Shape of Universe: Is Flatness Approved? Causes of Big Crunch

[usmhot]

That's stating it too strongly. Isotropy and homogeneity are likely to hold at scales significantly larger than our horizon, but this doesn't mean that they hold infinitely-far.

[snip]

Also, if you think it's weird that our universe may be infinite in space, bear in mind that it seems to be infinite in time: in the future, our universe is likely to expand forever.

I don't think it's weird that it may be infinite in space ... I think it's impossible. Not least because infinity is not a number.

And, in fact, it's good that you said that it seems to be infinite in time, because that exemplifies the problem with using 'infinity'. The Universe will not exist for an infinite time. At any point at which one would care to measure it no matter how far in the future the measurement will be a finite number. There is an important difference between existing for an indefinite amount of time and existing for an infinite amount of time.

So, to get back to its size - if it was ever finite in size then it will always be finite in size, as, no matter how much space you add to it, as long as you add a finite amount it will still be finite. Which implies that if it is infinite in size now then when it came into existence it must have been infinite in size. Which, I believe, is technically absurd. (Which is why I’d appreciate an explanation of a model in which an ‘infinite’ Universe can come from a Big Bang.)

So, the Cosmological Principle is extremely important. If it holds, then, as far as I can see, the only possible topology is a sphere (albeit with such a large radius that it is close to flat on the scale that we can measure). But, if the Cosmological Principle does not hold then either the Universe is a torus or it has boundaries.

 

[Chalnoth]

I don't think it's weird that it may be infinite in space ... I think it's impossible. Not least because infinity is not a number.

Infinity is a number on the extended number line. It's a slightly weird number, but it is a number.

And, in fact, it's good that you said that it seems to be infinite in time, because that exemplifies the problem with using 'infinity'. The Universe will not exist for an infinite time. At any point at which one would care to measure it no matter how far in the future the measurement will be a finite number. There is an important difference between existing for an indefinite amount of time and existing for an infinite amount of time.

This is playing word games. The dimension of time for our universe is likely to be infinite in extent.

So, to get back to its size - if it was ever finite in size

Why do you think it was ever finite in size?

So, the Cosmological Principle is extremely important. If it holds, then, as far as I can see, the only possible topology is a sphere (albeit with such a large radius that it is close to flat on the scale that we can measure). But, if the Cosmological Principle does not hold then either the Universe is a torus or it has boundaries.

Why not some sort of irregular blobby shape?

 

[Chronos]

I think the answer to the flatness question is undefinable. We know it is practically zero, but, will never never know if it is perfectly flat. I prefer to think it fluctuates around zero, but, is never exactly zero due to quantum uncertainty.

 

[Chalnoth]

I think the answer to the flatness question is undefinable. We know it is practically zero, but, will never never know if it is perfectly flat. I prefer to think it fluctuates around zero, but, is never exactly zero due to quantum uncertainty.

Or put another way, it can only be measured if it is significantly non-zero.

 

[TrickyDicky]

Infinity is a number on the extended number line. It's a slightly weird number, but it is a number.

Infinity is not considered a number(weird or not) in mathematics. It's more like a concept

Why do you think it was ever finite in size?

Well, usmhot has a point there, if it was infinite from the first instant after t=0, it doesn't make much sense to talk about changes of spatial size, or inflationary epochs, IOW how is something infinite comparable in size at different times, it seems logical that it would be equally infinite everytime.

 

[Chalnoth]

Well, usmhot has a point there, if it was infinite from the first instant after t=0, it doesn't make much sense to talk about changes of spatial size, or inflationary epochs, IOW how is something infinite comparable in size at different times, it seems logical that it would be equally infinite everytime.

It isn't clear at all that there was an absolute beginning, before which there was nothing. And certainly there was no singularity.

Furthermore, changes in size are not done with regard to the whole, but with regard to changes of distance within the universe. There is no problem whatsoever for an infinite universe to expand: it means that average distances between things in the universe are getting larger.

I would like to point out that the flat FRW metric that is generally used to examine these things is infinite in extent.

 

[George Jones]

Well, usmhot has a point there, if it was infinite from the first instant after t=0, it doesn't make much sense to talk about changes of spatial size

Spatial scale is not determined by topology (i.e., whether space is compact or non-compact), it is determined by an additional structure, the metric (as noted by Chalnoth). FLRW universes have time-dependent metrics (no timelike Killing vectors).

 

[chill_factor]

Infinity is not considered a number(weird or not) in mathematics. It's more like a concept

Sure it is. Make a graph of numbers where the X axis is defined by 1/R.

(0,0) is then infinity defined to be a single point on this graph.

