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Galaxy and halo angular clustering in LCDM and Modified Gravity cosmologies

Using N-body simulations and mock sky catalogs, this study demonstrates that angular clustering statistics of galaxies and halos in $f(R)$ and nDGP modified gravity models can be distinguished from General Relativity with high significance (up to 5 $\sigma$ ) at small scales, highlighting the potential of future high-precision surveys to test gravity theories.

Original authors: Paweł Drozda, Wojciech A. Hellwing, Maciej Bilicki

Published 2026-05-22

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CC BY 4.0

Original authors: Paweł Drozda, Wojciech A. Hellwing, Maciej Bilicki

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, invisible ocean. In this ocean, there are two main types of "weather": the standard weather we expect (called Lambda Cold Dark Matter or ΛCDM, which relies on Einstein's General Relativity), and some "weird weather" patterns that might happen if the laws of gravity are slightly different (called Modified Gravity or MG).

This paper is like a team of meteorologists who built a massive, virtual weather simulation to see if they can spot the "weird weather" by looking at how the stars and galaxies cluster together.

Here is a breakdown of what they did and found, using simple analogies:

1. The Setup: Building a Virtual Sky

The researchers didn't just look at a real photo of the sky; they built a virtual universe inside a computer using a supercomputer.

The Simulation: They created a digital box filled with billions of particles representing dark matter (the invisible glue holding galaxies together).
The Models: They ran this simulation twice: once with the standard rules of gravity (Einstein's rules) and twice with "tweaked" rules (Modified Gravity).
- The Tweaks: They tested two specific ways gravity might be different:
  1. f(R) Gravity: Imagine gravity has a "chameleon" suit. In dense areas (like our solar system), it hides its extra powers and acts normal. In empty space, it shows off.
  2. nDGP Gravity: Imagine gravity is like a leaky hose. Usually, water (gravity) stays in the pipe (our 3D world), but in this model, some water leaks out into a 5th dimension.

2. The Method: Counting Stars in a Bucket

Instead of measuring the distance between every single star (which is hard because we don't know their exact distances), the researchers looked at the sky from a specific angle, like looking at a map.

The "Cell" Analogy: Imagine throwing a giant net over the sky. The net is made of circular holes (cells).
Counting: They counted how many galaxies or dark matter clumps fell into each hole.
The Pattern: If gravity works exactly as Einstein predicted, the counts in these holes should follow a very specific, predictable pattern (like a bell curve). If gravity is "weird" (Modified Gravity), the pattern changes, especially in the "tails" of the distribution (the rare, extreme cases where a hole has way too many or way too few stars).

3. The Search: Finding the Sweet Spot

The researchers realized that the "weird weather" signals change depending on how far back in time (or how deep into the universe) they looked.

The Goldilocks Zone: They found that looking at a specific slice of the universe, between redshifts 0.15 and 0.3 (which is a specific distance in cosmic terms), gave them the clearest view of the differences. Looking too close or too far away made the signal too weak or too noisy.

4. The Results: What They Saw

They compared the "Standard Gravity" simulation against the "Modified Gravity" simulations.

The Difference: They found that the Modified Gravity models did indeed create different clustering patterns. In some cases, the difference was up to 20% compared to the standard model.
The Best Tool: They discovered that looking at third-order statistics (a fancy way of measuring the "skewness" or asymmetry of the star clusters) was the most sensitive tool. It was like using a high-precision scale instead of a bathroom scale.
- For Dark Matter, the differences were huge (over 5 times the noise level), but this is hard to see directly because we can't see dark matter with our eyes.
- For Galaxies, the differences were smaller but still detectable (2 to 4 times the noise level).
The "Faint" Signal: The researchers noted that their simulated galaxy catalogs were "sparse" (like a few stars in a huge patch of sky). Real telescopes in the future (like the Vera Rubin Observatory) will see millions of times more stars. Because they found a signal even with such a sparse sample, they are very optimistic that future telescopes will be able to catch these gravity "weirdness" signals with incredible clarity.

5. The Conclusion

The paper concludes that:

It works: We can use the way galaxies cluster on the sky (angular clustering) to test if Einstein's gravity is perfect or if it needs a tweak.
Higher orders matter: Looking at simple averages isn't enough; you need to look at the complex shapes and asymmetries of the clusters to see the difference.
Future is bright: Even though their computer models used a "low-resolution" view of the universe, they still found the signal. This suggests that when we look at the real, high-resolution sky in the near future, we might finally be able to prove if gravity behaves differently than Einstein thought.

In short: The team built a cosmic video game to test if the rules of gravity have hidden settings. They found that by looking at how galaxies bunch up in specific patterns, they can spot those hidden settings, and future telescopes will be powerful enough to confirm if those settings actually exist in our real universe.

Technical Summary: Galaxy and Halo Angular Clustering in $\Lambda$ CDM and Modified Gravity Cosmologies

Problem Statement
The standard cosmological model, $\Lambda$ CDM, relies on General Relativity (GR) to describe gravitational instability and structure formation. While $\Lambda$ CDM has passed numerous observational tests, the nature of dark energy and potential deviations from GR on cosmological scales remain open questions. Modified Gravity (MG) theories, such as $f(R)$ gravity and the normal branch of the Dvali-Gabadadze-Porrati (nDGP) brane-world model, propose additional scalar degrees of freedom that manifest as a "fifth force." These forces are typically suppressed in high-density environments (e.g., the Solar System) via screening mechanisms (chameleon or Vainshtein) but may alter structure formation on large scales.

