Lensing, Clustering, and Gravitational Slip

Nonrelativistic motion responds mainly to the Newtonian potential $\Psi$, while weak lensing responds to the Weyl combination $\Phi+\Psi$. Comparing clustering, velocities, and lensing can therefore test gravitational slip and the effective coupling to matter. The cancellation of galaxy bias in an ideal ratio is only approximate: redshift kernels, stochastic or scale-dependent bias, magnification, shear and redshift calibration, intrinsic alignment, nonlinear physics, and cross-survey covariance determine what is actually measured.

Required background. Probability and Statistical Inference supplies joint likelihoods, covariance, nuisance marginalization, and calibrated comparisons between correlated probes.

Helpful background. Weak-Field Gravity supplies the scalar-potential and gauge conventions; Relativistic Cosmology supplies lensing kernels and structure transfer functions; and Tests of Gravity supplies the operational distinction between a phenomenological coordinate and a theory.

Two potentials and two effective couplings

Use conformal Newtonian gauge, $a_0=1$, conformal time $\tau$, and comoving spatial Fourier modes $e^{i\mathbf k\cdot\mathbf x}$:

$$ds^2=a^2(\tau)\left[-(1+2\Psi)d\tau^2 +(1-2\Phi)d\mathbf x^2\right],$$

For a late-time linear matter source, define

$$-k^2\Psi=4\pi G a^2\mu(a,k)\rho_m\Delta_m, \qquad \eta(a,k)=\frac{\Phi}{\Psi}.$$

Light deflection is conveniently parameterized by

$$-k^2(\Phi+\Psi) =8\pi G a^2\Sigma(a,k)\rho_m\Delta_m, \qquad \Sigma=\frac{\mu(1+\eta)}{2}.$$

Here $\rho_m$ is the background matter density and $\Delta_m$ its comoving density contrast. The symbol $\eta$ denotes slip, not conformal time. GR gives $\mu=\eta=\Sigma=1$ when the neglected species’ perturbations and anisotropic stress are negligible. Relativistic free-streaming species can produce slip even in GR. These definitions are convention-dependent: some sources invert the ratio or attach the modified coupling to $\Phi$. Ishak’s Eqs. (86)–(91) present both $Q,R$ and $\mu,\eta$ bases, with Eq. (92) giving the conversion to $\Sigma$ Ishak 2019, §5.2, Eqs. (86)–(92). The ratio $\Phi/\Psi$ is undefined at a zero of $\Psi$; a code should evolve potentials or nonsingular combinations there.

The distinction between the probes follows from their trajectories. Slow matter obeys $\dot{\mathbf u}+H\mathbf u=-\nabla\Psi/a$. In the Born approximation in a spatially flat background, the weak-lensing convergence for sources at comoving distance $\chi_s$ is

$$\kappa(\hat n)=\frac12\int_0^{\chi_s}d\chi\, \frac{\chi(\chi_s-\chi)}{\chi_s} \nabla_\perp^2(\Phi+\Psi).$$

The potentials are evaluated along the unperturbed light ray and $\nabla_\perp$ is a comoving transverse derivative. The geometric kernel weights foreground structures, so lensing measures a projected sum, not $\Phi$ at one point. Tomographic source bins and cross-correlations help locate that information in redshift Bartelmann and Schneider 2001, §6.1.

Cross-correlations and the E_G construction

Galaxy positions trace $b\,\delta_m$ in the simplest deterministic linear model. Galaxy–galaxy lensing correlates that tracer with an integral of $\Phi+\Psi$, clustering measures approximately $b^2P_m$, and redshift-space distortions constrain $\beta=f/b$. Their ratio can reduce the leading bias dependence. A theoretical form of the statistic is

$$E_G(k,z) =\frac{-k^2(\Phi+\Psi)} {3H_0^2a^{-1}f(z)\Delta_m},$$

where $f$ is the growth rate and the velocity-to-density relation uses the same growing matter mode. Substituting $\rho_m=3H_0^2\Omega_{m0}a^{-3}/(8\pi G)$ gives

$$E_G(k,z)=\frac{\Omega_{m0}\Sigma(k,z)}{f(k,z)}.$$

It becomes $\Omega_{m0}/f(z)$ in GR under the stated assumptions. This also shows why $E_G$ alone is not a direct measurement of slip: both the lensing coupling and the growth response enter. Observational estimators replace the fields by projected galaxy–lensing and galaxy–velocity or clustering correlations. At matching redshift and scale, the numerator scales as $b\Sigma P_m$, while $\beta P_{gg}=(f/b)b^2P_m=fbP_m$; the leading $bP_m$ cancels. Ishak derives the estimator, GR limit, and known projection and scale-cut qualifications Ishak 2019, §5.5, Eqs. (110)–(113).

Bias does not vanish automatically. Lensing and clustering samples have different redshift kernels; magnification changes observed counts; stochasticity breaks a single $b$; intrinsic alignment contaminates shapes; and nonlinear bias and baryons alter small scales. The statistic also depends on the assumed $\Omega_{m0}$ and distance conversion. A scale-independent measurement can be shared by GR, smooth dark energy, and some modified-gravity models, while scale dependence can arise from survey effects as well as gravity.

