Weak-Lensing Gravity Statistics

Weak lensing tests gravity statistically: correlated galaxy shapes measure the projected Weyl potential, while galaxy density, redshift-space motion, and cosmic-microwave-background lensing supply complementary source and growth information. No shear catalog alone separates gravity from intrinsic alignments, redshift error, baryonic structure, and image response. The durable result is a forward-modeling strategy, not an undated preference among gravity models. The evidence assessment here is current through 2026-08-19.

Required background. Geodesics and Jacobi Fields supplies null-bundle convergence and shear; Probability and Statistical Inference supplies normalized likelihoods, covariance, and nuisance marginalization.

Helpful background. Weak-Field Gravity supplies the two metric potentials; Relativistic Cosmology supplies distances, structure growth, and tomographic projection.

Shear as a projected gravity observable

Convergence $\kappa$ changes a small image’s angular size, while the two shear components $\gamma_1,\gamma_2$ stretch it in different directions. The local source-to-image map is described by

$$\frac{\partial\boldsymbol\beta}{\partial\boldsymbol\theta} =\begin{pmatrix} 1-\kappa-\gamma_1&-\gamma_2\\ -\gamma_2&1-\kappa+\gamma_1 \end{pmatrix}.$$

A single galaxy has an unknown intrinsic shape. Averaging many galaxies with calibrated responses estimates the coherent distortion. Shapes respond to the reduced shear $g=\gamma/(1-\kappa)$; identifying $g\simeq\gamma$ requires the weak regime. Tomography repeats the comparison in source-redshift bins, so different foreground distances contribute with different weights. The two-potential calculation explains why this distortion probes $\Phi+\Psi$.

For source-redshift bin $i$, the convergence power spectrum has the schematic Limber form

$$C_\ell^{\kappa_i\kappa_j} =\int d\chi\, \frac{W_i(\chi)W_j(\chi)}{f_K^2(\chi)} \Sigma^2(k,z)P_{\delta\delta}(k,z), \qquad k=\frac{\ell+1/2}{f_K(\chi)}.$$

Here $\ell$ labels angular multipoles, $f_K$ is the comoving angular-diameter function, and $P_{\delta\delta}$ is the three-dimensional matter-density power spectrum, with dimensions of comoving volume. In a flat background a useful explicit normalization is

$$W_i(\chi)=\frac{3H_0^2\Omega_{m0}}{2} \frac{\chi}{a(\chi)} \int_\chi^\infty d\chi_s\,n_i(\chi_s) \frac{\chi_s-\chi}{\chi_s}, \qquad \int_0^\infty n_i(\chi_s)d\chi_s=1.$$

$H_0$ and $\Omega_{m0}$ are today’s Hubble rate and matter-density fraction; the source distribution $n_i$ is per unit comoving distance. In $c=1$ units $W_i$ has inverse-length dimensions, making $C_\ell$ dimensionless. The separate factor $\Sigma$ maps matter to the Weyl potential; it must not also be hidden inside $W_i$. Gravity changes both $\Sigma$ and growth in $P_{\delta\delta}$. The formula uses the Born and Limber approximations, neglects lens–lens coupling, and needs broad enough radial kernels and sufficiently large $\ell$. Extending a linear $\Sigma(k,z)$ to nonlinear wavenumbers requires a validated prescription or scale cuts. The projection and statistical formalism are developed in Bartelmann and Schneider 2001, §§6.3–6.6.

An image pipeline returns an ellipticity estimator, approximately

$$\widehat{\gamma}=(1+m)\gamma+c+e_{\rm int},$$

with multiplicative response $m$, additive bias $c$, and intrinsic shape $e_{\rm int}$. Tomographic two-point functions $\xi_\pm^{ij}(\theta)$ add intrinsic-alignment terms and may receive baryonic and reduced-shear corrections. Galaxy–galaxy lensing and angular clustering add a galaxy-bias model; cross-correlation with CMB lensing adds a higher-redshift kernel and a separately calibrated map. The useful gravity information comes from their different kernels and nuisance dependences, not simply from adding signal-to-noise.

Release, selection, and likelihood

A reproducible example is the Dark Energy Survey Year 3 release: the shape catalog, redshift distributions, masks, response quantities, covariance products, and scale cuts must be identified together. The METACALIBRATION catalog determines ensemble shear response by artificially shearing images, with object and redshift selections propagated into the response Gatti et al. 2021, §§2–6. The official Y3A2 products, rather than an unversioned plot, define the data boundary.

