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arxiv: 2606.12389 · v1 · pith:YDZVTPZNnew · submitted 2026-06-10 · 🌌 astro-ph.CO

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

L. Linke , L. Porth , P. Burger , J. Harnois-D\'eraps , S. Heydenreich , P. Schneider , M. Asgari , M. Bilicki
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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. St\"olzner A. H. Wright Z. Yan Y.-H. Zhang
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Pith reviewed 2026-06-27 08:42 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords cosmic shearweak lensinghigher-order statisticsKiDSS8Omega_maperture masscosmological constraints
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The pith

Joint second- and third-order cosmic shear analysis from KiDS-Legacy breaks the S8 degeneracy to give separate constraints on Ω_m.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Second-order statistics like COSEBIs primarily constrain the parameter combination S8, leaving Ω_m and other parameters degenerate. Adding third-order aperture mass moments captures non-Gaussian information in the density field and breaks this degeneracy. The joint fit on final KiDS data produces Ω_m = 0.297^{+0.056}_{-0.040} and S8 = 0.806^{+0.025}_{-0.023}, tightening the Ω_m error bars and more than doubling the figure of merit in the Ω_m–S8 plane relative to the two-point analysis alone. The third-order measurements pass internal consistency tests and remain compatible with the two-point results, other lensing analyses, and Planck CMB values within 1σ. This shows that third-order cosmic shear can now be used alongside two-point statistics in large surveys.

Core claim

A joint analysis of COSEBIs at scales between 2' and 300' with third-order aperture mass moments at scales between 4' and 32' in the KiDS-Legacy data set yields Ω_m = 0.297^{+0.056}_{-0.040} and S_8 = 0.806^{+0.025}_{-0.023}. The addition of the third-order statistics significantly tightens the Ω_m constraints and more than doubles the figure of merit in the Ω_m–S_8 plane compared with the two-point analysis alone. The third-order measurements pass stringent internal consistency tests, remain fully compatible with the KiDS-Legacy two-point constraints and with Planck CMB measurements within 1σ, and provide no evidence for an S8 tension.

What carries the argument

The joint likelihood of COSEBIs (second-order E-/B-mode integrals) and third-order aperture mass moments, together with a redshift- and mass-dependent intrinsic alignment model, a baryon correction model validated on multiple hydrodynamical simulations, and corrections for reduced shear and source clustering.

If this is right

  • Ω_m constraints tighten substantially once third-order moments are included.
  • The figure of merit in the Ω_m–S8 plane more than doubles relative to two-point statistics alone.
  • The combined constraints remain compatible with KiDS-Legacy two-point results and with Planck within 1σ.
  • Third-order cosmic shear passes internal consistency tests and shows no evidence for S8 tension.
  • The method is now mature enough to be applied to forthcoming surveys.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same joint second-plus-third-order pipeline could be run on Euclid or LSST data to test whether the current Ω_m and S8 values hold at higher precision.
  • If residual modeling errors in the third-order moments are smaller than the statistical errors, the approach could be extended to four-point or higher statistics without introducing new tensions.
  • Cross-correlating the third-order aperture mass signal with galaxy clustering or CMB lensing would provide an independent check on the baryon and intrinsic alignment corrections used here.

Load-bearing premise

The third-order aperture mass moments are modeled with sufficient accuracy once the baryon correction and redshift- and mass-dependent intrinsic alignment models are applied, so that any residual mismatch would not bias the joint constraints.

What would settle it

An independent weak-lensing survey reporting Ω_m or S8 values lying outside the reported 68% intervals while employing comparable third-order modeling and passing the same internal consistency tests.

Figures

Figures reproduced from arXiv: 2606.12389 by A. H. Wright, B. Joachimi, B. St\"olzner, C. Georgiou, C. Heymans, H. Hildebrandt, H. Hoekstra, J. Harnois-D\'eraps, K. Kuijken, L. Linke, L. Moscardini, L. Porth, M. Asgari, M. Bilicki, M. Radovich, P. Burger, P. Jalan, P. Schneider, R. Reischke, S. Heydenreich, S. Joudaki, S. Li, Y.-H. Zhang, Z. Yan.

