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Predicting galaxy bias using machine learning

Using data from the IllustrisTNG300 simulation, this study employs machine learning models, particularly Normalizing Flows, to successfully predict individual galaxy bias by identifying overdensities and cosmic-web distances as key predictors while effectively capturing the intrinsic stochastic variance of the matter-halo-galaxy connection.

Original authors: Catalina Riveros-Jara, Antonio D. Montero-Dorta, Natália V. N. Rodrigues, Pía Amigo, Natalí S. M. de Santi, Andrés Balaguera-Antolínez, Raul Abramo, Neill Guzmán, M. Celeste Artale

Published 2026-02-06

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CC0 1.0

Original authors: Catalina Riveros-Jara, Antonio D. Montero-Dorta, Natália V. N. Rodrigues, Pía Amigo, Natalí S. M. de Santi, Andrés Balaguera-Antolínez, Raul Abramo, Neill Guzmán, M. Celeste Artale

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 Big Picture: Why Do Galaxies Clump Together?

Imagine the universe as a giant, invisible ocean of dark matter. Galaxies are like fish swimming in this ocean. Sometimes, the fish swim alone; other times, they form massive schools.

Astronomers want to understand why the fish school up in certain spots. They use a concept called "bias." Think of bias as a "clumping score."

A galaxy with a high bias is like a fish that always swims in the densest parts of the school.
A galaxy with a low bias is like a fish that wanders into the empty, quiet parts of the ocean.

For a long time, scientists tried to predict this "clumping score" using simple math formulas. But the universe is messy, and these formulas often missed the mark because they couldn't handle the randomness (or "chaos") of how galaxies form.

The New Approach: Teaching a Computer to Guess

This paper is about teaching a computer (using Machine Learning) to predict a galaxy's "clumping score" based on its surroundings and its own history.

The researchers used a super-powerful digital simulation of the universe called IllustrisTNG. It's like a video game that has been running for billions of years, creating a fake universe with millions of galaxies, dark matter, and gas.

They gave the computer three different "brains" to learn from this fake universe:

The Random Forest (RF): Imagine a committee of 2,000 different experts. Each expert looks at the data and makes a guess. The computer takes the average of all their guesses.
The Neural Network (NN): Imagine a single, very complex brain with many layers of neurons. It tries to find a straight line (or a smooth curve) that connects the data points.
The Normalizing Flow (NF): This is the star of the show. Imagine a master chef who doesn't just guess one recipe for a dish, but understands the entire range of possible flavors. Instead of saying, "This galaxy will definitely be salty," the chef says, "There's a 70% chance it's salty, a 20% chance it's spicy, and a 10% chance it's bland." This captures the randomness of the universe.

What Did They Feed the Computer?

To make a prediction, the computer needed to know two things about each galaxy:

Internal Properties: What is the galaxy made of? (How heavy is it? How old is it? How fast is it spinning?)
External Environment: Where is the galaxy sitting? (Is it in a dense cluster? Near a giant cosmic wall? Or floating in a vast empty void?)

They used a special tool called DisPerSE to map the "cosmic web." Think of the cosmic web as a giant spiderweb of dark matter. The computer measured how far each galaxy was from the "knots" (dense nodes), the "strings" (filaments), and the "holes" (voids) in this web.

The Results: Who Won the Contest?

The researchers tested the three "brains" to see which one could best predict the galaxy's clumping score.

1. The Deterministic Winners (RF and NN):
The Random Forest and the Neural Network were good at predicting the average behavior. If you asked them, "What is the typical clumping score for a galaxy in this spot?" they got it right.

The Flaw: They failed to capture the spread. They tried to force every galaxy into a single, neat number. But in reality, two galaxies in the exact same spot can have very different clumping scores due to random chance. These models smoothed out the chaos, losing important details.

2. The Probabilistic Winner (Normalizing Flows):
The Normalizing Flow (NF) model was the clear winner.

Why? Because it didn't try to give a single answer. It gave a distribution (a range of possibilities).
The Analogy: If you ask a deterministic model, "Will it rain tomorrow?" it might say "Yes." If you ask the NF model, it says, "There's a 60% chance of rain, 30% chance of clouds, and 10% chance of sun."
The Result: The NF model was the only one that could perfectly recreate the "messy" histogram of the real data. It captured the fact that the universe is inherently random.

The Most Important Clues

The computer also told the researchers which clues were most important for making a prediction:

The #1 Clue: Overdensity (specifically $\delta_8$ ). This is simply a measure of how crowded the neighborhood is. The computer learned that if a galaxy is in a crowded neighborhood, it almost certainly has a high clumping score. This was by far the most important factor.
The #2 Clue: Distance to the Cosmic Web. How close the galaxy is to the "strings" or "knots" of the universe mattered next.
The Surprising Clue: Formation Time ( $z_{1/2}$ ). Interestingly, when a galaxy formed was more important than its mass or how fast it spins. Older galaxies tend to behave differently than younger ones.

The Bottom Line

This paper proves that to understand how galaxies cluster, we can't just use simple averages. We need to embrace the randomness.

By using a probabilistic machine learning model (Normalizing Flows), the researchers successfully taught a computer to predict not just where galaxies will cluster, but to understand the variability and chaos of the universe. It's like moving from a black-and-white sketch of the universe to a full-color, 3D movie that captures the unpredictable nature of how galaxies form and group together.

This method sets the stage for future telescopes (like DESI) to analyze real galaxy data, helping us understand the invisible dark matter that holds our universe together.

