An Empirical Mass Function Distribution

The halo mass function, encoding the comoving number density of dark matter halos of a given mass, plays a key role in understanding the formation and evolution of galaxies. As such, it is a key goal of current and future deep optical surveys to constrain the mass function down to mass scales that typically host L_∗ galaxies. Motivated by the proven accuracy of PressSchechter-type mass functions, we introduce a related but purely empirical form consistent with standard formulae to better than 4% in the medium-mass regime, 10¹⁰-10¹³ h^-1 M_⊙. In particular, our form consists of four parameters, each of which has a simple interpretation, and can be directly related to parameters of the galaxy distribution, such as L_∗. Using this form within a hierarchical Bayesian likelihood model, we show how individual mass-measurement errors can be successfully included in a typical analysis, while accounting for Eddington bias. We apply our form to a question of survey design in the context of a semi-realistic data model, illustrating how it can be used to obtain optimal balance between survey depth and angular coverage for constraints on mass function parameters. Open-source Python and R codes to apply our new form are provided at http://mrpy.readthedocs.org and https://cran.r-project.org/web/packages/tggd/index.html respectively.