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 [184] 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,
112,
114,
201,
209]).
The largest survey so far, the CLASS survey, has looked at
about 7000 radio sources at the moment (the goal is 10000). In
total CLASS found 12 new lens systems so far. Interestingly, all
the lenses have small separations (
arcsec), and all lensing galaxies are detected [34,
79]. 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 [98].
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 [36
]). This was first pointed out 1990 [63,
183]. 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 [36]. 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
[99] and
[113,
152]. 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,
125,
163]).
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Gravitational Lensing in Astronomy
Joachim Wambsganss http://www.livingreviews.org/lrr-1998-12 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |