Couldn't the universe be finite if Omega =1?

[Athanasius]

I am not a physicist or a cosmologist, just a science layman who has been doing a lot of reading and thinking. I have been reading a lot in popular literature that if Omega =1, then the universe must also be infinite. Do you think this is just an over-generalization intended for the general public? I can see expanding space as becoming infinite in volume when it reaches infinite time, but unless it was infinite to begin with, how could it have become infinite within a finite amount of time? Furthermore, since Omega can equal 1 with a finite amount of mass, it seems that the universe could have begun with a finite amount of mass. If were the case, how could it be flat, infinite, homogenous, and isotropic without an average density near zero?

The only way I can see the universe as currently being flat, infinite, homogenous, and isotropic while having infinite mass is if it was infinite in mass and space before expansion began. But that idea gives me some headaches, too.

Wouldn't infinite mass require a quantum fluctuation of infinite magnitude, something highly improbable?

An what about Mach's principle? If the universe had infinite mass, wouldn't all matter have infinite inertia?

Lastly, couldn't the volume of finite flat space be expanding in the direction of time, so that it is currently finite, but is infinite at t=∞? Isn't that the more reasonable idea? if that is what has been meant by an infinite universe all along, wouldn't it be a good idea to clarify this to the general public?

I am very interested in hearing your thoughts on this. If I have missed something, please forgive me and fill in the gaps in my knowledge.

 

[Simon Bridge]

Welcome to PF;
The cosmological $\Omega$ belongs to the FLRW model of the Universe.
http://en.wikipedia.org/wiki/Shape_of_the_universe#FLRW_model_of_the_universe

In order for the Universe to be finite, it must have some curvature - so if you keep going in one direction you end up back where you started.

$\Omega=1$ means zero curvature.
So the pop-science guys are correct that it means an infinite flat universe.
Although - I am a bit uncomfortable with this parameter being used to draw global conclusions.
You have already noticed that it is not a good idea to get your science from pop-science shows.

You intuition about being infinite in space and mass before expansion is correct.
In that situation, the expansion is understood in terms of density. The big bang would, therefore, be a rapid expansion from a hot dense state, not a small-volume one.

The primal energy density need not have been infinite so you don't need infinite fluctuations. Besides, the improbability of our Universe coming into being is irrelevant - put simply: we do not know how many "trials" there were, so we don't know the overall odds. We do know that the probability of getting the Universe we are in is currently 1: you are looking at it.

Mach's principle just asserts that local laws are influenced by the large scale structure of the Universe. The important word to notice here is "structure". It not how much mass there is but how it is distributed.
ifaik, there are not that many people taking Mach's principle, in it's simple forms, seriously - except maybe ans an exercise.
i.e. we understand the centrifugal effect in terms of non-inertial vs inertial reference frames - just like we do with gravity - and not in terms of a cosmological force where all the matter in the Universe pulls your arms out when you spin.

If the Universe is finite and flat, then it must have an edge ... which means it is not isotropic. The laws of physics near the edge would lose their symmetry in the direction that you run out of Universe. But certainly you could postulate a Universe like that - you could say that we don't see the edge because it is retreating from us faster than the speed of light or something.

But no - that is not what was meant by an infinite Universe all along, and it is not a more reasonable idea in the sense that it runs foul of Occam's Razor.

Aside: I am being a bit loose with the terminology here - but I think you'll get the idea even if I'm making some topologists twitch a bit.

 

[Athanasius]

Hi Simon,

Thanks very much for the friendly welcome.

Although - I am a bit uncomfortable with this parameter being used to draw global conclusions.

Good to know that I am not the only one who feels some discomfort with this.

Your intuition about being infinite in space and mass before expansion is correct. In that situation, the expansion is understood in terms of density. The big bang would, therefore, be a rapid expansion from a hot dense state, not a small-volume one.

It's good to hear you confirming my reasoning regarding this. Are there compelling observational reasons for assuming infinite space and mass prior to expansion if space has zero curvature? Or are they simply philosophical, or just an integral part of the model?

As an alternative scenario, what if we start out with finite space and mass prior to expansion, so that we begin with a primeval singularity that actually is smaller than a proton, as it is often described to have been in popular literature? As long as the expansion is exactly enough to prevent eventual recollapse due to gravity, then in this scenario Omega would equal one and we would have a finite universe with zero curvature, correct? Or if it is more than enough to prevent recollapse, we have a negatively curved finite universe, correct?

One additional thought along these lines. Suppose we start out with infinite space and infinite mass. But the expansion is not enough to prevent recollapse, so that Omega is greater than one. How do we compress infinite space into finite space? And how do we fit infinite mass into finite space? If that seems irreconcilable, then perhaps we would need to assume starting out with finite mass and space if Omega is greater than one. But if we can assume they are finite if Omega is greater than one, then what's to prevent us from assuming they are finite if omega is equal to or less than one?

 

[DrStupid]

$\Omega=1$ means zero curvature.
So the pop-science guys are correct that it means an infinite flat universe.

How about exotic geometries like a toroidal universe? It would be flat but finite.

 

[Athanasius]

How about exotic geometries like a toroidal universe? It would be flat but finite.

Yes, I agree that flat does not mean infinite. A Picard Horn, and a Poincaré dodecahedral space would also have flat local geometry and no locally curved space. But what my post is about, is that as long as we start out with finite matter and space before expansion, it seems logical to me that we should have a finite universe even with zero global curvature.

What do you think?

 

[DrStupid]

Yes, a finite universe will remain finite.

 

[Bill_K]

A Picard Horn, and a Poincaré dodecahedral space would also have flat local geometry and no locally curved space.

A Picard Horn is negatively curved, a Poincare dodecahedron is positive.

as long as we start out with finite matter and space before expansion, it seems logical to me that we should have a finite universe even with zero global curvature.

