Effort to get us all on the same page (balloon analogy)

[Larkus]

I would say try to think of the EXPERIENCE of being 2D and living in a 2D sphere. Don't picture the sphere as if you are a God, outside and looking from outside at the sphere. Using some new type of lightrays that travel in 3D rather than 2D. Think of a sphere as the experience of living in it. And also think of a hypersphere that way.

Let's say that you and your brothers discover a remarkable fact about the world namely that there is a special area K which you have determined experimentally which allows you to reliably predict the area of any triangle!
You just have to sum the angles, subtract π, and multiply by that area K!
this always turns out to give the area (if you take the trouble to measure the area carefully.

The rule used by Euclid, namely 1/2 the base times the height does not work for you, it is only approximately right for small area triangles and gets progressively wronger for larger ones.

That's part of what I mean by the experience. It would apply also to living in a hypersphere. It does not involve postulating an extra dimension which we don't experience and cannot access. It just involves experimenting with triangles and determining the value of the area K.

Circumnavigating is another aspect of the experience which you (as creature living in sphere or hypersphere) might have. You can think of various others.

This comic does a good job visualising it in an entertaining way:

'The Adventures of Archibald Higgins: Here's Looking at Euclid'

 

[marcus]

This comic does a good job visualising it in an entertaining way:

'The Adventures of Archibald Higgins: Here's Looking at Euclid'

That's a really nice piece of work! Thanks for the link, Larkus.

I hope you keep us informed about more clever cosmology stuff from Petit and his "learning without boundaries" project.

Today in the "How to prove the stretching of space" thread, I noticed a neat explanation by Brian Powell of how the wavelengths of light get stretched out as distances expand.

From general relativity (specifically, the geodesic equation), it is seen that the momentum of a particle is inversely proportional to the expansion (the scale factor, a(t)). From de Broglie, this becomes a statement about the wavelength of photons -- as space expands, the wavelength of light must increase.

Timmdeeg's reaction says it:
"...your explanation why λ goes with a(t) is very convincing and new to me, thanks."
I think this is an especially nice way to look at it, which doesn't exclude others as well.

 

[marcus]

The South Pole Telescope (SPT) has given us new narrowed-down ranges for the cosmological parameters.

At the highest confidence level these correspond to a cosmos which is NOT "Euclidean flat" and NOT spatially infinite but is the 3D hypersphere analog of the 2D spherical balloon surface model.

The SPT curvature estimates translate into an estimated range of the "radius of curvature" namely from 140 to 320 billion light years.

This may not be right, the U may not be spatially finite, or it might be finite and these numbers might subsequently be revised. But let's take them at face value and see. After all it is a fine instrument, a respected team, and these are the most recent published estimates. Here's what I posted earlier about it:
==quote post #448==
http://arxiv.org/pdf/1210.7231v1.pdf
Scroll to Table 3 on page 12 and look at the rightmost column which combines the most data:

Ω[SUB]Λ[/SUB]     0.7152 ± 0.0098
H[SUB]0[/SUB]     69.62 ± 0.79
σ[SUB]8 [/SUB]    0.823 ± 0.015
z[SUB]EQ[/SUB]    3301 ± 47

Perhaps the most remarkable thing is the tilt towards positive overall curvature, corresponding to a negative value of Ω_k

For that, see equation (21) on page 14
Ω_k =−0.0059±0.0040.
Basically they are saying that with high probability you are looking at a spatial finite slight positive curvature. The flattest it could be IOW is 0.0019, with
Ω_total = 1.0019
And a radius of curvature 14/sqrt(.0019) ≈ 320 billion LY.
Plus they are saying Omega total COULD be as high as 1.0099 which would mean
radius of curvature 14/sqrt(.0099) ≈ 140 billion LY.

For Jorrie's A27 calculator the important parameters as estimated by the SPT report are
current Hubble time = 14.0 billion years
future Hubble time = 16.6 billion years
matter radiation balance S_eq = 3300
==endquote==

Since I posted that, Jorrie upgraded calculator from A25 to A27, so I made that change in the quote.

2 pi ≈ 6
so you can, if you wish, estimate the CIRCUMFERENCE of the universe simply by multiplying the "radius of curvature" figures by 6. The smallest it could be is 140 x 6 billion lightyears
and the largest it could be is 320 x 6 billion lightyears.
So if you could stop the expansion process, to make circumnavigation possible, you would have to travel in a straight line for six times 140-320 Gly before you'd be back at starting point.
If you sent a laser flash off in some direction it would be six times 140-320 billion years before it came back at you from the opposite direction.

This is just a way of understanding equation (21) on page 14 of the SPT report.

Ω_k =−0.0059±0.0040.

It's a way to get an intuitive feel in your imagination for what it means.
Here, again, is the link to the technical paper itself:
http://arxiv.org/abs/1210.7231

 

[RayYates]

In non technical language.. OMG the scale is mind-blowing!

 

[TalonD]

That's a very disappointing development. Infinite would have been much more aesthetically pleasing to me. Oh well.

 

[marcus]

That's a very disappointing development. Infinite would have been much more aesthetically pleasing to me. Oh well.

Sorry about your sense of disappointment, but hey! it's not settled yet! Uncertainty about that could last a decade, or a generation.

I love the hypersphere S³ the threedimensional analog of the surface of a balloon, so I'm certainly pleased by the South Pole Telescope report, but I have no sense that the thing is finally decided.

But for the sake of an example, if we take the SPT findings at face value then (with 95% confidence) the SMALLEST the circumference could be is 6 times 140 billion LY.
In other words 840 billion light years. quite a big balloon, so to speak. Would take an awfully long time to circumnavigate, if you could stop it from expanding so that circumnavigation would be possible.

 

[TalonD]

So it's not carved in stone yet? There is still hope for infinity? :D YAY

 

[Mordred]

I have a lot of catching up to do on this thread. Thus far the info contained in it has been insightful. Creedos to Marcus on it. I look forward to the finalized draft.

That being said I found the suggestion of thinking that inside the balloon being the past and outside the future useful. The one concern I have with it is in the case of Black holes. The analogy may lead to misconception that due to its infinite density the singularity may reside in the past at the big bang. I know that's not likely lol but its often the way laymen like myself tend to misconstrue analogies.

 

[Jorrie]

That being said I found the suggestion of thinking that inside the balloon being the past and outside the future useful. The one concern I have with it is in the case of Black holes. The analogy may lead to misconception that due to its infinite density the singularity may reside in the past at the big bang.

Yes, it is a rather troublesome way of viewing the analogy: there could have been a contracting phase in the distant past, followed by a 'bounce'. During such a phase, the past would have been 'outside' and the future 'inside' the balloon.

In any case, if the cosmos happens to be spatially flat or slightly hyperbolic, there can't be a notion of 'inside' or 'outside''. However, the balloon analogy would still yield all the correct answers by just considering the observable universe as the surface patch 'visible' to us.

The motto seems to be: use the analogy to get our brains around the expansion/contraction issue of the surface; then ignore it and rather use the simple mathematics of the LCDM cosmic model (or use one of the many available calculators to play around).
--
Regards
Jorrie

 

[GeorgeDishman]

Sorry about your sense of disappointment, but hey! it's not settled yet! Uncertainty about that could last a decade, or a generation.

I think it is significant that all the results in Fig. 7 are consistent zero curvature other than those that include the BAO result, and BAO is significant in Figure 8 as well. Given the poor agreement between H0 and BAO, I would prefer to resolve the discrepancy before we claim we have strong evidence either way.

 

[marcus]

I don't think I posted anything about the WMAP9 report (Hinshaw et al.) what it said about Ω_k . I'll get a link.
http://arxiv.org/abs/1212.5226
Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results
G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley, M. R. Nolta, M. Halpern, R. S. Hill, N. Odegard, L. Page, K. M. Smith, J. L. Weiland, B. Gold, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. Wollack, E. L. Wright
(Submitted on 20 Dec 2012 (v1), last revised 30 Jan 2013 (this version, v2))

On page 19 you see:
Ω_k= −0.0027^+0.0039_-0.0038 this is using pretty much all the major data sets: WMAP +eCMB+BAO+H
eCMB or "externalCMB" includes SPT but not the LATEST SPT. But that is details.
When you translate their plus/minus stuff it leads to a confidence interval of
-0.0065 < Ω_k < 0.0012

So that is lopsided on the negative Ω_k side, which means FINITE but it also has some zero and positive territory which means spatial INFINITE. Thats how several recent major reports have been going. You can't exclude spatial infinite, at this point.

On page 20 they also have a figure −0.0065 ± 0.0040 which I don't take as seriously but which ostensibly is based on even more data namely WMAP +eCMB+BAO+H + SNe.
That would correspond to a purely negative interval:
-0.0105 < Ω_k < -0.0025
That would exclude the spatial infinite case, at whatever the confidence level is. But this ball is still up in the air.

