## 4. Measurements of the mass distributionIt is not difficult, at least in principle, to calculate the error
on from the two-point correlation function of
## 4.1. Weak lensing regimeIn this case we can use either Eq. (8) or Eq. (12). A
rather surprising result is that both methods lead to the same mean
values and errors for . The result for the mean
value has already been stated in Eq. (7), i.e. the measured mass
distribution is the convolution of the weight
function For an "isolated lens" (a case where is taken
to vanish outside a certain domain ) the
ambiguity associated with Eq. (16) is resolved and the concept of
total mass of the lens becomes meaningful. In the weak lensing limit,
from any reconstructed one can in principle
accept also . Now if we know that the density
vanishes outside , the constant where is the part of the field not contained in . Therefore, the appropriate density to be used is The associated total mass is Therefore: where is the true mass of the lens. In other
words, the The covariance of the lens distribution can be shown to be equal to (for both Eqs. (5) and (6)): In comparing Eq. (31) to Eq. (26), one should note that the similarity of results refers statistically to average errors, but not to the individual errors of one reconstruction. The variance in the measure of the total mass is the integral of the covariance of : where, we recall, The results of this subsection can be clarified by a simple
example. Instead of introducing the weight function Here we may come back to the issue of the Poisson noise, only mentioned at the beginning of Sect. 3. Strictly speaking, the relation in the previous paragraph should be replaced by , with following a Poisson distribution. We now consider the variance of as a function of . An estimate of the effect of the Poisson noise can be obtained in the limit by expanding Averaging over the ensemble thus yields The effect of the Poisson noise is here contained in the second term in brackets, which is negligibly small. Additional discussion is postponed to the end of Appendix B. ## 4.2. The general caseThe situation is, in principle, quite similar to the weak lensing
limit, but, in practice, the calculations are much more difficult. If
the angular scale of is much greater than the
angular scale of Difficulties in the calculation of the covariance of
mainly arise from the form of the covariance of
## 4.3. Edge effectsFinite boundaries introduce interesting effects, and make the
errors depend on the kernel used in
Eq. (17). For simplicity we take two different sets for
(see Fig. 1). The first set is
, i.e. the observation area that includes all
the lensed galaxies used in the reconstruction. The second set is
, i.e. the set where we measure
. We suppose that every point in
has a neighborhood with radius of the order of
the angular scale of In general, the covariance depends on the kernel used, and in particular on its divergence-free component (see Appendix C). [We recall that a vector field can be decomposed as , where has vanishing curl and has vanishing divergence (as usual, in the above notation and decomposition, the emphasis is on the variable , since is taken to be fixed).] The result is where is the quantity defined in Eq. (11). The first term is clearly independent of the kernel used, while the second term can be shown to be positive definite, i.e.
We now investigate the class of kernels that satisfy the following properties:
From the second property we can write Thus, if we find As in Sect. 2, is the unit vector
orthogonal to . The last relation shows that,
in order to satisfy point If is not known on the boundary of , we should also consider: In this case the kernel to be used is simply the one obtained from , i.e. (cfr. Eqs. (18-21)). Otherwise, Eq. (40) is to be dropped and the kernel is determined up to a term , where is a harmonic function (). This free function can be used to dispose of the contribution that would arise from the boundary term in Eq. (38). As, in general, the measured field is not curl-free, the inversion can only be approximate. The best inversion can thus be found by searching for the function that minimizes the functional If we vary the distribution , the functional would in general change to , with By setting we readily find the associated Euler-Lagrange equation This equation should be supplemented by unless we fix the value of on , so that the boundary term in Eq. (42) vanishes because on . Eqs. (43) and (44) define a Neumann boundary problem equivalent to Eqs. (39) and (40) above, in the sense that is precisely the Green function associated with it. This clarifies the interesting properties of . ## 4.4. Power spectrumIn order to express in a simple manner the errors involved in the reconstruction process, Seitz & Schneider (1996) introduce a "power spectrum" . In the weak lensing limit, their definitions are where the mean in Eq. (46) is also over the various directions
of . As a result, the power spectrum
is simply the variance of the complex map
, i.e. the Fourier transform of the
reconstruction error. Thus, for example, the value of
is proportional to ,
the error on the total mass, while its behavior for larger values of
Within our framework it is not difficult to evaluate the relevant power spectrum. A simple calculation (see Appendix D) for a Gaussian weight function gives i.e. a simple Gaussian with variance . One might anticipate a significant contribution to the power spectrum coming from the error associated with the difference between and , but we argue in Appendix D that such contribution is negligible in the weak lensing limit. © European Southern Observatory (ESO) 1998 Online publication: June 12, 1998 |