Selections include footprint and bright-star masks, signal-to-noise and size requirements, successful shape measurement, tomographic assignment, lens-sample quality, and angular scale cuts. Completeness is therefore a function of position, observing conditions, morphology, and redshift. Calibration parameters shared by bins must remain correlated. Photometric-redshift shifts, intrinsic-alignment amplitudes, galaxy bias, baryonic feedback, magnification, blending, point-spread leakage, and covariance uncertainty enter as nuisance parameters.

For data vector $\boldsymbol d$ and prediction $\boldsymbol t(\vartheta,\eta)$, a common normalized approximation is

$$\mathcal L= \frac{\exp\!\left[-\tfrac12 (\boldsymbol d-\boldsymbol t)^{T}C^{-1} (\boldsymbol d-\boldsymbol t)\right]} {(2\pi)^{N/2}\det(C)^{1/2}}.$$

The gravity parameters $\vartheta$ and nuisances $\eta$ require declared priors. If $C$ is estimated from simulations or depends materially on parameters, treating it as fixed and exact understates uncertainty. Posterior predictive residuals, sampler convergence, coverage on end-to-end mocks, catalog splits, B modes, and alternative scale cuts are part of the result. The DES three-probe construction and its validation structure are documented in DES Collaboration 2022, §§II–VI and Appendices A–C.

Results and limitations through 2026-08-19

Evidence or position	Gravity information retained	Limitation or scope
Tomographic cosmic shear	Combines Weyl-potential projection with scale- and redshift-dependent growth	Intrinsic alignment, source redshifts, shear response, baryons, and amplitude degeneracies remain
Galaxy clustering plus galaxy–galaxy lensing	Uses a common lens population to constrain bias and matter–light relations	Galaxy bias, magnification, lens selection, and nonlinear modeling can imitate gravity dependence
CMB-lensing cross-correlations and null splits	Extend the redshift lever arm and test survey-specific contamination	CMB-map response, foregrounds, shared sky covariance, and kernel broadness limit interpretation

The conservative position is that current combinations test conditional growth-and-lensing models and show no source-independent metric anomaly. A competing interpretation may assign a cross-probe mismatch to scale-dependent gravity, but it must outperform flexible intrinsic-alignment, baryon, redshift, and calibration alternatives on withheld data. A null result constrains only the chosen parameterization and its scale cuts.

The adversarial check is a blinded recovery of injected gravity parameters through the full image, selection, redshift, and summary-statistic pipeline. It should pass B-mode and observing-condition splits, retain frequentist coverage or calibrated posterior coverage, and remain stable when intrinsic-alignment and baryonic families are changed. Failure identifies a pipeline or model limitation before it becomes a gravity claim. The conclusion remains model-dependent because no finite nuisance family proves that all relevant astrophysics has been spanned.

What could change the conclusion. The interpretation could change after a superseding shear, redshift, CMB-lensing, calibration, covariance, or simulation release, or when an analysis demonstrates that the blinded injection and alternate-nuisance tests materially change the gravity-sensitive combinations.

Exercise

Suppose one tomographic shear amplitude scales as $A_{\rm obs}=(1+m)^2\Sigma^2A_m$, where $A_m$ is the matter-amplitude prediction. Explain why a measured excess does not by itself determine $\Sigma$.

Solution

Taking logarithms gives $\delta\ln A_{\rm obs}\simeq2\,\delta m+2\,\delta\ln\Sigma+\delta\ln A_m$. Therefore a shear-response error, altered structure growth, or a change in the primordial matter amplitude can reproduce the same excess. Cross-correlations help because their response and kernels differ, but they add bias and redshift nuisances. A gravity conclusion follows only after these shared parameters are fitted with their covariance and the model predicts held-out bins or scales.

For example, a $2\%$ amplitude excess with fixed $A_m$ is approximately reproduced either by $m=0.01$ or by $\Sigma=1.01$. These alternatives have different physical meanings but indistinguishable effects on this one amplitude to first order. The expansion is around $m=0$; at finite $m$ its first term is $2\,\delta m/(1+m)$.

References

• Bartelmann, M., and Schneider, P. (2001), “Weak Gravitational Lensing,” Physics Reports 340, 291–472. DOI.
• Dark Energy Survey Collaboration (2022), “Dark Energy Survey Year 3 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing,” Physical Review D 105, 023520. DOI.
• Gatti, M., et al. (2021), “Dark Energy Survey Year 3 Results: Weak Lensing Shape Catalogue,” Monthly Notices of the Royal Astronomical Society 504, 4312–4336. DOI.