3 Calculations

 

A major difference between the analysis of an optical survey (see Paper I and references therein) and a radio survey is that in the latter one does not know the redshifts of all the unlensed sources. One can still use the formalism of Paper I, however, substituting for the non-lensed objects in the sample a subsample with known redshifts, multiplying the logarithm of this contribution from the non-lenses to the likelihood by the ratio of the size of the parent sample to that of the subsample. Alternatively, one can take the redshifts from a sample selected according to the same criteria, assigning these randomly to objects in (a subsample of) the parent sample for a similar flux density range. Similarly, one does not know the number-magnitude relation for the sample and for its extension to fainter flux densities (needed to allow for the lens amplification). Again, this can be estimated from either a subsample (through extrapolation) or from another sample selected according to the same criteria (either through extrapolation or by having a fainter flux density limit in this other sample; in the latter case obviously the selection criteria should be identical to that of the original sample except for the lower flux density limit).

For this analysis, due to the paucity of the observational data, we have made rather stark assumptions: the redshift distribution of the sample is assumed to be identical to that of the CJF sample (Taylor et al. 1996), independent of flux density, and the number-magnitude relation is assumed to be identical to that of CLASS (Cosmic Lens All-Sky Survey, Myers et al. 1999), independent of redshift.

Otherwise, we have followed the procedure outlined in Paper I, calculating the a priori likelihood of obtaining the observational data as a function of $\lambda_{0}$ and $\Omega_{0}$ and the a posteriori likelihood for the three different choices of prior information used in Paper I. We present results both for the JVAS lens survey and for the combination of the JVAS results with those from the optical surveys analysed in Paper I.