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

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5. Results and discussion

5.1. Information content

Given some observational data D, some model parameters [FORMULA], and some prior and posterior probability density functions [FORMULA] and [FORMULA], the amount of information obtained from the data (e.g. Bernardo & Smith 1994) (on a logarithmic scale) is


The amounts of information obtained from our sample data are given in the caption of Fig. 3.

5.2. Results

The left panel of Fig. 2 shows the constraints on the cosmological parameters [FORMULA] and [FORMULA] based only on the information obtained from the lens statistics.

[FIGURE] Fig. 2. Left panel: The likelihood function [FORMULA]. 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 [FORMULA] and [FORMULA]. Right panel: Exactly the same as the left panel, but the parameter [FORMULA] is increased by two standard deviations

Quite good constraints can be placed on [FORMULA], more or less independent of [FORMULA]. It is a well-known fact (see K96 and references therein) that lensing statistics can provide a good upper limit on [FORMULA]. While in the past this has mainly been discussed in the context of flat cosmological models, it is of course more general (Carroll et al. 1992; Falco et al. 1998). Although no unexpected effects are seen, it is important to note that this is the first time [FORMULA] and [FORMULA] have been used as independent parameters in conjunction with a non-singular lens model in an analysis of this type.

Our analysis shows for the first time that gravitational lensing statistics can place a quite firm lower limit on [FORMULA] as well, again more or less independent of [FORMULA]. The constraint is not as tight since the gradient in the probability density is not as steep towards negative [FORMULA] as towards positive [FORMULA]. If this lower limit can be improved enough, it could provide an independent confirmation of the detection of a positive cosmological constant (see Sect. 5.3). On the other hand, this might be difficult, since Poisson errors in the number of lenses and uncertainties in the normalisation of the luminosity density of galaxies introduce relatively large uncertainties in this region of parameter space (K96, Falco et al. 1998). The latter effect is illustrated in the right panel of Fig. 2, where [FORMULA], the galaxy luminosity density normalisation, is increased by two standard deviations: the derived lower limit on [FORMULA] changes much more than does the upper limit. Nevertheless, our robust lower limit is much better than the -7 mentioned in Carroll et al. (1992).

Our results place no useful constraints on [FORMULA]. It is interesting to note the fact, however, that likely values of [FORMULA] and [FORMULA] are positively correlated. This is similar to most cosmological tests, a notable exception being constraints derived from CMB anisotropies (see Sect. 5.3). Fortunately, constraints on [FORMULA] from other sources are quite good (Sect. 3.2). Often, this is cast in the form of a constraint on [FORMULA] (e.g. Cooray et al. 1999) or, perhaps more practical, [FORMULA]. This is a reasonable way or reducing the information to one number, at least when one is concerned with upper limits on [FORMULA] (or [FORMULA]) in a relatively low-density universe. Besides the obvious dependencies on confidence levels and assumptions made, when comparing constraints on [FORMULA] from different investigations one should keep in mind whether they are approximations, like [FORMULA] in lensing statistics, and whether a value for a particular scenario (for example, for a flat universe) is the `obvious' definition or in fact describes the intersection of the [FORMULA] line with the corresponding 2-dimensional confidence contour, which in general will give a different number. Also, some authors plot `real' confidence contours while some actually plot contours at values which would correspond to certain confidence contours were the likelihood distribution in the parameter space in question Gaussian.

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 [FORMULA] and the age of the universe (see Sect. 3.2). Although neither method alone sets useful constraints on [FORMULA], their combination does, since the constraint involving [FORMULA] and the age of the universe only allows large values of [FORMULA] for [FORMULA] values which are excluded by lens statistics. Even though the 68% contour still allows almost the entire [FORMULA] range, it is obvious from the grey scale that much lower values of [FORMULA] are favoured by the joint constraints. The upper limit on [FORMULA] changes only slightly while, as is to be expected, the lower limit becomes tighter. Also, the change caused by increasing [FORMULA] by 2 standard deviations is less pronounced, with regard to both lower and upper limits on [FORMULA], as demonstrated in the right plot in top row of Fig. 3.

