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Constraining Galaxy Cluster Triaxiality via Weak Lensing -- I. Preparation for the Rubin Data Beyond Leading Order

This study utilizes weak lensing data from the Dark Energy Survey to measure the projected triaxiality of galaxy clusters, finding an ensemble ellipticity of approximately 0.31 with no significant dependence on mass or redshift, thereby validating a methodology for constraining cluster shapes in future large-scale surveys like LSST.

Original authors: Shenming Fu, Radhakrishnan Srinivasan, Tae-hyeon Shin, Rance Solomon, Deric Jones, Camille Avestruz, Yuanyuan Zhang, Michel Aguena, Céline Combet, Anthony Englert, Benjamin Levine, Alex I. Malz, Const

Published 2026-05-08

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Original authors: Shenming Fu, Radhakrishnan Srinivasan, Tae-hyeon Shin, Rance Solomon, Deric Jones, Camille Avestruz, Yuanyuan Zhang, Michel Aguena, Céline Combet, Anthony Englert, Benjamin Levine, Alex I. Malz, Constantin Payerne, Marina Ricci, Anja von der Linden, 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

The Big Picture: The Shape of Cosmic "Islands"

Imagine the universe is a giant ocean. In this ocean, there are massive islands made of invisible "dark matter" and regular stars. These islands are called galaxy clusters.

For a long time, scientists thought these islands were perfect spheres, like billiard balls. But this paper argues that they are actually tri-axial. Think of a tri-axial shape like a rugby ball or a potato: it has a long side, a medium side, and a short side. It's not a perfect sphere, and it's not a flat pancake; it's a 3D blob that looks different depending on which angle you view it from.

The problem? We can't see these islands in 3D. We are floating in space looking at them from the outside, so we only see a 2D shadow (a projection). It's like trying to guess the shape of a potato just by looking at its shadow on a wall. If the potato is lying on its side, the shadow looks long and thin. If it's standing up, the shadow looks round.

The Tool: Cosmic Magnifying Glasses

To figure out the shape of these invisible islands, the scientists used Weak Gravitational Lensing.

Imagine the galaxy cluster is a heavy bowling ball sitting on a trampoline. The trampoline represents space. The bowling ball curves the fabric. If you roll marbles (light from distant galaxies) past the bowling ball, their paths will bend.

The Effect: The light from the background galaxies gets stretched, just like a rubber band.
The Clue: If the bowling ball (the cluster) is round, the rubber bands stretch evenly in a circle. If the bowling ball is a rugby ball, the rubber bands stretch more in one direction than the other. By measuring how the background galaxies are stretched, scientists can infer the shape of the invisible cluster.

What This Paper Did

The authors used data from the Dark Energy Survey (DES), which is like a giant camera that has been taking pictures of the southern sky. They looked at thousands of these galaxy clusters.

Here is their step-by-step process, explained simply:

Finding the "North": To measure the shape, they first needed to know which way the cluster was facing. They looked at the visible galaxies inside the cluster (the "satellites") to find the long axis. It's like looking at a flock of birds to see which way the whole group is flying.
Stacking the Shadows: One cluster is too small to measure accurately. So, they took thousands of clusters, rotated them so they all faced the same way (like aligning a deck of cards), and stacked them on top of each other. This creates a super-clear, average "shadow" of a galaxy cluster.
The "Leading Order" vs. "Beyond": Previous studies looked at the main, obvious stretching (the "leading order"). This paper went a step further. They looked at the second-order details (the "beyond leading order").
- Analogy: Imagine looking at a blurry photo of a face. The first step is realizing it's a face. The second step is noticing the specific curve of the nose or the shape of the eyes. This paper looked at those finer details to get a more precise shape.
Cleaning the Data: They had to remove "noise."
- Boost Factor: Sometimes, galaxies that are actually part of the cluster get mistaken for background galaxies. This is like a guest at a party wearing the host's name tag. The scientists had to correct for this.
- Dilution Factor: Sometimes the visible galaxies don't perfectly line up with the invisible dark matter shape. It's like trying to guess the shape of a cloud by looking at the birds flying through it; the birds might be slightly off-center. They calculated how much this "off-center" view blurs the shape and corrected for it.

The Results: What Did They Find?

The Shape: They found that the average galaxy cluster is not a sphere.
- They measured an "ellipticity" (a number describing how stretched it is) of about 0.31.
- In terms of an axis ratio (the width of the short side divided by the long side), it is about 0.53.
- Simple Translation: If the long side of the cluster is 100 miles, the short side is only about 53 miles. It is definitely squashed, like a rugby ball.
Does Size or Age Matter? They checked if bigger clusters or older clusters looked different.
- Result: No. Whether the cluster was massive or small, or young or old, they all looked roughly the same shape. The "rugby ball" shape seems to be a universal rule for these clusters.
Accuracy Check: They tested their method using fake data (simulations) and found that their method works very well, with very little error.

Why Does This Matter?

The paper states that knowing the true shape of these clusters is crucial for two main reasons:

Weighing the Universe: Scientists use these clusters to weigh the universe and understand Dark Energy. If you assume a cluster is a perfect sphere when it's actually a rugby ball, you might get the weight wrong. This paper helps fix that scale.
Understanding the "Cosmic Web": These clusters form where giant threads of the universe (filaments) cross. The fact that they are rugby-ball shaped tells us how the universe pulls on them as they grow.

Summary

In short, this paper is a "preparation manual" for future, even bigger telescopes (like the Rubin Observatory). It says: "We have a new, more precise way to measure the shape of galaxy clusters. We tested it on current data, and we found that clusters are rugby-ball shaped, not spheres. This method is ready to be used on the massive amount of data coming from future space surveys to help us understand the universe better."