There is no reason whatsoever to impose those assumptions. Many interesting and intuitively appealing cosmologies have been considered in the past, but the universe has ignored our preference and stuck with the simplest cosmology imaginable - flat, infinite, and perpetually expanding. I think we lack the authority to overturn this decision.

 

[Tanelorn]

Isn't truly infinite an impossibility? I think we say that even for a true singularity we draw the line at the Planck length? Perhaps we should do the same for the size of the complete U eg 10^35 times O.U.?

 

[Simon Bridge]

How about exotic geometries like a toroidal universe? It would be flat but finite.

This is where I was being a bit sloppy with the terminology ;)

Isn't truly infinite an impossibility?

The trick is to verify that empirically - otherwise this is just a "no true Scotsman" argument.

I think we say that even for a true singularity we draw the line at the Planck length?

Nope - that is not what the "plank length" means.
Do you know of any publication that draws such a line?

Perhaps we should do the same for the size of the complete U eg 10^35 times O.U.?

Why would you pick that number? Why not OUx10^36 or OUx10^34?

Basically - such arguments run foul of Occam's Razor.
There is no need to make such an assumption.

What we are talking about here is not so much the way the Universe is or is not, but how we choose the model we use to describe the Universe. Sure you could use a model which is flat and finite ... but those describe exactly the same stuff as the flat-infinite ones with harder maths. Therefore we choose to use the ones with easier maths.

 

[Simon Bridge]

Back to the OP:

Are there compelling observational reasons for assuming infinite space and mass prior to expansion if space has zero curvature?

Yes. There are good scientific reason for using an infinite-flat model for the Universe. Oversimplifying: it is the model with the easiest maths which also agrees the most with what we can see. There are a lot of models that also agree with what we see - they have harder maths.

i.e. you could have a toroidal geometry. Go look it up :)

Also see: https://www.physicsforums.com/showthread.php?t=10542

Or are they simply philosophical, or just an integral part of the model?

Lets be plain: all science involves taking a philosophical position - you can look it up under "philosophy of science". What we are doing in these responses is telling you what's what in terms of that position. The position itself is not up for debate in these forums - that kind of discussion never gets anywhere. However, it is not "just" anything - the position taken here is a position which has been, and continues to be, immensely useful and relevant to understanding how Nature works where the other historically competing positions have not.

As an alternative scenario,...

There are a great many alternative scenarios - possibly infinitely many. The difficulty is not in coming up with alternatives but in choosing between them.

If you mean that omega=1 need not necessarily, by itself, mean an infinite Universe, even in the FLRW model, you are correct. The pop science show was, indeed, not being entirely accurate in it's depictions - charitably: it was making a bunch of assumptions without stating them.

Well spotted. You will come to realize that this is a bit like realizing that a politician may be lying or that your lawyer may just be looking out for his own interests ahead of yours.

Outside the realms of pop-science shows: there are good reasons for using an infinite flat model even though it is not the only model that fits observation.

I believe this is a complete answer to the question you stated in post #1.
Further reading:
http://en.wikipedia.org/wiki/Doughnut_theory_of_the_universe

 

[Athanasius]

A Picard Horn is negatively curved, a Poincare dodecahedron is positive.

Please note that I said the local geometry would be flat, not the global topology. Added later: Oh, I see your point. Because of the negative curvature, one end of the horn is finitely curved but the other is open. I am curious though. Why does the Wikipedia article say it has finite volume if one end of it is open?

There is no reason whatsoever to impose those assumptions. Many interesting and intuitively appealing cosmologies have been considered in the past, but the universe has ignored our preference and stuck with the simplest cosmology imaginable - flat, infinite, and perpetually expanding. I think we lack the authority to overturn this decision.

I am not trying to impose assumptions on anyone, and agree that as a science layman, I am certainly no authority on the subject. I am nothing more than a self-study student. But sometimes (though rarely) students can ask penetrating questions. I am just trying to see through and discern some of the over-generalizations I have heard in the pop-sci media. An infinite universe has a lot of philosophical and theological implications. If a universe that matches observations need not be infinite, that's pretty important to know, for me at least.

 

[Athanasius]

Lets be plain: all science involves taking a philosophical position - you can look it up under "philosophy of science". What we are doing in these responses is telling you what's what in terms of that position. The position itself is not up for debate in these forums - that kind of discussion never gets anywhere. However, it is not "just" anything - the position taken here is a position which has been, and continues to be, immensely useful and relevant to understanding how Nature works where the other historically competing positions have not.

There are a great many alternative scenarios - possibly infinitely many. The difficulty is not in coming up with alternatives but in choosing between them.

I am not asking these questions because I the kind of fellow who relishes controversy. I am just someone who is trying to discern the truth after hearing a lot of over-generalizations in popular science media. I agree that the Standard Model is called just that because of it's utility. Another model that I cannot help but wonder if it has a bright future is Carmeli's Cosmological Relativity, since it explains observations so well, and without the need for dark matter, dark energy or a cosmological constant. (It does require a fifth dimension, however!) What do you think of it's potential?

If you mean that omega=1 need not necessarily, by itself, mean an infinite Universe, even in the FLRW model, you are correct. The pop science show was, indeed, not being entirely accurate in it's depictions - charitably: it was making a bunch of assumptions without stating them.

Well spotted. You will come to realize that this is a bit like realizing that a politician may be lying or that your lawyer may just be looking out for his own interests ahead of yours.

Outside the realms of pop-science shows: there are good reasons for using an infinite flat model even though it is not the only model that fits observation.

I believe this is a complete answer to the question you stated in post #1.

Yes, it is. Thanks very much for your generosity in taking the time to answer it, Simon.

 

[Simon Bridge]

I am not trying to impose assumptions on anyone, ...

... he means there is no reason to impose those assumptions on a cosmological model, not a person.
It is a standard turn of phrase in science discussions - the word "impose" is taken in it's mathematical context.

