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

marcus

The last 5 pages or so have been largely devoted to discussing the development of what I think is a really fine cosmology teaching/learning calculator, by Jorrie, and also working out some simplified equations that approximately reproduce part the expansion history (after the radiation energy density stopped being a major factor in the early U.) I should stress that Jorrie's calculator is what I would call professional grade--it reproduces the standard cosmic model--whereas the other thing we were working on is different, more of a "toy model".

Now the calculator (currently version A25) has its own sticky thread "Look 88 billion years into the future..." and I'd like to find a way to get this thread back into the groove of helping to "get us all on the same page."

One thing that could be highlighted, that we haven't discussed much here so far, is that if a massive particle or object is given a kick so that it has its own individual motion relative to the universe rest frame---the "Hubble flow"---CMB rest, it will gradually slow down relative to CMB rest and given enough time will REJOIN the "Hubble flow", or come approximately to a STOP relative to the ancient light.

This is rather un-Newtonian and could be unintuitive to newcomers. It violates conventional conservation notions. But it is really basic to understanding so we should talk about it. It is analogous to the redshifting of light. the light loses energy and momentum as it travels across cosmological distances, although its speed doesn't change.
With a massive object, the mass doesn't change but the speed does, so there is the same loss of energy and momentum.

A thing's individual velocity relative to the ancient light is called its peculiar velocity (meaning "special to itself", not weird). The basic message is if a massive object is given a kick so it acquires some peculiar velocity (relative to CMB rest) then over a long period of time that velocity will tail off to zero and it will asymptotically come to rest.

This figures in discussions of the "tethered galaxy problem". We were discussing that in another thread and Jorrie plotted some informative curves . For now, at least, I'll just give a link to his post:
http://physicsforums.com/showthread....96#post4082096

BTW when thinking about the balloon analogy it's good to remember that fixed points on the balloon surface represent points at CMB rest. A galaxy at some fixed point like that sees the CMB the same temperature in all directions, instead of having a doppler hotspot caused by its own peculiar motion. In Ned Wright's balloon animation all the galaxies are at rest (no peculiar motion) shown by their staying always at the same latitude and longitude on the balloon surface. The photons on the other hand have motion. Each moves at constant speed in constant great-circle direction. However if you watch carefully will see their wavelengths enlarge to symbolically show redshift.

Jorrie

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.

[itex]L = r^2 m d\phi/dt [/itex] = constant. Since [itex]v = r d\phi/dt [/itex], it means that for non-relativistic speeds, the particle speed scales with [itex]1/r[/itex].

If the balloon is being inflated, the particle must lose surface speed, just like a Kepler orbit that is losing orbital speed at larger radius. If the increase in balloon radius is kept up, the particle’s surface speed will eventually approach zero, as radius tends to infinity. In cosmology, this is usually described as 'joining the Hubble flow'.

The analogy seems to hold even for relativist particles. The relativistic Keplerian equation for the conservation of orbital angular momentum is:

[itex]L = (1-v^2/c^2)^{-0.5} r^2 m d\phi/dt [/itex] = constant (e.g. MTW eq. 25.18).

This simple scheme can be shown to reproduce the curves of figure 3.5 obtained by Davis (2004) [http://arxiv.org/abs/astro-ph/0402278] (with the exception of [itex]v=c[/itex]).



I think that the [itex]v=c[/itex] case can also be handled by the analogy; the particle’s momentum must then be expressed in terms of the de Broglie wavelength.

Edit:
Relativistic de Broglie wavelength is given by: [itex]\lambda=\gamma h/(m v)[/itex], where [itex]\gamma[/itex] is the Lorentz factor.

If we write the angular momentum of the 'balloon particle' in terms of surface velocity, it is simply [itex]L = \gamma r m v [/itex]. Taking [itex]\gamma[/itex] from the de Broglie wavelength, gives the conservation of angular momentum as

[itex]L = r h / λ =[/itex] constant, valid for photons and matter.

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PeterJ

I have no ability to understand a sphere with nothing off the surface. I'd rather have Plotinus's hypersphere. This is more or less a balloon analogy, with the spacetime universe on the surface as usual, but it has a centre and at least one extra dimension. Then we have the ability to include an extra dimension (or bundle of them) at each point in spacetime, since this would be represented as a connection to the centre point. Plotinus was not talking physics exactly, but his model seems more sophisticated than a 3D balloon representing a 2D spacetime.

But I am way out my depths here.

marcus

Instructive example! A mystic will postulate additional details (like the "edge" of the universe, or extra spatial dimensions) because they appeal to his imagination. Or even that they are "required" by his imagination.

the type of person we could call pragmatic or perhaps "Occamite" will avoid adding features which lack an operational meaning---i.e. some way to experience, even if very tenuous or indirect.

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.

PeterJ

No it's okay. I'm happy with my way of thinking about it. I'd say that the topography of the universe is unrepresentable as a visual model, and so within quite wide limits it would be a matter of personal preference how we do it. A Klein bottle or Necker cube would also be relevant images.

I'm not sure what you mean about 'mystics' and the stuff that appeals to their imagination. It has nothing to do with imagination. If Plotinus is to believed he is trying to describe what he is seeing, and he was not seeing any edges, nor any inside or outside. Perhaps he is not to be believed, but his model does at least allow for the idea that distance is arbitrary, which seems to make it useful, and it's only one more dimension, making it more economical than string theory. He even adds the proviso 'it is as if'.

I'm not suggesting that his description is 'true', just something to consider.

marcus

New narrowed-down values of the cosmological parameters, coming out of the SPT (south pole telescope)
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:
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.

So the idea which is traditionally favored of perfect flatness and spatial infinite is hanging on by its 2 sigma fingernails. It is still "consistent" with the data at a 2 sigma level.

But the hypersphere ( abbreviated S³ for the 3D analog of the 2D surface of a ball ) is looming realer and realer as a kind of ignored elephant in the room. It could still go away of course. We avert our eyes and hope it will have the politeness to do so.

For Jorrie's A25 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

or with more precision put these into Google calculator:
1/(69.62 km/s per Mpc)
1/(69.62 km/s per Mpc)/.7152^.5

hagendaz

On the subject of space expansion....
If I built a spaceship with a machine (an engine if you will) on the tail end of the ship that eliminated all effects of gravity at the tail end of the ship , so as to counter act any gravitational pull on the tail end of the ship, would my spaceship thus be capable of traveling at light speed as I ride the expansion of space in what ever direction my ship is pointed?
Or can I only travel away from the center of the universe?

Would I instantly be moving at the same speed as the space is expanding the moment I turn on my spaceship, but feel no acceleration?

marcus

The expansion of distances doesn't GO anywhere. No person or object approaches any destination. Simply put: Things that aren't held together by their own gravity or molecular forces just get farther apart.

There is no "center" that anyone can point to.

So you cannot "ride" the expansion of space in any direction. Since there is no center you cannot " travel away from the center" as you say, either.

Typical very largescale distances grow several times faster than the speed of light but nothing travels anywhere.

The balloon analogy is intended to illustrate those things to make them easy to visualize. You might try studying the brief animation movie of it, reading some of this thread, or the FAQs.

You could start your own thread with this question, since it does not fit in so well in this balloon analogy thread.

Larkus

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

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

marcus

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.
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:
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

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 thats not likely lol but its often the way laymen like myself tend to misconstrue analogies.

Jorrie

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