Mass and Light distributions in Clusters of Galaxies - Henry A ...

Weak Lensing Dilution in A1689
To see this effect, we expand the reduced shear with respect to the convergence
κ as
g = γ(1 − κ) −1 = wγ ∞ (1 − wκ ∞ ) −1 = wγ ∞
∞
∑
k=0
(wκ ∞ ) k (2.33)
where κ ∞ and γ ∞ are the lensing convergence and the gravitational shear, respectively,
calculated for a hypothetical source at an infinite redshift. Hence,
the reduced shear averaged over the source redshift distribution is expressed
as
〈g〉 = γ ∞
∞
∑
k=0
〈w k+1 〉κ k ∞. (2.34)
In the weak lensing limit where κ ∞ , |γ| ∞ ≪ 1, then 〈g〉 ≈ 〈w〉γ ∞ . Thus,
the mean reduced shear is simply proportional to the mean lensing strength,
〈w〉. The next higher-order approximation for eq. (2.34) is given by
〈g〉 ≈ γ ∞
(
〈w〉 + 〈w 2 〉κ ∞
)
≈
〈w〉γ ∞
1 − κ ∞ 〈w 2 〉/〈w〉 . (2.35)
Seitz & Schneider (1997) found that eq. (2.35) yields an excellent approximation
in the mildly non-linear regime of κ ∞ 0.6. Defining f w ≡ 〈w 2 〉/〈w〉 2 ,
we have the following expression for the mean reduced shear valid in the
mildly non-linear regime:
〈g〉 ≈
〈γ〉
1 − f w 〈κ〉
(2.36)
with 〈κ〉 = 〈w〉κ ∞ and 〈γ〉 = 〈w〉γ ∞ (Seitz & Schneider 1997). For lensing
clusters located at low redshifts of z l 0.2, 〈w 2 〉 ≃ 〈w〉 2 or f w ≈ 1, so that
〈g〉 ≈ 〈γ〉/(1 − 〈κ〉).
The ratio of tangential shear estimates using two different populations B
and G of background galaxies, in the mildly non-linear regime, is given as
〈g (G)
T 〉
〈g (B)
T 〉 ≈ 〈w(G) 〉
〈w (B) 〉
≈ 〈w(G) 〉
〈w (B) 〉
1 − f w
(B) 〈w (B) 〉κ ∞
1 − f (G)
(2.37)
w 〈w (G) 〉κ ∞
{ (
1 − f
(B)
w 〈w (B) 〉 − f w (G) 〈w (G) 〉 ) κ ∞ + O(〈κ〉 2 ) } .
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