= Solution
<gravitational lensing> measures the gravitational influence of matter without requiring that matter to shine. In <general relativity> a <point mass> produces small-angle deflection
$$
\hat\alpha=\frac{4GM}{bc^2}.
$$
For a thin lens, with angular-diameter distances $D_d,D_s,D_{ds}$,
$$
\boldsymbol\beta=\boldsymbol\theta-
\frac{D_{ds}}{D_s}\hat{\boldsymbol\alpha}(D_d\boldsymbol\theta).
$$
Image multiplicity, distortion and magnification therefore constrain projected gravitating <mass>, with source and lens geometry measured separately.
Strong lensing gives multiple images, arcs and <Einstein rings>. Their positions and, where available, time delays constrain a central <mass> distribution. Weak lensing measures correlated small image distortions over a larger field. Using the <critical surface density for gravitational lensing> and <reduced gravitational shear>, the convergence and shear satisfy
$$
\kappa=\frac{\Sigma}{\Sigma_{\rm crit}},\qquad
\Sigma_{\rm crit}=\frac{c^2}{4\pi G}\frac{D_s}{D_dD_{ds}},\qquad
g_{\rm red}=\frac{\gamma}{1-\kappa}.
$$
Because intrinsic galaxy shapes are unknown, weak shear is inferred statistically using many sources. Point-spread-function anisotropy, shear calibration, photometric redshifts, intrinsic alignments and foreground contamination must be controlled.
An inferred lensing <mass> can be compared with the independently estimated stellar and hot-gas <mass>. The <bullet Cluster dark-matter separation> provides a particularly informative separation of components: the collisional X-ray-emitting gas is displaced from the galaxies, while lensing <mass> peaks approximately follow the less-collisional galaxy component. The primary analysis combined wide-field and Hubble imaging rather than inferring lensing <mass> from X-ray hydrostatic equilibrium.<A direct empirical proof of the existence of dark matter by Douglas Clowe and others>[arXiv:astro-ph/0608407, abstract and lensing/X-ray component comparison]{ref} This favors a dominant relatively collisionless unseen component. Translating a merger offset into a particle self-interaction <cross section> also requires a merger model and projection uncertainties; lensing does not directly reveal a <WIMP>'s <mass> or quantum numbers.
Microlensing tests a different possibility: <dark matter> in compact objects. A compact foreground lens can create an unresolved, time-dependent magnification of a background star. Event frequency and duration constrain the abundance and <mass> range of compact lenses only after survey cadence, detection efficiency, blending, source sizes and halo geometry are included. Achromaticity and the expected light-curve shape help distinguish simple lensing from intrinsic variability. A null survey result does not exclude every compact-object <mass> at every location.
Weak-lensing correlations and small-scale strong-lens structure also test the <matter power spectrum> and halo substructure, potentially distinguishing cold from warm free-streaming behavior. Baryonic feedback can change the same small-scale <mass> distribution. There are intrinsic inversion ambiguities: the mass-sheet transformation
$$
\kappa\mapsto\lambda\kappa+(1-\lambda),\qquad
\gamma\mapsto\lambda\gamma
$$
leaves <reduced gravitational shear> unchanged. Source-redshift information, absolute magnification or other data can restrict such degeneracies. \b[Lensing establishes where gravitating matter is and tests its distribution and interactions; identification of its particle species requires complementary evidence.]
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