Gravitational lenses in Cosmology

The extension of the lens equations to a cosmological context raises some technical difficulties because the background in which the objects are embedded is not flat. In particular one must pay attention to the distances that should be employed. The other change is that lenses will now have extended mass distribution. 

In cosmology one distinguishes between the radial distance, % (distance actually travelled on a light cone) and the angular distance Dq (defined from the apparent size of distant objects). Reminding that 

The distance x is directly related to the time dependence of the expansion factor. It is given by, 

The angular distances can be obtained from the radial ones and are given by, 

In the lens equation the distance to use is the angular distance. This can be unambiguously established by the geometric optic equations. 

---