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

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.

## Answers and Replies

Simon Bridge
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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.

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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?

##\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.

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?

Yes, a finite universe will remain finite.

Bill_K
Science Advisor
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. 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
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DrStupid said:
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 ;)

Tanelorn said:
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
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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

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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.

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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.

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Simon Bridge
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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.

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

Bill_K
Science Advisor
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!

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)

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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.

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
Science Advisor
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.

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!

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)

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.

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Simon Bridge
Science Advisor
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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.

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.