MODERN COSMOLOGY

414 Gravitational lensing

formulation of lensing theory (see, for instance, [65]). In the following we will,
however, not use it.
The reduced sheargis, in principle, observable over a large region. What
we are really interested in, however, is the mean curvatureκ, which is related to
the surface mass density. Since

g=

γ

1 −κ

(14.118)

we first look for relations between the shearγ=(γ 1 ,γ 2 )andκ.
From equation (14.37) we get that

+= 2 k (14.119)

or if, instead, we use the notationθ = (θ 1 ,θ 2 )for the image position
equation (14.119) can be explicitly written as

k(θ)=

1

2

(

∂^2 +(θ)

∂θ 12

+

∂^2 +(θ)

∂θ 22

)

. (14.120)

Using the definition forγias given in equation (14.42) we find

γ 1 (θ)=

1

2

(

∂^2 +(θ)

∂θ 12

−

∂^2 +(θ)

∂θ^22

)

≡D 1 + (14.121)

and

γ 2 (θ)=

∂^2 +(θ)

∂θ 1 ∂θ 2

≡D 2 +. (14.122)

where

D 1 :=

1

2

(

∂ 12 −∂ 22

)

, D 2 :=∂ 1 ∂ 2. (14.123)

Note the identity
D 12 +D^22 =^14 ^2. (14.124)

Hence
κ= 2

∑

i= 1 , 2

Diγi. (14.125)

Here, we can substitute the reduced shear, given by equation (14.118), on the
right-hand side forγi. This gives the important equation

κ= 2

∑

i

Di[gi( 1 −κ)]. (14.126)

For a given (measured)gthis equation does not determine uniquelyκ, indeed
equation (14.126) remains invariant under the substitution

κ→λκ+( 1 −λ) (14.127)