MODERN COSMOLOGY

Lens equation 387

usingξ=Ddθ. In this case the lens equation (14.13) is linear, which means that
βis proportional toθ:

β=θ−β=θ−

4 πG

c^2

DdsDd

Ds

&θ=θ−

&

&cr

θ. (14.30)

From equation (14.30) we immediately see that for a lens with a critical surface
mass density we get for all values ofθ:β=0. Such a lens would perfectly focus,
with a well-defined focal length. Typical gravitational lenses behave, however,
quite differently. A lens which has&>&crsomewhere in it is defined as
supercritical, and has, in general, multiple images.
Definingk(θ):=&(θDd)/&crwe can write the lens equation as

β=θ−α ̃(θ), (14.31)

with

α ̃(θ)=

1

π

∫

R^2

d^2 θ′k(θ′)

θ−θ′

|θ−θ′|^2

. (14.32)

Moreover,

α ̃(θ)=∇θ+(θ) (14.33)

where

+(θ)=

1

π

∫

R^2

d^2 θ′k(θ′)ln|θ−θ′|. (14.34)

The Fermat potential is given by

(θ,β)=^12 (θ−β)^2 −+(θ) (14.35)

and we then obtain the lens equation from

∇θ(θ,β)= 0. (14.36)

Note that
+= 2 k≥ 0 (14.37)

(usingln|θ|= 2 πδ^2 (θ)), sincekas a surface mass density is always positive
(or vanishes).
The flux of a source, located inβ, in the solid angle d
(β)is given by

S(β)=Iνd
(β). (14.38)

Iνis the intensity of the source in the frequencyν.S(β)is the flux one would see
if there were no lensing. However, the observed flux from the image located inθ
is
S(θ)=Iνd
(θ). (14.39)