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

[DaveC426913]

You said there's a special place on the balloon "diametrically opposite us". Which means it can't be diametrically opposite any other point. That means our spot on the balloon is special - unlike any other point on it.

Try reviewing your extension to the balloon analogy.

Where are we on the balloon? Call it point A
Where is this diametrically opposite point? Call it point A'.
How does A' behave such that it affects A? Does it contract ('suck' things)?

OK, now. Pick a point 90 degrees around the balloon from us. Call it point B.
Does it see our point (A')? Or its own (B')?
Does it see exactly the same phenomena there as we do here? (equal 'sucking' in all directions?)

If the former, then we see a unique view of the universe, shared by no other point,
If the latter, then you have two x' points. Indeed, you have infinite n' points, one for each x.

 

[Perduta]

... If the latter, then you have two x' points. Indeed, you have infinite n' points, one for each x.

Correct. Every point in the universe sees everything else falling away from them just like we do. That is because the centre of gravity of the universe is unique to every observer.

Observers at your point A will see point B falling towards your A'. Observers at B will see A falling towards B' ...and so on. Thus the explanation for accelerated expansion of the universe and redshift is amazingly simple and consistent with Einstein's equivalence principle. It also explains why the space/time each observer sees and the apparent relative movement is different too.

p.s. also it explains why the acceleration was greater in the past and will keep getting smaller in the future: In the past the balloon was smaller so the centre of gravity was closer and we all know that gravity decreases with distance.

 

[Drakkith]

p.s. also it explains why the acceleration was greater in the past and will keep getting smaller in the future: In the past the balloon was smaller so the centre of gravity was closer and we all know that gravity decreases with distance.

The expansion is accelerating and will become greater in the future. However this did not start to happen until recently. Prior to this point in time the expansion was slowing down because of gravity.

 

[DaveC426913]

Correct. Every point in the universe sees everything else falling away from them just like we do. That is because the centre of gravity of the universe is unique to every observer.

Observers at your point A will see point B falling towards your A'. Observers at B will see A falling towards B' ...and so on. Thus the explanation for accelerated expansion of the universe and redshift is amazingly simple and consistent with Einstein's equivalence principle. It also explains why the space/time each observer sees and the apparent relative movement is different too.

Then you have not described anything new at all. In order for each point to see every other point as receding, the balloon must be expanding consistently, at every point on its surface, which is exactly what the model shows in the first place.

 

[Perduta]

The expansion is accelerating and will become greater in the future. However this did not start to happen until recently. Prior to this point in time the expansion was slowing down because of gravity.

Meh - I read somewhere that the Hubble constant was decreasing. I.e in the future galaxies at a certain distance will be accelerating slower than the ones we can see currently at that same distance.

OTOH if it is true that the universe is breeding more and more dark matter/energy to fill it's expansion then the total mass will keep growing in proportion to the volume i.e R³ while gravity decreases with R², so that would make sense too as the acceleration would then be proportional to the Radius of the universe and eventually we would get to the point where evereything is just ripped apart. It all depends on this dark matter malarkey.

p.s. Note: In the 2D balloon model the radius of the universe woulod correspond to half the circumference of the balloon.

 

[bapowell]

=
while gravity decreases with R²

Oh goodness. Are you really thinking Newtonian gravity here?

 

[Perduta]

Then you have not described anything new at all. In order for each point to see every other point as receding, the balloon must be expanding consistently, at every point on its surface, which is exactly what the model shows in the first place.

Except that I give an explanation for what makes the balloon expand.

 

[Perduta]

Oh goodness. Are you really thinking Newtonian gravity here?

Meh you're right, I can't be arsed with a forum full of know-it-alls.
I was just asking a question.

 

[bapowell]

Meh you're right, I can't be arsed with a forum full of know-it-alls.

Just want to be sure we are using the right tools to answer the questions. No offense intended. You are bound to lead yourself astray if you are applying Newtonian physics to the balloon analogy...