Hinshaw et al also had other interesting stuff about other issues, like the number of neutrino species and what Dark Matter clouds might possibly consist of. That tended to get people's attention so what they had to say about curvature was less noticed.

 

[RayYates]

Yes, it is a rather troublesome way of viewing the analogy: there could have been a contracting phase in the distant past, followed by a 'bounce'. During such a phase, the past would have been 'outside' and the future 'inside' the balloon.

In any case, if the cosmos happens to be spatially flat or slightly hyperbolic, there can't be a notion of 'inside' or 'outside''. However, the balloon analogy would still yield all the correct answers by just considering the observable universe as the surface patch 'visible' to us.

The motto seems to be: use the analogy to get our brains around the expansion/contraction issue of the surface; then ignore it and rather use the simple mathematics of the LCDM cosmic model (or use one of the many available calculators to play around).
--
Regards
Jorrie

I may have been one of those that suggested looking at the inside as the past and the outside as the future. This was in response to someone trying to calculate the center of the balloon (aka the universe). It was a way of getting him to visualize that everything in the universe is on the surface, and there is no center on the surface.

The balloon analogy also assumes the balloon is always expanding, which as you point out may not be the case.

As for Black Holes, since time and space are the same fabric and space seems to have been compressed out of existence, time must also have stopped, (or nearly so). In my mind I see this as a point/line extending out away from the balloon into infinity (the future). So in a way, its not part of the balloon but still attached at a single point.

 

[GeorgeDishman]

As for Black Holes, since time and space are the same fabric and space seems to have been compressed out of existence, time must also have stopped, (or nearly so). In my mind I see this as a point/line extending out away from the balloon into infinity (the future). So in a way, its not part of the balloon but still attached at a single point.

If "time stops" at the event horizon, the radius would then be fixed, so it's more like a thread attached to the inside of the balloon with the other end fixed at the centre. Once it goes taut, the rest of the balloon expands as usual but that small patch gets left behind.

Like any analogy, you can only take it so far.

 

[RayYates]

If "time stops" at the event horizon, the radius would then be fixed, so it's more like a thread attached to the inside of the balloon with the other end fixed at the centre. Once it goes taut, the rest of the balloon expands as usual but that small patch gets left behind.

Like any analogy, you can only take it so far.

Good "point". (pun intended). I like visual image created by the thread analogy. The beginning point becomes fixed in space-time and the balloon keeps expanding into the future dragging the event horizon with it. That's a very helpful way to imagine it.

 

[Naty1]

Jorrie posts:

Actually, there is a way in which the balloon analogy can make cosmic particle momentum decay intuitive. Simply consider a massive, frictionless particle that moves along the surface of the spherical balloon as a Kepler orbit around the center of a balloon. This particle must conserve angular momentum relative to the center of the balloon, i.e.

s:

Very nice!. And here I thought we exhausted all the 'analogies' with Phinds BALLOON ANALOGY last year. Phinds...you should Add this and Marcus' prior post to your discussion!//////////////////

Quote by RayYates
As for Black Holes, since time and space are the same fabric and space seems to have been compressed out of existence, time must also have stopped, (or nearly so). In my mind I see this as a point/line extending out away from the balloon into infinity (the future). So in a way, its not part of the balloon but still attached at a single point.

Georgedishman:

If "time stops" at the event horizon, the radius would then be fixed..

I'm pretty sure there is some misunderstanding here:

The 'singularity', not the horizon, is believed to be a point in time where space has been compressed out of existence.

'time stopping' at the horizon is a local, coordinate effect. Only for an accelerating not an inertial [free falling] observer.

The event horizon is a global construct...

Illustrations : From Kip Thorne in BLACK HOLES AND TIME WARPS
Finkelstein’s Reference Frame

when the star forms a black hole:

Finkelstein's reference frame was large enough to describe the star's implosion ...simultaneously from the viewpoint of far away static observers and from the viewpoint of observers who ride inward with the imploding star. The resulting description reconciled...the freezing of the implosion as observed from far away with (in contrast to) the continued implosion as observed from the stars surface...an imploding star really does shrink through the critical circumference without hesitation...That it appears to freeze as seen from far away is an illusion...General relativity insists that the star's matter will be crunched out of existence in the singularity at the center of the black...

Kruskal–Szekeres coordinates
http://en.wikipedia.org/wiki/Kruskal_coordinates

These coordinates have the advantage that they cover the entire spacetime manifold of the maximally extended Schwarzschild solution and are well-behaved everywhere outside the physical singularity…..The location of the event horizon (r = 2GM) in these coordinates is given by

So, a light cone drawn in a Kruskal-Szekeres diagram will look just the same as a light cone in a Minkowski diagram in special relativity.

'well behaved' here means using Kruskal-Szekeres coordinates instead of Schwarzschild time does not 'stop' at the horizon.

for further discussion, a separate thread would be appropriate.

 

[GeorgeDishman]

Good points, and that is why I said you could only take it so far. Can you create an exact correspondence between coordinates on the balloon surface any any of those you listed?

 

[Naty1]

Can you create an exact correspondence between coordinates on the balloon surface any any of those you listed?

Good question, but above my paygrade for now...

Oddly that issue did not come up in a very long thread started by phinds "Balloon Analogy'...

But the endpoints [coordinates] of paths seems to not be the only issue:

In another discussion I tried unsuccessfully to sort out the idea that the Einstein Field Equations, used in cosmology, deal with geodesics in 4D spacetime. So what does a geodesic of 4D spacetime look like in 3D space? And what does that look like on a 2D balloon surface??

 

[marcus]

Before I lose track of the links, I'll try to get something together about current measurments of the spatial mean curvature.
Page 40 of the relevant Planck report ( http://arxiv.org/abs/1303.5076 )
==quote==
With Planck we detect gravitational lensing at about 26σ through the 4-point function (Sect. 5.1 and PlanckCollaborationXVII 2013). This strong detection of gravitational lensing allows us to constrain the curvature to percent level precision using observations of the CMB alone:

100Ω_K= −4.2^+4.3_-4.8 (95%; Planck+WP+highL);
100Ω_K= −1.0^+1.8 _-1.9 (95%; Planck+lensing + WP+highL)

These constraints are improved substantially by the addition of BAO data. We then find

100Ω_K = −0.05^+0.65_-0.66 (95%; Planck+WP+highL+BAO)
100Ω_K = −0.10^+0.62_-0.65 (95%;Planck+lensing+WP+highL+BAO)
==endquote==
Here's an earlier post on the topic of mean spatial curvature:

I don't think I posted anything about the WMAP9 report (Hinshaw et al.) what it said about Ω_k . *I'll get a link...

==quote==
http://arxiv.org/abs/1212.5226
Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results
G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley, M. R. Nolta, M. Halpern, R. S. Hill, N. Odegard, L. Page, K. M. Smith, J. L. Weiland, B. Gold, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. Wollack, E. L. Wright
(Submitted on 20 Dec 2012 (v1), last revised 30 Jan 2013 (this version, v2))

On page 19 you see:
Ω_k= −0.0027^+0.0039_-0.0038 * * this is using pretty much all the major data sets: * * * WMAP +eCMB+BAO+H
eCMB or "externalCMB" includes SPT but not the LATEST SPT. But that is details.
When you translate their plus/minus stuff it leads to a confidence interval of
-0.0065 < Ω_k < 0.0012

So that is lopsided on the negative Ω_k side, which means FINITE but it also has some zero and positive territory which means spatial INFINITE. *Thats how several recent major reports have been going. *You can't exclude spatial infinite, at this point.

On page 20 they also have a figure −0.0065 ± 0.0040 which I don't take as seriously but which ostensibly is based on even more data namely *WMAP +eCMB+BAO+H + SNe.
That would correspond to a purely negative interval:
-0.0105 < Ω_k < -0.0025
That would exclude the spatial infinite case, at whatever the confidence level is. But this ball is still up in the air.
==endquote==
And here was an earlier post that mentioned the curvature measurement by the SPT:

The South Pole Telescope (SPT) has given us new narrowed-down ranges for the cosmological parameters...

http://arxiv.org/pdf/1210.7231v1.pdf
==quote==
Perhaps the most remarkable thing is the tilt towards positive overall curvature, corresponding to a negative value of Ω_k

For that, see equation (21) on page 14
Ω_k =−0.0059±0.0040.
Basically they are saying that with high probability you are looking at a spatial finite slight positive curvature. The flattest it could be IOW is 0.0019, with
Ω_total = 1.0019
And a radius of curvature 14/sqrt(.0019) ≈ 320 billion LY.
Plus they are saying Omega total COULD be as high as 1.0099 which would mean
radius of curvature 14/sqrt(.0099) ≈ 140 billion LY.
==endquote==

 

[marcus]

I just saw a really classic handling by Bandersnatch of newcomer questions by TigerDave. It is so clear and concise I want to save it as part of "effort to get us on same page"

Here is TigerDave's original post

Regarding the creation of the universe and the current model:...

