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

[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)

====================
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.

 

[RayYates]

@Ray,
...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...

marcus. I get it! I've read the entire posting twice and the concept has become clear. Thank you.

The question I'm currently grappling with is, "What is the balloon? What is space?"

I get that space time is stretchy, bendable, compressible. In regions of higher mass time moves slower, so conversely in the regions between galaxies time moves faster. So what is space? Is space inside an atom the same as the rarefied space between galaxies clusters.

I've seen it theorized that on a scale many times smaller than sub atomic particles, space is granular like some twisted up dimensions tied into a knot and arranged on a grid. Is the VACUUM nothingness of space really something?

 

[Farahday]

Marcus -
EXCELLENT explanation of expansion. It does; however, leave me with the same pertinent questions raised by RayYates:

Unless the dispersion of the elemental particles that physicists deem to comprize the observable cosmos is a local event, then the entire cosmos, itself, is expanding.

Space 'exists' - if it did not we would all be set ablaze by Sol. Just because our technology seems unable to determine its composition doesn't mean it doesn't exist - and an infinite expanse of space sans fundamental particles would require no less justification than an infinite expanse of fundamental particles sans space.

If the cosmos (everything that exists everywhere) is expanding then either existing space must be increasing in volume (which would lead us to believe it is decreasing in density) or new space is being manufactured (conjured into existence, if you will).

How would you address this issue?

 

[CCWilson]

Isn't another way the balloon analogy helps is to demonstrate how spacetime is different from three dimensional space? Most people picture the big bang as an expanding balloon, except with all kinds of stuff spread within it. If that were the case, then our universe would be unbounded - there would always be empty space beyond any position specified which could be filled later. But if someone can get his mind around the idea that three dimensional space is represented by the 2D surface of the balloon, it makes sense that moving or shining a light in any direction will still be within the limits of the spacetime of our universe.

Question: Although at this point I assume that a beam of light sent out could never theoretically make the circuit and return, because of the extreme expanision of the universe, weren't there times when a beam of photons could have circled around the smaller spacetime at that point and returned to (non-human) sender?

 

[RayYates]

...If the cosmos (everything that exists everywhere) is expanding then either existing space must be increasing in volume (which would lead us to believe it is decreasing in density) or new space is being manufactured (conjured into existence, if you will)...

You put your finger on the point exactly.

 

[marcus]

...
If the cosmos (everything that exists everywhere) is expanding then either existing space must be increasing in volume (which would lead us to believe it is decreasing in density) or new space is being manufactured (conjured into existence, if you will).

How would you address this issue?

The standard view is that density is decreasing. The "steady state" idea of constant density went out of style by around 1960-1970, anyway a long time ago.

Cosmologists pretty much all accept 1915 Gen Rel as the currently best most reliable equation for how geometry/gravity evolves and is influenced by matter. Virtually all research is based on the 1915 Gen Rel equation.

According to that picture there are distances between real physical stuff, events etc. The network of distances (angles areas etc) is geometry. But distances are not made of anything, they are RELATIONS, not material substance.

In 1915 Einstein put it concisely: Dadurch verlieren Zeit und Raum den letzten Rest von physicalische Realität. (thereby lose time and space the last vestige of physical reality.)

The geometric relations among things are not a physical substance. "Space" is a word which does not refer to a material. It refers to a bunch of geometric relationships.

So it's misleading to talk about it being "manufactured".

And of course density declines as physical stuff gets farther apart.

... But if someone can get his mind around the idea that three dimensional space is represented by the 2D surface of the balloon, it makes sense that moving or shining a light in any direction will still be within the limits of the spacetime of our universe.

Question: Although at this point I assume that a beam of light sent out could never theoretically make the circuit and return, because of the extreme expanision of the universe, weren't there times when a beam of photons could have circled around the smaller spacetime at that point and returned to (non-human) sender?

You got it! Shining a light in any direction within our 3D world is like the 2D creatures shining a light in any direction along the curved 2D (infinitely thin) world they live in. It stays within the defined limits.

