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

[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.

 

[TrickyDicky]

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.

If time to the singularity is considered finite it can never require an infinite degree of fine tuning, but I can see how this could conflict with a spatially infinite geometry together with a global CP as I mentioned in my previous post.
But I just can't understand how exactly the horizon problem came to be considered a problem in the first place(unless the above mentioned conflict was also evident at the time), the three possible space geometries allowed by the FRW model are both geometrically and physically homogeneous by definition, so if one is going to use this model one is assuming that homogeneity and shouldn't worry about causal contact.

 

[Chalnoth]

But I just can't understand how exactly the horizon problem came to be considered a problem in the first place(unless the above mentioned conflict was also evident at the time),

Because there's no a priori expectation of the cosmological principle necessarily being true: it only makes sense if the physics of the early universe set things up that way, and the horizon problem points out that if you just take GR with the observed components of the current universe, it is impossible for any physical process to set up a universe that is approximately homogeneous and isotropic.

This is one of the reasons why inflation was proposed, but inflation doesn't extend the distance at which we expect the cosmological principle to hold out to infinity.

the three possible space geometries allowed by the FRW model are both geometrically and physically homogeneous by definition, so if one is going to use this model one is assuming that homogeneity and shouldn't worry about causal contact.

I think you're too attached to the FRW model. It's just a model. It's not reality.

 

[usmhot]

... There are no trivial questions, feel free to ask anything.

OK. Well, here goes ... I'm prepared for some scoffing.

The wiki article Flatness_problem proceeds to this equation
(Ω$^{-1}$ - 1)ρa$^{2}$ = -3kc$^{2}$/8∏G
with the claim that all of the terms on the RHS are constants.

Now, here comes the silly part ... are they? Constant, I mean. Really? Are we sure?

Take, for example, ∏. Is this supposed to be exactly and only the value of ∏ as calculated mathematically? Or is it supposed to be the ratio of circumference to radius of a circle in the universe in question AND at the time in question? So, if the Universe is spatially curved then wouldn't ∏ potentially be different to the value that we have calculated in a flat (Euclidean) geometry and use in our current equations? And, in particular, wouldn't it change value as the Universe expands causing the curvature to change?

So, take k as well. The wiki article says

k is the curvature parameter — that is, a measure of how curved spacetime is

but, if the Universe is finite and expanding then wouldn't that change the value of k over time as well?

And, then, onto the big one - c. Isn't it conceivable that the speed of light has changed as the Universe has expanded.
I completely accept the important position of the theories of relativity to modern physics (and I even understand the theories to a limited extent myself - particularly special relativity). But, as I understand it, the constancy of the speed of light as used in (special) relativity is with respect to different inertial frames of reference. But that doesn't mean that the speed of light measured in an earlier epoch(?) of the Universe would have to be the same value as now, does it? In one sense, it reads to me as different frames of reference in a 'static' universe.
Is it possible that the speed of light is a function of the curvature of space? So, that in the early Universe, when the curvature was extremely high the speed of light would have been much different (smaller?) to now. Obviously that would greatly affect our measures for the age and size of the Universe, but it might also provide an alternative to 'inflation' and it would also explain why the speed of light is a limit, as now the limit is actually imposed by the topology/structure of the Universe.
Is there any evidence to suggest that the speed of light is different for different values of curvature? For example, light is 'bent around' very massive objects such as galaxies - is this not the same as saying that light is refracted by very massive objects? Does such 'refraction' of light imply a velocity change in the region of the massive object, i.e. the region of (locally) different spatial curvature?

 

[Chalnoth]

Take, for example, ∏. Is this supposed to be exactly and only the value of ∏ as calculated mathematically? Or is it supposed to be the ratio of circumference to radius of a circle in the universe in question AND at the time in question?

The value of $\pi$ is independent of the universe. It's just a transcendental mathematical number, and is no less constant than the 3 or 8 in that formula.

The speed of light is, to the best of our knowledge, also constant.

The spatial curvature, k, is a constant that is a way of encapsulating the relationship between the expansion rate and the energy density of the universe. The value of k doesn't change because of how we define the term.

 

[TrickyDicky]

OK. Well, here goes ... I'm prepared for some scoffing.

The wiki article Flatness_problem proceeds to this equation
(Ω$^{-1}$ - 1)ρa$^{2}$ = -3kc$^{2}$/8∏G
with the claim that all of the terms on the RHS are constants.

Now, here comes the silly part ... are they? Constant, I mean. Really? Are we sure?

Take, for example, ∏. Is this supposed to be exactly and only the value of ∏ as calculated mathematically? Or is it supposed to be the ratio of circumference to radius of a circle in the universe in question AND at the time in question? So, if the Universe is spatially curved then wouldn't ∏ potentially be different to the value that we have calculated in a flat (Euclidean) geometry and use in our current equations? And, in particular, wouldn't it change value as the Universe expands causing the curvature to change?

So, take k as well. The wiki article says
but, if the Universe is finite and expanding then wouldn't that change the value of k over time as well?

And, then, onto the big one - c. Isn't it conceivable that the speed of light has changed as the Universe has expanded.
I completely accept the important position of the theories of relativity to modern physics (and I even understand the theories to a limited extent myself - particularly special relativity). But, as I understand it, the constancy of the speed of light as used in (special) relativity is with respect to different inertial frames of reference. But that doesn't mean that the speed of light measured in an earlier epoch(?) of the Universe would have to be the same value as now, does it? In one sense, it reads to me as different frames of reference in a 'static' universe.
Is it possible that the speed of light is a function of the curvature of space? So, that in the early Universe, when the curvature was extremely high the speed of light would have been much different (smaller?) to now. Obviously that would greatly affect our measures for the age and size of the Universe, but it might also provide an alternative to 'inflation' and it would also explain why the speed of light is a limit, as now the limit is actually imposed by the topology/structure of the Universe.
Is there any evidence to suggest that the speed of light is different for different values of curvature? For example, light is 'bent around' very massive objects such as galaxies - is this not the same as saying that light is refracted by very massive objects? Does such 'refraction' of light imply a velocity change in the region of the massive object, i.e. the region of (locally) different spatial curvature?

The three parameters are constant in that formula, c and $\pi$ are obviously constant, now k here is referring to the normalized curvature that is normally used in the Friedmann equations and in the FRW line element it can only be 1, 0, or -1. The evolution of positive or negative spatial curvature is then integrated in the scale factor a.