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

[Athanasius]

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

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

 

[bobie]

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

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

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

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

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

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

 

[Athanasius]

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

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

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

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

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

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

 

[Simon Bridge]

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[bobie]

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

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

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

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

These are only the main obscure points.

 

[Athanasius]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[Athanasius]

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

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

 

[Mordred]

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

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

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

These are only the main obscure points.

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

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

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

 

[bobie]

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

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

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

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

 

[Mordred]

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

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

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

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

the event horizon however accounts for expansion

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

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

 

[bobie]

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

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

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

 

[Mordred]

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

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

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

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

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

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

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

 

[bobie]

I did not confirm that the universe is finite.

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

 

[Mordred]

This is from the link you provided

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

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

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

no worries this is another common confusion

 

[bobie]

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

 

[Mordred]

correct the hot big bang model has no clue how the universe started, there is no agreement on that issue

the hot big bang model only states we had a hot dense state near the beginning after 10^-43 sec

 

[Athanasius]

Since Simon may not have time to answer, would someone else well acquainted with cosmology please answer this question, which is related to my first post? Simon said that the math is simpler with a flat infinite universe. I suppose that would completely do away with having to deal with an expanding edge. Is it primarily the problem of the edge that makes the math easier with an infinite flat universe, as opposed to a finite flat universe? If there are other aspects that make the math easier, I am also curious to know. And would others agree that since that results in a model with simpler math, Occam's razor inclines us to prefer an infinite flat model over an finite flat one? Lastly, besides the easier math, are there any observational reasons to prefer an infinite flat universe over a finite flat one?

 

[bobie]

since that results in a model with simpler math, Occam's razor inclines us to prefer an infinite flat

Occam's razor has nothing to do with maths:

It states that among competing hypotheses, the one with the fewest assumptions should be selected.

It refers to similar , simpler, equivalent theories

Philosophers also point out that the exact meaning of simplest may be nuanced.

It is an issue of philosophy and philosophy of Science

 

[Athanasius]

From the Wikipedia article at http://en.wikipedia.org/wiki/Occam's_razor#Science_and_the_scientific_method

In science, Occam's Razor is used as a heuristic (rule of thumb) to guide scientists in the development of theoretical models rather than as an arbiter between published models.

I would qualify this to say that is how it usually should be used, as most of us would typically prefer the simplest and most elegant explanation when choosing between two equally plausible models. That does not mean that the simpler model is the correct one, however.

I have often heard Occam's razor being applied wrongly (especially in web forums), as though it were evidence against a model.

Also from the article:

However, appeals to simplicity were used to argue against the phenomena of meteorites, ball lightning, continental drift, and reverse transcriptase.

The simplest answer is not always the correct one. Like any razor, you can nick yourself with it quite badly if you don't use it right!

I believe this relates to the post, since the post is about the less widely accepted and lesser known idea of a finite universe where Omega equals one.

 

[nuclearhead]

It could be torus shaped. i.e. like the game asteroids. Which is flat but if you go off one side of the screen you end up on the other side. Right?

 

[Simon Bridge]

It is hard to fathom why this is still going... these questions have been answered several times in the previous thread but ho well, obviously not well enough. On last go...

as it was stated in post #2, the choice of a flat and infinite U(niverse) has been made to dodge the problem of the edge, on the false assumption that a curved U must have an edge while flat one may not.

Please quote the bit of post #2 where I write that a curved spacetime must have an edge.

But the remedy is worse than the cure, ...

Since a cure has yet to be proposed, we are pretty much stuck with the remedy.

The context of post #2 was the FLRW Universe (spelled out int hat post, and follows from post #1).

Please see:
http://preposterousuniverse.com/grnotes/grtinypdf.pdf
... this is a crash-course in GR, the final chapter deals with FLRW spacetime and the implications of the omega factor in more rigorous terms.

I hope Mordred would care to clarify these simple points of the theory:

I'm sure Mordred won't mind if I have a go too?

- infinite U means infinite space or also infinite mass?

Please read up about the FLRW Universe (Link above.)
... a uniform density over infinite volume implies the mass is, in common-language - infinite.