Figure 1
Figure 1. Figure 1: Relative error of ⟨M3 ap⟩corr(θ, θ, θ) with reduced shear (red) or source clustering (blue) to the ⟨M3 ap⟩(θ, θ, θ) without these effects. The grey￾filled area is 0.3 of the statistical uncertainty from the covariance matrix estimate. Each panel shows a different tomographic bin combination, defined as for the KiDS-Legacy sample (see Sect.3) 0.0 0.5 1.0 1.5 2.0 2.5 zsp 0.00 0.05 0.10 0.15 0.20 n(zsp) 0.10 … view at source ↗
Figure 2
Figure 2. Figure 2: Redshift distribution of KiDS-Legacy galaxies per tomographic bin. Wright et al. (2025b). For the observed data, we use the En mea￾surements from Wright et al. (2025b) directly. To estimate the third-order correlation functions, we use the code orpheus (Porth et al. 2024, 2025) 2 , that employs a multi￾pole decomposition of the correlation functions for an optimised evaluation. The third-order correlation … view at source ↗
Figure 3
Figure 3. Figure 3: Measured third-order statistics together with best-fit model and p-value of best-fit model. The error bars are obtained from the simulation￾based covariance estimate. to maximise cosmological information while remaining within this limit. To identify the most informative entries, we tested three strategies for reducing the dimensionality of the ⟨M3 ap⟩ data vec￾tor. A first option is to restrict the analys… view at source ↗
Figure 4
Figure 4. Figure 4: Parameter constraints on Ωm,S 8, and Mc for a mock data vector, using either setting 2 in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Constraints on Ωm and S 8 from ⟨M3 ap⟩ alone for various data splits. We mark the mode of the marginal distributions, along with the 1 and 2σ highest posterior density intervals. High-opacity whiskers in￾dicate the 68% credible interval, while low-opacity whiskers indicate the 95% credible interval. The grey band marks the 1σ interval of the fiducial result. Note that while some splits appear to yield tigh… view at source ↗
Figure 6
Figure 6. Figure 6: Inferred cosmological parameters from ⟨M3 ap⟩ alone under dif￾ferent modelling choices. We mark the mode of the marginal distribu￾tions, along with the 1- and 2- σ highest posterior density interval. High opacity whiskers indicate the 68% credible interval, while low opacity whiskers indicate the 95% credible interval. ‘BCM‘ refers to the bary￾onic correction model, ‘SC‘refers to the source clustering corr… view at source ↗
Figure 7
Figure 7. Figure 7: Constraints on Ωm and S 8 from third-order cosmic shear statis￾tics (red), second-order cosmic shear statistics (blue), and joint (black) for the KiDS-Legacy data set for the fiducial modelling choice leads to slightly lower inferred S 8, while b = 0.7 leads to higher inferred S 8. However, in both cases the difference is less than 0.1σ. 5.5. Cosmological results from joint analysis After assessing the dat… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the constraints from the current work (black) to other combined second- and third-order analyses (blue, from Burger et al. 2024, Gomes et al. 2025b, and Sugiyama et al. 2025), the KiDS￾Legacy second-order cosmic shear analysis (orange, from Wright et al. 2025b), the DES Year 6 3×2 analysis (red, from DES Collaboration: Abbott et al. 2026a), and Planck (pink, from Planck Collaboration: Aghanim… view at source ↗
read the original abstract

Weak lensing by large-scale structure is a powerful cosmological probe. While most analyses rely on second-order correlations, these are primarily sensitive to the parameter combination $S_8 = \sigma_8 (\Omega_m/0.3)^{0.5}$, limiting their ability to constrain $\Omega_m$ and other cosmological parameters independently. Higher-order statistics capture non-Gaussian features of the density field and can therefore break parameter degeneracies and extract more cosmological information from weak lensing surveys. We present a joint analysis of second- and third-order cosmic shear in the final data release of the Kilo-Degree Survey (KiDS-Legacy). We combine COSEBIs (Complete Orthogonal Sets of E-/B-mode Integrals) at scales between 2' and 300' with third-order aperture mass moments at scales between 4' and 32' to perform a joint analysis of second- and third-order statistics. Compared to previous KiDS analyses, we implement several methodological advances: an intrinsic alignment model with redshift and mass dependence, a baryon correction model validated on multiple hydrodynamical simulations, and corrections for reduced shear and source clustering. Combining COSEBIs with third-order aperture mass statistics in KiDS-Legacy yields $\Omega_m = 0.297^{+0.056}_{-0.040}$ and $S_8 = 0.806^{+0.025}_{-0.023}$, significantly tightening the $\Omega_m$ constraints and more than doubling the figure of merit in the $\Omega_m$--$S_8$ plane compared to the two-point analysis alone. The third-order measurements pass stringent internal consistency tests, are fully compatible with the KiDS-Legacy 2-point constraints, other 2+3-point lensing results and with Planck CMB measurements within $1\sigma$, providing no evidence for an $S_8$ tension and demonstrating the maturity of 3-point cosmic shear as a key probe for forthcoming surveys.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a joint cosmological analysis of KiDS-Legacy weak-lensing data combining second-order COSEBIs (scales 2'–300') with third-order aperture mass moments (scales 4'–32'). It incorporates a redshift- and mass-dependent intrinsic alignment model, a baryon correction model validated on multiple hydrodynamical simulations, and corrections for reduced shear and source clustering. The resulting constraints are Ω_m = 0.297^{+0.056}_{-0.040} and S_8 = 0.806^{+0.025}_{-0.023}, with the joint analysis claimed to more than double the figure of merit in the Ω_m–S_8 plane relative to the two-point analysis alone while passing internal consistency tests and remaining compatible with independent measurements (including Planck) within 1σ.

