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Cosmology with one galaxy: An analytic formula relating $Ω_{\rm m}$ with galaxy properties

This paper presents a novel analytic formula derived from hydrodynamical simulations that directly links the cosmological matter density parameter ( $\Omega_m$ ) to intrinsic properties of individual galaxies, establishing a new, interpretable pathway for cosmological inference that bypasses traditional population-level statistics.

Original authors: Kito Liao, Francisco Villaescusa-Navarro, Romain Teyssier, Natalí S. M. de Santi

Published 2026-05-05

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CC BY 4.0

Original authors: Kito Liao, Francisco Villaescusa-Navarro, Romain Teyssier, Natalí S. M. de Santi

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

Imagine you are trying to figure out the recipe for a giant, invisible cake that makes up the entire universe. Traditionally, cosmologists (scientists who study the universe) have tried to guess the ingredients by looking at the whole cake at once—measuring how the frosting is spread out across the entire table or how the crumbs cluster together in big piles. They look at billions of galaxies to find patterns.

This paper proposes a radical new idea: You can figure out the recipe by tasting just one single crumb.

Here is the breakdown of how the authors did this, using simple analogies:

1. The Big Idea: One Galaxy is Enough

Usually, scientists think a single galaxy is too messy and chaotic to tell them anything about the universe's big rules. They assume the "noise" of how stars form and gas moves hides the signal.

However, the authors found that a galaxy is like a fossilized memory. Just as a single tree ring can tell you about the climate of a specific year, the internal properties of a single galaxy (how heavy it is, how fast it spins, and how "dirty" or metal-rich its stars are) actually contain a hidden code that reveals the density of matter in the universe (a number called $\Omega_m$ ).

2. The Detective Work: Symbolic Regression

To find this code, the researchers didn't just guess. They used a special type of artificial intelligence called Symbolic Regression.

Think of this AI not as a "black box" that gives you an answer, but as a mathematical detective that tries to write a simple sentence (an equation) to explain the data.

They fed the AI data from thousands of simulated galaxies.
The AI tried millions of different mathematical combinations (adding, multiplying, dividing, using curves).
It eventually found a compact, simple formula that worked. Unlike complex neural networks that are hard to understand, this formula is like a clear, readable sentence that explains why the connection exists.

3. The Formula: A "Galactic Thermometer"

The formula they found links the universe's matter density to three main things inside a galaxy:

Stellar Metallicity: How "polluted" the stars are with heavy elements (metals).
Mass and Speed: How heavy the galaxy is and how fast it spins.
Compactness: How tightly packed the galaxy is.

The Analogy:
Imagine a galaxy as a kitchen.

The universe's matter density is the amount of raw ingredients (flour, sugar) available in the pantry.
The galaxy's feedback (explosions from stars and black holes) is the chef throwing ingredients out the window.
The metallicity is the smell of the food cooking.

The authors found that if you know how much the chef threw away (feedback) and how much food is left in the pot (metallicity), you can mathematically reverse-engineer exactly how much flour was in the pantry to begin with. The formula acts like a thermometer that reads the "temperature" of the universe just by looking at the "smell" of one galaxy.

4. Testing the Recipe

The researchers tested this "one-crumb" recipe on four different types of simulated universes (IllustrisTNG, ASTRID, SIMBA, and Swift-EAGLE). These simulations use different rules for how stars explode and how black holes behave.

The Result: The same basic formula worked for all of them!
The Catch: They had to tweak a few numbers (like adjusting the salt in a recipe) depending on which simulation they were using, but the structure of the formula stayed the same.
Time Travel: They also checked if this worked for galaxies in the past (high redshift). They found that the formula still works, but the "taste" changes slightly as the universe ages, so they added a simple time-adjustment factor.

5. Why Does This Work? (The Physical Explanation)

The paper explains that this isn't magic; it's physics.

Galaxies are like sieves. They try to hold onto gas (ingredients) to make stars.
The universe's gravity (matter density) tries to pull gas in.
The galaxy's internal explosions (feedback) try to push gas out.

The balance between these two forces determines how "metal-rich" the galaxy becomes. Because the universe's gravity sets the initial stage, the final balance of ingredients in the galaxy's "kitchen" leaves a permanent mark. The formula simply decodes that mark.

Summary

This paper demonstrates that cosmology doesn't always require a telescope looking at billions of galaxies. By understanding the physics of how a single galaxy retains its gas and creates stars, we can derive a simple mathematical rule that tells us the density of matter in the entire universe.

It's a bridge between the very small (a single galaxy) and the very large (the whole cosmos), proving that the history of the universe is written in the chemical makeup of the stars we see today.

Technical Summary: Cosmology with One Galaxy

Problem Statement
Standard cosmological analyses typically decouple galaxy formation from cosmological parameter inference, relying on population-level statistics such as clustering, lensing, or halo abundances to constrain the matter density parameter, $\Omega_m$ . While recent machine learning studies on the CAMELS simulation suite have demonstrated that individual galaxy properties can encode cosmological information, the physical origin of this signal has remained unclear, and previous approaches relied on "black-box" neural networks lacking physical interpretability. This work addresses the challenge of deriving a physically transparent, analytic relationship that links $\Omega_m$ directly to intrinsic, single-galaxy observables, bypassing the need for large-scale clustering statistics.