You asked a good question. Assuming the finite volume sphere-like model, could a light beam ever have gotten around, made the full circuit? The standard answer is NO. At least after the first fraction of a second I'm not sure about the first few instants. there are different scenarios. But apart from some very early business I can only speculate about, expansion has ALWAYS been too rapid for that to have happened. If something had made expansion pause long enough, sometime in the past, it could have happened. But it didn't. Or if expansion had been much slower than we think it was. But the standard reconstruction of expansion history implies that it was always outpacing the ability of a flash of light to make the full circuit.

It has been calculated what the maximum distance some photons, a flash of light, could now be from the sender, if the flash is sent at start of expansion or as close to then as you like.
So the flash has been traveling for the whole 13.7 billion year history of expansion. (Today some cosmo models go back before start of expansion into a contraction phase, but we arent including that, just the usual 13.7 billion year expansion age.)

That maximum distance is called the PARTICLE HORIZON and it is calculated to be about 46 billion lightyears. The farthest a flash of light can have gotten (with the help of expansion) in the whole 13.7 billion year history is only 46 billion lightyears. We're fairly sure now that the circumference of the entire U, if it isn't actually infinite, is considerably bigger than that, by over a factor of 10.

There was something in a NASA report from the WMAP mission about this, by a bunch of authors: Komatsu et al. I can get the link if you want. Maybe somebody else has something more recent, I'm not entirely sure and would be happy to be corrected if there's some better information.

 

[CCWilson]

I'm sure this is a silly question, Marcus, but if indeed shortly after the big bang photons and particles could return to sender, because initial expansion hadn't been fast enough to disallow it, is it possible that this would have caused some sort of chain reaction, and could that have been the cause of the hyperinflationary period that is supposed to have occurred?

 

[marcus]

To recap the main content of this thread, the balloon analogy is to help understand and imagine the GEOMETRY of expansion. Not the physics. It helps to watch the animation carefully
http://www.astro.ucla.edu/~wright/Balloon2.html
notice the galaxies stay in the same place while getting farther apart.
The photons always move at the same speed, say a centimeter per second depending on the size of the image on your computer screen.
In the analogy, all existence is concentrated on the infinitely thin 2D sphere.
The distance between two galaxies can be increasing faster than light (faster than one cm per second) and yet a photon may be able to get from one to the other. You can see this kind of thing happen even before you understand conceptually how it happens. And the analogous thing happens in the real 3D universe. (The distances to most of the galaxies which we observe today are increasing >c and were increasing >c when they emitted the light which we are now getting from them. Watch the animation to get an intuitive feel for these things, which may at first seem paradoxical.
========

Since the balloon thread is about the geometry of Hubble law expansion I will say a bit about Hubble law. If you are a beginner you should experiment with one of the online cosmology calculators, like this one:
http://www.einsteins-theory-of-relativity-4engineers.com/cosmocalc_2010.htm
Put various redshifts in and study the results. Redshift z=1 for nearby galaxies, z=9 for the most distant galaxies confirmed so far, 1090 or thereabouts for the ancient light (the CMB or background).
Put sample redshifts in and find the corresponding distances and the rates those distances are increasing. It will also tell what the Hubble rate was in the past, back when the light we are now seeing was emitted.

The most recent official figure for the current Hubble rate itself is 70.4 km/s per Mpc. You are encouraged to learn to calculate with it. For example, as exercise paste this into the google search window (which doubles as a calculator) and press return:

70.4 km/s per Mpc in percent per million years

Google calculator knows how to express a rate of change as a percent per million years.

You will get 1/139 of one percent per million years.
It will actually say "0.00719973364 percent per million years"
but 0.0072 is very nearly the same as 1/139

That is the percentage rate that distances (beyond the immediate neighborhood of our group of galaxies) are currently growing. According to standard cosmo picture the rate is destined to continue declining approaching about 1/160 in the limit.

