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.

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 $2{\int }_0^{{\chi }_s}({\chi }_s-\chi )\Psi /({\chi }_s\chi )d\chi$.