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

2.B Non-linear effect in the reduced shear estimate
expectation value for the mean tangential shear estimate is
〈g (G)
T
(r)〉 = g T,∞(r)
N (G)
bg
N cl + N (G)
bg
〈w (G) 〉 z>zl . (2.29)
As for our background sample (denoted with B), including the red and blue
samples, this is
〈g (B)
T (r)〉 = g T,∞(r)〈w (B) 〉 z>zl . (2.30)
By taking the ratio of the two tangential shear estimates, we obtain the
following expression:
〈g (G)
T
(r)〉/〈g(B)
T (r)〉 = N (G)
bg
N cl + N (G)
bg
〈w (G) 〉 z>zl
〈w (B) 〉 z>zl
. (2.31)
Alternatively, we have the expression for the cluster galaxy fraction as
f cl (r) ≡
N cl
N cl + N (G)
bg
= 1 − 〈g(G) T
(r)〉
〈g (B)
T
〈w (B) 〉 z>zl
(r)〉 = 1 − 〈g(G) T
(r)〉 〈D (B) 〉 z>zl
〈w (G) 〉 z>zl 〈g (B) (r)〉 .
〈D (G) 〉 z>zl
T
(2.32)
This is the desired formula for the cluster galaxy fraction from the weak
lensing dilution effect. In order to take into account different populations of
background galaxies in the two samples, one needs to estimate the correction
factor, 〈D (B) 〉 z>zl /〈D (G) 〉 z>zl .
2.B Non-linear effect in the reduced shear estimate
In Appendix A, we assume that the observable reduced shear is linearly proportional
to the lensing strength factor, w(z). However, the reduced sear, defined
as g = γ/(1−κ), is non-linear in κ, so that the averaging operator with
respect to the redshift generally acts non-linearly on the redshift-dependent
components in g.
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