The functional form of galaxy and halo luminosity and mass functions

TL;DR

This paper uses symbolic regression to automatically discover optimal functional forms for galaxy and halo luminosity and mass functions, outperforming traditional models and providing a framework for future astrophysical data analysis.

Contribution

It introduces a novel application of symbolic regression to derive better fitting functions for astrophysical quantities, surpassing traditional models.

Findings

Many functions outperform Schechter and double Schechter fits for LF and SMF.

Functions outperform Press--Schechter and Warren/Tinker models for HMF.

Framework enables automated discovery of optimal functions for astrophysical data.

Abstract

The galaxy luminosity and stellar mass function (LF, SMF), and halo mass function (HMF), are fundamental quantities in astrophysics and crucial inputs to a range of astrophysical and cosmological analyses. They are typically parametrised by fitting functions that have been chosen "by eye" to match observed or simulated data. We apply symbolic regression -- specifically the Exhaustive Symbolic Regression (ESR) algorithm -- to automate the search for optimal LF, SMF and HMF functional forms. ESR scores all functions up to a maximum complexity composed of a user-defined basis set of operators using the description length, an approximation to the Bayesian evidence that balances accuracy with complexity. We find many functions outperforming the Schechter and double Schechter functions for the LF and SMF, and that outperform the Press--Schechter and Warren/Tinker functions for the HMF. By additionally imposing "physicality checks" on functions' extrapolation and integration properties, we identify the optimal, low-complexity functional forms in terms of accuracy, simplicity and behaviour beyond the data range. As well as providing drop-in replacements for literature LF, SMF and HMF fitting functions, and identifying robust behaviour across well-fitting functions, we present a framework with which symbolic regression may be used to automate the discovery of optimal functions for any astrophysical dataset.