Methodology
The authors utilize the CAMELS (Cosmology and Astrophysics with MachinE Learning Simulations) project, specifically the Latin Hypercube (LH) datasets and the extended TNG-SB28 dataset. These suites contain over 16,000 hydrodynamic simulations varying cosmological parameters ( $\Omega_m$ , $\sigma_8$ , $\Omega_b$ , $H_0$ , etc.) and astrophysical subgrid parameters (supernova and AGN feedback efficiencies).

The core methodology employs Symbolic Regression (using the PySR algorithm) to discover compact, analytic functional forms relating galaxy properties to $\Omega_m$ .

Data Preparation: The training set consists of galaxies sampled from IllustrisTNG simulations at $z=0$ . Key dimensionless variables are constructed, including stellar metallicity ( $Z_\star$ ), stellar mass fraction ( $f_\star$ ), effective dynamical velocity ( $V_{\text{eff}}$ ), and a compactness parameter ( $R_{\text{compact}}$ ).
Symbolic Search: The algorithm searches for an expression minimizing the mean squared error between the predicted and true $\Omega_m$ , subject to parsimony penalties and constraints on operator complexity (e.g., limiting nesting depth and exponent ranges).
Validation: The resulting analytic formula, trained exclusively on IllustrisTNG, is tested against independent simulation suites (ASTRID, SIMBA, Swift-EAGLE) and across different redshifts ( $z=0$ to $z=3$ ).
Extended Parameter Space: The framework is further tested on the TNG-SB28 dataset, which varies 28 parameters simultaneously, including $\Omega_b$ and $H_0$ , to assess robustness under broader cosmological and feedback variations.

Key Contributions

First Analytic Single-Galaxy Tracer: The paper presents the first physically interpretable analytic formula that relates $\Omega_m$ directly to individual galaxy properties. The derived expression takes the form:
$\Omega_m \approx \ln \left[ \sigma \left( \frac{Z_\star}{M_\star} \frac{\Omega_b V_{\text{eff}}^3}{G H_0} \frac{\Omega_b}{k} \right) + c_0 \right] - \frac{a_0}{R_{\text{compact}}}$
where $\sigma(x)$ is a sigmoid function.
Cross-Suite Robustness: The functional form derived from IllustrisTNG successfully recovers $\Omega_m$ in ASTRID, SIMBA, and Swift-EAGLE simulations with only minor recalibration of numerical coefficients ( $k, c_0, a_0$ ). This suggests the underlying physical coupling is universal across different numerical implementations of galaxy formation physics.
Physical Interpretation: The authors provide a transparent physical interpretation of the latent variable. The term inside the sigmoid encodes a baryon-cosmology coupling mediated by the Mass-Metallicity Relation (MZR). It represents the interplay between the gravitational potential depth (determined by $V_{\text{eff}}$ and cosmology) and the efficiency of baryon retention and chemical enrichment regulated by feedback.
Redshift Evolution: The authors derive a redshift-dependent extension of the formula using power-law scalings of $(1+z)$ , which maintains predictive accuracy up to $z=3$ for IllustrisTNG and ASTRID.
Generalization to High-Dimensional Parameter Space: In the TNG-SB28 suite (where $\Omega_b$ and $H_0$ vary), the analytic form adapts to include an explicit dependence on the Hubble parameter ratio ( $H_0/H_{0,\text{fid}}$ ) and a refined retention efficiency term $E(Z_\star, f_\star)$ , consistent with a gas-regulator framework.

Results

Predictive Performance: At $z=0$ , the model achieves coefficients of determination ( $R^2$ ) of 0.77 for IllustrisTNG and 0.73 for ASTRID. SIMBA and Swift-EAGLE show slightly degraded performance ( $R^2 \approx 0.6-0.7$ ) but still exhibit a strong monotonic correlation, indicating the signal is not an artifact of overfitting to a specific simulation.
Saturation Behavior: In the high-dimensional TNG-SB28 dataset, the predictor exhibits a saturation regime at high $\Omega_m$ ( $\Omega_m \gtrsim 0.43$ ). The authors attribute this to strong degeneracies between feedback physics and cosmology in the high-dimensional parameter space, where changes in cosmology are increasingly offset by baryonic processes, creating an information ceiling.
Physical Mechanism: The analysis confirms that the "knee" of the $\Omega_m$ –metallicity relation corresponds to the transition between cosmology-dominated enrichment (low metallicity) and feedback-saturated enrichment (high metallicity). The slope of this transition is governed by the feedback prescription, explaining why different simulations require different coefficient calibrations.

Significance and Claims
The paper claims to establish a direct, interpretable bridge between small-scale galaxy physics and large-scale cosmology. By demonstrating that $\Omega_m$ is locally encoded in the baryonic cycle of individual galaxies, the work opens a complementary pathway for cosmological inference that does not rely on traditional large-scale structure statistics.

The authors emphasize that this approach does not aim to replace standard probes but to provide a novel framework for understanding how cosmological information survives baryonic processing. The discovery of a compact, analytic relation suggests that the imprint of the matter density on galaxy formation is robust and governed by fundamental principles of baryon retention and gravitational potential depth. The work highlights that while machine learning can identify correlations, symbolic regression is essential for uncovering the underlying physical laws governing the galaxy-cosmology connection.