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The Splashback Mass Function of Galaxy Clusters from Photometric Data

This paper presents a fully photometric framework to measure individual galaxy cluster splashback radii and masses, successfully constructing the first observational splashback mass function from SDSS data that aligns with simulation predictions at high masses and demonstrates the viability of splashback-based cosmological studies without spectroscopic or lensing data.

Original authors: Lucas Gabriel-Silva, Laerte Sodré Jr

Published 2026-05-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lucas Gabriel-Silva, Laerte Sodré Jr

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, expanding ocean. In this ocean, massive islands of invisible "dark matter" form, pulling in galaxies like fish swimming toward a reef. These islands are called galaxy clusters.

For a long time, astronomers tried to measure the size and weight of these islands by drawing an invisible circle around them. They said, "Everything inside this circle where the density is X times the average is part of the island." But this method had a flaw: as the universe expands, the "average" changes, making the islands look like they are growing even if they aren't. It's like trying to measure a balloon's size by comparing it to a shrinking rubber band; the measurement gets messy.

This paper introduces a better way to measure these cosmic islands using a concept called the Splashback Radius.

The "Splashback" Analogy

Think of a cluster like a giant whirlpool in a river. As water (galaxies and dark matter) rushes in, it doesn't just stop at the edge of the whirlpool. It overshoots, swings around the center, and then falls back in. The point where this material reaches its furthest point before turning around is the Splashback Radius.

This isn't an arbitrary line drawn by humans; it's a physical boundary created by the motion of the material itself. It's the "splash" zone where the incoming stuff hits the edge of the orbit and turns back. Because it's based on actual motion, it doesn't suffer from the "shrinking rubber band" problem of the old methods.

The Challenge: Seeing Without a Telescope's "Zoom"

To find this splashback point, you usually need to know exactly how fast every single galaxy is moving toward or away from us (spectroscopy). This is like needing a high-speed camera to see a splash in slow motion. But getting this data for thousands of clusters is slow, expensive, and difficult.

Most modern surveys, like the Sloan Digital Sky Survey (SDSS), only have "photometric" data. This is like looking at a photo of the splash. You can see the shape and the color, but you don't have the high-speed motion data.

The Solution: A Smart "Membership" Filter

The authors of this paper developed a clever, probabilistic method to figure out which galaxies belong to the cluster and which are just background noise, using only the "photo" data.

The Filter: They created a mathematical filter that looks at two things:
- Location: Is the galaxy near the cluster?
- Color/Distance: Does the galaxy's color match the cluster's distance?
The Adaptive Cut: Instead of using a rigid rule (like "only include galaxies within 50% probability"), they let the rule change based on the cluster. A rich, crowded cluster gets a stricter filter to avoid false alarms. A smaller, lonely cluster gets a softer filter to catch every member. It's like a bouncer at a club who adjusts the entry rules depending on how busy the night is.

What They Did

Using this new method, the team analyzed 499 galaxy clusters from a database called CoMaLit. They mapped out the density of galaxies around each cluster to find the "splashback" point—the spot where the density of galaxies drops off sharply.

Once they found the splashback radius, they used a mathematical relationship to calculate the Splashback Mass (the total weight of the cluster up to that splash point).

The Results

It Works: They found that the splashback radius is typically about 1.1 times larger than the old "standard" radius. This matches what other scientists have found using more expensive data, proving their "photo-only" method is accurate.
No Evolution: They checked if this relationship changed over time (as the universe got older). They found it didn't. The rules for how big a splashback is relative to the cluster's mass stay the same, regardless of when in the universe's history you look.
The First "Splashback" Census: They applied their method to over 15,000 clusters in the Northern sky. They created the first-ever "Splashback Mass Function." Think of this as a census report that counts how many clusters of different sizes exist in the universe, but using this new, physically meaningful definition of size.

The Conclusion

The census matched perfectly with computer simulations at the high-mass end (the biggest clusters). At the lower end, the numbers dropped off, but the authors explain this isn't because their method failed; it's because the catalog they used simply doesn't detect the smaller, fainter clusters well (a known limitation of the survey, not the method).

In short: The authors proved that you don't need expensive, slow-motion data to measure the true physical boundaries of galaxy clusters. By using a smart statistical filter on standard photos, they can accurately weigh these cosmic islands and count them, opening the door for future cosmological studies using data from massive, upcoming sky surveys.