1/139 percent per million years is the same as
a millionth of a percent in 139 years.
So if you want to picture how rapidly distances in our universe are expanding think of a distance, and think of waiting 139 years, and then finding that it has increased by a millionth of a percent.
===================

The socalled HUBBLE RADIUS is the distance which, today, is increasing at exactly the speed of light. It is 13.9 billion lightyears. It's just the same number as 1/139 but flipped, with the decimal point moved.
Saying that according to standard model 1/139 will go down to around 1/160 in the longrun is the same as saying that the Hubble radius which is now 13.9 will increase to 16 billion lightyears in the long term. But these are glacially slow changes really unimaginably slow. So we think of the Hubble rate as constant, for the time being. (It has been much larger in the past, though.)
===================
There are a lot of questions to ask about the Hubble rate. Notice that it is a fractional rate of distance increase, not an absolute rate. How can the fact that it has been and will be declining be compatible with the talk one hears about "acceleration"?

Well, fractional or percent rate is not the same as absolute rate. If you take a given distance and plot the CURVE OF WHAT IT WILL BE IN THE FUTURE you will find the curve has increasing slope. That's true even though at every future time the PERCENTAGE INCREASE will be getting less and less. Percentage increase is not the same as slope. One can be steadily decreasing while the other increases. Your bank savings account can grow by an increasing absolute dollar amount each year even though the bank is gradually reducing the percentage interest they give you, because the PRINCIPLE is larger each year. Just how it is, no contradiction.
===================

There are still lots of questions about Hubble expansion law. It involves distances now and the rate they are increasing now. Likewise at some earlier moment in time, or later. How is that "now" instant defined? How are distances defined? (Imagine stopping the expansion process everywhere at once, at a definite instant to make it possible to measure distances without them changing while you measure them. How is "everywhere at once" defined?) We use the idea of observers at CMB rest. There's an FAQ about that in the FAQ section.
====================

Still lots of questions that can and should be asked. But I hope that some readers will try using the google calculator, watch the balloon animation observantly, and experiment with one of the online redshift calculators. Jorrie's for instance:
http://www.einsteins-theory-of-relativity-4engineers.com/cosmocalc_2010.htm

 

[marcus]

I'm sure this is a silly question, Marcus, but if indeed shortly after the big bang photons and particles could return to sender, because initial expansion hadn't been fast enough to disallow it, is it possible that this would have caused some sort of chain reaction, and could that have been the cause of the hyperinflationary period that is supposed to have occurred?

I'm not sure it is a silly question. I'm not an expert on the very brief inflation era that is widely supposed to have occurred. It's still speculative and there are a variety of scenarios.
One type model that is gaining attention involves a BOUNCE. In the main model of this type you get a brief period of faster than exponential growth of distances. Normally what is called "inflation" by cosmologists is exponential and slightly slower growth.

a(t) = e^Ht with H either steady or slowly declining.

In stark contrast to this, in socalled Loop cosmology you get this but with H increasing very rapidly to extremely high (Planck scale) values and because this is faster than the unsual exponential growth called inflation it is called "superinflation" by Loop cosmolgy researchers. I don't remember hearing the term "hyperinflation" in cosmology.

For me it is completely speculative what conditions could have been like and what could have been happening at such extremely high densities. In Loop cosmology, according to their equation model, gravity becomes repellent at near Planck density, which is what causes the bounce. It is a quantum gravity effect. Quantum nature doesn't like to be pinned down too tightly, so resists extremely dense compression.

One option is not to try to understand the very beginning of expansion but only start thinking about it a few blinks after it started. Wait until there is more agreement among the real experts before trying to understand. sorry so unhelpful.

BTW CCWilson, I'm curious to know. Have you done any of these things?
Watched the wright balloon animation
http://www.astro.ucla.edu/~wright/Balloon2.html
Experimented with the cosmology calculator
http://www.einsteins-theory-of-relativity-4engineers.com/cosmocalc_2010.htm
Read the LineweaverDavis SciAm article
http://www.mso.anu.edu.au/~charley/papers/LineweaverDavisSciAm.pdf
I'd be curious to know any impressions, what you may have learned etc.

 

[CCWilson]

Marcus, I had previously done the calculator and animation.

