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Impact of projection-induced optical selection bias on the weak lensing mass calibration of galaxy clusters

This paper quantifies how projection-induced optical selection bias, driven by halo triaxiality and large-scale structure, causes a significant 20–80% overestimation of weak lensing masses in optically selected galaxy clusters, thereby necessitating its inclusion in future cosmological analyses to ensure accuracy.

Original authors: Titus Nyarko Nde, Hao-Yi Wu, Shulei Cao, Gladys Muthoni Kamau, Andrius Tamosiunas, Chun-Hao To, Conghao Zhou

Published 2026-04-10

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CC BY 4.0

Original authors: Titus Nyarko Nde, Hao-Yi Wu, Shulei Cao, Gladys Muthoni Kamau, Andrius Tamosiunas, Chun-Hao To, Conghao Zhou

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 you are trying to weigh a collection of heavy suitcases (galaxy clusters) by looking at how much they bend a sheet of rubber (space-time) placed underneath them. This bending is called gravitational lensing. The more the rubber bends, the heavier the suitcase is.

Now, imagine you are also trying to guess the weight of these suitcases by counting how many people are standing on top of them. The more people you see, the heavier you assume the suitcase is. This is called richness.

In the real universe, astronomers do exactly this: they count the galaxies in a cluster to guess its mass, and then they check that guess by measuring how much the cluster bends light from background galaxies.

The Problem: The "Crowded Hallway" Effect

The paper by Titus Nyarko Nde and colleagues reveals a sneaky trick of the universe that messes up this weighing scale. It's called projection-induced selection bias.

Here is a simple analogy: Imagine you are standing in a long, crowded hallway looking down the line of sight. You see a group of people (a galaxy cluster) standing together. But because the hallway is long, you can't tell who is actually in the group and who is just standing behind them or in front of them, further down the hall.

The "Fake" Crowd: Sometimes, a group of people who aren't actually part of the cluster happen to line up perfectly behind it from your perspective. This makes the cluster look like it has more people (higher richness) than it really does.
The "Fake" Weight: At the same time, all those extra people (and the empty space they occupy) have mass. So, they also bend the rubber sheet underneath. This makes the cluster look heavier (stronger lensing signal) than it really is.

The Trap: Because the "fake" crowd and the "fake" weight happen at the same time, the universe tricks you. You see a cluster with a lot of people and a lot of bending, so you think, "Wow, this must be a super-heavy suitcase!"

But in reality, it's just a medium-weight suitcase with a bunch of strangers standing behind it.

What the Paper Found

The authors used two massive supercomputer simulations (MiniUchuu and Cardinal) to act out this scenario millions of times. They wanted to see how much this "hallway trick" messes up our calculations.

Here is what they discovered:

We are overestimating the weight: Because of this trick, astronomers are currently thinking galaxy clusters are 20% to 50% heavier than they actually are.
The further out, the worse it gets: If you look at the area far away from the center of the cluster (the "large scales"), the error gets even bigger, sometimes up to 80%. It's like trying to weigh a suitcase by looking at the people standing 50 feet away from it; the signal gets very noisy and misleading.
It's the biggest problem: There are other things that mess up measurements (like gas inside the cluster or errors in measuring distance), but this "hallway trick" is currently the biggest source of error in our understanding of the universe's expansion and dark energy.

Why Does This Matter?

Galaxy clusters are like the "canaries in the coal mine" for cosmology. By counting them and weighing them, scientists try to figure out the secrets of Dark Energy (the force pushing the universe apart) and how much matter exists in the universe.

If we think the suitcases are heavier than they are, our entire calculation of how the universe is expanding is wrong. It's like trying to calculate the speed of a car but using a speedometer that is stuck on "fast."

The Solution: How Do We Fix It?

The paper suggests we can't just ignore this problem. We need new strategies:

Look deeper: Use spectroscopy (like a high-tech scanner) to tell exactly how far away every single galaxy is, so we know who is actually in the group and who is just a background stranger.
Compare different views: Look at the same clusters using different methods (like X-rays or radio waves) to see if the "weight" matches up.
Simulate the trick: Use supercomputers to model exactly how this "hallway trick" works, so we can mathematically subtract the error from our real data.

In a nutshell: The universe is playing a game of optical illusion with us. It makes galaxy clusters look heavier and more crowded than they are because of things lining up in the background. This paper proves that this illusion is the biggest reason our current measurements of the universe's weight are off, and we need to fix our math to get the true picture of our cosmos.

1. Problem Statement

In optical cluster cosmology, the number counts of galaxy clusters as a function of mass are a powerful probe for constraining cosmological parameters (e.g., $\Omega_m$ , $\sigma_8$ , dark energy). However, the accuracy of these constraints relies heavily on the precision of mass calibration, typically achieved via weak gravitational lensing.

The paper identifies a critical systematic error: projection-induced selection bias.

Mechanism: Optical surveys identify clusters based on "richness" ( $\lambda$ ), defined as the number of member galaxies. Due to distance uncertainties, galaxies physically unassociated with a cluster but located along the same line of sight (LOS) are mistakenly counted as members.
Coherent Boost: This projection effect simultaneously boosts the measured richness and the lensing signal (which depends on projected matter density). Halos with triaxial shapes or those aligned with large-scale structure (filaments) along the LOS appear richer and have stronger lensing signals than their true mass would suggest.
Consequence: A sample selected by a specific richness range is contaminated by low-mass halos with "up-scattered" richness. Because up-scattered richness correlates with up-scattered lensing, the mean lensing signal for a richness bin is biased high relative to the true mean mass. This leads to an overestimation of cluster masses, which is currently the dominant systematic uncertainty limiting optical cluster cosmology.

