Hoekstra et al., 2013).
Several algorithms exist to correct for the above effects. Some correct for instrument depen- dent effects such as the charge transfer efficiency effect observed in Advanced Camera for Survey images from the Hubble Space Telescope (Massey et al., 2010), and other techniques correct for the observational impact of the PSF (e.g. Kaiser et al., 1995; Luppino and Kaiser, 1997; Rhodes et al., 2000; Hoekstra et al., 2000).
The other main challenge in relating the mean ellipticity of the weakly lensed galaxies to the mean surface mass density of the lens is theoretically based and relates to finding an optimal method of reconstructing the mass distribution using a combination of the shear γ or reduced shearg and/or the magnification µ. There are director inversemethods to do this. The former are based on the convolution of γ, the integration of the gradient ofg, or parametrically fitting observables directly (Fahlman et al., 1994; Fischer and Tyson, 1997; Clowe et al., 1998; Clowe and Schneider, 2002; Kneib et al., 2003; Hetterscheidt et al., 2005; Hoekstra, 2007; Okabe et al., 2010). The inverse approaches involve the derivation of both the convergenceκand the lensing potentialϕ, using either maximum likelihood or maximum entropy methods. Examples of these inverse techniques can be found in Kuijken (1999); Bridle et al. (2002); Kitching et al. (2008);
and Refregier and Bacon (2003).
5.4.2 Shape deformations to first order: shear
The previous section dealt with problems relating to accurately measuring the shape of back- ground galaxies experiencing weak lensing. Here we consider how to measure this shape to first order, assuming that the contaminating factors have been corrected for. There are higher order effects that can be measured, e.g. flexion (Bacon et al., 2005; Leonard et al., 2006; Okura et al., 2007; Goldberg and Leonard, 2007; Leonard et al., 2007; Leonard and King, 2010; Leonard et al., 2011; Munshi et al., 2011; Cain et al., 2011; Fedeli et al., 2012; Er and Bartelmann, 2013), but these are not discussed here.
To first order, the light distribution of a background galaxy can be approximated as a source with elliptical isophotes. In this manner the shape and size of a galaxy is defined in terms of the
ellipse axis ratio and the area enclosed by a defined boundary. However, since the true shape of a galaxy is often irregular and not well approximated by an ellipse, the moments of the pixelised galaxy surface brightness can be used to define the shape instead.
Assume a weakly lensed image whose surface brightness distribution is described by I(θ) whereθ = (θi, θj). The centre of the imageθC = (θCi , θjC)is defined as the first order moment ofI(θ):
θC =
R W(I(θ))θdθ
R W(I(θ))dθ , (5.24)
where the window functionW(I)serves the purpose of ensuring the integrals are finite in the case of noisy data. Common choices for the weighting function are (i) the Heavyside step function W(I) = H(I −Iiso) =
1, I(θ)> Iiso 0, otherwise
, whereIiso is the minimum isophote of the galaxy detection, and (ii)W(I) =I×H(I−Iiso)where the integral is weighted by the light distribution within the detection isophote.
The second order moment matrix, centred onθC, is given by
Mij =
R R W(I(θ))(θi−θCi )(θj−θjC)dθidθj R R W(I(θ))dθidθj
. (5.25)
ThatMij encapsulates the information about the galaxy size, axis ratio and orientation is readily apparent when written in terms of its principal axes:
Mij =Rθ
a2 0
0 b2
R−θ. (5.26)
HereRθis the rotation matrix of the position angleθ, andaandbare the major and minor axes respectively. It is possible to define a complex ellipticity = ||e2iθ which encodes both the shape and orientation information. Over the course of lensing studies there have been several different forms for the norm of the complex ellipticity, however the current standard choice is
||= a−b
a+b (5.27)
as it is a direct estimator of the measurable quantity, that being the reduced shearg. This is easily
shown using the shape deformations shown in Figure 5.2, where the major axis of the lensed source becomesa= 1/(1−κ− |γ|), and minor axis becomesb= 1/(1−κ+|γ|).
In the weak regime, the amplification matrix doesn’t vary significantly across the image (Kochanek, 1990; Miralda-Escude, 1991), a simplification which doesn’t hold for strong lensing.
Combining this with Etherington’s (1933) discovery that gravitational lensing conserves surface brightness, i.e. I(θI) = I(θS), the above definitions can be used to mathematically express the shape deformation of a background galaxy due to lensing.
The lens mapping between the shapes of the source and the image is described by MS =A−1MI A−1T
(5.28) for a singular amplification matrix, or
MI =AMSAT (5.29)
for a non-singular amplification matrixA−1. In the above relations,MSandMIare the moment matrices for the source and the image respectively, and XT denotes the transpose of matrixX.
Thus, for a measured image size ofσI, the size of the sourceσS is given by σS2 = detMS = detMI detA−12
=σI2µ−2. (5.30)
The image size is therefore a factor of the magnification µlarger than that of the source. Simi- larly, in the region outside of the critical lines, the complex ellipticity maps as
S = I −g 1−gI
, for|g|<1 (5.31)
between the source and image planes. For the region inside the critical lines, where|g|> 1, the source ellipticity is
S = 1−g∗∗I
∗I−g∗ , (5.32)
where∗denotes the complex conjugate. In the weak lensing region, i.e. where|g| 1, equation 5.31 simplifies to
I =S +g. (5.33)
Thus in the weak lensing limit the shape of the image is a linear sum of the source shape and the lensing distortion. If the intrinsic shapes of the background galaxies were known before hand, they would be an excellent tracer of the deformations caused by gravitational lensing.
However, galaxies exhibit a copious amount of individual shapes and a measurement of the lensing deformation is only obtainable via averaging over a large number of sources. Under the assumption that intrinsic source shapes are randomly oriented, their ensemble average hSi is zero, and the weak lensing mapping in equation 5.33 reduces to
hIi=hgi. (5.34)
Therefore for the above form of the complex ellipticity (eqn. 5.27), when averaged over many sources, the image ellipticities are a direct measurement of the shearγ, asκ1. Note that this depends on the complicating factors mentioned in the previous section having been dealt with prior to making image shape measurements.