## 2. Modeling depletion curves## 2.1. The magnification bias
The projected number density of objects magnified by a factor
where is the number density of
objects in an empty field and is the
logarithmic slope of the galaxy number counts. Note first that this
relation applies only for background galaxies. This means that we
implicitly work at faint magnitude where the foreground counts are
reduced and do not affect the slope .
Note also that the magnification An exact writing of the magnification bias, which also takes into
account the local effect of magnification, changing with radius
where is the density of galaxies
of apparent magnitude smaller than It is clear that the radial behavior of the ratio is strongly related to . It decreases from the center, up to a minimum at when magnification goes to infinity (Einstein radius) and then increases up to 1 again, when magnification goes to 0. This is the so-called radial depletion curve which has been detected in a few clusters already (Fort et al., 1997; Taylor et al., 1998; Athreya et al., 2000). From the lensing point of view, the magnification where is the magnification matrix. The convergence and the shear are expressed in cartesian coordinates as a function of the reduced gravitational potential by: with: All these factors depend on the lens and source redshifts, as is related to the true projected potential by: where is a characteristic scale of the lens. In the weak lensing regime we have the approximation (Broadhurst et al., 1995). So the potential interest of the depletion curves is that they trace directly the convergence distribution, or equivalently the surface mass density distribution. In principle, they may allow an easy mass reconstruction of lenses in the weak regime. We will see below what are the main limitations of such curves, already explored by Athreya et al. (2000). ## 2.2. The cluster lensDifferent sets of models are used, with increasing complexity. In all cases we suppose that the lens redshift is 0.4, with reference to the cluster Cl0024+1654 studied by Fort et al. (1997). With the adopted cosmology this means that the scaling corresponds to 6.4 kpc for 1". For each model we will use an analytic expression and develop it to compute analytically the magnification and its dependence with redshift and radius. All the models are scaled in terms of the Einstein radius . ## 2.2.1. The singular isothermal sphere (SIS)This model is the simplest one which can describe a cluster of galaxies and is very useful for the analytical calculations of the gravitational magnification. However it has also some physical meanings: -
it corresponds to a solution of the Jeans equation and thus it can be written as a function of the observed velocity dispersion; -
in a non-collisionnal description of the collapse of a self-gravitating system, the violent relaxation, during which particles exchange energy with the average field, leads to an isothermal distribution (Binney & Tremaine, 1987).
But, this model is only valid inside certain limits due to the divergence of the central mass density and of the total mass to infinity. This divergence has no consequences on our work because the validity limits of the model correspond to the limits of the depletion regime. The density of matter can be written as: where is the Einstein radius: and is the velocity dispersion along the line of sight. The gravitational magnification is simply: ## 2.2.2. The isothermal sphere with core radiusThis model avoids the divergence of the mass density in the inner part of the cluster with an internal cut-off of the density distribution. We chose the following distribution, as described in Hinshaw & Krauss (1987) or Grossman & Saha (1994). The density of matter is given by: where is the core radius, and is the central value of the gravitational potential. In this case, the Einstein radius is , where: The magnification can be written as: Physically, in most cases the core radius is smaller than or comparable to the Einstein radius, so we do not expect strong effects in the outer parts of the depletion curves. ## 2.2.3. A power-law density profileThis model is a generalization of the SIS. It can be used in order to test the departure from an isothermal profile far away from the cluster center. The density is given by: where is the logarithmic slope of the density profile. In order to keep a physical model for the mass distribution, we must have . The Einstein radius can be written as: where is the integrated mass inside the Einstein radius and . The magnification is then: ## 2.2.4. The singular isothermal ellipsoidThis type of model is interesting as a lot of clusters have elliptical shapes in their galaxy distribution or their X-ray isophotes (Buote & Canizares, 1992; Buote & Canizares, 1996; Lewis et al., 1999; Soucail et al., 2000). Thus, an elliptical lens represents a more realistic model although it is still reasonably simple. In order to study the effect of the ellipticity of the potential on the depletion, we introduced the singular isothermal ellipsoid in our simulations (Kormann et al., 1994). Using polar coordinates in the lens plane we introduce which is constant on ellipses with minor axis and major axis . The surface mass density can then be written as: where is the velocity dispersion along the line of sight. The convergence is: The magnification writes then quite simply: ## 2.2.5. NFW profileWe introduced in our simulations the universal density profile of Navarro et al. (1996) for dark matter halos which was found from cosmological simulations of the growth of massive structures. This profile is a very good description of the radial mass distribution inside the virial radius . Wright & Brainerd (2000) have compared it with a SIS for several cosmological models. They find that the assumption of an isothermal sphere potential results in an overestimate of the halos mass which increases linearly with the value of the NFW concentration parameter. This overestimate depends upon the cosmology and is smaller for rich clusters than for galaxy-sized halos. The NFW density profile is given by: where is a characteristic radius
and is the critical density.
If we take , the surface mass density and the shear of a NFW lens can be written as (Wright & Brainerd, 2000): where is the critical surface mass density. and express as: while and express as: The magnification by the NFW lens is then: ## 2.3. The galaxy redshift distribution## 2.3.1. Analytical distributionWe used the analytical redshift distribution introduced by Taylor et al. (1998) to fit the redshift distribution of the galaxies in the range for : with and
. Here, we extend the redshift range
between and
and we take
objects per arcmin ## 2.3.2. The magnitude-redshift distributions in the U to K photometric bandsWe also used a model of galaxy number counts from which the magnitude-redshift distribution in empty field can be computed with a large set of photometric bands. This model was developed by Bézecourt et al. (1998) and largely inspired by Pozetti et al. (1996). It includes the model of galaxy evolution developed by Bruzual & Charlot (1993), with the upgraded version so-called GISSEL, and standard parameters for the initial mass function (IMF) and star formation rates (SFR) for different galaxy types. In order to reproduce the deep number counts of galaxies, evolution of the number density of galaxies is included to compensate for the smaller volumes in an universe, following the prescriptions of Rocca-Volmerange & Guiderdoni (1990). This model reproduces fairly well deep number counts up to or , as well as redshift distributions of galaxies up to or (Bezecourt et al., 1998). The advantage of this model is that the galaxy distribution can be computed for any photometric band, and the effects of the color distribution of sources can be explored. In practice we will limit our study in this paper to B, I and K bands (Table 1).
© European Southern Observatory (ESO) 2000 Online publication: October 2, 2000 |