...if mass is finite, is it distributed on infinite radius?

The term "radius" implies there must be a center to have a radius from. Since there is no center, the question is meaningless. It is a common trap.

Let me help: the infinite mass is distributed over an infinite volume in such a way as to have a finite mass density. Better?

The FLRW Universe starts out by making assumptions about that density.
It's kinda the whole point. (see the link above for details)

- infinite means no-shape?

Define "shape".
In GR - the Universe is a 4D object not embedded in any larger dimensionality - which should make the concept of shape quite tricky even for finite Universes.

An infinite universe has a geometry described by it's metric. (see the link above for details)

- an infinite but curved U[niverse] is worse than flat one?

For a scientific definition of "worse" yes. The maths is harder to get the same value predictions: why bother?

But more to the point: curved spacetime is not ruled out (re post #2 say) by being inconvenient but by being excluded as a solution to the Friedman equations which have gamma=1.

If you want to discuss curved spacetimes, you should start a new thread (after a bit of reading - see link blah blah.)

- infinite + c = infinite or not?, if so, how can U expand?

Define "c".
If c is taken to be a finite number, then the question is meaningless: you cannot add a constant to infinity like that without more care.

naively: infinity+1 = infinity.
This causes the kind of mess that Cantor worked on.
http://www.c3.lanl.gov/mega-math/workbk/infinity/inbkgd.html

We avoid this in cosmology by never adding a finite anything to anything infinite.

...if the rate of expansion is over 3c at a certain distance what is that rate at infinite distance?

Cosmological expansion is a local phenomenon. In the FLRW Universe, the rate of expansion is the same everywhere.

- if U is (flat and) infinite right now, what is the meaning of the radius of U being now 14 Gly?

That's the age of the Universe. Big-bang cosmology proposes that the Universe had a beginning in time but may be infinite in space.

The infinite-flat FLRW Universe models one such geometry.

- was [the] U[niverse] infinite even before Big-Bang?

In the FLRW model, an infinite-flat Universe would have had to be infinite in the plank epoch ... so simple answer: "yes" - with reservations depending on what you think the words "Big Bang" mean.
Your questions suggest you may not be thinking of the same thing as me.

... if [the Universe] was [infinite before the big bang]: what is the use of this theory if it says that "even space and time did not exist before BB" , then:
... if [the Universe] was not [infinite before the big bang]: how can it be infinite now after only 14 G-years?

It is difficult to parse this question... I have put in square brackets and redone some punctuation to see if that helps - please let me know if this is not what you intended.

The second part is how can something finite become infinite in a finite time ... this is not impossible with maths - consider: y=tan(πt/2) starts from 0 at t=0 and becomes infinite in the finite time t=1. However, I don't need to go into this in more detail since the model being discussed does not propose that the Universe started out finite.

The first part seems to be asking how an infinity could have existed before the big bang if space and time did not exist before than.

This has some issues.
1. the big bang is usually taken to be the start of the rapid expansion: the Universe already existed then. i..e there was space and time before the big bang.
http://en.wikipedia.org/wiki/Big_Bang#Timeline_of_the_Big_Bang

2. by your own arguments, getting any finite something from nothing is as big-a problem as getting infinite something from nothing - they both involve, for example, an infinite percentage increase in the amount of matter and energy and spacetime. (Though "before" and "after" are problematical concepts when you are talking about time itself.) Mathematically it does not matter.

3. GR is a theoretical framework for describing space-time once it exists, it tells us nothing about conditions in the absence of a Universe.

In order to deal with the transition from nothing to space-time we need a theoretical framework beyond that supplied by GR which is off-topic for this discussion. This discussion concerns the ratio "omega" and it's relation to a particular set of theoretical models.

I quite like closed models for the Universe myself - sadly the Universe I find myself in does not care what I find appealing.

These are only the main obscure points.

These points represent common confusions experienced by beginning students of maths and cosmology and mostly come from mixing up different models, and not appreciating what happens to arithmetic when infinity is involved. These are confusions that get repeated a lot in the junk-science writings so it is important to take care with this sort of reasoning.

