# Accelerating universe

1. Nov 13, 2008

### spidey

What is the value of acceleration of universe? like Earth's gravitational acceleration is around 9.8 m/s2..has anyone found this value?

2. Nov 13, 2008

### marcus

An important thing to notice is that the universe does not have a definite speed at which it expands. It has a percentage growth rate which is currently about 1/140 of a percent per million years. All longrange distances increase by about that much every million years---the longer distances increase more, of course, because it is a percentage or fractional growth rate. They grow in proportion to their length.
A less hokey and more technical way to put it is that the universe has a fractional growth rate a'(t) the time derivative of the scalefactor a(t).

The number you are asking about is a''(t) the second time-derivative of the scalefactor a(t)
and it is given by one of the two Friedmann eqns.
Try wikipedia Friedmann_equations.
http://en.wikipedia.org/wiki/Friedmann_equations

You will see that the second Friedmann gives you an explicit formula to compute a"(t)/a(t)
and it happens by convention that the scale factor is normalized to equal one at the present time!
so a(now) = 1
and therefore the second Friedmann gives you an explicit formula for a"(now). You just have to evaluate the righthand side of the equation, what it is at the present point in history.

To make it easier, I will massage that equation a little. The overall energy density including dark is 0.85 joule/cubic km and 73 % of that is dark energy so that is 0.62 joules/cubic km.

so on the RHS we have (-4 pi G/3 c^2) (0.85 - 3*0.62 nanojoules/m^3)

But 3*0.62 = 1.86, so what we have in the parens there is (- 1.01 nJ/m^3). Is that clear, if not, please say. And the minus signs cancel so we have (4 pi G/(3 c^2)) (1.01 nJ/m^3)

Let's put that into the window at Google, and press return. What I get from Google is:
(((4 * pi) * G) / (3 * (c^2))) * (1.01 (nJ / (m^3))) = 3.14115742 × 10^-36 s^-2

Which means 3.14 x 10-36 per second per second.

Last edited: Nov 13, 2008
3. Nov 13, 2008

### marcus

The challenge associated with this is how to interpret it in everyday language for people familiar with percentage growth rates. A good way to remember the current rate of expansion of the universe is that it is about 1/140 percent every million years.

In a million years, each longrange distance will increase about 1/140 of a percent. That is the present rate, roughly. We are going to improve the accuracy in a moment.

Maybe the increase is easier to imagine if we use a timestep of a billion years. In a billion years the fractional increase is about 1/14. Picture each distance increasing by roughly 1/14 of its length, or about 7 percent, in the course of a billion years. Again this is a crude approximation.

If you know Hubble parameter is 71 km/s per Megaparsec (Mpc)---which you may since a lot of people have it memorized!---then you can get that on your own with the Google calculator. Just put this into the ordinary search window and press return:
1/(71 km/s per Mpc)
Google will say it equals 13.77 billion years.
What that means is each distance increases fractionally by 1/13.77 of itself per billion years, which is 7.3 percent per billion years

When we talk about acceleration, we can ask how much will that percentage grow in a billion years. It won't be 7.3 any more, what will it be? Will it be 7.4, or 7.5, or 8.1, or what? That's what acceleration means in this context, the increase in the expansion rate.

So we are going to use a billion year timestep instead of seconds. I multiply what I had before by the square of a billion years---that is, put this into Google:
(10^9 year)^2(4 pi G/(3 c^2)) (1.01 nJ/m^3)

and interpret the result I get (which is 0.00313) as 0.00313 per billion years per billion years
or, in other words, as 0.313 percent per billion years per billion years

So if the rate starts out at 7.3 percent per billion years, then after a billion years have gone by
it will be that same percent plus an additional 0.313 percent. The new rate of expansion of distances will be 7.6 percent per billion years.
What I have done here is a quickdirty to get a feel for the sizes of the numbers.
And remember it is the Friedmann acceleration equation that gives us a"(t=now) in those terms as 0.00313 per billion years per billion years. Except for adjusting units it is a straightforward calculation.
==================

The gist is that the current expansion rate for distances is 7.3 percent per billion years, and the rate that is increasing is by 0.313 (or more simply 0.3) percentage points per billion years.

So that a billion years in the future the expansion rate will be 7.6.

People can quibble about the exact numbers. Exponential growth, compounding. I'm not trying to be terribly precise and I'm just using linear approximations.

so does this make sense to you?

Last edited: Nov 14, 2008
4. Nov 14, 2008

### spidey

Thanks Marcus..You have given information more than i asked..Why i asked this is, i want to compare unruh effect and CMBR..since universe is accelerating, the galaxies should be getting the thermal unruh radiation as per their acceleration and so if i get the acceleration of galaxies, i can get the unruh temperature and i wanted to see whether this is same as CMBR temperature 2.7K. Just a thought. You have any information with unruh temperature and CMBR temperature..

5. Nov 14, 2008

### marcus

Unruh temperature doesn't apply to cosmological expansion of distances.
For unruh temp, you need real motion and some quite substantial acceleration in a local inertial frame.

In the first place, the distant galaxies are not moving much, negligible speeds of just a few hundred km/s as far as we've been able to tell.

And on top of that, they have trivial acceleration, essentially zero, in any local inertial frame.

Of course the distances from us to them are increasing, often at rates several times the speed of light, but that is just General Relativity change in geometry, has nothing to do with Unruh effect.

It might help if you had a look at the cosmo basics sticky thread in cosmo forum

6. Nov 14, 2008

### spidey

Thank you very much for your clear explanation..

7. Nov 14, 2008

### marcus

You are more than welcome! I thought your question was a valuable one, whereever it came from. The basic function of time in cosmology is the scalefactor a(t)
which evolves according to the Friedmann equations (that's at the heart of cosmological models)

We all know that a'(t) is positive, that's what expanding universe means, the scalefactor is increasing, distances are getting bigger. And we have a handle on a'(t) in the form of the Hubble parameter H(t) = a'(t)/a(t) by definition.

OK, now what "accelerating expansion" means mathematically is that a"(t) is positive. But does anyone bother to calculate it?