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

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Appendix A: notation

We collect here the main symbols used in this paper.

  • [FORMULA] Superscript s identifies source (unlensed) quantities.
  • [FORMULA] Subscripts refer to the complex representation, e.g. [FORMULA] or, when indicated, to the vector representation, e.g. [FORMULA].
  • [FORMULA], [FORMULA] Unlensed and observed position of a point source.
  • [FORMULA] Critical density.
  • [FORMULA] Projected mass distribution of the lens.
  • [FORMULA] Dimensionless mass distribution: [FORMULA].
  • [FORMULA], Q Unlensed and observed quadrupole moment of an extended image.
  • [FORMULA], [FORMULA] Unlensed and observed (complex) ellipticity of an extended image: [FORMULA], and similarly for [FORMULA].
  • [FORMULA] Complex shear.
  • g Complex reduced shear: [FORMULA].
  • c Covariance of the source ellipticity for an isotropic distribution: [FORMULA].
  • Id, [FORMULA] Identity matrix.

Appendix B: the shear

In this Appendix we derive Eq. (23) and Eq. (25) of Sect. 3. We assume that the reduced shear g has been measured through Eq. (5). Calculations based on Eq. (6) are very similar. As explained in Appendix A of Paper I, the mean value of g obeys the relation

[EQUATION]

Here [FORMULA] is the mean value of [FORMULA] in [FORMULA] and thus depends on [FORMULA]. We now assume that [FORMULA] does not change significantly on the angular scale of [FORMULA]. This implies that we can expand the previous equation to first order in [FORMULA]. We choose, as starting point, the value [FORMULA] given by

[EQUATION]

Then we find easily

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] is the expected value of the ellipticity when the reduced shear is equal to [FORMULA]. Eq. (B3) has the obvious solution [FORMULA], because the first term, [FORMULA], vanishes by definition and the second vanishes due to the choice of [FORMULA] (notice that the partial derivatives in the latter do not depend on [FORMULA]). When moving to a continuous description, we have to calculate the average expected value of g over all possible positions [FORMULA] of the source galaxies

[EQUATION]

The weight function [FORMULA] depends on all [FORMULA] because of the adopted normalization. However, in the limit [FORMULA], the weight function [FORMULA] can be considered to depend only on [FORMULA] alone, so that the above integral can be approximated by

[EQUATION]

This proves Eq. (23). Our result simply states that the use of the first order expansion in g reduces every mean to a weighted arithmetic mean.

Calculations for the covariance of g are much more difficult but basically repeat those given for the unweighted situation in App. A.1 of Paper I. In particular, if we call [FORMULA] the function defined in the l.h.s. of Eq. (5), we have

[EQUATION]

[EQUATION]

where

[EQUATION]

All functions have to be calculated in the mean value of their arguments. Some calculations then lead to

[EQUATION]

The term in brackets in Eq. (B6) can be written in the form

[EQUATION]

[EQUATION]

Here "linear terms" means additional terms linear with respect to the quantity [FORMULA], based on the same expansion defined by (B2).

By averaging over the source positions and by moving to a continuous description (basically following the same steps indicated in (B4) and (B5)), we thus obtain Eq. (25). Notice that the "linear terms" in Eq. (B10) do not give any contribution when averaged over the source positions. We stress that the results stated here are valid only if the weight function is even (property 1 of Sect. 2.1).

Finally, we point out that the approximation that takes us from (B4) to (B5) is precisely associated with neglecting the Poisson noise.

Appendix C: the lens mass distribution

In this Appendix we will derive Eq. (7) and the results stated in Sect. 4, assuming a weight function invariant upon translations (case 2 of Sect. 2.1).

C.1. Weak lensing limit

Calculations in the weak lensing limit are not difficult. As explained in Sect. 2.2, we can use either Eq. (8) or Eqs. (11) and (12) to convert the reduced shear into the mass distribution.

In the case of Eq. (8) we can write

[EQUATION]

Here the star denotes convolution, while [FORMULA] is the component i of the true shear map [FORMULA]. The second step is justified by Eq. (23) applied in the weak lensing limit. By using the associative and commutative properties of the convolution and by noting that [FORMULA], we can write

[EQUATION]

This proves Eq. (7).

About the covariance of [FORMULA] we can write

[EQUATION]

As [FORMULA] and [FORMULA] are even functions, this is simply a double convolution, and thus the result depends only on the difference between [FORMULA] and [FORMULA]. Therefore we can write for [FORMULA] the expression

[EQUATION]

i.e. Eq. (31). Here we have used again the commutative property of convolutions and the relation [FORMULA] given by Eqs. (8) and (10). [Hereafter [FORMULA] means the Dirac delta distribution.]

The results are the same if we use Eqs. (11) and (12). In fact we can write Eq. (11) as a convolution between [FORMULA] and the operator

[EQUATION]

where [FORMULA]. Thus we are allowed to use the properties of convolutions. It is obvious then that the convolution with the weight function W in [FORMULA] can be moved to the true lens distribution [FORMULA], and we find again the result of Eq. (C2). The covariance matrix of [FORMULA] can be calculated using the operator (C5). As a result, we find

[EQUATION]

We then have

[EQUATION]

A simple calculation shows that [FORMULA]. This leads again to Eq. (31).