You would have more chance of sorting out your confusion if you stayed on topic: re: FLRW Universe with omega=1. Discuss other models in other threads - there are many already.

Other models say different things.
Don't mix them up.

Applying the FLRW Model to this Universe... which is where the interest lies after all:
The FLRW Universe is telling us that the flatness of the observed Universe suggests that we are in an epoch close to the threshold curvature between open and closed global topologies.

Simple-closed topology is "spherical".
Simple open topology is "hyperbolic"
Right between those two you have an infinite plane.
Happens to have easy(er) maths.
It's not hard to understand.

Ergo - it is a reasonable thing for a pop-science show to say.

If you prefer flat and finite, then you are welcome to do all your maths in toroidal spacetime if you really want to.
http://arxiv.org/pdf/gr-qc/0411014.pdf

... also see:
https://www.physicsforums.com/showthread.php?t=237353
Enjoy.

 

[Athanasius]

Right between those two you have an infinite plane.

Or, as we have discussed, a flat but finite space with more difficult math.

Simon, I was hoping to get your opinion regarding this. Is it primarily the problem of an expanding edge that makes the math easier with an infinite flat universe, as opposed to a finite flat universe? And what particular aspect of the math of an edge is so challenging?

(If you are not American, please pardon my use of the word "math" rather than "maths". I must be true to myself. It just does not look right when I write "maths"!)

 

[bobie]

. Is it primarily the problem of an expanding edge that makes the math easier with an infinite flat universe, as opposed to a finite flat universe?

I do not know about math(s), but the edge raises formidable conceptual problems.
U is being, what is, all-that-exists, beyond the edge of what-is can only be what-is-not, The Nothing, which does not exist and cannot do anything, let alone bind.

Apart from philosophical formulation: what happens when a spaceship reaches the edge? can it approach it? will it rebound? can it trespass? what happens to energy/matter?, does it simply vanish? what keeps energy/matter inside the edge? ... ...
and so on and so forth.
Most likely both theoretical and technical problems have very simple solutions because they are ill-framed, they are false problems, because U has indeed an edge, but man is unable to conceive a different formulation, cannot conceive the absolute.
The parallel with the Earth is misleading, as the surface has indeed no edge, but only on 2 dimensions. Downward the edge is the crust and upwards ditto when man couldn' t fly, now is the edge of U.

It is simple to dodge all these unsolvable problems, just with a handwave and say: U is infinite, (or , when pressed)... but we really do not know.

As to math(s), wait for Simon.

 

[PeterDonis]

If the Universe is finite and flat, then it must have an edge

Is this actually true? A flat torus universe would have finite spatial volume but no edge.

I think you may be implicitly making an analogy with the case of a 2-dimensional surface embedded in 3-dimensional space. If the Earth's surface were flat but finite in area, it would have to have an edge, because there's no way to embed a flat 2-torus in 3-dimensional space. But there's no reason to impose that kind of restriction on the spatial slices of the universe as a whole, because the universe doesn't have to be embedded in any higher dimensional space.

[Edit: a flat torus universe with no edge would still not be isotropic, as Bill_K confirms in a later post.]

 

[julcab12]

I do not know about math(s), but the edge raises formidable conceptual problems.
U is being, what is, all-that-exists, beyond the edge of what-is can only be what-is-not, The Nothing, which does not exist and cannot do anything, let alone bind.
It is simple to dodge all these unsolvable problems, just with a handwave and say: U is infinite, (or , when pressed)... but we really do not know.

As to math(s), wait for Simon.

...I think much has been answered (Simon/Modred/etc). They are talking about model dependent/ local of which we have 'time'/cosmic time/time-clock relation/ global time linked to motion modeled to the best of what we can in SR interpreted in GR and extrapolate to observational data's.. U as infinite is what the model predicts based from what they have so far. It's a good thing though. Infinity is just a way of telling that a model is incomplete or a hint to something new. ... I wonder how would you define an edge?