The problem with the cosmology calculator is that the terms - the omegas and the Hubble constant and even some of the calculated results - were unfamiliar or unclear to me, so it wan't that helpful. Would be much improved for laypeople with instructions and definitions.

The balloon animation was kind of interesting but not compelling. I'm not sure it added to my understanding, but I already had a fair grasp of it, having ruminated on the balloon analogy before, which was tremendously helpful; I'd have been lost in trying to understand universe expansion without it. One problem for me with the animation's redshift changes was that we are not outside the sphere watching the whole thing, our two-dimensional selves are presumably somewhere on the balloon skin, in which case the redshift for us should not be the same for all the galaxies and other sources of photons, right?

I just now went through the "Misconceptions about the Big Bang" and thought it was great. I learned some new things. One is that the red shift is not Doppler but related to expansion of space, which sort of makes sense. Question: When Hubble noticed the red shift, from which he deduced that the universe was expanding, was he aware that it was not, strictly speaking, the Doppler effect? In fact, was he aware at first that it indicated expansion of space time, or did he think it meant that galaxies were moving away from each other within a three dimensional universe?

Also, it made clearer the concept of how some galaxies can be moving away from us faster than the speed of light. The idea that we can see some of those galaxies moving away from us faster than the speed of light, I'm still working on that one.

These ideas are a lot for anybody to get our minds around, especially laymen. But the more you read such articles and think about them, the more sense they make. Are there people who can actually think in four dimensions, or do even you smarties have to rely on balloon analogies and such, and are resigned to doing the mathematical calculations without having a clear visualization in your minds?

 

[marcus]

Thanks for your reply!

Marcus, I had previously done the calculator and animation.

The problem with the cosmology calculator is that the terms - the omegas and the Hubble constant and even some of the calculated results - were unfamiliar or unclear to me, so it wan't that helpful. Would be much improved for laypeople with instructions and definitions.I just now went through the "Misconceptions about the Big Bang" and thought it was great...Also, it made clearer the concept of how some galaxies can be moving away from us faster than the speed of light. The idea that we can see some of those galaxies moving away from us faster than the speed of light, I'm still working on that one...

I agree that Jorrie's calculator could be much improved, by what you say and also by editing. There's too much there. You have to train yourself to look just at what you want to know.

There's a much simpler calculator, by a university prof in Iowa, that only gives the very basic stuff, so less confusing. Also a lot fewer decimal places--she gives distances in billions of LY instead of millions, and rounds off to just a few digits. It's more user friendly.
Google "cosmos calculator".

When you go there you first have to type in three numbers: the present matter fraction (.27) the cosmological constant equivalent (.73) and the current Hubble parameter (70.4). It makes you aware how important those three numbers are---the results depend on them!
The age of the universe now and when the light was emitted, the distances, the recession speeds all depend on the values of these parameters.

Jorrie saves one the trouble of typing in those three numbers. But maybe Prof. Morgan's calculator is better pedagogically because it makes you type them in---and then gives you only the simplest basic output, in numbers of only 3 digits or so. Here's the link in case you or anyone might be interested:
http://www.uni.edu/morgans/ajjar/Cosmology/cosmos.html

Oh, Prof Morgan does give a couple of things to ignore: "distance modulus" and "luminosity distance". A general purpose tool always tends to have features you need mentally to filter out, nothing is perfect .

And Prof. Morgan has a short paragraph of explanations and directions right there under the calculator. Pedagogically I don't see how it could be better. It's just inconvenient when you are in a hurry because you have to type in .27, .73, 70.4 before you can calculate.

 

[marcus]

...
I just now went through the "Misconceptions about the Big Bang" and thought it was great...

Also, it made clearer the concept of how some galaxies can be moving away from us faster than the speed of light. The idea that we can see some of those galaxies moving away from us faster than the speed of light, I'm still working on that one...

That's not too hard to understand when you appreciate how much the Hubble rate has declined over the years. And it is still declining though not as rapidly as it did in earlier times.
Prof. Morgan calculator gives the Hubble rate at times in the past, so you can see this.

