Weak Lensing and Magnification

Cosmological weak lensing infers a line-of-sight tidal field from coherent distortions of many galaxy images. The signal is geometric, but the inference is statistical: galaxies have unknown intrinsic shapes, redshifts are probabilistic, the point-spread function changes across the focal plane, and nonlinear structure, intrinsic alignments, and baryons enter the prediction. This page connects convergence and shear to a tomographic two-point likelihood and then makes the calibration and reproducibility boundary explicit for a current public release.

Required background. Geodesics and Jacobi Fields supplies the screen Jacobi map whose trace and trace-free parts define convergence and shear.

Helpful background. Einstein Equations relates lensing potentials to matter, Relativistic Matter and Compact Objects supplies source-frame shape concepts, Relativistic Cosmology supplies distance and power-spectrum evolution, and Probability and Statistical Inference supplies nuisance marginalization and posterior checks.

Convergence, shear, and magnification

For a weakly perturbed ray in a small sky patch, let $\boldsymbol\theta$ be the observed image direction and $\boldsymbol\beta$ the unlensed source direction. The image-to-source Jacobian, $\mathcal A_{ab}=\partial\beta_a/\partial\theta_b$, is

$$\mathcal A= \begin{pmatrix} 1-\kappa-\gamma_1 & -\gamma_2\\ -\gamma_2 & 1-\kappa+\gamma_1 \end{pmatrix}.$$

Convergence $\kappa$ changes area, while complex shear $\gamma=\gamma_1+i\gamma_2$ changes shape. Image ellipticities respond to the reduced shear and magnification,

$$g=\frac{\gamma}{1-\kappa}, \qquad \mu=\frac{1}{(1-\kappa)^2-\lvert\gamma\rvert^2},$$

so $g\simeq\gamma$ and $\mu\simeq1+2\kappa$ only in the weak limit. Shape alone cannot distinguish $\gamma$ from the isotropic factor $1-\kappa$; magnification adds information through sizes, fluxes, or number counts but brings a selection model. The reduced-shear transformation and its mass-sheet degeneracy are derived in Bartelmann and Schneider 2001, § 4.2, printed pp. 58–60, eqs. (4.5)–(4.9).

For sources in tomographic bin $i$, let $n_i(\chi)$ be a distribution per unit comoving distance with $\int n_i(\chi)d\chi=1$. Write $f_K(\chi)=S_K(\chi)$ for the transverse comoving distance. Using the general-relativistic matter Poisson relation on subhorizon scales, the Born- and Limber-level convergence power is

$$C_\ell^{ij} =\int_0^{\chi_H}d\chi\, \frac{W_i(\chi)W_j(\chi)}{f_K^2(\chi)} P_{\mathrm m}\!\left(\frac{\ell+1/2}{f_K(\chi)},z(\chi)\right),$$

with

$$W_i(\chi)=\frac{3H_0^2\Omega_{\mathrm m}}{2a(\chi)}f_K(\chi) \int_\chi^{\chi_H}d\chi'\,n_i(\chi') \frac{f_K(\chi'-\chi)}{f_K(\chi')}.$$

This also gives the shear $E$-mode power in the flat-sky limit; the exact full-sky conversion is $C_\ell^{EE}=[(\ell+2)(\ell-1)/(\ell(\ell+1))]C_\ell^{\kappa\kappa}$ for $\ell\geq2$. The Limber approximation requires sufficiently high multipoles and kernels that vary slowly over a radial wavelength. The line-of-sight density kernel and its source-distribution average follow Bartelmann and Schneider 2001, §§ 6.3–6.4, printed pp. 121–124, eqs. (6.14)–(6.22), PDF. With $W_i$ of dimension inverse length and $P_m$ of dimension volume, the displayed spectrum is dimensionless. Intrinsic-alignment power must be added to the gravitational $GG$ term; it is not detector noise.

From pixels to a tomographic statistic

A schematic first-order measurement model for calibrated ellipticity component $a$ is

$$\epsilon_a^{\mathrm{obs}} =(1+m_a)(\epsilon_a^{\mathrm{int}}+\gamma_a) +\alpha_a\epsilon_a^{\mathrm{PSF}}+c_a.$$

This model assumes that the estimator has already been normalized to unit ensemble shear response; the exact transformation of an individual galaxy’s ellipticity is nonlinear. The residual multiplicative response $m_a$ is calibrated with image simulations; additive $c_a$ and PSF leakage $\alpha_a$ are constrained using stars, null correlations, and survey-property splits. Random intrinsic orientations average toward zero, but their dispersion produces shape noise. Correlated intrinsic alignments do not average away and require a physical nuisance model.

Two common real-space summaries are

$$\xi_\pm^{ij}(\theta) =\left\langle\gamma_t^i\gamma_t^j\right\rangle \pm\left\langle\gamma_\times^i\gamma_\times^j\right\rangle,$$

with $E/B$ combinations or band powers used to control angular response. Detection and shape-quality cuts, blending flags, a sky mask, tomographic boundaries, and angular scale cuts are part of the estimator. Changing them changes both the selected population and its shear response; a catalog row is not an unbiased shear sample merely because it contains an ellipticity.

