# Critical rate of expansion ?

1. Feb 25, 2010

### waynexk8

critical rate of expansion ???

Could any of you tell me in laymans terms, the little olde Universes started out with so nearly the critical rate of expansion. And even now Billions of years after what you call the Big Bang it is still expanding at nearly the critical rate.

What is the critical rate; well not what is it, but how slow fast is it ??? Is the rate of expansion say 50,000kph and if it was 10,000kph less each way, Planets would not have formed. Or is the number bigger or smaller in KPH either way ???

Wayne

2. Feb 26, 2010

### Chalnoth

Re: critical rate of expansion ???

Basically, using GR, one can write down the expansion as follows (neglecting units):

$$H^2 = \rho - \frac{k}{a^2}$$

Here the expansion rate is $$H$$, $$\rho$$ is the total energy density of the universe at a given time, and the parameter $$k$$ is the spatial curvature.

Now, these three components are not independent: k is a function of the other two. Specifically, if, in the right units, $$H^2 = \rho$$ at any time, then $$k = 0$$. And since the spatial curvature is independent of time, $$H^2 = \rho$$ for all time.

3. Feb 26, 2010

### waynexk8

Re: critical rate of expansion ???

Hi there,

Thx for the answer, but could you not give it a little more in laymans terms, as thats a bit over my head.

Could you not please expland on the below ???

How slow fast is it ??? Is the rate of expansion say 50,000kph and if it was 10,000kph less each way, Planets would not have formed. Or is the number bigger or smaller in KPH either way ???

Wayne

4. Feb 26, 2010

### marcus

Re: critical rate of expansion ???

Wayne it sounds like you may have been reading some bad science popularization, or watching Discovery channel, and gotten some wrong ideas.
There is no definite speed (like 50,000 kph) that the universe is expanding.

There is a percentage rate that distances are growing at.

Currently the rate is 1/140 percent every million years.

So if you look at really largescale distances, like between independent clusters of galaxies, each of those distances will increase about 1/140 of one percent, over the next million years. How much it grows depends on how long it is. (There is no one particular speed.)

This percentage rate has been larger in the past. It has steadily decreased. And according to the standard model it is destined to continue decreasing and kind of level out around 1/160 of one percent.

Don't let the sloppy journalism at Discover channel or in the newspaper confuse you. When they say "accelerated" expansion they do not mean that this percentage rate is increasing. They mean something else. The percentage rate is slated to decrease indefinitely.

Again, about planets forming. There were already stars and planets forming back in the days when this percentage was over tenfold larger.
So it is just a pile of baloney if they tell you that the present expansion rate is somehow critical.

There certainly are some critical parameters in the picture, but popularizers on TV tend to oversimplify. One critical parameter is called Lambda, the cosmological constant. It is not a speed.
It can be thought of as an energy density. The density of a certain energy called 'dark' which has been around since the start and as far as we know is always constant everywhere at a value of
0.6 nanojoules per cubic meter. Or 0.65 nanojoules per cubic meter. I forget exactly.

That density is critical to our existence. If that density of special energy had been much larger then back when the stars and planets were supposed to form there would have been too much expansion and they would not have formed.

If you know metric units, a joule of energy is about the amount released when you drop a book on the table. Drop a kilogram about a tenth of a meter. Thud. It is a modest amount of energy.

0.6 nanojoules per cubic meter is the same as 0.6 joules per cubic kilometer.

So take a cube of space which is one kilometer on an edge, and put a modest thud of energy into it, all evenly distributed like in the form of a very very faint light.

It seems kind of wimp negligible, like it shouldnt matter. But it matters. That is Lambda.

There are a few things that kind of had to be in the right range of values for stars and planets to work out. Lambda is one of them.

The thing is, it is not a speed. I don't know of any actual speed (like you mentioned 50,000 kph) which had to be just right. It's other things that aren't speeds. Frustrating I know.

