# Confusion about the accelerating expansion of the universe and Hubble

1. Jun 18, 2013

### question dude

My school textbook says that discovery of the 1a supernova was what led to the understanding that the universe expansion is accelerating

but doesn't hubble's equation already suggest that the universe would expand at an accelerating rate?

V = Hd (V = velocity, H = hubble's constant, d = distance from the observer)

a constant rate of expansion would be if the galaxies move away at constant velocity, but hubble's equation already shows than velocity would increase (so galaxies would accelerate) with increased distance moved, so I don't get why they didn't know the universe expansion was accelerating just from looking at hubble's equation

I know I'm missing something....

2. Jun 18, 2013

### marcus

You're right! IF Hubble rate stayed constant, but it had been declining as a percentage growth rate.

Here's what I think is a good way to think of it. If you work out all the units (kilometers, seconds, megaparsecs) you find that according to current estimate of H(now), it corresponds to distances growing 1/144 of one percent per million years.

That's equivalent BTW to the "Hubble distance" (i.e. the size of a generic distance growing at speed of light) being 14.4 billion ly.

As long as the percentage growth rate is constant, or even gradually declining but only very slowly, we have exponential growth.

But the catch is for much of the expansion history of the universe the Hubble rate was declining so fast that the growth of a generic distance was actually DEcelerating!

And then in 1998 with some more precise measurements, that you know about, we discovered "HEY, the percentage growth rate is NOT DECLINING AS FAST as we thought and is actually going to level out at 1/173 % per million years! We are very gradually approaching an exponential growth pattern and there is this very slight acceleration!"

Last edited: Jun 18, 2013
3. Jun 18, 2013

### Bandersnatch

4. Jun 18, 2013

### marcus

Here's a picture of the growth of a generic distance over time. You can see that the slope was clearly declining until around year 7 billion. Then it looks almost constant for a while as it begins very gradually to steepen. Acceleration started around year 7 billion, but so far has been almost imperceptibly gradual. We are now at year 13.8 billion--steepening becomes more visible on the graph around there.

This plot is what you get if you open Jorrie's calculator and do the following clicks:
click on "chart" and on "set sample chart range"
click "column definition" to get the menu and UNcheck everything except for Time and Scale factor.
click the white dot at the Time menu item, to make that the independent variable
press "calculate".

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Last edited: Jun 18, 2013
5. Jun 18, 2013

### question dude

so if hubble's constant has stayed the same through all time, then the expansion of the universe would indeed be accelerating?

and when you say its been declining as % growth rate, that means the actual constant value is still increasing, right? so that means the constant isn't staying the same, so the expansion of the universe is accelerating? - how does this work, I thought you said I was right that the expansion would be accelerating if the hubble's constant stayed the same

6. Jun 18, 2013

### marcus

This is a good example of an intelligent question caused by bad terminology. We should not say "Hubble constant", because the percentage expansion rate has not been constant over time.
For much of history the Hubble rate has declined rapidly enough that we have NOT had accelerated growth of distances. But the decline in the Hubble rate is easing off and we are entering into the exponential growth regime that one would expect from a constant percentage growth rate---as per dude's question.

Remember that you can always convert the Hubble distance to a reciprocal growth rate like this:
Hubble distance R = 14.4 Gly ⇔ Hubble rate H = 1/144% per million years. So we can read either one off this table: $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline T (Gy)&R (Gly) \\ \hline 0.0674&0.1021\\ \hline 0.0956&0.1445\\ \hline 0.1354&0.2044\\ \hline 0.1917&0.2890\\ \hline 0.2713&0.4086\\ \hline 0.3839&0.5775\\ \hline 0.5430&0.8160\\ \hline 0.7676&1.1522\\ \hline 1.0847&1.6251\\ \hline 1.5315&2.2869\\ \hline 2.1589&3.2044\\ \hline 3.0346&4.4534\\ \hline 4.2427&6.0957\\ \hline 5.8756&8.1265\\ \hline 8.0089&10.3976\\ \hline 10.6648&12.5991\\ \hline 13.7872&14.3999\\ \hline 17.2617&15.6499\\ \hline 20.0000&16.2548\\ \hline 22.8231&16.6519\\ \hline 25.7011&16.9035\\ \hline 28.6133&17.0597\\ \hline \end{array}}$$This says that around year 67 million, distances grew at rate 1% per million years.

And around year 13.8 billion (the present) they are growing 1/144% per million years.

It's easy to read other percentages off the table. For instance around year 1 billion (more exactly 1.08 billion) distances were growing 1/16 of one percent per million years.

To get that table from J's calculator is easy and a good exercise for getting used to it.
Open http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html
click "set sample chart range"
open menu "column definition and selection" and UNcheck everything except Time and Hubble radius R
then press "calculate"

That will give the same table except with too many rows! The default number of steps, in this particular case, is 60 which way more than you need. So go back up top where it says STEPS and number 60 is in the box, and change 60 to 20---or whatever smaller number you like.

then press "calculate" again.

Last edited: Jun 18, 2013
7. Jun 18, 2013

### marcus

I think before I answer I should suggest you try the "google calculator" as a way of converting the same physical quantity into different units.

Please try this: go to google and put "2 hours in seconds" into the box, press return.
You should get 7200 seconds. (Type or paste it in, but without the quotes.)
Put "300000 km/s in miles per second" in box and press return.
You should get around 186,000 miles per second.
It's the same physical quantity just expressed in different units.

"(100 km/s per Mpc) in percent per million years"
Just an example of a Hubble rate somewhat bigger than what we have today. Paste it in (without the quotes) and see what you get.

The idea is: if a megaparsec distance is growing at 100 km/s, what percent growth is that per million years? It is the same physical quantity just expressed in different units. Growth speed is proportional to size of [cosmic-scale] distances. Where the Hubble law applies, the same percentage works across the board [at any one time in history].
Does the google calculator function work on your computer? If so, you can convert any quantity (length, time, mass, speed, growth rate, energy...) into different units.

In this case I think a Hubble rate of 100 km/s per Mpc converts to something like 0.0102 % per million years. You could also write that as something like 1/99 of one percent.

The latest Planck satellite report gave a figure around 67.9 km/s per Mpc. That is very recent: March 2013.
What does that convert to?
When I do it, I get 0.006944, which is essentially the same as 1/144.
This means that saying "67.9 km/s per Mpc" is describing the same physical quantity as saying "1/144% per million years."

So the "percent per million years" and the "km/s per Mpc" are the same physical quantity and if one of them decreases the OTHER has to decrease also. They are equivalent ways of desribing the quantity H.

H can be written H(t) to show that it depends on time. It has been decreasing throughout the expansion history and according to standard cosmic model it is expected to continue decreasing (albeit more gradually, tending to level out eventually). And this is true whether one describes it in one set of units or some other set of units.

Last edited: Jun 18, 2013