists do not jump to premature conclusions, do not gloss over different cases, and instead take the Omega confidence interval very seriously.

Just as in pre-Einstein days, they took the confidence interval on c very seriously too.

Since the confidence interval has a substantial range above 1 that necessarily requires a spatial finite (but "nearly" flat) version of LCDM.

And what is the confidence interval on c today? It's not infinitely narrow, right? So does that require we have a "very nearly relativistic" version of physics that we also have to bear in mind, and put in every textbook on relativity theory? Models are intended to be simplifications, there's no "conclusions" that are drawn when we adopt one, certainly not that we are announcing that we are convinced the model should suddenly be regarded as "correct," ignoring the fate of all "correct models" for time immemorial. All it means, when we adopt a particular idealization in some model, is that we are tired of doing extraneous and unnecessary work tracking what is much simpler to just remove from the model. We have simply reached the point of diminishing returns for tracking the complexity, relative to just adopting the simpler postulate. I'm saying cosmology is at that point, but it might take it a little while to make the transition.

Edit: let me rephrase that, I'm not trying to tell cosmologists how to do their business, I'm pointing out that we may very well be approaching a time when we need to give very serious consideration to treating the flatness of our models as a physical principle. Note this still does not represent a claim that the universe is actually flat, any more than relativity is a claim that the photon is exactly massless, it is merely a recognition of the value in adopting a particular mathematical simplification in our best models. That is also an accurate description of the theory of relativity, despite how it is often framed in less scientifically careful terms!

 

[ViewsofMars]

Picking up where I left off.

Nature 404, 955-959 (27 April 2000) article:

A flat Universe from high-resolution maps of the cosmic microwave background radiation
P. de Bernardis et al
[. . .]
The blackbody radiation left over from the Big Bang has been transformed by the expansion of the Universe into the nearly isotropic 2.73 K cosmic microwave background. Tiny inhomogeneities in the early Universe left their imprint on the microwave background in the form of small anisotropies in its temperature. These anisotropies contain information about basic cosmological parameters, particularly the total energy density and curvature of the Universe. Here we report the first images of resolved structure in the microwave background anisotropies over a significant part of the sky. Maps at four frequencies clearly distinguish the microwave background from foreground emission. We compute the angular power spectrum of the microwave background, and find a peak at Legendre multipole lpeak = (197 plusminus 6), with an amplitude DeltaT200 = (69 plusminus 8) microK. This is consistent with that expected for cold dark matter models in a flat (euclidean) Universe, as favoured by standard inflationary models.
[. . .]
http://www.nature.com/nature/journal/v404/n6781/full/404955a0.html

 

[marcus]

...
Actually I have seen curved models invoked many times, my point is that none of those models ever gave us the value, the bang for the buck, that the flat model does. Indeed those models can now be seen to be largely a source of unnecessary complication. Almost all cosmology textbooks, for example, start out with the three possible geometries, and go to great lengths describing their differences, only to throw it all away when they come to describing the currently favored model! It's so much wasted overhead...

Ah. This is where your personal attitude comes in. I remember in another thread you were urging that students not be exposed to the spatially curved versions of the model. You are campaigning for a kind of educational reform, in effect. Cosmology textbooks and curriculum should not WASTE STUDENT'S TIME by introducing the slightly curved case, or cases. It is "unnecessary complication"

The course outline, in effect, should focus exclusively on the flat case.

But not because flat is BELIEVED by any kind of mainstream majority or consensus.

Indeed to illustrate, in a central paper like the 2010 WMAP5 report by Komatsu et al they were keeping their options open and calculated up front with THREE versions of LCDM showing their results already on page 3 as I recall, Table 2, I think. A central paper with a dozen big name cosmologists reporting on a flagship project. Not fringe.

You are advocating a curriculum reform, to save "overhead", which would render students incapable of undertanding the options being kept open by core top professionals in the field.

It strikes me as a bit short sighted, a false "economy". It seems to have no logical basis, since we do not KNOW curvature is zero, and we may in future discover that it is on the positive or negative side of today's rather broad 95% confidence interval.

There is no logical basis for you to insist on this change in the course outline. It seems to have more to do with PERSONAL AESTHETIC.

I guess if we are going to talk at the level of personal aesthetics, prejudices etc. I will state my own, about what beginning cosmology students should be taught.

I would wish the course to present and explain the current confidence interval for Ω_k from the WMAP7 report (also Komatsu et al) and, assuming today's best estimate for the cosmological constant, describe the two basic kinds of universe contained in that confidence interval, both indefinitely expanding, one with slight positive curvature and the other with zero or slight negative. One model spatially finite (now and at the start of expansion) and the other infinite (now and at the start) or topologically rather intricate.

I'll go get that confidence interval for Ω_k Just google "komatsu wmap 7" and you get
http://arxiv.org/abs/1001.4538 and page 3 says:
−0.0133 < Ω_k < 0.0084

which means:
0.9916 < Ω < 1.0133

 

[Ken G]

Yes, thanks for that ViewsofMars. The take of that paper is that the observed curvature is consistent with a flat universe, which is of course all any observation could ever say. The authors of the paper take this observed fact and add an interpretation that this should be taken as evidence that the universe is Euclidean (i.e., actually perfectly flat), presumably because a Euclidean model is seen as a kind of conceptual watershed that should be given special attention if it is an allowed possibility. Probably that same conclusion could be framed in more uncontrovertibly scientific language by simply saying that these results call into question the usefulness of continuing to propagate non-flat models throughout the theoretical literature, and certainly more recent results further refine the confidence interval while reaching that same conclusion. None of this asserts that we should block out from our minds the possibility of curvature, it just means, as I said, the observations may be trying to tell us that we have reached a point of vanishing returns for continuing to carry around the mathematical excess baggage of nonzero curvature in the models, unless the question of curvature is explicitly the target of some investigation.

