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Astron. Astrophys. 335, 1-11 (1998)

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4. Measurements of the mass distribution

It is not difficult, at least in principle, to calculate the error on [FORMULA] from the two-point correlation function of g. The error on [FORMULA], of course, depends on the reconstruction method used. For this reason, following Sect. 2, we consider different methods separately. For simplicity, we suppose that the weight function W is invariant upon translations. Moreover, we suppose that the angular scale of the weight function W is much smaller than the angular scale of [FORMULA] (i.e. the scale where [FORMULA] varies significantly). In general, if we ignore edge effects, the relation between the error on g and that on [FORMULA] is given by [FORMULA]. Here we show only the results obtained, referring to Appendix C for a derivation. As in Sect. 2, we first refer to the case where [FORMULA] is identified with the whole plane (finite field effects will be addressed in Sect. 4.3 below).

4.1. Weak lensing regime

In 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 [FORMULA]. The result for the mean value has already been stated in Eq. (7), i.e. the measured mass distribution [FORMULA] is the convolution of the weight function W with the true mass distribution [FORMULA].

For an "isolated lens" (a case where [FORMULA] is taken to vanish outside a certain domain [FORMULA]) 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 [FORMULA] one can in principle accept also [FORMULA]. Now if we know that the density vanishes outside [FORMULA], the constant C can be determined by requiring

[EQUATION]

where [FORMULA] is the part of the field not contained in [FORMULA]. Therefore, the appropriate density to be used is

[EQUATION]

The associated total mass is

[EQUATION]

Therefore:

[EQUATION]

where [FORMULA] is the true mass of the lens. In other words, the smoothing effect associated with W does not change the measured total mass M of the lens.

The covariance of the lens distribution [FORMULA] can be shown to be equal to (for both Eqs. (5) and (6)):

[EQUATION]

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 [FORMULA]:

[EQUATION]

[EQUATION]

where, we recall, A is the area used. 1 Obviously, the latter approximate expression holds when [FORMULA]. Curiously, this result does not depend explicitly on the weight function W. The derivation given in Appendix C assumes that the weight function is of the form of Eq. (3), but a similar expression for the variance of M is expected to hold in the more general case.

The results of this subsection can be clarified by a simple example. Instead of introducing the weight function W, we consider the unweighted Eqs. (5) or (6) on small patches of the sky. For simplicity, we refer to a square set [FORMULA] of length L divided into [FORMULA] equal square patches: thus we expect [FORMULA] galaxies per patch. In this case the expected variance of [FORMULA] is (see Paper I) [FORMULA], and the variance of [FORMULA] is of the same order of magnitude. The expected variance of M is then [FORMULA], where the first factor is the variance of [FORMULA] in every patch, the second factor is the area of every patch (the square is necessary because we are dealing with variances), and the third factor arises because we must add [FORMULA] independent variables. The final result for the variance of M is [FORMULA], exactly as stated by Eq. (32).

Here we may come back to the issue of the Poisson noise, only mentioned at the beginning of Sect. 3. Strictly speaking, the relation [FORMULA] in the previous paragraph should be replaced by [FORMULA], with [FORMULA] following a Poisson distribution. We now consider the variance of [FORMULA] as a function of [FORMULA]. An estimate of the effect of the Poisson noise can be obtained in the limit [FORMULA] by expanding

[EQUATION]

Averaging over the ensemble thus yields

[EQUATION]

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 case

The 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 [FORMULA] is much greater than the angular scale of W, then we can prove that the mean value of the measured mass distribution given by Eq. (7) holds unchanged.

Difficulties in the calculation of the covariance of [FORMULA] mainly arise from the form of the covariance of g given by Eq. (25), because of the dependence on [FORMULA] and [FORMULA] of the first factor. However, if the lens has [FORMULA], the covariance given by Eq. (25) is smaller than that of the weak lensing limit of Eq. (26) and thus we can consider all the results given in the weak lensing regime as upper limits for the errors.