His paragraphs are interspersed in the following by Bander.

Hello, TigerDaveJr. Welcome to PF! ...

==quote Bandersnatch classic response to newcomer==

Is it assumed that the universe, at the time of creation was finite in size (or at least more finite than it is now) prior to the rapid expansion, or was the protoexistance finite in size in an infinite universe? So, did the universe AND its contents expand, or did a collection of mass within the universe expand, creating the physicality we know today?
​

The universe was either finite or infinite, and it still is one of those. We cannot say which one it is, but if it's finite, then it has got a very large curvature radius(~88 billion ly was the minimum estimate, iirc).
The key part to understand is that whenever you hear of the universe's expansion, it does mean the entirety of it. It's not about some matter expanding into a preexisting space, but space WITH matter and energy, expanding.

I have seen the expansion explained like a balloon. However, if this were true, would not most mass be on the 'outside' of the balloon? Is there content in the middle of the universe, or is there a hollow center that is getting bigger as we get further from the center? I've read that asking about the center is impossible, and that the universe has infinite shape, but if that's true can we say we're expanding? Would there not be an origin point, or is that one of the problems that a physics-uneducated person like myself would be unable to grasp (re: Plato's allegory of the cave).
​

The balloon analogy is not perfect, as it creates this erroneous intuition that there is something outside(or inside) the balloon, due to the way we imagine it being a three dimensional object.
The analogy requires you to think of only the surface of the balloon as the universe.
There is no centre to a 2d surface(but there is curvature), and the expansion is still easily observable by comparing the distances between any two points on that surface at two different times.

These two pages go into more detail about the balloon analogy, its aims and limitations, all in layman's terms:
http://www.mso.anu.edu.au/~charley/papers/LineweaverDavisSciAm.pdf (first page is blank)
http://www.phinds.com/balloonanalogy/

Can we not use red shift in order to determine the relative center of this expansion? I understand that we observe red shift based upon where we're standing, but should we not be able to calculate from all that where the overall center is? Where are we in regards to this?
​

You should see from the above links that it is impossible to define a centre of uniformly expanding space.
You can easily define the centre of the observable universe, which is wherever you are standing.

Was expansion more like bread dough? Did the pre-expansion material tear? Was that tearing uneven, that left behind general emptiness in some spots and densely clumped matter in others that led to our original star nurseries?
​

You are taking the analogy too far. Of course the universe is not made of dough, so it does not tear like dough does. It is important to limit yourself to only what the analogy is trying to convey(i.e., the expansion of space) and not to go overboard with drawing conclusions from it.

Are galaxies considered expanding or collapsing? I've heard that there's supposed to be black holes in the center, so is this local mass "going down the drain" or is this mass being spun off from the center? Is it both? Do we consider galaxies to be generally "on par" with each other in the creation of more complex atomic structures, or do we expect each birth/nova/collapse/rebirth cycle of stellar material to continually generate more complex material, and that individually from galaxy to galaxy?
​

Galaxies are stable structures, with no significant amount of mass going down the black hole or escaping.
Sure there might be some rogue star gaining enough speed from random gravitational interactions to fling itself into the intergalactic space, and there tends to be some gas falling down the black hole - mostly because it takes so long to actually get there.
But overall, there is no expansion or collapse. The expansion of space does not affect small scale structures(like galaxies), and the black holes are not the voracious vacuum cleaners of doom that you might sometimes see in the popular media. Most stars stay in pretty much stable orbits around the galactic centre, and it's not going to change much, barring collisions with other galaxies.
All the galaxies coalesced from the same primordial gas, and the laws of physics governing them are the same, so it stands to reason that they are similar.
The difference is in the time scale. As you look farther away, you see younger galaxies, and the younger the galaxy, the less time its stars have had to go through their life cycles and produce heavier elements.
Generally the longer the universe exists, the more heavy elements it contains(in the early universe there was only hydrogen, helium and some lithium).

Second to last, is it possible, in the same way that we view time against the overall amazingness of deep time, that this initial universal expansion was just one bubble in an even larger sea of expanding pockets that we have yet to get close enough to see the evidence of? Not getting into dimensions, but is our universe just one in an entire "hyper-universe" of immense activity, that we can't directly "observe" in the same way that our tiny blip of existence fits in the concepts of deep time?
​

It's a kind of a vague and dangerously philosophical-sounding question, but I suppose it asks about the multiverse hypothesis?
As you say, it's not observable, therefore not falsifiable, which makes it an empty question really.
The first half an hour or so of this talk by Lee Smolin:
http://pirsa.org/13020146/
touches on the subject.

Finally and most importantly, where should I be aiming myself educationally in order to learn the answers to these questions, and to ask even more?​

I'd recommend starting here:
http://www.astro.ucla.edu/~wright/cosmolog.htm
and going through either/both tutorial or/and FAQ.

Stephen Weinberg's "First three minutes" is a classic book concerning the early expansion of the universe. It's a bit dense at times, and getting somewhat old, but still worth reading.

Alan Guth's "The Inflationary Universe" talks about the birth of the idea of inflation, that is a major(if still somewhat dodgy) part of current cosmology.

Finally, understanding Relativity might be necessary. This popular treatment by Einstein himself is a good start:
http://www.gutenberg.org/files/30155/30155-pdf.pdf

You should be able to understand the ideas without any maths knowledge, but once you dig deeper into cosmology, you'll notice that it's at its heart a mathematical science, requiring you to learn higher mathematics to truly understand what's going on.
Unless you do that, you'll have to do with imperfect analogies, so if you have such an option, take calculus and algebra courses.

Finally, you might find the courses/videos on these sites relevant to your interests:
http://www.perimeterinstitute.ca/video-library (you probably want the "public lectures" section)
http://www.academicearth.org/ (try astronomy section)
https://www.coursera.org/ (actual online courses; physics section covers cosmology as well)
https://www.khanacademy.org/ (not a lot on cosmology, but good for learning maths and basic physics concepts)
==endquote Bandersnatch==

 

[Ron Bert]

Forgive me for I am new and not an expert but I do desire that this "same page" concept enables us to advance in the unity of thought, for the betterment of all mankind. It really comes down to application over endless fantasy. I believe Thomas Edison said something to the effect that if an idea proved to be wrong or fruitless, don't linger on it, for it will only leave you fruitless. Unquestionably, it is the fruit of agreement that initiates real advancement and what we as a civilization so desperately need. It seems that the balloon analogy is a prelude to a larger all-encompassing truth. The dualistic nature of our universe must be incorporated, up-down, top-bottom, beginning-end, start-finish; universal clarity.

Can I assume the balloon analogy prefers rising characteristics over that of falling? Both rising and falling are components of our reality. Perhaps an "invaloom" at all real points would more directly address the reality we hope to know. Black holes certainly throw a loop into the pot we stew in. I wonder if not all black holes have a common intersection thereby assisting a more uniform quantum foundation.

In this balloon analogy, does it exist in one single space as it flows energetically forward, as a function of time? It seems these simple concepts lack clear ultimate definition(s). Time, Space, Energy; we struggle to know what encapsulates these age old concepts or could we say that we hope to soon recognize obvious equations and principles that simply and absolutely explain all, (inclusively), which again, would enhance the clarity of all human life. Certainly, what defines all of the most elemental structure must also be strikingly universal so as to achieve the widest angle of opportunity for scientific inclusion and technical implementation. Including up-large (macro ~ relativity) and down-small (micro ~ quantum mechanics), understandable location and dimension theories (laws) are key components for significant applications to be realized. We attempt to explain that which we can perceive and many capable individuals leave the rest of us behind, but, unarguably, we must accept our unperceivable universe, which can never be seen NOW, for humanity to rise to the greatness that is the truth. This would involve jumping out of the box, to see all of the balloon for what it is. Modern science observes the tangible producing hypothesis to be scrutinized rigorously to verify truth. ~This “Scientific Method” is unable to deliver unseen truths. I do wonder what forum, or even if there is one, that can rise to the occasion to deliver truth over speculation; genuine potential applications over bickering posturing. I can only say that may all of our struggles to achieve truth be blessed.

 

[marcus]

Can I assume the balloon analogy prefers rising characteristics over that of falling? Both rising and falling are components of our reality. Perhaps an "invaloom" at all real points would more directly address the reality we hope to know. Black holes certainly throw a loop into the pot we stew in. I wonder if not all black holes have a common intersection thereby assisting a more uniform quantum foundation.
...

