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Astron. Astrophys. 342, 15-33 (1999)

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2. Generation of realistic [FORMULA]-maps

2.1. Lensing effects, displacement and amplification matrix

Any 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 [FORMULA] will be observed at position [FORMULA], with


where [FORMULA] is the radial coordinate ([FORMULA] is the one of the source), [FORMULA] the co-moving angular diameter distance and [FORMULA] 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 [FORMULA], and an anisotropic distortion given by the complex shear [FORMULA]. The so-called amplification matrix [FORMULA] 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 [FORMULA] via,


and where [FORMULA] is given by,


where [FORMULA] is the angle such that [FORMULA]. Here [FORMULA] depends on [FORMULA] since the potential is integrated from a given source plane [FORMULA] 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 matrices

A source galaxy may be described by a complex ellipticity defined as


where [FORMULA] is the galaxy orientation, and r is the square root of the ratio of the eigenvalues of the shape matrix [FORMULA] of the (centered) surface brightness profile of the galaxy,


In presence of lensing, the shape matrix of the image of the galaxy is given by [FORMULA], 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 [FORMULA] is the complex reduced shear. The observable is the distortion [FORMULA], but for sub-critical lenses like those we discuss in this work, g is also directly observable,


If the orientation of the source galaxies is random ([FORMULA]), the observed mean ellipticity of galaxies is an unbiased estimate of the reduced shear (Schramm & Kayser 1995, Schneider & Seitz 1995),


The weak lensing approximation ([FORMULA]) 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 reconstruction

The adopted procedure to build and analyze projected density maps is:

1) Generation of a density map, which directly provides the convergence [FORMULA] given the source redshifts:
In order to make a precise analysis of the cosmic variance and noise properties in the reconstructed mass maps it is necessary to have a large set of simulations of large scale structure. Instead of CPU intensive N-body codes we use numerical fast second order Lagrangian dynamics (Moutarde et al. 1992) which has been shown to accurately reproduce the statistical properties of LSS (Munshi et al. 1994, Bernardeau et al. 1994, Bouchet et al. 1995). This allows us to build a large amount of projected mass maps for different cosmological models. The Appendix A gives details about the generation of these maps and compares the non-Gaussian features with expectations from real dynamics.

2) Calculation of the associated reduced shear map g and addition of a given level of noise to the reduced shear map:
From the previous projected mass maps, a gravitational distortion map is computed, and the noise due to intrinsic ellipticities is added, as described in Appendix B. A "high" and "low" noise levels are simultaneously considered which correspond respectively to a mean number density of galaxies of [FORMULA]gal/arcmin2 and [FORMULA]gal/arcmin2. For the I-band, these number densities are reachable with respectively 1.5 or 4.5 hours exposure at CFHT which is expected to provide galaxies of redshift of about unity. At this stage we have at our disposal a large number of maps that are supposed to mimic accurately what can be observed with large CCD cameras.

3) Reconstruction of the original [FORMULA] map from [FORMULA]:
Many mass reconstruction methods already exist (Seitz et al. 1998, Squires & Kaiser, 1996, and references therein). In this work, we use a non-parametric least square method (see Bartelmann et al. 1996 for details) which overfits data if no regularization process is used. The hope by doing so is to preserve all the noise properties intact, so that a detailed noise analysis can be done. The Appendix B gives more technical details about the reconstruction algorithm.

Most of our generated images have an angular size of [FORMULA] degrees, [FORMULA] pixels, each pixel having a [FORMULA] angular side (hereafter, the superpixel size)  3. These are typical scales that a MEGACAM survey at CFHT could probe. Images of size [FORMULA] degrees ([FORMULA] pixels) have also been generated, in order to estimate how the cosmic variance depends on the survey size in the non-linear regime, and to compare the merit of deep-small survey area versus shallow-large survey. For each cosmological model and observational context series of 60 compact maps are produced. Table 1 summarizes the cases that investigated.


Table 1. List of simulations which have been carried out. The power spectrum BG corresponds to the formula (A13). It is the same for [FORMULA] and [FORMULA]. The CDM model corresponds to a standard CDM with [FORMULA]. [FORMULA] is the ratio between the local convergence and the projected normalized density contrast, [FORMULA] (see Appendix A).

Fig. 1 shows two examples of initial [FORMULA] 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 [FORMULA] pixels on Fig. 1, which is clearly the required minimum area to detect large scale structure features like super clusters, filamentary structures or voids.

[FIGURE] Fig. 1. Example of reconstructions of projected mass maps. The top panels show the initial noise-free [FORMULA] map for either [FORMULA] (left panel ) or [FORMULA] (right panel ) with the same underlying linear random field (see Appendix A) and the same rms distortion. The bottom panels show the reconstructed [FORMULA] maps with noise included in the shear maps. The maps cover a total area of 25 degrees2. Each pixel has an angular size of 2.5 arcmin2 and averages the shear signal expected from deep CCD exposures (about 30 galaxy/arcmin2). The sources are assumed to be all at redshift unity and to have an intrinsic ellipticity distribution given by Eq. (B.5). Such a survey is easily accessible to MEGACAM at CFHT. The precision with which the images can be reconstructed and the striking differences between the two cosmological models demonstrate the great interest such a survey would have.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998