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I am trying to solve a Cosmology problem, but a certain quantity $h$ appears in it, of which I do not know the definition (I have never seen it mentioned anywhere before). So I thought maybe someone here could tell me what this $h$ is?

The problem goes as follows:

If $a$ is the scale factor in a FLRW-universe and $1+z=1/a$ is the redshift, then the luminosity distance of a far away object that is emitting light is given by $d_L=a_0r(1+z)$ (where $a_0$ is the scale factor today and can be scaled to $a_0=1$). Knowing the luminosity distance, we can get the angular diameter distance $d_A=l/\theta=d_L/(1+z)^2$, where $l$ is the proper size of the source and $\theta$ the apparent size. Knowing $l$, what is the minimum angular size of the radiating object? Be sure to first express your finding in terms of $h$, and then use $h=0.7$ to get a numerical value.

So, the problem asks to find the minimum angular size (which is $\theta$ I assume), and I can do that. But I have no idea what $h$ is supposed to be! Can someone clarify? Thanks for any suggestion!

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2 Answers 2

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This goes back to the history of the Hubble constant. It's easy enough to write most cosmological formulas in terms of this value, but measuring it was something of a challenge for a while.

For several decades, we were confident it was between $50$ and $100\ \mathrm{km/s/Mpc}$. Many results scale with this value (to some power), so what people did was write $$ H_0 = h \times 100\ \mathrm{km/s/Mpc}. $$ This way, you could put any value of $h$ you personally believed into the final result, but since this factor is of order unity you can also remove it from the result (replace it with $1$) to get a quick sense of what the answer is. Today we know $h = 0.7$ well enough that this practice of leaving $h$ in equations is disappearing... slowly. It's still a useful exercise, though, if you are comparing results that assume different values of $H_0$, in order to see how the change in $H_0$ affects the results.

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  • $\begingroup$ I do know of $H_0$ but did not suspect a connection between $H_0$ and $h$. Maybe I should have guessed, since $H_0\approx 68 \frac{km}{Mpc\cdot s}$. Anyway, now I know. Thank you! $\endgroup$
    – Kagaratsch
    Feb 9, 2015 at 23:05
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This is explained in the Wikipedia article on Hubble's Law: http://en.wikipedia.org/wiki/Hubble%27s_law

In particular,

"Dimensionless Hubble parameter

Instead of working with Hubble's constant, a common practice is to introduce the dimensionless Hubble parameter, usually denoted by h, and to write the Hubble's parameter $H_0$ as 100 h km s −1 Mpc−1, all the uncertainty relative of the value of $H_0$ being then relegated on h."

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