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

[usmhot]

OK. Had to do some reboning ... in my distant memory I had mixed up the fact that a torus degenerates into a sphere to the belief that it is equivalent, which, of course it isn't.

And, indeed, I understand now how a torus is topologically equivalent to a flat plane, but is a closed unbounded surface. So, a torus (or any topologically equivalent surface?) could be a valid shape for a finite unbounded flat Universe.

I accept that there might be some question of the actual validity of the Cosmological Principle, but, for the sake of discussion, let's assume it holds at the very large scale. Is a torus shaped Universe consistent with the Cosmological Principle? If not, what shape(s) are?

 

[Chalnoth]

OK. Had to do some reboning ... in my distant memory I had mixed up the fact that a torus degenerates into a sphere to the belief that it is equivalent, which, of course it isn't.

And, indeed, I understand now how a torus is topologically equivalent to a flat plane, but is a closed unbounded surface. So, a torus (or any topologically equivalent surface?) could be a valid shape for a finite unbounded flat Universe.

I accept that there might be some question of the actual validity of the Cosmological Principle, but, for the sake of discussion, let's assume it holds at the very large scale. Is a torus shaped Universe consistent with the Cosmological Principle? If not, what shape(s) are?

Almost, but not quite. A torus shape is fundamentally anisotropic, because you can return to your previous position in a rather short distance only if going in very specific directions.

To fully satisfy the cosmological principle, you need a sphere or a plane (or a much more complicated shape for negative curvature).

 

[usmhot]

... A torus shape is fundamentally anisotropic, because you can return to your previous position in a rather short distance only if going in very specific directions.

To fully satisfy the cosmological principle, you need a sphere or a plane (or a much more complicated shape for negative curvature).

Right. That makes sense.

So, if the cosmological principle holds then there are only two possibilites - the Universe is either a sphere or it is infinite?

 

[George Jones]

And, indeed, I understand now how a torus is topologically equivalent to a flat plane

A flat torus is not topologically equivalent to a flat plane. A torus is formed from a plane by topological identifications (i.e. a torus is a quotient space). See my attachment. The identifications change the topology.

A flat plane is simply connected, while a flat plane is not simply connected. Every closed curve in the plane is contractible to a point. A closed curve that loops around the "small" circumference of the torus is not contractible to a point.

A torus is a homogeneous space, but not isotropic. A flat plane is homogeneous and isotropic.

Even though the torus in my attachment looks curved, it is actually flat.

[edit]Chalnoth also posted about this.[/edit]

 

[George Jones]

Almost, but not quite. A torus shape is fundamentally anisotropic, because you can return to your previous position in a rather short distance only if going in very specific directions.

To fully satisfy the cosmological principle, you need a sphere or a plane (or a much more complicated shape for negative curvature).

So, if the cosmological principle holds then there are only two possibilites - the Universe is either a sphere or it is infinite?

Topologically, yes, i.e., the plane and the space of negative curvature used in FLRW universes are topologically equivalent.

 

[usmhot]

Topologically, yes, i.e., the plane and the space of negative curvature used in FLRW universes are topologically equivalent.

OK. Thanks.

So, can a spatially infinite Universe come from a Big Bang?

 

[Chalnoth]

OK. Thanks.

So, can a spatially infinite Universe come from a Big Bang?

Depends upon the model.

But whichever way you slice it, there's still no reason to believe that the cosmological principle holds at scales much larger than our horizon.

 

[usmhot]

Depends upon the model.

But whichever way you slice it, there's still no reason to believe that the cosmological principle holds at scales much larger than our horizon.

Well, there's one reason to suggest it - if doesn't hold everywhere then it's likely not to hold in our observable Universe, right? (As you implied yourself

We expect that the cosmological principle must hold significantly beyond the observable universe primarily because if it didn't, we would expect to see some deviation within it as well.

But, anyway, just exploring some thoughts ... assume the following axioms (as well as the laws and constants as measured in our observable Universe)
1. The Universe started from a Big Bang singularity
2. The Cosmological Principle holds

can you explain how an infinite Universe can exist?

 

[Chalnoth]

Well, there's one reason to suggest it - if doesn't hold everywhere then it's likely not to hold in our observable Universe, right? (As you implied yourself

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.

I would also like to point out that the cosmological principle is most certainly not a precise description of even our own, visible universe: there are differences in density from place to place.

But, anyway, just exploring some thoughts ... assume the following axioms (as well as the laws and constants as measured in our observable Universe)
1. The Universe started from a Big Bang singularity
2. The Cosmological Principle holds

can you explain how an infinite Universe can exist?

This is insufficient. The problem is that a Big Bang singularity is ill-defined (and also nonsensical).

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.

 

[Passionflower]

I don't think it is a simple assumption that is probably wrong, it is "the" assumption that sustains the LCDM model(including the flat space, inflation and dark matter and dark energy assumptions) and GR's FRW metrics.