I'm sorry to have to disappoint you. The title of the thread is to be narrowly construed in the context of this specialized Cosmology sub-forum. We discuss mainstream (observational, quantitative) cosmology here. In this context the balloon analogy is a familiar means of visualizing the Hubble Law pattern of expanding distances between stationary or near-stationary objects. We do not discuss philosophical matters or mankind's moral/religious aspirations in this particular sub forum.

Can you get this little animation to run on your computer?
http://www.astro.ucla.edu/~wright/Balloon2.html
It is a simple short movie of an expanding sphere surface with galaxies painted on it.
A college teacher at UCLA put it on line.

 

[Ron Bert]

I'm sorry to have to disappoint you. The title of the thread is to be narrowly construed in the context of this specialized Cosmology sub-forum. We discuss mainstream (observational, quantitative) cosmology here. In this context the balloon analogy is a familiar means of visualizing the Hubble Law pattern of expanding distances between stationary or near-stationary objects. We do not discuss philosophical matters or mankind's moral/religious aspirations in this particular sub forum.

Can you get this little animation to run on your computer?
http://www.astro.ucla.edu/~wright/Balloon2.html
It is a simple short movie of an expanding sphere surface with galaxies painted on it.
A college teacher at UCLA put it on line.

I thank you for your response. I'm sorry if my discussion offended you or you felt it inappropriate. I read and enjoyed much of your most recent posts and did review your Balloon2.html link. I am not sure I inferred any religious aspirations over that of hopeful and feel a bit puzzled, believing that the goal of being on the "same page" would include an open mind to embrace anything that would enhance or achieve more clarity for the whole, including what I perceived as constructive input for the sheer sake of truth. I guess I am trying to look at the book while you review the page.

 

[pullmanwa]

To have an enjoyable cosmo forum we needed a balance between mental freedom on the one hand and a shared knowledge base on the other.
People should be free to imagine the universe the way they want, but everybody should try to understand the standard LCDM (Lambda-cold-dark-matter) model as a starting point.

I'm going to try to avoid mathematical equations in this thread because they put many people off and also to avoid using too many abbreviations like LCDM. This thread should be at the entry-level for the Cosmo forum. You are welcome to contribute ideas and comments.

The LCDM is based on a more general mathematical model called FRW or FLRW (Friedmann, Lemaître, Robertson, Walker) which is built into Ned Wrights calculator
Everybody who comes in and posts here should have played some with that calculator or one like it because in practical terms that is what a mathematical model is. Cosmology is mathematical (not verbal) and observational---it fits a mathematical model to data.
The galaxy counts, redshift surveys, supernova brightness, microwave background data and so on are all supposed to check out and match what the model says they should be.

When you use Wright's calculator you have specified three parameters (the default values are 0.73 for dark energy fraction, 0.27 for matter fraction, 71 for Hubble).
If you don't change the default settings, you get the standard LCDM. If you change them you get some other version of FLRW.

You can think of the LCDM as the fine-tuned version of the general FLRW where the parameters are chosen to get the best possible fit to our universe--to match the observational data.

So the existence of these models is always in the background but what we need to focus on here in this thread is the INTUITION. How to picture it so that if you were playing around with one of the calculators, changing the parameters and finding how far away various things were when they emitted the light we are getting etc, you would kind of know what to expect. Intuition about how the parameters effect things, and how redshifts relate to distance and recession speed.

So what I hope for is that those of us asking questions and discussing here at Cosmo forum all have a shared basic intuition---which is a kind of home base---and probably the most convenient way to get that is to properly understand the balloon analogy.

In my experience many of the misconceptions people have when they first come to this forum stem from misunderstanding what that analogy is intended to teach us. And a lot of the confusion we occasionally experience comes from getting that analogy somehow crossed up. So in this thread what I propose we do is, at least for starters, simply discuss the balloon analogy. Get clear on it. Find out any problems people have with it, if there are some.

We can do that without having to use a lot of math formulas, I think, and a minimum of technical jargon. Don't get me wrong---I'm all in favor of jargon, we simply won't need much of it here.

For people who want to get some hands-on experience with Wright's cosmo calculator it is here
http://www.astro.ucla.edu/~wright/CosmoCalc.html
the homepage for his other cosmo resources is
http://www.astro.ucla.edu/~wright/cosmolog.htm
you can always get these links just by googling "ned wright"
In my sig I have a link to MORGAN's cosmo calculator which has some valuable features and is harder to get by googling.
You might want to try that one too, it gives recession speeds.

It appears you are trying to teach us "conjectured" information. Nothing in Cosmology has ever been proven true... such as the big bang theory and other theories. Face it, there is only one Universe, it is infinite with no start and no end. It has always existed. There are some things that have no concrete explanation and the Cosmos is one of them. Please accept this and your life will be simpler.

 

[Drakkith]

It appears you are trying to teach us "conjectured" information. Nothing in Cosmology has ever been proven true...

Nothing in science has ever been proven true. Proofs only exist in math. We merely have enough evidence to say that our theories and models are accurate to a certain extent.

such as the big bang theory and other theories.

Cosmology is as much constrained by the normal rules of science as any other branch. As such, theories in cosmology are based upon evidence and observations and subject to peer-review. Personally I find it absolutely amazing that we know as much as we do without being able to get off of this little rock and get out into interstellar/intergalactic space.

Face it, there is only one Universe, it is infinite with no start and no end. It has always existed.

That is one possibility, yes. It is certainly not the only one.

 

[GeorgeDishman]

It appears you are trying to teach us "conjectured" information. Nothing in Cosmology has ever been proven true...

Nothing in any branch of science is "proven true", the best it can be is "not falsified".

Face it, there is only one Universe, it is infinite with no start and no end. It has always existed. There are some things that have no concrete explanation and the Cosmos is one of them. Please accept this and your life will be simpler.

I could accept that the universe is Galilean Invariant and life would be much simpler, no relativistic effects to worry about. I could accept that all physics is classical, no quantum effects, and life would be even simpler still. I would be wrong though, and many things I could observe would behave differently to my beliefs. Imagining an infinitely old universe might be simpler for you, but that is not what we see through our telescopes, and I have no desire to believe things that are untrue no matter how much simpler life might be.

 

[ViperSRT3g]

As I explained to someone else here on these forums in the past, here was as simple of an analogy as I could put it.

Imagine a meter stick, the space it occupies is expanding at a tiny rate due to the expansion of the universe. Of course it literally is not growing, but the space it occupies grows at a steady rate.

Now imagine a measuring stick a light year long. This too is taking up space that is also expanding with the rate of the universe. But imagine it being made up of the meter sticks. They are all expanding at that same rate as a single meter stick, except the effect is amplified because the light-year long measuring stick can be said to have been made up of meter sticks.

 

[marcus]

In March 2013 we got a new set of cosmological parameters, when the ESA Planck mission released its report.
I want to call attention to TABLE 5 on page 20 of http://arxiv.org/pdf/1303.5076.pdf
which is their main cosmological parameters paper.

The rightmost two columns are where they combine their data with a complete set of other recent studies including
WP : WMAP polarization data
HighL: a combination of Atacama Cosmology Telescope and South Pole Telescope (ACT+SPT)
BAO: Baryon acoustic oscillation data obtained from galaxy census figures aka redshift surveys.

BACK WHEN NASA's WMAP mission was publishing we saw that these COMBINED reports, where they merged data with selected other studies in progress at the same time, often had a narrower range of uncertainty and tended towards broader acceptance. So I want to recommend the results in the two rightmost columns, which are labeled
"Planck+WP+HighL+BAO"

Hubble parameter 67.80 ± 0.77
Omega_Lambda 0.692 ± 0.010
Assuming zero average curvature this corresponds to a benchmark matter density of 0.308
The two best fit numbers given for baryonic and dark matter are 0.02216 and 0.11889 totaling 0.14105
meaning about 15.7% of matter is ordinary and 84.3% is dark.

This Hubble parameter corresponds to a Hubble time of 14.42 billion years. In Jorrie's Lightcone calculator that's rounded for convenience to 14.4 billion years.
Critical density can be expressed in a variety of units. Google calculator insists on calling a joule per cubic meter a "pascal" because of course it is also one Newton per square meter.
The energy density and the pressure units are algebraically equivalent.

So if you go to google and paste in 3c^2/(8 pi G)/(14.42 billion years)^2
you get 0.7763 nanopascal
making overall matter density 0.239 nanopascal
and dark matter density, gotten by pasting in "0.7763*0.308*0.843", comes out 0.20156 nanopascal. Let's say 0.202 nanojoule per cubic meter.