The idea is that the assumptions we include in a scientific model should come from someplace other than inside our own heads.

Another model that I cannot help but wonder if it has a bright future is Carmeli's Cosmological Relativity, since it explains observations so well, and without the need for dark matter, dark energy or a cosmological constant. (It does require a fifth dimension, however!) What do you think of it's potential?

I think that is a topic for another thread :)

Also see:
https://www.physicsforums.com/showthread.php?t=541783
Looks like a non-starter.

 

[Athanasius]

Isn't truly infinite an impossibility?

Infinities do result in some paradoxes, such as Hilbert's paradox of the grand hotel. It seems similar to expanding infinite space. Is it a solution to the problem of expanding infinite space, or rather a show-stopper? With a universe of both infinite space and infinite mass, the volume of the mass is infinite, but it still must be smaller than the still larger infinite volume of space. Even if the same metric is used to measure the volume! What if we subtract the infinite volume of space from the infinite volume of matter? We end up with another infinite number, though the "size" of it ought to have some relationship to the density of space!

Another paradox associated with infinities is Gabriel's Horn. Though infinite, it's volume can be shown to be finite. Does anyone know, is this why a Picard's horn global topology of the universe would have finite volume?

 

[Simon Bridge]

Cantor can teach you how to handle infinities - hint: not the way you are doing.

 

[Bill_K]

With a universe of both infinite space and infinite mass, the volume of the mass is infinite, but it still must be smaller than the still larger infinite volume of space.

Really strange comment. Are you imagining the universe to be a relatively small clump of matter expanding into a preexisting empty space?? Because that's not the case - not at all! The matter uniformly fills all of space, at all times, whether infinite or not. Both the matter and the space expand together. They are always the same size!

 

[Tanelorn]

Simon, a British cosmologist (Penrose?) came up with this size 10^35 or was it 36?
Anyway it was just an example number for some thing extremely big but less than infinity.

Also regarding the smallest possible size for anything I thought that I had read this was called the plank length, so the only thing smaller would have to be a true S. (I thought just by logic)

 

[Athanasius]

Really strange comment. Are you imagining the universe to be a relatively small clump of matter expanding into a preexisting empty space??

No. I am imagining both space and mass to be infinite to begin with, and the average density decreasing as space expands. At any point in time, you could subtract the infinite volume of mass from the infinite volume of space, and the difference would be infinite, too. So you would end up with one infinity being larger than the other, though they are both measured by the same metric.

 

[DrStupid]

At any point in time, you could subtract the infinite volume of mass from the infinite volume of space

What is the "volume of mass" and how do you subtract infinite volumes?

 

[Bill_K]

No. I am imagining both space and mass to be infinite to begin with, and the average density decreasing as space expands. At any point in time, you could subtract the infinite volume of mass from the infinite volume of space, and the difference would be infinite, too. So you would end up with one infinity being larger than the other, though they are both measured by the same metric.

Ok, well that doesn't make any sense either. As I said, the standard FRW cosmologies describe the behavior of a continuous distribution of matter. Matter fills all space uniformly in these cosmologies, and there is no meaningful way to split things into "mass volume" and "empty space volume". And if you could, the metric certainly would be different.

 

[Athanasius]

Cantor can teach you how to handle infinities - hint: not the way you are doing.

Thanks. As you advised, Simon, I just did some reading on Cantor. In light of Cantor's contributions, would you not say there is still a paradox, but just that I did not state it as I should have? So I will make another go at it. Please let me know if I make any mistakes that you can see (which is quite likely).

Cantor said that two sets are equal in magnitude (i.e. size) if their elements can be put into one-to-one correspondence with each other.

In an expanding universe with an infinite volume of space and an infinite volume of baryonic matter, at any given instant in time, the volume of space seems to our minds to logically be greater than the volume of baryonic matter.

However, let's take a random point in infinite space and from there, building an ever growing cube one square meter at a time, start counting the volume of space and the volume of baryonic matter. Both sets (the set of cubed meters of the volume of baryonic matter, and the set of cubed meters of the volume of space), though infinite, are listable or countable (denumereable, as Cantor called them), and therefore they can be matched up on a one-to-one correspondence. Therefore according to Cantor, both infinite sets have the same cardinality, which is Aleph-null.

So don't we still have a seeming paradox? If they both have the same cardinality at any given time, how could the density of matter be less than one?

I am not saying that this paradox, if I actually am correct in thinking that there is one, necessarily invalidates the idea of an infinite universe. But it certainly makes me less inclined to accept the idea so quickly.

If you agree that there is an apparent paradox, I'm not going to try to use that as a basis to debate that a finite universe is therefore the best idea; I think it is a waste of time to argue about paradoxes like that. I simply came here seeking to get some questions answered that I have been grappling with, and you have been of great help. Thanks!

 

[Athanasius]

PS. I just read on the Wolfram MathWorld website that in 2005 Renteln and Dundes gave us this cool little song: "Aleph-null bottles of beer on the wall, Aleph-null bottles of beer, Take one down, and pass it around, Aleph-null bottles of beer on the wall!" (http://mathworld.wolfram.com/Aleph-0.html)

 

[Athanasius]

Ok, well that doesn't make any sense either.

Well, isn't that a characteristic of paradoxes?

And if you could, the metric certainly would be different.

We could imagine placing all of the baryonic matter in my rephrased paradox above under the same environmental conditions (such as temperature, pressure and surrounding gravitational pull) so that we could use the same metric (volume of space occupied). Or we could just count volume of space occupied under the current environmental conditions, assuming it would all average out to some particular group of values of environmental conditions. Either way, the volume of baryonic matter should be less than the volume of space if the density is less than one. But according to Cantor (if I understand him correctly) both infinities would have the same cardinality of Aleph-null.

