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

WL Determination of the Distance-Redshift Relation
4.4 Weak lensing measurements
To make the WL catalogs, we use the IMCAT package developed by N.
Kaiser 7 to perform object detection and shape measurements, following the
formalism outlined in Kaiser, Squires, & Broadhurst (1995, hereafter KSB).
Our analysis pipeline is described in Umetsu et al. (2010). We have tested
our shape measurement and object selection pipeline using STEP (Heymans
et al. 2006) data of mock ground-based observations (see Umetsu et al. 2010,
§ 3.2). Full details of the methods are presented in Umetsu & Broadhurst
(2008), Umetsu et al. (2009) and Umetsu et al. (2010).
The shape distortion of an object is described by the complex reducedshear,
g = g 1 + ig 2 , where the reduced-shear is defined as:
g α ≡ γ α /(1 − κ). (4.1)
The tangential component g T is used to obtain the azimuthally averaged
distortion due to lensing, and computed from the distortion coefficients g 1 , g 2 :
g T = −(g 1 cos 2θ + g 2 sin 2θ), (4.2)
where θ is the position angle of an object with respect to the cluster center,
and the uncertainty in the g T measurement is σ T = σ g / √ 2 ≡ σ in terms of the
RMS error σ g for the complex shear measurement. To improve the statistical
significance of the distortion measurement, we calculate the weighted average
of g T and its weighted error, as
∑
i
〈g T (θ n )〉 =
u g,i g
∑ T,i
, (4.3)
i u g,i
σ T (θ n ) =
√ ∑
i u2 g,i σ2 i
( ∑ i u g,i) 2 , (4.4)
where the index i runs over all of the objects located within the n-th annulus
with a median radius of θ n , and u g,i is the inverse variance weight for i-th
7 http://www.ifa.hawaii.edu/~kaiser/imcat
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