Now let us prove Eq. (32). From Eq. (C4) and Eq. (26) we find

[EQUATION]

In the limit [FORMULA] explained in Sect. 4.1, the main contribution to the variance of M derives from a double integration of [FORMULA]. A simple change of variables gives

[EQUATION]

Here [FORMULA] is the Fourier transform of W and the last equality holds because of the normalization (22) of the weight function.

C.2. The general case

In the general case we restrict ourselves to estimating the mean value of the lens distribution because calculations for the covariance are too difficult. Under the hypothesis that the angular scale of W is much smaller than the angular scale of [FORMULA] (or g), the situation is much like that of the weak lensing limit. As shown in Appendix B, this basically implies that all averages are weighted arithmetic averages. Simple calculations show that we have [FORMULA], and hence [FORMULA]. As usual the assumed ordering of scale lengths leads again to Eq. (C2).

C.3. Edge effects

For simplicity we refer to [FORMULA]. We rewrite Eqs (17), (11) and (10) with a different notation

[EQUATION]

Here [FORMULA] is the characteristic operator for the set [FORMULA]:

[EQUATION]

Eq. (C10) is equivalent to Eq. (17) with [FORMULA] if we redefine the kernel [FORMULA] for every [FORMULA] and [FORMULA] so that [FORMULA] if either [FORMULA] or [FORMULA]. With this simple definition we can extend the integration domain (usually [FORMULA]) to the whole plane. Notice that while [FORMULA] and [FORMULA] are used in convolutions, [FORMULA] is a generic linear operator. From these equations we have

[EQUATION]

and thus we find the identity

[EQUATION]

Eq. (12) with the new notation is

[EQUATION]

This, together with Eqs. (C11) and (C12), gives us another identity:

[EQUATION]

Using Eqs. (C10), (C12), and the relation [FORMULA], we can easily obtain the mean value for measures of [FORMULA]:

[EQUATION]

As usual, subscript 0 indicates the true value of a quantity. This equation, rewritten in the more standard notation, is Eq. (7) for [FORMULA].

Let us calculate the covariance of [FORMULA]. First of all note that, while Eq. (C14) implies Eq. (C15), from Eq. (C17) we cannot deduce that [FORMULA] is the identity. This happens because the two components of [FORMULA] are not functionally independent, as one can see from the relation [FORMULA]. In fact, using Fourier transforms it is easy to prove that the operator [FORMULA] selects the curl-free component of a vector field. Its Fourier transform is

[EQUATION]

From (C10) we have

[EQUATION]

Every vector field can be written as the sum of two vector fields, of which one is curl-free and the other is divergence-free. Hence, if we consider [FORMULA] a vector field with respect to [FORMULA], we can write

[EQUATION]

where

[EQUATION]

Thus [FORMULA] and [FORMULA] can be written as the gradient and the "curl" of two scalar fields:

[EQUATION]

There is some freedom in the choice of [FORMULA] and [FORMULA] (or equivalently of [FORMULA] and [FORMULA]). However, it is always possible to choose [FORMULA] and [FORMULA] so that they vanish for [FORMULA]. With the decomposition (C21) we have [FORMULA] and thus

[EQUATION]

Recalling now the definition of [FORMULA] and using Eq. (C15) we find

[EQUATION]

If the kernel [FORMULA] has vanishing curl, then [FORMULA] and we find the final result

[EQUATION]

[EQUATION]

In general however we must evaluate three additional terms. Two of them are of the form

[EQUATION]

where we have used Eq. (C24). By the change of variable [FORMULA] and after integrating by parts we find

[EQUATION]

where the last relation holds in virtue of Eq. (C23). We finally rewrite Eq. (C27) in a simplified form:

[EQUATION]

Here the first term is independent of the specific kernel [FORMULA] used, while the second term depends only on [FORMULA]. As, by definition, [FORMULA] is positive definite, the last term in Eq. (C31) is also positive definite. In other words if [FORMULA] the error on [FORMULA] will increase.

Appendix D: power spectrum

The power spectrum reported in Eq. (47) can be deduced from the expression of the mean and covariance of the measured mass distribution. In particular we have

[EQUATION]

[EQUATION]

The subscripts in the exponentials denote real ([FORMULA]) and imaginary ([FORMULA]) parts. The power spectrum [FORMULA] is directly related to the covariance of [FORMULA]. In fact, we have

[EQUATION]

with summation implied on i and mean over all directions of [FORMULA]. For a large set [FORMULA] we can perform integrations over the whole plane. Thus we find

[EQUATION]

Here, as usual, hats indicate Fourier transform. In the weak lensing limit the second term of this expression can be dropped. Hence, if W is a Gaussian of the form of Eq. (3) we find Eq. (47). [Note that no averaging over the direction of [FORMULA] is needed in the weak lensing limit if the weight function has the form (3).]

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998

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