That means for any given size of distance, like 1 Megaparsec (3.26 million LY) the rate distances that size have been increasing has gotten less and less over time.

Suppose a galaxy it at some distance that is increasing at rate 2c and it emits a photon in our direction. the photon is tryng to get to us but it keeps receding at rate c. However the galaxy is receding at rate 2c. so after a while the photon is a lot closer to us than the galaxy!
And by that time that might be close enough because the recession rates of various size distances keep declining. At any given range it keeps getting easier.
===============

The way to make that argument mathematically clear is to define a distance threshold called Hubble radius which at any given time t is the size of distances that are increasing exactly at rate c. If a photon gets within that range it will begin to make progress, because distances less than that range increase slower than c.

But the Hubble radius is c/H(t) ---just by simple algebra from the Hubble law v = H(t) D.
Set the distance growth rate v equal to c and solve for D, in the equation c = H(t) D.

Since H(t) has been decreasing throughout history, the threshold range c/H(t) has been INCREASING. It has been REACHING OUT to photons trying to get to us.

So we are seeing light from a lot of galaxies which themselves have been receding faster than c all during the time the light has been traveling on its way to us.

 

[marcus]

I'm thinking a tabulation like this might be useful.

 standard model using 0.272, 0.728, and 70.4 (the % is per million years)
timeGyr   z        H-then   H-then(%)   dist-now(Gly)  dist-then(Gly) 
   0      0.000       70.4   1/139 
   1      0.076       72.7   1/134
   2      0.161       75.6   1/129
   3      0.256       79.2   1/123
   4      0.365       83.9   1/117
   5      0.492       89.9   1/109
   6      0.642       97.9   1/100
   7      0.824      108.6   1/90  
   8      1.054      123.7   1/79
   9      1.355      145.7   1/67
  10      1.778      180.4   1/54
  11      2.436      241.5   1/40
  12      3.659      374.3   1/26
  13      7.190      863.7   1/11

You know that the present Hubble rate is put at 70.4 km/s per Mpc which means distances between stationary observers increase 1/139 percent per million years. And the Hubble radius (a kind of threshold of safety within which distances are expanding slower than c) is currently 13.9 billion LY.
So by analogy you can see how the Hubble rate has been greater in the past, and has been declining, while the Hubble radius (reciprocally) has been increasing. That means reaching out farther to struggling photons and welcoming them inclusively into safe water where the current is not so strong.

So you can see that 4 billion years ago distances were increasing 1/117 percent per million years, and the Hubble radius (safe harbor threshhold) is 11.7 billion LY.
In the intervening time, in other words, the Hubble radius has extended farther out from 11.7 billion LY to 13.9 billion LY.

That extension of the threshold simply reflects the fact that the Hubble expansion rate has declined from 1/117 percent to 1/139 percent per million years.

(It is not like the physical expansion of a distance between two stationary observers, the kind of expansion decribed and governed by Hubble law. It's more like gradual revision of a criterion of admittance: a convenient scale we calculate from other parameters in the system.)

 

[CCWilson]

I think I get the Hubble radius concept. As long as space between observer and galaxy isn't expanding faster than the speed of light, a photon will succeed in reaching the observer. At exactly at the Hubble radius, that photon would remain suspended at the exact same distance (maybe that's not quite the right word) from the observer but if the Hubble radius increases, would slowly start making headway. Is that correct?

 

[marcus]

I think I get the Hubble radius concept. As long as space between observer and galaxy isn't expanding faster than the speed of light, a photon will succeed in reaching the observer. At exactly at the Hubble radius, that photon would remain suspended at the exact same distance from the observer but if the Hubble radius increases, would slowly start making headway.

Yes! You get it exactly and express it clearly in just a few words. I will use that visual way of putting it next time I need to explain this, if you don't beat me to it.

It will hang there right on the edge, and then (when the radius increases) it will slowly start to make headway.

 

[Farahday]

The standard view is that density is decreasing. The "steady state" idea of constant density went out of style by around 1960-1970, anyway a long time ago.

