3. Measurements of the reduced shear map and of the two-point correlation function
In this section we will give an expression for the reduced shear map measured using Eqs. (5) or (6). In Paper I we have calculated the statistics associated with a local shear measurement under the hypothesis that the probability distribution for the source ellipticity is sharp, i.e. most source galaxies are nearly round. Now we consider situations where the reduced shear is a function of the position , but we assume that , a smooth function of , does not change significantly on the angular scale of the weight function . An important new aspect of the analysis that has to be addressed here, in view of the goal of determining the error on the reconstructed mass, is the calculation of the two-point correlation function for the shear map.
So far we have considered source galaxies with random orientation but with fixed position on the sky (corresponding to on the observer's sky). It is interesting to average all results by assuming that galaxies have random positions. The result of the average can be approximated by considering a continuous distribution of galaxies with density (number of galaxies per steradian). This leads us to ignore, for the moment, the effects of Poisson noise associated with the finite number of source galaxies (further comments are given at the end of Sect. 4.1). Following Seitz & Schneider (1996), we consider a homogeneous distribution of galaxies in , i.e. in the observer's plane. If is independent of , we may change summations with integrals using the rule . Here, to simplify the derivations and the discussion, we adopt, for every , the normalization
for the weight function. For an infinite field or for the case described in Fig. 1, this normalization does not break the translation invariance of W. Then as shown in Appendix B, the relation between expected and true value of g, corresponding to Eq. (7), is
As is intuitive, "near" galaxies give the most important contribution to the measured value of g.
The correct generalization of the covariance matrix when g is a function of the position is a two-point correlation function:
Note that the knowledge of the "diagonal" values is not sufficient to calculate the error on other variables, such as the density distribution , determined from g (cf. Eq. (32) and Eq. (35)).
If we assume that the weight function is even (property 1 of Sect. 2.1), then the two-point correlation function of g can be written in the simple form (see Appendix B)
Here c is the covariance of the ellipticity distribution of the source galaxies. In this equation, as noted for Eq. (23), we suppose the weight function to be normalized.
In the weak lensing limit Eq. (25) then reduces to
The last relation holds for a Gaussian weight function of the form given in Eq. (3); here is a simple Gaussian with variance and depends only on . The variance of is simply (without summation on i) and thus increases if decreases. This behavior can be explained by considering that the number of galaxies used for a single point is of the order of . Notice also that sets the scale length of the covariance of : measurements of and are uncorrelated if .
© European Southern Observatory (ESO) 1998
Online publication: June 12, 1998