Astron. Astrophys. 322, 1-18 (1997)

2. The physical model for the galaxy and mass distributions

The gravitational lensing effects makes intervene both the redshift distributions of the lenses and the sources. In the last section we discuss in more detail the knowledge we have or need to have on the sources, but at this stage we simply describe the redshift distribution of these objects by with the normalization,

To illustrate the results we will either assume that the sources are all at the same redshift or are given by a broad distribution. In the latter case we adopt a reasonnable analytic model which reproduces the redshift distributions observed in the faint redshift surveys and those expected from models of galaxy evolution (Charlot & Fall in preparation),

in which it is assumed that the average redshift of the sources is and that the width of the distribution is about (see Fig. 1, thin lines in the upper panel).

Fig. 1a and b. Shapes of the efficiency functions in different cases. In the upper panel the selection functions are given by the thin lines (corresponding to Eq. [2] for and =2) and we assume an Einstein-de Sitter Universe. In the lower panel the sources are assumed to be all located at and the cosmological parameters are varied, thick solid line for , , thin solid line for and thin dashed line for and .

We are interested in the lensing effects induced by the large-scale density fluctuations that are assumed to have emerged from Gaussian initial conditions. In the linear regime we thus assume that the local density can be written ¹,

where is the time dependence of the linear growing mode. Note that the function depends on the cosmological parameters as well. It is proportional to the expansion factor only for an Einstein-de Sitter universe. The Fourier transforms of the initial local density contrast, , are assumed to form a set of Gaussian variables. In such a case the power spectrum entirely determines the statistical properties of the initial density field. It is defined by

From the observed density fluctuation in the APM galaxy survey it is possible to construct a realistic power spectrum for both its magnitude and its shape. Using the results of Baugh & Gaztañaga (1996) we can use the shape (we have set ),

with

that reproduces the observed density fluctuations in the linear regime ². Baugh & Gaztañaga compared the observed fluctuations in the APM galaxy survey with the prediction of this power spectrum and found a good agreement in the linear regime. Results of numerical simulations also show that the nonlinear evolution of the density fluctuations induced with such a spectrum are in good agreement with the observed full shape of the angular two-point correlation function.

In this paper, and contrary to previous studies, we do not make any assumption on both the density of the Universe and on the cosmological constant, .

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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