[1] Luminosity functions are used to study the properties of large groups or classes of objects, such as the stars in clusters or the galaxies in the Local Group.
It is used to compare star formation and death rates, and evolutionary models, with observations.
Main sequence luminosity functions vary depending on their host galaxy and on selection criteria for the stars, for example in the Solar neighbourhood or the Small Magellanic Cloud.
As this is determined by the rates at which these stars form and cool, it is of interest for the information it gives about the physics of white dwarf cooling and the age and history of the Galaxy.
provides an approximation of the abundance of galaxies in a luminosity interval
The luminosity function has units of a number density
is a characteristic galaxy luminosity controlling the cut-off, and the normalization
Equivalently, this equation can be expressed in terms of log-quantities[6] with The galaxy luminosity function may have different parameters for different populations and environments; it is not a universal function.
[7] It is often more convenient to rewrite the Schechter function in terms of magnitudes, rather than luminosities.
If one assumes that every dark matter halo hosts one galaxy, then the Press-Schechter model yields a slope
The reason for this failure is that large halos tend to have a large host galaxy and many smaller satellites, and small halos may not host any galaxies with stars.
See, e.g., halo occupation distribution, for a more-detailed description of the halo-galaxy connection.