408 Gravitational lensing
events at redshiftsz 1 andz 2 (z 1 <z 2 ),isgivenby
D(z 1 ,z 2 )= 2 c√
1 + 0 z 1 ( 2 − 0 + 0 z 2 )−√
1 + 0 z 2 ( 2 − 0 + 0 z 1 )
H 0 ^20 ( 1 +z 2 )^2 ( 1 +z 1 ).
(14.109)
Equations (14.107)–(14.108) provide the basis for the determination of the
Hubble parameter with gravitational lensing. One should also take into account
that the universe might have a clumpy structure, which then affects the light
propagation (for details on this problem see [57, 58]).
From equation (14.108) we obtain the cosmological lens mapping using
Fermat’s principle, which implies that∇θφ(ˆθ,β) =0 and gives an equation
identical to equation (14.22), but, with the present meaning of the symbols, it
holds for arbitrary redshifts.
Consider two images at the (observed) positionsθ 1 , θ 2 , with separation
θ 12 ≡θ 1 −θ 2 and time delayt 12. Using the lens equation we obtain
θ 12 = 2 RSDds
DdDs[
∂ψ ̃
∂θ∣
∣
∣
∣
θ 1−
∂ψ ̃
∂θ∣
∣
∣
∣
θ 2]
. (14.110)
The time delayt 12 =φ(ˆθ 1 ,β)−φ(ˆθ 2 ,β)contains the unobservable angleβ,
but this can be eliminated with the lens equation and equation (14.110):
t 12 = 2 RS( 1 +zd){
1
2
(
∂ψ ̃
∂θ∣
∣
∣
∣
θ 1+
∂ψ ̃
∂θ∣
∣
∣
∣
θ 2)
·θ 12 −(
ψ( ̃ θ 1 )−ψ( ̃ θ 2 ))
}
.
(14.111)
Given a lens model (i.e.&( ̃ θ)), then equations (14.110) and (14.111) give a
relation between the observablesθ 12 ,t 12 andH 0 , provided that 0 ,zd,zsare
also known. Fortunately, the dependence on 0 is, in practice, not strong.
Consider as an example a point source lensed by a point mass (Schwarzschild
lens). Thenψ( ̃θ)=ln|θ|and equation (14.110) gives
θ 12 = 2 RSDds
DdDs(
1
θ 1−
1
θ 2)
, (14.112)
However, equation (14.111) becomes
t 12 = 2 RS( 1 +zd){
θ 22 −θ 12
2 |θ 1 θ 2 |+ln∣
∣
∣
∣
θ 2
θ 1∣
∣
∣
∣
}
. (14.113)
We write this in terms of the ratioνof the magnifications. Using equation (14.74)
one findsν=ln(θ 2 /θ 1 )^2 and thus
t 12 =Rs( 1 +zd){ν^1 /^2 −ν−^1 /^2 +lnν}. (14.114)