MODERN COSMOLOGY

The lens equation in cosmology 407

We recall that the expression for the time delay in an almost Newtonian
situation is given by equation (14.19) with equations (14.20), (14.21):

ct=

DdDs

2 Dds

(

ξ

Dd

−

η

Ds

) 2

−ψ(ˆ ξ)+constant. (14.102)

Note that (
ξ
Dd

−

η

Ds

)

=(θ−β).

If the distances involved are cosmological, we must multiply the whole expression
by( 1 +zd),wherezdis the redshift of the lens. In addition all distances must be
interpreted as angular diameter distances. (For a detailed derivation we refer to
the book by Schneideret al[4] or [56]). With these modifications we obtain for
the time delay,

ct=( 1 +zd)

[

DdDs

2 Dds

(θ−β)^2 −+(ˆ ξ)

]

+constant, (14.103)

where the prefactor of the first term is proportional to 1/H 0 (H 0 is the present
Hubble parameter).
For cosmological applications, it is convenient to rewrite the potential term
using the length scaleξ 0 =Ddas defined in equation (14.18) andθ =ξ/Dd.
This way we get

ψ(ˆ ξ)= 4 G

∫

d^2 θ′Dd^2 &(Ddθ′)ln|θ−θ′|= 2 RSψ( ̃ θ), (14.104)

whereRS= 2 GMis the Schwarzschild radius of the total massMof the lens,
and

ψ( ̃ θ)=

∫

d^2 θ′&( ̃ θ′)ln|θ−θ′|, (14.105)

with

&( ̃ θ):=&(Ddθ)

M

D^2 d. (14.106)

This quantity gives the fraction of the total massMper unit solid angle as seen
by the observer. We can now write equation (14.103) in the form

ct=φ(ˆθ,β)+constant, (14.107)

whereφˆis thecosmological Fermat potential:

φ(ˆθ,β)=^1

2

( 1 +zd)

DdDs

Dds

(θ−β)^2 − 2 RS( 1 +zd)ψ( ̃ θ). (14.108)

For a Friedmann–Lemaitre model with density parameter
0 and vanishing
cosmological constant, the angular diameter distanceD(z 1 ,z 2 )between two