## 2. From the shear map to the mass distributionWe consider a field of the sky with ## 2.1. Spatial weight functionSource galaxies located close to a given position will better constrain the value of the reduced shear at such location. In order to describe this effect, we may thus introduce a suitable weight function . The first argument of the weight function, , represents the point of the sky under consideration and for which we want to measure the shear , while the second argument represents the location of one observed galaxy. The weight function should penalize galaxies far from , i.e. should decrease for increasing . Some additional "natural" conditions can be given to further characterize a specific choice of weight function and these are most convenient when applied (beginning with Eq. (22)) to a spatially continuous distribution of source galaxies with density . Here we list a few possible assumptions, where the first is obviously the least restrictive: 1. The weight function is 2. The weight function is said 3. One natural choice is that of a where the angular scale should be sufficiently large to ensure the presence of an adequate number of galaxies in a disk of radius centered on the generic point . The value of the weight function at a given point
, of course, has no particular meaning: only
for every . Of course, such a normalization
will remove the translation invariance property, if initially present
in The spatial weight function operates much like the "shape" weight functions considered in Paper I (see Eqs. (23) and (21) there). In particular, using the isotropy condition, we can obtain the shear map either from or from In Paper I we discussed the different merits of the two options. In the limit of "sharp" distributions for the source galaxies (), they both lead to the same determination of the true reduced shear (apart from the ambiguity associated with the invariance). As we will see the angular scale of the weight function This relation, similar to that found when an image is degraded by a PSF, suggests the possible application of deconvolution techniques (see Lucy 1994) to the present context. ## 2.2. Weak lensing regimeWe first consider the case where is very large, so that the field is identified with the whole plane (the effects of the boundaries will be discussed soon, in Sect. 2.4). There are basically two ways to reconstruct the mass distribution from the shear map (Seitz & Schneider 1996). The first, more natural method is based on the integral relation where the kernel is given by (Kaiser & Squires 1993) In the weak lensing limit , and thus the
reduced shear map can be used directly in Eq. (8) to derive
. Note that the inverse relation holds with the
This relation will turn out to be useful in Appendix C. A second possibility, which can be proved to be mathematically equivalent to the first, is based on the exact relation (see Kaiser 1995) which is a direct consequence of the thin lens equations. Here . By analogy with the condition used to derive Eq. (8), if we assume that vanishes for large values of , Eq. (11) can be inverted to give with the kernel In Eq. (12) the shear map enters through the vector , which, in the weak lensing limit, involves the derivatives of . This second method thus introduces undesired differentiations, but it has the advantage that it is more easily generalized to include the effects of the boundaries (see Sect. 2.4 below). ## 2.3. The general caseWhen the lens is not weak, Eq. (8) can be solved by iteration (Seitz & Schneider 1995). The second method, related to Eq. (12), has been generalized by Kaiser (1995) for strong lenses. If we introduce and the new vector then it is possible to show that the relation holds. As a result can be obtained from via the same integral equation (12) used earlier. The fact that is determined only up to a constant here translates into a non-trivial invariance for the density distribution , under the transformation (see Schneider & Seitz 1995) consistent with Eq. (14). ## 2.4. Effect of the boundariesThe methods described so far assume an infinite domain of integration. In practice, one can measure the shear only in a finite area (e.g. the CCD area), which is often small compared to the angular size of the lensing cluster. Therefore, the relations given earlier should be properly modified. We briefly noted that the second method is better suited for the
purpose. In the following, for simplicity, we consider only the weak
lensing limit, but the equations that we will provide can be easily
extended to the general case by replacing . The
relations suggested by Seitz & Schneider (1996) for mass
reconstruction in a field of finite area
Here is a constant representing the average
of , while is a suitable
kernel. The kernel is chosen so as to give the correct mass
distribution if could be measured with no
errors (see Eq. (C14)). There is however some freedom left in the
choice of the kernel, mainly because it returns a scalar field
() from a vector field ().
This freedom will be further discussed later on. One interesting
kernel, called where is the solution of Neumann's boundary problem ( is the unit vector orthogonal to ) The term related to the area © European Southern Observatory (ESO) 1998 Online publication: June 12, 1998 |