MODERN COSMOLOGY

Galaxy clusters as lenses 415

whereλis a real constant. This is the so-calledmass-sheet degeneracy(a
homogeneous mass sheet does not produce any shear).
Equation (14.126) can be turned into an integral equation, by making use of
the fundamental solution

G=

1

2 π

ln|θ| (14.128)

for whichG=δ^2 (δ^2 is the two-dimensional delta function). Then we get

k(θ)= 2

∫

R^2

d^2 θ′G(θ−θ′)

∑

i= 1 , 2

(Diγi)(θ′)+k 0. (14.129)

After some manipulations we can bring equation (14.129) into the following form

k(θ)=

1

π

∑

i= 1 , 2

∫

R^2

d^2 θ′[D ̃i(θ−θ′)γi(θ′)]+k 0 , (14.130)

or, in terms of the reduced shear,

k(θ)=k 0 +

1

π

∑

i= 1 , 2

∫

R^2

d^2 θ′[D ̃i(θ−θ′)(gi( 1 −k))(θ′)], (14.131)

where

D 1 ln|θ|=

θ 22 −θ 12

|θ|^4

≡D ̃ 1 , D 2 ln|θ|=−

2 θ 1 θ 2

|θ|^4

≡D ̃ 2. (14.132)

The crucial fact is thatγ(θ)is an observable quantity and thus using
equation (14.130) one can infer the matter distribution of the considered galaxy
cluster. This result is, however, fixed up to an overall constantk 0 (problem of the
mass-sheet degeneracy).
As discussed in section 14.2.4 we can define the ellipticityof an image of
a galaxy as

= 1 +i 2 =

1 −r

1 +r

e2iφ, r≡

b

a

(14.133)

whereφis the position angle of the ellipse andaandbare the major and minor
semi axis, respectively.aandbare given by the inverse of the eigenvalues of the
matrix defined in equation (14.42). If we take the average on the ellipticity due to
lensing and make use of equation (14.133) as well as of the expressions foraand
bwe find the relation

〈〉=

〈

γ

1 −k

〉

. (14.134)

The angle bracket means average over a finite sky area. In the weak lensing
limitk  1and|γ|1 the mean ellipticity directly relates to the shear:
〈γ 1 (θ)〉〈 1 (θ)〉and〈γ 2 (θ)〉〈 2 (θ)〉. Thus a measurement of the average