 

[Ken G]

Ah. This is where your personal attitude comes in. I remember in another thread you were urging that students not be exposed to the spatially curved versions of the model. You are campaigning for a kind of educational reform, in effect. Cosmology textbooks and curriculum should not WASTE STUDENT'S TIME by introducing the slightly curved case, or cases. It is "unnecessary complication"

Precisely. For some reason, you seem to disagree, though I can give you countless examples where we do precisely that in virtually every textbook, with no less justification. Let me choose a random example for illustrative purposes. A textbook wants to calculate the effect of the Moon on Earth tides. It's first step will be to choose a model for the Moon's gravity. Will the textbook:
a) cite a look-up table of precise measurements of the mass distribution of the Moon, or
b) treat the Moon as a sphere.
Seriously, I'm asking you-- which do you think that book is going to do? Surely you must be appalled if they choose (b), if they do it because they fear it would waste the student's time by using approach (a), right? You must say we cannot use a model that treats the Moon as a sphere, that would be blocking out of our minds any other possibility, while leaving our students incapable of understanding anything but spheres.

But it's just exactly the same issue with a model of cosmology. So why is everyone so happy to see a model of the Moon as just that (a model of the Moon), but suddenly when it's a cosmology model, we invoke some kind of religious devotion to the model? Such that it would be some kind of awful oversight to simply recognize that it's silly to do a bunch of extraneous math when a much simpler calculation will give us results well within the observational error bars? That's what I would like to know.

The course outline, in effect, should focus exclusively on the flat case.

Of course it should. The course outline is going to focus exclusively on the case where the speed of light is a constant in a vacuum, will it not? But that would be terrible, the idea that it would just waste the student's time to consider all the other possible ways that c might vary that are perfectly consistent with the observational constraints on the actual precision to which we can claim that c is constant in a vacuum.

Yes, I'm being a bit sarcastic, in response to yours, to demonstrate why your criticism is baseless. You simply put cosmology on a kind of pedastol for different treatment from every other subject you have ever seen in physics, when of course all we ever have anywhere in physics is observational constraints that are consistent with the idealizations in our models. Yet we make no apologies for not wasting student's time by including all those other possibilities in the course. But doing the exact same thing in cosmology, that would just be awful, you are saying.

Indeed to illustrate, in a central paper like the 2010 WMAP5 report by Komatsu et al they were keeping their options open and calculated up front with THREE versions of LCDM showing their results already on page 3 as I recall, Table 2, I think. A central paper with a dozen big name cosmologists reporting on a flagship project.

And what of it? It's hardly surprising that the flatness simplification must be examined closely before it is adopted, but it is inevitable that it will as the precision narrows more and more, as soon as we get tired of carrying around what is starting to seem like more and more useless baggage. We're already close enough that even if curvature is detected, the most commonly used model won't even use it, just as the most common treatments of gravity in astronomy still treat objects as spheres even when we have detected deviations. This is because models are designed to be simplifications, and they only need to be tailored to a reasonable accuracy target, never claims on the reality.

You are advocating a curriculum reform, to save "overhead", which would render students incapable of undertanding the options being kept open by core top professionals in the field.

And you see that as such a terrible thing? Why? Don't you realize we already do that all over the map? When is the last time you saw a cosmology book include the overhead of a rotating cosmology? Does that mean you think the observations have constrained the rotation of the universe to be zero? Of course not, it's exactly the same issue-- our observations are consistent with no rotation, so nobody bothers to waste the student's time by putting in all kinds of rotating cosmologies because they just have no reason to include all that unnecessary mathematical overhead. So you must be arguing this is a terrible choice that is rendering our students "incapable of understanding the options being kept open" by core top professionals who are working hard to observationally constrain the upper bounds on the rotation of our universe! So it's fine for rotation, but a terrible oversight for curvature? I have a good idea why you think that-- because curvature is ingrained in our cosmological upbringing, and rotation is not, by purely happenstance historical reasons. The ultimate irony would be if we never detect any curvature, but do someday detect a tiny rotation, and all the old cosmology textbooks get thrown in the garbage for spending all that time on curvature and completely ignoring rotation.

It seems to have no logical basis, since we do not KNOW curvature is zero, and we may in future discover that it is on the positive or negative side of today's rather broad 95% confidence interval.

It's not that broad, really it isn't.

There is no logical basis for you to insist on this change in the course outline. It seems to have more to do with PERSONAL AESTHETIC.

There is nothing "personal" in the aesthetic of removing extraneous mathematical baggage from our models, this is quite central to every chapter of every physics book everywhere in the world.

I'll go get that confidence interval for Ω_k Just google "komatsu wmap 7" and you get
http://arxiv.org/abs/1001.4538 and page 3 says:
−0.0133 < Ω_k < 0.0084

which means:
0.9916 < Ω < 1.0133

Like I say, really not that broad at all.

 

[marcus]

Physics is to a large extent the art of making the right simplifying assumptions (but not holding to one simplification exclusively) and choosing the right approximation (but not always the same) in order to calculate.

You gave the example of the tangent plane to a manifold. The tangent plane is a good approximation for some purposes. It is flat and infinite. It works fine for some things. But not for everything.

I happen to disagree with you about pedagogy in a beginning cosmo course, that's about all. I think you passionately overstate the case that students should be introduced at the start to no case except the infinite flat one. (Because nowadays for many calculations we use the flat approx.) I think they should meet the uncertainty up front and be prepared to read and understand mainstream calculations that use, say, the spatial finite endlessly expanding LCDM.

BTW that interval which you say is not broad has an upper limit of 1.0133 which means a radius of curvature of about 120 billion LY and a circumference of about 750 billion LY.

That would mean nothing in the whole wide universe is more than 380 billion LY from us. That strikes me as fairly close quarters given that the particle horizen, the most distant stuff we can see, is over 45 billion LY.

The upper limit of the 95% interval, IOW, says that the most distant stuff is less than a factor of 10 farther away than the stuff we can see.

I don't know if I'd WANT the upper limit to be larger than 1.0133
it would make things even tighter quarters, more closed in.

So you say the interval is not broad. To me it seems quite generous. And so we wait, and see if and how much the Planck observatory mission narrows it down.

Personally I think it is the wrong time to campaign for reforming the college course outline.

 

[Ken G]

Physics is to a large extent the art of making the right simplifying assumptions (but not holding to one simplification exclusively) and choosing the right approximation (but not always the same) in order to calculate.

Yes, I agree.

You gave the example of the tangent plane to a manifold. The tangent plane is a good approximation for some purposes. It is flat and infinite. It works fine for some things. But not for everything.