 

[bapowell]

Well feel free to explain what gravitational force and the direction thereof that YOU think the sum total mass of the universe exerts on us.

In a perfectly homogeneous and isotropic universe, the net gravitational attraction from all other objects in the universe on our galaxy would be zero. We would have zero velocity relative to the expansion. This is not the case, however. Local gravitational sources (galaxies in our local group) dominate any net force on our galaxy, leading to a nonzero peculiar velocity. But this has nothing to do with the expansion of the universe.

The expansion of the universe is not due to a gravitational field in the Newtonian sense, which is why it's important not to think in those terms.

 

[bapowell]

We don't use GR because gravity travels at the speed of light; we use it because it is the modern theory of gravity. You have not demonstrated in any rigorous way (beyond just using words) that Gauss's law gives you the observed expansion of the universe. In know this can't work, because, as I've stated to you, the net gravitational field in a homogeneous universe is zero. This is not a matter of opinion -- it can be calculated -- there simply is no potential gradient across a uniform energy density.

 

[Drakkith]

I agree one should probably apply relativistic effects (e.g. assuming gravity travels at the speed of light) but Newtonian physics already gives a better explanation than simply claiming "expansion of space" where we can't even identify the physical thing that is expanding.

Newtonian physics is already known to be incorrect (or inaccurate if you prefer). Why would you use it? It does not explain many things that GR does.

 

[Perduta]

Newtonian physics is already known to be incorrect (or inaccurate if you prefer). Why would you use it? It does not explain many things that GR does.

Well none of the geniuses here answered the simple Newtonian question in respect of the centre of gravity and the consequent accelleration we should see towards it... so what's the point in alluding to general relativity and more complex models. I thought we were discussing the balloon analogy. My bad.

 

[Drakkith]

Well none of the geniuses here answered the simple Newtonian question in respect of the centre of gravity and the consequent accelleration we should see towards it... so what's the point in alluding to general relativity and more complex models. I thought we were discussing the balloon analogy. My bad.

The balloon analogy has nothing to do with Newtonian physics, so of course people either aren't going to be able to answer it or won't have much reason in trying. In Newtonian physics the balloon cannot expand. It doesn't even make any sense because there is no such thing as spacetime geometry in that theory. Don't take the analogy and use it where it doesn't apply, as it will not work. And please, enough with the attitude. You came here and asked us, we did not seek you out.

 

[Perduta]

The balloon analogy has nothing to do with Newtonian physics, so of course people either aren't going to be able to answer it or won't have much reason in trying. In Newtonian physics the balloon cannot expand. It doesn't even make any sense because there is no such thing as spacetime geometry in that theory. Don't take the analogy and use it where it doesn't apply, as it will not work. And please, enough with the attitude. You came here and asked us, we did not seek you out.

I think the replies were condescending, dismissive and confrontational as well as going off at a tangent from my original question which is WHAT physical entity are we to understand is "expanding" with respect to the way measure it.

Secondly, the Equivalence principle clearly states that acceleration is equivalent to gravity which made me think on the spur of the moment, where could that gravity be coming from. Gauss clearly says that the volume integral of mass is equal to the change of flux through the surface of said volume so instead of thinking about it we should just poo-poo the whole of Newtonian physics and allude to some vastly more complicated explanations that only "real" scientists can understand.

OK, I got the gist of these forums, which is what I thought they would be like.

 

[DaveC426913]

OK, I got the gist of these forums, which is what I thought they would be like.

Well, all I can tell you is that I see three types of people on PF.
1] There are professionals, who do this for a living.
2] There are laypeople, with varying degrees of knowledge, who come here to ask questions and learn from 1] and 2],
3] There are laypeople, with varying degrees of knowledge, who come here to voice their own opinions, and aren't interested in learning.

You're a 2] but if you're not open to corrections then you risk being treated as if you are a 3].

The key is to put forth your ideas as questions, not assertions.

I'm a 2]. I hope you stick around. We need more 2]s.

