# Rarely asked question: converting 1/(Hubble time) to km/s per Mpc

1. Sep 9, 2014

### marcus

A couple of questions came up from different people and I thought I'd respond here in case anyone else might be wondering.

First, not to confuse age of universe with Hubble time. Just coincidence that they are near.
When you look at Jorrie's calculator (Lightcone link in my sig) you see we are in year 13.8 billion of the expansion
and the Hubble time (a reciprocal measure of expansion rate) is 14.4 billion years. they are different.

Secondly, suppose you do one over 14.4 billion years. You get 0.069444 per billion years

Now you want to convert to the older clunkier traditional units: "km/s per Mpc"

So to do the conversion you paste this into google window (without the quotes) and press return:

"0.069444 per billion years in (km/s per Mpc)"

See what you get. It works for me, if it doesn't work for you please let us know.

Last edited: Sep 9, 2014
2. Sep 9, 2014

### Drakkith

Staff Emeritus
I get approximately 67.9 Km/s per Mpc.

3. Sep 9, 2014

### nick1o2

Same here.
I also tried for 0.473 billion years where Ht was "1.0176"
∴ H = 1/Ht
∴ H = 1/1.0176
∴ H = 1.407459536

And when i place "1.407459536 per billion years in (km/s per Mpc)" in Google i get this (screenshot):
http://gyazo.com/1688b0bbcd25a12d9f9a3ec1956df195

I am probably wrong but wouldn't this suggest that the hubble constant is getting smaller over time. In other words the universe is slowly expanding less, which doesn't make sense. Could you shed some light on this?

4. Sep 9, 2014

### Chronos

Traditionally, the Hubble time is based on the assumption expansion has been constant over time. If the rate of expansion has constantly increased over time, the Hubble time will underestimate the age of the universe. If expansion has constantly decreased over time, the Hubble time overestimates the true age of the universe. Considering expansion has been accelerating for about 5 billion years, the traditional value for Hubble time should underestimate the age of the universe. Alas, as in most cases involving cosmology, the truth is never quite so simple. Evidence suggests the rate of expansion decelerated for about 9 billion years before reversing course about 5 billion years ago. Depending on the rate of early deceleration, the traditional value for the Hubble time could be low, high, or just right compared to the age of the universe. With this in mind, it is unsurprising the Hubble time could be somewhat inconsistent with the age of the universe [or vice versa].

5. Sep 9, 2014

### marcus

That's right. That's the default value of H0 that Lightcone uses, corresponds to a value reported by ESA Planck mission in 2013, as I recall. thanks for confirmation Drakkith!

Good! You got the same answer as Drakkith and I did.
Of course it does, according to the standard cosmic model (so called LCDM or Lambda CDM) the hubble constant HAS BEEN DECLINING up until now and is EXPECTED TO CONTINUE DECLINING for the indefinite future. Your reasoning is not wrong. It's straightforward.

H(t) is the fractional (or percentage) distance growth rate. Like the interest rate on a bank savings account. If it declines slowly enough then your savings account will still grow (almost) exponentially. IOW it will "accelerate" in annual dollar growth terms.

There is nothing inconsistent or puzzling about the fact that the H(t) percentage growth rate has been declining (almost since the very start) and will continue to decline. That's just how it is. Why did you imagine otherwise? You can see that from Jorrie's Lightcone calculator, which shows the Hubble radius R(t) increasing (but gradually leveling off in the distant future. that means H(t) declines but the decline tends to level off in the distant future.

6. Sep 10, 2014

### nick1o2

Thankyou both. I did get a bit confused. My reasoning behind my question was that i know the universe has been gradually expanding faster and faster. And i believed that the hubble constant is the rate at which the universe is expanding (67.9 Km/s per Mpc). Wouldn't this mean that the hubble constant should be getting bigger not smaller? I was referring to the hubble constant slowly getting smaller and i think i confused myself with what hubble constant really is (i though it was the rate at which the universe expands).

7. Sep 10, 2014

### nick1o2

Also is there an exact conversion we could just times the hubble constant with to get the km/s per Mpc units?
I was playing around with some numbers and got 9.78x10^2. It isn't exactly precious so was wondering if there is one.

8. Sep 11, 2014

### Jorrie

It is actually 977.8, but then, the Hubble constant is only known to within a few percent, so 978 is quite accurate enough.

9. Sep 11, 2014

### Staff: Mentor

Sure, this is just a simple unit conversion. Both Mpc/km and seconds per billion years give a factor, divide both and you are done. As those values don't have an uncertainty (they follow from definitions), the factor is exact.

10. Sep 11, 2014

### marcus

You may have been confusing percentage growth RATE on the one hand with a particular distance's SPEED. The speed can increase even while the percentage rate is declining.
The percentage growth rate can decline and yet the size of any given distance you choose to watch can increase faster and faster, i.e. accelerate. There is no contradiction.

By an historical accident, the Hubble rate is customarily expressed in misleading units. You can use google to convert it to a percentage rate. Paste this in google window and press return:

"67.9 km/s per Mpc in (percent per million years)"

It will convert the rate from the old units and express it IN NEW TERMS, namely as a percent growth per million years.
It will say something like 0.007 of a percent per million years, more precisely 1/144 percent per million years.

So while we humans have existed as a species, distances in the cosmos have grown by somewhere around 1/144 of one percent.

Last edited: Sep 11, 2014
11. Sep 11, 2014

### nick1o2

I still don't think i catch your drift marcus. Sorry for being stubben i just want to be 100% clear.
So Hubble constant is the measurement used to describe the expansion of the universe. And Hubble Time is how old the universe is, but things like inflation and dark energy make it more than it really is? So dark energy speeds up the process of the hubble time in a way?

And the example would be that the current time is 13.787By but Hubble time it is 14.44by. So is this saying that if dark energy didn't exist then the universe slowly took it's time to expand (no interruptions), it would take another 0.74By to get to the same point as we are now (starting at 13.787By and moving 0.74By to 14.44).
-Thanks everyone for the answers above as well.

12. Sep 12, 2014

### Chronos

No, it only says the Hubble time is inconsistent with other indicators of the age of the universe. I think that is unsurprising for reasons already stated.

13. Sep 12, 2014

### Jorrie

No, you seem to misunderstand. Hubble time is not really a time - it is simply the inverse of the Hubble constant and is not directly related to the age of the universe. As has been explained by Marcus, the Hubble constant is the rate of expansion at a specific distance and can be seen as the % growth in cosmological distances.

The time it took to reach the present distances from an initial very dense state depends on how this % changed over time. The % was high in the beginning and has been declining ever since to around 1/144 % per million years today. Dark energy's accelerated expansion is working against this decline in % and will eventually stop the decline and settle the constant to some 1/173 % per million years.

This means that the Hubble time will eventually become constant at 17.3 billion years, while the age of the universe will move on, presumably forever.

Last edited: Sep 12, 2014
14. Sep 12, 2014

### nick1o2

Right this makes sense now.
And would this mean Hubble radius and Hubble time are inverse of each other as well? So Hubble radius has been increasing while Hubble time has been decreasing?

15. Sep 12, 2014

### Jorrie

No, they are the same in units where c=1. Both are presently 14.4 Gyr or Glyr.