Cosmologists pretty much all accept 1915 Gen Rel as the currently best most reliable equation for how geometry/gravity evolves and is influenced by matter. Virtually all research is based on the 1915 Gen Rel equation.

According to that picture there are distances between real physical stuff, events etc. The network of distances (angles areas etc) is geometry. But distances are not made of anything, they are RELATIONS, not material substance.

In 1915 Einstein put it concisely: Dadurch verlieren Zeit und Raum den letzten Rest von physicalische Realität. (thereby lose time and space the last vestige of physical reality.)

The geometric relations among things are not a physical substance. "Space" is a word which does not refer to a material. It refers to a bunch of geometric relationships.

So it's misleading to talk about it being "manufactured".

And of course density declines as physical stuff gets farther apart.

Of course if Einstein was wrong about space not being a substance (ethereal vs material), then GR is flawed - and so are any other theories using GR as a basis.

It would be like a ballistics engineer who dismisses air as 'nothing'. His calculations over short distances would be accurate within "tolerable limits", but the farther the distance, the less accurate his figures would become. He, too, might chalk the errors up to some speculative unknown factor(s) affecting the empirical circumstances.

 

[RayYates]

...According to that picture there are distances between real physical stuff, events etc. The network of distances (angles areas etc) is geometry. But distances are not made of anything, they are RELATIONS, not material substance.

In 1915 Einstein put it concisely: Dadurch verlieren Zeit und Raum den letzten Rest von physicalische Realität. (thereby lose time and space the last vestige of physical reality.)

The geometric relations among things are not a physical substance. "Space" is a word which does not refer to a material. It refers to a bunch of geometric relationships.

So it's misleading to talk about it being "manufactured".

And of course density declines as physical stuff gets farther apart...

If space is geometry, where dies dark energy come from? I had the impression it came from space itself or is that just speculation?

 

[marcus]

It's not clear that the cosmological constant is an "energy" in any reasonable sense of the word. The evidence so far is of a nonzero curvature constant that occurs naturally in the Einstein 1915 GR equation.

As you suggest might be the case, a lot of the talk one heard following the discovery of acceleration finding was "speculative".

My impression is that the speculative hubub has died down quite a bit in the past 3 years of so. There is now a greater tendency to simply consider that the 1915 law of gravity has TWO constants, Newton G and Einstein Lambda.

Less tendency now to speculate about some mysterious "energy". Ten years ago people were excitedly talking about "phantom" energy and "quintessence" and "Big Rip", not just in the popular media but in the scientific literature.

More data came in in the past 10 years and it was all consistent with the simple idea that Einstein 1915 law of gravity (GR) has two constants: G and Λ. Nature was behaving consistently as if there was this constant Λ which was just a constant and not doing anything funny.

I think to some extent it is simply a matter of taste and personal inclination. If you LIKE to think of space as being filled with some kind of mysterious "energy field" that causes the curvature constant Λ, then that's cool. You should believe what you want. It has not been proved or disproved.

Many people are skeptical of that however. Since there is no scientific reason to believe it, so far, then until there is evidence the simplest thing seems to be to stay with the original GR idea. Two constants appear naturally in the theory and for many years most of us thought one of them was zero, but it turned out not to be.

Here's a presentation of the skeptical viewpoint:
http://arxiv.org/abs/1002.3966/
Why all these prejudices against a constant?

 

[marcus]

I think I get the Hubble radius concept. As long as space between observer and galaxy isn't expanding faster than the speed of light, a photon will succeed in reaching the observer. At exactly at the Hubble radius, that photon would remain suspended at the exact same distance (maybe that's not quite the right word) from the observer but if the Hubble radius increases, would slowly start making headway. Is that correct?

This is good. I want to take this a step farther and focus on the LIMITING value of the Hubble radius. It's slated to continue increasing and ultimately approach an upper bound of about 16 billion lightyears.

Reciprocally, the Hubble (fractional growth) rate is slated to decline from 1/139 to about 1/160 if you express it as a percent per million years.

