# Is the expansion rate of the universe slowing?

1. Aug 23, 2011

### Peter Watkins

In a reply to a correspondents question, it was stated that the expansion rate of the universe is slowing. This would seem in direct contrast to the accelerating rate of expansion that we have been hearing of since '98. An invisible form of repellent energy that is able to overcome gravity, (Dark Energy), is said to be the mechanism behind this acceleration.
It should perhaps be pointed out that the notion of acceleration came about because careful measurement of the rate at which galaxies are receding shows that this rate is increasing, ie. they are moving apart at ever faster rates.
Has there now been a publication stating that the expansion rate is slowing, and if so, by whom, on what is this assertion based?

2. Aug 23, 2011

### cmb

I'd mention one thing, which appeared in a New Scientist article in 2008, I think:

Measurements of far off objects, on which such 'conclusions' are based, presume we sit in a homogenious and isotropic background medium. If we happen to be sitting in a local 'void', all our observations of the external universe would be skewed. We'll only know if this is affects our observations once we get far enough away to look back and see if there is an earth-centric void.

On my personal 'observations', I expect the universe's expansion to be accelerating.

3. Aug 23, 2011

### phinds

Seems to me you should ask the poster who made the assertion. If you are not willing to do that, you should at least post here a reference to where you found the assertion.

4. Aug 23, 2011

### Peter Watkins

The assertion was made by none other than Chalnoth in reply #14 to my question; "How does the expanding universe stretch light?"
RE "New Scientist" article; read that one. The cover said something along the lines,"Does dark energy really exist?" As you say, there was speculation about our being in a sparcely populated void, with no firm conclusion.

5. Aug 23, 2011

### marcus

Just a problem with words. The Hubble rate is decreasing, but the scale factor is increasing at an increasing rate.

The math is clear but when it gets translated into English it sounds contradictory.

a(t) is the scale factor. Distances between stationary points increase in proportion to a(t).

To say the U is expanding just means the time derivative a'(t) is positive.

To say this expansion is speeding up just means that a'(t) is increasing. In other words the second deriv a"(t) is positive.

However the fractional or percentage rate of expansion, by definition, is H(t) = a'/a.

This is decreasing and according to the model in general use (the LCDM, based on the Friedmann equation) it is expected to continue decreasing indefinitely, gradually leveling out to an asymptotic value which is about sqrt(0.73) of the present one.

Chalnoth is certainly right about that, since most people when they say "expansion rate" are talking about the Hubble parameter, and it has been steadily decreasing since very early times and is slated to continue doing so.

The reason is simple arithmetic. Yes we learned in 1998 that a'(t) is increasing! But in a given period of time a(t) the universes scale factor increases more markedly. a'(t) increases proportionately less. So therefore the Hubble H(t) = a'(t)/a(t) actually declines.
It's a fraction where the numerator is making only modest gains percentagewise compared with the denominator.

Peter, if you are really interested in cosmology you would, I think, benefit from going right to the heart of the matter and studying the Friedmann equation. Wikipedia has something. It is a rather simple equation that gives you explicitly how H(t) evolves. Actually it tells you about the square of H(t), but that's just as good, just take square root.

You can see at first glance that if the "dark energy" fraction of total density is 0.73 then since sqrt(0.73) = 0.85 the eventual or asymptotic value of the Hubble rate must be 85% of today's value.

Last edited: Aug 23, 2011
6. Aug 23, 2011

### Peter Watkins

I realise that the above reply is couched in kinder-garden maths, but it illustrates the difference between you an I, (education apart). You and other PF. contributors think in numbers, formulae, tables and so on, that, for the most part, to me are gibberish and do not aid my understanding. I think that what you are saying above is that the amount of expansion as a percentage of the distance between two galaxies is decreasing, but that the rate at which it is doing it is increasing, ie. moving apart at an increasing rate, for any given length of time. The question is; "is the universe still regarded as expanding at an accelerating rate, or is it now realised that the universe is simply doing what it has always done?

7. Aug 23, 2011

### marcus

Definitely! The acceleration was discovered in 1998 and has been born out by subsequent observations. It is a very gradual acceleration.

This is consistent with the fact that the Hubble expansion rate which has always been decreasing is expected to continue to decrease---as Chalnoth indicated.

The Hubble parameter is a fractional or percentagewise rate of growth of distance.

The current value of the Hubble is that distances increase by 1/140 of one percent every million years.

This percentage is expected to continue gradually declining. It might eventually get down to 1/160 of one percent every million years. I forget the exact level it is tending towards.
Multiply 0.85 times 1/140 to get the answer.

What is increasing (when we talk of acceleration) is the absolute rate of growth.

Absolute rate of growth is different from percentage rate of growth.

Say you have 100,000 dollars in the bank and every year someone is adding 1000 dollars to it. The absolute rate is 1000/year steady but the percentage rate is going down.

Now suppose each year he increases the amount he adds by one penny. The first year he adds 1000, the next he adds 1000.01, the next year 1000.02, and so on. Now there is acceleration in absolute terms, each year the amount added gets bigger.

But percentagewise speaking your percent yield is going down.

english language does not readily distinguish between different kinds of rates. fractional versus absolute.
"expansion rate" is ambiguous in spoken English. Have to live with this and be alert to it.

Last edited: Aug 23, 2011
8. Aug 23, 2011

### phinds

Since Marcus didn't address this particular thing specifically, I'll add this: your question implies a dichotomy that does not exist. That is, you posit and "either / or" between two things that are NOT either or, they are both yes. YES the universe is expanding at an accelerating rate and YES the universe is simply doing what it has always done. As Marcus explained, it is doing it a a slightly decreasing rate of acceleration, but it is still accelerating, as it has been for at least many billons of years.