 

[Simon Bridge]

Simon, a British cosmologist (Penrose?) came up with this size 10^35 or was it 36?
Anyway it was just an example number for some thing extremely big but less than infinity.

Citation please? Could this have been in an interview on a pop-science show?
Anyway - the context will tell us how this number was picked out of all the possible numbers available.

Of course the Universe can have any topology you can think of so long as the troublesome bits are well outside the observable Universe.

Also regarding the smallest possible size for anything I thought that I had read this was called the plank length, so the only thing smaller would have to be a true S. (I thought just by logic)

The Plank length is $\sqrt{hG/c^3}$ ... there is nothing there to say you cannot have half a plank length. It is just that this is the sort of scale you would need to be able to explore to directly verify some theory of quantum gravity.

OTOH: multi-stellar quantities of matter matter compressed by gravity to smaller than the size of a proton will probably end up as a black hole. Small quantities compressed by some other means - say, an unobtainium field - would be more likely to form some exotic state of matter.

No. I am imagining both space and mass to be infinite to begin with, and the average density decreasing as space expands. At any point in time, you could subtract the infinite volume of mass from the infinite volume of space, and the difference would be infinite, too. So you would end up with one infinity being larger than the other, though they are both measured by the same metric.

You are thinking something like Cantor's hotel, where every other room is occupied ... there are an infinite number of guests and twice as many rooms as guests. If you subtracted the number of guests from the number of rooms to get the number of empty rooms, the resulting number is still infinity ... but that makes perfect sense when you take into account the way that the infinities have come about. The matter becomes clear when you talk about the hotel's guest density instead of the total number of guests and rooms.

Well, isn't that a characteristic of paradoxes?

No, it isn't.

Paradoxes are supposed to make sense all the way, in terms of the initial axioms, but lead to conflicting conclusions.

Your difficulty in subtracting two infinite numbers shows sloppy thinking, not a paradox.
Basically you neglected to include the different kinds of infinity, and how the situations come about, in your description so it sounded paradoxical. Anything can sound like a paradox if you miss stuff out. Pop-sci shows love to do this as a way of illustrating the counter-intuitive nature of modern theories - sometimes going out of their way to use confusing descriptions - try not to fall into that trap.

We could imagine placing all of the baryonic matter in my rephrased [description] above under the same environmental conditions (such as temperature, pressure and surrounding gravitational pull) so that we could use the same metric (volume of space occupied).

Stop there - you are no longer talking about an FLRW model of the Universe.

If you mix up the models you are going to get nonsense.

The model you are talking about has discrete lumps of matter and lots of vacuum.
You are trying to describe the situation where there is a lot more volume than there is stuff to fill it - in the manner of a bucket that is half full. FLRW Universe you started out talking about has a classical mass field which is continuous rather than discrete. (Bear in mind that the mass-energy relation means that mass and energy are the same thing so matter is a form of energy.)

I have a feeling that this new model does not describe the present Universe very well (what about energy fields? what do you mean by "empty space"?)

Whatever - isn't this a topic for another thread.

 

[Athanasius]

You are thinking something like Cantor's hotel, where every other room is occupied ... there are an infinite number of guests and twice as many rooms as guests. If you subtracted the number of guests from the number of rooms to get the number of empty rooms, the resulting number is still infinity ... but that makes perfect sense when you take into account the way that the infinities have come about. The matter becomes clear when you talk about the hotel's guest density instead of the total number of guests and rooms.

No, it isn't.

Paradoxes are supposed to make sense all the way, in terms of the initial axioms, but lead to conflicting conclusions.

Your difficulty in subtracting two infinite numbers shows sloppy thinking, not a paradox.
Basically you neglected to include the different kinds of infinity, and how the situations come about, in your description so it sounded paradoxical.

You mean Hilbert's Hotel? The scenario I have read multiple times is of moving all of the guests to the next even-numbered room to create an infinite number of vacancies in a full hotel for an infinite number of new guests.

And I was speaking of two infinities of the same type. Both were cardinal Aleph-null infinities.

Stop there - you are no longer talking about an FLRW model of the Universe.

If you mix up the models you are going to get nonsense.

The model you are talking about has discrete lumps of matter and lots of vacuum.
You are trying to describe the situation where there is a lot more volume than there is stuff to fill it - in the manner of a bucket that is half full. FLRW Universe you started out talking about has a classical mass field which is continuous rather than discrete. (Bear in mind that the mass-energy relation means that mass and energy are the same thing so matter is a form of energy.)

In my modified paradox I spoke specifically about the density of baryonic matter per cubic meter. Not often discussed, but there would be an average density of baryonic matter in an FLRW Universe, would there not be, since it amounts to 4.6 of the total mass. I intentionally left out other forms of mass to make the argument simpler, not because I was proposing a new model.

 

[bobie]

In order for the Universe to be finite, it must have some curvature - so if you keep going in one direction you end up back where you started.

$\Omega=1$ means zero curvature.

If the universe has some curvature and light follows geodesics with same curvature how do you realize it is finite and is not flat? What is the absolute parameter that makes you conclude that when you are pointing a telescope you are not looking at the back of your head?

The magic of a sphere is just that: being finite but boundless, ergo practically infinite.
Had there been no oceans , primitive man could have wandered forever and think the planet is flat and infinite.

Is this (flat and infinite) nonsense or: is there some truth in it?

 

[Simon Bridge]

If the universe has some curvature and light follows geodesics with same curvature how do you realize it is finite and is not flat?

Please look up "FLRW Universe" - your questions are answered in the construction of that model.

If you are having trouble understanding that model, then please open a new thread to ask your questions.

What is the absolute parameter that makes you conclude that when you are pointing a telescope you are not looking at the back of your head?

Simple answer - I know what the back of my own head looks like.

The absolute parameter in the FLRW model is omega.

Had there been no oceans , primitive man could have wandered forever and think the planet is flat and infinite.

