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