It has a simple mathematical form, easy to use in practice, and approximates well a manifold over some domain. That's a pretty good description of a flat cosmological model applied to observations of our universe, in the current state of affairs. So it doesn't have to work for everything, it just has to work for that thing.

I happen to disagree with you about pedagogy in a beginning cosmo course, that's about all. I think you passionately overstate the case that students should be introduced at the start to no case except the infinite flat one.

Do you think they should be introduced to rotating cosmology models, with all the equations and so forth? Why or why not?

BTW that interval which you say is not broad has an upper limit of 1.0133 which means a radius of curvature of about 120 billion LY and a circumference of about 750 billion LY.

That would mean nothing in the whole wide universe is more than 380 billion LY from us. That strikes me as fairly close quarters given that the particle horizen, the most distant stuff we can see, is over 45 billion LY.

It's a factor of 10 away from anything we can see, yes. That's why I say that we would never be able to use such a model to actually conclude that the universe was closed, it would never be observationally constrained as such. And then poof, there goes the whole main distinguishing feature of that model-- all that extra complexity and no payoff in terms of being able to say anything concrete about the universe's global geometry.

I don't know if I'd WANT the upper limit to be larger than 1.0133
it would make things even tighter quarters, more closed in.

Well, chances are, that upper limit will just keep dropping with time. It might not, but I'd bet good money it will. Anyway, if I'm right, eventually you will come around-- it's just a question of how long you will hold out! That's pretty much how I feel about the whole question-- everyone has to have a kind of personal limit where they finally decide the overhead just isn't worth it any more. I'm there now-- how much smaller does that upper limit need to be before you would go there too?

Personally I think it is the wrong time to campaign for reforming the college course outline.

Then we wait. I'm a patient man.

 

[twofish-quant]

That's the problem with a peak curvature that just happens to be what we can barely measure, why on Earth would life come along at just the time when it can barely measure the curvature?

The dark energy "cosmic coincidence problem" is a totally different problem. Inflation was never designed to fix that problem, and I think that's a different problem that irrelevant to inflation. Also, if you set flatness to zero, the "cosmic coincidence problem" also doesn't go away.

That's the "fine tuning problem" that you would be staring at if curvature is detected, and that's what would steal most of the wind from inflation's sails.

It wouldn't. The "flatness problem" is in fact a rather weak reason to support inflation. If we found that inflation didn't address the flatness problem then we'd still have the horizon problem and the CMB perturbations, which are far stronger pieces of evidence in support of inflation.

Exactly, and if curvature is detected, then we will have the fine tuning problem that dark energy is taking over at exactly the point when the curvature is barely detectable by intelligent life.

And if curvature isn't detected we have this fine tuning problem that dark energy is taking over at exactly the point at which we are making observations. Setting flatness to zero doesn't help you.

That's just the fine tuning that Weinberg argued is evidence for a multiverse, in relation to the amount of dark energy-- you would be in the exact same boat, but now in regard to curvature instead.

If you have two holes in a boat, that's not much worse than one.

In any case, it wouldn't affect the validity of inflation. The strongest evidence for inflation is that it predicts very well CMB fluctuations.

I agree completely, I don't think resorting to multiple universes is a fair way to make a theory seem palatable or plausible.

Except we have an example in which that happens. If you ask Steven Weinberg why he takes multiple universes and the anthropic principle seriously, the answer he will give is exoplanets.

Exoplanets provide an example of the anthropic principle in action. It turns out that solar systems with circular orbits are rare and hot Jupiters are common, but we didn't know about hot Jupiters because of the anthropic principle. If there were any hot Jupiters in our solar system, we wouldn't see them, because we wouldn't be here.

Also, exoplanets provide an example of how you can deduce something you can't observe. People first deduced the existence of exoplanets in the 1600's. They were only first observed in the 1990's, and they were detected using technology that was unimaginable in the 1600's. Weinberg would argue that trying to deduce the existence of multiple universes today is no difference than deducing the existence of exoplanets in the 1600's.

That's exactly why I claim any inflation proponent should be hoping we never detect curvature, and indeed, should probably even be confident we never will.

Disagree. The physics of inflation are sufficiently complex that it's not that hard to create an inflationary model that produces large amounts of curvature. During the 1990's, it appeared that the universe was open, and there were a flurry of plausible scenarios in which you could naturally create universes with curvature of -0.7 look up "open inflation". People stopped doing that in 1998, but there was nothing physically wrong with those models, and if we find curvature then we can dust off those models.

The other thing is that inflationary models predict curvature. The universe is not flat, it's wrinkly. All you have to do is to set up inflation so that one of the "wrinkles" is larger than the Hubble distance, and bammm, you have a small amount of local curvature.

The problem with inflation is that the detailed physics is sufficiently unknown and complex that we can't rule out curved inflationary models. Look at what happened in the early 1990's, the observers thought that omega=0.3, and the theorists came up with models that produced omega=0.3. Contrast that with the reaction of theorists when people came up with FTL neutrinos. The reaction of the theorists was "do your measurements again, you did something wrong." Whereas, no cosmologist that I know of reacted to the 1990's CDM measurements with "you did your measurements wrong" and they didn't because the theory is just not firm enough to make that statement.

People stopped working on open inflation models once the data looked like omega is close to one. But if it turns out that we have our dark energy models wrong, then people will work on them again.

There's just no reason for the parameters of a working inflationary model to be so well perched at that arbitrary tipping point that would suddenly seem very special indeed.

Well there it is. Also this has nothing to do with inflation. The cosmic coincidence problem is there if you assume flatness. The fact that the cosmic coincidence problem exists (and I don't know why) is why I reject "this can't happen because it would mess up our simple theories" arguments.

I don't agree, I think that for the vast majority of ways to set up that universe, the curvature will remain way too small to detect, because the one-two punch of inflation and dark energy will insure that.

You are invoking multiverse arguments. Once you talk about "alternative ways of setting up a universe" you are invoking a multiverse argument. It's a philosophical problem. If you assert there is one universe, then you can't really talk about "alternative ways of setting up a universe."

In the vast majority of universes, we wouldn't see a dark energy omega that isn't either 0 or 1, but we see it and it's 0.7.

You have to really fine tune the combination of inflation and dark energy to both have a universe that inflates enough to be anything like what we see (and, dare I say it, to support life), but still leave a window for detectable curvature for a few billion years out of that vastly aging universe-- exactly when life comes along.

