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$4\times3$ Point Correlation Functions in Galaxy Surveys: Impact of Baryonic Feedback

This paper investigates the impact of baryonic feedback on galaxy density and weak lensing correlation functions using simulated LSST data, finding that three-point statistics (specifically the $4\times3$ PCFs) are significantly more sensitive to baryonic effects at small scales than two-point functions, thereby offering enhanced leverage for constraining both cosmological and baryonic feedback parameters.

Original authors: Avijit Bera, Joachim Harnois-Déraps, Juan Mena-Fernández, Mike Jarvis, Cyrille Doux, Katrin Heitmann, Mustapha Ishak, The LSST Dark Energy Science Collaboration

Published 2026-05-04

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CC BY 4.0

Original authors: Avijit Bera, Joachim Harnois-Déraps, Juan Mena-Fernández, Mike Jarvis, Cyrille Doux, Katrin Heitmann, Mustapha Ishak, The LSST Dark Energy Science Collaboration

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 web made of dark matter. This web holds everything together, but we can't see it directly. Instead, we look at how it bends the light from distant galaxies (like looking at a straw in a glass of water) and how galaxies cluster together. This is how cosmologists try to understand the universe.

For a long time, scientists have used simple "two-point" measurements to study this web. Think of this like measuring the distance between just two points on a map. It's great for seeing the big picture, but it misses the messy, complex details in the middle.

This paper introduces a more advanced method: measuring three-point correlations. Instead of just two points, we look at triangles formed by three galaxies. This is like looking at the shape of a triangle rather than just a line. It reveals the "clumpy" nature of the universe, showing us filaments, clusters, and voids that the simple two-point method misses.

Here is the core story of the paper, broken down with simple analogies:

1. The "Baryonic Feedback" Problem: The Cosmic Sprinkler

The universe isn't just dark matter; it also contains normal stuff (gas, stars, black holes), which scientists call "baryons." These baryons act like a powerful cosmic sprinkler system.

When stars form or black holes eat matter, they blast out huge amounts of energy. This energy pushes gas out of galaxy clusters, changing the shape of the dark matter web.

The Paper's Analogy: Imagine you have a perfect, smooth mound of sand (the dark matter). Now, imagine someone comes in with a high-pressure hose (baryonic feedback) and sprays the sand around. The mound changes shape. If you try to measure the mound without knowing about the hose, your measurements will be wrong.

2. The Experiment: Simulating the Universe

The authors didn't just look at the sky; they built a massive virtual universe inside a computer.

The Setup: They started with a simulation of a universe that only had gravity (no sprinklers). This is their "control group."
The Twist: They then used a clever mathematical trick called the Baryonic Correction Model (BCM). Instead of running a super-expensive simulation that includes every gas particle and star, they took the "gravity-only" simulation and mathematically "sprayed" the sand around to mimic the effect of the cosmic sprinklers.
The Variables: They turned the "sprinkler" up and down by changing two knobs:
1. How much gas gets pushed out? (The amount of water from the hose).
2. How far does the gas fly? (The range of the spray).

3. The Discovery: Triangles Feel the Spray More Than Lines

The team compared their "gravity-only" universe with their "sprinkled" universe using both the old two-point method (lines) and the new three-point method (triangles).

The Big Finding:
The "triangles" (three-point statistics) are much more sensitive to the sprinkler than the "lines" (two-point statistics).

The Analogy: If you have a calm pond and you throw a rock in, the ripples (lines) spread out. But if you look at the chaotic splash pattern right where the rock hit (the triangles), you see the immediate, violent impact of the splash.
The Numbers: On very small scales (like looking at a single galaxy cluster), the "sprinkler" effect could change the three-point measurements by up to 90%. For the older two-point methods, the change was significant but smaller (around 20–40%).

4. Why This Matters: The "Goldilocks" Zone

The paper argues that these new triangle measurements are a double-edged sword, but a useful one:

The Problem: If you want to measure the fundamental properties of the universe (like how much dark energy there is), you usually want to avoid the small, messy scales where the "sprinkler" messes things up.
The Opportunity: Because the triangles react so strongly to the sprinkler, they are actually excellent tools for studying the sprinkler itself.
- If you use the triangles, you can figure out exactly how strong the "cosmic sprinkler" is and how far it sprays.
- Once you understand the sprinkler, you can subtract its effect and get a cleaner view of the dark matter web.