 

[Drakkith]

I think the replies were condescending, dismissive and confrontational as well as going off at a tangent from my original question which is WHAT physical entity are we to understand is "expanding" with respect to the way measure it.

I'm sorry you feel that way. I think the replies were entirely free of all of that and were simply explaining what was incorrect with your line of thinking. This is an internet forum, and nonverbal and verbal cues as to how each person is talking to you do not exist. Most of the time people are simply giving facts and trying to explain the correct concepts and are not trying to be condescending or anything else at all. If you are not used to being corrected then it can seem like peoples posts are far harsher than they really are.

Secondly, the Equivalence principle clearly states that acceleration is equivalent to gravity which made me think on the spur of the moment, where could that gravity be coming from.

That's fine, but the expansion of the universe is not an acceleration in local space. IE you would never feel an acceleration because from your frame of reference, you are not accelerating. A galaxy 5 billion light years away is also not accelerating in it's frame of reference either. However the distance between us and that galaxy is increasing.

OK, I got the gist of these forums, which is what I thought they would be like.

I hope that gist is of a forum with people willing to help people understand the current mainstream theories of science. Otherwise there is a misunderstanding between us.

 

[StandardsGuy]

Evidently our rigid conceptual measuring sticks are not "expanding" and 3 meters in any direction remains 3 meters in that direction regardless.

So my question is...would the equivalence principle not allow us to attribute it to a gravitational field that is "outward" bound?

Show me the evidence for that.

No the equivalence principle would not.

 

[StandardsGuy]

Well none of the geniuses here answered the simple Newtonian question in respect of the centre of gravity and the consequent accelleration we should see towards it... so what's the point in alluding to general relativity and more complex models. I thought we were discussing the balloon analogy. My bad.

According to current theory, the universe has no center, so there is no center of gravity. You came here with an agenda (you are preaching). This forum has rules against that. I have been warned, so I am warning you.

The balloon analogy is 2-dimensional (the surface of the balloon only). You apply imagination to make it 3-dimensional (or more). It gets bigger with time. That represents the expansion of space itself. Objects get farther apart like the dots on the balloon. It's just a theory, but it's the main-stream one. You can't propose another one here.

 

[GeorgeDishman]

I think the replies were condescending, dismissive and confrontational as well as going off at a tangent from my original question which is WHAT physical entity are we to understand is "expanding" with respect to the way measure it.

What is expanding is the vacuum.

Gauss clearly says that the volume integral of mass is equal to the change of flux through the surface of said volume so instead of thinking about it we should just poo-poo the whole of Newtonian physics and allude to some vastly more complicated explanations that only "real" scientists can understand.

The equivalent to that is to draw a line round an arbitrary region on the surface of the balloon. As it inflates, the region grows while the mass contained within it remains the same (neglecting small amounts crossing due to local 'proper motion'). A region like that is called a "co-moving volume".

OK, I got the gist of these forums, which is what I thought they would be like.

This particular thread was intended to produce a common explanation of a well-known model for newcomers which is somewhat different to other threads where various alternative ideas are often discussed. If you come with an expectation, you will probably only see what fits it.

 

[marcus]

I think you put the right spin on it, George.
This thread is essentially an adjunct to this short animated "movie" which I hope everyone has watched.
http://www.astro.ucla.edu/~wright/Balloon2.html
Watching it for a few minutes is a valuable exercise of geometrical imagination.

You see galaxies (white) remaining in the same place while distances between them increase.
You see photons (colored) traveling among the galaxies always at the same constant speed.

To learn from the balloon model you need to think of it as simply telling you about the changing geometry of space. It is not a physical analogy. It doesn't mean that space is a physical substance like rubber, it is about imagining geometric relationships changing over time, not explaining why they do.

To learn you need to imagine that there is no inside or outside to the balloon. All existence is concentrated on the surface and there is no 'anywhere else'. It is a 2D analog of our 3D space. There is no time-coordinate in the picture. Spacetime is not shown. You just see change happen as the movie plays.