... I suspect that ending up back in the same place every now and again would have been a bit of a giveaway though.

In science we do not rely on "wandering" about to work out what model to use. We make deliberate tests.
Nobody had to go all the way around the Earth to determine that the best model was spherical.
You only needed to go from Syrene to Alexandria - no oceans in the way.

 

[bobie]

The absolute parameter in the FLRW model is omega.

... I suspect that ending up back in the same place would have been a bit of a giveaway though.
.

My question referred to all possible models, Simon. Omega is not an absolute reference point, is something you assume at the ground of that model.

As to your other ( I suppose humorous) remarks , I , for one, am not able to look at my neck.
Secondly, it would have taken more than a lifetime to a primitive man to end up in the same place and , even if he lived that long , it would have taken an exceedingly clever navigation to come to the very starting point.
Modern man is as helpless in universe as primitive man on Earth. That was the sense of my remark, no sun to shine into a well at Syene, there! That is the absolute parameter you are missing, Simon.
All models about the universe are mere speculations. Choosing a model because it has (alleged) simpler maths may not be the right choice. Moreover, 'zero', 'nothing' and 'infinite' are empty words as they have no 'signified' and should not be an option in the foundation of any scientific theory.

If you think this is off topic in FLRW, I'll consider starting a new thread.
Thanks for your attention

 

[Mordred]

how energy-density Ω relates to universe geometry is covered here, feel free to ask any questions on it in another thread (this thread is specified for the Ω=1 condition)

the question of whether Ω=1 equals infinite has already been answered succinctly so I have nothing further to add in those regards

http://cosmology101.wikidot.com/universe-geometry page 2 for the FLRW metric portion
http://cosmology101.wikidot.com/geometry-flrw-metric/

more info can be found in these articles

http://arxiv.org/pdf/hep-ph/0004188v1.pdf :"ASTROPHYSICS AND COSMOLOGY"- A compilation of cosmology by Juan Garcıa-Bellido
http://arxiv.org/abs/astro-ph/0409426 An overview of Cosmology Julien Lesgourgues
http://arxiv.org/pdf/hep-th/0503203.pdf "Particle Physics and Inflationary Cosmology" by Andrei Linde

 

[bobie]

feel free to ask any questions on it in another thread (this thread is specified for the Ω=1 condition)

Thanks, Mordred,
probably I was too abstract and did not express my thoughts clearly: I am not saying that Ω is/should not be =1.
I am simply asking:
if light were following geodesics (as , if I am not wrong, GR says) would you conclude that universe is flat , or is there something that could make you detect the fact the universe only seems flat?

 

[Mordred]

geodesics are determines by the geometry, in the curved scenarios you can have curved geodesics. and angles in a triangle do not necessarily add up to 180 degrees. The universe geometry article I posted covers how geodesics are determined due to energy-density relations. The CMB measurements allowed us to reliably determine that our universe is extremely close to flat. please take the time to read the article the distance relations I used in the FLRW metrics show the geodesic relations as well as the images.

see the triangles on images 1.0, 1.1, and 1.2 on the second page

http://cosmology101.wikidot.com/geometry-flrw-metric/

 

[bobie]

The CMB measurements allowed us to reliably determine that our universe is extremely close to flat. please take the time...

I'll read your articles in depth, but if it is not covered there, could you suggest an article that explains how CMB implies that universe is nearly flat. Or , if it ieasier to answer, if geodesics were curve, what would happen to CMB?
Could you also be more precise on the adverb 'close' to? as any deviation, however small, from absolute flat leads to a circle , however huge, I suppose. Or not?

 

[Mordred]

CMB measurements for light paths involve looking for distortions, the mathematics behind it is extremely complex.

http://map.gs
fc.nasa.gov/mission/sgoals_parameters_geom.html

" By measuring the apparent (angular) size of these spots we can infer which sets of lines the light followed to reach us. We use the location of the main peak in the temperature spectrum to determine the average apparent spot size "

here is a generalized article with some images, a full paper would be highly technical as it extensively involves sound wavelengths ( if you want the actual metrics they are covered in the 3 textbook style articles I posted)

http://www.astro.virginia.edu/class/whittle/USEM/Week8/Diagnostics_bw.pdf

there is potential for an extremely large circle, by being not perfectly flat or a saddle horn depending on if the offset from the critical density is positive or negative. A prime reason to try and tighten down any form of errors in analysis

 

[bobie]

there is potential for an extremely large circle, by being not perfectly flat

Thanks , Mordred

 

[timmdeeg]

I am simply asking:
if light were following geodesics (as , if I am not wrong, GR says) would you conclude that universe is flat , or is there something that could make you detect the fact the universe only seems flat?

The angular power spectrum of the CMB shows that the universe is very close to spatial flatness. There are certain density fluctuations (called baryonic acoustic oscillations) within this spectrum whose true diameter is known. These peaks are observed at an angle of 1 degree. This combined with their diameter yields a sum of angles of around 180°. So, the local geometry seems Euclidean.
As discussed, this observation is not related to any conclusion regarding the topology of the universe and to whether or not it is infinite.

 

[Athanasius]

The angular power spectrum of the CMB shows that the universe is very close to spatial flatness. There are certain density fluctuations (called baryonic acoustic oscillations) within this spectrum whose true diameter is known. These peaks are observed at an angle of 1 degree. This combined with their diameter yields a sum of angles of around 180°. So, the local geometry seems Euclidean.
As discussed, this observation is not related to any conclusion regarding the topology of the universe and to whether or not it is infinite.

