## 6. Summary and conclusionsWe have re-analysed optical gravitational lens surveys from the
literature, using the techniques described in Kochanek (1996), for the
first time allowing both the cosmological constant
and the density parameter
to be free parameters while also
using a non-singular lens model. We confirm the well-known results
that gravitational lensing statistics can provide a good upper limit
on but are relatively insensitive to
. We have presented the new result of
a robust lower limit on , which is a
substantial improvement on previously known Using lens statistics information alone, at 95% confidence, our
lower and upper limits on are
respectively -3.17 and 0.3. For a flat universe, this corresponds to
lower and upper limits on of
respectively -1.09 and 0.65. Keeping in mind the difficulties of a
quantitative comparison, this is in good agreement with other recent
measurements of the cosmological constant. This value was calculated
from Table 5 and assuming a degeneracy in
as in Cooray et al. (1999) and
Cooray (1999). For comparison, from Table 4, the corresponding
value for the upper limit on is 0.62
and the value from K96 is
0.66.
For detailed comparison of cosmological tests, one needs to compare confidence contours-calculated in the same, preferably in the `real', way-in the same parameter space. Of course, this makes it difficult to meaningfully reduce the results of a given cosmological test to one or even a few single numbers. Unless a cosmological test is developed which can measure independently of any other parameters, there is not much point in quoting unqualified `limits on '. Presently tentative claims of the detection of a positive cosmological constant, if true, would rank among the great discoveries of cosmology. Even though there are serious difficulties involved, it seems worthwhile to be able to confirm this result by improving the lower limit on derived from gravitational lensing statistics. Targetting the two primary sources of uncertainty calls for improving our knowledge of the normalisation of the local luminosity density of galaxies as well as increasing the size of gravitational lens surveys. As far as the latter goes, the CLASS survey (Browne et al. 1998; Myers et al. 1999) looks the most promising at the moment. In a companion paper (Helbig et al. 1999), we have shown that comparable constraints to the ones presented in this work can be obtained from the JVAS gravitational lens survey; this gives us hope that the much larger CLASS survey will offer improvement in this area. Cosmological tests which set tight upper limits on
imply, for a flat
universe, a value of
which is ruled out by lensing
statistics. For a non-flat universe, many tests are indicating
, and at present a cosmological
model with and
seems compatible with all known
observational data, with neither a flat universe nor a universe
without a positive cosmological constant being viable alternatives.
The simplest case, the Einstein-de Sitter universe with
and
, both flat and without a
cosmological constant, had been abandoned long before the new
observational data cited in this work came to light (see, e.g.,
Ostriker & Steinhardt 1995, and references therein); this trend
has continued, with the next-most-simple cases also no longer viable.
For and
, we have in a sense reached the least
simple case; it will be interesting to see if this trend continues
with regard to the other cosmological parameters, in particular those
which can be measured by the © European Southern Observatory (ESO) 1999 Online publication: March 29, 1999 |