Until I see some mathematical proof that a mostly void space with extremely small but massive lumps of matter can accurately be modeled by an FRW metric I would stay skeptical.

 

[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.

 

[Chalnoth]

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.

The cosmological principle obviously holds, to a high degree of accuracy, within our own horizon. That is all that is required to apply FRW.

The statement that the cosmological principle constrains the possible GR solutions is just a statement of the fact that we know how to solve the GR equations in that situation: GR is such that only a few simple space-times with a high degree of symmetry have been solved. But just because we don't yet know how to solve the equations for more complicated space-times doesn't mean that more complicated space-times don't exist.

 

[TrickyDicky]

The cosmological principle obviously holds, to a high degree of accuracy, within our own horizon. That is all that is required to apply FRW.

In the FRW model the CP holds everywhere except obviously at the singularity, mathematically the model doesn't make a distinction about any observational horizon in that respect. A different thing is that you may decide to apply the model only to the observable universe for practical reasons.
But the LCDM parametrization of the FRW model includes cosmic times much earlier than the CMB radiation observable limit, and in those early cosmic times the CP also must hold if only for the sake of the logical congruence of the mathematical model.
If you think otherwise please cite a textbook or peer-reviewed journal reference where it is explicitly stated that the CP doesn't hold outside our horizon.

 

[Chalnoth]

In the FRW model the CP holds everywhere except obviously at the singularity,

Yes, but any deviation from pure FRW that is beyond our cosmological horizon cannot be measured. This means that FRW can be used no matter what the state of the universe at scales beyond the horizon.

But the LCDM parametrization of the FRW model includes cosmic times much earlier than the CMB radiation observable limit, and in those early cosmic times the CP also must hold if only for the sake of the logical congruence of the mathematical model.

Extending the model back in time doesn't extent the in principle observable universe infinitely. But where do you get the idea that the cosmological principle must hold at distances beyond our observable horizon for "logical congruence of the mathematical model"? Where did you get that idea from?

If you think otherwise please cite a textbook or peer-reviewed journal reference where it is explicitly stated that the CP doesn't hold outside our horizon.

Nobody is going to say that it certainly doesn't hold, because there has been no detection of any deviation on super-horizon scales. There probably can't be either. But many cosmologies have been proposed that violate the cosmological principle globally to varying degrees, such as eternal inflation and the string theory landscape.

And I'd also like to mention that the cosmological principle is only approximate within our own universe anyway: there are deviations from homogeneity and isotropy at all length scales within our observable universe.

 

[usmhot]

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".

With all due respect, I don't think you answered my question. I'm aware of the use of 'infinity' as a limit in Calculus, and as such how it would often be found in the equations that describe the Universe. However, I thought you implied that there were specific observable or describable infinities or infinitesimals and wanted to know of some examples. The use of the 'infinity' shorthand in mathematics does not imply the actuality of a real infinity.

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.

I'm having a problem understanding this. In the context that you're describing here I have always been under the impression that the physicists were talking about the curvature of the 4d space-time surface. The first Freidmann equation referenced in the wiki article involves the rate of expansion which (I presume) involves the time dimension.
As I said before, I have no problem with a universe that expands indefinitely in a perfect balance between the rate of expansion and the density of the total energy, and how this is 'flat'.
However, surely the spatial shape of such a universe could well be spherical.
Which brings me to another question ... is it conceivable that the processes / mechanisms used to determine the curvature are, necessarily, determining that of space-time rather than just space?

 

[Chalnoth]

As I said before, I have no problem with a universe that expands indefinitely in a perfect balance between the rate of expansion and the density of the total energy, and how this is 'flat'.
However, surely the spatial shape of such a universe could well be spherical.

Actually, it's the other way around. The expansion itself is a manifestation of space-time curvature. So a universe with an energy density equal to the critical density (meaning the energy density is in some sense balanced by the rate of expansion) has no spatial curvature, but has significant space-time curvature related to the expansion.

Basically, when you compute the space-time curvature of a FRW universe, you get two terms. One is related to the expansion rate, while the other is related to the spatial curvature.

 

[TrickyDicky]

Yes, but any deviation from pure FRW that is beyond our cosmological horizon cannot be measured. This means that FRW can be used no matter what the state of the universe at scales beyond the horizon.

Right, this was conveyed in the sentence following the one you quoted.

Extending the model back in time doesn't extent the in principle observable universe infinitely. But where do you get the idea that the cosmological principle must hold at distances beyond our observable horizon for "logical congruence of the mathematical model"? Where did you get that idea from?

Again, from the FRW metrics, the Copernican principle and the Friedmann equations that govern the dynamics of expansion for homogeneous and isotropic spacetimes at any cosmic time, that is what the model says, your comments about when the CP should hold and when it shouldn't are your personal opinion and purely speculative. you haven't mentioned a single reason that allows us to depart from the mathematical model other than your preferences about what happens in regions we cannot measure, that is what I call not being congruent with the model.