5. The Tool: A New Way to Count Triangles

To do this, the authors used a new feature in a software tool called TreeCorr.

The Analogy: Imagine you have a billion marbles (galaxies) scattered on the floor. Counting every possible triangle they form by hand would take forever. TreeCorr is like a super-fast robot that can instantly group these marbles into triangles and measure their shapes, even when some are galaxies and some are distorted shapes of light (shear). This paper is the first time this specific tool was used to measure these complex "mixed" triangles (involving both galaxy positions and distorted shapes).

Summary

In short, this paper says: "We built a virtual universe and added a 'cosmic sprinkler' to see how it messes up our measurements. We found that looking at triangles of galaxies is much better at detecting this sprinkler than looking at simple lines. This helps us understand the messy physics of gas and stars, which in turn helps us get a clearer picture of the invisible dark matter holding the universe together."

They conclude that while these new measurements are harder to model, they are essential for the next generation of telescopes (like the Rubin Observatory) to avoid being fooled by the "sprinklers" of the universe.

1. Problem Statement

Future Stage-IV galaxy surveys (e.g., LSST, Euclid, Roman) aim to constrain cosmological parameters using weak gravitational lensing and galaxy clustering. The standard approach utilizes 3 × 2-point correlation functions (3 × 2PCFs), combining galaxy-galaxy clustering ($gg$), galaxy-shear ($gG$), and shear-shear ($GG$) statistics.

Limitation: 2PCFs primarily probe linear scales and miss higher-order information encoded in small-scale, non-linear structures (filaments, clusters).
The Challenge: To access this small-scale information, researchers turn to 3-point correlation functions (3PCFs). However, small scales are heavily contaminated by baryonic feedback (astrophysical processes like AGN jets and stellar winds that expel gas from halos).
The Gap: While baryonic effects on 2PCFs are well-studied, their impact on the full suite of 4 × 3PCFs (combinations of galaxy density $g$ and shear $G$ : $ggg$, $ggG$, $gGG$, $GGG$) remains largely unexplored. Furthermore, analytical models for baryonic effects on these higher-order statistics are currently unavailable.

2. Methodology

The authors employ a simulation-based approach to quantify baryonic impacts on 3PCFs without running expensive full hydrodynamical simulations for every parameter variation.

Simulation Framework:
- Base Simulation: A high-resolution gravity-only (Dark Matter Only, DMO) N-body simulation using the HACC code ( $2048^3$ particles, $600 \, h^{-1}$ Mpc box).
- Mock Catalogs: Generated to mimic the LSST Year-10 (Y10) survey (5000 deg $^2$ , 5 tomographic redshift bins, $30 \, \text{gal/arcmin}^2$ ).
- Lightcones: Constructed using the Pollux code, projecting particle data onto Healpix maps ( $N_{side}=8192$ ) to generate convergence ( $\kappa$ ) and shear ( $\gamma$ ) maps.
Baryonic Correction Model (BCM):
- Instead of hydrodynamics, the authors use the BCM to "baryonify" the DMO snapshots. This model applies physically motivated particle displacements to mimic the redistribution of matter caused by baryonic feedback.
- Key Parameters: Two parameters are varied to explore the parameter space:
  1. $M_c$ : Characteristic mass scale governing the slope of the gas profile (amount of gas expelled).
  2. $\theta_{ej}$ : Maximum ejection radius relative to the halo boundary ( $r_{ej} = \theta_{ej} r_{200}$ ).
- Shell Baryonification: The BCM is applied to density shells (projected maps) rather than 3D particle data, allowing for efficient generation of baryon-influenced lensing maps.
Measurement Tools:
- TreeCorr (v5.1.3): A newly updated code capable of computing mixed-spin 3-point correlations. This allows for the first-time measurement of cross-correlations between spin-0 fields (galaxy positions) and spin-2 fields (galaxy shapes).
- Configuration: The study focuses on equilateral triangle configurations to simplify the analysis to a single scale variable ( $\vartheta_{eq}$ ).