Bobie, related to timmdeeg's post, I read that the measurements were accurate within an error range of 0.4 percent. While a universe close to the size of the Hubble volume could fall within that error range, I also read that Bayesian averaging applied to the measurements conservatively indicates a universe of at least 251 Hubble volumes. We cannot conclude that it is infinite based on that alone (because the curvature could be so slight that we just can't detect it, or the universe could have a closed shape that makes it appear to be flat), but we can say that it is huge. So, regarding the analogy you were drawing about primitive man not being to detect the curvature of the earth, that would apply if the universe is so huge that it is only very slightly curved. Then our instruments would not be able to detect the curvature. Or if it has one of certain finite global topologies such as a Poincaré Dodecahedral Space, it would be finite but we would still be getting flat measurements. Then we could be looking at the "back of the head" of many galaxies, as you put it, when we look at the night sky, but not be able to discern it yet because they would be in an earlier stage of development and position.

 

[bobie]

that would apply if the universe is so huge that it is only very slightly curved. ...

. Then we could be looking at the "back of the head" of many galaxies, as you put it, when we look at the night sky, but not be able to discern it yet because they would be in an earlier stage of development and position.

Hi Athanasius,
- of course the universe is huge and very slightly curved and that applies, but it applies a fortiori if it were more than huge, as they suggest, i.e. infinite.
But my remark went even beyond that, and imagined a plausible scenario in which, whatever curvature, the geodesìcs match or other factors intervene (see next) so that the signal comes straight to you and you cannot detect any curvature.

- of course we would not be able to discern it, we would be long dead: it was a metaphore or, rather, a hyperbole I used to forcibly express my thought. As an example (not completely fitting), imagine a transmitter sending a unidirectional radio signal westward. If on the eastern side of the building you point a directional aerial to the east you'll catch "a signal from the back of your head", and that because the wave is reflected by the ionosphere. I do not know if you are able to find out that the wave is not coming from the east, but that shows the principle I was referring to.

As to infinite, I do not think religion plays any role, as I said, that word should be written off the vocabulary of any man of science: it is 'meaning'- less and any theory based on or including it is not falsifiable. I am sure it is not infinite and I cannot prove it, but I am on the same footing as the greatest scientist: he can't prove it is infinite.

I do not see , as an outsider, the necessity of getting into trouble, why not use a vague term like 'immense', 'boundless' or just humbly admit: we have no clue on how big it is.

 

[Athanasius]

Hi Athanasius,
- of course the universe is huge and very slightly curved and that applies, but it applies a fortiori if it were more than huge, as they suggest, i.e. infinite.
But my remark went even beyond that, and imagined a plausible scenario in which, whatever curvature, the geodesìcs match or other factors intervene (see next) so that the signal comes straight to you and you cannot detect any curvature.

- of course we would not be able to discern it, we would be long dead: it was a metaphore or, rather, a hyperbole I used to forcibly express my thought. As an example (not completely fitting), imagine a transmitter sending a unidirectional radio signal westward. If on the eastern side of the building you point a directional aerial to the east you'll catch "a signal from the back of your head", and that because the wave is reflected by the ionosphere. I do not know if you are able to find out that the wave is not coming from the east, but that shows the principle I was referring to.

As to infinite, I do not think religion plays any role, as I said, that word should be written off the vocabulary of any man of science: it is 'meaning'- less and any theory based on or including it is not falsifiable. I am sure it is not infinite and I cannot prove it, but I am on the same footing as the greatest scientist: he can't prove it is infinite.

I do not see , as an outsider, the necessity of getting into trouble, why not use a vague term like 'immense', 'boundless' or just humbly admit: we have no clue on how big it is.

So you are referring to some sort of interference that would make space appear to be flat when it is not?

Regarding your last comment regarding religion, I would gladly comment, but since this is a cosmology forum and not a philosophy of science forum I will abide by the wishes of the moderators, who have made it clear to me following my last post that they do not want such things discussed here. I will therefore strictly abide by the rules of methodological naturalism in my comments here. You can send me a private message if you want to know my thoughts.

 

[Simon Bridge]

In the event of some long-range effect interfering with our ability to see the backs of our heads, so to speak, we would have to rely on short range measurements. Historically this is what was done to find out the overall shape of the Earth. There were things making long-range measurements difficult - mountains and oceans and so forth.

Fortunately, it was not necessary to circumnavigate the globe to demonstrate that it was a globe.

But it did need sufficiently accurate measurements in order to tell the difference from flat. Fortunately the Earth is very round so the measurement accuracy needed was within the scope of the ancients. Before that accuracy was achieved, it would have been quite reasonable to model the Earth as flat - indeed, some quite fanciful models emerged.

It would have been trickier proving the Earth was infinite and flat though, even if it was.

Imagine that mountains and atmospheric effects make long-distance measurements difficult.
We could lay out a really big triangle with lasers and carefully measure the angles and add them up.
The uncertainty in the measurements would act as limits on the curvature and so allow us to assign probabilities to different models.

If we are also limited in how much of the world we can see, which implied by the presence of the mountains etc. then we only need a model of the world that matches what we see ... there will be many. How do we choose between them.

Thus, in the context of FLRW, we do not say that omega=1 but that omega is approximately 1.
To the best we can measure.[but see footnote]

That's the same with pretty much anything -
- we also say that photons are massless and electrons are point particles.

With all these parameters, we have available a range of possible models that could give rise t them.
Which ones are we best advised to choose?

We could pick the one that gets us paid the most, or the most fame or power. Historically, these methods have been used to select from the models available. But we are scientists so we pick the one which has the least trouble with Occam's razor - preferring the ones with easy maths. Well, as easy as we can get away with. Where several are really close, we get to argue about them.

This could go some way to explain the lack of rich scientists compared with say, politicians or lawyers.

-------------------------------

footnote:
This diverges from the topic of the thread though ... off post #1 the question is not about the flatness or otherwise of our Universe is but about what it means for the shape of the FLRW universe if omega=1.

 

[bobie]

So you are referring to some sort of interference that would make space appear to be flat when it is not? .

