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KiDS-Legacy: The consistency test of the large-scale structure with Bernardeau-Nishimichi-Taruya transform

This paper presents the first $k$ -cut cosmic shear analysis of the KiDS-Legacy survey using the Bernardeau-Nishimichi-Taruya transform, demonstrating that while removing small scales yields an $S_8$ consistent with standard results, the choice of covariance matrix significantly impacts the inferred value and reveals a mild scale-dependent trend that serves as a diagnostic for weak-lensing inference pipelines.

Original authors: Shiming Gu, Ziang Yan, Ludovic van Waerbeke, Francis Bernardeau, Hendrik Hildebrandt, Angus H. Wright, Maciej Bilicki, Christos Georgiou, Shun-Sheng Li, Laila Linke, Lauro Moscardini, Robert Reischke

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shiming Gu, Ziang Yan, Ludovic van Waerbeke, Francis Bernardeau, Hendrik Hildebrandt, Angus H. Wright, Maciej Bilicki, Christos Georgiou, Shun-Sheng Li, Laila Linke, Lauro Moscardini, Robert Reischke, Benjamin Stölzner

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: Measuring the Universe's "Clumpiness"

Imagine the universe is a giant, invisible soup made of matter (mostly dark matter, which we can't see directly). Over billions of years, gravity has caused this soup to clump together, forming galaxies and huge cosmic webs.

Cosmologists want to measure exactly how "clumpy" this soup is. They use a specific number called $S_8$ to describe this. Think of $S_8$ as a "clumpiness score."

High $S_8$ : The universe is very clumpy (lots of big galaxies, tight clusters).
Low $S_8$ : The universe is smoother (galaxies are more spread out).

There is a famous disagreement in science right now. When we look at the very early universe (the "baby picture" taken by the Planck satellite), it predicts a certain clumpiness score. But when we look at the current universe using weak gravitational lensing (the "adult picture"), we often get a lower score. This is called the $S_8$ tension.

The Problem: The "Fuzzy Lens"

To measure the current universe's clumpiness, scientists use a technique called Weak Gravitational Lensing.

The Analogy: Imagine looking at a distant streetlight through a slightly wavy, dirty window. The light looks distorted. By measuring how the light bends, you can figure out how dirty or wavy the window is.
The Reality: Massive objects in space bend the light from distant galaxies. By measuring these tiny distortions, scientists can map the invisible dark matter.

The Catch: The "window" of space isn't perfect. On very small scales (tiny clumps), the physics gets messy and complicated because of things like exploding stars and black holes (astrophysical feedback). These messy effects can trick the scientists into thinking the universe is smoother or clumpier than it really is.

The New Tool: The "BNT Transform"

This paper introduces a new mathematical tool called the Bernardeau-Nishimichi-Taruya (BNT) transform.

The Analogy: Imagine you are listening to a noisy orchestra. You want to hear the cellos, but the violins are too loud and the drums are rattling the floor.
- Old Method: You just turn down the volume on the whole room. You lose some cello sound, but you also lose some violin sound you might have wanted.
- The BNT Method: This is like a super-smart noise-canceling headphone. It specifically targets the "drum noise" (the messy small scales) and cancels it out without silencing the cellos (the useful large scales). It reorganizes the data so that scientists can ignore the messy parts of the universe while keeping the clear, useful parts.

What They Did

The authors applied this new "BNT noise-canceling" tool to data from the KiDS-Legacy survey, which is a massive map of the sky containing millions of galaxies.

They did two main things:

The "Clean" Test: They used the BNT tool to cut out the messy, small-scale data and measured the clumpiness score ( $S_8$ ).
The "Consistency" Test: They checked if the result changed depending on which scales they cut out. If the universe is behaving normally, the answer should be the same whether they look at big scales or medium scales.

The Results: A Tale of Two Math Methods

Here is where it gets interesting. The scientists tried two different ways to calculate the "uncertainty" (the margin of error) in their measurements.

1. The "Theoretical" Way (TDC):
They calculated the uncertainty based on what the standard laws of physics say should happen.

Result: When they used this method, the BNT tool worked perfectly. No matter which scales they looked at, the clumpiness score ( $S_8$ ) stayed consistent. It matched the "adult picture" of the universe well.

2. The "Observational" Way (ODC):
They calculated the uncertainty based on the actual data they collected from the sky.

Result: This is where things got weird. When they used the real data to calculate the error bars, the results started to drift.
- When they looked at large scales (big, smooth areas), the clumpiness score ( $S_8$ ) dropped significantly.
- When they looked at smaller scales, the score went back up.
- This created a "trend" where the bigger the area they looked at, the smoother the universe seemed to be.

The Conclusion: It's Not the Universe, It's the Math

The authors ran computer simulations (mock tests) to see if this weird trend was real.

The Finding: When they fed the computer a "perfect" universe (no real noise), the weird trend disappeared. Both the "Theoretical" and "Observational" math methods agreed.
The Takeaway: The weird trend seen in the real data isn't because the universe is actually changing. It's likely a glitch in the math when you mix the real data with the real-data-based error calculation.

In simple terms:
The paper says, "We built a super-precise tool (BNT) to measure the universe's clumpiness. When we used it, we found that our universe looks consistent and normal. However, we discovered that if you calculate your 'margin of error' using the real data itself, you might get a slightly biased result that makes the universe look smoother on large scales. This isn't a new discovery about the universe; it's a warning sign that tells us how to be more careful with our math."

Why This Matters

This study proves that the BNT tool is a great way to filter out messy physics. More importantly, it acts as a diagnostic tool. It helped the scientists realize that their own method of calculating errors might be introducing a small bias. It's like a mechanic using a new diagnostic scanner to realize that the car isn't broken, but the speedometer was calibrated wrong.