...I think the confusion is the way you conceptualized nothing/edge. It is much easier if you're imagining 'everything' is contained in everything(no edge/beginning or origin) and focus on its context and DISCOVER its dynamics(like what they do in physics). It will come in handy when dealing with infinity. So far to make an intuitive sense on the concept of infinity. It must contain some variable to make a bound system work. The only thing i could think of is BOUNCE.
... Things change relative to a/ time(conventional understanding) as a construction(to some)-if you will. We understand the universe for what we think it should be in hopes that it will obey our interpretation of nature/math. And for what it really is, remains neutral or unknown for now.

 

[julcab12]

.

Apart from philosophical formulation: what happens when a spaceship reaches the edge? can it approach it? will it rebound? can it trespass? what happens to energy/matter?, does it simply vanish? what keeps energy/matter inside the edge? ... ...
and so on and so forth.
Most likely both theoretical and technical problems have very simple solutions because they are ill-framed, they are false problems, because U has indeed an edge, but man is unable to conceive a different formulation, cannot conceive the absolute.

.

... Nature won't allow that or you can't make an absolute postulate. We construct a mental picture of edge/center bec we put constraint to any given medium. E.x. An object such as pencil is a bounded thing. We identify it's edge as it's head/tip limited to constraint of the structure which is from the tip to the head. The universe doesn't apply to this principle. However we can assume a formulation of an edge 'IF' we put 'constraint'(the same as we did with the pencil) on the OBSERVABLE PART(not the whole isotropic and homogeneous universe) in relation to observer. The center would be any observer and the edge is in any point in the Observable universe or observable universe itself.

 

[Bill_K]

Is it primarily the problem of an expanding edge that makes the math easier with an infinite flat universe, as opposed to a finite flat universe? And what particular aspect of the math of an edge is so challenging?

I don't know why on Earth anyone would find a finite flat universe more appealing than an infinite one. By making it finite, you lose isotropy. And that is a heavy price to pay.

A finite universe is constructed from an infinite one by identifying points, so that if I travel in any particular direction I will eventually come back to my starting point. The galaxies seem to repeat themselves, like in a crystal lattice. But just like a solid crystal, the universe must then have principal axes - there is no way to make the periodicity the same in all directions.

 

[old dog]

I don't know why on Earth anyone would find a finite flat universe more appealing than an infinite one. By making it finite, you lose isotropy. And that is a heavy price to pay.

If you have an infinite universe you will have an infinite number of 15 Gly regions of approximately the same composition. An infinite number will be very, very similar to the one we inhabit. There will be some where a close, if not exact, copy of you and I exist. It may well be true scientifically but I find it philosophically disturbing.

Isotropy? I don't keep up on everything but I don't think the "Axis of Evil" anomaly has been completely resolved.

There are still a lot of loopholes before we declare the universe infinite.

Do we have an adequate theory of fundamental physics to declare which "constants" are truly immutable and eternal or merely slowly varying. Omega could well be asymptotically approaching 1 but never get there. On an experimental level, can we ever distinguish 1.0000000000000001, 0.9999999999999999 and 1.0000...?

 

[marcus]

I don't know why on Earth anyone would find a finite flat universe more appealing than an infinite one. By making it finite, you lose isotropy. And that is a heavy price to pay.

A finite universe is constructed from an infinite one by identifying points, so that if I travel in any particular direction I will eventually come back to my starting point. The galaxies seem to repeat themselves, like in a crystal lattice. But just like a solid crystal, the universe must then have principal axes - there is no way to make the periodicity the same in all directions.

Good point about loss of isotropy! Also nice to hear the issue of how one thinks of the U treated as a matter of taste, which model one finds *more appealing*.

I don't believe it makes any different to real world computations (where there is always a limit on precision) whether one assumes Ω exactly = 1, or instead something like 1.000001.

That would correspond to a U which is spatially a 3D sphere. And the 3D spatial slice (a "hypersphere") would currently have a radius of curvature of 14400 billion light years.

In effect, no one could tell it from flat And with that one still has isotropy.

So it is a question of personal taste. Do you like the mathematical exactitude of Ω=1 and hopefully have the self-restraint to avoid drowning in philosophical infinity?
Or do you prefer to imagine a slightly curved very nearly flat space, while doing all your calculations as if space were perfectly flat. A very slight positive curvature is not going to change the answers--since its effect on the equations will be too small to include.

