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

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3. Observational data and prior information

We use the observational data of the optical multiply imaged quasar surveys by Crampton et al. (1992), Jaunsen et al. (1995), Kochanek et al. (1995), Yee et al. (1993) and the observational data of the HST Snapshot Survey compiled by Maoz et al. (1993), including Q 0142-100, Q 1115+080 and Q 1413+117. If applicable, we replace the apparent quasar V magnitude catalog data found in Crampton et al. (1992), Jaunsen et al. (1995) and Yee et al. (1993) with more current data from Veron-Cetty & Veron (1996). We estimate the Kochanek et al. (1995) apparent quasar V magnitude data by adding the survey average V-R and V-I colours to the observational R and I magnitude data. Following K96, we only include quasars with redshift [FORMULA]. In all, our sample contains 807 singly and 5 multiply imaged quasars. The observational data of the multiply imaged quasars are summarised in Table 1. Our complete input data can be obtained from

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

This follows K96 for purposes of comparison. Since much larger surveys (i.e. CLASS) will be considered in a future paper, there is little point in increasing the number of lenses for its own sake. Since radio observations are considered in more detail in a companion paper (Helbig et al. 1999), we restrict ourselves to optical surveys in this paper. We use the Crampton et al. (1992), HST Snapshot Survey and Yee et al. (1993) survey selection functions proposed in Kochanek (1993), the Jaunsen et al. (1995) survey selection function at [FORMULA] seeing and the preliminary Kochanek et al. (1995) survey selection function.


[TABLE]

Table 1. Observational data of multiply imaged quasars contained in the sample. The magnitudes are V magnitudes unless otherwise specified. The image separations are taken from Kochanek et al. (1997)


Before considering prior information in more detail, one must first decide which region of the [FORMULA]-[FORMULA] plane is to be investigated. Clearly, this region should be defined by either exact constraints or conservative estimates, as opposed to current `best fit' values (and their errors), in order to avoid excluding any possibly viable cosmological models. Also, it is desirable for the region to be on the large side, so that in addition the sensitivity of the test (i.e. what regions of the [FORMULA]-[FORMULA] plane can be ruled out at a high confidence level) can be investigated.

3.1. The range of [FORMULA] and [FORMULA]

The mass clustered with galaxies on smaller scales, [FORMULA], is 0.1 within a factor of two (e.g. Peebles 1993). This lower limit is small compared to our full [FORMULA] range so we do not assume any prior lower limit on [FORMULA] except, of course,

[EQUATION]

Especially for comparison with other work it is important to note that, within the framework of cosmological models based on general relativity with which we (and almost everyone else at present) are working, [FORMULA] is a requirement . Results reported which include [FORMULA] within the errors, or even as a best-fit value, do not indicate `implausible results' but merely improper statistics. Often, confidence contours are assumed to be ellipses and these are extended, if applicable, to [FORMULA]. (Of course, it is possible that [FORMULA] is within the errors or even the best fit value for a certain set of results.)

An extremely conservative upper limit comes from dynamical tests on larger (though still cosmologically small) scales; when this work was started, we assumed an (again, extremely conservative) upper limit [FORMULA] (Czoske 1995). Since then, these methods have started to indicate smaller values of [FORMULA], (e.g. da Costa et al. 1998) more in line with both a long tradition of low [FORMULA] values (e.g. Gott et al. 1974; Coles & Ellis 1994, 1997) (albeit with somewhat larger errors) as well as new determinations (often with quite small errors), examples of which are mentioned in Sect. 3.2.

We have assumed no prior upper or lower limits on [FORMULA] per se. This has two reasons:

  • `Direct' measurements of [FORMULA] (as opposed to measurements of a combination of parameters involving [FORMULA]) are virtually nonexistent.

  • We obtain a small enough range in [FORMULA] from the values obtained from joint constraints on the range of [FORMULA] and [FORMULA].

