MODERN COSMOLOGY

Lens equation 385

t=0. It will then proceed straight until it reaches the lens, located at the point I,
and where it will be deflected and then proceed again straight to the observer in
O. We thus have

t=

1

c

∫ (

1 −

2 φ

c^2

)

dl=

l

c

−

2

c^3

∫

φdl, (14.15)

wherelis the distance SIO (Euclidean distance). The term containingφhas to be
integrated along the light trajectory. From figure 2.1 we see that

l=

√

(ξ−η)^2 +D^2 ds+

√

ξ^2 +D^2 d

Dds+Dd+

1

2 Dds

(ξ−η)^2 +

1

2 Dd

ξ^2 , (14.16)

whereηis a two-dimensional vector in the source plane If we takeφ=−GM/|x|
(corresponding to a point-like lens of massM)weget

∫I

S

2 φ

c^3

dl=

2 GM

c^3

[

ln

|ξ|

2 Dds

+

ξ·(η−ξ)

|ξ|Dds

+O

(

(η−ξ)^2

Dds

)]

(14.17)

and similarly for

∫O

I^2 φ/c

(^3) dl.
Only the logarithmic term is relevant for lensing, since the other ones are of
higher order. Moreover, instead of a point-like lens we consider a surface mass
density&(ξ)(as defined in equation (14.9)) and so we obtain, for the integral
containing the potential term (neglecting higher-order contributions)
2
c^3

∫

φdl=

4 G

c^3

∫

d^2 ξ′&(ξ′)ln

|ξ−ξ′|

ξ 0

, (14.18)

whereξ 0 is a characteristic length in the lens plane and the right-hand side term
is defined up to a constant.
The difference in the arrival time between the situation which takes into
account the light deflection due to the lens and without the lens, is obtained
by summing equation (14.16)–(14.18) and by subtracting the travel time without
deflection from S to O. This way one obtains

ct=φ(ˆξ,η)+constant, (14.19)

whereφˆis theFermat potentialdefined as

φ(ˆξ,η)=DdDs

2 Dds

(

ξ

Dd

−

η

Ds

) 2

−ψ(ˆξ) (14.20)

and

ψ(ˆ ξ)=^4 G

c^2

∫

d^2 ξ′&(ξ′)ln

(

|ξ−ξ′|

ξ 0

)

(14.21)