Not clear. If inflation and dark energy are connected then you can try to come up with a natural way of connecting the two.

And in any case, this problem doesn't go away if you get rid of curvature.

That's the problem I've been talking about, this bizarre "glimpse of curvature" phenomenon, which has no "natural" explanation at all, and would sorely tax the whole spirit of using inflation to recover a "natural" feel.

1) If there isn't an obvious natural explanation, then we look for one.

2) Even if we can't find one, then it doesn't kill inflation. There are enough pieces of evidence for inflation independent of flatness that if it turns out that it requires weird coincidences to have inflation work, then that is just the way the universe works.

 

[twofish-quant]

Such that it would be some kind of awful oversight to simply recognize that it's silly to do a bunch of extraneous math when a much simpler calculation will give us results well within the observational error bars? That's what I would like to know.

The problem with this is that it dumps out the reason for thinking that LCDM is the current model of the universe. Assuming the universe is flat is the "zeroth order" calculation. It's a Newtonian model of the universe, and people *do* use it for pedogogy. Adding curved space is a first order calculation, that gets you Friedmann-Walker metrics.

The thing about LCDM that makes is a spectacularly good model is that it makes very detailed and correct predictions about the distribution of matter in the universe (i.e. the first/second/third acoustic peaks) and those calculations are not simple ones. Without doing those calculations, there is no reason to trust LCDM.

One problem with teaching cosmology is that people don't realize that we are long past "spherical cows." Our current models are good enough so that we can make complex and detailed predictions about the early universe.

Yet we make no apologies for not wasting student's time by including all those other possibilities in the course. But doing the exact same thing in cosmology, that
would just be awful, you are saying.

Most introductory cosmology courses introduce the mathematics of cosmology through a Newtonian cosmology. You assume that the universe is flat, and then with simple math you can derive things like the Hubble relations.

The Newtonian cosmology is a perfectly good toy model that is great for teaching cosmology, but it is *NOT* LCDM. Comparing LCDM with Newtonian cosmology is like comparing a Boeing 747 with a paper airplane.

If you want to introduce cosmology through simple Newtonian models, that's great, but it's important to emphasize that this is *NOT* LCDM. LCDM contains all of the messy details that aren't in Newtonian cosmology.

It's hardly surprising that the flatness simplification must be examined closely before it is adopted, but it is inevitable that it will as the precision narrows more and more, as soon as we get tired of carrying around what is starting to seem like more and more useless baggage.

As precision increases, our models get more complicated.

We're already close enough that even if curvature is detected, the most commonly used model won't even use it, just as the most common treatments of gravity in astronomy still treat objects as spheres even when we have detected deviations.

Part of the reason I'm jumping up and down is that I don't think you understand what LCDM is.

LCDM contains curved space. Even if it turns out that we set the average curvature to zero, you still have a parameter in LCDM which measures the variation of that curvature. LCDM contains some detailed physics describing particle interactions, which let's you calculate acoustic peaks.

If you drop curvature, you still have a serviceable cosmological model, but it is *NOT* LCDM. It's something else. If you drop the interaction model, you end up with FLRW. If you drop curvature, you end up with Newtonian cosmology. People *do* use Newtonian cosmology for some rough calculations, but it's *NOT* LCDM.

When is the last time you saw a cosmology book include the overhead of a rotating cosmology?

When I was in graduate school? It's going to be in any course in GR.

Of course not, it's exactly the same issue-- our observations are consistent with no rotation, so nobody bothers to waste the student's time by putting in all kinds of rotating cosmologies because they just have no reason to include all that unnecessary mathematical overhead.

The danger is that you end up with students that think that they understand more than they do. Also, graduate courses are very different from undergraduate ones.

I have a good idea why you think that-- because curvature is ingrained in our cosmological upbringing, and rotation is not, by purely happenstance historical reasons. The ultimate irony would be if we never detect any curvature, but do someday detect a tiny rotation, and all the old cosmology textbooks get thrown in the garbage for spending all that time on curvature and completely ignoring rotation.

Cosmology changes very rapidly. Any textbook that is more than two years old is hopelessly out of date.

Also we do detect curvature. CMB background flucutations are the result of spatial curvature. Whether there is average *global* curvature, is another question.

There is nothing "personal" in the aesthetic of removing extraneous mathematical baggage from our models, this is quite central to every chapter of every physics book everywhere in the world.

If you want to do cosmology past the "toy model" Newtonian stage, you have to do GR. If you do GR, you have to include curvature.

My concern is that you need to present the material in a way that doesn't mislead students. I'm concerned because you *think* you understand what LCDM is and isn't, but you don't, and I'm trying to present the material in a way that doesn't lead to the misconceptions that you have. (Again, I apologize for being harsh, but it has to be said).

The issue is that the gravity model and curvature is probably the *least* mathematically messy parts of LCDM. The more messy parts are the parts dealing with particle interactions.

 

[twofish-quant]

Do you think they should be introduced to rotating cosmology models, with all the equations and so forth? Why or why not?

Who is they?

In any sort of graduate cosmology course that's theory based, then absolutely. If you want to do theoretical work in cosmology, you need to understand how to handle rotating frames.
The whole point of graduate physics courses is to train students to do complex math, so the more messy math, the better. It builds character.

For undergraduate courses, it's sufficient to mention why we think the universe isn't rotating. The Newtonian cosmology is something that's good to introduce in undergraduate courses, but when talking about the Newtonian cosmology, it's important to explain how that is similar and different from LCDM.

Whether to introduce the mathematics of GR depends on the level of the class.

For graduate students, I'm teaching them to fly a Boeing 747. For undergraduates, I can show them a paper airplane and take them on a tour of the 747.

Also for graduate students, it's really important to go through "failed" models and why they failed. For undergraduates, it's less important.

That's pretty much how I feel about the whole question-- everyone has to have a kind of personal limit where they finally decide the overhead just isn't worth it any more. I'm there now-- how much smaller does that upper limit need to be before you would go there too?

With LCDM you *need* curvature in order to calculate the CMB fluctuations and the location of the acoustic peaks.

 

[ViewsofMars]

The universe is not flat, it's wrinkly.

