Lensing, Clusters, and Cosmological Structure

Galaxy kinematics constrain how slow matter responds, but lensing, clusters, and cosmological structure ask for more of a theory. Light probes the sum of two metric potentials; clusters test nonspherical and partially relativistic systems; early-universe structure tests the background and perturbations long before galaxies enter a low-acceleration equilibrium. This page develops the theory-side requirements. Statistical fits, survey systematics, and bounds belong to Tests of Gravity.

Required background. Weak-Field Gravity supplies weak metric potentials and null deflection.

Helpful background. Einstein Equations supplies the relation between geometry and stress energy; Gravitational Actions and Boundary Terms supplies covariant conservation; Formulations of General Relativity supplies initial-value and constraint language; Gravity Beyond General Relativity supplies branch and mode checks; and Relativistic Matter and Compact Objects supplies the matter models needed when pressure and self-gravity cannot be neglected.

Lensing needs both metric potentials

In conformal Newtonian gauge, write the physical matter metric as

$$d\tilde s^2=-(1+2\Psi)dt^2 +a^2(t)(1-2\Phi)d\boldsymbol x^2.$$

For a quasistatic lens and $c=1$, the leading transverse deflection is

$$\boldsymbol\alpha =\int d\chi\,\nabla_\perp(\Phi+\Psi).$$

Here $\chi$ is comoving distance along the unperturbed ray and $\nabla_\perp$ is the comoving transverse gradient; both potentials are dimensionless. The sign follows the usual lens-equation deflection convention. This is the local bending contribution of a localized lens in the Born approximation. Converting it to a source-plane angular map in an expanding geometry also requires the observer–lens–source distance weights. It is not an unweighted formula for cosmological shear at arbitrary source distance.

Slow orbital dynamics principally determines $\Psi$; lensing determines the Weyl combination $(\Phi+\Psi)/2$. A nonrelativistic acceleration law therefore cannot predict lensing until a relativistic action and physical matter metric fix $\Phi$, $\Psi$, and any gravitational slip $\eta=\Phi/\Psi$ on the same solution branch.

In TeVeS, the disformal scalar–vector–tensor matter metric was designed so that the potential enhancing quasistatic dynamics also contributes to light deflection. Bekenstein derives this equality in the weak, quasistatic regime in Bekenstein 2004, § VI, Eqs. (109) and (114)–(116). That derivation is conditional on the theory, background, and approximations; one may not combine an AQUAL orbit with a GR dark-halo lens by declaration.

Why clusters are a separate scale

An isolated spherical galaxy can collapse the field equation to one algebraic relation. A cluster generally contains galaxies, hot gas, pressure support, substructure, and an environment-dependent external field. A valid calculation must solve the chosen relativistic or nonrelativistic field equations for that mass distribution and compare several potentials inferred from the same solution.

The important theoretical result is negative: the elementary low-acceleration scaling does not generically remove the full cluster mass discrepancy. In particular, the central regions can lie outside the deepest modified regime, and lensing reconstructions can still require an additional unseen component. The review synthesis and its dependence on interpolating function and cluster environment are given in Famaey and McGaugh 2012, §§ 6.6.4 and 8.3.

This does not logically restore one unique dark-matter model. It says that a specific modified-gravity theory must declare any added massive species or unseen baryons and evolve them consistently. Calling such a component “effective” is not enough if it clusters, carries stress energy, or changes the lensing metric.

Cosmological perturbations are not galaxy equilibria

Before recombination, baryons are tightly coupled to photons. They cannot play the role of a freely clustering pressureless component that preserves gravitational potentials on small scales. A relativistic low-acceleration theory must therefore supply background expansion, constraint equations, and additional perturbations capable of producing the required potential evolution. The galaxy relation $g\sim\sqrt{a_0g_N}$ is silent about all three.

For each Fourier mode, a complete calculation must evolve at least

$$\{\Phi_k,\Psi_k,\delta_{b,k},\theta_{b,k}, \delta_{\gamma,k},\theta_{\gamma,k},\text{extra-field modes}\}$$

from specified primordial data. The scalar and vector perturbations in a relativistic completion may provide an effective dustlike sector, anisotropic stress, or both. In the Skordis–Zlosnik construction, a background scalar charge produces a component whose density scales approximately as $a^{-3}$, and the authors evolve the linear perturbations through CMB and matter-power calculations (Skordis and Zlosnik 2021, “Cosmological observables,” equations (7)–(12)).

Agreement for linear spectra is an existence result for that parameterized model. It does not derive nonlinear halo formation, cluster collisions, or the galaxy acceleration relation from the same simulation.

Results and open questions through 2026-08-19

The evidence below is organized by what the cited calculation actually establishes, not by whether one prefers a modified-gravity or dark-sector interpretation.

Evidence or mechanism	Supported result	Scope or limitation
Weak-field TeVeS lensing calculation	The physical metric can make the same quasistatic potential govern slow dynamics and light bending	Holds in the analyzed weak branch; it does not prove stability or fit every lens configuration
Cluster hydrostatic and lensing analyses summarized in the review literature	The elementary low-acceleration modification can leave residual missing mass in clusters	Size and distribution depend on the field equation, baryonic model, environment, and any added component
Relativistic perturbation theory	Extra scalar/vector modes can sustain cosmological potentials unavailable to photon-coupled baryons alone	Behavior is formulation-, parameter-, background-, and initial-data-dependent
Skordis–Zlosnik linear calculation	One fixed covariant construction can reproduce low-acceleration quasistatics and viable linear CMB/matter spectra	Does not establish nonlinear structure, generic backgrounds, compact objects, or a unique theory

A permissive position regards these calculations as proof that no simple lensing-or-cosmology no-go theorem excludes all relativistic modified dynamics. A conservative position emphasizes that residual cluster mass and model-dependent cosmological extra modes weaken the claim that one new force law replaces the entire dark sector. Which position survives depends on the fixed action, matter coupling, solution branch, and scale being tested; a universal empirical verdict remains unresolved here.

The conclusions could change after an independent end-to-end simulation evolves one fixed action from early-universe initial data into nonlinear galaxies and clusters, after a new lensing analysis changes the inferred residual component under controlled baryonic systematics, or after a stability result removes a branch used by the cosmological calculation.

Diagnostic exercise

Two relativistic theories produce the same nonrelativistic potential $\Psi$ around a lens. Theory A has $\Phi=\Psi$; theory B has $\Phi=0$. Compare their leading lensing deflections without changing the matter distribution.

Solution

The deflection is proportional to the line-of-sight integral of $\nabla_\perp(\Phi+\Psi)$. Theory A supplies $2\Psi$ inside that integral, whereas theory B supplies only $\Psi$. Under the stated assumptions, theory A’s deflection is twice theory B’s even though slow bodies infer the same $\Psi$. The exercise isolates why a rotation curve cannot determine lensing. The ratio would cease to be this simple if the theories produced different background distances, matter distributions, nonlinear branches, or line-of-sight potentials; those inputs must be held fixed in an actual comparison.

References

• Bekenstein, Jacob D. (2004), “Relativistic Gravitation Theory for the MOND Paradigm,” Physical Review D 70, 083509; erratum 71, 069901. DOI. Open PDF.
• Famaey, Benoît, and Stacy S. McGaugh (2012), “Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions,” Living Reviews in Relativity 15, 10. DOI. Open PDF.
• Skordis, Constantinos, and Tom Zlosnik (2021), “A New Relativistic Theory for Modified Newtonian Dynamics,” Physical Review Letters 127, 161302. DOI. Open PDF.