Trying to understand Dark Energy

[phinds]

Thanks for all that, Marcus.

 

[marcus]

You are very welcome Phinds! No problemo.
A picture of exponential growth at a constant rate might give Rede additional insight.
I have to go out, unfortunately don't have time to paste one in.

Exponential growth at a constant rate is our classical picture of acceleration i.e. distance growth at increasing speed.
It's kind if iconic, so a graph of growth at some constant percent rate might be a good reminder.

Also the lightcone calculator can plot curves, like the declining leveling out curve of the universe's expansion rate H(t). Maybe we should use more graphics in answering beginner questions.

Hope Rede is getting clearer--he says he is--and is OK with this, so far.

 

[rede96]

Rede, it might help you get over the confusion if you would practice doing something:
when you mean speed, say speed, not "rate"
use the word "accelerate" only when talking about a speed. That's the root meaning of "accelerate".

Sure, I'd like to understand this all properly, so your help is much appreciated.

A galaxy that today is 14.4 billion LY from us is receding at speed c.

Like the current rate of distance growth is 1/144 percent per million years.

Ok, that makes sense now, so a galaxy that is 14.4 billion years away from me now, will be in 1 million years be 14.401 billion years away? (14.4 billion * 0.00694%, where 0.00694 = 1/144) And a galaxy that is half that distance away from me (7.2 billion years) will be receding at 0.5c

And if you pick two widely separated galaxies the size of the distance between them will be changing, of course, over time. In the long run it will be growing exponentially at 1/173 % per million years.

So in that case, a galaxy that is 14.4 billion LY away from me would be receding at a speed of 0.8c

You can talk about the expansion rate INCREASING or DECREASING over time. Actually it has always been decreasing since very early times near start of expansion.

So if the expansion rate is decreasing, why do people say that universe is accelerating? That's really confusing!

 

[Bandersnatch]

So if the expansion rate is decreasing, why do people say that universe is accelerating? That's really confusing!

Yes, this is confusing at first. I see we even have a featured thread in the cosmology section about this very subject and how to avoid this confusion.

In any case, there are two distinct things being mixed up here: the rate of expansion, which is by what fraction all distances increase every time unit, and the change in distances between actual objects carried by the expansion of space. The first is decreasing at an ever slowing rate, the latter increasing at an accelerated rate.

A very handy analogy to use here that may illuminate the difference is the analogy with a savings bank account. I know marcus is also fond of using it, and perhaps he even already did (the lazy me only skimmed the thread).

As you know, when you put a given amount of money on the account, it'll usually be made to grow at some specified yearly or monthly rate.
If you were to think about the amount of money as analogous to distances between some objects in the universe, and the percentage rate at which your money grows as the Hubble parameter, you'll get a pretty good picture.
Turns out, you can make the amount of money on your account grow at an accelerated pace, even though the percentage rate by which it grows may be decreasing.
Imagine you've got an account in a bank that makes your deposited money grow by (1+1/n)%, where n is the number of months since deposit. You get 2% increase after the first month, 1.5% after the second, 1.3% after the third and so on. The farther away in time you go, the less the rate decreases, but it never stops decreasing.
The first months, when the drop in the percentage rate was significant, will make the amount of money grow at a slowing pace, but after a while when the rate flattens out and asymptotically approaches 1%, you'll get an accelerated, or even exponential growth of money.

Try it out on paper. Imagine you've got 10000 bucks or so deposited. The first month, you get 200 bucks. The second month, 153 bucks - the rate is going down, and the 'speed' at which you're gaining money is going down as well.
But if it's, say a 10th month, and you have at that time say, 1 000 000 bucks, on the tenth month you'll get 1.1% of 1 000 000 = 11000$, the next, eleventh month, 1.09% of 1 011 000 = 11020$. So your money (distances) are already accelerating in their growth.

Here, I believe in visual aids. marcus' numbers can be plotted:

The above is the graph of evolution of the Hubble parameter, or expansion 'rate' with time (analogous to percentage rate on your account)

while the following one:

Shows the increase of distances between given objects (i.e. the 'scale factor'; the total amount of money in the analogy) over the same period of time.

You should be able to easily see that the first is always going down, while the second always increases (i.e. universe expands, your savings grow), but at first the curve was sloping down (decelerating expansion, or less money gained every month), while after a while it begun curving up (accelerating expansion).

 

[marcus]

Yes! Thanks for those graphs, Bander! The changeover happens right around year 7 billion, where the curve crosses the 0.6 line.

Rede, can you see that? until year 7 billion the sample distance growth curve is CONVEX upwards (with declining slope going forward) and after year 7 billion it is CONCAVE upwards (with steepening slope going forward)

This curve is basically the growth history of some particular distance which we say has SIZE EQUAL ONE
at the present time. You can see roughly where the present year 13.8 billion is, on the time axis and at that point the height of the curve is 1. It's as if we took two galaxies and made the present size of their separation our unit and focused on that one specified distance, as it grows. If they are currently 10 billion LY apart we take that as our unit and quantify all the other distance sizes in those terms.

So in year 13.8 billion the curve is at height one. It's just a convention, to get the history of a particular distance that we can focus on.

 

[marcus]

Meanwhile the other curve is the universe's EXPANSION RATE. For convenience you can see it is compared with the present rate of 1/144% per million years. You can see it leveling off right about 0.8 of 1/144%
Which is 1/173%

Rede, recall the 0.8 you calculated in your post #38? that was 144/173 and that was correct. Or (1/173)/(1/144) however you calculated it.

That is the 0.8 which the curve H(t)/H(now) is tending to. All this stuff fits together in basically simple ways.