I thought I'd chime in with this comment of yours. Berkeley Lab had an interesting article Clocking an Accelerating Universe: First Results from BOSS dated March 30, 2012. Here's a quote from it:

“All the data collected by BOSS flows through a data-processing pipeline at Berkeley Lab,” says Stephen Bailey of the Physics Division, who describes himself as the “baby sitter of the pipeline.” Working with Schlegel at Berkeley Lab and Adam Bolton at the University of Utah, Bailey “turns the data into something we can use – catalogues of hundreds of thousands of galaxies, eventually well over a million, each identified by their two-dimensional positions in the sky and their redshifts.” The data are processed and stored on the Riemann computer cluster, operated by Berkeley Lab’s High-Performance Computing Services group.

The current crop of BOSS papers is based on less than a quarter of the data BOSS will continue to collect until the survey ends in 2014. So far, all lines of inquiry point toward the so-called “concordance model” of the universe: a “flat” (Euclidean) universe that bloomed from the big bang 13.7 billion years ago, a quarter of which is cold dark matter – plus a few percent visible, ordinary, baryonic matter (the stuff we’re made of). All the rest is thought to be dark energy in the form of Einstein’s cosmological constant: a small but irreducible energy of puzzling origin that’s continually stretching space itself.

But it’s way too soon to think that’s the end of the story, says Schlegel. “Based on the limited observations of dark energy we’ve made so far, the cosmological constant may be the simplest explanation, but in truth, the cosmological constant has not been tested at all. It’s consistent with the data, but we really have only a little bit of data. We’re just beginning to explore the times when dark energy turned on. If there are surprises lurking there, we expect to find them.”
http://newscenter.lbl.gov/news-releases/2012/03/30/boss-first-results/

 

[twofish-quant]

Yup, and one reason *not* to adopt flatness as a principle just yet is that the calculations of omega make assumptions about dark energy. If it turns out that dark energy is "something odd" then the numbers are going to change.

 

[Ken G]

The dark energy "cosmic coincidence problem" is a totally different problem. Inflation was never designed to fix that problem, and I think that's a different problem that irrelevant to inflation. Also, if you set flatness to zero, the "cosmic coincidence problem" also doesn't go away.

I'm not talking about dark energy, I'm talking about an analogous issue that is all about inflation and curvature. The curvature problem most definitely does go away if you have unobservable curvature, exactly the way the cosmic coincidence problem you are talking about would not have appeared had there been no dark energy.

It wouldn't. The "flatness problem" is in fact a rather weak reason to support inflation. If we found that inflation didn't address the flatness problem then we'd still have the horizon problem and the CMB perturbations, which are far stronger pieces of evidence in support of inflation.

Yes, that's a good point, it means that inflation has a lot of reasons to be here and probably isn't going away any time soon. Still, it would be a cool person in its armor to lose its "one stop shopping" flavor, and end up still having to address a fine tuning problem after all that.

If you have two holes in a boat, that's not much worse than one.

It is if you have two different boats!

In any case, it wouldn't affect the validity of inflation. The strongest evidence for inflation is that it predicts very well CMB fluctuations.

I don't dispute that, indeed that's exactly why I claim we should expect the flatness precision to only increase with more observations. The inflation phenomenon has good support, and should not lead to fine tuning problems like the "glimpse of curvature" conundrum, so that is the argument for expecting a flat model to continue to be excellent. A separate argument is that it is already known to be good enough for all but the most stringent accuracy needs.

Except we have an example in which that happens. If you ask Steven Weinberg why he takes multiple universes and the anthropic principle seriously, the answer he will give is exoplanets.

Exoplanets provide an example of the anthropic principle in action. It turns out that solar systems with circular orbits are rare and hot Jupiters are common, but we didn't know about hot Jupiters because of the anthropic principle. If there were any hot Jupiters in our solar system, we wouldn't see them, because we wouldn't be here.

There is a great deal of confusion about what the anthropic principle is. There is a weak version of it which is actually pretty obvious, and that is all that is being invoked by hot Jupiters. It's perfectly normal science to be able to observe some distribution, like planets, and have some special selection criterion, like life, which cuts the distribution in a highly non-generic way. That's quite a yawn, actually. But what makes it science is that we can indeed observe those hot Jupiters! Then there's a strong version, where we feel the need to invent a distribution of other universes that is completely untestable because the other universes cannot be observed, simply for the purposes of being able to feel better about fine tuning issues that nobody knows are even a problem in the first place.

Weinberg would argue that trying to deduce the existence of multiple universes today is no difference than deducing the existence of exoplanets in the 1600's.

And what that argument misses badly is that what makes exoplanets interesting is just one thing: they've actually been detected! Few people gave a hoot about the "deductions" of the 1960s, or the speculations of Bruno in the 1500s either for that matter. It's not even a remotely good analogy-- we saw stars out there, they look a lot like the Sun, it is perfectly natural to speculate that they might have planets around them. But if there was never any way to detect those planets, then the whole issue would never have been science at all.

Disagree. The physics of inflation are sufficiently complex that it's not that hard to create an inflationary model that produces large amounts of curvature. During the 1990's, it appeared that the universe was open, and there were a flurry of plausible scenarios in which you could naturally create universes with curvature of -0.7 look up "open inflation". People stopped doing that in 1998, but there was nothing physically wrong with those models, and if we find curvature then we can dust off those models.

Except for one thing-- they will of course be vastly finely tuned! So there goes the hope that inflation models will seem generic or inevitable. What's more, doesn't it bother you at all the "all things to all people" aspects of inflationary theory that you keep alluding to? If we need flatness, poof, inflation explains it. If we need curvature, poof, inflation explains it. If we need the model to seem generic, poof, inflation will make it all seem generic. If we need to explain some finely tuned result (like barely detectable curvature), poof, inflation does that too. Now, there's nothing wrong with a versatile theory, but I think we need a little truth in advertising-- I feel like putting my hand on my wallet when people start telling me all the conflicting advantages of these "all things to all people" inflationary theories.

other thing is that inflationary models predict curvature. The universe is not flat, it's wrinkly. All you have to do is to set up inflation so that one of the "wrinkles" is larger than the Hubble distance, and bammm, you have a small amount of local curvature.

