4 Results and discussion

  The left panel of Fig. 1 shows the constraints on the cosmological parameters $\lambda_{0}$ and $\Omega_{0}$ based only on the information obtained from the JVAS lens statistics, while the right panel shows the joint constraints from the JVAS lens sample and the optical samples from Paper I. Figure 2 is identical except that one of the input parameters, the normalisation of the galaxy luminosity function, was increased by two standard deviations. This gives an idea of the magnitude of systematic uncertainties. (See the discussion in Paper I.)

  

Figure 1: Left panel: The likelihood function $p(D\vert\lambda_0,\Omega_0,\vec{\xi}_0)$ based on the JVAS lens sample. All nuisance parameters are assumed to take precisely their mean values. The pixel grey level is directly proportional to the likelihood ratio, darker pixels reflect higher ratios. The pixel size reflects the resolution of our numerical computations. The contours mark the boundaries of the minimum 0.68, 0.90, 0.95 and 0.99 confidence regions for the parameters $\lambda_{0}$ and $\Omega_{0}$. Right panel: Exactly the same as the left panel, but the joint likelihood from the JVAS lens sample and the optical samples from Quast & Helbig (1999, Paper I)

  

Figure 2: Exactly the same as Fig. 1, but the parameter $n_{\mathrm{e}}$ is increased by two standard deviations. This parameter, the normalisation of the luminosity function of the lens galaxies, is one of the more uncertain input parameters, thus one can get a rough estimate of the overall uncertainty by comparing this figure and Fig. 1. See the discussion in Paper I

   Figure 3: Left column: The posterior probability density functions $p_1(\lambda_0,\Omega_0\vert D)$ (top panel), $p_2(\lambda_0,\Omega_0\vert D)$ (middle panel) and $p_3(\lambda_0,\Omega_0\vert D)$ (bottom panel). All nuisance parameters are assumed to take precisely their mean values. The pixel grey level is directly proportional to the likelihood ratio, darker pixels reflect higher ratios. The pixel size reflects the resolution of our numerical computations. The contours mark the boundaries of the minimum 0.68, 0.90, 0.95 and 0.99 confidence regions for the parameters $\lambda_{0}$ and $\Omega_{0}$. The respective amounts of information obtained from our sample data are I₁=1.42, I₂=1.32 and I₃=1.45. Right column: Exactly the same as the left panel, but the joint likelihood from the JVAS lens sample and the optical samples from Quast & Helbig (1999). The respective amounts of information obtained from our joint sample data are 1.98, 1.95 and 1.96. See Paper I for definitions

The left plot in the top row of Fig. 3 shows the joint likelihood of our lensing statistics analysis and that obtained by using conservative estimates for H₀ and the age of the universe (see Paper I). Although neither method alone sets useful constraints on $\Omega_{0}$, their combination does, since the constraint from H₀ and the age of the universe only allows large values of $\Omega_{0}$ for $\lambda_{0}$ values which are excluded by lens statistics. Even though the 68% confidence contour still allows almost the entire $\Omega_{0}$ range, it is obvious from the grey scale that much lower values of $\Omega_{0}$ are favoured by the joint constraints. The upper limit on $\lambda_{0}$ changes only slightly while, as is to be expected, the lower limit becomes tighter. Right plot: exactly the same, but including optical constraints from Paper I. The upper limits on $\lambda_{0}$ decrease slightly, while the lower limits improve considerably. The latter is probably due to the fact that, in addition to just using more data the JVAS sources are at significantly different redshifts than those from the optical surveys analysed in Paper I (the JVAS sources are generally at lower redshift). The former is consistent with the slightly higher optical depth for radio surveys found by FKM and will be discussed more below.

The middle row of Fig. 3 shows the effect of including our prior information on $\Omega_{0}$ (see Paper I). As is to be expected, (for both the JVAS and combination data sets) lower values of $\Omega_{0}$ are favoured. This has the side effect of weakening our lower limit on $\lambda_{0}$ (though only slightly affecting the upper limit). This should not be regarded as a weakness, however, since including prior information for $\lambda_{0}$ and $\Omega_{0}$ from the constraint from H₀ and the age of the universe as well as for $\Omega_{0}$ itself, as illustrated in the bottom row of Fig. 3, tightens the lower limit again (without appreciably affecting the upper limit).