3. Key Contributions

First Measurement of Mixed 3PCFs: This paper presents the first simulation-based measurements of the $ggG$ (galaxy-galaxy-shear) and $gGG$ (galaxy-shear-shear) 3PCFs using the TreeCorr code.
Comprehensive Baryonic Impact Study: It quantifies the suppression of all four 3PCF types ($ggg$, $ggG$, $gGG$, $GGG$) due to baryonic feedback, comparing them against the standard 3 × 2PCFs.
Scale-Dependent Sensitivity: The study identifies specific angular scales where baryonic effects become dominant for each statistic, providing crucial "scale cuts" for future cosmological analyses.
Demonstration of Complementarity: It establishes that 3PCFs offer enhanced sensitivity to baryonic physics compared to 2PCFs, suggesting a pathway to break degeneracies between cosmological parameters and astrophysical feedback.

4. Key Results

Enhanced Suppression in 3PCFs:
- Baryonic feedback suppresses 3PCFs more severely than 2PCFs at small scales.
- $ggg$ (Clustering): Significant suppression (>30%) begins at scales < 4 arcmin.
- $ggG$: Suppression becomes significant at < 10 arcmin.
- $gGG$: Suppression becomes significant at < 40 arcmin.
- $GGG$ (Shear): Suppression becomes significant at < 1 degree.
- Magnitude: At the smallest scales, 3PCFs can be suppressed by up to 90% depending on the BCM parameters, whereas 2PCFs typically show 10–40% suppression.
Distinct Features:
- The ratio of Baryonified (DMB) to DMO measurements shows a distinct "bump" or upturn at intermediate scales (before the deep suppression) for $ggg$ and $ggG$. This feature is linked to the ejection radius parameter ( $\theta_{ej}$ ) and the cooling/condensation of stellar components within halos.
- The $GGG$ statistic is affected at much larger angular scales than clustering statistics due to the projection of the lensing kernel, which integrates matter over a wider redshift range.
Scale Cuts for Cosmology:
- To avoid baryonic bias in cosmological parameter estimation (assuming a DMO model), the study derives scale cuts where the DMO model can be rejected at 99% confidence.
- For the fiducial model, $w(\vartheta)$ (2PCF) requires a cut at ~~2 arcmin, while $\zeta(\vartheta)$ (3PCF) requires a similar cut (~~2 arcmin). However, shear-based 3PCFs (like $\Gamma$ ) require much larger cuts (~40 arcmin) compared to their 2PCF counterparts ( $\xi_-$ at ~30 arcmin), indicating that 3PCFs are more sensitive to baryonic effects at a given scale.
Parameter Constraints:
- The results suggest that while 3PCFs are noisier (larger error bars) than 2PCFs, their enhanced sensitivity to baryonic physics makes them powerful tools for constraining the BCM parameters ( $M_c, \theta_{ej}$ ) when combined with 2PCFs.

5. Significance and Future Outlook

Cosmological Synergy: The paper argues that a joint analysis of 3 × 2PCFs and 4 × 3PCFs is essential for the LSST era. The 2PCFs constrain cosmology on large scales, while the 3PCFs provide leverage to constrain baryonic feedback parameters on small scales. This joint approach can break degeneracies between cosmology and astrophysics.
Systematic Mitigation: By understanding the specific scale-dependent signatures of baryonic feedback in 3PCFs, future analyses can either apply tighter scale cuts or use the 3PCFs themselves to marginalize over baryonic uncertainties, leading to more robust dark energy constraints.
Theoretical Development: The work highlights the urgent need for analytical models of 3PCFs that include baryonic feedback. Currently, the field relies on simulation-based emulators (like the BCM used here) because analytical predictions for these statistics are computationally challenging and incomplete.

In conclusion, this paper demonstrates that 4 × 3PCFs are not just a theoretical extension but a practical necessity for next-generation surveys to control baryonic systematics and fully exploit the information content of the non-linear universe.