Weak gravitational lensing ($\mathbf{\beta }=\mathbf{\theta }-\mathbf{\alpha }$)

When gravitational lensing gives small angular distortions without generic multiple imaging, its deflection field changes source positions, magnification and shear. Angular derivatives of the lensing potential determine these distortions. Ensemble galaxy-shape distortions and cosmic microwave background remapping probe the foreground gravitational field.

A correlation of galaxy shapes caused by their formation environment or tidal fields rather than by the gravitational lensing of their light. It can contaminate a weak-shear correlation used to infer structure growth and dark energy. Redshift separation, population dependence and physically motivated modelling help distinguish it from a lensing contribution.

Cosmic shear is the statistically correlated lensing shear of distant galaxies produced by the intervening large-scale matter distribution. Correlations across angle and source redshift constrain clustering, the growth of structure and distance geometry, thereby testing dark matter and dark energy. Intrinsic galaxy alignments and measurement errors must be separated from the lensing signal.

The trace-free part of the angular lensing-potential Hessian stretches an image anisotropically. The source-image Jacobian matrix has eigenvalues $1-\kappa \pm |\gamma |$. Thus a circular source becomes an ellipse in the local linear approximation; the orientation follows the shear axes.

The combination of lensing shear and lensing convergence governing the local shape distortion of background sources. Galaxy ellipticities estimate it statistically under a specified ellipticity convention and intrinsic-orientation model. It is invariant under the mass-sheet degeneracy, so shape information alone does not generally fix absolute projected mass normalization.

The shear-to-isotropic-focusing ratio inferred from background image shapes. The mass-sheet transformation rescales numerator and denominator equally, leaving this quantity unchanged.

The trace of the angular derivative of the reduced deflection defines convergence in the standard thin gravitational lens equation convention. It equals projected surface mass density divided by the critical surface density, ${\Sigma }_{\mathrm{crit}}=c^2{D}_s/(4\pi G{D}_d{D}_{ds})$. If deflection is instead defined as source minus image angle, the divergence relation has a minus sign.

The geometrical projected density scale converting a thin lens surface mass density into convergence. Its value depends on the angular-diameter distances to lens and source, not only on lens matter.

For a source at comoving distance ${\chi }_s$ in a spatially flat universe, define deflection as apparent minus unlensed source direction. The transverse Weyl lensing potential gradient along the ray receives the distance weight $({\chi }_s-\chi )/{\chi }_s$. Integrating the transverse photon acceleration twice and dividing the source displacement by ${\chi }_s$ gives the formula. Reversing the definition of the deflection reverses its sign. Its angular lensing potential is $2{\int }_0^{{\chi }_s}({\chi }_s-\chi )\Psi /({\chi }_s\chi )d\chi$.