Historically, positive [FORMULA] values have been considered more than negative ones, probably because positive values can have a wide range of relatively easily observable effects, while negative ones are more difficult to measure. Many cosmological tests have a degeneracy such that [FORMULA] and [FORMULA] are correlated, so that increasing [FORMULA] can be compensated for in some sense by increasing [FORMULA] as well. Thus, effects of negative values of [FORMULA] for a given value of [FORMULA] are hard to differentiate from the effects of larger values of [FORMULA] for larger (less negative) values of [FORMULA] or even [FORMULA].

Here, we consider negative values of [FORMULA] as well. There is no a priori reason why they cannot exist. If one believes that the `source' of [FORMULA] are zero-point fluctuations of a quantum vacuum, this would lend support to the idea that [FORMULA]. However, it is not clear that this must be the only source of [FORMULA], and indeed it has been argued that, if this source of [FORMULA] exists, there must be an additional contribution with a negative value (e.g. Martel et al. 1998, though the assumption that this is possible is so obvious to the authors it is barely stated!).

In spatially closed ([FORMULA]) models, the antipode is required to be at [FORMULA], the redshift of the most redshifted multiply imaged object currently known (Gott et al. 1989; Park & Gott 1997). 2 The light grey shaded area in Fig. 1a marks the right side of the region thus enclosed. This gives us a slightly [FORMULA]-dependent upper limit on [FORMULA] which is slightly stronger than that obtained by merely excluding models with no big bang. (This can be done because these models have a maximum redshift which is less than the redshift of high-redshift objects, the only exception being some cosmological models which have [FORMULA], the robust lower limit discussed above (e.g. Feige 1992).)

[FIGURE] Fig. 1. a  The cosmological parameter plane. The four curved lines in a are the isochrones [FORMULA]. The straight line marks spatially flat world models. In the white region, the antipodal redshift falls below [FORMULA], the redshift of the most redshifted multiply imaged object currently known (Gott et al. 1989; Park & Gott 1997). b -d The prior probability distributions [FORMULA] b , [FORMULA] c and [FORMULA] d . The pixel grey level is directly proportional to the probability density ratio, darker pixels reflect higher ratios. The pixel size reflects the resolution of our numerical computations. The contours mark 0.61, 0.26, 0.14 and 0.036 of the peak likelihood for the parameters [FORMULA] and [FORMULA], which would correspond to the boundaries of the minimum 0.68, 0.90, 0.95 and 0.99 confidence regions if the distribution were Gaussian

The age of the universe in units of the Hubble time, [FORMULA], is

[EQUATION]

where [FORMULA] is given by Eq. (2) and thus depends on [FORMULA] and [FORMULA]. (There are world models in which the maximum redshift is not infinite but these are all models without a big bang and are excluded by the constraint from the antipodal redshift or the lower limit on [FORMULA] as discussed above and are thus not relevant for this work.) Clearly, in any physically realistic world model, [FORMULA] exceeds the age of the oldest galactic globular clusters:

[EQUATION]

Following Carroll et al. (1992), we take a robust lower limit on [FORMULA] from conservative lower limits on the Hubble constant and age of the universe. This gives a lower limit on [FORMULA] from the value at [FORMULA]; at larger values of [FORMULA] the constraint on [FORMULA] is not as strict-by assuming the lower limit of [FORMULA] independent of [FORMULA] we are being conservative. We choose [FORMULA] instead of [FORMULA] as in Carroll et al. (1992) since no published current constraints examine this region in detail. (Were this the case, then including this area would be helpful if only to aid a direct comparison.) This value corresponds roughly to the one-sided 99% confidence level in Fig. 1b (see Sect. 3.2), which is also a reason not to extend the area to more negative [FORMULA] values.

3.2. Prior probability for [FORMULA] and [FORMULA]

We have assumed no prior knowledge of [FORMULA] per se, apart from the upper and lower limits discussed above. This has three reasons:

  • `Direct' measurements of [FORMULA] (as opposed to measurements of a combination of parameters involving [FORMULA]) are virtually nonexistent.

