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KiDS-Legacy: Joint analysis of second- and third-order cosmic shear

The KiDS-Legacy collaboration presents a joint analysis of second- and third-order cosmic shear statistics using the final Kilo-Degree Survey data, which significantly tightens constraints on the matter density $\Omega_m$ and the parameter $S_8$ while demonstrating the maturity of three-point statistics as a robust cosmological probe consistent with Planck CMB measurements.

Original authors: L. Linke, L. Porth, P. Burger, J. Harnois-Déraps, S. Heydenreich, P. Schneider, M. Asgari, M. Bilicki, C. Georgiou, C. Heymans, H. Hildebrandt, H. Hoekstra, P. Jalan, B. Joachimi, S. Joudaki, K. Kuijk

Published 2026-06-11

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CC BY 4.0

Original authors: L. Linke, L. Porth, P. Burger, J. Harnois-Déraps, S. Heydenreich, P. Schneider, M. Asgari, M. Bilicki, C. Georgiou, C. Heymans, H. Hildebrandt, H. Hoekstra, P. Jalan, B. Joachimi, S. Joudaki, K. Kuijken, S. Li, L. Moscardini, M. Radovich, R. Reischke, B. Stölzner, A. H. Wright, Z. Yan, Y. -H. Zhang

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: Mapping the Invisible Universe

Imagine the universe is a giant, invisible ocean. We can't see the water itself, but we can see how floating leaves (galaxies) drift and twist as they move through currents. In astronomy, this "twisting" is called cosmic shear. It happens because massive structures (like dark matter) bend the light from distant galaxies, slightly distorting their shapes.

For a long time, scientists have studied these distortions using second-order statistics. Think of this like measuring the average speed of the leaves. It tells you a lot about the general flow, but it only gives you a blurry picture of two specific things mixed together: how much "stuff" (matter) is in the universe and how "clumpy" that stuff is. It's like trying to guess the weight of a suitcase and the density of its contents just by looking at how fast it slides down a ramp; you can't easily separate the two.

The New Approach: Listening to the "Crunch"

This paper introduces a new way to look at the data by using third-order statistics. If second-order statistics measure the average speed, third-order statistics measure the skewness or the "lumpiness" of the flow.

Imagine a crowd of people walking through a hallway.

Second-order: You measure the average distance between people.
Third-order: You measure how often three people bump into each other at the exact same time.

The authors argue that while the average distance tells you about the crowd's density, the "three-way bumps" (non-Gaussian features) happen because of complex interactions that the average distance misses. By listening to these "bumps," they can untangle the mix of matter and clumpiness, giving a much sharper picture of the universe.

The Toolkit: The KiDS-Legacy Survey

The team used data from the KiDS-Legacy survey, which is like a massive, high-resolution map of the sky covering a huge area. They didn't just look at the "average speed" (two-point data); they combined it with the "three-way bumps" (three-point data).

To do this, they built a sophisticated neural network emulator. Think of this as a super-smart video game AI. Instead of running a slow, real-time physics simulation every time they wanted to test a theory (which would take years), they trained this AI to instantly predict what the universe should look like under different rules. This allowed them to test millions of possibilities quickly.

The Challenges: Noise and Distortions

Measuring these subtle "bumps" is incredibly hard because the universe is messy. The paper had to account for several "noise" factors:

Intrinsic Alignments: Sometimes galaxies aren't twisted by gravity; they just naturally grow in the same direction, like trees in a forest growing toward the sun. The team had to build a model to tell the difference between a "gravity twist" and a "natural growth twist."
Baryonic Feedback: This is like the "wind" in our ocean analogy. Normal matter (gas and stars) explodes and pushes around the dark matter, changing the shape of the currents. The team used a new, flexible model to account for this wind so it didn't mess up their measurements.
Blinding: To ensure they didn't accidentally "cheat" by looking at the answer before finishing the math, they used a strict "blinding" procedure. They hid the true answer behind a digital lock, did all their work, and only unlocked the answer at the very end.

The Results: A Clearer, Sharper Image

When they combined the "average speed" (second-order) with the "three-way bumps" (third-order), the results were impressive:

Breaking the Lock: They successfully separated the "amount of stuff" from the "clumpiness." This is a big deal because previous methods struggled to do this.
Tighter Constraints: Their measurements became significantly more precise. The uncertainty on the amount of matter in the universe shrank by about 36%, and the uncertainty on how clumpy it is shrank by 24%.
No "Tension": There has been a long-standing mystery called the " $S_8$ tension," where measurements of the local universe disagreed with measurements of the early universe (from the Planck satellite). This new analysis found that there is no tension. The KiDS-Legacy results now agree perfectly with the Planck satellite data. The "missing piece" of the puzzle was likely the extra information provided by the third-order statistics.

The Conclusion

The paper concludes that looking at the universe through this "third-order" lens is a mature and powerful technique. It proves that we can extract much more information from the same data by listening to the complex interactions of the cosmic web, not just the simple averages. This method is ready to be used by future, even bigger telescopes (like Euclid and LSST) to map the universe with unprecedented clarity.

In short: By adding a new layer of complexity to their analysis, the team turned a blurry, mixed-up picture of the universe into a sharp, high-definition image that finally agrees with our best theories of how the cosmos began.