A versioned observational example

Scientific evidence in this example is assessed through 2026-08-19. The versioned Wright et al. 2025 analysis reports 40,894,394 sources in six photometric-redshift bins spanning $0.10\lt z_B\leq2.00$, a 967.4-square-degree effective lensing area, and the 1,347-square-degree parent imaging footprint.

The mutable KiDS-Legacy public product page, inspected 2026-08-21 was checked after that cutoff only to document the current retrieval path. At that inspection it listed a 125 MB high-level package containing data vectors, covariance matrices, redshift distributions, Nautilus chains, and CosmoSIS configurations; the catalog, masks, and DR5 photometry were separate products. These post-cutoff portal observations do not revise the scientific evidence assessment or establish the portal state on 2026-08-19.

The release-specific chain is:

• Calibration: lensfit shapes are calibrated with multi-band SKiLLS image simulations; each bin carries multiplicative response uncertainty. PSF leakage and additive terms are tested empirically. Self-organizing-map redshift distributions are checked with clustering redshifts, and correlated redshift-bias parameters are retained.
• Selections: the gold sample requires reliable shape and redshift information, uses six declared tomographic bins and survey masks, and applies the published statistic and scale response. The fiducial analysis uses angular support from 2 to 300 arcminutes; alternative scale and bin removals are validation tests, not interchangeable data vectors.
• Statistical model: a Gaussian likelihood compares COSEBI or band-power vectors with the full covariance. Cosmological parameters, baryonic feedback, intrinsic alignment, correlated redshift shifts, and shear calibration enter with the published priors; the covariance includes Gaussian, connected non-Gaussian, super-sample, and multiplicative-calibration pieces.

These ingredients, including catalog properties in Table 1, covariance construction, null tests, and the fiducial likelihood and priors, are documented in Wright et al. 2025, §§ 2.1–2.6, 3.1–3.5, 4–5 and Table 3. The quoted credible intervals are posterior statements under that model; shape noise and finite-volume covariance are statistical, whereas shear response, PSF leakage, redshift calibration, intrinsic alignments, baryonic feedback, nonlinear power, reduced shear, magnification, and selection response are systematic or model-dependent contributions.

Likelihood and posterior checks

For released vector $\boldsymbol d$, theory $\boldsymbol t(\vartheta,\nu)$, covariance $\mathbf C$, cosmological parameters $\vartheta$, and nuisance parameters $\nu$,

$$-2\ln\mathcal L =(\boldsymbol d-\boldsymbol t)^{\mathsf T} \mathbf C^{-1}(\boldsymbol d-\boldsymbol t) +\ln\det\mathbf C+\text{constant}.$$

A stable posterior is not enough. One should whiten residuals, inspect $E/B$ nulls and survey-property splits, repeat conservative angular cuts, vary intrinsic-alignment and baryon models, shift redshift and shear calibrations within their correlated priors, and perform posterior-predictive checks on each tomographic pair. Holding a nuisance parameter fixed because its posterior is broad understates model uncertainty.

Spatial averages on a constant-time slice do not directly equal weak-lensing observables. The inference uses null propagation, a source redshift distribution, image selection, and a detector response; an averaged density must pass through that full forward map.

Reproduction path and present boundary

No KiDS-Legacy product tarball, shape catalog, or pinned scientific runtime is stored in this repository, and no likelihood or headline constraint was recomputed for this page. The public CosmoPipe repository was inspected at snapshot 37680f3695cedb9d09f6fe30e748a1044e114a14 dated 2026-08-03, but its moving master branch is not asserted to be the exact code state that generated the published chains.

A defensible reproduction would download the official high-level package and catalog, record checksums and vector ordering, verify $n_i(z)$ normalization and mask/scale metadata, instantiate the release configuration in a pinned environment, recompute at least one published two-point or COSEBI component, and evaluate one likelihood point with the full covariance. It would then repeat conservative scales and alternate alignment, baryon, redshift, and shear-calibration choices. Until those steps are executed, the public release supplies a reproducible route, not a reproduced result here.

Inference diagnostic

Suppose multiplying every galaxy ellipticity by $1.02$ changes the measured shear vector by two percent. May the analyst correct the vector but leave the covariance and theory response unchanged?

Solution

No. A coherent multiplicative response changes each shear factor and therefore a two-point signal by approximately four percent at first order. The transformation must be applied consistently to the estimator, its calibration nuisance model, and any covariance contribution that depends on signal or response. If the response uncertainty is marginalized in theory space, manually rescaling the data as well would double count the correction. The correct choice depends on the release convention and must be verified from its configuration and metadata.

References

• Bartelmann, Matthias, and Peter Schneider. 2001. “Weak Gravitational Lensing.” Physics Reports 340, 291–472. DOI. Open PDF.
• KiDS Collaboration. n.d. “KiDS Science Data Products: KiDS-Legacy Cosmic Shear.” Official data portal, accessed 2026-08-21.
• Wright, Angus H., et al. 2025. “KiDS-Legacy: Cosmological Constraints from Cosmic Shear with the Complete Kilo-Degree Survey.” Astronomy & Astrophysics 703, A158. DOI.