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
and the redshift-distribution (3.8) with and . The galaxies are randomly distributed in the source plane, and the number density of galaxy images is . The cluster is placed at a redshift so that , and .
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 , and , 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.
However, in practice we do not know and we therefore choose 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 ) is that the former quantity is much more affected by noise. Choosing 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 .
In Fig. 4 we show the mass distribution obtained using and . The mean mass density 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 and for the reconstruction and obtain a mean mass density of . Common to the Figs. 3 to 5 is that the substructure is nicely recovered, regardless of the assumed redshift distribution.
In Fig. 6 we show the mean mass density obtained for different assumed redshift distributions. We find that this is approximately proportional to , i.e., it is inversely proportional to the assumed mean effective lensing strength.
6.2. Breaking the mass invariance in practice
We now use Eq. (5.12) to break the mass invariance. First we choose the parameter 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 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 galaxy images for our simulations.
Next we perform the mass reconstruction according to Eq .(4.9) for different values of . From the resulting mass and shear map we calculate the map , where we assume to know the true redshift distribution of the sources and the true slope . Finally, we derive from Eq. (5.12) the number of expected galaxies for the assumed value of and calculate from that with Eq. (5.13) the function . The resulting function is shown in Fig. 7 for different values of the parameter .
We find that all curves show a minimum close to , but the minima are narrower the more the slope
deviates from one. This is because the larger , the stronger the magnification bias (or anti-bias) and the stronger the dependence of is on . Hence, for the same number density of galaxy images, the significance of rejecting values of increases with increasing . From our simulations we find that the mean surface mass density is ( level) for , for and for . The true value of the mean surface mass density is . Increasing the number density of the observed galaxy images improves the significance levels, because the function is roughly proportional to the number of observed galaxies. We note that the galaxies used for determining 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 would reduce the confidence levels to for and to for .
In practice, the redshift distribution of the sources is poorly known and the mean redshift can be over- or underestimated in a cluster mass reconstruction. If the true mean redshift is and if we assume for the reconstruction [1.5], then we obtain the confidence limits [ ]. To derive these limits we used a slope for the number counts of the faint galaxies.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998