## Quick guide to h dependence in cosmology

Astronomers often calculate the distance to a galaxy assuming that
 distance = f(z,Ωm) / H0
where f() is a function that depends on the type of cosmological distance (luminosity, comoving, angular diameter). Since the scaling of distance with H0 is simple, we can use the Hubble parameter
 h = H0 / (100 km s-1 Mpc-1)
and include this dependence in estimated distances and in related quantities. Even now, with precision cosmology, the difference between h=0.66 and h=0.73 can be significant so there is some merit in leaving the parameter h in distance values. Other astrophysicists prefer to assume a value of h and give distance values without the parameter h. See Darren Croton's paper for further discussion, pitfalls and examples.

The following table compares these approaches with some examples for various observational quantities (tables in both directions here):

 Value assuming h=0.7 Value with h dependence line-of-sight distance, and transverse distance from distance times angular separation 1.00 Mpc 0.70 h-1 Mpc volume from distance cubed 1.00 Mpc3 0.343 h-3 Mpc3 luminosity that is proportional to distance squared 1.00 L☉ 0.49 h-2 L☉ number density of galaxies from inverse volume 1.00 Mpc-3 2.92 h3 Mpc-3 luminosity density from combining previous two 1.00 L☉ Mpc-3 1.43 h L☉ Mpc-3 absolute magnitude from -2.5 log luminosity -20.00 -19.23 + 5 log h mass from scaled luminosity 1.00 M☉ 0.49 h-2 M☉ mass from velocity squared times transverse distance 1.00 M☉ 0.70 h-1 M☉

Other h dependencies:

 Value assuming h=0.7 Value with h dependence critical density of the universe from dynamical equations 1.36 x 1011 M☉ Mpc-3 2.775 x 1011 h2 M☉ Mpc-3 Hubble time from inverse H0 13.97 Gyr 9.78 h-1 Gyr mass in N-body gravitational simulations, allowed scaling 1.00 M☉ 0.70 h-1 M☉

And now it can get slightly confusing. The h dependence can be defined by adjusting a variable, or by placing the dependence after the variable's value or in front of the units. The following table shows examples of these variations in style.

 Adjusting variable After value In front of units Tabulating number density values φ h-3 / Mpc-3 = 0.1 φ / Mpc-3 = 0.1 h3 φ / (h3 Mpc-3) = 0.1 Assigning absolute magnitude values M - 5 log h = -20 M = -20 + 5 log h Physical baryon density measurement using density parameter Ωb h2 = 0.022 Ωb = 0.022 h-2

Written by Ivan Baldry, 2015 November.