What this means is that there is a COSMIC EVENT HORIZON out there at around 16 billion LY in the sense that news can never reach us from an event that occurs today in a galaxy at that distance. The photons trying to get to us, from that event, are forever beyond the reach of our Hubble radius because it can never extend more than 16 Gly.

I would like to call that limiting Hubble rate by the name H_∞. We know that H(t) declines with time and the standard notation for PRESENT value is H₀. So it seems natural to denote the eventual asymptotic value in the far future by H_∞.

Then we have a nice easy to remember equation relating Einstein's cosmological constant Λ to that H_∞. Assuming a universe that is spatially flat, or nearly, as by all accounts it seems to be, we have:

Λc²/3 = H_∞²

Morally speaking the c² and the 1/3 factor are just accidental features of how Einstein originally defined Λ and put it in his GR equation. Morally, I would say, the cosmological constant simply is H_∞².

Remember that H_∞ is a number per unit time (a fractional growth rate). And Einstein defined the constant Λ to be a number per unit area (a type of curvature unit).
So multiplying Λ by c² changes number per (length)² into number per (time)² which is (number per unit time)² in other words the SQUARE of a fractional growth rate.

So the units match.

How best to write H_∞? It's a bit longwinded to say 1/16 percent per million years. How about 1/16 ppb per year?

I want a format that communicates well to newcomers to forum. The present H₀ is about 1/14 ppb per year.
"Distances between stationary observers are increasing by about 1/14 parts per billion per year."
Actually closer to 1/13.9, but 1/14 is close enough.
I like this format because it has the Hubble time (13.9 billion years) and Hubble radius (13.9 billion lightyears) built into it.

So then this important H_∞ (an important constant of nature, which is morally a form of the cosmological constant) becomes
H_∞ = 1/16 ppb per yr = 1/16 of a part per billion per year.

It is the lowest the fractional growth rate of distance is ever destined to get, according to todays best understanding.

See what you think. Would you prefer scientific notation, with a 10^-9 or 10^-10?
Can you figure out a better way to say it that can communicate to newcomers?

 

[CCWilson]

Marcus, it seems to me that 1/16 parts per billion per year is such a tiny percentage that using 1/16 rather than 0.0625 doesn't help anyone visualize it. So I'd prefer 0.0625 parts per billion per year, or whatever it really is. That seems more in keeping with scientific use. So you could say that the Hubble/cosmological "constant" - the percentage rate of increase of distance between two distant points - will decrease from the current 0.071 ppb per year to 0.0625 ppb per year, at which point it will truly remain constant.

Please keep in mind that I struggle to figure out these concepts and many years have passed since any math courses, so my preference may not be everybody else's.

Is the reason we believe that the Hubble constant will eventually slow down as calculated is that we think that the curvature of the universe is flat, and those calculations flow from that assumption, right?

 

[marcus]

You are probably right, esp about respecting conventional notation. I like the idea that the Hubble radius (that threshold of admission for photons trying to get to us) is so important and one can just flip the Hubble constant and get it

H₀ = 1/13.9 ppb per year ≈ 0.072 ppb per year
Hubble radius (now) = 13.9 billion lightyears.

You get to remember two quantities for the price of remembering one. But it does jar a little to write 1/13.9.

I will calculate the limiting value of the Cosmic Event Horizon and of the Hubble radius and try writing it in the style you suggest. Let's use the current estimates of 70.4 km/s per Mpc and 0.728

Put this into the google search window:
1/(sqrt(0.728)*70.4 km/s per Mpc)
16.279 billion years

So, without doing much round-off, what we get for for the longterm Hubble radius is
16.279 billion lightyears.

And what we get for the longterm Hubble rate is 1/16.279 or written as you suggest:
H_∞ = 0.0614 ppb per year.

As I say, you are probably right. But for a while, at least, to see how it goes, I will keep trying to think of it as
Radius = 16 billion lightyears
H_∞ = 1/16 ppb per year,
also let's keep the option of adding another digit of accuracy---e.g. say 16.3 and 1/16.3