2. Probability of multiply imaged sources
In this section we briefly review the statistical concepts introduced in K96; this also serves to define our notation. Note that with regard to cosmogical notation we follow that of Kayser et al. (1997), repeating here only 2 equations needed for discussion in this paper: the comoving spherical volume element at redshift z reads
Following the K96 approach, we assume that the light deflection properties of the gravitational lenses can be modelled with a particular type of circularly symmetric lens models with a monotonically declining radial mass profile. Such lens models generally create three images and have two critical radii on which the magnification diverges (e.g. Schneider et al. 1992). It is possible to estimate the probability of the event
A source at redshift is triply imaged. The total apparent magnitude of the three images ism. The image configuration meets the selection criteriaSand, particularly, shows the propertiesC.
If the outer and inner critical angular radii of the lens potential are respectively and , the image magnification at radial angular position r is , the total magnification of the three images of a source at angular position y is , the functions and are valued selection functions, the comoving density of lenses of luminosity L is and the number-magnitude counts of sources are , then
The critical radii, the image magnifications and the source position are functions of the lens model, the luminosity of the lens galaxy and the redshifts of the source and the lens galaxy. If the underbraced functions are dropped, Eq. (3) yields the optical depth - the fraction of the sky included within the caustics of all lenses between us and the sources at redshift . The inclusion of these functions accounts for magnification bias, survey selection effects (including what is defined as a lensing event) and allows the observed image separation to be taken into account.
Eq. (3) parametrically depends on and through Eq. (1) and through the angular size distances, 1 which are needed for calculating observable quantities from the lens model (these also depend on the source and lens redshifts). Eq. (3) additionally depends on parametric submodels required to model the lens population and the number-magnitude counts of sources. Since throughout this paper we are principally interested in and , hereafter we refer to the submodel parameters as nuisance parameters (although technically they are on the same footing with and , there are not of as much interest here and thus a nuisance). In principle, one could also incorporate other observables into the parametric model; the reasons for not doing so are practical.
Assuming the survey selection function S is known, we can numerically compute Eq. (3) and reasonably estimate the probability that the quasar i is singly imaged or the probability that the quasar i is multiply imaged and its images (within some tolerance) are separated by . If the survey data D contains M singly and N multiply imaged quasars, we can estimate the probability of the event
In a model universe fixed by the cosmological parameters , and the nuisance parameters , a multiply imaged quasar survey collects the observational data D.
where the logarithm was expanded to first order. We can combine surveys of different objects by adding the logarithms of the likelihood functions for the individual surveys, and can combine surveys containing the same objects by applying their joint selection function.
In Bayesian theory the model parameters , , are regarded as random quantities with known joint prior probability density function . Applying Bayes's theorem, the appropriate posterior probability distribution given the observational data D is
yields the (marginal) posterior probability density function for the parameters and . In the limit where all nuisance parameters take a precise value, , the joint prior probability density function factorises into and a delta distribution , and Eq. (7) simplifies to
On the basis of Eq. (7) or Eq. (8), we can calculate confidence regions for two parameters or perform further marginalisations and calculate mean values, standard deviations and marginal confidence intervals for one parameter.
© European Southern Observatory (ESO) 1999
Online publication: March 29, 1999