# Is there a limit to the universe's expansion?

1. Feb 20, 2013

### lmoh

I've known for a while that the universe is expanding, and that objects within the universe are moving farther and farther away from each other. But one thing that fascinates me is the rate of this expansion, which is said to be faster than light (but that doesn't violate relativity). What's even more interesting is that the rate of expansion continues to accelerate, but as far as I know, there is no limit. That last part is really just speculation on my part, but it really makes me wonder, is there a limit to the rate of expansion? It already moves faster than light, so we can't apply the same limit as we do to material objects.

2. Feb 20, 2013

### marcus

The best way to understand "rate of expansion" is as a percentage growth rate.
According to standard cosmology this percentage growth rate is declining.

It is now measured at 1/140 of a percent per million years.
this is slated to go down to 1/165 of a percent per million years.

This is too slow to affect structures like galaxies or small groups of galaxies---structures that are only a million lightyearsx apart. So according to the standard cosmic model that pretty much every professional cosmologist works with these days, these structures will continue to hold together by their own gravity. You will see no "acceleration" or any effect of expansion at all.

A million lightyears distance expands only at speed 1/140 of a percent of the speed of light---so slow it can be overwhelmed and canceled by things being bound together in orbits by their own gravity and tending to fall together.

However a distance like 14 billion lightyears is expanding at the speed of light. You can work that out by simple arithmetic. One percent of 14 billion is 140 million. 1/140 of a percent of 14 billion lightyears is a million lightyears. It grows that much in a million years. So it is growing at the speed of light.
So two galaxies separated at that distance would obviously not be bound together by their gravity and they would continue to get farther apart. As the distance between them grew the SPEED of growth would increase, of course, because it is a PERCENTAGE growth. Bigger distances increase at a greater speed. That is the "acceleration" people talk about. What they are really talking about is constant (or slightly declining) percentage growth rate. In a certain sense that does "accelerate". Just like the money in your savings account at the bank "accelerates", even though it is earning only a small fixed percentage rate of interest.

Last edited: Feb 20, 2013
3. Feb 20, 2013

### phinds

And as a direct answer to your subject-line question "is there a limit", that is unknown but it would seem a bit weird if there were.

There are two possibilities

1) the universe is infinite.
2) the universe is finite but unbounded.

In neither case does there seem to be any reason why there should be a limit to the expansion

4. Feb 20, 2013

### lmoh

I'm not entirely sure what you are saying here. By percentage growth rate, are you referring to the current universe as a whole? And when you mention that this rate is going down, are you suggesting that the expansion is not accelerating?

5. Feb 20, 2013

### marcus

Cosmologists don't know the overall diameter of the universe so of course they do not know the speed that is increasing. All they have measured is a percentage growth rate.

the speed of increase of any given large distance is proportional to the size of the distance

The percentage growth rate is all we know. There is no one "speed" of expansion. There is a widely observed PATTERN of expansion discovered by Hubble, called the Hubble Law. Which is that if you look at large enough scale you see distances increasing 1/140 of one percent per million years.

This is the same rate at which WAVELENGTHS of light increase during their travels over large distances.

The words that popularizers and journalists use tend to confuse the public because they are vague and impressionistic. Lay folks get the idea there is one particular "speed" that the whole universe is expanding. Like some miles per hour or some multiple of the speed of light. Different distances expand at different speeds. You have to think in terms of a percentage rate.

And of course astronomers have determined that the percentage rate has been declining over the course of history, and according to the standard model, based on Einstein's GR equation, it will continue to decline (but more slowly in future).

If you want a simple numerical picture, click on "TabCosmo" in my signature. It is Jorrie's online realization of the standard cosmology model.

It goes out to year 88 or 89 billion, in the future. So you can see the expansion rate decline with your own eyes, and level out at 1/165 of one percent per million years.

Mathematically that is the same as saying that the "Hubble Time" will level off at 16.5 billion years. It is the reciprocal of the percentage rate. So as the rate slowly declines the T_Hub slowly increases. Look at the table, you will see T_Hub slowly increasing to 16.5 billion years.

Last edited: Feb 20, 2013
6. Feb 20, 2013

### marcus

I said already in what sense "expansion is accelerating".

==quote from post #2==
However a distance like 14 billion lightyears is expanding at the speed of light. You can work that out by simple arithmetic. One percent of 14 billion is 140 million. 1/140 of a percent of 14 billion lightyears is a million lightyears. It grows that much in a million years. So it is growing at the speed of light.
So two galaxies separated at that distance would obviously not be bound together by their gravity and they would continue to get farther apart. As the distance between them grew the SPEED of growth would increase, of course, because it is a PERCENTAGE growth. Bigger distances increase at a greater speed. That is the "acceleration" people talk about. What they are really talking about is constant (or slightly declining) percentage growth rate. In a certain sense that does "accelerate". Just like the money in your savings account at the bank "accelerates", even though it is earning only a small fixed percentage rate of interest.
==endquote==

That is exactly the sense in which "expansion is accelerating":
like the money in a bank savings account deposited at a very small, slowly declining, rate of interest.

Verbal description you get in popularized accounts tends to be vague and abstract, to see what is involved you have to look at concrete numbers. The best I know is this:
http://www.einsteins-theory-of-relativity-4engineers.com/TabCosmo6.html

Put the mouse on a blue dot to make an explanation of the number pop up. Look at the table (which shows the kind of "expansion" we're talking about) and THEN ask questions. Find out what terms like "Hubble Time" mean. Thats the key to understanding the prevailing cosmic model.

7. Feb 20, 2013

### Garth

Any limit to the universe's expansion, given its present mass content and expansion will depend on the nature of dark energy (DE) and its equation of state.

If there were no DE and the universe is finite and unbounded, which would happen if the average density of the universe was higher than a critical value beyond which the gravitational attraction of objects within the universe would also overcome the expansion, then there would be a limit. When such a universe reached this limit it would then contract, eventually all the way back to a 'Big Crunch'.

However an acceleration of the expansion has been observed so there is DE of some form, which might well mean the universe will expand for ever even if the average density is above the critical value.

We can discover its equation of state by studying the expansion of the universe at various epoch of its life.

It might be already slowing down: Is cosmic acceleration slowing down?.
(emphasis mine)

Or the DE could be simply a Cosmological Constant, a property of space-time allowed by Einstein's equations in which case the Universe would expand for ever, or even lead to a 'Big Rip' if the DE is of an even more exotic variety!

Garth

Last edited: Feb 20, 2013
8. Feb 20, 2013

### lmoh

Okay Marcus, after reading your edited post, I think I see what you mean. So the rate of expansion is proportional to the distance. I guess that makes sense on the bigger scheme of things, but with regards to my original question, does this mean that there is no limit to expansion, putting aside the fact that the percentage growth rate is decreasing? As I can imagine galaxies moving infinitely farther from each other, then it seems that the rate in which they are receding will become infinitely large as well.

9. Feb 20, 2013

### phinds

The rate of expansion is finite. Finite things do not become infinite.

10. Feb 20, 2013

### marcus

Wait Phinds, as I interpret it, I think Imoh has a valid point. It could be something worth discussing.
It depends on whether or not there is a bound on distances themselves, Imoh.

According to the present standard "consensus" model (officially called LCDM) that 99 out of 100 cosmologists are using, the expansion process is slated to continue indefinitely, getting closer to that limiting rate of 1/165 % per million years.

EVEN IF THE UNIVERSE IS SPATIALLY FINITE that would lead to distances eventually reaching any finite size you can imagine, and so distances would achieve arbitrarily large finite growth speeds.

As Phinds points out, there is no such thing as an "infinite" growth speed, but distances can grow (in this scenario) at any arbitrarily large finite multiple of the speed of light.

And if the universe is spatially INFINITE, which is a possibility, then EVEN TODAY there are arbitrarily large distances, which are therefore growing at arbitrarily high finite speeds. If some distance is e.g. 14 billion ly then it is now growling at rate c.
So therefore if a distance is 14 billion billion lightyears then it is now growing at a billion times the speed of light.

I tend not to worry much about the spatially infinite case. Recent data suggests, but does not prove, that there could be a very slight positive spatial curvature indicating that space is the 3D analog of the 2D surface of a balloon and therefore space has a finite volume and there is some current limit on today's distances. The issue has not been decided. I tend to think in terms of space being finite. That is, there would not be any distance that is really growing today at a billion times the speed of light. But we have to keep an open mind about the spatial finiteness issue until more data on the curvature is in. We might get some more news about that in March when the European "Planck" spacecraft mission reports.