The message of this 2D toy model of changing geometry is not that our real 3D space is LIKE that in the sense of being finite, or having a sphere topology. It might be infinite or finite--we don't know. It's just an exercise to help one learn to imagine how galaxies can stay in the same place (i.e. same latitude longitude on the balloon) while distances between them increase. In the 2D space of the balloon surface, and that is all the space there is, they are not moving. Meanwhile the photons of light are moving (across the surface) always at the same speed.

It's a simple message to absorb. Once a newcomer gets it into his or her imagination it is up to them to extend the visualization up a dimension from 2D toy analogy to real 3D. Then there's more, that's only a beginning.

I think your post captured this way of looking at things, so what I am doing is amplifying on what you said. It sounds like we are on the same page about this.

 

[GeorgeDishman]

To learn from the balloon model you need to think of it as simply telling you about the changing geometry of space. It is not a physical analogy. It doesn't mean that space is a physical substance like rubber, it is about imagining geometric relationships changing over time, not explaining why they do.

I think you need to a little careful about that. Whether spacetime is like a substance or not is somewhat philosophical ("substantivalism"), IMHO it would be better to rephrase that in a more neutral style or it may sound like "preaching" which was mentioned in a previous post.

To learn you need to imagine that there is no inside or outside to the balloon. All existence is concentrated on the surface and there is no 'anywhere else'. It is a 2D analog of our 3D space. There is no time-coordinate in the picture. Spacetime is not shown. You just see change happen as the movie plays.

That is one of the problems I highlighted with the model, however you can visualise the radial distance as comoving time.

I think your post captured this way of looking at things, so what I am doing is amplifying on what you said. It sounds like we are on the same page about this.

Mostly, I've already voiced my concerns about the model in an earlier post and I don't agree with your view regarding "only geometry", there is a view that the gravitational field described by the metric can have as much claim to an existence as the quantum field. I think it's better to leave that as an open question in philosophy, the interpretation is separate from the science.

 

[marcus]

George Jones made a point concisely in the Phinds' Balloon Critique thread in a quote stressing the purely geometric message of a pattern of "increasing distances between observers at rest with respect to" the CMB or words to that effect.

This reminds me of some things I've been wanting to say. Analogies are ideally used in a controlled way with a clear intent. In this thread I've tried (subject to limitations of time and energy) to make clear how I intend the analogy to be taken. Above all this thread is an adjunct to the short animated movie I've frequently linked to:
http://www.astro.ucla.edu/~wright/Balloon2.html
The most important thing is to watch the animation--essentially that's the "balloon analogy" topic of the thread. That and the takeaway geometry message.

A. The analogy is not intended to suggest the U is spatially finite. And it might or might not be. It might or might not be topologically a 3-sphere. We don't know about those things. The analogy is not about finite/nonfinite. I think that has been made explicit from time to time.

B. the main purpose is to illustrate a pattern of increasing distances between (approximately) stationary observers. In the animation one sees photons moving at constant speed among the observers, represented by little galaxies. Some distances between galaxies are obviously increasing more rapidly than the photons move. In a given amount of time the photons can obviously cover more distance with the help of expansion than they would in a static setting.
The individual galaxies stay at the same lati-longitude positions.

C. the message is geometrical, learning to visualize changing geometry. It is not meant to suggest anything about what space is made of or not made of If people start talking about whether space is a "substance" or not then AFAICS the analogy has gone out of control. It MAY OR MAY NOT be some kind of substance. I don't want to say anything about that with the balloon imagery. The analogy, as I think it is ideally presented, is geometrical not physical.

D. the idea is to think of all existence concentrated on the 2D surface. No inside or outside of the balloon. No "center" which would have to be in some higher-dimensional 3D space that the 2D creatures living on the surface do not know about. The analogy is not intended to say anything about whether there is a higher dimensional space in which ours is embedded. It is a sketch or diagram illustrating changing geometric relations among stationary observers and light, within a 2D toy model of our 3D space, so you then have to mentally extend your comprehension to the analogous 3D geometry.