To make the thing simpler to remember, just in round numbers, I guess you can say that a cubic kilometer of the universe contains, on average, 0.2 joules worth of dark matter, and 0.04 joules worth of ordinary, for a total 0.24 joules of matter overall.

 

[GeorgeDishman]

The extended parameters in Table 10 are also interesting, the addition of the HighL data seems to add a bit of bias.

 

[marcus]

The extended parameters in Table 10 are also interesting, the addition of the HighL data seems to add a bit of bias.

Table 10 is definitely interesting! To save folks trouble searching, it is on page 39 and it shows
"extensions" of the standard LCDM (lambda cold dark matter) model where you allow each one of several parameters to vary separately.

It is interesting how nearly FLAT the large-scale geometry turns out to be. The curvature number is very close to zero. Its 95% confidence interval has shrunk.

It is also interesting to look at the 95% confidence interval for the "eqn of state of D.E." denoted "w". If w is anything but -1, then the cosmological constant Lambda is not a true cosmological constant, which would make things a lot more complicated.

Anyway thanks to George D. for pointing out this additional table of cosmic parameters!

 

[megacal]

Pullmanwa wrote:

There are some things that have no concrete explanation and the Cosmos is one of them. Please accept this and your life will be simpler.

Maybe true, but what fun would that be?
Besides being insatiably curious, I think they enjoy the complexities of cosmology, the mental challenge and the comradery. I wish I had the math skills, IQ, and time to delve deeper with them. To each his (or her) own.

Marcus,
you are remarkably patient with us mathematically-challenged folks, and I appreciate the time you spend helping us to come to some understanding of cosmology.

re: Balloon Analogy
you tried to help me to get on the same page in Dec 2011...I just reviewed our posts back on page 12 of this thread.

I basically let the Balloon Analogy go as something I just couldn't get, even if I squinted at it until
my eyes crossed and rolled up in my head.

But recently I came back to ask about Pre-Big Bang <<< Proton?. Mordred, JesseM, and Chalnoth all gave me good help and links to relevant articles which I downloaded.

However it was Phinds's 2 page article on the The Balloon Analogy that I found especially helpful. And I see you also evidently agree as you gave a link to it in a post above. As well as to Misconceptions About the Big Bang by Lineweaver & Davis which Mordred also gave me.

So I think I'm finally on the same page, at least as far as understanding what the BA is and is not. I'm not yet able to say I fully agree with it, but at least I think I understand what you & Phinds are saying about it.

Quoted from Phinds's article:

What the Balloon Analogy is intended to describe:

(1) The universe is expanding OUTSIDE of systems that are gravitationally bound, or bound by other local forces (e.g. strong and weak forces) That is, things the size of a local cluster of galaxies and smaller (like, the Milky Way, Earth, you, me, atoms, and so forth), do NOT expand.

(2) The expansion has no center, and everything is moving away from everything else, with things farther from each other receding faster from each other than things closer together

This is the part I have most difficulty comprehending or visualizing now...

things farther from each other receding faster from each other than things closer together

...just found your post #9

2. to picture distances increasing at a percentagewise rate. Like one percent per minute.
So the longer the distance the faster (inches per minute) it increases. This is Hubble Law.

I'll look it up.

 

[GeorgeDishman]

This is the part I have most difficulty comprehending or visualizing now...

things farther from each other receding faster from each other than things closer together

Here is a map of "nearby" galaxies:

http://www.sdss.org/includes/sideimages/sdss_pie2.html

Print it out, then put it through a photocopier set to 110%, that's roughly 1.4 billion years worth of expansion. Now choose any pair of galaxies in the original map and measure how far apart they are. Find the same pair on the larger map and measure. You should find the separation has increased by 10% of course.

Two galaxies 10mm apart will now be 11mm apart so if we are one, the other has moved 1mm in 1.4 billion years.

Two galaxies 20mm apart will now be 22mm apart so if we are one, the other has moved 2mm in 1.4 billion years, and that is twice the speed of the previous example.

 

[megacal]

Two galaxies 20mm apart will now be 22mm apart so if we are one, the other has moved 2mm in 1.4 billion years, and that is twice the speed of the previous example.

Thanks, that makes sense.

 

[somebodyelse]

To my mind the balloon analogy is a nuisance, gallaxies ect are not stuck to a surface, once one has read about the BA it takes some getting rid of.

My comments as an interested layman (who has not yet read the entirety of this thread):
I find the balloon analogy as very helpful in explaining how objects expanding at the speed of sound can be receding at apparently greater "speeds" because of the expansion of the space they are traveling on. But it has another major problem. To a layman, a balloon is a 3D object, not a flat surface. So it inevitably provokes the question of where the center of the expanding balloon is (read where the big bang was physically located).

I know there is no radius to this mythical balloon and that it should only be viewed as a surface analogy, but as a visualization tool used to explain things to a layman I feel it would be helpful to emphasize this peculiarity *at the very onset* before going much further. Otherwise it is normal to assume the balloon has a center.

FWIW, I only offer this as a teaching comment to explain (one of many errors) a non physicist might make when visualizing the balloon analogy.

 

[Jorrie]

I know there is no radius to this mythical balloon and that it should only be viewed as a surface analogy, but as a visualization tool used to explain things to a layman I feel it would be helpful to emphasize this peculiarity *at the very onset* before going much further. Otherwise it is normal to assume the balloon has a center.

I found two ways to soften the peculiarity, depending on the skill of the layman. One is to view the balloon as "infinitely large", making whatever the surface dweller can observe to appear spatially flat (which is more or less the status of our universe).

The other one is to emphasize that if the universe is closed (positively curved), there may be a "center" in an extra hyper-spherical dimension. The balloon analogy is just dropping one of the normal spatial dimensions...

 

[Frank Weyl]

It appears that there are a lot of very interesting models for the 'shape' or 'topological' forms of the universe:
Riemann, twistor (Penrose), mobius, etc.
My take is below...
http://en.wikipedia.org/wiki/Ricci_flow
Although, and I think, we will all have a big surprise, when it is finally resolved!

 

[Frank Weyl]

Interesting video of the expansion from the plenum to...!
https://www.youtube.com/watch?v=NjSFR40SY58#t=113

A picture (video) paints a thousand words.

 

[marcus]

It might help get us all "on the same page" if we could share in common some intuitive understanding of the basic equation used in Cosmology. This is called the Friedman equation or (when people list two equations) the first Friedman equation. (His name is spelled variously, often with another final n, as Friedmann.)

I've been thinking about what might be the best way to provide a layman's intuitive understanding of the equation (in a simplified case where overall spatial curvature is negligible and matter is dispersed enough so overall average pressure is negligible as well.)

Since the Friedman equation is a radically simplified version of the Einstein GR equation, getting some intuition about it could be a good way to get a grasp of the GR equation it is derived from.

Here's a thread where I've been working on some explanations relating to this:
https://www.physicsforums.com/showthread.php?t=760988
Here's the post in that thread that has a good many of the essentials:
https://www.physicsforums.com/showthread.php?p=4793450#post4793450

It might be useful to summarize it here.

 

[marcus]

It's said that the Einstein GR equation relates change in geometry (on the left) to matter (on the right)--matter tells geometry how to curve, geometry tells matter how to flow. Let's try to be more specific. How are the two measured--in what units?--and what constant converts between the two sorts of quantities?

Just as frequency and fractional growth rate are commonly measured in units of reciprocal time (e.g. some number per second or per year) so a common measure of curvature is reciprocal area--some number over a unit length squared--some number over a unit area.

Common measures of matter are pressure and energy density. It happens that if you multiply a curvature by a FORCE you get a pressure. In metric terms: "per square meter" (curvature) multiplied by "Newton" (force) gives "Newton per square meter" which is the metric unit of pressure called the "pascal". It is also equivalent to the metric unit of energy density "joules per cubic meter". You just have to multiply N/m² by meter in both numerator and denominator to get Nm/m³, which is "joules per cubic meter".

So it's not too surprising that the central constant in the Einstein GR equation is a FORCE, namely c⁴/8πG. You can think of it as multiplying a 4x4 array of curvatures, to give a 4x4 array of pressures and energy densities. Or reciprocally as dividing each pressure or density in the array describing matter, to give an array of curvatures describing what geometry is doing around that particular point in space and time. In the iconic form of GR equation the force appears as its RECIPROCAL, one over the force, namely 8πG/c⁴:

G_{\mu\nu} + \Lambda g_{\mu\nu} = {8\pi G \over c^4} T_{\mu\nu}In fact the central constant 8πG/c⁴ is the reciprocal of an actual intrinsic quantity of force that is, so to speak, built into nature. This universal constant force is what relates MATTER (expressed by the T_** tensor on the right) to the dynamically responding GEOMETRY, expressed by G_** the so called "Einstein tensor". If you want to know the size (in quarter pound metric force units) of that innate force woven into the fabric of existence just paste this into google:
c^4/(8pi G)
You should get some large number of Newtons.

