Some slight confusion regarding the dimensionless Hubble parameter h.

[taylrl3]

Hi,

I am just writing up my MSc thesis and want to explain the dimensionless Hubble parameter that I have been using through my work. I understand that you take the valuefor the Hubble "constant" and then divide by 100km/sec/Mpc to leave get a value which has no units. There seems to be surprisingly little literature on something that is so commonly used and fundamental a concept in cosmology. I just want to check that I am right in thinking this value is independent of time and position (hence dimensionless). Also I read in "Peebles: The large-scale Structure of the Universe 1981" that this dimensionless value also reflects the uncertainty in H. I can't see how. Can anyone explain? Also why do we use units of h^2 sometimes and h^-1 at other times? Your help is most appreciated.

Cheers
Taylrl

 

[marcus]

When Peebles wrote that in the 1980s I believe there was a lot of uncertainty about the value of H₀ (the present value of H(t) )

Since a lot of things that you calculate depend on what you choose for H₀, and since different people preferred different values, ranging from 50 to 150 in the usual units!
it made sense to write your answers down in terms of that conventional "h" number.

Then anybody could pick the h they preferred and plug it in and get the answer compatible with their idea of H₀.

In that sense, the widespread use of this "h" convention DID reflect the uncertainty about
H₀. Since in the 1980s the value was basically just a wild guess, or at least the focus of a lot of controversy, you sort of HAD to leave it undetermined in your published numbers derived from observational data.

Now the value of H₀ is more reliably known. But people still carry on the old practice. It is easy, you just stick in a 0.71 or a 0.75 as you go along reading the article.

I just want to check that I am right in thinking this value is independent of time and position (hence dimensionless)...

I am not sure I understand what you mean. "Dimensionless" means no units.
It does not mean constant over space and time.
The units (like meters or kilometers-per-second or Parsecs or kilograms) are the dimensions of a physical quantity.

The Hubble parameter H(t) is constant over all space (in standard cosmology) but varies over time. It used to be much larger than it is today.
It is dimensionful because it must be given in terms of some units, like "km/s per Megaparsec".

If you divide H₀ by the standard 100 km/s per Mpc, then the units cancel out and you are left with a pure number. A pure number is dimensionless.

However a pure number, a dimensionless number, can change over time. The important thing is not whether it is constant or not but whether or not you need units to express the quantity.

EDIT: Nick, I don't mean to suggest that one divides H(t) by 100 km/s/Mpc. Or that h is an example of a dimensionless number that changes with time. h is always H₀ divided by that conventional quantity, and it is constant. I agree with your following comment. My point was that some dimensionless numbers do or can change (logically dimensionless does not imply constant.)

 

[nicksauce]

Well, I think what's usually done is not writing H(t) = 100h km/s/Mpc, but rather H_0 = 100h km/s/Mpc. In that case, h is independent of time because, H_0 is.