## 2. Generation of realistic -maps## 2.1. Lensing effects, displacement and amplification matrixAny mass concentration deflects light beams by an angle proportional to the gradient of the local gravitational potential. This effect induces an apparent displacement of the sources, so that a source that was at the angular position will be observed at position , with where is the radial coordinate ( is the one of the source), the co-moving angular diameter distance and the 3D gravitational potential. The differential displacement of the images induces an image distortion, which depends on the second derivatives of the gravitational potential, i.e. on the mass density and the tidal field. This gravitational lensing effect of a thin lens is therefore characterized by an isotropic stretching, described by the convergence , and an anisotropic distortion given by the complex shear . The so-called amplification matrix describes the change of local coordinates between the source and image planes and can be written as Its elements are related to the first derivative of the displacement field, and are therefore related to the projected gravitational potential of the lens via, where is the angle such that . Here depends on since the potential is integrated from a given source plane to the observer. This can be trivially generalized to a source redshift distribution. This equation is valid only if the lens-lens coupling is dropped and the Born approximation is used. The validity of this assumption has been discussed by BvWM and Schneider et al. 1997 (hereafter SvWJK). ## 2.2. The galaxy shape matricesA source galaxy may be described by a complex ellipticity defined as where is the galaxy orientation,
and In presence of lensing, the shape matrix of the image of the galaxy is given by , and provided that the amplification matrix does not vary over the galaxy's area, the observed ellipticity is still described by Eq. (9), and it writes (Schneider & Seitz 1995), where is the complex reduced
shear. The observable is the distortion
, but for sub-critical lenses like
those we discuss in this work, If the orientation of the source galaxies is random (), the observed mean ellipticity of galaxies is an unbiased estimate of the reduced shear (Schramm & Kayser 1995, Schneider & Seitz 1995), The weak lensing approximation () is generally used in the case of lensing by large scale structures. However, we will not use that since we want to analyze the noise propagation in mass map reconstructed in the full non-linear regime (Eq. (10)). ## 2.3. Construction of the synthetic projected mass maps, and their reconstructionThe adopted procedure to build and analyze projected density maps is: 1) Generation of a density map, which directly provides the
convergence given the source
redshifts: 2) Calculation of the associated reduced shear map 3) Reconstruction of the original
map from : Most of our generated images have an angular size of
degrees,
pixels, each pixel having a
angular side (hereafter, the
Fig. 1 shows two examples of initial maps, and the reconstructed mass of the noisy distortion maps. This panel illustrates what MEGACAM should be able to get during only 5 nights (!): 25 exposures in the I band, 1.5 hour each. A single MEGACAM field corresponds to a size of pixels on Fig. 1, which is clearly the required minimum area to detect large scale structure features like super clusters, filamentary structures or voids.
© European Southern Observatory (ESO) 1999 Online publication: December 22, 1998 |