🔭 astrophysics

The functional form of galaxy and halo luminosity and mass functions

This paper introduces a framework using Exhaustive Symbolic Regression (ESR) to automatically discover optimal functional forms for galaxy and halo luminosity and mass functions, identifying new mathematical models that outperform traditional standard fits like the Schechter or Press–Schechter functions.

Original authors: Amelia Ford, Harry Desmond, Deaglan J Bartlett, Pedro G Ferreira

Published 2026-04-28

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CC BY 4.0

Original authors: Amelia Ford, Harry Desmond, Deaglan J Bartlett, Pedro G Ferreira

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Cosmic "Best Fit": Finding the Perfect Recipe for the Universe

Imagine you are a chef trying to write down a single, perfect recipe that describes how much salt, sugar, and flour goes into every single cake ever baked in the history of the world.

You have millions of data points—millions of cakes—but you don't want a recipe that is a mile long. You want something elegant, short, and accurate. For decades, astronomers have been trying to do exactly this, but with the "ingredients" of the universe: galaxies and dark matter haloes.

The Problem: The "Eyeball" Method

In astronomy, we want to know the "census" of the universe. How many massive galaxies are there? How many tiny ones? How many giant clouds of dark matter (haloes) exist?

To describe these populations, scientists use mathematical formulas. But for a long time, these formulas were chosen "by eye." It’s like a chef looking at a thousand cakes and saying, "Hmm, it looks to me like most cakes have about two eggs. Let's just write down '2 eggs' as our rule."

This works okay, but it’s not perfect. Sometimes the "rule" is too simple and misses the details, and sometimes it’s so complicated it becomes useless.

The Solution: The Mathematical "Auto-Chef" (Symbolic Regression)

This paper introduces a new way to find these rules using a super-smart tool called Symbolic Regression (specifically an algorithm called ESR).

Think of ESR as an "Auto-Chef." Instead of a human guessing the recipe, you feed the millions of "cake" data points into a computer. The computer then tries out every possible combination of ingredients (math operators like plus, minus, exponents, and logarithms) to build a recipe.

But there’s a catch: the computer isn't just looking for the most accurate recipe; it’s looking for the most efficient one. It uses a principle called "Description Length."

The Analogy: Imagine you are playing a game of Charades.

Player A describes a "Golden Retriever" by saying: "It is a mammal, it has four legs, it is yellow, it has floppy ears, it wags its tail, and it barks." (Very accurate, but way too much talking!)
Player B just says: "A Golden Retriever." (Perfectly accurate and incredibly efficient!)

The ESR algorithm is looking for "Player B"—the simplest possible math formula that still explains the data perfectly.

What did they find?

The researchers tested this "Auto-Chef" on three major cosmic census lists:

The Luminosity Function (LF): How much light galaxies give off.
The Stellar Mass Function (SMF): How much "stuff" (stars) galaxies are made of.
The Halo Mass Function (HMF): How much dark matter is in the invisible "cradles" that hold galaxies.

The Results:

Better Recipes: The computer found new formulas that are more accurate than the "classic" recipes astronomers have used for 50 years.
The "Super-Exponential" Surprise: The computer discovered that when you get to the really, really massive galaxies, they don't just fade away slowly; they drop off much more sharply than we previously thought. It’s like realizing that while there are many medium-sized cakes, there are almost zero cakes the size of a skyscraper.
Robustness: They tested the recipes against 100 different "universes" (simulations) to make sure the recipes weren't just a fluke. The best recipes worked every single time.

Why does this matter?

If we want to understand how the universe grew from the Big Bang to today, we need to know exactly how many galaxies were born at different times. If our "census formulas" are slightly wrong, our entire understanding of gravity, dark matter, and the evolution of the cosmos will be slightly off.

By using this "Auto-Chef," astronomers can stop guessing and start using the most efficient, mathematically perfect recipes to describe the grand architecture of the cosmos.

Technical Summary: The Functional Form of Galaxy and Halo Luminosity and Mass Functions

1. Problem Statement

In astrophysics and cosmology, the abundance of galaxies and dark matter haloes is characterized by three fundamental 1-point statistics: the Galaxy Luminosity Function (LF), the Stellar Mass Function (SMF), and the Halo Mass Function (HMF). Traditionally, these functions are parameterized using analytic fitting formulae (e.g., the Schechter, Bernardi, Tinker, or Warren functions) that were selected "by eye" or through trial-and-error to match observed or simulated data.