You could also be thinking about a parameter called Omega which is a density ratio. The real density compared to a density needed for spatial flatness. If Omega is much bigger than one, the universe collapses while it is still young. If Omega is much less than one then it leads to a funny spatial geometry and uncomfortably rapid expansion. Again, Omega is not a speed. But it is possible to discuss it in terms of energy densities.
Imagine converting all the matter in the universe into energy and add it all up so you get one number. Some number of joules per cubic kilometer. Or nanojoules per cubic meter. The critical density currently needed for spatial flatness is about 0.83 nJ/m3. And what measurements show we actually have is very very close to that, which I guess is something to be happy about.

Last edited: Feb 26, 2010
5. Feb 26, 2010

### waynexk8

Re: critical rate of expansion ???

Hi there,

So its not speed.

Actually, it was from Stephen hawking book.

Why did the universe start out with so nearly the critical rate of expansion...[and]....even now, ten thousand million years later, it is still expanding at nearly the critical rate of expansion? If the rate of expansion one second after the big bang had been smaller by even one part in a hunderd thousand million million, the universe would have recollapsed before it ever reached its present size.

And it said, as you know, if it were to expand slightly faster, Stars and Planets would not have formed.

I cannot understand the; one part in a hundred thousand million million. That is why I said some of what I did, could you explain that in laymans terms somehow please.

Wayne

6. Feb 26, 2010

### marcus

Re: critical rate of expansion ???

Now I see you are talking about Omega (and its relation to the Hubble expansion rate, which is not a speed, it's that percentage thing I was talking about, but we still call it expansion rate.)

There is some math that relates the Hubble expansion rate to the critical density needed for spatial flatness.

At any point in history the real density has to be very close to critical, or else things will get progressively wronger.

Omega is the ratio of the two. The actual measured density and the theoretical perfect density (for perfect spatial flatness). Saying that that the two must be close (or things will go progressively worse) is the same thing as saying Omega must be close to one.

Maybe in a laymans explanation we should just forget about omega and talk in terms of the two densities, the real and the ideal.

I don't think I ever tried to give a layman's explanation of this before. Maybe someone else like Chalnoth or Silas or whoever can give a better one. I'll try though. Have to do something else now, but in a few hours.

=============EDIT==============
I'VE GOT SOMETHING but it is not, at least yet, a layman's explanation. It is a way that you yourself can calculate the critical expansion rate, given the observed density which we see to be around 0.83 nJ/m^3.

The way is you copy or paste whatever is blue here into the window at Google and press search:
sqrt((8*pi*G/(3*c^2))*(.83 nJ/m^3))*10^6 years in percent

Google has a calculator which knows that c is the speed of light and that G is newton's gravity constant, and it knows how to express a number as a percent, if you tell it to.

If you look at the standard model cosmo equation, the Friedman, you will see that to have spatial flatness the expansion rate H has to be equal to
THE SQUARE ROOT OF THIS QUANTITY:
(8 pi G/3c^2) times the overall density of 0.83 nanojoule/m^3

That is actually very simple. It is some coefficient made of physics constants multiplied by the density itself.

We can always remember the coefficient out front by thinking "ate the pig over three see squares".

Now that gives you the fractional increase in a distance that occurs in ONE SECOND, but we want the fractional increase that occurs in A MILLION YEARS so we just have to tell Google calculator to multiply the first thing by "10^6 years".

And that will give 0.00007. which is the fractional amount of a distance that gets added to it. But it is easier to look at if we say "in percent".

And then it will give "0.007 percent". and that is 1/140 percent!

So for every density, like 0.83 nJ/m^3 or something else more dense or less dense, you can plug it in, and the calculator will tell you the percentage rate that distances have to be increasing in order for everything to be approximately level and all right.

Last edited: Feb 26, 2010
7. Feb 26, 2010

### waynexk8

Re: critical rate of expansion ???

K.

Thx for your time, look forward hereing a laymans explanation.

Wayne

8. Feb 27, 2010

### Chalnoth

Re: critical rate of expansion ???

Well, a sort of simple way of putting it is in terms of Earth's gravity. If we neglect the cosmological constant, this analogy holds very well.

