Angular deflection kernel for cosmological lensing (${\alpha }^i=2{\int }_0^{{\chi }_s}\frac{{\chi }_s-\chi }{{\chi }_s}{\partial }_{\perp }^i\Psi d\chi$)

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