Last edited: Feb 20, 2013
11. Feb 20, 2013

### phinds

So, if you are right, doesn't that imply that the rate of expansion is ALREADY infinite (infinitely far away from us) ?

I still contend that finite things do not become infinite. Do you disagree w/ that? I'm certainly open to being wrong about that but I just can't see it.

12. Feb 20, 2013

### marcus

there is an ambiguity of language. In an infinite universe there are no infinite distances. there are arbitrarily large distances.

Just like on the real number line there is no number that is infinitely far from zero.
There are only numbers that are as LARGE AS YOU PLEASE.

So, with a percentage expansion rate as we have now (1/140%) or expect in future (1/165% according to most recent NASA WMAP report) there are no distances increasing with infinite speed.

If the universe is spatially infinite then there exist arbitrarily large finite distances
and therefore there exist distances which are increasing with arbitrarily high finite speed.

Imoh said: "...then it seems that the rate in which they are receding will become infinitely large as well."

I think what he meant was "...then it seems that the rate in which they are receding will become *arbitrarily* large as well." the English language is ambiguous---that is one of the ways to interpret what Imoh said.
IOW speeds will grow without bound. They will grow *infinitely*. Yes, that is right. They will *grow infinitely large*.
Well, yes, if that means the same thing as "they will grow infinitely", i.e. grow without bound.

It's just a trouble with our common language. One reason people use mathematics is it eliminates some of the ambiguity in ordinary spoken language.

Last edited: Feb 20, 2013
13. Feb 20, 2013

### phinds

Not sure I agree w/ you in this case, but I do get your point.

Thanks

14. Feb 20, 2013

### lmoh

Okay, there seems to be alot of confusion over what I said. Phinds, when I used the term "infinite", I meant unbounded, like Marcus said. My whole point was that, since I cannot see a limit to the space between two objects, then there shouldn't be a limit to how fast they are moving away from each other (i.e. expansion).

15. Feb 20, 2013

### marcus

That is correct. So far we don't have sufficient evidence of slight positive mean curvature that we can conclude there is a bound on size of distances.
therefore (with the standard cosmology assumption of percentage increase) we cannot conclude that distance increase speed is bounded.

However you might nevertheless be interested in some recent reports that put a 95% confidence interval on the large-scale mean curvature which was pretty lopsided on the positive side. So one can calculate what range circumference that corresponds to. And thus get some kind of preliminary handle on what the largest distances might be, in space, at this time.

See for example:

==quote==

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

Ωk =−0.0059±0.0040.

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

Imoh, the convention is (don't ask me why) that negative Ωk corresponds to positive overall mean curvature.
And the rule of thumb is is simple. To find the circumference you DIVIDE 88 billion lightyears by the square root of |Ωk|

That'll give slightly more accurate numbers than the approximation I made in that earlier post.

this doesn't mean you can BELIEVE what it says in the south pole telescope (SPT) report. this is still controversial. More data will be reported soon from the European "Planck" spacecraft mission. I just suggest watching for what new confidence intervals for Ωk are reported, and knowing how to translate them into spatial sizes. to get an idea what the numbers mean.

Another thing to check is the Hinshaw et al report from WMAP 9th year. I should put something in the sticky thread about that---the "balloon model" sticky.

Last edited: Feb 20, 2013
16. Feb 21, 2013

### Chronos

There is little evidence to suggest the universe is finite, or infinite. I doubt it is mathematically possible to clearly prove either conjecture. The best we can do is to constrain the minimum size of the universe.

17. Feb 21, 2013

### Jorrie

AFAIK, it comes from the first Friedmann equation for the time dependent Hubble constant, when written in the form:

$$H(t) = \sqrt{\frac{1-\Omega}{a^2}+\frac{\Omega_m}{a^3}+\frac{\Omega_r} {a^4} +\Omega_\Lambda}$$
The $1-\Omega$ was often replaced by $\Omega_k$, a 'curvature density parameter' ( e.g. Peebles textbook 1993), thereby giving a negative $\Omega_k$ for a positive curvature ($\Omega > 1$). Peebles labeled it $\Omega_R$ though.

18. Feb 21, 2013

### phinds

Thanks for that clarification. THAT I can agree w/, as I do not see "unbounded" (finite but growing without limit) as equivalent to "infinite" so now I agree w/ Marcus that this was a terminology issue.