Astron. Astrophys. 342, 15-33 (1999) Appendix A: second order Lagrangian dynamicsIn order to investigate the statistical properties of the reconstructed mass field that contains a significant amount of non-linearities, a non-linear evolution model of large-scale structures is required. Second order Lagrangian perturbation theory has the advantage to be extremely fast to compute, and rather accurate compared to N-body simulations. It gives correct values for the skewness and we expect it to gives correct estimation of the cosmic variance (see Munshi et al. 1994, Bernardeau et al. 1994, Bouchet et al. 1995). Actually we do not even need to do 3D simulations. At the level of perturbation theory it is equivalent to perform 2D second order Lagrangian evolutions of the structures on an initial linear map of the projected mass fluctuations. A.1. Construction of the initial linear mapThe local convergence map is given by where is the horizon distance, the density parameter, a the expansion factor and the number density of sources as a function of the distance . For clarity we introduce the lens efficiency function so that the local convergence is simply given by the integral over the line of sight of, The projected density contrast would be, It is important to keep in mind that the local convergence is related to the actual density contrast with a constant that depends on the cosmological parameters. In the following we will assume that all sources are at the same redshift (it is not realistic but of no consequences on our results here). In the linear approximation the field is expected to be a 2D Gaussian field, characterized by a power spectrum, . with (see Kaiser 1992, 1996, and SvWJK) ^{11}, as a result of the relation between and the 3D density contrasts. The initial conditions for are therefore generated by a 2D Gaussian field in Fourier space where the complex random variables verify A.2. The 2D second order Lagrangian dynamicsIn order to introduce a significant amount of nonlinearities in the maps, we apply the 2D second order Lagrangian dynamics to the projected density, . Following the notations of Bouchet et al. (1992), transposed in the 2D case, let us write the 2D Eulerian coordinates as a perturbation series over the 2D displacement field , where is the angular Lagrangian coordinate and a small dimension-less parameter. The first order term reduces to the Zel'dovich (Zel'dovich 1970) approximation. The divergence of the second order displacement field can be written in terms of the first order solutions, This result is exact for an Einstein-de Sitter Universe. The values of the coefficient is only slightly altered (about 1%) for other cosmological models (Bouchet et al. 1992, Bernardeau 1994) and in the following we did not take into account this dependence. Once the second order displacement field has been computed, the local 2D density contrast can then be written in terms of the Jacobian of the transform between the Lagrangian coordinates and the Eulerian coordinates, where . The local convergence is then given by The linear density maps are built on a regular grid from random modes following a given power spectrum. The different quantities up to the Jacobian are generated on this same grid by successive Fast Fourier Transforms. There is then a technical difficulty to solve in order to get the resulting values of on a regular grid. This is done via a local triangulation and an interpolation of the values (from standard IDL packages). Note that the continuity equation provides us with and not with the projected potentiel. The latter will have to be computed from a subsequent Fourier transform (see Appendix B). The amplitude of the fluctuations is such that the displacement field does not induce shell crossings ^{12}. Finally, bands of sufficient width along the edges were cut out to avoid edge effects induced by the displacement filed. A.3. The skewness with this approximate dynamicsThe skewness of the projected density is given by the skewness of the 2D dynamics (Munshi et al. 1997), that is, The skewness for the convergence would then simply be This result has to be compared with the results obtained in BvWM. Their general formula (67) contains some extra geometrical factors of order unity (coming from a different averaging procedure along the line of sight). Eq. (A12) actually corresponds to the approximate form of Eq. (75) of BvWM. A.4. Shapes and normalizations of the power spectraThe 3D power spectrum given by Baugh & Gaztañaga (1996) (BG spectrum) is used in most of our simulations, where . In a series of simulations we compared this model also with the standard CDM spectrum. In most cases the fluctuations are normalized according to the convergence field . The value of that has been chosen thus depends on in such a way that it is equal to 0.6 for a flat Universe (following the normalization inferred from galaxy cluster counts (Eke et al. 1996, Oukbir & Blanchard 1997). As a result we take, It implies that the value of grows for low values of . Note that since , this growth is only slightly more important in our case than for the galaxy cluster counts (the exponent would be about 0.5 to 0.6). Appendix B:This appendix gives the details of the reconstruction algorithm, how the g map is generated from an initial map, and the noise model. B.1. Reconstruction algorithmThe lensing quantities of interest are defined in the Eqs. (6), and the observable is the reduced shear . The reconstruction problem is how to infer the map from an observed ellipticity field , knowing that is an unbiased estimate of the reduced shear? Note that this problem is under-constrained since a change in the potential of the form, where is a constant, leaves the reduced shear g invariant, but will transform the convergence as (see Seitz et al. 1998). This is the so called mass-sheet degeneracy. Thus in order to get a realistic convergence map, has to be determined by forcing at the survey scale. The estimate of is obtain by a non-parametric least method (Bartelmann et al. 1995). The image is sampled on a grid, and the potential on a grid. Starting from a guess on , a guess on the reduced shear is obtained and the following function is minimized with respect to , The finite difference schemes which are used in order to calculate the second derivatives of the potential at pixel are, where is the pixel size. We found that these schemes give the best regularization at small scales and avoid the usual high frequency oscillations in crude reconstruction schemes. At the edges of the field the shift of 2 pixels in Eqs. (B.3) must be only one pixel in the direction perpendicular to the border since the potential is sampled on a grid only. This has the consequence to slightly increase the noise at the boundaries but does not produce any bias. Once Eq. (B.2) is minimized, the map is found, and then the condition is imposed on it. B.2. Construction of the initial g mapA problem that we have to solve in this work is how to get the shear pattern of a projected mass distribution? Namely from the simulated we want to get the corresponding distortion map, put a noise on it, and reconstruct . The construction of a distortion map from a convergence map is unfortunately an under-constrained problem again. For example any transformation of the potential with leaves the convergence unchanged, but may change the shear (such a solution is for example ). A peculiar solution for the potential can be obtained by minimizing the function: The resulting reduced shear is then , where is the unphysical solution given by , but it is possible to reconstruct the convergence using Eq. (B.2), since is unchanged by the presence of the term . Unfortunately it is no longer the case when the noise is included via Eq. (10), because explicitly comes in the denominator. Fortunately, the role of the denominator in Eq. (10) is weak (this is a one percent effect on each galaxy if we take and ), this is in particular one of the reasons why the weak lensing approximation works so well. We thus did not try to correct for the presence of a spurious contribution in all the calculations, since it gives a negligible contribution to the signal (in terms of power spectrum and moments). B.3. Noise generationOnce the true g maps are obtained the noise is introduced in a realistic way, using Eq. 10: a sample of background galaxies with random intrinsic orientations is sheared, from which the map is reconstructed. The galaxies are observed in a grid of superpixels of 2.5 arcmin size (the minimum size of the map simulations), in which the number of galaxies per superpixel i is known. In principle this number suffers of the amplification bias (depending on the line of sight matter quantity), but this effect is neglected here. follows a Gaussian distribution of mean and variance . Each pixel i of the image gives a local estimate of the reduced shear from the galaxies, each of them having an intrinsic ellipticity which contributes as a source of noise for the shear signal. The distribution of is a truncated normalized Gaussian defined over the range , where we choose . (which corresponds to a typical axis ratio of 0.8). From Eq. (10), the observed mean ellipticity of a number of galaxies in the image plane is given by Where is an unbiased estimate of the reduced shear g in the superpixel i (Schramm & Kayser 1995, Schneider & Seitz 1995). A realistic estimate of depends on the observational context, the telescope used, the optical filter, the atmospheric conditions. By choosing gal/arcmin^{2} or gal/arcmin^{2} we adopted a conservative and reasonable assumption about the telescope time: the former is accessible in the I-band with 1.5 hour integration at CFHT, while the later is accessible for 4 hours in the same conditions. © European Southern Observatory (ESO) 1999 Online publication: December 22, 1998 |