## Quick guide to *h* dependence in cosmology

Astronomers often calculate the distance to a galaxy assuming that

distance = f(z,Ω_{m}) / H_{0} |

where f() is a function that depends on the type of cosmological distance
(luminosity, comoving, angular diameter).
Since the scaling of distance with H_{0} is simple, we can use the Hubble parameter
*h* = H_{0} / (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 Mpc^{3} | 0.343 *h*^{-3} Mpc^{3} |

**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 *h*^{3} 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 10^{11} M_{☉} Mpc^{-3}
| 2.775 x 10^{11} *h*^{2} M_{☉} Mpc^{-3} |

**Hubble time** from inverse H_{0}
| 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 *h*^{3}
| φ / (*h*^{3} 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
| Ω_{b} *h*^{2} = 0.022
| Ω_{b} = 0.022 *h*^{-2} |

Written by Ivan Baldry, 2015 November.

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