⚛️ general relativity

Disentangling modified gravity and galaxy bias with field-level inference

This paper presents a field-level inference framework that jointly constrains modified gravity and galaxy bias parameters by analyzing the full three-dimensional galaxy distribution, demonstrating that leveraging non-Gaussian and phase information—particularly from under-dense regions—significantly breaks the degeneracies inherent in traditional power-spectrum analyses.

Original authors: Sophie Hoyland, Daniela Saadeh, Kazuya Koyama, Harry Desmond

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sophie Hoyland, Daniela Saadeh, Kazuya Koyama, Harry Desmond

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 the universe as a giant, three-dimensional loaf of bread rising in an oven. Inside this loaf, the "dough" is dark matter, and the "raisins" are galaxies. For decades, astronomers have tried to understand how this bread rises by looking at the average distance between the raisins. They measure how clumpy the dough is, but they throw away the specific recipe of how the raisins are arranged. They only look at the "clumpiness score," ignoring the unique shapes, twists, and turns of the dough itself.

This paper introduces a new way to look at the universe: Field-Level Inference (FLI). Instead of just measuring the average clumpiness, FLI looks at the entire 3D map of the raisins, preserving every tiny detail about their specific arrangement.

Here is a breakdown of what the paper does, using simple analogies:

1. The Problem: The "Blindfolded" Detective

Astronomers want to know if the laws of gravity work exactly as Einstein predicted (General Relativity) or if they are slightly tweaked (Modified Gravity). They also want to know how "sticky" the raisins (galaxies) are to the dough (dark matter).

The Old Way (Power Spectrum): Imagine trying to identify a song by only listening to its volume at different frequencies. You know it's loud or quiet, but you miss the melody, the lyrics, and the rhythm. In the universe, this method measures the "volume" of the galaxy distribution but throws away the "rhythm" (the specific shapes and phases).
The Confusion: Because the old method throws away the rhythm, it creates a massive mix-up. A change in the "stickiness" of the raisins (galaxy bias) looks exactly the same as a change in the gravity recipe (Modified Gravity). It's like trying to tell if a cake is sweet because of more sugar or because of a different type of flour, but you can only taste the total sweetness, not the texture.

2. The Solution: The "Full-Map" Detective

The authors built a new tool that looks at the entire 3D map of the galaxies, keeping every detail of their arrangement.

The Analogy: Instead of just listening to the volume of the song, FLI listens to the entire melody and lyrics. It looks at the specific shapes of the cosmic web: the empty spaces (voids), the thin threads (filaments), the walls, and the dense clusters.
The Result: By keeping the "rhythm" (the Fourier phases) and the "texture" (the non-Gaussian shapes), the new method can tell the difference between a change in gravity and a change in galaxy stickiness. It breaks the confusion that plagued the old method.

3. How They Tested It

Since we can't run the universe in a lab, the authors created a virtual universe (a simulation).

They built two versions of this virtual universe: one where gravity works normally (General Relativity) and one where gravity is slightly stronger in empty spaces (a theory called $f(R)$ gravity).
They filled these universes with virtual galaxies using a complex rulebook (bias model) that decides where galaxies form based on the density of the dough.
They then ran their new "Full-Map" detective tool on these simulations and compared it to the old "Average Clumpiness" tool.

4. The Big Discovery

The results were clear:

The Old Tool: Could not tell the difference between the two gravity theories. It was stuck in a "degeneracy," meaning it couldn't decide if the changes were due to gravity or the galaxies' stickiness.
The New Tool (FLI): Successfully distinguished between the two. It provided much tighter, more accurate answers about the strength of gravity and the stickiness of the galaxies.

5. Where the Clues Were Hiding

The authors asked: Where exactly in the universe did the new tool find the clues?

They discovered that the empty spaces (voids) and the thin walls surrounding them were the most important.
The Analogy: Imagine trying to figure out how a sponge absorbs water. The old method looked at the whole sponge. The new method realized that the holes in the sponge (the voids) tell you the most about how the sponge works.
Specifically, the tool found that counting galaxies in these empty regions (where there is usually 0 or 1 galaxy) was the key to breaking the confusion. If gravity is different, the way galaxies populate these empty holes changes in a specific way that the old method missed.

6. The "Secret Ingredient": The Phase

The paper highlights that the secret sauce of this new method is Fourier Phases.

The Analogy: Think of a picture. The "amplitude" is how bright the pixels are. The "phase" is the specific arrangement of those pixels that creates the image.
The old method only looked at the brightness (amplitude). The new method looked at the arrangement (phase).
The authors showed that even if you only looked at the arrangement (ignoring the brightness), you could still tell the difference between the gravity theories, though not as well as looking at both. This proves that the shape of the universe holds the key to understanding gravity.

Summary

This paper demonstrates that by looking at the full, detailed 3D map of the universe—rather than just averaging the data—scientists can finally untangle the mystery of whether gravity is changing or if galaxies are just behaving differently. It turns out that the empty spaces in the universe hold the most critical clues, and the specific shape of the cosmic web is the missing piece of the puzzle that previous methods threw away.

Technical Summary: Disentangling Modified Gravity and Galaxy Bias with Field-Level Inference

Problem Statement
Traditional cosmological analyses of Large-Scale Structure (LSS) rely on summary statistics, primarily the power spectrum ( $P(k)$ ), which compresses the three-dimensional matter distribution into two-point correlations. While effective for Gaussian initial conditions, this approach discards non-Gaussian information and Fourier phase correlations generated by non-linear gravitational evolution. This loss of information creates a critical degeneracy between Modified Gravity (MG) parameters and galaxy bias. Specifically, the enhanced clustering power driven by MG effects (e.g., a fifth force in $f(R)$ theories) can be mimicked by changes in the galaxy bias parameters. Consequently, two-point statistics struggle to distinguish between deviations from General Relativity (GR) and variations in how galaxies trace the underlying dark matter distribution.

