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Astron. Astrophys. 318, 687-699 (1997)

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6. Application to simulated data

For testing the reconstruction technique we use the same numerically modelled cluster as in Paper II. Using its surface mass density shown in Fig. 2, we generate a distribution of synthetic galaxy images using the source ellipticity distribution

[EQUATION]

and the redshift-distribution (3.8) with [FORMULA] and [FORMULA]. The galaxies are randomly distributed in the source plane, and the number density of galaxy images is [FORMULA]. The cluster is placed at a redshift [FORMULA] so that [FORMULA], [FORMULA] and [FORMULA].

[FIGURE] Fig. 2. The surface mass density distribution of the numerically modelled cluster. The sidelength is about [FORMULA] corresponding to 3.8 Mpc for [FORMULA] km/s/Mpc and an EdS universe. The levels of the contour lines are [FORMULA].

6.1. The derived mass distribution for different assumed redshift distributions of the sources

We show in Fig. 3 the resulting surface mass density map for [FORMULA], [FORMULA] and [FORMULA], i.e., the assumed redshift distribution is the true one. We find a good agreement with the original mass density displayed in Fig. 2. Compared to the mass reconstruction shown Paper II - where all sources were at the same redshift - the noise is increased due to the dependence of the lensing strength of the cluster on the redshift of the sources.

[FIGURE] Fig. 3. The reconstructed mass density obtained solving Eq. (4.9) iteratively, assuming that the redshift distribution is known and assuming [FORMULA]. We obtain almost the same result if we choose [FORMULA] such that the minimum of the locally-smoothed mass density is zero. This assumption leads to [FORMULA]. The levels of the contour lines are the same as in Fig. 2

However, in practice we do not know [FORMULA] and we therefore choose [FORMULA] such that the minimum of the - locally smoothed - reconstructed surface mass density is zero. The reason for not requiring the minimum of the reconstructed mass map to be zero but that of the locally smoothed mass density (gaussian smoothing function with smoothing length of [FORMULA]) is that the former quantity is much more affected by noise. Choosing [FORMULA] in that way leads to a good reconstruction only if the minimum of the cluster mass distribution is indeed close to zero, which is valid in practice if the observed field is several Mpc wide.

With that procedure we obtain almost the same mass density as that shown in Fig. 3 and the mean mass density detected inside the field is [FORMULA].

In Fig. 4 we show the mass distribution obtained using [FORMULA] and [FORMULA]. The mean mass density [FORMULA] is chosen such that the minimum of the locally-smoothed mass density is zero. Overestimating the mean redshift of the galaxies leads to an underestimation of the overall mass distribution and vice versa. This is shown in Fig. 5, where we use [FORMULA] and [FORMULA] for the reconstruction and obtain a mean mass density of [FORMULA]. Common to the Figs. 3 to 5 is that the substructure is nicely recovered, regardless of the assumed redshift distribution.

[FIGURE] Fig. 4. The reconstructed mass density obtained solving Eq. (4.9) iteratively assuming that the redshift distribution given through (3.8) with [FORMULA] and [FORMULA]. The true redshift distribution is described by [FORMULA] and [FORMULA]. For the mean mass density we obtain [FORMULA]. The levels of the contour lines are the same as in Fig. 2
[FIGURE] Fig. 5. Same as Fig. 4 but assuming that the redshift distribution is given through (3.8) with [FORMULA] and [FORMULA]. For the mean mass density we obtain [FORMULA]. The levels of the contour lines are the same as in Fig. 2. Note that the scale of the z -axis is increased in this figure

In Fig. 6 we show the mean mass density obtained for different assumed redshift distributions. We find that this is approximately proportional to [FORMULA], i.e., it is inversely proportional to the assumed mean effective lensing strength.

[FIGURE] Fig. 6. The mean mass density [FORMULA] averaged over the data field [FORMULA] as a function of the assumed redshift distribution of the sources. The mean mass density is obtained such that the minimum of the locally averaged mass density is zero. For redshift distributions with [FORMULA] we use crosses, for [FORMULA] triangles and for [FORMULA] dashes. The left panel shows the mean mass density obtained from the reconstruction as a function of the mean redshift [FORMULA], the right panel as a function of [FORMULA] defined in (A2.1). For producing the synthetic galaxy images we use for their redshift distribution [FORMULA] and [FORMULA], i.e., [FORMULA]. The true mean mass density of the cluster is [FORMULA]

6.2. Breaking the mass invariance in practice

We now use Eq. (5.12) to break the mass invariance. First we choose the parameter [FORMULA] describing the number counts versus flux of the faint galaxies. Again, the source ellipticity- and redshift distribution is given through (3.8) and (6.1) with parameters as given in Table 1. The spatial distribution of the galaxy images is given through Eq. (5.5), where [FORMULA] is calculated from integrating the `true' local magnification weighted with the redshift distribution of the sources. We keep the number of galaxy images (lens plane) fixed and use [FORMULA] galaxy images for our simulations.

Next we perform the mass reconstruction according to Eq .(4.9) for different values of [FORMULA]. From the resulting mass and shear map we calculate the map [FORMULA], where we assume to know the true redshift distribution of the sources and the true slope [FORMULA]. Finally, we derive from Eq. (5.12) the number of expected galaxies for the assumed value of [FORMULA] and calculate from that with Eq. (5.13) the function [FORMULA]. The resulting function [FORMULA] is shown in Fig. 7 for different values of the parameter [FORMULA].

[FIGURE] Fig. 7. The [FORMULA] as a function of the assumed value of the mean mass density [FORMULA] used for the mass reconstruction. We use different parameters [FORMULA] to describe the dependence of the number counts on the flux ([FORMULA]) of the faint galaxies. We assume to know the redshift distribution of the sources and the parameter [FORMULA]. The true values of the mean mass density is [FORMULA]

We find that all curves [FORMULA] show a minimum close to [FORMULA], but the minima are narrower the more the slope

[FORMULA] deviates from one. This is because the larger [FORMULA], the stronger the magnification bias (or anti-bias) and the stronger the dependence of [FORMULA] is on [FORMULA]. Hence, for the same number density of galaxy images, the significance of rejecting values of [FORMULA] increases with increasing [FORMULA]. From our simulations we find that the mean surface mass density is [FORMULA] ([FORMULA] level) for [FORMULA], [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA]. The true value of the mean surface mass density is [FORMULA]. Increasing the number density of the observed galaxy images improves the significance levels, because the [FORMULA] function is roughly proportional to the number of observed galaxies. We note that the galaxies used for determining [FORMULA] can be different from that used for the mass reconstruction, since the shape of the galaxies has not to be measured for the former purpose, increasing the number density of available galaxy images considerably. A number density of [FORMULA] would reduce the [FORMULA] confidence levels to [FORMULA] for [FORMULA] and to [FORMULA] for [FORMULA].

In practice, the redshift distribution of the sources is poorly known and the mean redshift [FORMULA] can be over- or underestimated in a cluster mass reconstruction. If the true mean redshift is [FORMULA] and if we assume for the reconstruction [FORMULA] [1.5], then we obtain the [FORMULA] confidence limits [FORMULA] [ [FORMULA] ]. To derive these limits we used a slope [FORMULA] for the number counts of the faint galaxies.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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