 

[hitchiker]

the common balloon analogy and bread-raisin analogy about bigbang creates more doubts in the mind of people than they already had

1.if bigbang created space ,then where did Bigbang happen in the first place? if space is nothing how it can be created?is space finite or infinite?

there bigbang have no answer and are open questions in astronomy and philosophyI think the best way is to make an animation starting with present view of movement of bodies in universe and then going backwards in time drawing galaxies coming together and then congregating into denser and tinier space

"Bigbang created "spacetime" and bigbang itself are best description based on current understanding of universe which varies from time to time as we venture out farther in space but many people take these theories for granted without giving much intellectual thought

i don't really like this animation but something like this with smoother transition back to bigbang ..but what i love here is before BB there's still space but with no matter&energy but an empty space that has the potential for a bigbang
http://resources.schoolscience.co.uk/STFC/bang/bang.swf

unsolved problems
http://en.wikipedia.org/wiki/Unsolved_problems_in_physics

 

[marcus]

the common balloon analogy and bread-raisin analogy about bigbang creates more doubts in the mind of people than they already had

1.if bigbang created space ,then where did Bigbang happen in the first place? if space is nothing how it can be created?...

Here's an essay about the different meanings of the words "Big Bang" and some of the confusions in the mind of the public.
http://www.einstein-online.info/spotlights/big_bangs
It's part of the outreach program of a research institute in Europe.

Ordinary mainstream cosmology only kicks in after expansion has already started. It is not about scientists' guesses as to how the expansion began.

The balloon analogy is intended to help imagine the expansion process AFTER we already had matter and light, and expansion of distances was under way. It is not intended to help picture the "Very Beginning" of the process.

There certainly are a prominent minority of cosmologists who devote time to speculating about the Very Beginning. That involves guesswork and unverified assumptions---it is fine for their colleagues and the general public as well to regard conjectures about the start of expansion in a critical and skeptical light, if they pay any attention at all. Doubt is a good scientific attitude.

Most cosmologists spend their time gathering data and checking to see how well it fits the standard cosmic model (a model which does NOT go back to the very start of expansion.)

There is some confusion that arises because "Big Bang" is also used in a second sense: to mean the whole expansion process over the course of billions of years, as described by the standard cosmic model. That is supported by a lot of observational data and has been scrutinized and checked by a lot of people. It does not say anything about what happened at the very start. The confusion that comes from people using the words in two different senses is picked apart in that "Two Big Bangs" essay I linked to.

There's also a good article that corrects common misconceptions about the standard cosmic model, "Big Bang" in the second sense:
http://www.mso.anu.edu.au/~charley/papers/LineweaverDavisSciAm.pdf
The first page of the PDF file is blank, so scroll down.

Since the nature of the very start of expansion has not yet been determined, one is free to choose how one imagines it. I personally picture it as a rebound from a prior contracting phase. It has not been proven impossible for there to have been space and matter fields existing for 100s of years before the very start of expansion. Perhaps thousands of years or even more--I won't assume or suggest an upper limit. It's one possibility and it doesn't oblige you to worry about tricky stuff like "nothing" and "fluctuations in nothing". It is one of the alternatives being seriously studied by people whose specialty is called quantum cosmology (a small branch of research which is different from ordinary standard cosmology.) If you want a toy model to picture (simply to imagine, not to believe! ) think of the balloon deflating until it is fairly small and quantum effects resist further shrinking, and then rebounding and starting to swell up again. But no surrounding 3D existence, in this toy model, only the 2D surface.

It has not been scientifically shown that the very start of expansion must have also been a beginning of space or time. So one can choose to believe that or not, as one wishes.

But these sorts of of "Big Bang" issues are really OFF TOPIC in this thread, which is about understanding the geometry of expansion (after it got started) using a simple 2D model, namely:
http://www.astro.ucla.edu/~wright/Balloon2.html
So if you would like to discuss these things some more I'd like to suggest that you start a regular discussion thread (about the start of expansion and such matters) for that purpose.