 

[WannabeNewton]

Sure it is. Make a graph of numbers where the X axis is defined by 1/R.

(0,0) is then infinity defined to be a single point on this graph.

What?

I repeat...what?? I don't even...O.O

 

[Chalnoth]

Sure it is. Make a graph of numbers where the X axis is defined by 1/R.

(0,0) is then infinity defined to be a single point on this graph.

This isn't true. The value at 0 in such a graph is undefined. This can be understood as due to the fact that if you take the limit as x approaches 0 for 1/x, you get different answers if you approach zero from the positive direction vs. the negative direction (+\infty and -\infty, respectively).

Anyway, what you call infinity is somewhat irrelevant. It does behave differently from other numbers in a few fundamental ways (that is, it behaves differently under various operations than other numbers). But the fact of the matter is, none of this has any bearing on whether or not the concept of infinity can be applied to reality. Even if we claim that space-time is fully-described by the real numbers and not the extended reals, and if our space-time maps onto all of the reals, then it is infinite in extent (in both time and space). There is nothing nonsensical about this statement.

 

[micromass]

I think we should realize that there is no such thing as a "number". Saying that infinity is or is not a number is an ambigous statement until we specifiy what we mean with number.
It is absolutely true that infinity is not a real number. But it is also true that infinity is an extended real number.

 

[TrickyDicky]

It isn't clear at all that there was an absolute beginning, before which there was nothing. And certainly there was no singularity.

I'm not talking about the singularity, I'm referring to the usual narrative in the LCDM model of the first instants after whatever it was that you are sure was not a singularity. That narrative compares the size of the universe at different times, I just was wondering what that could mean if the universe is Infinite at all those moments.

Furthermore, changes in size are not done with regard to the whole, but with regard to changes of distance within the universe.

The scale factor produces changes to the whole spatial metric.

There is no problem whatsoever for an infinite universe to expand: it means that average distances between things in the universe are getting larger.
I would like to point out that the flat FRW metric that is generally used to examine these things is infinite in extent.

Certainly.

 

[Chalnoth]

I'm not talking about the singularity, I'm referring to the usual narrative in the LCDM model of the first instants after whatever it was that you are sure was not a singularity. That narrative compares the size of the universe at different times, I just was wondering what that could mean if the universe is Infinite at all those moments.

I generally expect that this kind of thing is generally sensible if only a small fraction of the universe inflated at that time, or if we're living in some sort of eternal inflation scenario. There may also be other possibilities.

The scale factor produces changes to the whole spatial metric.

Just because it happens everywhere within the metric doesn't mean the impact we measure isn't a local impact. The fact that it is measured locally, in fact, is critically important, because global measurements are not possible (due to our horizon).

 

[TrickyDicky]

I generally expect that this kind of thing is generally sensible if only a small fraction of the universe inflated at that time,

I am asking about what the BB cosmological model parametrized by LCDM states, are you saying you don't consider sensible its description of the first minutes of the universe? Assumptions like spatial flatness, inflation and cold dark matter all follow from them and the cosmological principle that in its mainstream version certainly is valid for the largest scales, not only for the observable part.

or if we're living in some sort of eternal inflation scenario. There may also be other possibilities.

Not much interested in this kind of speculation either.

Just because it happens everywhere within the metric doesn't mean the impact we measure isn't a local impact. The fact that it is measured locally, in fact, is critically important, because global measurements are not possible (due to our horizon).

I'm not concerned with my question with the local metric or local measures at all.

 

[Chalnoth]

Not much interested in this kind of speculation either.

Therein lies the problem. There are many possible models for the early universe, and for the universe as a whole. We don't yet know which is accurate, and it doesn't make sense to a priori assume that certain things (e.g. an infinite universe) are automatically out of bounds. It may just be that you haven't considered the right model yet.

 

[TrickyDicky]

Therein lies the problem. There are many possible models for the early universe, and for the universe as a whole. We don't yet know which is accurate, and it doesn't make sense to a priori assume that certain things (e.g. an infinite universe) are automatically out of bounds. It may just be that you haven't considered the right model yet.

Sure, I'm not discarding spatially infinite FRW metrics, just trying to understand how statements from the LCDM model make sense in the context of a spatially infinite universe. For instance, a common statement would be:"Approximately 10^−37 seconds into the expansion, a phase transition caused a cosmic inflation, during which the Universe grew exponentially." Now, how can something infinite in extent grow exponentially? No matter how I try to interpret it I can't find a sensible mathematical meaning for it.

