## Appendix A: notationWe collect here the main symbols used in this paper. - Superscript s identifies
*source*(unlensed) quantities. - Subscripts refer to the complex representation, e.g. or, when indicated, to the vector representation, e.g. .
- , Unlensed and observed position of a point source.
- Critical density.
- Projected mass distribution of the lens.
- Dimensionless mass distribution: .
- ,
*Q*Unlensed and observed quadrupole moment of an extended image. - , Unlensed and observed (complex) ellipticity of an extended image: , and similarly for .
- Complex shear.
*g*Complex reduced shear: .*c*Covariance of the source ellipticity for an isotropic distribution: .- Id, Identity matrix.
## Appendix B: the shearIn this Appendix we derive Eq. (23) and Eq. (25) of
Sect. 3. We assume that the reduced shear 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 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 Calculations for the covariance of where 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 distributionIn 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 limitCalculations 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 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 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 Here is the Fourier transform of ## C.2. The general caseIn 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
## C.3. Edge effectsFor 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 where 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 spectrumThe 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 Here, as usual, hats indicate Fourier transform. In the weak
lensing limit the second term of this expression can be dropped.
Hence, if © European Southern Observatory (ESO) 1998 Online publication: June 12, 1998 |