I do think I read that the acceleration of the rate of expansion may not have been present right after the age of inflation (because dark energy had not yet overcome gravity) but I may be mis-remembering that.

9. Aug 23, 2011

### marcus

phinds, this is a fine point but as I recall the inflection point in a(t) occurs around 6 billion years.

That is, for the first 7 billion years (or 6-8, I forget) the slope a'(t) is decreasing because matter is slowing the process. Then by that time matter is so thinned out that the effect of the cosmo constant dominates, and the slope a'(t) increases.

So the scale factor history is always increasing but not like a straight line, seen from above it is first convex and then after 7-or-so billion concave.

there is a picture of the history of a(t) in a paper by Lineweaver from around 2003 or 2004. the paper is called *Inflation and the Cosmic Microwave Background*.
It is a paper with a lot of unusually educational pictures. Very intuitive and teacherly.
See if you can find it on arXiv, if you want to see a picture of how the scalefactor has grown over time.

The picture is Figure 14 "Size and Destiny of the Universe". He does not actually mean size, he means the scalefactor. We don't know that the U actually has a size. The scalefactor a(t) is the relative size of distances, with the present normalized to equal 1,
i.e. a(present)=1.

So a = 0.01 was when distances were 100 times smaller than now and a = 100 will be when distances between the same stuff will be 100 times bigger than now. Assuming the stuff is at rest relative to background, as usual.

I see the Lineweaver article is so famous you can just google "Lineweaver inflation and"
and google jumps to the conclusion that you are looking for "Lineweaver inflation and the cosmic microwave background". Google knows a good thing, now and then

Last edited: Aug 23, 2011
10. Aug 23, 2011

### phinds

Thanks Marcus, I'll check that out. It MIGHT be what I read in the first place and remembered about half-right (I did vaguely recall that the inflection point was about about 6 or 7 billion years but I had the situation before the inflection wrong).

11. Aug 24, 2011

### Chalnoth

The Friedmann equations relate the expansion rate of the universe to the contents of the universe. If we have a universe that only has a cosmological constant and normal/dark matter, then the expansion rate is given by:

$$H^2(a) = H_0^2\left({\Omega_m \over a^3} + \Omega_\Lambda\right)$$

Here $H_0$ is the current expansion rate (the Hubble Constant), and $\Omega_x$ is the current density fraction of that kind of stuff. Right now, $\Omega_m$ is around 0.27, while $\Omega_\Lambda$ is around 0.73. As time goes forward, the factor $\Omega_m/a^3$ will decrease, meaning that the expansion rate $H(a)$ will decrease.

However, it will not decrease indefinitely: it is going to approach a constant in the far future. And when $H(a)$ is a constant, we have a very simple differential equation:

$$H(a) \equiv {1 \over a}{da \over dt} = H_0$$

This can be rewritten as:

$${da \over dt} = aH_0$$

If you know your differential equations, you'll know that this is the equation for exponential growth. When the Hubble rate reaches a constant, the scale factor will be growing exponentially.

12. Aug 24, 2011

### keepit

Galaxies become increasingly farther apart from each other with each passing year (accelerating expansion). That being said, would it be fair to say that this increasing expansion is only because expansion of the universe is a function of how far apart things are. However, this percentage increase in distance from each other will not be as great in the future as it was last year (Hubble constant).
?

13. Aug 24, 2011

### WannabeNewton

The scale factor a(t) that increases the "size" of each successive space - like hypersurface is a function of time. The friedmann metric can be written as $ds^{2} = -dt^{2} + a^{2}(t)d\Sigma ^{2}$ where $d\Sigma ^{2}$ represents the geometry of the members in the family of space - like hypersurfaces. You can see that expansion of the universe is related to the scale factor a(t) as a function of time.

14. Aug 24, 2011

### keepit

will the Hubble parameter go to zero in the future?

15. Aug 24, 2011

### marcus

No, not according to the standard model (LCDM) that almost everybody uses.

According to the LCDM adjusted to have best fit to the data we can expect the Hubble parameter to decline towards 85% of its present value.

To continue declining indefinitely, more and more slowly, and approach something like that as a limit.

Nothing is certain. It's just a model. But it fits the data collected so far extremely well, and thats what it says to expect.

Essentially leveling out at 85% of present value.

So if you think of the present value as saying that a distance will grow by 1/140 of one percent in the next million years, then you multiply that by 0.85 and you get a slower rate of 1/160 of a percent every million years.

Last edited: Aug 24, 2011
16. Aug 25, 2011

### edgepflow

We can look at the magnitude of the acceleration of the universe with the deceleration parameter (REF: "Introduction to Modern Cosmology," Liddle, Eqn. 7.11)

q = $\frac{\Omega}{2}$ - $\Omega$$_{\Lambda}$

If q < 0, then the universe is accelerating:

At the current time:

$\Omega$ = density parameter $\approx$ 0.3

For a flat universe: $\Omega$ + $\Omega$$_{\Lambda}$ = 1

Thus,

$\Omega$$_{\Lambda}$ = 0.7

And,

q = 0.3 /2 - 0.7 = -.55

In the Far Future (based on current models)

$\Omega$ $\approx$ 0
$\Omega$$_{\Lambda}$ $\approx$ 1

and q $\approx$ -1

So the magnitude of the acceleration universe is still increasing but is bounded.