Methodology
The authors develop a Field-Level Inference (FLI) framework to address these limitations by performing a Bayesian likelihood analysis directly on the three-dimensional galaxy number counts field ( $n_g$ ), rather than compressing the data.

Theoretical Framework: The study utilizes the Hu-Sawicki $f(R)$ gravity model, characterized by the parameter $f_{R0}$ , which introduces a chameleon screening mechanism. This is coupled with a non-linear, local-in-density galaxy bias model (Neyrinck et al. 2014) featuring parameters $\beta$ (primary bias), $\rho$ , and $\epsilon$ .
Simulation Pipeline: Non-linear structure formation is modeled using the COmoving Lagrangian Acceleration (COLA) method, extended for modified gravity (FML-COLA). The pipeline generates dark matter overdensity fields ( $\delta_{DM}$ ) which are mapped to mock galaxy catalogues ( $n_g$ ) via the non-linear bias prescription.
Inference Frameworks: The authors compare two distinct likelihood approaches:
1. Power Spectrum Analysis: A traditional Gaussian likelihood applied to the shot-noise-subtracted galaxy power spectrum $P_g(k)$ .
2. Field-Level Analysis: A voxel-by-voxel Poisson likelihood (and a truncated Poisson likelihood for thresholded data) applied directly to the 3D galaxy number counts field. This preserves both Fourier amplitudes and phases.
Parameter Estimation: A grid of theoretical predictions is generated across the parameter space of $\beta$ and $|f_{R0}|$ . A linear regular grid interpolator is used to create a continuous likelihood surface for Markov Chain Monte Carlo (MCMC) sampling.
Robustness Checks: The pipeline is tested against variations in Poisson noise realizations, initial conditions (cosmic variance), and data thresholding (excluding low-count voxels).

Key Contributions and Results
The study applies this framework to mock data representing two scenarios: a GR fiducial model ( $f_{R0}=0$ ) and an $f(R)$ model with $|f_{R0}| = 10^{-6}$ (F6).

Breaking Degeneracies: The field-level approach successfully breaks the strong degeneracy between $|f_{R0}|$ and $\beta$ inherent in power spectrum analyses.
- In the GR case, the power spectrum analysis yields a 95% upper limit of $\log_{10}|f_{R0}| < -5.698$ , whereas the field-level analysis tightens this to $\log_{10}|f_{R0}| < -6.760$ .
- In the F6 case, the power spectrum analysis produces a broad, degenerate posterior ( $\log_{10}|f_{R0}| = -6.223^{+1.032}_{-0.411}$ ), while the field-level analysis recovers the true value with high precision ( $\log_{10}|f_{R0}| = -5.958^{+0.100}_{-0.107}$ ).
- Constraints on the bias parameter $\beta$ are also significantly tighter in the field-level analysis ( $\pm 0.003$ ) compared to the power spectrum ( $\pm 0.004$ to $0.007$).
Role of Fourier Phases: By isolating Fourier phases (keeping amplitudes fixed to the fiducial model), the authors demonstrate that phase information alone provides constraints on $|f_{R0}|$ superior to amplitude-only (power spectrum) analyses, though the combination of both yields the tightest constraints. This confirms that the morphological configuration of the cosmic web, encoded in phases, carries distinct signatures of gravity that bias cannot replicate.
Cosmic Environment Sensitivity: Through cosmic web classification (voids, walls, filaments, clusters), the authors identify that under-dense regions are the primary drivers for distinguishing gravity models.
- Voxels with a single galaxy ( $n_{obs,g}=1$ ) are highly sensitive to increases in bias or gravity strength.
- Crucially, these informative $n_{obs,g}=1$ voxels are predominantly located within voids and walls of the underlying dark matter field.
- The field-level likelihood leverages the specific spatial arrangement of these environments, which two-point statistics fail to capture.
Robustness: The results remain consistent across 100 different Poisson noise realizations and five different initial condition realizations. The method also demonstrates resilience to data thresholding; even when excluding voxels with $n_{obs,g} < 5$ , the field-level approach retains significant constraining power, though the information gain from low-density regions is partially lost.

Significance and Claims
The paper claims that field-level inference offers a powerful path forward for next-generation surveys (e.g., DESI, Euclid, LSST) by exploiting the full information content of the galaxy distribution. The authors assert that their approach:

Disentangles MG and Bias: It uniquely resolves the MG-bias degeneracy by utilizing non-Gaussian morphological information and Fourier phases, which are inaccessible to traditional two-point statistics.
Identifies Information Sources: It provides a physical understanding of where in the cosmic web (specifically voids and walls) the information distinguishing gravity models resides.
Validates Forward Modeling: It demonstrates that forward-modeling the full 3D field allows for the simultaneous, robust constraint of gravitational and bias parameters.

The authors note limitations, specifically that their current implementation fixes the initial phases of the dark matter field. They acknowledge that applying this to real data will require marginalizing over these initial phases (as done in frameworks like BORG) and incorporating more complex bias models and observational systematics. However, the study establishes the theoretical and practical viability of using field-level likelihoods to test gravity in the non-linear regime.