## 3. Measurements of the reduced shear map and of the two-point correlation functionIn 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 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 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 As is intuitive, "near" galaxies give the most important
contribution to the measured value of
The correct generalization of the covariance matrix
when 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 If we assume that the weight function is Here 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 © European Southern Observatory (ESO) 1998 Online publication: June 12, 1998 |