Technical Summary: Constraining Galaxy Cluster Triaxiality via Weak Lensing — I. Preparation for the Rubin Data Beyond Leading Order

Problem Statement
Galaxy clusters, the largest gravitationally collapsed structures in the universe, possess triaxial 3D mass distributions rather than spherical ones. This geometry introduces significant scatter (approximately 20%) in weak lensing (WL) mass measurements of individual clusters if a spherical density model is assumed. Furthermore, triaxiality coupled with projection effects can induce detection and selection biases, where clusters aligned along the line of sight appear richer and easier to detect, while those perpendicular appear rounder and harder to detect. Accurately constraining the projected shapes (ellipticities) of cluster ensembles is crucial for reducing mass measurement scatter, correcting selection biases, and probing cosmological parameters and dark matter physics (e.g., self-interacting dark matter constraints). While previous studies have measured cluster shapes, a comprehensive ensemble analysis of the Dark Energy Survey (DES) Year 3 (Y3) data using advanced multipole modeling had not been performed.

Methodology
The authors utilize the Cluster Lensing Mass Modeling (CLMM) software, developed by the Dark Energy Science Collaboration (DESC), to analyze the DES Y3 public data release. The analysis pipeline involves:

Data Selection:
- Clusters: 9,962 clusters from the redMaPPer catalog (richness $20 < \lambda < 222$ , redshift $0.2 < z < 0.65$ ) with a high centering probability ( $p_{cen} > 0.9$ ) to ensure relaxed systems and accurate centering.
- Source Galaxies: Shapes from the DES Y3 metacalibration catalog and photometric redshifts (photo-z) from the Directional Neighborhood Fitting (DNF) catalog.
- Validation: Mock catalogs simulating elliptical Navarro–Frenk–White (NFW) halos and a subset of DES Y1 redMaPPer clusters for cross-validation.
Orientation Determination:
- The major-axis orientation of each cluster is determined by the distribution of its satellite member galaxies (using weighted second moments), rather than the central galaxy alone, to minimize miscentering effects.
- The weak lensing shear estimates are rotated to align with this major axis before stacking.
Theoretical Modeling (Beyond Leading Order):
- The surface mass density is expanded using a multipole decomposition up to the second order (quadrupole).
- Monopole ( $\Sigma_0$ ): Represents the circular component. The authors include a second-order correction term ( $\epsilon^2$ ) to the monopole, which accounts for the mass redistribution due to ellipticity.
- Quadrupole ( $\Sigma_2$ ): Represents the elliptical deformation.
- The analysis simultaneously fits the monopole and quadrupole signals to constrain the mass, concentration, and projected ellipticity ( $\epsilon$ ).
Systematic Corrections:
- Dilution Factor: Corrects for the misalignment between the satellite galaxy distribution (used to define orientation) and the true dark matter halo orientation. This is estimated via Monte Carlo sampling based on cluster richness.
- Boost Factor: Corrects for the contamination of background source galaxies by unlensed cluster members (photometric redshift errors), estimated using the redshift distribution of galaxies in radial bins.
- Radial Cuts: Sources within 300 kpc are excluded to mitigate blending and strong lensing effects.

Key Contributions

First DES Y3 Ensemble Shape Measurement: This work presents the first measurement of the effective projected ellipticity for a large ensemble of clusters from the DES Y3 redMaPPer catalog using weak lensing.
Beyond Leading Order Analysis: The analysis extends the multipole expansion to the second order of ellipticity for the monopole term. This provides a percent-level correction to the mass profile, improving the accuracy of mass measurements compared to leading-order approximations.
Robust Systematic Handling: The study implements and validates a pipeline that explicitly corrects for dilution (misalignment) and boost (contamination) factors, demonstrating the readiness of the CLMM tool for future large-scale surveys.
Validation: The methodology is validated against mock catalogs with known input shapes and against DES Y1 data, showing consistency and robustness against variations in measurement methods (per-object vs. per-HEALPix response).

Results

Projected Ellipticity: For the baseline sample, the measured projected halo ellipticity is $\epsilon = 0.310^{+0.017}_{-0.016}$ , corresponding to an axis ratio of $q = 0.527^{+0.018}_{-0.019}$ . This represents a $\sim 19\sigma$ detection of non-zero ellipticity.
Dependence on Properties: The analysis finds no statistically significant dependence of the projected ellipticity on cluster mass (richness) or redshift within the current sample size and statistical precision. The differences between high/low richness and high/low redshift bins are approximately $0.4\sigma$ .
Mass Bias: When comparing the triaxial model fit to a spherical model, the triaxial model yields a mass estimate approximately 5% lower for the same monopole signal, consistent with theoretical expectations for elliptical NFW profiles.
Mock Validation: The measurement algorithm recovers input axis ratios with a bias of less than 5% for true axis ratios $q > 0.35$ . The analysis is found to be insensitive to halo concentration variations (tested for $c=2, 4, 6$ ).

Significance and Claims
The paper positions this work as a critical preparatory step for the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST). The authors claim that:

The CLMM tool, now equipped with a triaxiality module, is capable of efficiently and consistently analyzing cluster weak lensing data in large volumes.
The demonstrated methodology is directly applicable to upcoming wide-area surveys (LSST, Euclid, Roman Space Telescope).
Future datasets with larger sample sizes and deeper observations will significantly tighten constraints on cluster ellipticities (estimating an 82% reduction in error bars for LSST Y10 compared to DES Y3), allowing for more precise tests of the dependence of halo shape on mass and redshift, and better control of systematics in cluster cosmology.
This paper serves as the first in a series, with future work planned to utilize cosmological simulations (SkySim5000) to further quantify selection biases, miscentering, and the relationship between halo triaxiality and the surrounding large-scale structure.