Technical Summary: Predicting Galaxy Bias Using Machine Learning

Problem Statement
Understanding how galaxies trace the underlying matter density field is fundamental to observational cosmology and galaxy formation models. This connection is quantified by the galaxy bias ( $b$ ). While traditional analytical approaches often rely on population-averaged bias measurements (e.g., via power spectra or correlation functions), these methods struggle to disentangle specific dependencies on halo internal properties and environmental factors due to signal-to-noise limitations and the inherent stochasticity of the matter-halo-galaxy connection. Recent literature emphasizes the need for probabilistic frameworks to capture the intrinsic variance in this relationship. This work addresses the challenge of predicting the individual linear bias ( $b_i$ ) of galaxies based on their host halo properties, local environment, and cosmic web location, while explicitly modeling the stochastic nature of the bias parameter.

Methodology
The study utilizes data from the IllustrisTNG300 magnetohydrodynamical simulation. The dataset comprises 174,607 central galaxies selected with halo masses $M_{vir} > 10^{10.5} h^{-1}M_\odot$ and stellar masses $M_* > 10^{8.75} h^{-1}M_\odot$ .

Target Variable: The individual large-scale linear bias ( $b_i$ ) for each galaxy is computed using the object-by-object estimator formalism introduced by Paranjape et al. (2018) and refined by Balaguera-Antolínez et al. (2024). This method assigns a specific bias value to each galaxy based on the ratio of galaxy-matter to matter-matter power spectra in the large-scale limit ( $k \leq 0.2 \, h \, \text{Mpc}^{-1}$ ).
Input Features: The models are trained on a set of features including:
- Internal Halo Properties: Virial mass ( $M_{vir}$ ), formation redshift ( $z_{1/2}$ ), concentration ( $c_{vir}$ ), and spin ( $\lambda_{halo}$ ).
- Galaxy Properties: Stellar mass ( $M_*$ ) and color ( $g-i$ ).
- Local Environment: Overdensities ( $\delta_3, \delta_5, \delta_8$ ) calculated on scales of 3, 5, and $8 \, h^{-1}\text{Mpc}$ .
- Cosmic Web Location: Distances to critical points (voids, nodes, saddle points, and filaments) identified via the DisPerSE algorithm.
Machine Learning Framework: Three distinct models are evaluated on a split dataset (58% training, 30% test, 12% validation):
1. Random Forest Regressor (RF): A deterministic ensemble method.
2. Neural Network (NN): A deterministic single-output deep learning model.
3. Normalizing Flows (NF): A probabilistic, generative model (specifically Neural Spline Flows) designed to learn the conditional probability distribution $p(b_i | \{x\}_i)$ , where $\{x\}_i$ represents the input features.
Evaluation Metrics: Performance is assessed using the Pearson Correlation Coefficient (PCC), the Kolmogorov–Smirnov (K–S) test, and the Wasserstein Distance (WD). For the probabilistic NF model, metrics are computed both for a single random realization and for the mean of 1,000 realizations.

Key Contributions

Object-by-Object Bias Integration: The work successfully incorporates the individual bias formalism into a machine learning pipeline, treating $b_i$ as a specific property of each galaxy rather than a population average.
Probabilistic Modeling of Bias: The study demonstrates that deterministic models (RF and NN) fail to reproduce the full width and tails of the true bias distribution, whereas Normalizing Flows can accurately capture the intrinsic stochasticity and joint distributions of bias with other galaxy properties.
Feature Importance Analysis: Through permutation feature importance (PFI) and correlation analysis, the work quantifies the relative influence of environmental versus internal properties on galaxy bias.

Results

Feature Dependencies: Correlation analysis reveals that local overdensities, particularly $\delta_8$ , are the most strongly correlated features with galaxy bias. This is followed by distances to cosmic-web structures and, to a lesser extent, internal halo properties. Among internal properties, formation redshift ( $z_{1/2}$ ) shows the strongest correlation, surpassing halo mass in the full sample.
Model Performance:
- Deterministic Models: RF and NN achieve high PCC values (0.811 and 0.797, respectively) but exhibit high K-S test statistics and Wasserstein distances, indicating they accurately predict the mean trend but fail to reproduce the distribution's scatter and tails.
- Probabilistic Model (NF): When predicting a single random realization, the NF model achieves a K-S statistic of 0.010 and a WD of 0.046, significantly outperforming deterministic models in reproducing the true distribution shape. While the PCC for a single realization is lower (0.652) due to the intrinsic scatter, the model correctly captures the global behavior without systematic bias. The mean prediction of the NF performs comparably to the RF in terms of PCC (0.807) but retains the ability to model the distribution width.
Joint Distributions: The NF model successfully reproduces the scatter in joint distributions, such as bias versus stellar mass and bias versus color, with 2D K-S statistics of 0.017 and 0.029, respectively.
Conditional Distributions: The study shows that the NF model's ability to constrain individual probability distributions depends heavily on the input features. Conditioning on low-information features (e.g., halo spin) results in broad distributions similar to the global prior, whereas conditioning on high-information features (e.g., $\delta_8$ ) yields tightly constrained, object-specific distributions.

Significance
The paper claims that its framework provides a foundation for future efforts to measure individual galaxy bias using upcoming spectroscopic surveys, such as the Dark Energy Spectroscopic Instrument (DESI). By demonstrating that probabilistic methods like Normalizing Flows are necessary to capture the intrinsic variance of the galaxy bias, the work offers a more robust tool for cosmological inference. The results suggest that while deterministic models are sufficient for predicting mean trends, they are inadequate for characterizing the full stochastic nature of the galaxy-halo connection, which is critical for precision cosmology. The study also highlights that environmental properties, specifically large-scale overdensities, dominate the prediction of galaxy bias over internal halo properties.