 

[GeorgeDishman]

D. the idea is to think of all existence concentrated on the 2D surface. No inside or outside of the balloon. No "center" which would have to be in some higher-dimensional 3D space that the 2D creatures living on the surface do not know about. The analogy is not intended to say anything about whether there is a higher dimensional space in which ours is embedded. It is a sketch or diagram illustrating changing geometric relations among stationary observers and light, within a 2D toy model of our 3D space, so you then have to mentally extend your comprehension to the analogous 3D geometry.

Of course I agree that the 2D surface has to represent 3D space and there is no implication of embedding in a higher dimensionality. The radial coordinate in the balloon analogy is strictly plotted as the scale factor which, for a matter-dominated universe, goes as

a(t) = (t/t₀)^2/3

or equivalently

t = t₀a^3/2

It is therefore reasonable to identify the radial coordinate with cosmological time albeit on a non-linear scale. A section through the centre then can look like this:

http://www.astro.ucla.edu/~wright/omega_2.gif

though the light cones would not be exactly at 45 degrees to the radials. Conversely, they can be exact if distances on the surface of the balloon are considered scaled relative to proper distance by a factor dependent on radius.

http://en.wikipedia.org/wiki/Comoving_distance#Uses_of_the_proper_distance

This has the advantage that a small patch on the above diagram can be seen to be an approximation of the usual Minkowski spacetime diagram again bearing in mind some scaling issues.

 

[hitchiker]

It has not been scientifically shown that the very start of expansion must have also been a beginning of space or time. So one can choose to believe that or not, as one wishes.

thanks for saying it as it is .. you are a real science adviser :!)

 

[RayYates]

No it is not impossible at all in fact it is nearly inevitable. It is quite simply the plain old concept of centre of gravity.

To understand this, reflect first on how we would calculate the Earth's gravitational force at a point inside the Earth. The answer comes from Gauss' law.

Then replace that with doing the equivalent for being inside the universe: First think of our balloon again. Pick any point you like to represent us. What would you say is going to be the centre of gravity of the entire 2D balloon for that point? How would you apply Gauss' law on the balloon? Which way does the gravity suck things?

As an interested layman, earlier in this thread it helped me get my head around the balloon analogy when I was reminded that the 2D surface of the balloon is the entire universe. There is no inside or outside so there is no "center". Rather think of the inside of the balloon as the past, the outside of the balloon is the future.

Everything, energy, matter, space and time exist ONLY on the surface of the balloon which extends in 3 directions of space and 1 of time. That 4D space time may be closed (wrapped around on itself; exit one side you come back in the other) so there can be no CG calculation, or open (infinite) and again there can be no CG calculation.

I hope that helps a bit.

As for "What is expanding?", as I understand it, empty space has a tiny amount of energy (vacuum energy) so small that compared to gravity, it has only recently been calculated. Gravity slowed the expansion of the universe until the amount of space grew large enough that vacuum energy was able to overcome gravity between galaxies and the expansion began to accelerate.

Inside a galaxy, gravity is large enough that the galaxy is self contained and does not expand. Galaxy clusters also do not expand because they are close enough that they are gravitational bound.

Think of a rising loaf of bread. The air pockets in the bread expands, not the flour.

Is vacuum energy creating MORE space or is space stretching? Is the a meter the same between galaxies as outside galaxies. I think the consensus is that it is creating more space, otherwise the speed of light would depend on the space it was traveling through. There are some people who argue against that idea, but that is another discussion.

 

[bapowell]

Keep in mind that the expansion of space is a classical phenomenon, one that we should not need quantum mechanics (and, hence, vacuum energy) to understand.

 

[RayYates]

Keep in mind that the expansion of space is a classical phenomenon...

Sorry, I don't know what that means.

 

[bapowell]

Sorry, I don't know what that means.

I just mean that the expansion of space occurs as a solution to the classical equations of general relativity, i.e. the solution exists for classical matter sources (like, say, a matter-dominated FRW universe.) So, we should be able to understand the expansion of space without recourse to the vacuum energy of quantum fields -- expansion exists even in perfectly classical universes.