Luckily, as Mordred said, there is margin of possibility of a tiny curvature. If there were no proof that would not exclude our inability to detect it due to unknown factors. That was my point.

But the problem is more complex and profound :
as it was stated in post #2, the choice of a flat and infinite U(niverse) has been made to dodge the problem of the edge, on the false assumption that a curved U must have an edge while flat one may not. But the remedy is worse than the cure, as flat is not enough, it must be infinite, and 'infinite' opens a Pandora's box of infinite trouble.
I hope Mordred would care to clarify these simple points of the theory:

- infinite U means infinite space or also infinite mass?
...if mass is finite, is it distributed on infinite radius?
- infinite means no-shape?
- an infinite but curved U is worse than flat one?
- infinite + c = infinite or not?, if so, how can U expand?
...if the rate of expansion is over 3c at a certain distance what is that rate at infinite distance?
- if U is (flat and) infinite right now, what is the meaning of the radius of U being now 14 Gly?
- was U infinite even before BigBang?
... if it was: what is the use of this theory? if it says that even space and time did not exist before BB,then:
... if it was not: how can it be infinite now after only 14 G-years?

These are only the main obscure points.

 

[Athanasius]

In the event of some long-range effect interfering with our ability to see the backs of our heads, so to speak, we would have to rely on short range measurements. Historically this is what was done to find out the overall shape of the Earth. There were things making long-range measurements difficult - mountains and oceans and so forth.

Fortunately, it was not necessary to circumnavigate the globe to demonstrate that it was a globe.

But it did need sufficiently accurate measurements in order to tell the difference from flat. Fortunately the Earth is very round so the measurement accuracy needed was within the scope of the ancients. Before that accuracy was achieved, it would have been quite reasonable to model the Earth as flat - indeed, some quite fanciful models emerged.

It would have been trickier proving the Earth was infinite and flat though, even if it was.

Imagine that mountains and atmospheric effects make long-distance measurements difficult.
We could lay out a really big triangle with lasers and carefully measure the angles and add them up.
The uncertainty in the measurements would act as limits on the curvature and so allow us to assign probabilities to different models.

If we are also limited in how much of the world we can see, which implied by the presence of the mountains etc. then we only need a model of the world that matches what we see ... there will be many. How do we choose between them.

Thus, in the context of FLRW, we do not say that omega=1 but that omega is approximately 1.
To the best we can measure.[but see footnote]

That's the same with pretty much anything -
- we also say that photons are massless and electrons are point particles.

With all these parameters, we have available a range of possible models that could give rise t them.
Which ones are we best advised to choose?

We could pick the one that gets us paid the most, or the most fame or power. Historically, these methods have been used to select from the models available. But we are scientists so we pick the one which has the least trouble with Occam's razor - preferring the ones with easy maths. Well, as easy as we can get away with. Where several are really close, we get to argue about them.

This could go some way to explain the lack of rich scientists compared with say, politicians or lawyers.

-------------------------------

footnote:
This diverges from the topic of the thread though ... off post #1 the question is not about the flatness or otherwise of our Universe is but about what it means for the shape of the FLRW universe if omega=1.

A very nice explanation, Simon, thanks. I am curious, do you agree with bobie that is it primarily the problem of the edge that makes the math easier with an infinite flat universe, as opposed to a finite flat universe? If there are other aspects that make the math easier, I am also curious to know.

 

[Athanasius]

...there is margin of possibility of a tiny curvature.

As I understand it, an undetectable curvature is not just a small possibility. There are a lot of vast but finite sizes that our universe could be between 251 Hubble volumes and infinity. All that we can say based on the measurements, which (as Simon said) indicate that it is approximately flat, is that it is very big, infinite, or of a global topology that makes it look larger than it really is.

 

[Mordred]

Luckily, as Mordred said, there is margin of possibility of a tiny curvature. If there were no proof that would not exclude our inability to detect it due to unknown factors. That was my point.

But the problem is more complex and profound :
as it was stated in post #2, the choice of a flat and infinite U(niverse) has been made to dodge the problem of the edge, on the false assumption that a curved U must have an edge while flat one may not. But the remedy is worse than the cure, as flat is not enough, it must be infinite, and 'infinite' opens a Pandora's box of infinite trouble.
I hope Mordred would care to clarify these simple points of the theory:

- infinite U means infinite space or also infinite mass?
...if mass is finite, is it distributed on infinite radius?
- infinite means no-shape?
- an infinite but curved U is worse than flat one?
- infinite + c = infinite or not?, if so, how can U expand?
...if the rate of expansion is over 3c at a certain distance what is that rate at infinite distance?
- if U is (flat and) infinite right now, what is the meaning of the radius of U being now 14 Gly?
- was U infinite even before BigBang?
... if it was: what is the use of this theory? if it says that even space and time did not exist before BB,then:
... if it was not: how can it be infinite now after only 14 G-years?

These are only the main obscure points.

to be honest I don't really concern myself with scenarios beyond the cosmic event horizon, I find the subject too conjectural. We have no scientific data of what occurs outside our observable universe, nor are we likely to do so. The question of whether or not the overall universe is finite or infinite may never be answered with 100% accuracy.
To me the question is largely meaningless except in the question of how the universe started, to my way of concern. In that arena, we only know that the universe had a hot, dense state of unknown size and origin.
For our observable universe, its sufficient to know and understand the observable universes geometry, as this affects our measurements in terms of light paths and expansion.

An infinite universe does mean infinite energy and matter, a finite universe will not become infinite, if its finite in the past it will be finite in the future, and vise versa.
An infinite universe can expand, in that the overall density can decrease.

as far as what infinite means in regards to the other questions I'll leave those questions in the hands of others

 

[bobie]

...to be honest I don't really concern myself with scenarios beyond the cosmic event horizon,... The question of whether or not the overall universe is finite or infinite may never be answered ...
For our observable universe, its sufficient to know and understand the observable universes geometry,

You are right when you say that it is better not to speculate beyond the observable universe, that is not scientific.
But the problem is that someone ventured to state that U is flat and infinite.
And the big problem is that same people went on elaborating a theory based on the assumption that U is curved and finite , ignoring the founding axioms.