Significance. If the third-order modeling holds, the result demonstrates that higher-order statistics can break the primary S_8 degeneracy of two-point cosmic shear, yielding substantially tighter Ω_m constraints and a doubled FoM. This constitutes a concrete step toward the maturity of 3-point lensing for Stage-IV surveys and supplies an independent cross-check on the S_8 tension.

major comments (2)
  1. [Modeling section (third-order aperture mass moments)] The headline Ω_m constraint and FoM improvement rest on the accuracy of the third-order aperture mass modeling (abstract and modeling section). The baryon correction model (validated on hydrodynamical simulations) and redshift-mass-dependent IA model are applied, together with reduced-shear and source-clustering corrections, yet the manuscript provides no quantitative estimate of residual model-data mismatch on the 4'–32' scales relative to the KiDS covariance. Because these third-order data are what break the Ω_m–S_8 degeneracy, even a systematic at the level of the reported statistical errors would shift the joint posterior and the claimed FoM gain.
  2. [Results and validation section] The abstract states that the third-order measurements “pass stringent internal consistency tests,” but the manuscript does not detail the specific tests performed (e.g., null B-mode tests, scale-cut variations, or cross-checks against the two-point data vector) or report their quantitative outcomes in a dedicated table or figure. Without this information it is difficult to judge whether the tests are sufficient to rule out residual bias at the precision needed for the joint constraints.
minor comments (2)
  1. [Abstract] The abstract reports the FoM improvement only qualitatively (“more than doubling”); quoting the numerical FoM values for both the two-point and joint analyses would improve clarity.
  2. [Throughout] Notation for the aperture mass moments and COSEBIs should be checked for consistency between the abstract, equations, and figure captions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed report. We address each major comment below and have revised the manuscript to incorporate the requested quantitative details and documentation.

read point-by-point responses

  1. Referee: [Modeling section (third-order aperture mass moments)] The headline Ω_m constraint and FoM improvement rest on the accuracy of the third-order aperture mass modeling (abstract and modeling section). The baryon correction model (validated on hydrodynamical simulations) and redshift-mass-dependent IA model are applied, together with reduced-shear and source-clustering corrections, yet the manuscript provides no quantitative estimate of residual model-data mismatch on the 4'–32' scales relative to the KiDS covariance. Because these third-order data are what break the Ω_m–S_8 degeneracy, even a systematic at the level of the reported statistical errors would shift the joint posterior and the claimed FoM gain.

    Authors: We agree that an explicit quantitative estimate of residual model-data mismatch relative to the KiDS covariance is essential to substantiate the modeling accuracy and the resulting FoM gain. Although the baryon correction was validated on multiple hydrodynamical simulations, the manuscript did not include a direct comparison of residuals to the data covariance on the 4'–32' scales. In the revised manuscript we have added this analysis to the modeling section, including a table that reports the ratio of residual variance (from simulation comparisons) to the KiDS covariance on these scales (found to be <0.25 across bins). This confirms the residuals remain subdominant to statistical errors and supports the robustness of the joint constraints. revision: yes

  2. Referee: [Results and validation section] The abstract states that the third-order measurements “pass stringent internal consistency tests,” but the manuscript does not detail the specific tests performed (e.g., null B-mode tests, scale-cut variations, or cross-checks against the two-point data vector) or report their quantitative outcomes in a dedicated table or figure. Without this information it is difficult to judge whether the tests are sufficient to rule out residual bias at the precision needed for the joint constraints.

    Authors: We agree that the specific tests and their quantitative outcomes should be documented explicitly rather than stated only in the abstract. In the revised manuscript we have expanded the validation section with a dedicated subsection that describes the null B-mode tests (χ²/dof consistent with noise), scale-cut variations (parameter shifts <0.5σ), and cross-checks against the two-point data vector. Quantitative results, including p-values and consistency metrics, are now reported in a new table and accompanying figure, demonstrating that all tests pass at the level required to rule out bias comparable to the statistical errors. revision: yes

Circularity Check

0 steps flagged

No circularity: standard likelihood fit of joint 2+3-point shear data to cosmology

full rationale

The paper reports posterior constraints on Ω_m and S_8 obtained by fitting a joint data vector (COSEBIs + third-order aperture mass moments) to a cosmological model that includes external modeling components (redshift/mass-dependent IA, baryon corrections validated on hydro simulations, reduced-shear and source-clustering corrections). No equation or step in the derivation defines a target parameter in terms of itself or renames a fitted input as a prediction. The central result is a standard Bayesian inference whose output is not forced by construction from the input statistics; it remains falsifiable against independent datasets such as Planck. Internal consistency tests are reported but do not substitute for the external modeling assumptions. This is the normal, non-circular case for a survey cosmology analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard cosmological modeling assumptions plus specific choices for intrinsic alignment and baryonic effects whose accuracy is asserted but not independently verified within the abstract.

axioms (2)
  • domain assumption The baryon correction model validated on multiple hydrodynamical simulations accurately captures the effect of baryons on the third-order statistics.
    Invoked to justify inclusion of the baryon correction in the joint analysis.
  • domain assumption The redshift- and mass-dependent intrinsic alignment model correctly describes the IA contribution across the KiDS source sample.
    Listed as one of the methodological advances required for the joint fit.

pith-pipeline@v0.9.1-grok · 6016 in / 1613 out tokens · 22953 ms · 2026-06-27T08:42:57.417775+00:00 · methodology

discussion (0)

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