Oh, I guess it makes a difference to modeling the early universe. I had forgotten about that.

 

[bobie]

nice to hear the issue of how one thinks of the U treated as a matter of taste, which model one finds *more appealing*.

Hi marcus, just imagine what your favourite wiseman, Anassagoras, would say to that. (what? me worry?)
Ancient sophoi discovered the truth because they followed the laws of necessity, the laws of Being: it is so because it must be so, it can only be so, and it's a miracle that it can be at all. The basic laws of Nature contrast whit what Popper said, there is ony one solution, nay, sometimes there is no solution, and Nature finds the impossible solution.

That would correspond to a U which is spatially a 3D sphere. And the 3D spatial slice (a "hypersphere") would currently have a radius of curvature of 14400 billion light years.
In effect, no one could tell it from flat And with that one still has isotropy.

That's amazing, marcus, can you expand on that, how did you find that magic value? the likely size of U ≈10³¹ cm. That's just what I was describing a few posts ago!
That would solve many, almost all problems.
Would that explain also the fact that it was impossible to detect CMB going round and round, and solve the problem of inflation, too?
Can you give me some links to learn the details?
Thanks a lot!

 

[marcus]

Hi marcus, just imagine what your favourite wiseman, Anassagoras, would say to that. (what? me worry?)
Ancient sophoi discovered the truth because they followed the laws of necessity, the laws of Being: it is so because it must be so, it can only be so, and it's a miracle that it can be at all. The basic laws of Nature contrast whit what Popper said, there is ony one solution, nay, sometimes there is no solution, and Nature finds the impossible solution.

That's amazing, marcus, can you expand on that, how did you find that magic value? ...
Would that explain also the fact that it was impossible to detect CMB going round and round,..

I explained how to find the radius of curvature in a special tutorial thread that was in part thinking of you as reader. "Friedman for the lay learner". You have described yourself as a "LAY READER". If you sincerely want to learn standard cosmology, in good faith, then I would call you a "lay learner" and that thread is for you.

Anaxagoras (born circa 500BC) used verbal reasoning to conclude things like that the sun was a hot stone about the size of the Peloponnese section of Greece. But Aristarchus (born around 280BC) used trigonometric math reasoning and measurement of an angle to determine that the sun was much larger than the Earth!

Now our understanding is based on quantitative relations (equations) involving change, and when you read a verbal description of some finding that is only a translation into less suitable language. So if you truly want to understand cosmology, I would urge you to become acquainted with the Friedman equation.

And as an experiment to see if it helps, I will try to put the equation in a more intuitively assimilable form. And I will try out different ways of explaining to see if we can find one that works.

==========
If we conceive of a spatial section as hypersphere then it turns out that the 3D sphere is expanding so fast that it will always be impossible for light (like the CMB) to go all the way around. So indeed as you intelligently point out, that is a non-problem.

However the question of whether to treat the spatial slice as very large expanding 3-sphere or, instead, as INFINITE with zero curvature, is in a sense a merely VERBAL or artificial problem, not to be taken too seriously. Because for the practical purposes of calculating there is essentially no difference between zero curvature and a negligible amount of positive curvature.

Given the projected expansion history there's a limit on the size of the region we will ever be able to observe and it makes hardly any difference, mathematically, whether that region is part of a huge 3-sphere or an infinite extent with average curvature precisely zero.

Mathematics is an art of controlled approximation.

So I think your first language must have been Italian. BTW I thought I detected a note of mildly humorous pride in a previous post when you mentioned the famous tenor Luciano Pavarotti. You are right to be proud. Why do you think "Anassagoras" would be a favorite sage, for me? Should he be the mascot of all who look for rational law in Nature?

 

[bobie]

I detected a note of mildly humorous pride in a previous post Why do you think "Anassagoras" would be a favorite sage, for me?

I misquoted

But apparently Anaximander thought a little more deeply and said: Yes the round Earth is situated in the midst of empty space but it does not fall...because there is no preferred direction for it to fall in!