The Lambda in the GR equation is a curvature constant, a reciprocal area. Multiplying it by the metric tensor little g_** gives again an array of curvatures, ready to add to G_**.
The Friedman equation derives from the GR equation and is a much-simplified form of it. Let's define Φ = 3c²/(8πG). It is LIKE the central force constant in the GR equation, but with a factor of 3 thrown in, and missing a factor of c². The reciprocal of this force-like quantity, 1/Φ = 8πG/3c², is the central constant in the Friedman equation:
H(t)^2 - H_\infty^2 = {1 \over \Phi} \rho (t)This is what the equation boils down to in the case where overall spatial curvature and average pressure are negligible. Comparing the two equations shows that they have analogous parts.

On the right, instead of the full T_** tensor (essentially a 4x4 matrix varying over space and time, describing the concentrations of energy and momentum that matter represents) we just have the one density quantity rho of t varying over time: the average energy density ρ(t).
On the left, instead of the Einstein G_** tensor (basically a 4x4 matrix expressing the changes in geometry felt by probing in various spacetime directions), all we have is the square of a simple expansion rate H(t), and a constant squared expansion rate term.
And the Lambda term in the original GR equation is reflected in what is written here as H_∞ squared. In fact H_∞² = Λc²/3, so it is an almost verbatim transcription of the Lambda term in the original.

So in the Friedman equation, instead of 4x4 arrays of geometric and matter quantities we just have one quantity. On the left it is a SQUARED GROWTH RATE---not reciprocal length squared but pretty close, reciprocal time squared. The H quantities are fractional growth rates---some number per unit time (per second, per year, per million years). The force-like constant is just what we need to multiply the squared growth rates by to get an energy density. Or reciprocally, it is what we need to divide the density by, on the right, to get a squared growth rate.

 

[marcus]

Part of the idea here is to argue for everybody having at least a certain level of knowledge of basic cosmic model parameters---including a notion of what the current and longterm Hubble radii are estimated to be.
Someone, having heard the earlier WMAP estimates and recalling those, might say R(now)=14.0 Gly and R_∞ = 16.5 Gly. On the other hand another person, who's been using Jorrie's calculator (where Planck estimates are the default), might say R(now) = 14.4 Gly and R_∞ = 17.3 Gly. The exact figures, as long as they're reasonably recent, are not critical. Future missions may revise them. In examples here, I will use the Planck mission 14.4 and 17.3 Gly. Hopefully any reader of this thread will be using those or will have his or her own figures in mind.

As an example, those two Hubble radius tell me that the present and future values of the Hubble growth rate are
H(now) = 1/14400 per My = 1/14400 per million years, and
H_∞ = 1/17300 per My.
Those are the fractional growth rates of the distance between two unmoving points. Picture two points painted on an expanding balloon surface (with all existence concentrated on that 2D surface). They are not moving in any direction that exists in their universe. They are not going anywhere, the distance between them is simply increasing..

Since I have H²(now) and H_∞², the Friedman equation tells me the present-day density of matter in space
ρ(now) = Φ(H²(now) - H_∞²) = 3c^2/(8pi G)*(1/14400^2 - 1/17300^2) per (million years)^2 = 0.239 nanopascal

I highlighted what you would paste into google to get it to do the calculation.

 

[marcus]

Here's another example. Suppose you want to know the rate distances were growing at a time in the past when distances (between unmoving points) were HALF what they are today. Well then volumes were 1/8 of present size and the matter density then ρ(then) = 8*0.239 nanopascal.

We can use Friedman equation again

H(then)² = H_∞² + (1/Φ)8*0.239 nanopascal

H(then) = (1/17300^2 per (million years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5

I paste that in, and it tells me H(then) = 3.9 x 10^-18 Hz
in other words a number per second. But I want a number per million years so I multiply the answer by a million years:
(1/17300^2 per (million years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5*million years
and it tells me H(then) = 0.000123231935 per million years
So I tell it 1/0.000123231935 and it says 8115
The answer therefore is H(then) = 1/8115 per million years. The Hubble growth rate, which is now 1/144% per million years WAS back then 1/81% per million years.
And the Hubble radius, which is now 14.4 Gly, was 8.1 Gly.

 

[Jorrie]

...
H(then) = (1/17300^2 per (million years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5
I paste that in, and it tells me H(then) = 3.9 x 10^-18 Hz
in other words a number per second. But I want a number per million years so I multiply the answer by a million years:
(1/17300^2 per (million years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5*million years
and it tells me H(then) = 0.000123231935 per million years
So I tell it 1/0.000123231935 and it says 8115
The answer therefore is H(then) = 1/8115 per million years. The Hubble growth rate, which is now 1/144% per million years WAS back then 1/81% per million years.
And the Hubble radius, which is now 14.4 Gly, was 8.1 Gly.

I know you are fond of the % per million years growth rate. My question is, with everyone around here being used to think billions of years (Gy) in large scale cosmology, why not stick to it. One then uses the Hubble radii as we talk about them, i.e. your paragraph "paraphrased":

"H(then) = (1/17.3^2 per (billion years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5
I paste that in, and it tells me H(then) = 3.9 x 10^-18 Hz,
in other words a number per second. But I want a number per billion years so I multiply the answer by a billion years:
(1/17.3^2 per (billion years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5*(billion years)
and it tells me H(then) = 0.123231935 per billion years
So I tell it 1/0.123231935 and it says 81.15
The answer therefore is H(then) = 1/81.15 per billion years. The Hubble growth rate, which is now 1/14.4 per billion years WAS back then 1/81.15 per billion years."

I think this may avoid any confusion about the units used.

 

[marcus]

Hi Jorrie, thanks for the comment! Please keep me apprised of other things you notice. I'll think about switching to a coarser timescale and play around with it, but I probably won't shift over at least right away at this point. Very used to the 1/144 % per million year format, now. It has become a habit. But as you show it wouldn't be difficult to edit over to the coarser timescale format--just by moving the decimal point at strategic places. I'll try a kind of compromise edit in this post to see how it goes (but am not promising to shift permanently.)

Here's another example. Suppose you want to know the rate distances were growing at a time in the past when distances (between unmoving points) were HALF what they are today. Well then volumes were 1/8 of present size and the matter density then ρ(then) = 8*0.239 nanopascal.

We can use Friedman equation again

H(then)² = H_∞² + (1/Φ)8*0.239 nanopascal

H(then) = (1/17300^2 per (million years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5

I paste that in, and it tells me H(then) = 3.9 x 10^-18 Hz

in other words a number per second. But I want a number per billion years so I multiply the answer by a billion years. That means I paste in:
(1/17300^2 per (million years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5 * billion years
and it tells me H(then) = 0.12323... per billion years
Now 0.123… is about one eighth. Actually a bit less than 1/8, more like 1/8.1
The answer therefore is H(then) = 1/8.1 per billion years, and taking it in smaller steps that is 1/81 of a percent growth per million years.
The Hubble growth rate, which is now 1/144% per million years WAS back then 1/81% per million years.
And the Hubble radius, which is now 14.4 Gly, was 8.1 Gly.

Actually I like that kind of hybrid explanation. Let's try "take two" of the hybrid:

in other words a number per second. I want to know the Hubble time back then in billion years so I multiply the answer by a billion years. That means I paste in:(1/17300^2 per (million years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5 * billion years
and it tells me H(then) = 0.12323... per billion years
That 0.123… is about one eighth--actually a bit less than 1/8, more like 1/8.1
So the Hubble time back then was 1/H(then) = 8.1 billion years. As we know from regularly converting Hubble times and Hubble radii to percentage growth rates, this corresponds to H(then) being a distance growth rate of 1/81 of a percent per million years.
The Hubble growth rate H(t), which is now 1/144% per million years was 1/81% per million years, back then, and the Hubble radius, which is now 14.4 Gly, was 8.1 Gly.


You might be right. It might be better to completely switch over to billion years.
Then one gets fractions with a decimal point in the denominator: e.g. 1/14.4
and 1/17.3 but and one loses touch with the language of percentage growth rates. The percentages would be
1/0.144 percent per billion years and 1/0.173 % per billion years.
But to compensate, some calculations like this one would be considerably trimmer! We can keep this open and continue considering it.