  • Based on general knowledge from the literature and our own low-resolution calculations, we expect lens statistics itself to constrain [FORMULA] quite well.

  • Although recent measurements are encouraging (see Sect. 5), the value of [FORMULA] is observationally not as well established as that of [FORMULA].

Regarding [FORMULA] and [FORMULA] as independent random quantities with known prior probability density functions [FORMULA] and [FORMULA], the probability that Eq. (11) is satisfied is

[EQUATION]

A cosmological world model is compatible with the absolute age of the oldest galactic globular clusters as long as the above expression does not vanish. Reasonably, we assume a prior probability density function that is proportional to this expression

[EQUATION]

The best estimate of the absolute age of the oldest galactic globular clusters currently is [FORMULA] (Chaboyer et al. 1998). We choose to formulate this prior information in the form of a lognormal distribution that meets these statistics

[EQUATION]

Similarly, we roughly estimate [FORMULA] and choose to formulate this prior information in form of a normal distribution

[EQUATION]

where the notation for L and N is such that the two arguments correspond to the mean and standard deviation.

This estimate is compatible with `small' values of the Hubble constant, which is conservative in the sense that it restricts our region of the [FORMULA]-[FORMULA] plane less than would `large' values. By the same token we neglect any time between the big bang and the formation of the oldest globular clusters. Inserting Eq. (14) and Eq. (15) in Eq. (13) one obtains a well-founded a priori probability distribution for the parameters [FORMULA] und [FORMULA].

Although observational evidence has always indicated a low value of [FORMULA] (e.g. Gott et al. 1974; Coles & Ellis 1994, 1997), the inflationary paradigm (e.g. Guth 1981), coupled with a prejudice against a non-negligible value of [FORMULA], has created a prejudice in favour of [FORMULA]3 unfortunately too often to the extent where this prior belief has been elevated to the status of dogma (see, e.g., Matravers et al. 1995, for an illuminating account) even though there are serious fundamental problems with the inflationary idea (e.g. Penrose 1989) and even though there might be other solutions to the problems it claims to solve (e.g. Barrow 1995; Collins 1997). What is more, some current inflationary thinking (e.g. Turok & Hawking 1998) seems able to predict values for [FORMULA] and [FORMULA] similar to current observationally determined values, though it would have been more interesting had this prediction been made before the recent improvements in the observational situation. (To be fair, many leading practitioners of inflation consider a flat universe to be a robust prediction and its observational falsification essentially a falsification of the entire paradigm.) Recently, in the light of overwhelming observational evidence in favour of a low value of [FORMULA] (e.g. Carlberg et al. 1998b; Carlberg 1998; Carlberg et al. 1998c; Bahcall 1998; Bahcall et al. 1997; Fan et al. 1997; Bartelmann et al. 1998; Lineweaver 1998), whether determined more or less independently or in combination with other parameters, this prejudice is starting to weaken. Conservatively, these results can be summarised as

[EQUATION]

A prior constraint on [FORMULA] is useful since lensing statistics alone, as expected and as our results show, cannot usefully constrain [FORMULA].

In addition, we also consider the product of [FORMULA] with the age constraint [FORMULA],

[EQUATION]

3.3. General discussion of prior information

Using harsher constraints would mean that results would reflect almost exclusively the prior information as opposed to the information derived from lensing statistics. It is not the purpose of this paper to do a joint analysis of several cosmological tests, 4 but rather to examine lens statistics as a cosmological test. For practical reasons, an upper limit on [FORMULA] and upper and lower limits on [FORMULA] are required. On the other hand, it is sensible to combine the results with conservative constraints from other well-understood cosmological tests where there is general agreement and little room for debate. Within our upper and lower limits, we present our results both with and without the constraints discussed above. The density values and confidence contours of the three prior probability density functions are shown in Fig. 1b-d.

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Online publication: March 29, 1999
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