[FIGURE] Fig. 3. Left column: The posterior probability density functions [FORMULA] (top panel), [FORMULA] (middle panel) and [FORMULA] (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 [FORMULA] and [FORMULA]. The respective amounts of information (Eq. (29)) obtained from our sample data are [FORMULA], [FORMULA] and [FORMULA]. Right column: Exactly the same as the left column, but the parameter [FORMULA] is increased by two standard deviations

The middle row of Fig. 3 shows the effect of including our prior information on [FORMULA] (see Sect. 3.2). As is to be expected, (for both values of [FORMULA]) lower values of [FORMULA] are favoured. This has the side effect of weakening our lower limit on [FORMULA] (though only slightly affecting the upper limit).

We believe that the left plot of the bottom row of Fig. 3 represents very robust constraints in the [FORMULA]-[FORMULA] plane. The upper limits on [FORMULA] come from gravitational lensing statistics, which, due to the extremely rapid increase in the optical depth for larger values of [FORMULA], are quite robust and relatively insensitive to uncertainties in the input data (compare the left and right columns of Fig. 3) as well as to the prior information used data (compare the upper, lower and middle rows of Fig. 3). The upper and lower limits on [FORMULA] are based on a number of different methods and appear to be quite robust, as discussed in Sect. 3.2. The combination of the relatively secure knowledge of [FORMULA] and the age of the universe combine with lens statistics to produce a good lower limit on [FORMULA], although this is to some extent still subject to the caveats mentioned above.

If one is interested in the allowed range of [FORMULA], one can marginalise over [FORMULA] to obtain a probability distribution for [FORMULA]. This is illustrated in Fig. 4 and Table 2.

[FIGURE] Fig. 4. Left column: The top panel shows the normalised marginal likelihood function [FORMULA] (light grey curve) and the marginal posterior probability density functions [FORMULA] (medium grey curve), [FORMULA] (dark grey curve) and [FORMULA] (black curve). All nuisance parameters are assumed to take precisely their mean values. The bottom panel shows the respective cumulative distribution functions. These can be used to construct any desired [FORMULA]-averaged upper or lower limits on [FORMULA]. Right column: Exactly the same as the left column, but the parameter [FORMULA] is increased by two standard deviations


Table 2. Marginal mean values, standard deviations and 0.95 confidence intervals for the parameter [FORMULA] on the basis of the marginal distributions shown in the top row of Fig. 4; `information' refers to Eq. 29

5.3. Comparison with other results

For comparison with other results, as a first step one can examine the allowed range of [FORMULA] for the current `best-fit' value for [FORMULA], which we take, based on the work cited in Sect. 3.2, to be [FORMULA]. (A more conservative estimate is reflected by using the prior probability distribution [FORMULA] as shown by the dark grey curve in Fig. 4 and in Table 2.) On the other hand, previous limits on [FORMULA] have often been quoted for a flat universe (K96 and references therein). We consider both cases in Tables 3 and 4.