Technical Summary: KiDS-Legacy Joint Analysis of Second- and Third-Order Cosmic Shear

Problem and Context
Weak gravitational lensing by large-scale structure is a primary probe for cosmological parameters. However, standard analyses relying on second-order statistics (two-point correlation functions) are predominantly sensitive to the parameter combination $S_8 = \sigma_8(\Omega_m/0.3)^{0.5}$ . This sensitivity creates a degeneracy between the matter density parameter ( $\Omega_m$ ) and the clustering amplitude ( $\sigma_8$ ), limiting the ability to constrain them independently. Furthermore, a persistent discrepancy known as the " $S_8$ tension" exists between late-time weak lensing measurements (which often yield lower $S_8$ values) and early-time Cosmic Microwave Background (CMB) observations from Planck. While higher-order statistics (HOS) can capture non-Gaussian features of the density field to break these degeneracies, their application has been limited by computational complexity and modelling challenges. This paper addresses the need to test whether incorporating non-Gaussian information via third-order statistics improves cosmological constraints and resolves the $S_8$ tension within the final data release of the Kilo-Degree Survey (KiDS-Legacy).

Methodology
The authors perform a joint analysis of second- and third-order cosmic shear using the KiDS-Legacy catalogue, which covers 967.4 deg² with an effective galaxy number density of $n_{eff} = 8.79$ arcmin $^{-2}$ .

Statistics:
- Second-order: Complete Orthogonal Sets of E-/B-mode Integrals (COSEBIs) are used on scales between 2′ and 300′.
- Third-order: Third-order aperture mass moments ( $\langle M_{ap}^3 \rangle$ ) are utilized on scales between 4′ and 32′. The analysis includes both equal-scale and unequal-scale configurations.
Modelling Advances: Compared to previous KiDS analyses (e.g., Burger et al. 2024), this work implements several methodological improvements:
- Intrinsic Alignments (IA): An IA model with explicit dependencies on redshift and stellar mass (NLA-M model) is adopted, moving beyond scale-independent or purely redshift-dependent models.
- Baryonic Feedback: A new baryon correction model (Burger et al. 2026) is employed, validated against hydrodynamical FLAMINGO simulations. This model interpolates between multiple feedback scenarios and focuses on two key parameters: hot gas mass ( $M_c$ ) and ejected gas density ( $\eta$ ).
- Higher-Order Corrections: Corrections for reduced shear and source clustering are applied using simulated shear and convergence maps.
- Emulation: To accelerate inference, neural-network emulators are developed. Specifically, a redshift-resolved emulator for $\langle M_{ap}^3 \rangle_z$ is trained, decoupling the source redshift distribution from the emulator and allowing flexible application to arbitrary surveys. Separate emulators are trained for COSEBI modes ( $E_n$ ) tailored to specific IA models.
Inference:
- A Bayesian framework is used with a likelihood formulation (Percival et al. 2014) that accounts for the noise in the covariance matrix estimated from 1,944 Takahashi simulations.
- Hamiltonian Monte Carlo (NUTS) is employed for efficient posterior sampling.
- A strict blinding procedure was implemented to prevent confirmation bias during the analysis of the third-order data.
- The data vector dimensionality was optimized (reducing to 291 entries) to balance cosmological information gain against the limitations of the simulation-based covariance estimate.

Key Results

Cosmological Constraints: The joint analysis yields $\Omega_m = 0.297^{+0.056}_{-0.040}$ and $S_8 = 0.806^{+0.025}_{-0.023}$ .
Improvement in Constraints: Combining third-order statistics with second-order COSEBIs significantly tightens the constraints on $\Omega_m$ (reducing the marginal error by ~36%) and improves constraints on $S_8$ by ~24% compared to the two-point analysis alone.
Figure of Merit (FoM): The FoM in the $\Omega_m$ – $S_8$ plane more than doubles, increasing from 429 (two-point only) to 895 (joint analysis). This demonstrates that third-order statistics break the $\Omega_m$ – $\sigma_8$ degeneracy effectively.
Consistency:
- The third-order measurements pass stringent internal consistency tests (Tier 2 and Tier 3) across various data splits (North/South, scale cuts, tomographic bins), showing no significant deviations.
- The joint constraints are fully compatible with the KiDS-Legacy two-point results and consistent with Planck CMB measurements within $1\sigma$ .
- The results are consistent with other combined 2+3-point analyses from KiDS-1000, DES, and HSC, though the KiDS-Legacy results favor a slightly higher $S_8$ value, aligning better with Planck.
Modelling Sensitivity: The analysis confirms that neglecting intrinsic alignments or baryonic feedback leads to biased low values for both $\Omega_m$ and $S_8$ . However, the choice of specific IA model (NLA, NLA-z, NLA-M) does not significantly bias the final cosmological constraints within $1\sigma$ .

Significance and Claims
The paper claims that the KiDS-Legacy joint analysis delivers competitive cosmological constraints that are consistent with other weak lensing results and Planck, providing no evidence for an $S_8$ tension within the adopted modelling framework. The work demonstrates the maturity of third-order cosmic shear as a robust probe, capable of extracting non-Gaussian information to break parameter degeneracies that limit two-point analyses.

The authors emphasize that the inclusion of unequal aperture scales significantly enhances the constraining power of third-order statistics, particularly for baryonic feedback parameters and $\Omega_m$ . Furthermore, the development of redshift-resolved emulators and flexible baryon correction models establishes a pipeline ready for application to upcoming Stage-IV surveys such as Euclid and LSST. The study concludes that robust joint modelling of second- and third-order statistics, including corrections for systematics like reduced shear and source clustering, is now feasible and essential for maximizing the cosmological potential of current and future weak lensing surveys.