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WEAK LENSING
Outline


Lensing in the WeakDistortion Regime
Introduction
Gravitational lensing can be categorised into two broad categories. Strong lensing occurs in the regime in which the convergence is large, κ >~ 1, causing the production of multiple images. However, even in the weak regime, where κ << 1, a great deal of cosmological information can be extracted. Furthermore since the vast majority of galaxies will lie along a line of sight in which there are no large over densities we may expect that the cosmological information inherent in the weakly distorted galaxy images coupled with the overwhelming number of weakly lensing galaxies may result in a powerful cosmological probe.
In the absence of a concentrated compact object close to the path taken by light, lensing occurs along the entire path from the background object due to density perturbations δ along the line of sight. Since this lensing occurs over the entire journey taken by the light, we can no longer use the thin lens approximation. Instead we need to develop the idea of an effective convergence, by integrating along the line of sight, which allows us to continue using the distortion matrix, discussed in general lensing, to study the lensing properties. The distortion matrix causes both a change in size and shape of the lensed object. Figure WL1 shows the type of distortion that occurs when a circle of unit radius is lensed.
Figure WL1 (taken from Munshi et al., 2006) Shows the effects that convergence and shear have on a circular image. Convergence acts to increase the average radius of the image and shear acts to stretch the circle into an ellipse.
The circle becomes an ellipse with a major axis, a=1/(1κ γ), and minor axis, b=1/(1κ +γ). There are many ways we can look at the properties of an ellipse. For example, if we look at the quantity g=(ab)/(a+b) we find
WL1
This quantity is known as the reduced shear. By rearranging the distortion matrix to
WL2
we see that the change in the size of an image is completely described by a (1κ) term and that shape distortion is completely described by the reduced shear. By understanding the change in distortion and/or the magnification of lensed objects, we can hope to understand the lensing medium.
Weak Lensing
Weak lensing produces weakly distorted, single, images of sources outside of caustics. For weak lensing both κ<< 1 and γ<< 1, equivalently g<< 1.
As previously stated there are two ways in which weak lensing can produce a measurable effect. Firstly background galaxies will be weakly lensed by foreground structure, either by LSS or by foreground galaxy clusters (see Villumsen, 1996). Secondly the weak magnification effect can change
the observed number density of source background galaxies or change the size of images of a given surface brightness (see Schneider, 2006
for a review). The use of the weak distortion of background galaxies for cosmology is known as cosmic shear measurements.
Real galaxies are not perfect circles (as is Figure WL1) but instead, they have structure and their own intrinsic shape that must be taken into consideration. Since galaxies are generally noncircular in shape the shear effect induces an extra, additional ellipticity. It is this additional ellipticity that is induced by lensing foreground structure. We must now come up with a formalism that can describe this additional ellipticity. Or put another way we want to measure the shape of a galaxy and determine statistically how significant the the shapes of the galaxies are with respect to what we expect the average shape to be.
The typical starting point for this is to define ellipticity ε as
WL3
An alternative definition that one can use is
WL4
however we will use the definition of equation (WL3) throughout.
To relate the definition of ellipticity to the shear we can follow the standard route (see Bartelmann & Schneider, 2001) of considering the second order surface brightness moments of a galaxies image.
Consider a galaxy for with the surface brightness profile I(θ) is well defined for all angular separations θ from the centre of the image, so that:
WL5
where w[I(θ)] is a suitably chosen weight function such that the integrals converge. The tensor of second brightness moments
is now:
WL6
The trace part of Q contains the size information whilst the traceless part contains ellipticity information; for a circular image Q_{11}=Q_{22} and Q_{12}=Q_{21}=0. From the definition of Q_{ij} a complex ellipticity can be defined (in direct analogy with complex shear  see general lensing):
WL7
where
WL8
or
WL9
Both these equally valid definitions have the same phase but different amplitudes. Here we will use the definition from equation (WL9) will be used. Note for a circular image ε_{1}=ε_{2}=0. Schneider (2006) showed that the original (intrinsic) ellipticity ε^{S} of a galaxy is transformed under lensing, using equation (WL9) like:
WL10
the inverse transformation is obtained by interchanging ε with ε^{*} and replacing g with g. In the weak lensing case g<< 1 the inverse of equation (WL10) reduces to:
WL11
This is a very simple and intuitive result. The observed ellipticity of a galaxy is simply the original/intrinsic ellipticity plus some extra ellipticity component due to lensing structure along the line of sight.
