The second approach is of statistical nature: find out how many (what fraction of) quasars are multiply imaged by gravitationally lensing, determine their separation and redshift distributions  and deduce the value of (or limits to) – matter in clumps of, say, – and to – the value of the cosmological constant.
The first approach has already been treated in Section 4.1. Here we will concentrate on the statistical approach. In order to determine which fraction of a certain group of objects is affected by strong lensing (i.e. multiply imaged), one first needs a well-defined underlying sample. What needs to be done is the following:
Since quasars are rare objects and lensing is a relatively rare phenomenon, steps 1 and 2 are quite difficult and time-consuming. Nevertheless, a number of systematic quasar surveys with the goal to find (many) lens systems with well defined selection criteria have been done in the past and others are underway right now (e.g. [34, 116, 118, 201, 209]).
The largest survey so far, the CLASS survey, has looked at about 7000 radio sources at the moment (the goal is 10,000). In total CLASS found 12 new lens systems so far. Interestingly, all the lenses have small separations (), and all lensing galaxies are detected [34, 82]. That leaves little space for a population of dark objects with masses of galaxies or beyond. A detailed discussion of lens surveys and a comparison between optical and radio surveys can be found in .
The idea for the determination of the cosmological constant from lens statistics is based on the fact that the relative lens probability for multiple imaging increases rapidly with increasing (cf. Figure 9 of ). This was first pointed out 1990 [63, 181]. The reason is the fact that the angular diameter distances , , depend strongly on the cosmological model. And the properties that determine the probability for multiple lensing (i.e. the “fractional volume” that is affected by a certain lens) depend on these distances . This can be seen, e.g. when one looks at the critical surface mass density required for multiple imaging (cf. Equation (16)) which depends on the angular diameter distances.
The consequences of lensing studies on the cosmological constant can be summarized as follows. The analyses of the frequency of lensing are based on lens systems found in different optical and radio surveys. The main problem is still the small number of lenses. Depending on the exact selection criteria, only a few lens systems can be included in the analyses. Nevertheless, one can use the existing samples to put limits on the cosmological constant. Two different studies found 95%-confidence limits of  and [117, 154]. This is based on the assumption of a flat universe (). Investigations on the matter content of the universe from (both “macro-” and “micro-”) lensing generally conclude that the fractional matter in compact form cannot exceed a few percent of the critical density (e.g. [35, 45, 129, 165]).
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