Table 3. 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]. Except where noted, the ranges quoted are the projections of the corresponding confidence contours in the [FORMULA]-[FORMULA] plane onto the [FORMULA] axisa (as opposed to [FORMULA]-independent estimates, which of course would always give a smaller range), and are of course two-sided, not one-sided, bounds. Values are either those quoted in the references given and/or obtained from figures in those references; inequalities mean that the corresponding confidence contour is to be found in the range indicated by the inequality, e.g. [FORMULA] would mean that the corresponding contour level is to be found at [FORMULA], not that the constraint is [FORMULA] at the corresponding confidence level. This arises because the corresponding area of parameter space was not examined in the reference in question. If the confidence interval could not be determined from the reference, both values in the corresponding column are missing.
a) Note that some references quote confidence ranges for [FORMULA]-in general, these will be different than the projection of the intersection of the corresponding contour in the [FORMULA]-[FORMULA] plane onto the [FORMULA]-axis.
b) Falco et al. (1998)
c) contour at 95.4% not 95%
d)Falco et al. (1998)
e) contour at 95.4% not 95%
f) Falco et al. (1998)
g) contour at 95.4% not 95%
h) Perlmutter et al. (1998)
i) Riess et al. (1998)
j) Fig. 6, solid contours
k) contours at 68.3%, 95.4% and 99.7% instead of 68%, 95% and 99% respectively
l) Carlberg (1998)
m) Lineweaver (1998)
n) contours at 68.3%, 95.4% and 99.7% instead of 68%, 95% and 99%, respectively
o) Webster et al. (1998)
p) Guerra et al. (1998)

We do not do a comparison for the special case [FORMULA] since this analysis of gravitational lensing statistics does not usefully constrain [FORMULA] (any limits coming only from the prior information on [FORMULA]).

It is beyond the scope of this paper to do a full comparison of different cosmological tests. Except for a few general comments, we therefore restrict ourselves to comments on the similarities and differences between the results from this work without using prior information on [FORMULA] and [FORMULA], i.e. (the left plot in) Fig. 2, and the those from K96 and Falco et al. (1998) (using only optical data, i.e. the lower left plot in their Fig. 5).

Taking all results at face value and examining the [FORMULA] case first, we note that with `three-and-one-half ' exceptions (counting as one test each the four from this work and the three from Falco et al. (1998)) the 68% c.l. lower limit from Lineweaver (1998) is higher than all 68% upper limits from other tests, while the 95% lower and upper confidence levels from Lineweaver (1998) are higher than the corresponding limits from the other tests for all but one of these. Even at the 99.9% confidence level (not shown in Table 3), the Lineweaver (1998) result requires [FORMULA]. If one assumes [FORMULA], only Lineweaver (1998) requires [FORMULA], though all other tests (except Carlberg (1998)) are compatible with this. This is not surprising, since it is well-known that constraints from CMB anisotropies tend to run more or less orthogonal in the [FORMULA]-[FORMULA] plane to those from most other tests (e.g. White 1998; Eisenstein et al. 1998b; Tegmark et al. 1998a,b).

Examining the [FORMULA] case, it is interesting to note that the 68% (90%) confidence level lower limit on [FORMULA] from Carlberg (1998) is higher than all of the 68% (90%) c.l. upper limits from all other tests except Guerra et al. (1998). Otherwise, with `one-and-one-half ' exceptions all tests are compatible even at the 68% confidence level. If one assumes [FORMULA], then the evidence for [FORMULA] looks convincing: at the 68% confidence level, again with `one-and-one-half ' exceptions, all tests indicate [FORMULA]; even at 90% the evidence is still quite good, if one keeps in mind that the gradient towards smaller values of [FORMULA] is generally not as steep as towards larger values.

Again taken at face value, neither the [FORMULA] case nor the [FORMULA] case are compatible with all tests, even at the [FORMULA]90% confidence level. It appears the simplest solution to achieve concordance would be to have [FORMULA], which is within the error on [FORMULA] discussed in Sect. 3.2. For [FORMULA] this would imply [FORMULA], which seems to be ruled out, thus ruling out the flat universe altogether. For a non-flat universe, reducing [FORMULA] would, due to the CMB constraint, require a higher value of [FORMULA], and thus make the [FORMULA] case more unlikely, ruling out this special case as well.

On balance, a cosmological model with [FORMULA] and [FORMULA] seems compatible with all known observational data (not just those discussed here) at a comfortable confidence level.