Since the ellipticity of any individual source is unknown (we do not a priori know the shape of any galaxy) the above expressions (upto equation WL11) are of little use when applied to individual galaxies. However, the key realisation in weak lensing is that when a large statistical sample of galaxies is used the average intrinsic ellipticity should be zero as stated earlier i.e. since there should be no preferred orientation of galaxies in the Universe:
WL12
So the average additional ellipticity (which we now know is the reduced shear) in the weak lensing regime is:
WL13
By assuming that a galaxy sample covers a small angular patch on the sky, so that the light from each part of the galaxy experiences approximately the same gravitational field an estimator for the shear can be deduced
WL14
The result here is actually for sources all at a fixed redshift, however Bartelmann & Schneider, 2001 show that this result is in fact general for a
redshift distributed source population.
Equation (WL13) also implies that the variance in the shear is related to the variance in the ellipticity by σ^{2}_{γ}=σ^{2}_{ε}, or for the individual components γ_{1} and γ_{2}, σ^{2}_{γα}=σ^{2}_{ε}/2. Note that if the definition of complex ellipticity in equation (WL8) is used then<ε>=2<g> and γ~<ε>/2 with σ^{2}_{γα}=σ^{2}_{ε}/4.
To measure the ellipticity of a galaxy a number of complementary techniques have been developed. The first to be developed, and most
commonly used is known as the KSB test after Kaiser, Squires & Broadhurst (1995), which allows for the accurate removal of the smearing of a galaxies image due to an anisotropic point spread function (PSF) on an instrument. Shapelets, see Refregier, Chang & Bacon (2002), is an alternative where a galaxy's image is decomposed into spherical polar harmonics, the shear signal corresponding to particular `quantum numbers' in the devolution. More recently Bayesian model fitting methods have been shown to yield unbiased estimates of the shear, see Miller et al. (2007) and Kitching et al. (2008). For reviews of shape measurement methods see the Shear TEsting Programme (STEP1 and STEP2) and the GRavitational LEnsing Accuracy Testing 2008 (GREAT08) Pascal challenge.
Tangential Shear
Weak lensing is typically studied in two regimes. Either around large galaxy clusters, or in the field (i.e. away from any large over densities) this second area of weak lensing is known as cosmic shear and is review in cosmic lensing. When studying weak lensing around galaxy clusters the shear signal is referred to as tangential shear since the distortion effect of a cluster should be to introduce a alignment of galaxy images at tangential to the line from the galaxy image to the cluster center.
Figure WL2: A lensing cluster Abell 1689 showing tangentially aligned lensed galaxy images.
The components of complex shear γ_{1} and γ_{2} are defined relative to a local Cartesian coordinate frame. However it is often apt to consider the projected shear components in a rotated frame, particularly in the case of galaxy clusters where the centre of the polar coordinate frame can be defined as the centre of the cluster. For a lensing cluster the image distortions are aligned tangentially about the cluster as can be seen in Figure WL2. If φ_{c} specifies the angular position about the centre of the coordinate frame the the tangential and crosscomponent shears (aligned respectively perpendicular and parallel to the radius vector) are:
WL15
or using the definition for complex shear γ=γ_{1}+iγ_{2}
WL16
A perfect lensing cluster should only produce a tangential signal in the shear so the crosscomponent shear (which should be γ_{X}=0) can be used to estimate the noise on the measurement of the tangential shear.
Five Easy Steps
 Exercise 1
 Exercise 2
 Exercise 3
 Exercise 4
 Exercise 5
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