Technical Summary: The Splashback Mass Function of Galaxy Clusters from Photometric Data

Problem Statement
The halo mass function is a fundamental statistic in cosmology, linking theoretical structure formation to observations. However, robust mass estimates for galaxy clusters traditionally rely on expensive or complex techniques, such as spectroscopic virial measurements, X-ray observations, or weak gravitational lensing. Furthermore, standard definitions of halo mass and size based on spherical overdensity (e.g., $M_{200c}$ , $R_{200m}$ ) suffer from "pseudo-evolution," where halo masses appear to grow due to the decreasing reference density of the universe, even if the physical density profile remains unchanged. While the splashback radius ( $R_{sp}$ )—defined as the apocenter of recently accreted material—offers a physically motivated, pseudo-evolution-free boundary, its application to large-scale cosmological studies has been limited by the need for spectroscopic data or weak lensing measurements. This work addresses the challenge of measuring individual cluster splashback radii and constructing a splashback mass function using solely photometric data.

Methodology
The authors utilize Sloan Digital Sky Survey (SDSS) data to develop a fully photometric framework for measuring splashback properties. The methodology proceeds in three main stages:

Probabilistic Photometric Membership:
The authors adapt a Bayesian model to assign membership probabilities to galaxies based on photometric redshifts ( $z_{phot}$ ) and radial position. The total probability ( $P_f$ ) is the product of a redshift term ( $P_z$ ) and a radial term ( $P_r$ ).
- Redshift Term: Modeled as a Gaussian probability density function centered on the cluster redshift, with a magnitude-dependent scatter ( $\sigma_z$ ) calibrated using SDSS spectroscopic data.
- Radial Term: Derived from a S'ersic profile plus a constant background fitted to the surface number density of a "clean" sample (filtered by redshift and brightness cuts).
- Adaptive Thresholding: Instead of a fixed probability cut, the authors implement an adaptive threshold ( $P_{cut}$ ) optimized for each cluster. This threshold maximizes the detection significance ( $S_N$ ) of the cluster core relative to its outskirts, following the Li & Ma (1983) formalism. This approach accounts for variations in cluster richness and redshift.
Splashback Radius and Mass Estimation:
Using the selected member galaxies, the authors model the cumulative number profile of each cluster. They fit a "trunc-NFW" model (a projected NFW profile with a smooth exponential truncation and a power-law two-halo term) to the data using MCMC sampling.
- The splashback radius ( $R_{sp}$ ) is defined as the radius where the logarithmic slope of the density profile is steepest (the minimum of $d \log \Sigma / d \log R$ ).
- The splashback mass ( $M_{sp}$ ) is calculated by integrating the mass profile (derived from the fitted parameters and cataloged $M_{200c}$ values) out to $R_{sp}$ .
Scaling Relation and Mass Function Construction:
The authors recalibrate the $M_{sp}–R_{sp}$ scaling relation over a wide redshift range ( $0.01 < z < 0.8$ ) using a sample of 499 clusters from the CoMaLit weak-lensing compilation. This calibrated relation is then applied to the redMaPPer cluster catalog (SDSS Northern Galactic Cap) to derive splashback masses for over 15,000 systems, enabling the construction of the first observational splashback mass function based purely on photometry.

Key Contributions and Results

Photometric Membership Performance: Validated against a spectroscopic subsample, the probabilistic method achieves $\sim70\%$ completeness and purity within $3R_{200m}$ for a standard cut, improving to $\sim90\%$ completeness and $\sim85\%$ purity within $R_{200m}$ . The adaptive thresholding significantly reduces systematic bias in $R_{sp}$ recovery compared to fixed cuts or simple redshift windows.
Splashback Radius Measurements: For the CoMaLit sample, the median ratio is found to be $R_{sp}/R_{200m} \simeq 1.10 \pm 0.01$ , consistent with previous observational studies. The median splashback overdensity is $\Delta_{sp} \simeq 127 \pm 2$ .
Recalibrated $M_{sp}–R_{sp}$ Scaling Relation: The authors find a power-law relation with a slope $A = 2.63 \pm 0.05$ , which is significantly shallower than the constant-density expectation ( $A=3$ ). Crucially, after rescaling for cosmic expansion using the Hubble parameter $E(z)$ , the relation shows no statistically significant redshift evolution ( $B = 0.07 \pm 0.15$ ). The intrinsic scatter is low ( $\sim0.15$ dex), comparable to mass-richness relations.
Observational Splashback Mass Function: Applying the scaling relation to the redMaPPer catalog, the authors construct a mass function that agrees well with simulation-based predictions (Diemer 2020) at the high-mass end. Deviations at lower masses ( $M_{sp} \sim 2 \times 10^{14} h^{-1}_{70} M_{\odot}$ ) are attributed to the known completeness limits of the optical redMaPPer catalog rather than failures in the splashback methodology.

Significance
The paper demonstrates that splashback-based cluster sizes, masses, and abundances can be robustly measured using photometric surveys alone, without requiring spectroscopic or lensing data. By establishing a physically motivated mass definition free from pseudo-evolution and validating a method to recover it from photometric redshifts, the work enables cosmological studies using the vast volumes of current and future photometric surveys. The close agreement between the derived observational mass function and theoretical expectations validates splashback quantities as reliable probes of halo structure and evolution.