Technical Summary: KiDS-Legacy: The consistency test of the large-scale structure with Bernardeau-Nishimichi-Taruya transform

Problem Statement
The paper addresses the persistent tension in cosmological measurements of the amplitude of matter fluctuations, parameterized as $S_8 = \sigma_8(\Omega_m/0.3)^{0.5}$ . While the $\Lambda$ CDM model, constrained by Cosmic Microwave Background (CMB) data (e.g., Planck), predicts a specific $S_8$ value, weak gravitational lensing surveys often infer lower values. A primary concern in weak lensing is the potential bias introduced by modeling uncertainties in the nonlinear regime of structure formation, such as baryonic feedback. Conventional scale cuts in harmonic space ( $\ell$ -space) are imperfect due to the projection effect, which mixes physical scales ( $k$ -space) into angular scales. The authors aim to test the consistency of the KiDS-Legacy survey data with the $\Lambda$ CDM model while mitigating these nonlinear systematics and diagnosing potential biases in the inference pipeline.

Methodology
The authors perform the first $k$ -cut cosmic shear analysis of the KiDS-Legacy survey using the Bernardeau-Nishimichi-Taruya (BNT) transform.

BNT Transform: This method reorganizes the standard tomographic redshift bins into linear combinations that construct weak-lensing kernels more localized in physical space. When combined with an angular scale filter, the BNT transform effectively acts as a 3-D physical-scale cut in Fourier space, allowing for the removal of information from specific physical scales ( $k$ ) while retaining constraining power from the target range.
Data Vector: The analysis utilizes the pseudo- $C_\ell$ method to estimate angular power spectra from the weighted shear maps of the KiDS-Legacy dataset (the fifth data release). This approach directly handles mode coupling induced by the survey footprint and anisotropic weights, offering a faster alternative to catalogue-based correlation functions, particularly for covariance estimation.
Scale Cuts: The authors define specific $k$ -space cuts (e.g., removing $k \geq 0.33 \text{ Mpc}^{-1}$ ) and map them to corresponding $\ell$ -ranges using dimensionless ratios ( $\hat{R}$ ) to ensure the data vector remains accurate within a defined tolerance.
Covariance Matrices: Two distinct covariance prescriptions are tested:
1. TDC (Theoretical Data-Vector-based Covariance): The Gaussian component of the covariance is computed from the theoretical data vector.
2. ODC (Observational Data-Vector-based Covariance): The Gaussian component is computed from the observed data vector.
Analysis: Bayesian posterior estimations are performed using the Nautilus package, sampling cosmological parameters ( $S_8, \Omega_m$ , etc.) and nuisance parameters (intrinsic alignment, shear bias, redshift errors) against the KiDS-Legacy pseudo- $C_\ell$ data.

Key Results

Consistency with Fiducial Results: When no $k$ -cut is applied, the pseudo- $C_\ell$ analysis recovers $S_8 = 0.798^{+0.045}_{-0.045}$ (using TDC) and $0.795^{+0.030}_{-0.025}$ (using TDC with extended $\ell$ -ranges). These results agree with the fiducial KiDS-Legacy bandpower results and the no- $k$ -cut pseudo- $C_\ell$ posterior within $0.1\sigma$ , indicating no significant bias from nonlinear astrophysical feedback at the current precision of KiDS-Legacy.
Effect of $k$ -Cuts on TDC: Using the TDC covariance, constraints derived from different physical scale selections (e.g., $k < 0.33 \text{ Mpc}^{-1}$ vs. $k \in (0.33, 3.3) \text{ Mpc}^{-1}$ ) are highly consistent with each other (within $1.5\sigma$ ). This suggests the TDC pipeline is internally self-consistent.
Scale-Dependent Trend in ODC: A notable discrepancy arises when using the ODC covariance. In this setup, the $S_8$ constraint exhibits a mild scale-dependent trend where larger physical scales (lower $k$ ) prefer lower $S_8$ values. Specifically, the cut $k < 0.33 \text{ Mpc}^{-1}$ yields $S_8 = 0.717^{+0.047}_{-0.046}$ , while the range $k \in (0.33, 3.3) \text{ Mpc}^{-1}$ yields $S_8 = 0.833^{+0.044}_{-0.044}$ . The maximum deviation between these scales is $1.80\sigma$ .
Mock Tests: Tests using a noiseless theoretical mock data vector show that the ODC and TDC posteriors produce nearly identical results. This indicates that the covariance matrix alone or the data vector alone does not generate the large-scale deviation seen in the real ODC analysis. The trend appears to arise from the interplay between the observational pseudo- $C_\ell$ data vector and the observational-data-vector-based Gaussian covariance matrix.

Significance and Claims
The paper claims that the BNT $k$ -cut method serves a dual purpose:

Mitigation Strategy: It provides a robust method to remove information from nonlinear scales where astrophysical modeling is uncertain, thereby isolating cosmological information from linear scales.
Diagnostic Tool: The method acts as a sensitive diagnostic for weak-lensing inference pipelines. The observed scale-dependent trend in the ODC analysis (which is absent in mock tests and TDC analyses) suggests that using the observed data vector to construct the Gaussian covariance matrix introduces a data-dependent weighting of likelihood residuals. This effect, while not statistically significant enough to claim a departure from $\Lambda$ CDM, highlights a potential systematic sensitivity in the pipeline that warrants attention.

The authors conclude that the KiDS-Legacy results are consistent with the standard model and that the observed $S_8$ tension is not critically affected by nonlinear systematics at the current survey precision. However, the interplay between the data vector and the covariance prescription in the ODC setup reveals a subtle bias that the BNT framework successfully identified.