2. Methodology

The authors quantify this bias using two distinct mock catalogs derived from $N$ -body simulations, employing different galaxy-halo connection models and cluster finders to ensure robustness.

A. Simulations and Mock Catalogs

MiniUchuu-based Mock:
- Simulation: Based on the Uchuu suite ( $N$ -body, $\Lambda$ CDM).
- Galaxy Model: Uses a Halo Occupation Distribution (HOD) model to populate halos with central and satellite galaxies.
- Cluster Finding: Simulates richness using a "counts-in-cylinders" algorithm with a fixed LOS depth of $\pm 30 \, h^{-1}\text{cMpc}$ to mimic distance uncertainties.
- Purpose: To test a specific, simplified projection model.
Cardinal Mock:
- Simulation: A suite of lightcone mocks designed for the Dark Energy Survey (DES).
- Galaxy Model: Uses the ADDGALS algorithm, which assigns galaxies based on luminosity-density relations and includes tidal stripping effects.
- Cluster Finding: Applies the redMaPPer algorithm, which uses matched filters on galaxy colors and luminosities to identify clusters and determine photometric redshifts.
- Purpose: To study realistic, observationally driven cluster finding.

B. Analysis Procedure

Lensing Signal: The authors calculate the stacked excess surface mass density ( $\Delta\Sigma$ ) profiles for clusters binned by richness ( $\lambda$ ) and redshift ( $z$ ).
Mass Calibration: They perform a mock mass calibration by fitting an analytic model (based on the Navarro–Frenk–White profile and 2-halo term) to the simulated lensing data.
Likelihood Fitting: Using a likelihood function with a covariance matrix derived from DES Year 1 analyses, they fit for the mass ( $M$ ) and concentration ( $c$ ) parameters.
Scale Separation: The analysis splits the lensing signal into:
- Small scales: $0.2 - 3$ Mpc (dominated by the 1-halo term).
- Large scales: $3 - 30$ Mpc (dominated by the 2-halo term and LOS structure).
- Combined: All scales.

3. Key Results

The study quantifies the mass bias ( $M_{obs} / M_{true} - 1$ ) caused by the selection bias:

Magnitude of Bias:
- MiniUchuu: Shows a mass bias of ~25% on small scales and ~50% on large scales. The combined bias is approximately 25%.
- Cardinal (redMaPPer): Shows a more severe bias, ranging from 20–40% on small scales and 20–80% on large scales. The combined bias is 20–50%.
Scale Dependence: The bias is significantly larger on large scales ( $>3$ Mpc) because the 2-halo term is more susceptible to LOS projection effects (filaments and large-scale structure) than the 1-halo term.
Redshift and Richness Dependence (Cardinal):
- The bias is strongest for high-redshift clusters ( $z \geq 0.5$ ) due to larger photometric redshift uncertainties.
- The bias is strongest for low-richness clusters ( $\lambda \approx 20$ ), where the relative impact of projection is higher.
Comparison to Other Systematics: The projection-induced bias (20–50%) vastly exceeds other known systematics in optical cluster lensing, such as baryonic effects, miscentering, or source photometric redshift errors (which typically contribute 5–16% bias).

4. Key Contributions

Quantification of Dominant Systematic: The paper definitively establishes that projection-induced selection bias is the primary limiting factor for the accuracy of mass calibration in optical cluster cosmology, outweighing all other currently known systematics.
Multi-Model Validation: By using two fundamentally different simulation approaches (HOD vs. ADDGALS; counts-in-cylinders vs. redMaPPer), the authors demonstrate that the bias is a robust physical phenomenon, not an artifact of a specific simulation or algorithm.
Scale-Specific Insights: The work highlights that restricting lensing analysis to small scales ( $<3$ Mpc) mitigates but does not eliminate the bias, while large-scale analyses are severely compromised.
Redshift/Richness Trends: It provides specific trends showing that the bias worsens at higher redshifts and lower richness, crucial for designing future survey strategies.

5. Significance and Future Directions

Cosmological Impact: If uncorrected, this bias leads to an overestimation of cluster masses, which would systematically shift derived cosmological parameters (e.g., underestimating $\sigma_8$ or misinterpreting dark energy properties).
Mitigation Strategies: The authors advocate for a multi-probe, multi-wavelength approach:
- Spectroscopic Follow-up: Using spectroscopic redshifts of member galaxies to separate projected from physical members.
- Cross-Correlation: Combining cluster lensing with cluster-galaxy cross-correlations to self-calibrate the selection bias.
- Multi-wavelength Comparison: Comparing optical clusters with gas-selected clusters (X-ray or mm-wave) which are less affected by optical projection biases.
Modeling Recommendations: Future analyses should move beyond simple analytic corrections and adopt simulation-based forward modeling. This approach can self-consistently model the complex interplay between projection effects, richness, lensing, and gas observables, allowing for a flexible correction that accounts for the specific galaxy model and cluster finder used.

In conclusion, this paper serves as a critical warning and a roadmap for the next generation of cluster cosmology, emphasizing that accurate mass calibration requires explicitly modeling and correcting for projection-induced selection bias.