Sure, but note that's also exactly why I've claimed that detecting curvature would not imply anything about the global geometry of the universe! Note this is the whole fallacy of placing so much importance on detecting some tiny curvature, it doesn't matter much at all unless you think it constrains what exists way beyond what you can actually observe.

Whereas, no cosmologist that I know of reacted to the 1990's CDM measurements with "you did your measurements wrong" and they didn't because the theory is just not firm enough to make that statement.

Well, the cosmologists I knew in the 1990's very much did suspect that the observation was wrong, or more correctly, misinterpreted. Certainly it was a perfectly standard statement at the time that non-flatness was a big headache for inflation, and many inflation proponents were quite clearly saying that they suspected something wrong with the non-flat interpretation. Ironically, those who stuck to their guns had a lot less backpedalling to do later on when dark energy came around.

The fact that the cosmic coincidence problem exists (and I don't know why) is why I reject "this can't happen because it would mess up our simple theories" arguments.

I agree there, I think fine tuning is not nearly as much of a weakness of a theory as multiverse thinking is. I never think a theory can dictate to reality, it is always the other way around. But fine tuning is something you do not expect to see if you haven't already seen it, that's the whole point about the flatness issue. If I'm playing poker, I never expect my opponent to have 4 aces. But if he is betting the roof, and I don't think he's bluffing, only then do I need to adjust my expectations, and I do so without requiring the existence of a multiverse of other poker games in which I am winning!

You are invoking multiverse arguments. Once you talk about "alternative ways of setting up a universe" you are invoking a multiverse argument.

No, there is a huge difference, summed up in the analogy I just made. When you are playing poker, of course you imagine a range of possible deals, but when you get evidence that the deal you are in has very unusual properties, you just accept that at face value, and discard the vast numbers of hypothetical deals that don't fit the facts-- a multiverse argument is something different, it is the argument that "if my deal is special, then there has to actually be a bunch of generic deals somewhere else, but I couldn't exist in them so I'm in this one." It is purely a way to "feel better" about being in a very unusual deal, and it is strictly for people who want the laws of physics to make the universe seem inevitable or generic. Anyone who is just fine with an amazingly special universe has no use for a multiverse, but they still have every use for imagining a "range of deals" when addressing what is not already known to be unusual. That's the key difference.

Not clear. If inflation and dark energy are connected then you can try to come up with a natural way of connecting the two.

That's true, it would seem necessary in fact, if both curvature and dark energy seemed to come out very special. If we did detect curvature, and were then face to face with the "glimpse of curvature" conundrum (but we should not expect this, as it is not something that is already known to hold), I think we could make a strong case that we would need to kill both those birds with the same stone-- we'd need to connect inflation and dark energy to seek one explanation instead of two.

And in any case, this problem doesn't go away if you get rid of curvature.

No, but we already know we have that problem (or the Nobel committee thinks we already know that), whereas we do not already know we have a curvature problem. That's a crucial distinction. If you already know one opponent has a very unusual hand, you still expect the other opponent not to.

2) Even if we can't find one, then it doesn't kill inflation. There are enough pieces of evidence for inflation independent of flatness that if it turns out that it requires weird coincidences to have inflation work, then that is just the way the universe works.

I agree, I'm not arguing that inflation will be killed. Indeed, I'm arguing that inflation is probably pretty good, and that is the basis why we should expect curvature to remain undetected, just as we should expect rotation to remain undetected. I see no evidence those issues should be treated so vastly differently as they are.

 

[Ken G]

Who is they?

Advanced undergraduates would be a fine test bed for what I'm suggesting. Possibly also graduate courses in cosmology, it depends on whether or not the instructor has some particular reason to want to address rotation. I'd wager that most graduate, and virtually all undergraduate, cosmology courses say little or nothing about rotating models, but the vast majority go into great detail about the various curvature possibilities. Just why is that? I argue it's purely an accident of history, and is high time to correct. Certainly any course is not going to be able to cover everything, so you pick and choose what areas give you the greatest "bang for your buck." If you stick to flat models, it does not at all mean, as was suggested, that the students will be hopelessly crippled for thinking about anything else, what it means is that you can spend your energy instead on digging deeper into some other area, perhaps inflationary models, that has much more promise of being something important, and not just a minor correction in the second decimal place.

With LCDM you *need* curvature in order to calculate the CMB fluctuations and the location of the acoustic peaks.

Of course, but you don't need global curvature in your model to do that. Indeed, mixing global curvature with the local curvature that affects CMB fluctuations is exactly the kind of extraneous detail that obscures the important concepts, rather than brings them out-- unless one favors the "black box" school of education, where you just teach students to put everything but the kitchen sink into the computer, and see what comes out, without any real understanding entering the student's brain.

 

[Ken G]

I thought I'd chime in with this comment of yours. Berkeley Lab had an interesting article Clocking an Accelerating Universe: First Results from BOSS dated March 30, 2012. Here's a quote from it:

Yes, "the concordance model", I forgot about that tidbit of jargon. That is what I have been referring to as the "consensus best model", but whoever coined "concordance" is a PR genius! Thanks for showing abstracts of concordance model papers that quite clearly demonstrate that the concordance model is a flat model with a cosmological principle, i.e., an infinite model of the universe. Indeed I would personally not go so far as to take that as evidence that the universe is actually infinite-- anyone who does may extrapolate too much (infinitely too much?) to suggest that!

 

[ViewsofMars]

Thank you Ken. twofish-quant, and Marcus

As far as dark energy this is what I recently read:

How can we solve the mystery of dark energy?

Observations of light emitted near the horizon of the universe reveal that everything seems to be flying apart with increasing velocity. Big Bang cosmology attributes this to “dark energy” that fills the entire universe— an amazing phenomenon! Is the Big Bang model too simple? Should Einstein’s equations be modified? Is there an unknown fundamental force? As the answers emerge, I expect that in the next decade physicists will solve the mystery of dark energy.

Wilfried Buchmueller, DESY, Germany
http://www.interactions.org/beacons/twenty-first-century-questions

I must say that I absolutely love discussions about the universe.

 

[twofish-quant]

I'm not talking about dark energy, I'm talking about an analogous issue that is all about inflation and curvature. The curvature problem most definitely does go away if you have unobservable curvature, exactly the way the cosmic coincidence problem you are talking about would not have appeared had there been no dark energy.