 

[usmhot]

OK. Personally, from my understanding of any real definition of 'infinity' (and, to be blunt, I don't understand how any physicist would happily think of 'infinity' as a potentially real physical thing) I cannot accept that anything can be 'infinite' as this has no meaning (to me, other than as a useful mathematical shorthand).
I can accept that the Universe may expand indefinitely, but, as I say, there is a huge difference between the statement that something is infinite and something has no definite end.
However, I can't accept that the Universe started as 'infinite'. And, my understanding of the Big Bang theory would be that expansion is one of the supporting arguments, as reversing it leads to a singularity (or whatever - you can forgive a layman's lack of knowledge on why the Big Bang point might not be called a 'singularity').
So, it seems to me (in my naivete) that the Universe (in its entirety) started from a finite point and as such must be still finite. In this case, if the Cosmological Principle holds (and it seems that it is more likely to hold given that it seems to hold in the observable Universe and to suggest that this is somehow an argument against it holding is illogical), then what I'm trying to get at is does all of that imply that the only possible shape for the Universe is spherical?
In other words, in a Universe based on the following axioms
1. the constants are as measured in our Universe
2. the Cosmological Principle holds
3. the Universe is finite in spatial size
is a sphere the only possible spatial topology?
Could somebody answer that?

 

[TrickyDicky]

Or for instance all the considerations about density or temperature of the universe, how exactly an infinite universe can have changes in those quantities mandated by changes in global size if at any moment the size is equally infinite? Are they different kinds of infinites?

 

[TrickyDicky]

OK. Personally, from my understanding of any real definition of 'infinity' (and, to be blunt, I don't understand how any physicist would happily think of 'infinity' as a potentially real physical thing) I cannot accept that anything can be 'infinite' as this has no meaning (to me, other than as a useful mathematical shorthand).

I disagree, the concept of infinity is naturally found in physics in many situations, and is threrefore as "real" as any other concept can be.

I can accept that the Universe may expand indefinitely, but, as I say, there is a huge difference between the statement that something is infinite and something has no definite end.
However, I can't accept that the Universe started as 'infinite'. And, my understanding of the Big Bang theory would be that expansion is one of the supporting arguments, as reversing it leads to a singularity (or whatever - you can forgive a layman's lack of knowledge on why the Big Bang point might not be called a 'singularity').
So, it seems to me (in my naivete) that the Universe (in its entirety) started from a finite point and as such must be still finite. In this case, if the Cosmological Principle holds (and it seems that it is more likely to hold given that it seems to hold in the observable Universe and to suggest that this is somehow an argument against it holding is illogical), then what I'm trying to get at is does all of that imply that the only possible shape for the Universe is spherical?
In other words, in a Universe based on the following axioms
1. the constants are as measured in our Universe
2. the Cosmological Principle holds
3. the Universe is finite in spatial size
is a sphere the only possible spatial topology?
Could somebody answer that?

If you use the FRW metric to model the universe and you demand a spatially finite geometry with k=1, then yes the hypersphere is the only spatial geometry allowed.

 

[Chalnoth]

Sure, I'm not discarding spatially infinite FRW metrics, just trying to understand how statements from the LCDM model make sense in the context of a spatially infinite universe. For instance, a common statement would be:"Approximately 10^−37 seconds into the expansion, a phase transition caused a cosmic inflation, during which the Universe grew exponentially." Now, how can something infinite in extent grow exponentially? No matter how I try to interpret it I can't find a sensible mathematical meaning for it.

Because you're not looking at it locally.

 

[TrickyDicky]

Because you're not looking at it locally.

Because the description I quoted is not looking at the universe locally, it's a cosmological model. Locally we only need GR.

 

[TrickyDicky]

Because you're not looking at it locally.

You are probably remarking here that the word "universe" in the sentence I quoted above really means "observable universe" which is obviously finite. o it is the observable universe that grew exponentially, right?

I'm aware of that but what I wanted to make evident is that if one applies the cosmological principle only to the observable universe, the true geometry of the whole universe doesn't really matter, and the BB model reduces basically to a model of the observable universe. But again without a good theory about initial conditions we have no reason to say that the cosmological principle only applies to a part of the universe.

 

[Chalnoth]

But again without a good theory about initial conditions we have no reason to say that the cosmological principle only applies to a part of the universe.

A universe where the cosmological principle applies globally is an incredibly low-entropy universe, much lower in entropy than one where the cosmological principle is only a local phenomenon (local in this sense being at least a few Hubble volumes).

 

[TrickyDicky]

A universe where the cosmological principle applies globally is an incredibly low-entropy universe, much lower in entropy than one where the cosmological principle is only a local phenomenon (local in this sense being at least a few Hubble volumes).

