MODERN COSMOLOGY

386 Gravitational lensing

is thedeflection potential, which does not depend onη. The Fermat principle can
thus be written as dt/dξ =0, and inserting equation (14.19) one once again
obtains the lens equation

η=

Ds

Dd

ξ−Ddsα(ξ), (14.22)

whereαis defined in equation (14.10). (If we defineβ=η/Dsandθ=ξ/Dd
we obtain equation (14.13). One can also write equation (14.22) as follows.

∇ξ(ˆ ξ,η)= 0 , (14.23)

which is an equivalent formulation of the Fermat principle.
The arrival time delay of light rays coming from two different images (due
to the same source inη) located inξ(^1 )andξ(^2 )is given by

c(t 1 −t 2 )=(ˆ ξ(^1 ),η)−(ˆ ξ(^2 ),η). (14.24)

14.2.4 Remarks on the lens equation

It is often convenient to write (14.22) in a dimensionless form. Letξ 0 be a length
parameter in the lens plane (whose choice will depend on the specific problem)
and letη 0 =(Ds/Dd)ξ 0 be the corresponding length in the source plane. We set
x=ξ/ξ 0 ,y=η/η 0 and

κ(x)=

&(ξ 0 x)

&cr

, α(x)=

DdDds

ξ 0 Ds

αˆ(ξ 0 x), (14.25)

where we have defined a critical surface mass density

&cr=

c^2

4 πG

Ds

DdDds

= 0 .35 g cm−^2

(

1Gpc

D

)

(14.26)

withD≡DdDDsds(1 Gpc= 109 pc). Then equation (14.22) reads as follows

y=x−α(x), (14.27)

with

α(x)=

1

π

∫

R^2

x−x′

|x−x′|^2

κ(x′)d^2 x′. (14.28)

In the following we will mainly use the previous notation rather than that in
equation (14.28).
An interesting case is a lens with a constant surface mass density&. With
equation (14.11) one then finds, for the deflection angle,

α(θ)=

4 G

c^2 ξ

&πξ^2 =

4 πG&

c^2

Ddθ, (14.29)