But we have dark energy. That's why I don't buy this "let's assume that we won't observe this because it will lead to a weird coincidence" logic. We've already seen it fail once.

Then there's a strong version, where we feel the need to invent a distribution of other universes that is completely untestable because the other universes cannot be observed, simply for the purposes of being able to feel better about fine tuning issues that nobody knows are even a problem in the first place.

It's not untestable. The idea behind anthropic principle is that you can use this to estimate parameters that you haven't observed yet. If someone comes up with an anthropic argument for the mass of the electron to thirty digits, and they start matching, that's good evidence that we've got something.

Few people gave a hoot about the "deductions" of the 1960s, or the speculations of Bruno in the 1500s either for that matter. It's not even a remotely good analogy-- we saw stars out there, they look a lot like the Sun, it is perfectly natural to speculate that they might have planets around them. But if there was never any way to detect those planets, then the whole issue would never have been science at all.

We don't know that there isn't a way of detecting exoplanets or multiverses until you think about it for a long time. The problem with your definition of science is that it means that in 1590, Bruno should have given up thinking about exoplanets, because they are unobservable by the technology of the 16th century.

What's more, doesn't it bother you at all the "all things to all people" aspects of inflationary theory that you keep alluding to? If we need flatness, poof, inflation explains it. If we need curvature, poof, inflation explains it. If we need the model to seem generic, poof, inflation will make it all seem generic. If we need to explain some finely tuned result (like barely detectable curvature), poof, inflation does that too. Now, there's nothing wrong with a versatile theory, but I think we need a little truth in advertising-- I feel like putting my hand on my wallet when people start telling me all the conflicting advantages of these "all things to all people" inflationary theories.

As I mentioned before flatness is a very weak argument in favor of inflation. The two strong ones are CMB power spectrum and the horizon problem.

In some cases the theory is stronger the the observations. For example, when FTL neutrinos were observed people were pretty sure that the observations were wrong since the theory is strong. Inflation has some strong parts and some weak parts. The parts regarding flatness are one of the weaker parts. That means that if it turns out tomorrow that someone claims that we messed up dark energy, and omega=0.1, I'm more likely to redo inflation to fit the observations than to assume someone messed up the observations.

Note this is the whole fallacy of placing so much importance on detecting some tiny curvature, it doesn't matter much at all unless you think it constrains what exists way beyond what you can actually observe.

It matters quite a bit because changing curvature also changes the calculated power spectrum which also changes things like galaxy formation. It also eliminates possible inflation scenarios.

It's also important just to get the science right.

Well, the cosmologists I knew in the 1990's very much did suspect that the observation was wrong, or more correctly, misinterpreted.

Name three.

No, there is a huge difference, summed up in the analogy I just made. When you are playing poker, of course you imagine a range of possible deals, but when you get evidence that the deal you are in has very unusual properties, you just accept that at face value, and discard the vast numbers of hypothetical deals that don't fit the facts

I don't. If I flip a coin 50 times and it comes up heads, I don't just accept that.

I agree, I'm not arguing that inflation will be killed. Indeed, I'm arguing that inflation is probably pretty good, and that is the basis why we should expect curvature to remain undetected, just as we should expect rotation to remain undetected. I see no evidence those issues should be treated so vastly differently as they are.

I'm arguing that inflation is good for some things. Bad at others. Curvature is one of the things that 's bad at, so if we detect curvature, then it's not hard to tweak the model to explain why.

As far as why they are treated differently. LCDM contains a model for curvature variation so that if you do LCDM, you have to include curvature.

 

[twofish-quant]

I'd wager that most graduate, and virtually all undergraduate, cosmology courses say little or nothing about rotating models, but the vast majority go into great detail about the various curvature possibilities. Just why is that?

Because the heart of LCDM involves calculating density perturbations, and without going into GR models (which include curvature) you can't do that.

I argue it's purely an accident of history, and is high time to correct.

History is important. You have to go through the history of cosmology models to point out what didn't work. Any decent course either graduate or undergraduate has got to mention steady state and tired light.

If you stick to flat models, it does not at all mean, as was suggested, that the students will be hopelessly crippled for thinking about anything else

I think they would. They'll be stuck in a Newtonian world, and you need to go into GR.

What it means is that you can spend your energy instead on digging deeper into some other area, perhaps inflationary models, that has much more promise of being something important, and not just a minor correction in the second decimal place.Of course, but you don't need global curvature in your model to do that.

You need GR, which means that you need curvature.

The sequence is

Newtonian -> FLRW -> LCDM

Indeed, mixing global curvature with the local curvature that affects CMB fluctuations is exactly the kind of extraneous detail that obscures the important concepts, rather than brings them out

No it doesn't. It's the same theory of gravity. It's also not extraneous detail. It's the heart of LCDM. The global curvature affects the growth rate of local perturbations

http://arxiv.org/pdf/1106.0627.pdf

Unless one favors the "black box" school of education, where you just teach students to put everything but the kitchen sink into the computer, and see what comes out, without any real understanding entering the student's brain.

For non-major undergraduates, computers are useful, because they can illustrate what happens when you vary the parameters.

 

[twofish-quant]

Right, most likely they believe almost as many different things as their are cosmologists, and indeed they are welcome to hold any personal beliefs they wish, but believing it wouldn't make it science.

One thing that's grating on the nerves is that you are talking to several theorists and trying to advance your view of science as somehow gospel. Why should your definition of science be better than mine or Steven Weinberg's?

There is a tendency to use the euphemism "speculative" to mean crank, and "mainstream" to mean "non-crank" but this will not work in this situation. The anthropic principle and multiverse concept is an important part of mainstream cosmology. I dislike it, but that's my personal opinion (and Max Tegmark has come up with some clever ways of addressing my issues).

But there is also a clear consensus on what is currently regarded as the best model, the model that is often heard in a sentence with "precision cosmology", and it is a model with no reason to include any curvature, so it doesn't. There's always the interplay between consensus and contrariness in science, and nowhere did I ever say that there is only one cosmological model that ever gets looked at-- I said there is one widely regarded best model, and Nobel prizes have been awarded.

And you have several people with experience with cosmology telling you that you are wrong.

The reason that you have to use a flat LCDM model is that if you don't fix curvature then you can't get information on the time evolution of dark energy. Flat LCDM models are essential if you want to study the evolution of dark matter, but using a flat LCDM doesn't mean that someone thinks that the universe is in fact, flat.

