5 Conclusions and outlook

 

We have used the method outlined in Quast & Helbig (1999) to measure the cosmological constant $\lambda_{0}$ from the lensing statistics of the Jodrell Bank-VLA Astrometric Survey. At 95% confidence, our lower and upper limits on $\lambda_{0}-\Omega_{0}$,using the JVAS lensing statistics information alone, are respectively -2.69 and 0.68. For a flat universe, these correspond to lower and upper limits on $\lambda_{0}$ of respectively -0.85 and 0.84. Using the combination of JVAS lensing statistics and lensing statistics from the literature as discussed in Quast & Helbig (1999) the corresponding $\lambda_{0}-\Omega_{0}$ values are -1.78 and 0.27. For a flat universe, these correspond to lower and upper limits on $\lambda_{0}$ of respectively -0.39 and 0.64. Note that the lower limit is affected more than the upper limit with respect to the difference between the JVAS results and those in Paper I and with respect to combining the JVAS results with those from Paper I.

Our determination is consistent with other recent measurements of $\lambda_{0}$, both from lensing statistics and from other cosmological tests (see Quast & Helbig 1999, Paper I, for a discussion). We confirm the result of Falco et al. (1998, FKM) that radio surveys give higher values of $\lambda_{0}$ than optical surveys. Cooray et al. (1999) and Cooray (1999) obtain a 95% confidence upper limit on $\lambda_{0}$ in a flat universe of 0.79 from analyses of the Hubble Deep Field and CLASS. However, these analyses suffer from systematic effects due to our ignorance of the underlying flux density-dependent redshift distribution (or, equivalently, the redshift-dependent luminosity function) of the unlensed parent population. As discussed in Cooray (1999), the value of $\lambda_{0}$ obtained from CLASS will decrease if the mean redshift of the sample is lower than presumed. Thus, although there is no real conflict at present as the lower limits on $\lambda_{0}$ are not as tight, it seems not unlikely that a more detailed analysis of CLASS, incorporating more information about the unlensed parent population, will result in a value more in line with our value obtained from the JVAS analysis. Of course, the JVAS analysis also suffers from systematic effects, but the general agreement between the results obtained from the analysis of optical surveys (cf. Paper I and references therein) and radio surveys as presented here and in FKM suggests that these are not overwhelming. Also, the difference, a higher value of $\lambda_{0}$ from radio surveys, is what one would expect, as lens systems which go unnoticed will, all other things being equal, reduce the value of $\lambda_{0}$. This could be the case in optical surveys since it is possible that extinction in the lens galaxy and the fact that the resolution is only slightly better than the image separation could lead to lens systems being missed. Again, the general agreement does suggest though that these effects are not overwhelming.

Of course, one could imagine that the agreement is coincidental, the optical surveys being heavily affected by extinction and resolution bias and the radio surveys by our ignorance of the unlensed parent population. However, the fact that lens statistics in general gives results which are not in conflict with other cosmological tests (cf. Paper I) suggests that this is not the case. Moreover, extinction would bias the results from lens statistics and the m-z relation (e.g. for type Ia supernovae, cf. the results in Tables 3 and 4 of Paper I and in the references mentioned there) in the opposite direction. Thus, their agreement suggests that both methods have their systematics more or less under control.

The major source of uncertainty in radio lens surveys is the lack of knowledge about the redshift distribution and number-magnitude relation of the source sample (e.g. Kochanek 1996b). We are currently undertaking the necessary observations to reduce this source of systematic error. Since the time scale for this project is comparable to that for the followup of the CLASS survey, there seems little point in doing a better analysis of JVAS in the future, especially since CLASS is defined so that JVAS is essentially a subset of it. The larger size of the CLASS survey, coupled with better knowledge of the redshift distribution and number-magnitude relation of the source sample, should reduce both the random and systematic errors on our value of $\lambda_{0}$.

Acknowledgements

We thank our collaborators in the JVAS, CJF and CLASS surveys for useful discussions and for providing data in advance of publication and many colleagues at Jodrell Bank for helpful comments and suggestions. We also thank John Meaburn and Anthony Holloway at the Department of Astronomy in Manchester and the staff at Manchester Computing for providing us with additional computational resources. RQ is grateful to the CERES collaboration for making possible a visit to Jodrell Bank where this collaboration was initiated. This research was supported in part by the European Commission, TMR Programme, Research Network Contract ERBFMRXCT96-0034 "CERES''.