Saying that U is infinite and flat is like sweeping under the carpet the problem of the edge of the U, but the very Hubble law implies that U has an edge and that that edge is right now at 14.4 Gly from us and is expanding exactly at C.
The complete law is in fact: C/T₀*V_E
(where V_E is the velocity of the edge , and it needs be = C if the Hubble constant 1/T₀ must be the rate of expansion of 1cm, = 2.2*10^-18 cm/s)

If this is correct, the basic law of the theory itself proves that U is finite and curved, so the question you refer to has already been answered by the observable universe geometry. That was the point of my posts.

 

[Mordred]

my article applies only to the observable universe, much of that work is done with the examples from Barbera Ryden's "Introduction to cosmology" see the footnotes on the article.

The reason I chose her work is that it examines the FLRW metric in a very straight forward manner.
You cannot base the size of the observable universe as per Hubble's law. Based on Hubble's law the point at which redshift is greater than the speed of light is called the Hubble's sphere.
We know we have recessive velocities of 3c. the reason for this is due to the cosmological constant. The observable universe is larger than the Hubble sphere

"In cosmology, a Hubble volume, or Hubble sphere, is a spherical region of the Universe surrounding an observer beyond which objects recede from that observer at a rate greater than the speed of light due to the expansion of the Universe."

http://en.wikipedia.org/wiki/Hubble_volume

the event horizon however accounts for expansion

however we can retrieve information as far as our cosmological horizon (observable universe)
http://en.wikipedia.org/wiki/Cosmological_horizon

A good article to cover this is
http://tangentspace.info/docs/horizon.pdf :Inflation and the Cosmological Horizon by Brian Powell

 

[bobie]

You cannot base the size of the observable universe as per Hubble's law constant... The observable universe is larger than the Hubble sphere... we have recessive velocities of 3c.
l

If you are interested, we can discuss V_E>C later (or in another thread), but the fact that the observable U is a little larger doesn't affect the issue at hand.
If the value of the radius needs some adjustment, its' OK, the fact remains that the theory we are discussing is based on the assumption that U is finite and curved (perfectly spherical if the expansion is the same in every direction).

I do not know if the idea that even space and time were created at BB is generally shared, but in any case, once you have ascertained the exact value of the radius, you have confirmed that U is finite and curved.
To imagine even a speck of dust outside that (whatever) R would make all BB theory anf FLRW metric crumble down. You can't have it both ways.

 

[Mordred]

If you are interested, we can discuss it later (or in another thread), but the fact that the observable U is a little larger doesn't affect the issue at hand.
If the value of the radius needs some adjustment, its' OK, the fact remains that the theory we are discussing is based on the assumption that U is finite and curved (perfectly spherical if the expansion is the same in every direction).

I do not know if the idea that even space and time were created at BB is generally shared, but in any case, once you have ascertained the exact value of the radius, you have confirmed that U is finite and curved.
To imagine even a speck of dust outside that (whatever) R would make all BB theory anf FLRW metric crumble down. You can't have it both ways.

I did not confirm that the universe is finite. I stated that it is possible with a slight curvature from a critically dense universe. However those parameters rely upon accuracy of measurements or if the actual density is in fact slightly offset from a flat universe. Physicists never state 100% certainty in any measurement.

the slight offset from Ω=1 depends on the dataset. (this is the slight offset I referred to)
Ω_tot= 1.002±0.011

http://pdg.lbl.gov/2012/reviews/rpp2012-rev-cosmological-parameters.pdf

the question of accuracy is always an open question, however this value is incredibly close to a flat universe.

my article also stated,
"The observable universe which is the portion we can see is a finite, sphere with a radius of 46 Gly, which is equal to 46 billion light years. The 46 Gly particle horizon refers to the today's distance of objects, whose radiation emitted in the past we receive today. The overall size of the universe is not known, it could be infinite or finite. If its infinite now then it would be infinite in the past, a finite value can never become infinite"

 

[bobie]

I did not confirm that the universe is finite.

Probably you did not, Mordred, I am saying that the theory says it.
It says that at BB space and time were created and then it gives you a detailed chronology http://en.wikipedia.org/wiki/Chronology_of_the_universe that tells you roughly what the radius of the U was after a certain time.
That was one of my questions you ignored: how can U be infinite after a finite number of seconds?
how can anything exist outside the shockwave of BB be it 14 or 46 or 10¹⁰⁰Gly?

 

[Mordred]

This is from the link you provided

"This chronology of the universe describes the history and future of the universe according to Big Bang cosmology, the prevailing scientific model of how the universe developed over time from the Planck epoch, using the cosmological time parameter of comoving coordinates. The instant in which the universe is thought to have begun rapidly expanding from a singularity is known as the Big Bang. As of 2013, this expansion is estimated to have begun 13.798 ± 0.037 billion years ago.[1] It is convenient to divide the evolution of the universe so far into three phase"

the time prior to 10^-43 is the singularity they are referring to, in this specific case the singularity refers to a point at which to the best of our best knowledge of physics we can no longer describe what is occurring. The Planck epoch starts at this point. (this is not the same as a black hole singularity, a BH singularity is an infinitely dense, region with a pointlike volume, stating zero volume makes even less sense, some articles state that)

this is the hot dense state, that the hot big bang model covers, however any details prior to inflation depends on our understanding of particle physics. We can't observe this time due to the dark ages (see the dark ages on that same page), in this case singularity can be infinite or finite as it represents a breakdown in our current physics understanding, not of volume

no worries this is another common confusion

 

[bobie]

They are plainly saying, we don't have a clue!