Understanding the Scale Factor in the Friedman Equation | Cosmology Explained

[robertjford80]

I'm having trouble understanding the part in the Friedman equation where a'/a. I can't figure out why it's the derivative of a over a itself. If someone could give me some sort of analogy to something I already understand that would help. I'm pretty sure a is the rate at which the universe expands but not 100% sure. I've watched the Susskind lectures on Cosmology but he sort of glosses over that part.

 

[BillSaltLake]

a'/a is the fractional expansion per unit time. At present, it's about 2.5x10^-18 per second or 1/[14 billion] per year.

 

[robertjford80]

but why is it a'/a and not just a?

 

[twofish-quant]

but why is it a'/a and not just a?

Because a is a scale and you need the equations to be scale invariant.

a is an arbitrary number and you can pick any number of the current value of a (i.e. you can pick 1, 2, 3000, 0.333) since you don't care about a particular value of a, you are interested in the ratios of a over time.

Because of that the equations need to be such that if you multiply a by a constant number, the equation still works out, and that means that all derivatives of a have to have an "a" in the denominator. That way if you multiply a by a random number, the equation is still true.

 

[robertjford80]

can you think of another phenomenon in nature that obeys such a law so that I can maybe wrap my head around it.

 

[marcus]

I'm having trouble understanding the part in the Friedman equation where a'/a. I can't figure out why it's the derivative of a over a itself. If someone could give me some sort of analogy to something I already understand that would help. I'm pretty sure a is the rate at which the universe expands but not 100% sure. I've watched the Susskind lectures on Cosmology but he sort of glosses over that part.

Often the scalefactor a(t) is adjusted for convenience to equal ONE at the present time. It is called "normalizing the scale factor to equal 1 at present."

You realize that we do not know the size of the universe, maybe it is infinite so it has no finite size in lightyears*. Or maybe it IS finite but we do not know the circumference. So what are you going to do?

Very commonly you just say well WHATEVER the size is, today, we will call that ONE.
And then back when distances were just half the size they are today, that becomes a = 1/2.
And back when distances were 1/10 what they are today, the scalefactor was a = 1/10.

What's wrong with that? It seems like a reasonable way to encode the information.

Suppose you want to tell me your growth history from the time you were an infant up to when you reached full adult height. But suppose you did NOT want me to know your actual height, in centimeters or feet or whatever unit.

year   1    4    7      10       13      16          
a     1/6  2/6  3/6     4/6      5/6      1

Maybe you reached full adult height at age 16
Maybe your growth history was like in this table.
I'm not supposed to know your overall size but I'm supposed to understand the history
by which you reached that size, whatever it is.
You might be 6 miles tall, I don't know. Or 6 meters tall.

*If the U is infinite size then you could think of the scalefactor as describing the growth of the "average distance between galaxies" or you could arbitrarily PICK some distant galaxy as your distance scale indicator, assuming it was far enough away so its distance would increase in the same generic proportion with other longrange distances.

 

[robertjford80]

Ok, I sort of understand.

 

[marcus]

Notice that in that example you grow by 1/18 every year. 1/18 of your full adult height whatever that is.

So what is your FRACTIONAL GROWTH RATE, say at age 7 when your height is 0.5?

well it is 1/18 (your absolute annual growth) divided by 1/2 (your height at that age).

It is a'/a which is the fraction of your height at age 7 that you are going to increase by that year.

Fractional growth rate is a very handy quantity---a good handle on various processes. It happens to be a good handle on the Friedmann equation. To me the Friedmann eqn looks simplest when you work with the fractional growth rate a'/a and call that H and then the left hand side is simply H². That fractional growth rate H is really what you want to know anyway.

Sometimes people talk about fractional growth rate as PERCENTAGE growth rate. So if you are going to grow by 1/20 of your current height that year, it is a 5% growth rate. It is a way of talking about a'/a.

Does this make more sense now? It is just an extremely convenient way of encoding the growth process history. (Especially when you don't know the overall size in miles meters lightyears whatever)

 

[robertjford80]

could you give me a test question, maybe with say the population of the united states, to see if i really get it.

 

[Chalnoth]

could you give me a test question, maybe with say the population of the united states, to see if i really get it.

Well, in this case there's a very direct relationship. With population, we frequently talk in the US of the rate of population growth: around 2% per year. This rate of growth is directly analogous to the rate of expansion. In fact, the current rate of expansion at 70km/s/Mpc in the units usually quoted in astrophysics equates to 0.000000007% growth per year in the distance between galaxies.

Edit: Sorry, I'm not very good at making up test questions, so I don't know what to say there.

 

[marcus]

could you give me a test question, maybe with say the population of the united states, to see if i really get it.

I'm not sure what "it" is that you think you don't get. But I'll make up a test question. It can't hurt.

Let's imagine that the Usa population is growing at a steady 2% per year. We don't want to bother with actual numbers (or maybe we don't know the actual head-count) so let's just say that the population as of 1950 = 1 by definition.

So we use a scale factor for the population which is normalized so that:
a(1950) = 1

a'(1950) = ? [do you recognize that a' is the amount that a increases per year, or per unit time, whatever the time unit is? it is a notation used in some but not all calculus books]

Now what about a'/a ? What is that for the year 1950?

Now suppose I tell you that a(1960) = 1.219

Please tell me what a'(1960) is!

Also then what is the ratio a'/a for that year?

These are extremely simple things. It's possible you are impeded simply by the fact that the PRIME notation is unfamiliar to you. If f(t) is a function of time then f'(t) is the change per unit time.
And the FRACTIONAL CHANGE, the change f'(t) expressed as a fraction of the the whole, is obviously going to be f'(t)/f(t).

 

[robertjford80]

I think a' is .02 and a for 1960 is 1.22

 

[marcus]

a'(1960) = .02*1.219 = 0.02438, the absolute increase.

a'/a = 0.02438/1.219 = 0.02, the fractional increase (the absolute gain expressed as a fraction of the whole).

Hope the example was not too simple. You asked for an example involving the Usa population, I hope this was helpful.

 

[robertjford80]

ok, thanks

 

[twofish-quant]

can you think of another phenomenon in nature that obeys such a law so that I can maybe wrap my head around it.

There's an example in finance.

Suppose you find that bananas cost $1 and apples cost $2. If you convert to Euros, you'd expect the ratio of prices in bananas to be 1:2. So if bananas cost 0.75 Euros you'd expect that apples to also cost 1.50 Euros. You'd also expect that the equations for the price of apples not care if the apples are measured in Euros or dollars.

Another example, you don't care if a stock costs $4 or $40. What you care about is the ratio of stock prices (i.e the stock yesterday was $4, today it's $5) that's the same situation in which the stock yesterday was $40 and today it's $50. So when a company announces a stock split, you don't care, since you just care about the ratios of stock prices, not the prices themselves.

Using this fact you can write some equations for how stock prices behave, which is why banks hire cosmologists.