 

[RayYates]

...expansion exists even in perfectly classical universes.

Yes I see what you mean but does that also explain the observed increase in the rate of expansion. That's really what I was trying to get to.

 

[bapowell]

Yes I see what you mean but does that also explain the observed increase in the rate of expansion. That's really what I was trying to get to.

I see, no it doesn't. By "expansion" I took you to mean just general expansion, not the special case of accelerated expansion.

 

[marcus]

As an interested layman, earlier in this thread it helped me get my head around the balloon analogy when I was reminded that the 2D surface of the balloon is the entire universe. There is no inside or outside so there is no "center". Rather think of the inside of the balloon as the past, the outside of the balloon is the future.
... may be closed (wrapped around on itself; exit one side you come back in the other) so there can be no CG calculation, or open (infinite) and again there can be no CG calculation.

I hope that helps a bit.
...

@Ray,
I thought that part of your comment was well put and quite helpful, so wanted to emphasize it. Just because we use the balloon as a 2D toy model doesn't mean that actual 3D space has to be closed and finite. Space can be open and infinite. But as you said in either case one can't expect 3D space to have a central point.
=====================
I started a thread attempting to explain, in basic language and very briefly, the Cosmic Event Horizon (abbr. CEH). It may be useful to work it in here, as part of understanding the expansion process. Here's a second draft --- I corrected the title and added some material.
The existence of the Cosmic Event Horizon (CEH) depends on the fact that the scalefactor curve a(t) has a slope a'(t) which (although gradually decreasing for roughly the first half of the expansion age) is now slowly increasing. The scalefactor curve is very gradually getting steeper, and is expected to continue doing so.

Picture: http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg
the dark solid curve labeled (.27, .73).

This has the interesting consequence that the most distant galaxy which we could, today, send a message to and expect it to arrive is only about 16 billion lightyears away. We currently see galaxies much farther away than that, and if it were not for this gentle acceleration effect we could in principle flash messages which would eventually reach them. But because of the slight acceleration they are actually "beyond our event horizon". And it works both ways: they, as of today, could not send information to us. If a star exploded today in one of those galaxies, we would never see it, no matter how long we waited.

I want to try to explain where this figure of 16 billion lightyears comes from. This is a first attempt and comments are welcome. It goes in 8 steps starting from the most basic concept. For some readers much of this will be review:

CMB rest: There is an FAQ entry for this. An observer at rest relative to the CMB sees approximately the same temperature (of the ancient light) in all directions. There is no Doppler hotspot which would indicate that he or she was moving in that direction. It's like being at rest with respect to the ancient matter when it was more uniformly spread out, or with respect to the expansion process itself.

Universe time: Time as clocked by observers at CMB rest.

Proper distance at a particular time t: What you would measure by any conventional means (radar, tape measure...) if you could stop expansion at some given moment of universe time. Stopping expansion gives you time to measure---the distance won't change while you are sending the radar pulse, for example.

Scale factor a(t): This curve plots the expansion of distance as an increasing function of time. It is normalized to equal 1 at the present time. a(now) = 1. Back when distances between stationary observers were only half what they are today a(then) = 0.5. The slope a'(t) has not been constant so it's convenient to have the curve as a record of expansion history. Picture: http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg
The dark solid curve labeled (.27, .73) is the one to focus on.

Fractional rate of increase of a(t): A good handle on the rate distances are increasing is the fractional or percentage increase over time. Currently the scalefactor increases by about 1/140 of one percent per million years. So any largescale distance (e.g. between galaxies free of each other's gravity and each approximately at CMB rest,) will increase at that rate. (More precisely using the latest data 1/139 of one percent per million years.) The math expression for this rate, at any time t, is a'(t)/a(t). This is the absolute increase at that time, divided by the current size at that time, IOW a fractional or percentage increase rate.

