## 2. Galaxy clusters as lensesIn order to calculate the lensing rate for background galaxies due to foreground galaxy clusters, we model the lensing clusters as singular isothermal spheres (SIS) and use the analytical filled-beam approximation (see, e.g., Fukugita et al. 1992). In the case of the optical arcs, the lensed sources towards clusters have all been imaged in magnitude limited optical search programs. Such observational surveys are affected by the so-called "magnification bias" (see, e.g., Kochanek 1991), in which the number of lensed sources in the sample is larger than it would be in an unbiased sample, because lensing brightens sources that would otherwise not be detected. Thus, any calculation involving lensed source statistics should account for the magnification bias and associated systematic effects. We refer the reader to CQM for full details of our lensing
calculation involving foreground galaxies as lensing sources.
Following CQM, if the probability for a source at redshift where the integral is over all values of amplification The probability of strong lensing depends on the number density and typical mass of foreground objects, and is represented by the in above Eq. 1, where is the mean value over the lens redshift distribution, . For SIS models, this factor can be written by the dimensionless parameter (CQM): where where is the mean background
density at redshift when , and when (flat) and (open). Here is the linear growth factor and is the critical overdensity. For an open universe (), can be written as (Lacey & Cole 1993): where and . For a flat universe with , was parameterized in Mathiesen & Evrard (1998) as: which was derived by Kitayama & Suto (1997). The for a flat universe was calculated using the linear growth factor found in Peebles (1980), where , and
, the inflection point in the scale
factor. This function was integrated numerically to find the growth
factor at redshifts In addition to growth factors suggested by Mathiesen & Evrard (1998), our calculation uses power-spectrum normalizations deduced by Viana & Liddle (1996) for based on cluster temperature function: when , and when . We have also assumed a scale-free power spectrum with (), which corresponds to a power spectrum shape parameter of 0.25 in CDM models. In order to calculate the parameter
, we also require knowledge of
cluster velocity dispersion, , which
is the velocity dispersion of clusters in the SIS model. We assume
that the is same as the measured
velocity dispersion for galaxy clusters based on observational data.
To relate with cluster mass
distribution, we use the scaling relation between
and cluster temperature, and the relation between to derive a relation between and
cluster mass and was numerically calculated, in additional to the above, by weighing over the redshift distribution of galaxy clusters derived based on the PS theory to obtain , the mean value of , which is used in Eq. 1. In order to compare the predicted number of bright arcs towards
clusters with the observed number towards a X-ray luminosity,
where Ignoring various small changes due to the choice of cosmological model, we use a minimum mass of , corresponding to above . Using the numerical values for and , and performing numerical integrations we find to range from when to when . The error associated with is rather uncertain. For example, the quoted random uncertainty in from Viana & Liddle (1996) is . It is likely that has an overall statistical uncertainty of 50%, however, as we discuss later, there could also be systematic errors in our determination. © European Southern Observatory (ESO) 1999 Online publication: December 16, 1998 |