We believe that the right plot of the bottom row of Fig. 3 represents very robust constraints in the $\lambda_{0}-\Omega_{0}$ plane. The upper limits on $\lambda_{0}$come from gravitational lensing statistics, which, due to the extremely rapid increase in the optical depth for larger values of $\lambda_{0}$,are quite robust and relatively insensitive to uncertainties in the input data (cf. Fig. 2 and the discussion of the effect of changing the most uncertain input parameter by 2$\sigma$ in Paper I) as well as to the prior information used (compare the upper, lower and middle rows of Fig. 3). The combination of data from JVAS and optical surveys leads to much tighter lower limits on $\lambda_{0}$ than using either alone. The upper and lower limits on $\Omega_{0}$ are based on a number of different methods and appear to be quite robust (see Paper I). The combination of the relatively secure knowledge of H₀ and the age of the universe combine with lens statistics to produce a good lower limit on $\lambda_{0}$, although this is to some extent still subject to the caveats mentioned above.

If one is interested in the allowed range of $\lambda_{0}$, one can marginalise over $\Omega_{0}$ to obtain a probability distribution for $\lambda_{0}$. This is illustrated in Fig. 4

  

Figure 4: Left column: The top panel shows the normalised marginal likelihood function $p(\lambda_0\vert D)$ (light gray curve) and the marginal posterior probability density functions $p_1(D\vert\lambda_0)$(medium gray curve), $p_2(D\vert\lambda_0)$ (dark gray curve) and $p_3(D\vert\lambda_0)$ (black curve) derived from the JVAS analysis. All nuisance parameters are assumed to take precisely their mean values. The bottom panel shows the respective cumulative distribution functions. Right column: Exactly the same as the left panel, but the joint likelihood from the JVAS lens sample and the optical samples from Quast & Helbig (1999)

and Table 2.

  

Table 2: Marginal mean values, standard deviations and 0.95 confidence intervals for the parameter $\lambda_{0}$ on the basis of the marginal distributions shown in the top row of Fig. 4

The comparison values from this work corresponding to those in Tables 3 and 4 of Paper I are presented in Tables 3 and 4.

  

Table 3: Mean values and ranges for assorted confidence levels for the parameter $\lambda_{0}$ for our a priori and various a posteriori likelihoods from this work for $\Omega_{0}=0.3$. This should be compared to Table 3 in Paper I

  

Table 4: Mean values and ranges for assorted confidence levels for the parameter $\lambda_{0}$ for our a priori and various a posteriori likelihoods from this work for k=0. This should be compared to Table 4 in Paper I

For a "likely'' $\Omega_{0}$ value of 0.3 we have calculated the likelihood with the higher resolution $\Delta\lambda_{0}=0.01$. This is show in Fig. 5.

  

Figure 5: Left panel: The likelihood function as a function of $\lambda_{0}$ for $\Omega_{0}=0.3$ and with all nuisance parameters taking their default values, using just the JVAS data. Right panel: The same but plotted cumulatively

  

Figure 6: As Fig. 5 but combining optical and radio data. Left panel: The likelihood function as a function of $\lambda_{0}$ for $\Omega_{0}=0.3$ and with all nuisance parameters taking their default values. Right panel: The same but plotted cumulatively

From these calculations one can extract confidence limits which, due to the higher resolution in $\lambda_{0}$, are more accurate. These are presented in Table 5 and should be compared to those for $p(D\vert\lambda_0)$ from Table 3.

  

Table 5: Confidence ranges for $\lambda_{0}$ assuming $\Omega_{0}=0.3$. Unlike the results presented in Table 3, these figures are for a specific value of $\Omega_{0}$ and not the values of intersection of particular contours with the $\Omega_{0}=0.3$ line in the $\lambda_{0}-\Omega_{0}$ plane. These are more appropriate if one is convinced that $\Omega_{0}=0.3$ and have been calculated using ten times better resolution than the rest of our results presented in this work. See Figs. 5 and 6

As mentioned in Paper I, to aid comparisons with other cosmological tests, the data for the figures shown in this paper are available at

http://multivac.jb.man.ac.uk:8000/ceres
/data_from_papers/JVAS.html

and we urge our colleagues to follow our example.