What happens if you allow any curvature uncertainty is that you can't pull out some numbers that you'd like to get.

Edit: let me rephrase that, I'm not trying to tell cosmologists how to do their business

It comes across that way.

Part of what I'm trying to tell you is that there is a reason why cosmologists make the assumptions that they do, and they are good reasons. I'm being somewhat harsh because you keep making statements about what cosmologists do that are false.

I'm pointing out that we may very well be approaching a time when we need to give very serious consideration to treating the flatness of our models as a physical principle.

And several things have to happen before that point is reached.

1) we have to understand what dark energy is. First of all, in order to get omega = 1, we are making several assumptions about the nature of dark energy. If those assumptions are wrong, then the omega=1 calculation falls apart. Also, if the nature of dark energy changes as a result of a phase transition, that will change the value of omega.

2) we have to understand inflation better than we do. The omega=1 result can be achieved if you let inflation run for a large number of e-foldings, but we have to understand what starts and stops inflation.

3) we have to push the limits on omega to below what they are.

Note this still does not represent a claim that the universe is actually flat, any more than relativity is a claim that the photon is exactly massless, it is merely a recognition of the value in adopting a particular mathematical simplification in our best models.

That doesn't work.

Relativity and electroweak *is* a claim that the photon is *exactly* massless. If there are any differences from zero mass, then electroweak theory and much of relativity is wrong.

The point of physics is to make claims on the nature of the universe. I see no reason to claim that the universe if flat, unless and until we actually think that it is flat. We can be wrong, but making incorrect assertions is what pushes science further.

 

[twofish-quant]

Quoting myself:

One thing that's grating on the nerves is that you are talking to several theorists and trying to advance your view of science as somehow gospel. Why should your definition of science be better than mine or Steven Weinberg's?

This may have sounded harsher than it was intended, but it's in fact a serious question. One thing that is a reality is that "anthropic arguments" and "multiverse" are taken quite seriously in high energy physics and cosmology. So in arguing that those arguments are invalid and "not science" is arguing against the "scientific mainstream" on this issue.

Now what?

The reason I dislike anthropic arguments is that they involve sociological assumptions. You assume that with situation X, intelligent life could not evolve. How do you know that in situation, intelligence is impossible?

However, Max Tegmark has come up with a clever way around that issue. Instead of "counting" universes in which there is intelligent life, he counts universes in which stars form or galaxies form, which let's him take human beings out of the anthropic equation. Saying that under condition X, intelligent life is not possible is a statement I'm not willing to make. Saying that under condition X, stable self-gravitating objects are impossible, is.

 

[twofish-quant]

One other thing, I'm a fan of going to the original papers. Here is the paper for BOSS

http://arxiv.org/pdf/1203.6594v1.pdf

Something to point out is that they put their data through six different parameterization, and then they explain why they do it.

The reason why is this

http://arxiv.org/pdf/0802.4407v2.pdf

Essentially, you can get very impressive looking numbers if you assume that the cosmological constant is constant. However, once you assume that dark matter changes then it becomes difficult to tell what is evolving dark energy and what is curvature.

Since the BOSS people are observationalists, they run their data through several models.

One reason I think this is worth looking at is that we have no clue what dark matter is, and if it turns out that it is evolving, that gets rid of the cosmic coincidence problem.

Other practical point is that the minimum curvature that we can measure is 10^-4 to 10^-5. Remember that in LCDM, the universe is not flat. It's wrinkly. If global curvature goes below 10^-5, then it gets lost in the wrinkles.

 

[ViewsofMars]

[snip]
The reason that you have to use a flat LCDM model is that if you don't fix curvature then you can't get information on the time evolution of dark energy. Flat LCDM models are essential if you want to study the evolution of dark matter, but using a flat LCDM doesn't mean that someone thinks that the universe is in fact, flat.

[snip]
The point of physics is to make claims on the nature of the universe. I see no reason to claim that the universe if flat, unless and until we actually think that it is flat. We can be wrong, but making incorrect assertions is what pushes science further.

Twofish-quant, on the previous two pages we discussed a flat universe. I was wondering what you think about the comments by NASA Official: Dr. Edward J. Wollack
Page Updated: Monday, 04-02-2012- WAMP:

The Universe Content: the Ingredients

There are three ingredients in this universe: normal matter (or atoms), cold dark matter, and dark energy.

Atoms: The amount of ordinary matter (atoms) in your universe, the stuff you see around you: tables, chairs, planets, stars, etc. Expressed as a percentage of the "critical density".
Cold Dark Matter: The amount of cold dark matter in your universe, as a percentage of the critical density. Cold dark matter can not be seen or felt, and has not been detected in the laboratory, but it does exert a gravitational pull.
Dark energy: The amount of dark energy in your universe, as a percentage of the "critical density". Unlike dark matter, dark energy exerts gravitational push (a form of anti-gravity) that is causing the expansion of the universe to accelerate or speed up.

Note that the three ingredients can add up to more than or less than 100%. The sum is compared to a quantity that determines the Flatness of the universe. A "flat" universe is said to be at "critical density", having 100% of the matter and energy needed to be "flat". Euclidean geometry describes a flat universe, but non-Euclidean geometries are needed for the alternatives. If the ingredients add up to more than 100%, then the universe has positive curvature and said to be "closed". This means that it curves around on itself (like the surface of a ball), and that if you go in one direction long enough, you'll get back to where you started. If the ingredients add up to less than 100%, then the universe has negative curvature and is called "open". This is the type of curvature that you'd find (in 2 dimensions) on the surface of a horse's saddle, or a potato chip. In that case, space is curved, but it doesn't wrap back around on itself. (Footnote: Mathematicians can probably come up with pathological models where positively curved universes don't wrap around on themselves, and negatively curved ones do, by cutting and pasting various parts of the universe together. We just describe the simplest cases here.)

The Age of the universe is controlled by the amount of the ingredients and the flatness of the universe. By viewing the scale of the universe now, and using Einstein's General Relativity equations to compute the time, under these conditions, needed to reverse the universe to "zero" size, we have the age calculated for us.
http://map.gsfc.nasa.gov/resources/camb_tool/index.html