 

[marcus]

The first occurrence of the cosmological constant Lambda in the GR equation, around 1920 (actually as early as 1917), is of interest in this connection. A source:
Einstein, A. 1917. Kosmologischege Betrachtungen zur allgemeinen Relativitätstheorie. Sitzungsb. König. Preuss. Akad. 142-152, reprinted and translated in The Principle of Relativity (Dover, 1952) 175-188
This recent historical paper quotes a recently translated original source that is later (1931):
http://arxiv.org/abs/1402.0132

I found the first occurrence of Lambda (cosmological constant) I know of in a 1917 paper Cosmological Considerations on the General Theory of Relativity translated in the Dover book on page 179, equations 2 and 3.
the book is available online at Internet Archive (archive.org) so it doesn't require a trip to the library.

 

[marcus]

The issue of entropy gets raised from time to time in connection with bounce cosmologies. People who think of entropy as an ABSOLUTE physical quantitity, rather than an observer-dependent one, occasionally ask how it apparently got to be so low at the start of expansion (if the start was a rebound from prior contracting phase.) I responded in the context of separate discussion so, for convenience, I'll save the reply here where I can refer to it easily.

If you look at how entropy is defined you see it is observer-dependent because it depends on the observer's coarse-graining---the macrovariable versus microvariable distinction. Entropy is the logarithm of the number of microstates (based on degrees of freedom irrelevant to the observer) comprising one grand macrostate (based on d.o.f that he actually interacts with and which affect him).

Any observer has a coarse-graining map corresponding to the lumping together of microstates into macrostates (consolidating all those which don't make any difference to the observer). Entropy measures the "size" in the particular macrostate we're in. The amount of information in it, that we ignore.

There's a group of people who think of entropy as absolute, who don't think that when you talk about it you have to specify a coarse-graining map. It is difficult for them to accept bounce cosmology because it looks to them as if "the entropy" (an absolute quantity) was reset to zero at the bounce. And there are other people who don't have that problem.

If you think of entropy as defined for a particular coarse-graining, then you don't encounter that mental obstacle. There is a pre-bounce guy and according to his coarsegraining the entropy increases astronomically as you go into the bounce, and it never thereafter declines! Because everything post-bounce is irrelevant to him, like it was inside the horizon of a black hole, the whole universe.
The post-bounce guy has a DIFFERENT coarsegraining and he sees the entropy initially low, everything about the bounce matters to him, is of vital importance, affects him thru variables he interacts with. Then as the U expands and diversifies regions of phase space become indifferent and irrelevant to him and entropy (for the post-bounce guy) increases.

The second law holds for any particular guy's entropy---defined based on his coarse-graining of the world.

This has been pointed out by various people. I think probably it would have come up in your Abhay&Ivan interview documentary video. As I recall Thanu Padmanabhan stated it clearly. Entropy is observer-dependent, or words to that effect. I've lost track of all the people who have made that point. Recently it came up here:
http://arxiv.org/abs/1407.3384
Why do we remember the past and not the future? The 'time oriented coarse graining' hypothesis
Carlo Rovelli
(Submitted on 12 Jul 2014)
Phenomenological arrows of time can be traced to a past low-entropy state. Does this imply the universe was in an improbable state in the past? I suggest a different possibility: past low-entropy depends on the coarse-graining implicit in our definition of entropy. This, in turn depends on our physical coupling to the rest of the world. …

Some more reading, if curious:
http://arxiv.org/abs/gr-qc/9901033
http://arxiv.org/abs/hep-th/0310022
http://arxiv.org/abs/hep-th/0410168

To give a bit of the flavor I'll quote a passage from Don Marolf's 2004 paper

==quote http://arxiv.org/abs/hep-th/0410168 from conclusions==
the realization that observers remaining outside a black hole associate a different (and, at least in interesting cases, smaller) flux of entropy across the horizon with a given physical process than do observers who themselves cross the horizon during the process. In particular, this second mechanism was explored using both analytic and numerical techniques in a simple toy model. We note that similar effects have been reported³⁵ for calculations involving quantum teleportation experiments in non-inertial frames. Our observations are also in accord with general remarks^36,37 that, in analogy with energy, entropy should be a subtle concept in General Relativity.
We have concentrated here on this new observer-dependence in the concept of entropy. It is tempting to speculate that this observation will have further interesting implications for the thermodynamics of black holes. For example, the point here that the two classes of observers assign different values to the entropy flux across the horizon seems to be in tune with the point of view (see, e.g., Refs. 38,39,40,41,42) that the Bekenstein-Hawking entropy of a black hole does not count the number of black hole microstates, but rather refers to some property of these states relative to observers who…
==endquote==
For context see: https://www.physicsforums.com/showthread.php?p=4810929#post4810929

 

[marcus]

I got a PM letter yesterday from one of our members asking basic questions about the declining Hubble expansion rate, the increasing growth *speed* of the scale factor and related matters, so decided to respond here in case it could be useful for anybody else.
In words: H(t) is a fractional growth rate, conveniently expressed as percentage growth per million years, rate is different from *speed*. The speed a distance grows is proportional to its size, larger distances grow faster.
So the percentage RATE can be DECLINING even though if you watch a particular distance it's growth SPEED can be increasing.
Like if you have a savings account at the bank, the percent INTEREST on it can be constant or even slowly declining and yet, because your principal is growing your savings can still be increasing by a greater amount each year, in gross dollar terms.

I think it's good to look at a concrete example. A purely verbal description like the above leaves something missing. Let's look at some numbers, using Jorrie's Lightcone calculator:
This table runs from year 67 million, when distances were 1/40 their present size, out to year 28.6 billion when distances will be 2.5 times their present size. The present is year 13.787 billion where you see scale factor a(t) = 1 and its reciprocal S(t) = 1 indicating distances are exactly their present size.
The way to read the percentage growth rate is to mentally multiply the R column number by TEN and take ONE OVER THAT. So in year 67 million, the growth rate was one percent per million years
Is that clear? You multiply 0.1 by ten and get 1, and one over one is one.

I would advise getting used to reading the R column that way. Another example: in year 135 million distances were growing 1/2 percent per million years , you take 0.2, multiply by ten to get 2 and take one over two to get 1/2.

You can see from the table that at present, year 13.787 billion, distances are growing 1/144 percent per million years. The percentage rate has come down a lot over time and it is continuing to decline towards a limit of 1/173 %. The table extends into the future far enough to show it getting down to 1/171 %, which is getting close to where it is expected to end up.

{\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&amp;S&amp;T (Gy)&amp;R (Gly)&amp;a&#039;R_{0} (c) \\ \hline 0.025&amp;40.000&amp;0.067&amp;0.1&amp;3.53\\ \hline 0.031&amp;31.764&amp;0.096&amp;0.1&amp;3.14\\ \hline 0.040&amp;25.223&amp;0.135&amp;0.2&amp;2.79\\ \hline 0.050&amp;20.030&amp;0.192&amp;0.3&amp;2.49\\ \hline 0.063&amp;15.905&amp;0.271&amp;0.4&amp;2.22\\ \hline 0.079&amp;12.630&amp;0.384&amp;0.6&amp;1.97\\ \hline 0.100&amp;10.030&amp;0.543&amp;0.8&amp;1.76\\ \hline 0.126&amp;7.965&amp;0.768&amp;1.2&amp;1.57\\ \hline 0.158&amp;6.325&amp;1.085&amp;1.6&amp;1.40\\ \hline 0.199&amp;5.022&amp;1.531&amp;2.3&amp;1.25\\ \hline 0.251&amp;3.988&amp;2.159&amp;3.2&amp;1.13\\ \hline 0.316&amp;3.167&amp;3.035&amp;4.5&amp;1.02\\ \hline 0.398&amp;2.515&amp;4.243&amp;6.1&amp;0.94\\ \hline 0.501&amp;1.997&amp;5.876&amp;8.1&amp;0.89\\ \hline 0.631&amp;1.586&amp;8.009&amp;10.4&amp;0.87\\ \hline 0.794&amp;1.259&amp;10.665&amp;12.6&amp;0.91\\ \hline 1.000&amp;1.000&amp;13.787&amp;14.4&amp;1.00\\ \hline 1.259&amp;0.794&amp;17.262&amp;15.6&amp;1.16\\ \hline 1.495&amp;0.669&amp;20.000&amp;16.3&amp;1.32\\ \hline 1.774&amp;0.564&amp;22.823&amp;16.7&amp;1.53\\ \hline 2.106&amp;0.475&amp;25.701&amp;16.9&amp;1.79\\ \hline 2.500&amp;0.400&amp;28.613&amp;17.1&amp;2.11\\ \hline \end{array}}

This table is an implementation of the FRIEDMANN EQUATION as a table of numbers instead of as an equation. It's good to study the equation and understand it, but I think it also helps to mull over the actual numbers of the history of the universe which the equation generates when you plug in the observed values of the parameters and run it.