For a `likely' [FORMULA] value of 0.3 we have calculated the likelihood with the higher resolution [FORMULA]. This is shown in Fig. 5. From these calculations one can extract confidence limits which, due to the higher resolution in [FORMULA], are more accurate. These are presented in Table 5 and should be compared to those for [FORMULA] from Table 3.

[FIGURE] Fig. 5. Left panel: The likelihood function as a function of [FORMULA] for [FORMULA] and with all nuisance parameters taking their default values. Right panel: The same but plotted cumulatively. See Table 5

Again, a full discussion of joint constraints involving discussion of possible sources of error for each test, as well as comparing the full contours in the [FORMULA]-[FORMULA] plane, is beyond the scope of this paper. However, quick comparisons would be aided were the results of all tests available in an easy-to-process electronic form (see below); such quick consistency tests would enable one to spot areas of inconsistency much more quickly. Also, it should be emphasised that the projections onto the [FORMULA]-axis of the intersection of a particular confidence contour with the [FORMULA] or [FORMULA] axis are generally not the same as the corresponding confidence interval for the [FORMULA] or [FORMULA] special cases.

For a flat universe, our 95% confidence level upper limit on [FORMULA]-[FORMULA], i.e. the value of [FORMULA] where this contour crosses the [FORMULA] line, is [FORMULA]. This is essentially the same as the [FORMULA] of K96, as was to be expected considering we used essentially the same data and methods. Interpreted cautiously, one might conclude from this that the singular isothermal sphere model is a good approximation as far as determining the cosmological parameters from lens statistics is concerned, as was assumed in Falco et al. (1998). Our 99% confidence level upper limit on [FORMULA] is [FORMULA]. This is quite a tight upper bound on [FORMULA] and appears to be quite robust.

Perhaps more interesting is the comparison with (the results using only optical data in) Falco et al. (1998). Although a detailed comparison is complicated by the different plotting scheme and reducing the entire contour (or indeed grey-scale) plot to a few numbers throws away information, it is obvious that the plots are broadly similar. Our 68% contour is, for [FORMULA], roughly parallel to the [FORMULA]-axis at [FORMULA]. This is just at the edge of the Falco et al. (1998) plot, and as they provide no grey-scale, it is difficult to compare the lower limits on [FORMULA]. Thus, while our main goal was to explore a `large enough' region of parameter space, comparison in the areas where there is overlap shows consistency, which strengthens our faith in the conclusions pertaining to areas of parameter space where there is no overlap.

Recently, it has become quite fashionable to discuss joint constraints derived from a variety of cosmological tests. This has grown from plotting the overlap of likelihood contours (often in a space spanned by parameters other than [FORMULA] and [FORMULA]) (e.g. Ostriker & Steinhardt 1995; Turner 1996; Bagla et al. 1996; Krauss 1998; White 1998) to full-blown joint likelihood analyses, both detailed theoretical investigations of what will be possible in the future (e.g. Tegmark et al. 1998a,b; Eisenstein et al. 1998a,b) and more restricted analyses using present data (e.g. Webster et al. 1998). While in some cases it is quick and easy to calculate the likelihood as a function of [FORMULA] and [FORMULA] given the data, for example for tests using the m-z relation, in other cases such as the present one it is a major programming and computational effort to do so. To aid comparisons, all figures from this paper are available in the form of tables of numbers at


and we urge our colleagues to follow our example. We applaud the fact that most results are now presented in the [FORMULA]-[FORMULA] plane, as opposed to using other parameters such as [FORMULA] or [FORMULA]. A further aid in comparison would be a uniform choice of axes. We prefer to plot [FORMULA] on the y-axis and [FORMULA] on the x axis since up/down symmetry is less fundamental than left/right symmetry and this mirrors the fact that [FORMULA] has the physical lower limit [FORMULA] whereas no corresponding upper or lower limits for [FORMULA] exist. Square plots with the same range would further aid the comparison. Of course, if all data are publicly available, then they can be re-plotted to taste.

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Online publication: March 29, 1999