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

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6. Summary and conclusions

We 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 [FORMULA] and the density parameter [FORMULA] 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 [FORMULA] but are relatively insensitive to [FORMULA]. We have presented the new result of a robust lower limit on [FORMULA], which is a substantial improvement on previously known robust lower limits. Coupled with relatively conservative prior information about the Hubble constant [FORMULA], the age of the universe and the well-established value of [FORMULA], one can reduce the allowed parameter space in the [FORMULA]-[FORMULA] plane to a small, finite region, which is similar to the area allowed by joint constraints based on many other cosmological tests (see Fig. 3).

Using lens statistics information alone, at 95% confidence, our lower and upper limits on [FORMULA] are respectively -3.17 and 0.3. For a flat universe, this corresponds to lower and upper limits on [FORMULA] 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 [FORMULA] as in Cooray et al. (1999) and Cooray (1999). For comparison, from Table 4, the corresponding value for the upper limit on [FORMULA] is 0.62 and the value from K96 is 0.66. 5


Table 4. Mean values and ranges for assorted confidence levels for the parameter [FORMULA] for our a priori and various a posteriori likelihoods from this analysis and from other tests from the literature (using the latest publicly available results) for the special case [FORMULA]. Otherwise the same as Table 3, in particular the references are not listed in the footnotes to this table. X denotes the fact that there is no intersection of the confidence contour with the [FORMULA] line.
a) value for [FORMULA], not projection
b) contour at 95.4% not 95%
c) contour at 95.4% not 95%
d) contour at 95.4% not 95%
e) Fig. 6, solid contours
f) contours at 68.3%, 95.4% and 99.7% instead of 68%, 95% and 99% respectively
g) contour at 68.3% instead of 68%; other contours, and part of the 68.3% contour, lie partially in the [FORMULA] area of parameter space which was not examined for technical reasons in Lineweaver (1998)
h) value for [FORMULA], not projection


Table 5. Confidence ranges for [FORMULA] assuming [FORMULA]. Unlike the results presented in Tablef 3, these figures are for a specific value of [FORMULA] and not the values of intersection of particular contours with the [FORMULA] line in the [FORMULA]-[FORMULA] plane. These are more appropriate if one is convinced that [FORMULA] and have been calculated using ten times better resolution than the rest of our results presented in this work. See Fig. 5

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 [FORMULA] independently of any other parameters, there is not much point in quoting unqualified `limits on [FORMULA]'.

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 [FORMULA] 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 [FORMULA] imply, for a flat [FORMULA] universe, a value of [FORMULA] which is ruled out by lensing statistics. For a non-flat universe, many tests are indicating [FORMULA], and at present a cosmological model with [FORMULA] and [FORMULA] 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 [FORMULA] and [FORMULA], 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 [FORMULA] and [FORMULA], 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 Planck Surveyor mission. Larger gravitational lens surveys such as CLASS will be a step in this direction.

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Online publication: March 29, 1999