How is "little $h$" measured in cosmology? The dimensionless parameter from the Hubble constant, $H_0$

cosmology astronomy space-expansion physical-constants

5,510

The little $h$ is a historical artifact, one that will probably die out soon enough.

The thing is, $H_0$ was extremely difficult to measure precisely for many decades after its importance was realized. At some point, cosmologists were divided between the "$H_0 = 50\ \mathrm{km/s/Mpc}$" and the "$H_0 = 100\ \mathrm{km/s/Mpc}$" camps. Because the quantity appears as an overall scale factor to some power in many cosmological formulas, people adopted $h$ to be $H_0/(100\ \mathrm{km/s/Mpc})$ by definition. Rather than plugging in their preferred value of $H_0$, they quoted formulas in terms of $h$ and its powers, so that others using different values of $H_0$ could compare to them. All $h$ does is make undoing someone's erroneous value for $H_0$ easier (sort of).

Today, we know $H_0$ to a few percent or so, and few people lose sleep over the imprecision. Since there is nothing physically meaningful about the $100\ \mathrm{km/s/Mpc}$ scaling, it is $H_0$, not $h$, that is more fundamental.

5,510

ShanZhengYang

Updated on August 01, 2022

Comments

ShanZhengYang 5 months

Hubble's law has been well-know for close to a century now. It is written as

$v = H_0 d$

where the Hubble constant $H_0$ is the constant of proportionality between recession speed $v$ and distance $d$ in the expanding Universe.

The expression for the Hubble constant itself is normally written as

$H_0 = 100\,h\,{\rm km}\,{\rm s}^{−1}\,{\rm Mpc}^{−1}$

where $h$ is the dimensionless parameter expressing of ignorance. How is this "little h" measured?

Do astronomers measure the Hubble constant (using Cepheid distances, Type Ia Supernovae, etc.) and then calculate $h$?

Supposedly the value agreed upon today is around $h\approx0.7$. Give or take.
- Kyle Kanos over 7 years
  
  Note that $H_0h=100\cdot0.70=70\,\rm km/s/Mpc$ which is the one of the accepted values of Hubble constant, then extrapolate the definition/meaning of $h$.
- KidElephant over 7 years
  
  Why should $h$ be measured, when we can instead measure $H_0$ and use the above definition? $H_0$ has physical meaning; $h$ is a shorthand.
- Kyle Kanos over 7 years
  
  I wrote $H_0$ but it should just be the 100, sorry for confusion.
ShanZhengYang over 7 years

Ah, so "little h" is basically just a historical artifact at this point. This answers my question: today, we try to measure $H_0$ and then extrapolate from there. The Hubble constant $H_0$ has been measured fairly precisely. For a recent paper, read Efstathiou, 2014, arXiv:1311.3461v2 Planck (2013) measures $H_0 = 67.3 \pm 1.2 km s^{-1} Mpc^{-1}$, while direct measurements of Cepheid data comes up with $H_0 = 72.5 \pm 2.5 km s^{-1} Mpc^{-1}$.
ShanZhengYang over 7 years

Could you actually provide some more information as to when/why people started using "little h"? It would be good extra historical background.
Kyle Oman over 7 years

Was about to ask this question as I would have killed for a google-able answer to it when I was a fledgeling student, and came across this. Recent too, I guess I was a little bit too slow. Anyway, I took the liberty of editing in a link to a paper of immense use (imho). The "little h" rabbit hole is deeper than I could have guessed...
pela over 7 years

@KyleOman: I was just about to link to that very important paper when I was your comment. :)
Winther over 7 years

The rumours of little $h$'s death is greatly exagerated:) The use of $h$ is so ingrained in cosmology that I expect it to be with us long after we hit sub-percent accuracy. The value is also model dependent so as long as there exist viable alternatives to $\Lambda$CDM it will stick around. I bet it has at least $50$ more years left.