Normally, if you throw a ball up in the air, it falls back down. This is typically because we can't throw balls all that fast: if the ball and the Earth don't move fast enough away from one another, their mass attracts them back to each other and they slam together. In cosmology, if we had no cosmological constant, this would be the equivalent of a closed universe. Too much matter density and not enough expansion rate makes it just recollapse right back on itself.

On the other hand, if you give the ball enough kinetic energy when it's thrown, then it just escape's Earth's gravity, and keeps going forever. In cosmology, this is the equivalent of having an open universe: too much expansion and not enough energy density makes it so that the expansion goes on forever.

In between these two options is the limiting case. For a ball thrown up from the Earth, the limiting case is where the ball keeps slowing down, and approaches zero speed as time goes to infinity. The cosmological equivalent is similar: just the right amount of expansion for your matter density means that the expansion never stops, it just slows down more and more all the time, approaching zero in the limit of time going to infinity. This is a flat universe.

The reason why our universe appears to be extremely flat comes down to inflation. During inflation, the way that he stuff that drove inflation behaved is that the more the universe expanded, the more the expansion rate came into line with the energy density of that stuff. Since the universe had to expand by somewhere around a factor of 10^30, the expansion rate had more than enough time to get exceedingly close to the energy density of the inflaton that drove inflation.

9. Feb 27, 2010

### waynexk8

Re: critical rate of expansion ???

I have to say thank you to you both, and its getting more clear, however I still do not get the critical rate of expansion, not so much the critical rate of expansion, but what in some sort of numbers in is so close to the critical rate of expansion.

Could numbers and speed like I did in my first post ??? The rate of expansion say 50,000kph and if it was 10,000kph less each way, Planets would not have formed. Or is the number bigger or smaller in KPH either way ???

However, if you cannot use numbers like above, seems I am still not getting it. As it’s expanding at a rate of speed is it not ??? All matter from the Big Bang spewed outwards at a critical rate of expansion, {speed} and still now Billions of years later it’s still close to the critical rate of expansion, but must have slowed down.

Maybe God and the Devil just made friends one day, and put all their Gunpowder together, and it was just a lucky guess, ROL.

Wayne

10. Feb 27, 2010

### marcus

Re: critical rate of expansion ???

No it is not expanding at any particular rate of speed. In this context "rate of expansion" does not mean speed, it means something more like "percentage growth per year"

or "percentage growth per million years".

I added some stuff to my post #6, back a few. You might try looking back and see if it helps at all.

For any density you pick, there is an ideal percentage expansion rate. In post #6 I showed how to calculate it. If the expansion rate is even a tiny bit different things go haywire and you don't get normal foursquare level geometry like we appear to have.

Let's take a simpler problem. How do we say this in a clear way that would, for example, impress fraternity brothers and their dates, at a noisy beer party? I'm trying to think.

I told you the current expansion rate is 1/140 percent per million years. Multiply that by 1000. It is the same as saying that the current expansion rate is 7 percent every billion years.

If back near the start of expansion, that percentage had been even ONE PERCENT DIFFERENT from what it was supposed to be, the universe would have been føcked.

If anyone wants to check the exact statement I think that this is covered for example in a paper by Charles Lineweaver called "Inflation and the Cosmic Microwave Background" from some time around 2003, but it wasn't new then---that was just a pedagogical paper explaining wellknown stuff.

The density of the universe used to be much higher. Whatever it is, it is some number of nanojoules per cubic meter. You plug that in and you can find the critical rate of expansion needed at that density. Right now the density is 0.83 nJ/m^3. So plug that in:

Put this into google, it will give the billion year percentage cause it says 10^9 years.

sqrt((8*pi*G/(3*c^2))*(.83 nJ/m^3))*10^9 years in percent

When you put that in google, it will say "7.17 percent". You can try it for other densities. Like back when the universe was 1000 times denser than today, then you plug in
8.3 nJ/m^3 (which means 8.3 nanojoules per cubic meter)

and you will get a much larger percentage expansion rate. And that must have been the rate back then (or very very nearly it) or the world would look very different and we might not be here.

Last edited: Feb 27, 2010