Astron. Astrophys. 344, 721-734 (1999)

Appendix A: getting a feel for it

The likelihood of a given cosmological model for a given set of observational data, calculated using Eq. (3), is the result of the complex interplay of many factors. While this is necessary for a detailed analysis, it perhaps obscures the fact that the likelihood is basically the product of two terms, the likelihood that the non-lenses in our sample are not lenses (see Fig. A1) and the likelihood that the lenses in our sample (see Fig. A2) have the observed properties. ⁶ The latter in turn is the result of two basic effects: the dependency of the volume element on and (see Fig. A3) and the dependency on the lensing cross section on and (see Fig. A4). One can also use the probability that the non-lenses in our sample are not lenses (illustrated in Fig. A1) to calculate the expected number of lenses in our sample (see Fig. A5), although obviously just counting the number of lenses does not make use of as much of the available information as does using Eq. (3).

Fig. A1. Likelihood that the non-lenses in our sample are not lenses. The contour levels mark changes of a factor of ten in the probability, which is also indicated by the grey scale, darker values corresponding to higher values

Fig. A2. Likelihood that the lenses in our sample have the properties they do. The contour levels mark changes of a factor of ten in the probability, which is also indicated by the grey scale, darker values corresponding to higher values

Fig. A3. The volume element at the typical lens galaxy redshift . The contours indicate the fraction of the volume element in the limiting case of the de Sitter model (, ). This is also indicated by the grey scale, darker values corresponding to a larger volume. For smaller redshifts the contours are more vertical (and further apart), for larger redshifts more horizontal (cf. Fig. 3 of Tegmark et al. (1998b) but note their swapped axes)

Fig. A4. Cross section for the softened singular isothermal sphere model used in this work for a typical lens redshift and a typical source redshift for the fiducial values and (see Sect. 4). The contours indicate the fraction of the cross section in the limiting case of the de Sitter model (, ). This is also indicated by the grey scale, darker values corresponding to a larger cross section

Fig. A5. Expected number of lenses. Contours, from left to right, indicate 1, 2, 3, 4, 5, 10, 15, 20 and 25 lenses. Darker values of the grey scale correspond to higher values. Cf. Cooray et al. (1999) and Cooray (1999) where the number of lenses as a function of and has been calculated for the Hubble Deep Field and for CLASS

© European Southern Observatory (ESO) 1999

Online publication: March 29, 1999
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