You explained it with symmetry, I explain it with necessity. As to the tenor, I mentioned him because he is a famous figure and his chest reminds one of a low bass singer. And 'great guy' was meant for you! But yet your intuition was right, as to my language. I think it is also patent that English is not my language!

I'll find your thread and read it, but if you wish, tell me one thing:
I do not particularly want to know how to find the radius of curvature, but why with that very curvature U would be considerd flat and keep isotropy? does it depend on the sensibility of your instruments or it is a principle?how do you derive that figure? from Friedman equation? Do those properties apply also or a fortiori if the radius is twice that figure, say 30,000 Gly? what if that were the real, actual size of U? why would you rule that out?

Thanks for your attention, marcus, I suppose I ought to give you a break, now!

 

[timmdeeg]

[/QUOTE]

I don't believe it makes any different to real world computations (where there is always a limit on precision) whether one assumes Ω exactly = 1, or instead something like 1.000001.

That would correspond to a U which is spatially a 3D sphere. And the 3D spatial slice (a "hypersphere") would currently have a radius of curvature of 14400 billion light years.

Hi marcus, in my opinion we talk at least about a big philosophical issue. Any deviation from
Ω = 1 related to $k = +1$ supports a finite universe.

In the case of a 3-sphere said deviation should be much smaller than 1.000001, because during inflation the critical density was roughly constant, while the matter density decreased with $1/a^3$, whereby $a$ increased by about 10⁵⁰ during this period.

 

[marcus]

Bobie,
Your english is very good (not to worry) just at rare times slightly *different* from typical american. I don't have to say this, since you are already self-confident. As you should be!
My favorites are Anaximander (b. circa 600 B) and Aristarchus (b. circa 310 B).

But my knowledge is sketchy (partial) and you may know more about pre-Soc. and other classic topics.

I'll find your thread and read it, ...

Great! I hope you do! and that you find the explanation clear enough and somewhat helpful. It is an attempt to develop a new way of explaining the Friedman equation (or as some people say "FLRW" for friedman-lamaître-robinson-walker but it was really Alex Friedman's equation. He died in 1925 not long after finding it. Why don't they at least call it FLWR and pronounce it "flower"?)

You can look up "Friedmann equations" in Wikipedia and get a different presentation. It might be good to do. The spatial curvature term that I call "Q²" is there in the guise of "kc²/a²". I am just translating quantities into quantities I find more transparent such as radius of curvature and reciprocal expressed as growth rate. (I am ignoring the negative spatial curvature case, so my simplified Friedman is not fully general--it just covers the "flat" ie. zero curved and "hypersphere" i.e. small positive curved cases.)

 

[marcus]

...Hi marcus, in my opinion we talk at least about a big philosophical issue. Any deviation from
Ω = 1 related to $k = +1$ supports a finite universe.

In the case of a 3-sphere said deviation should be much smaller..., because during inflation...

Hi Tim, I like your use of the word "deviation" for the amount that current Ω differs from Ω=1.

So in the case I used as example, where present-day Ω=1.000001 has a positive deviation of 0.000001 or one millionth.

Another term for the same thing, which I find pedagogically clumsy, is "- Ω_k".
I think that notation, especially the minus sign, is an historical accident. People got into the habit of writing "Ω = 1 - Ω_k" and the usage stuck.

I don't want to argue about whether or not a millionth is a good size present-day deviation to consider as an example. It is just a convenient example to take, numerically. The square root is recognizably 0.001, namely a thousandth. And therefore the radius of curvature of the hypersphere (today) is a thousand times the present-day Hubble radius 14.4 Gly.

I am using Planck mission parameters, essentially, which is why I say 14.4 instead of, say 14.0 which is closer to the latest figure WMAP reported.

So just to have a pedagoguish example, multiply 14.4 by 1000 and there you are.

BTW did you ever look at Charles Lineweaver's 2003 paper called "Inflation and the CMB"? It has a page or so discussing how the "deviation" (as you and I call it) changes over time. Inflation can pull it down to be very very small, but then it can slowly creep back up again. anyway one cannot so easily nail down what range it ought to be in today. The WMAP reports showed upper limits on the order of 0.01, so I would say 0.000001 is not unrealistically large (but you may disagree )