According to Steinhardt and Penrose (see Steinhardt video lecture where he explains it googling "Steinhardt pirsa"), the strategy to fix this, the inflationary models that only assume the cosmological principle"locally", actually have a much more intense low entropy problem, so it seems switching from a global CP to a local one is of no use wrt the low entropy problem.

Besides by the Copernican principle it would seem that only a global CP is acceptable.

 

[MathematicalPhysicist]

How certainly is the universe flat? Is is absolutely approved or not?
If yes, what will cause the big crunch?

I don't want to be too skeptical or philosophical, but if something has some sort of shape, doesn't it assume a prescribed space to our own space such that with it regard our universe is flat?

I mean if the universe has some shape, then it means that someone or thing can look on it from outside, doesn't it?

I mean I just say my own intuitive view of my experiences in the world.

 

[Chalnoth]

According to Steinhardt and Penrose (see Steinhardt video lecture where he explains it googling "Steinhardt pirsa"), the strategy to fix this, the inflationary models that only assume the cosmological principle"locally", actually have a much more intense low entropy problem, so it seems switching from a global CP to a local one is of no use wrt the low entropy problem.

Besides by the Copernican principle it would seem that only a global CP is acceptable.

It's difficult for me to know without seeing the specific wording used, but I don't think this has any relevance to the point I made. Whether or not inflation itself has an entropy problem is completely orthogonal to the entropy of the whole universe with or without a cosmological principle. Simply put, there are many, many more ways a universe can fail to obey the cosmological principle than it can obey one, so the overall entropy is much higher without it.

 

[Chalnoth]

I don't want to be too skeptical or philosophical, but if something has some sort of shape, doesn't it assume a prescribed space to our own space such that with it regard our universe is flat?

I mean if the universe has some shape, then it means that someone or thing can look on it from outside, doesn't it?

I mean I just say my own intuitive view of my experiences in the world.

Curvature in General Relativity is fully-described from inside the space-time. The "outside" view is just a visualization used to try to get us to understand what's going on.

 

[TrickyDicky]

It's difficult for me to know without seeing the specific wording used, but I don't think this has any relevance to the point I made. Whether or not inflation itself has an entropy problem is completely orthogonal to the entropy of the whole universe with or without a cosmological principle. Simply put, there are many, many more ways a universe can fail to obey the cosmological principle than it can obey one, so the overall entropy is much higher without it.

That is obvious but we needed the CP to constrain possible GR solutions, remember? No CP means no FRW model and no Friedman equations. I guess one could question even the local CP on observational grounds after last January's discivery of the "Huge Large Quasar Group" but you seem to be willing to doubt even basic FRW cosmology to save your point about the significance of the CP.

 

[usmhot]

I disagree, the concept of infinity is naturally found in physics in many situations, and is threrefore as "real" as any other concept can be.

Really? I'd be very interested to have one or two such examples given to me.

If you use the FRW metric to model the universe and you demand a spatially finite geometry with k=1, then yes the hypersphere is the only spatial geometry allowed.

Right. So is the main counter evidence to a finite spatially spherical universe the curvature measurements? Or are there other reasons to think this may not be the shape? Is there other compelling evidence for a spatially open and flat universe?

 

[TrickyDicky]

Really? I'd be very interested to have one or two such examples given to me.

You probably are thinking of infinity as a quantity, but as commented above by others there are different things that are meant by "infinite", some are close to the also not well or uniquely defined concept of "number" and some that have nothing to do.
In physics infinities as quantities are a sign that something is wrong when you obtain them as a result of a calculation, they are taken as nonsensical. See for instance the problem with infinities in QFT that is dealt with thru renormalization.
In this case since we are discussing the shape of the universe I am referring to infinity as a concept from topology and analysis and from differential geometry. In that sense the infinity concept of calculus is all over physics in as much as physics uses calculus and similarly with its extention to geometry as in differential geometry and its applications to GR and cosmology.
And you are right that "infinite" has nothing to do with "indefinite".

Right. So is the main counter evidence to a finite spatially spherical universe the curvature measurements? Or are there other reasons to think this may not be the shape? Is there other compelling evidence for a spatially open and flat universe?

As Chalnoth said in #64 curvature can only be measured if it is significantly non-zero. As long as that curvature is not measured and that can happen if it is 0 or very small, there is no compelling evidence to choose a compact(spherical) or non-compact(flat or hyperbolic) topology
The fact is that we observe a universe that if it is not exactly flat, must have a quite small curvature, either positive or negative. This in the FRW model is related to a parameter called critical density, and a close to flat curvature observation corresponds to a value close to the critical density (this density is the energy density).
Cosmologists consider there's compelling evidence for a flat universe due to something called the "flatness problem" combined with the above mentioned observations. http://en.wikipedia.org/wiki/Flatness_problem.