Nobody is going to say that it certainly doesn't hold, because there has been no detection of any deviation on super-horizon scales. There probably can't be either.

Well you said that the CP was probably wrong well outside our horizon and that the model didn't require te CP to hold there. Both statements seem unwarranted and speculative to me just by looking at the math of the model.

But many cosmologies have been proposed that violate the cosmological principle globally to varying degrees, such as eternal inflation and the string theory landscape.

That's for sure, and many others even more exotic, but we are here discussing the mainstream model. AFAIK, the eternal inflation multiverse with no beginning nor end is not included in the LCDM model, that relies on new inflation.

And I'd also like to mention that the cosmological principle is only approximate within our own universe anyway: there are deviations from homogeneity and isotropy at all length scales within our observable universe.

Sure, this is understood. Only true breakdowns of the CP are considered here.

 

[TrickyDicky]

I thought you implied that there were specific observable or describable infinities or infinitesimals and wanted to know of some examples. The use of the 'infinity' shorthand in mathematics does not imply the actuality of a real infinity.

Ok, then please define "real infinity" and what you understand by its actuality.

As I said before, I have no problem with a universe that expands indefinitely in a perfect balance between the rate of expansion and the density of the total energy, and how this is 'flat'.
However, surely the spatial shape of such a universe could well be spherical.

It would be spherical if the ratio of current density to critical density was >1.

 

[usmhot]

Ok, then please define "real infinity" and what you understand by its actuality.

in response to

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).

you wrote ...

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

I'm afraid I took rather a literal interpretation of the word 'naturally' and assumed you were saying there were real occurring sets (of 'things') of infinite size.

I cannot define "real infinity" in any way outside of mathematics, because it is simply a mathematical shorthand - a concept, as you yourself have said. And, though any number is also a concept in a similar way, in a significantly different way it has a natural or real analog in physicality. As a trivial example, the concept '2' can be readily demonstrated with the aid of oranges (or indeed any fruit of the day) :).

My fundamental point is, I do not accept an infinite universe simply because I believe this is mixing two different things - a concept with an actual physical reality (like apples and oranges, if you will ;) ).

I have a problem with it on other scores too. One of them being that if the Universe has expanded from a hot, dense state (as so lyrically put on that well-known TV show) then going backwards from that was hotter and denser - implying smaller. But smaller than infinite is either still infinite, in which case how could it be denser, or finite, in which case how could it possibly become infinite?

Anyway, I've been reading, with interest, the description given in the wiki article referenced above (Flatness_problem) and I have some, probably quite naive, queries about the reasoning and assumptions - probably too trivial to bother the readers at large around here with, so I'd be very appreciative if someone could pm me for an offline conversation about it.

 

[Chalnoth]

Again, from the FRW metrics, the Copernican principle and the Friedmann equations that govern the dynamics of expansion for homogeneous and isotropic spacetimes at any cosmic time, that is what the model says, your comments about when the CP should hold and when it shouldn't are your personal opinion and purely speculative.

Hardly. The cosmological principle holding to infinity requires an infinite degree of fine-tuning: how did the universe, out to infinite distances, know to be the same density in all locations, with the appropriate time-slicing?

It's rather like the horizon problem, but expanded to infinite distances instead of merely being required to hold in our visible universe.

Another way of stating the problem is to look at the classic model of inflation. If inflation were extended infinitely into the past, then inflation could easily explain a global cosmological principle. However, we know that can't be the case: extending inflation infinitely into the past also requires infinite fine-tuning: inflation predicts a singularity somewhere in the finite past, and the further back you try to push that singularity, the more fine-tuning you need. And if inflation can only be extended a finite distance back into the past, then it isn't possible for the universe as a whole to have reached any sort of equilibrium density, as if you go far enough away, you'll eventually reach locations that have always, since the start of inflation, been too far for light to reach one another. Any regions of the universe that lie beyond this distance aren't likely to be remotely close to one another in density.

Of course, this argument is based upon the assumption that a simplistic model of inflation is true, but the argument is reasonably-generic among most inflation models.

 

[TrickyDicky]

... if the Universe has expanded from a hot, dense state (as so lyrically put on that well-known TV show) then going backwards from that was hotter and denser - implying smaller. But smaller than infinite is either still infinite, in which case how could it be denser, or finite, in which case how could it possibly become infinite?

As I said in a previous post I also find there is a difficulty explaining this and no one has answered it satisfactorily except to say that this should be approached locally, but I can't see how that approach can lead to infer that an infinite expanding space should get hotter and denser going back in time, since no matter how much closer geodesics get they will always be infinetely far from the singularity. While in the spatially finite case this difficulty doesn't come up.

Anyway, I've been reading, with interest, the description given in the wiki article referenced above (Flatness_problem) and I have some, probably quite naive, queries about the reasoning and assumptions - probably too trivial to bother the readers at large around here with, so I'd be very appreciative if someone could pm me for an offline conversation about it.

That's what these forums are for. There are no trivial questions, feel free to ask anything.