Joint likelihood and release boundary

For an angular data vector containing shear–shear, galaxy–shear, and galaxy–galaxy correlations, use

$$-2\ln\mathcal L =(\mathbf d-\mathbf m)^TC^{-1}(\mathbf d-\mathbf m) +\ln\det(2\pi C).$$

The model includes $\mu$ and $\Sigma$ or a theory map, galaxy bias, intrinsic alignment, photometric-redshift shifts, multiplicative shear calibration, magnification, baryonic and nonlinear corrections, neutrino mass, masks, window functions, and scale cuts. A common sky makes cross-covariance mandatory. Priors and covariance estimates must be released with the data vector; multiplying separate lensing and clustering posteriors would double count shared modes.

At the evidence cutoff 2026-08-19, the worked example uses dataset version DES Year 3, with a public 3-by-2-point data vector, covariance, calibration, selection, and nuisance framework DES Collaboration 2022, §§II–IV and Tables I–II. The chapter overview distinguishes this reusable example from the newer DES Y6 analysis and its product-release statement. A baseline $\Lambda$CDM or $w$CDM likelihood must be extended and revalidated before its posterior can be interpreted as a constraint on flexible $\mu,\Sigma$. The named Planck 2018 likelihood adds CMB primary and lensing information with its own foreground and calibration model Planck Collaboration 2020, papers V, VI, and VIII. DESI DR2 geometry results help constrain the distance background; they are not themselves a lensing–clustering slip likelihood DESI Collaboration 2025–2026, official DR2 products.

Reproduction records the exact release, masks, tomographic samples, selection/completeness weights, redshift distributions, shear response, blinding state, data vector, covariance realization, theory and sampler code commits, scale cuts, priors, and convergence. GR synthetic data should recover unit slip; alternate redshift, alignment, baryon, bias, and covariance models should not create a spurious deviation.

Results and limitations through 2026-08-19

Evidence or result	Position supported	Limitation or counterevidence
DES Y3 public 3-by-2-point likelihood	Jointly connects clustering and the Weyl potential with calibrated nuisance parameters	Photo-$z$, shear, alignment, bias, baryons, nonlinear scales, and covariance limit interpretation
Planck legacy primary and lensing likelihoods	Anchors early fluctuations and adds an integrated lensing field	Foregrounds, reconstruction response, cosmological model, and overlap covariance enter
E_G-like ratios	Reduce leading deterministic linear galaxy bias	Projection, magnification, stochastic bias, scale cuts, and $\Omega_{m0}$ prevent exact cancellation

These releases provide the ingredients for a joint potential comparison, but a baseline cosmological fit is not already a measurement that $\eta=1$. A slip result requires the actual modified-gravity parameterization, matter sector, priors, and scale validation used in the fit. An apparent slip can reflect modified gravity, anisotropic stress, redshift or shear calibration, bias evolution, baryonic modeling, or covariance error. Its physical interpretation remains model-dependent until those alternatives are distinguished for that likelihood and those scales.

An adversarial independent check generates common-sky GR mocks with $\mu=\eta=\Sigma=1$ and verifies recovery of unit slip, then injects a localized scale- and redshift-dependent slip. Repeating with independent theory codes and alternate bias, photo-$z$, alignment, baryon, magnification, and scale-cut choices would distinguish a gravitational pattern from survey leakage. The bias-cancellation claim must be tested rather than assumed.

What could change the conclusion. The interpretation could change when a superseding shear, clustering, CMB-lensing, or spectroscopic overlap release changes calibration or covariance, or when validated nonlinear and baryonic models move the safe scale boundary.

Exercises

At one scale and redshift, take $\Omega_{m0}=0.3$, $f=0.6$, and $\Sigma=0.9$. Find $E_G$. Could the same $E_G$ occur with $\Sigma=1$, and would that determine $\eta$?

Solution

The result is $E_G=0.3(0.9)/0.6=0.45$. With $\Sigma=1$, the same number follows from $f=0.3/0.45=2/3$. Therefore the statistic alone does not distinguish a change in the lensing coupling from a change in growth. Nor does it determine slip: $\eta=2\Sigma/\mu-1$ also requires $\mu$ in the same convention and regime. These numbers are an algebraic example, not measured cosmological constraints.

References

• DES Collaboration, “Dark Energy Survey Year 3 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing,” Physical Review D 105, 023520 (2022), DOI.
• DESI Collaboration, “DESI DR2 Cosmology Results and Supporting Data Products,” official publication list and cosmology-chain release (2025–2026), DESI Data.
• Matthias Bartelmann and Peter Schneider, “Weak Gravitational Lensing,” Physics Reports 340, 291–472 (2001), DOI.
• Mustapha Ishak, “Testing General Relativity in Cosmology,” Living Reviews in Relativity 22, article 1 (2019), DOI.
• Planck Collaboration, “Planck 2018 Results. V, VI, and VIII,” Astronomy & Astrophysics 641, A5, A6, and A8 (2020), official publication list.