4.3. Edge effects

Finite boundaries introduce interesting effects, and make the errors depend on the kernel [FORMULA] used in Eq. (17). For simplicity we take two different sets for [FORMULA] (see Fig. 1). The first set is [FORMULA], i.e. the observation area that includes all the lensed galaxies used in the reconstruction. The second set is [FORMULA], i.e. the set where we measure [FORMULA]. We suppose that every point in [FORMULA] has a neighborhood with radius of the order of the angular scale of W completely enclosed in [FORMULA]. This assumption greatly simplifies calculations and does not have major practical consequences, except that it leads to discarding a small strip [FORMULA] around the boundary [FORMULA] of [FORMULA]. With this hypothesis the expected measured mass distribution is again given by Eq. (7) as long as [FORMULA]. However this is strictly true only if we choose correctly the mean mass distribution [FORMULA] (see Eq. (17)).

In general, the covariance depends on the kernel [FORMULA] used, and in particular on its divergence-free component (see Appendix C). [We recall that a vector field [FORMULA] can be decomposed as [FORMULA], where [FORMULA] has vanishing curl and [FORMULA] has vanishing divergence (as usual, in the above notation and decomposition, the emphasis is on the variable [FORMULA], since [FORMULA] is taken to be fixed).] The result is

[EQUATION]

[EQUATION]

where [FORMULA] is the quantity defined in Eq. (11). The first term is clearly independent of the kernel [FORMULA] used, while the second term can be shown to be positive definite, i.e.

[EQUATION]

Thus the error on [FORMULA] is minimized if a curl-free kernel [FORMULA] is used. This suggests that only curl-free kernels should be used in weak lensing reconstructions. In fact, the kernels so far judged to be "good" by means of simulations, all have vanishing curl (see Sect. 5). For a curl-free kernel, such as the noise filtering kernel given in Eqs. (18-20), the result is independent of the kernel used and of the set [FORMULA].

We now investigate the class of kernels [FORMULA] that satisfy the following properties:

i. [FORMULA] inverts Eq. (11) when [FORMULA] is measured with no error;

ii. [FORMULA] is curl-free.

From the second property we can write

[EQUATION]

Thus, if [FORMULA] we find

[EQUATION]

As in Sect. 2, [FORMULA] is the unit vector orthogonal to [FORMULA]. The last relation shows that, in order to satisfy point i., we must have

[EQUATION]

If [FORMULA] is not known on the boundary of [FORMULA], we should also consider:

[EQUATION]

In this case the kernel to be used is simply the one obtained from [FORMULA], i.e. [FORMULA] (cfr. Eqs. (18-21)). Otherwise, Eq. (40) is to be dropped and the kernel [FORMULA] is determined up to a term [FORMULA], where [FORMULA] is a harmonic function ([FORMULA]). 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 [FORMULA] field is not curl-free, the inversion can only be approximate. The best inversion can thus be found by searching for the function [FORMULA] that minimizes the functional

[EQUATION]

If we vary the distribution [FORMULA], the functional would in general change to [FORMULA], with

[EQUATION]

By setting [FORMULA] we readily find the associated Euler-Lagrange equation

[EQUATION]

This equation should be supplemented by

[EQUATION]

unless we fix the value of [FORMULA] on [FORMULA], so that the boundary term in Eq. (42) vanishes because [FORMULA] on [FORMULA]. Eqs. (43) and (44) define a Neumann boundary problem equivalent to Eqs. (39) and (40) above, in the sense that [FORMULA] is precisely the Green function associated with it. This clarifies the interesting properties of [FORMULA].

4.4. Power spectrum

In order to express in a simple manner the errors involved in the reconstruction process, Seitz & Schneider (1996) introduce a "power spectrum" [FORMULA]. In the weak lensing limit, their definitions are

[EQUATION]

where the mean in Eq. (46) is also over the various directions of [FORMULA]. As a result, the power spectrum [FORMULA] is simply the variance of the complex map [FORMULA], i.e. the Fourier transform of the reconstruction error. Thus, for example, the value of [FORMULA] is proportional to [FORMULA], the error on the total mass, while its behavior for larger values of k is related to the angular scale of the weight function used.

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

[EQUATION]

i.e. a simple Gaussian with variance [FORMULA]. One might anticipate a significant contribution to the power spectrum coming from the error associated with the difference between [FORMULA] and [FORMULA], but we argue in Appendix D that such contribution is negligible in the weak lensing limit.

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998

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