Hubble rate H(t): By definition H(t) = a'(t)/a(t), just another name for the fractional rate of expansion. The current value of the Hubble rate is denoted H_o. Or you could say H(now), or a'(now)/a(now). It would all mean the same thing. Mathematically it is a fractional rate of increase the current value of which is 1/140 of one percent per million years. (Or 1/139 using the latest data)
That's the rate that distances (between observers at CMB rest) grow, at present. Using proper distance and the universe standard timescale.
In common astronomy units it is 70.4 km/s per Mpc. 70.4 km/s is the speed a distance of one Mpc is growing.
The Hubble rate is slated to decline in future to sqrt 0.728*H_o ≈ 60 km/s per Mpc.

Hubble radius c/H(t): This is the radius within which proper distances increase at speeds less than c. If a photon is trying to get to us and can manage to get within this radius then it will begin to approach. The photon's own speed is then faster than the remaining distance is increasing, so it can make progress towards us and narrow the gap.
The google search window doubles as a calculator. Try using it to find the current Hubble radius in lightyears. I invite you to copy this into the search window:
1/70.4 km/s per Mpc
When you press return, the calculator will say 13.9 billion years.
Multiply by c and you obviously get 13.9 billion lightyears.
This is the current Hubble radius.
Photons within that radius are going to make it.

Cosmic Event Horizon ≈ c/(sqrt 0.728*H_o) ≈ 16 billion lightyears.
Photons heading for us can still make it even if they are OUTSIDE the current Hubble radius as long as the radius itself is increasing fast enough and reaches out and takes them in.
What would make c/H(t) increase? The denominator H(t) decreasing would. The Hubble expansion rate has decreased sharply in the past which is why we can see such a lot of stuff that we know is receding faster than light.
But according to the standard cosmic model H(t) though still declining is not expected to go below sqrt(0.728) of its current value.
It is expected to level out at (sqrt 0.728)*70.4 km/s per Mpc
So what will the Hubble radius be then?
Try putting this in the google window
c/(sqrt 0.728 *70.4 km/s per Mpc) in lightyears
You will get the longterm value of the Cosmic Event Horizon (abbreviated CEH)

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The number 0.728 is technical and hard to explain, so I've had to put it in *ad hoc*. It represents a constant VACUUM CURVATURE contributing to the near flatness of space, which would otherwise be negatively curved (e.g. triangles adding to less than 180 degrees). Without such an inherent constant curvature bias, (or cosmological constant) the current density of matter/energy would only be 0.272 (or about 27%) of what was needed for the observed degree of flatness. So (although in my opinion it's a bit confusing to think this way) the number 0.728 could be imagined as a fictitious energy contribution making up the rest of what would be needed without a cosmological constant.
The square root of 0.728 gets into the picture for technical reasons when we want to talk about the longterm value of the Hubble rate, the level below which it is not expected to decline (because of the acceleration in the scalefactor.)

 

[megacal]

As an interested layman, earlier in this thread it helped me get my head around the balloon analogy when I was reminded that the 2D surface of the balloon is the entire universe. There is no inside or outside so there is no "center". Rather think of the inside of the balloon as the past, the outside of the balloon is the future.

Everything, energy, matter, space and time exist ONLY on the surface of the balloon which extends in 3 directions of space and 1 of time. That 4D space time may be closed (wrapped around on itself; exit one side you come back in the other) so there can be no CG calculation, or open (infinite) and again there can be no CG calculation.

I hope that helps a bit.

Ray,
yes. thanks for that relatively concise explanation...I think I grok it a bit better (but don't ask me to explain it). I'm used to things having a physical center or center of gravity,
i.e. I think in 3D & have a concept of time (though maybe it's an illusion?).

When out in the desert on top of a "massive" boulder, I know it could easily crush me, but
then I realize all I see is mostly empty space, and it's very disconcerting!

Marcus,
thanks for explaining the various constants and parameters, though I don't have the math skills to put them together. It would be very helpful though, if you could do a video tutorial
showing us the equation(s) and how you derive the answer. I think even I could follow that.

Just a thought.