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Astron. Astrophys. 335, 1-11 (1998)

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2. From the shear map to the mass distribution

We consider a field of the sky with N source galaxies located at [FORMULA] and characterized by observed quadrupole [FORMULA] and ellipticity [FORMULA] (see Appendix A for a summary of the adopted notation). Here we suppose that the galaxies are observed inside a field [FORMULA] of area A, with mean spatial density equal to [FORMULA]. In the rest of the paper we will reserve the term "weak lensing" to the limit of small lens density [FORMULA], i.e. [FORMULA].

2.1. Spatial weight function

Source galaxies located close to a given position [FORMULA] will better constrain the value of the reduced shear [FORMULA] at such location. In order to describe this effect, we may thus introduce a suitable weight function [FORMULA]. The first argument of the weight function, [FORMULA], represents the point of the sky under consideration and for which we want to measure the shear [FORMULA], while the second argument [FORMULA] represents the location of one observed galaxy. The weight function should penalize galaxies far from [FORMULA], i.e. [FORMULA] should decrease for increasing [FORMULA]. 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 [FORMULA]. Here we list a few possible assumptions, where the first is obviously the least restrictive:

1. The weight function is even with respect to [FORMULA], i.e.

[EQUATION]

2. The weight function is said invariant upon translations, if it is even (see above) and if it depends only on the difference [FORMULA]:

[EQUATION]

3. One natural choice is that of a Gaussian dependent only on the distance [FORMULA],

[EQUATION]

where the angular scale [FORMULA] should be sufficiently large to ensure the presence of an adequate number of galaxies in a disk of radius [FORMULA] centered on the generic point [FORMULA].

The value of the weight function at a given point [FORMULA], of course, has no particular meaning: only relative values are significant. Indeed all the following results can be shown to be unaffected if we merely multiply the weight function by a constant. Thus, we may always choose a normalized weight function, so that

[EQUATION]

for every [FORMULA]. Of course, such a normalization will remove the translation invariance property, if initially present in W. Still, the invariance may be retained when we will move to a continuous description (see comment after Eq. (22)).

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 [FORMULA] either from

[EQUATION]

or from

[EQUATION]

In Paper I we discussed the different merits of the two options. In the limit of "sharp" distributions for the source galaxies ([FORMULA]), they both lead to the same determination of the true reduced shear [FORMULA] (apart from the ambiguity associated with the [FORMULA] invariance).

As we will see the angular scale of the weight function W, i.e. the diameter of the set where [FORMULA] is significantly different from zero, determines a lower bound for the smallest details shown in the reconstructed map [FORMULA]. For example, in the weak lensing limit and for a weight function invariant upon translations, the mean value of the measured density [FORMULA] is related to the true density [FORMULA] through the expression (see Appendix C)

[EQUATION]

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 regime

We first consider the case where [FORMULA] 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 [FORMULA] from the shear map [FORMULA] (Seitz & Schneider 1996). The first, more natural method is based on the integral relation

[EQUATION]

where the kernel [FORMULA] is given by (Kaiser & Squires 1993)

[EQUATION]

In the weak lensing limit [FORMULA], and thus the reduced shear map can be used directly in Eq. (8) to derive [FORMULA]. Note that the inverse relation holds with the same kernel

[EQUATION]

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)

[EQUATION]

which is a direct consequence of the thin lens equations. Here [FORMULA]. By analogy with the condition used to derive Eq. (8), if we assume that [FORMULA] vanishes for large values of [FORMULA], Eq. (11) can be inverted to give

[EQUATION]

with the kernel

[EQUATION]

In Eq. (12) the shear map enters through the vector [FORMULA], which, in the weak lensing limit, involves the derivatives of [FORMULA]. 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 case

When the lens is not weak, Eq. (8)

[EQUATION]

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 [FORMULA] and the new vector

[EQUATION]

then it is possible to show that the relation [FORMULA] holds. As a result [FORMULA] can be obtained from [FORMULA] via the same integral equation (12) used earlier. The fact that [FORMULA] is determined only up to a constant here translates into a non-trivial invariance for the density distribution [FORMULA], under the transformation (see Schneider & Seitz 1995)

[EQUATION]

consistent with Eq. (14).

2.4. Effect of the boundaries

The 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 [FORMULA]. The relations suggested by Seitz & Schneider (1996) for mass reconstruction in a field [FORMULA] of finite area A are of the form

[EQUATION]

Here [FORMULA] is a constant representing the average of [FORMULA], while [FORMULA] is a suitable kernel. The kernel is chosen so as to give the correct mass distribution if [FORMULA] 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 ([FORMULA]) from a vector field ([FORMULA]). This freedom will be further discussed later on. One interesting kernel, called noise filtering, has been introduced by Seitz & Schneider (1996)

[EQUATION]

where [FORMULA] is the solution of Neumann's boundary problem ([FORMULA] is the unit vector orthogonal to [FORMULA])

[EQUATION]

[EQUATION]

The term related to the area A ensures the proper applicability of the Gauss theorem. Note that the kernel [FORMULA] has vanishing curl, i.e.

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998

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