Appendix A: notation
We collect here the main symbols used in this paper.
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
Here is the mean value of in and thus depends on . We now assume that does not change significantly on the angular scale of . This implies that we can expand the previous equation to first order in . We choose, as starting point, the value given by
Then we find easily
where is the expected value of the ellipticity when the reduced shear is equal to . Eq. (B3) has the obvious solution , because the first term, , vanishes by definition and the second vanishes due to the choice of (notice that the partial derivatives in the latter do not depend on ). When moving to a continuous description, we have to calculate the average expected value of g over all possible positions of the source galaxies
The weight function depends on all because of the adopted normalization. However, in the limit , the weight function can be considered to depend only on alone, so that the above integral can be approximated by
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 the function defined in the l.h.s. of Eq. (5), we have
All functions have to be calculated in the mean value of their arguments. Some calculations then lead to
The term in brackets in Eq. (B6) can be written in the form
Here "linear terms" means additional terms linear with respect to the quantity , 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
Here the star denotes convolution, while is the component i of the true shear map . 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 , we can write
This proves Eq. (7).
About the covariance of we can write
As and are even functions, this is simply a double convolution, and thus the result depends only on the difference between and . Therefore we can write for the expression
i.e. Eq. (31). Here we have used again the commutative property of convolutions and the relation given by Eqs. (8) and (10). [Hereafter 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 and the operator
where . Thus we are allowed to use the properties of convolutions. It is obvious then that the convolution with the weight function W in can be moved to the true lens distribution , and we find again the result of Eq. (C2). The covariance matrix of can be calculated using the operator (C5). As a result, we find
We then have
A simple calculation shows that . This leads again to Eq. (31).
Now let us prove Eq. (32). From Eq. (C4) and Eq. (26) we find
In the limit explained in Sect. 4.1, the main contribution to the variance of M derives from a double integration of . A simple change of variables gives
Here 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 (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 , and hence . As usual the assumed ordering of scale lengths leads again to Eq. (C2).
C.3. Edge effects
For simplicity we refer to . We rewrite Eqs (17), (11) and (10) with a different notation
Here is the characteristic operator for the set :
Eq. (C10) is equivalent to Eq. (17) with if we redefine the kernel for every and so that if either or . With this simple definition we can extend the integration domain (usually ) to the whole plane. Notice that while and are used in convolutions, is a generic linear operator. From these equations we have
and thus we find the identity
Eq. (12) with the new notation is
This, together with Eqs. (C11) and (C12), gives us another identity:
Using Eqs. (C10), (C12), and the relation , we can easily obtain the mean value for measures of :
As usual, subscript 0 indicates the true value of a quantity. This equation, rewritten in the more standard notation, is Eq. (7) for .
Let us calculate the covariance of . First of all note that, while Eq. (C14) implies Eq. (C15), from Eq. (C17) we cannot deduce that is the identity. This happens because the two components of are not functionally independent, as one can see from the relation . In fact, using Fourier transforms it is easy to prove that the operator selects the curl-free component of a vector field. Its Fourier transform is
From (C10) we have
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 a vector field with respect to , we can write
Thus and can be written as the gradient and the "curl" of two scalar fields:
There is some freedom in the choice of and (or equivalently of and ). However, it is always possible to choose and so that they vanish for . With the decomposition (C21) we have and thus
Recalling now the definition of and using Eq. (C15) we find
If the kernel has vanishing curl, then and we find the final result
In general however we must evaluate three additional terms. Two of them are of the form
where we have used Eq. (C24). By the change of variable and after integrating by parts we find
where the last relation holds in virtue of Eq. (C23). We finally rewrite Eq. (C27) in a simplified form:
Here the first term is independent of the specific kernel used, while the second term depends only on . As, by definition, is positive definite, the last term in Eq. (C31) is also positive definite. In other words if the error on 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
The subscripts in the exponentials denote real () and imaginary () parts. The power spectrum is directly related to the covariance of . In fact, we have
with summation implied on i and mean over all directions of . For a large set we can perform integrations over the whole plane. Thus we find
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 is needed in the weak lensing limit if the weight function has the form (3).]
© European Southern Observatory (ESO) 1998
Online publication: June 12, 1998