          Astron. Astrophys. 348, 38-42 (1999)

## 5. Examples of simulated reconstructions

In addition to the reconstructions from "synthetic" data as described in the previous section, we have performed several additional tests in order to demonstrate the reliability of our method. The tests, designed with the aim to reproduce the main features of a "real" reconstruction, have been made following a straightforward procedure (see, e.g., Lombardi & Bertin 1999 for similar simulations).

First we have generated a population of source galaxies using a pseudo-random number generator. Here a source galaxy is represented by its position and by its ellipticity (see Eq. (11) of Lombardi & Bertin 1999, following Seitz & Schneider 1997). Positions are drawn from a homogeneous distribution (with a density of 70 galaxies per square arcmin), while ellipticities are drawn from a truncated Gaussian distribution with variance . Sources are assumed to have all the same redshift . Source ellipticities are then transformed into observed ellipticities. For simplicity, the observed galaxy positions are assumed to be equal to the source positions: in other words, no depletion effects are included in the simulations.

Then the calculation of the observed ellipticities has been done by referring to a cluster of galaxies placed at with total mass inside the 10' ' field . For the purpose of introducing the lensing effects, we only need to specify the dimensionless projected mass map . For simplicity, we have used a density distribution made of three symmetrical components; each component is described by the analytical model outlined by Schneider et al. (1992, p. 244), which, at large radii, is approximately isothermal.

By averaging the observed ellipticities, we have then obtained a map of the reduced shear and, from that map, the vector field . The mass inversion has been performed using the direct method and the over-relaxation method. The two mass distributions have been then compared.

An example of typical results is shown in Fig. 2. Here, from top to bottom, we display the original cluster mass distribution, the reconstruction obtained using the direct method, and the residuals, i.e. the difference between the reconstructed maps from the direct method and from the over-relaxation method. As the figure clearly shows, differences are mainly confined to the boundary of the field where they are found to be of the order of 0.0002, well below the statistical errors of the reconstruction. In the inner field the differences are about one order of magnitude smaller. Note also that the wavy overall appearance of the reconstructed map is normal for weak lensing reconstructions, resulting from the relatively low number of source galaxies involved (see Lombardi & Bertin 1998b for a discussion of the statistical aspects of the problem). Fig. 2. A typical result of mass reconstruction; at the adopted distance for the lensing cluster, the side of the square field, 10 arcminutes, corresponds to approximately 2.88 Mpc. From top to bottom, true dimensionless mass distribution, reconstructed distribution (from the direct method), and difference between maps of the variable derived from direct and over-relaxation methods. The very small residuals show that the two methods are practically equivalent in terms of accuracy.

© European Southern Observatory (ESO) 1999

Online publication: July 16, 1999 