While three-dimensional clustering and redshift-space distortions have been extensively studied for MG constraints, the properties of angular clustering in light-cone projections for these scenarios remain under-explored. Angular clustering is particularly relevant for future photometric surveys (e.g., Vera Rubin Observatory, Euclid) which offer vast sky coverage but lack precise spectroscopic redshifts. This paper investigates whether higher-order angular clustering statistics can robustly distinguish MG scenarios from GR, specifically focusing on the sensitivity of non-Gaussian features in the matter distribution.

Methodology
The authors utilize the ELEPHANT suite of N-body simulations, performed with the ECOSMOG code, to model structure formation under different gravity models.

Simulations: The simulations evolve $1024^3$ $102 4^{3}$ particles in a box of $1024/h$ $1024/ h$ Mpc. The baseline is a flat $\Lambda$ $Λ$ CDM model (WMAP 9-year cosmology). Two MG families are tested:
- $f(R)$ gravity: Hu-Sawicki variants labeled F5 ( $f_{R0} = -10^{-5}$ ) and F6 ( $f_{R0} = -10^{-6}$ ), utilizing the chameleon screening mechanism.
- nDGP gravity: Variants N1 ( $H_0 r_c = 1$ Gpc/ $h$ ) and N5 ( $H_0 r_c = 5$ Gpc/ $h$ ), utilizing the Vainshtein screening mechanism.
Light Cones: To mimic observer-frame data, the authors construct light cones by replicating the simulation box. They generate 2D sky catalogs covering $\sim 1567$ deg $^2$ for dark matter (DM) particles, dark matter halos ( $M_{vir} \ge 10^{12} M_\odot/h$ ), and mock galaxies (generated via Halo Occupation Distribution).
Optimization: An optimization procedure identifies the redshift range maximizing the MG signal relative to GR. The optimal range is determined to be $0.15 < z < 0.3$ .
Statistics: The study employs the counts-in-cells (CIC) method to measure area-averaged angular correlation functions, $W_J(\theta)$ $W_{J} (θ)$ , and reduced cumulants (hierarchical amplitudes), $S_J \equiv W_J / W_2^{J-1}$ $S_{J} \equiv W_{J} / W_{2}^{J - 1}$ , up to the ninth order ( $J=9$ $J = 9$ ).
- $W_J$ represents the $J$ -th order central moment of the density contrast.
- $S_J$ (e.g., $S_3$ for skewness) normalizes these moments to remove scale dependence related to the variance.
Analysis: The authors compare simulation results against Perturbation Theory (PT) predictions (using both linear and non-linear Halofit power spectra) to validate the non-linear regime. They calculate the significance of deviations ( $\psi$ ) between MG models and the fiducial GR case, accounting for shot noise and cosmic variance.

Key Results

Optimal Redshift Range: The signal for MG deviations is maximized in the redshift range $0.15 < z < 0.3$ .
Magnitude of Deviations: Across various scales and statistics, MG models exhibit relative departures from GR of up to 20%.
Sensitivity of Statistics:
- Third-order statistics ( $W_3$ and $S_3$ ) are identified as the most sensitive probes for distinguishing structure formation scenarios for halos and galaxies.
- For halos, $W_3$ and $S_3$ reach significance levels of $2\sigma$ to $4\sigma$ at angular scales $\theta \sim 0.13^\circ$ .
- For smooth dark matter, deviations are even stronger ( $>5\sigma$ ) but occur at smaller scales ( $\theta \sim 0.08^\circ$ ), where baryonic physics and resolution limits become critical.
- Reduced cumulants ( $S_J$ ) often provide higher significance than raw correlation functions ( $W_J$ ) because they contain extra information regarding the shape and asymmetry of the distribution, which is less degenerate with bias parameters.
Model-Specific Features:
- $f(R)$ Models: The F5 variant shows significant deviations, while F6 (weaker background field) surprisingly shows stronger deviations in $W_3$ for halos due to its inherently more non-linear chameleon screening behavior for low-mass halos. However, for galaxies, F5 and F6 show opposite signs in $W_3$ .
- nDGP Models: The N1 variant exhibits a "peak-like" scale dependence with maximum significance at $\theta \sim 0.1^\circ$ . The N5 variant generally shows weak, insignificant deviations.
Order Limitations: Due to the relatively low surface number densities of the mock catalogs ( $\sim 15$ galaxies/deg $^2$ ), statistical noise (shot noise and cosmic variance) limits robust signal extraction to orders $J \le 5$ . Higher orders ( $J > 5$ ) are dominated by scatter.
Comparison with Observations: The mock catalogs have significantly lower number densities than current or future photometric surveys (e.g., DES, Euclid). The authors note that if similar deviations are found in these sparse mocks, future surveys with much higher densities should yield even higher precision and significance.

Significance and Claims
The paper claims to be the first to systematically investigate higher-order angular clustering in light-cone projections for MG scenarios. Its primary contributions are:

Demonstrating Feasibility: It establishes that angular clustering, specifically higher-order moments like skewness ( $S_3$ ), is a viable and sensitive probe for testing GR vs. MG, even in the absence of spectroscopic redshifts.
Highlighting Non-Gaussianity: The results reinforce that the late-time large-scale structure is non-Gaussian and that higher-order statistics capture information (asymmetry, shape) that two-point statistics miss, making them crucial for distinguishing MG models where linear growth rates might be similar.
Future Potential: The authors emphasize that their findings, derived from catalogs with low number densities, suggest an "optimistic outlook" for real-world applications. They argue that the much higher surface densities of upcoming surveys (e.g., Vera Rubin, Euclid) will suppress shot noise and cosmic variance, potentially allowing for the detection of MG signals with even greater significance than reported here.

The paper concludes that while angular correlations may not show deviations as large as redshift-space distortions, the sheer volume and density of photometric data can compensate, offering a unique window into scale-dependent modifications of gravity.