Past exam of the relativistic Astrophysics and Cosmology course of the University of Cambridge / 2012 / Paper 1 / 3 / e / Solution

A galaxy cluster deflects light from background galaxies according to its projected gravitational field. In a thin gravitational lens approximation,The distances are angular-diameter distances; only mass in the appropriate foreground/background geometry lenses the chosen source this way. Near critical curves, strong gravitational lensing produces highly magnified arcs and multiple images of the same source. Farther out, weak gravitational lensing produces small coherent shear distortions, inferred statistically because individual galaxies have unknown intrinsic ellipticities. The measured reduced shear is $g=\gamma /(1-\kappa )$. Transparent lensing obeys surface-brightness conservation in gravitational lensing at fixed source redshift while changing solid angle and integrated flux.

Strong-lensing image positions and source redshifts constrain the central projected mass; weak shear extends the map to larger radii. Multiple source planes help constrain geometry and degeneracies, but the mass-sheet degeneracy illustrates why image shapes alone do not uniquely fix absolute mass. Point-spread-function anisotropy, selection, shape calibration, cluster-member contamination, source redshift errors and unrelated line-of-sight structure require controls. Lensing does not require hydrostatic equilibrium, making it valuable for disturbed clusters, although interpretation still depends on the gravitational model.

An instructive observation is the merging Bullet Cluster: weak-lensing mass concentrations approximately follow the collisionless galaxies while the dominant visible baryonic plasma, traced by X-rays, is displaced. Clowe and collaborators combined ground-based and Hubble images for lensing reconstruction and compared them with the gas distribution, providing evidence for mass beyond the observed baryons.^[ref]