The authors identify two primary issues with this traditional approach:

Sub-optimality: Human-selected functions may not be the most efficient or accurate representations of the underlying data.
Lack of Rigor: There is no systematic way to balance the trade-off between a function's accuracy (likelihood) and its complexity (simplicity/interpretability), which is essential for avoiding overfitting and ensuring physical validity.

2. Methodology

The study employs Symbolic Regression (SR)—a machine learning technique that searches for the optimal mathematical expression to describe a dataset—to automate the discovery of these functions.

Algorithm: The authors use Exhaustive Symbolic Regression (ESR). Unlike stochastic methods (like genetic programming), ESR performs a complete, systematic search of all possible functional forms up to a defined maximum complexity (number of operators, variables, and parameters).
Optimization Criterion: Functions are ranked using the Minimum Description Length (MDL) principle. MDL acts as a proxy for Bayesian evidence, penalizing complexity to find the "simplest" function that explains the data with high accuracy.
Data Sources:
- LF/SMF: Data from the SDSS (Sloan Digital Sky Survey) via Bernardi et al. (2013), considering both Sérsic and cmodel photometry.
- HMF: Data from the Quijote suite of N-body simulations (100 different realizations to test robustness against cosmic variance).
Physicality Checks: To ensure the discovered functions are scientifically useful, the authors impose three constraints:
1. Asymptotic Behavior: Functions must vanish at high mass/luminosity and remain finite/non-negative at the low end.
2. Monotonicity: A check for consistency in single-component models.
3. Integrability: The total integrated mass/luminosity density must be physically plausible (e.g., the total stellar mass cannot exceed the cosmic baryon budget).

3. Key Contributions

Automated Discovery Framework: The paper presents a robust framework for using ESR to discover optimal functional forms for any astrophysical dataset.
New Functional Forms: The authors provide specific, high-performing analytic expressions for the LF, SMF, and HMF that outperform standard literature models.
Robustness Testing: By testing across 100 HMF realizations, the authors demonstrate that the discovered functions are not artifacts of specific initial conditions but are robust features of the data.

4. Results

The ESR-discovered functions consistently outperformed standard literature fits in terms of Description Length (DL).

Luminosity Function (LF): ESR identified functions that provide a better fit than the Schechter function. A notable finding is that the high-luminosity cutoff is more accurately described by a stretched exponential ( $\phi \sim e^{-cx^p}$ with $p \approx 0.3\text{--}0.5$ ) rather than the simple exponential used in the Schechter form.
Stellar Mass Function (SMF): While the highly complex Bernardi function remains the most accurate for the SMF, ESR discovered much simpler (lower complexity) alternatives that are statistically superior to the Schechter function.
Halo Mass Function (HMF): The ESR functions significantly outperformed the Press–Schechter, Warren, and Tinker functions. The best-performing HMF function is a logistic-type form:
$f(\sigma) = \frac{\theta_0}{\theta_1 + e^{\sigma\theta_2 + \sigma}}$
Extrapolation Behavior: The authors noted that while ESR functions are optimal within the data range, they tend to favor a "plateau" at the low-mass/luminosity end due to the simplicity bias of the MDL metric. This differs from some theoretical predictions (like the $1/\sigma$ scaling in HMF) but remains well-behaved and integrable.

5. Significance

This work marks a shift from "heuristic" fitting to "principled" function discovery in astrophysics.

Precision Cosmology: As cosmological analyses move into an era of sub-percent precision, using sub-optimal fitting functions can introduce systematic errors. The ESR functions provide more accurate "drop-in" replacements for high-precision modeling.
Impact on Derived Quantities: The authors demonstrate that while the impact on integrated quantities (like total stellar mass density) is modest within the data range ( $\lesssim 2\%$ ), the choice of function becomes critical (up to $5\text{--}10\%$ difference) when extrapolating to the extreme high-mass/luminosity ends.
Methodological Paradigm Shift: The study proves that symbolic regression can successfully navigate the complexity-accuracy trade-off, providing a path to automate the discovery of physical laws and statistical distributions across all of astrophysics.