2. Galaxy clusters as lenses
In 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 z to be strongly lensed is , as calculated based on the filled-beam formalism, we can write the number of lensed sources, , with amplification greater than as:
where the integral is over all values of amplification A greater than , and and are parameters of the HDF luminosity function at various redshifts as determined by Sawicki et al. 1997. Here, the sum is over each of the galaxies in our sample. The index i represents each galaxy; hence, and are, respectively, the redshift, and the rest-frame luminosity of the ith galaxy. The step function, , takes into account the limiting magnitude, , of a given optical search to find lensed arcs in the sky, such that only galaxies with lensed magnitude brighter than the limiting magnitude is counted when determining the number of lensed arcs. Since rest-frame luminosities of individual galaxies are not known accurately due to uncertain K-corrections, as in CQM, we estimate the average amplification bias by summing the the expectation values of , which were computed by weighting the integral in above equation by a normalized distribution of luminosities drawn from the Schechter function at redshift .
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 n is the number density of lensing objects, , and is the velocity dispersion in the SIS scenario. In general, the parameter F is independent of the Hubble constant, because the observationally inferred number density is proportional to . In the present calculation, the foreground objects are galaxy clusters, and thus n represent the number density of clusters and represent their velocity dispersion. Since the number density of clusters in different cosmological models are expected to vary, we calculate the number density of galaxy clusters, , between mass range using a PS analysis (e.g., Lacey & Cole 1993):
where is the mean background density at redshift z, is the variance of the fluctuation spectrum filtered on mass scale M, and is the linear overdensity of a perturbation which has collapsed and virialized at redshift z. Following Mathiesen & Evrard (1998; Appendix A), we can write as:
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:
where , and , the inflection point in the scale factor. This function was integrated numerically to find the growth factor at redshifts z and 0.
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, T, of the form (Girardi et al. 1996):
to derive a relation between and cluster mass M, . Finally, we can write the interested parameter F as a function of the lens redshift, :
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, L, selected sample of galaxy clusters, we also need a relation between M and L. We obtain this relation based on the observed relation recently derived by Arnaud & Evrard (1998):
where L is the X-ray luminosity in the 2-10 keV band, and relation in Eq. 13. Since we will be comparing the predicted number of lensed arcs to the observed number, we will be setting the minimum mass scale, , which corresponds to the minimum luminosity, , of clusters in optical search programs to find lensed arcs. The luminosity cutoff of the EMSS cluster arc survey by Le Fèvre et al. (1994) is , which is measured in the EMSS band of 0.3 to 3.5 keV. By comparing the tabulated luminosities of EMSS clusters in Nichol et al. (1997), Mushotzky & Scharf (1997), and Le Fèvre et al. (1994), we evaluate that this luminosity, in general, corresponds to a luminosity of in the 2 to 10 keV band. This luminosity is calculated under the assumption of , and for different cosmological parameters, it is expected that the value will change as the luminosity distance relation is dependent on and . However, for clusters in the EMSS arc survey, with a mean redshift of 0.32, such variations are small compared to the statistical and systematic uncertainties in the scaling relations used in the calculation.
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