The rightmost column is the growth *SPEED* of a chosen sample distance whose present size is 14.4 billion light years. You can see it starts out (way back in year 67 million) at 1/40 of its present size and growing at 3.53 times the speed of light.
And that speed declines until around year 8 billion.
And then it starts to increase.
And by now, in year 13.787 billion, it is increasing at exactly the speed of light.
So ever since year 8 billion it has been, in a manner of speaking, "accelerating".
But the word is not quite apt. Distance growth is not like ordinary motion. Nobody GETS anywhere by it, everything just becomes farther apart. So the word "accelerating" is just slightly misleading and can give a false mental image. It just means that the speed of distance growth is increasing.
Although of course as we noted earlier the percentage RATE of distance growth is declining.

 

[thedaybefore]

I said I would try to avoid abbreviations, but I need another one: CMB for cosmic microwave background.

The balloon analogy teaches various things, but sometimes you have to concentrate in order to learn them.

One thing it teaches is what it means to be not moving with respect to CMB.

the balloon is a spherical surface and as it gradually expands a point that always stays at the same longitude and latitude is stationary with respect to CMB.

Distances between stationary points can increase, and in fact they do. They increase at a regular percentage rate (larger distances increase more). In our 3D reality this is called Hubble Law. It is about distances between points which are at rest wrt CMB.

In our 3D reality you know you are at rest wrt CMB if you point your antenna in all directions and get roughly the same temperature or peak wavelength. There is no doppler hotspot or coldspot in the CMB sky. That means you are not moving with respect to the universe.

In cosmology being at rest is a very fundamental idea, we had it even before the 1960s when the CMB was discovered. Then it was defined as being at rest with respect to the process of expansion---you could tell you were at rest with respect to the universe if the expansion around you was approximately the same in all directions---not faster one one side of the sky and slower on the other, but balanced. It is the same idea but now we use the CMB to define it because it is much more accurate. Sun and planets are traveling about 380 km/s with respect to CMB in a direction marked by the constellation Leo in the sky. It is not very fast but astronomical observations sometimes need to be corrected for that motion so as to correspond to what an observer at CMB rest would see.

Now let's take another look at the balloon and see what else it can tell us.

The CMB is electromagnetic radiation and all non-accelerating reference frames are non-moving to the observer. Accordingly, how can a non-accelerating observer not be at rest with respect to the CMB? All CMB will be moving at C to every observer.

 

[marcus]

At rest wrt CMB MEANS temperature essentially the same in all directions.

Solar system we know is not at rest, for reason given in what you quote. There is a hot spot in constellation Leo. And a cold spot in the opposite direction.
What you quoted says 380 but a better figure is the solar system is moving about 370 km/s in the Leo direction, relative to the soup of ancient light. A recent report says 369 km/s
That is about 0.123 of a percent of the speed of light.
Therefore the temperature in that direction in the sky is 0.123 of a percent WARMER than the average CMB sky temperature. Something like 0.003 kelvin warmer than the average 2.725 kelvin

Another way to say observer "at rest" is to say ISOTROPIC observer. Isotropic means "universe looks the same in all directions" In particular the CMB temperature is the same in all directions.

An observer riding with the solar system is not an isotropic observer because there is a measurable temperature "dipole", a hotspot coldspot axis.

Hubble already discovered this motion, or dipole, before the CMB was known. The galaxies in the Leo direction are receding on average a LITTLE SLOWER than the overall Hubble rule predicts. This is because the solar system is not at rest wrt universal expansion process. And galaxies in the opposite direction are receding a little faster .

Discovering the CMB and the temperature hotspot only made this more accurate, but it was already known that the universe has a criterion of rest.

It does not depend on the electromagnetic field, it depends on the approximately uniform distribution of the ancient matter, the primordial gas.

 

[marcus]

A good recent report is
http://arxiv.org/abs/0803.0732
Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Data Processing, Sky Maps, and Basic Results
G. Hinshaw, J. L. Weiland, R. S. Hill, N. Odegard, D. Larson, C. L. Bennett, J. Dunkley, B. Gold, M. R. Greason, N. Jarosik, E. Komatsu, M. R. Nolta, L. Page, D. N. Spergel, E. Wollack, M. Halpern, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. L. Wright
(Submitted on 5 Mar 2008 (v1), last revised 17 Oct 2008 (this version, v2))
We present new full-sky temperature and polarization maps in five frequency bands from 23 to 94 GHz, based on data from the first five years of the WMAP sky survey. The five-year maps incorporate several improvements in data processing made possible by the additional years of data and by a more complete analysis of the ...
==quote==
samples from both methods to produce the conservative estimate shown in the bottom row. This approach, which enlarges the uncertainty to emcompass both estimates, gives

(d, l, b) = (3.355 ± 0.008 mK, 263.99◦ ± 0.14◦, 48.26◦ ± 0.03◦), (1)

where the amplitude estimate includes the 0.2% absolute calibration uncertainty. Given the CMB monopole temperature of 2.725 K (Mather et al. 1999), this amplitude implies a Solar System peculiar velocity of 369.0 ± 0.9 km s−1 with respect to the CMB rest frame.
==endquote==

See also this one:
http://arxiv.org/abs/1303.5087
Planck 2013 results. XXVII. Doppler boosting of the CMB: Eppur si muove
Planck Collaboration: N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, A. J. Banday, R. B. Barreiro, J. G. Bartlett, K. Benabed, A. Benoit-Lévy, J.-P. Bernard, M. Bersanelli, P. Bielewicz, J. Bobin, J. J. Bock, J. R. Bond, J. Borrill, F. R. Bouchet, M. Bridges, C. Burigana, R. C. Butler, J.-F. Cardoso, A. Catalano, A. Challinor, A. Chamballu, L.-Y Chiang, H. C. Chiang, P. R. Christensen, D. L. Clements, L. P. L. Colombo, F. Couchot, B. P. Crill, F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, J. M. Diego, S. Donzelli, O. Doré, X. Dupac, G. Efstathiou, T. A. Enßlin, H. K. Eriksen, F. Finelli, O. Forni, M. Frailis, E. Franceschi, S. Galeotta, K. Ganga, M. Giard, G. Giardino, J. González-Nuevo, v1), last revised 10 Nov 2014 (this version, v2))
Our velocity relative to the rest frame of the cosmic microwave background (CMB) generates a dipole temperature anisotropy on the sky which has been well measured for more than 30 years, and has an accepted amplitude of v/c = 0.00123, or v = 369km/s. In addition to this signal generated by Doppler boosting of the CMB monopole, our motion also modulates and aberrates the CMB...
...gnificant confirmation of the expected velocity.

 

[thedaybefore]

At rest wrt CMB MEANS temperature essentially the same in all directions.

Solar system we know is not at rest, for reason given in what you quote. There is a hot spot in constellation Leo. And a cold spot in the opposite direction.
What you quoted says 380 but a better figure is the solar system is moving about 370 km/s in the Leo direction, relative to the soup of ancient light. A recent report says 369 km/s
That is about 0.123 of a percent of the speed of light.
Therefore the temperature in that direction in the sky is 0.123 of a percent WARMER than the average CMB sky temperature. Something like 0.003 kelvin warmer than the average 2.725 kelvin

Another way to say observer "at rest" is to say ISOTROPIC observer. Isotropic means "universe looks the same in all directions" In particular the CMB temperature is the same in all directions.

An observer riding with the solar system is not an isotropic observer because there is a measurable temperature "dipole", a hotspot coldspot axis.

Hubble already discovered this motion, or dipole, before the CMB was known. The galaxies in the Leo direction are receding on average a LITTLE SLOWER than the overall Hubble rule predicts. This is because the solar system is not at rest wrt universal expansion process. And galaxies in the opposite direction are receding a little faster .

Discovering the CMB and the temperature hotspot only made this more accurate, but it was already known that the universe has a criterion of rest.

It does not depend on the electromagnetic field, it depends on the approximately uniform distribution of the ancient matter, the primordial gas.

What concerns me here is that it sounds suspicously like you are creating a preferred reference frame forhe Universe. It may be that there are differences in temperature of the microwave background and that the objects in the Universe have relative velocity to each other, I don't believe we can say that the CMB has some ultimate zero velocity. At best, I think we can say there is relative velocity between the objects that emitted the CMB 14 billion years ago and us.

 

[marcus]

the U represents a particular solution of GR. In the mother theory (GR) there is no preferred ref frame. But individual solutions to GR equation can have preferred frame specific to that solution.
So in cosmology we have a preferred frame.

It depends on initial conditions---eg a particular configuration of initial matter. even.

The solution is basically the Friedmann solution, the Friedmann metric. It has a preferred time called universe time or Friedmann time.

AFAIK everybody who does cosmology knows there is a preferred time, and a criterion of rest with respect to the universe (aka wrt CMB).

If you don't believe me, there is nothing I can say. This is basic.