## 3. Observational data and prior informationWe 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 . 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 seeing and the preliminary Kochanek et al. (1995) survey selection function.
Before considering prior information in more detail, one must first decide which region of the - 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 - plane can be ruled out at a high confidence level) can be investigated. ## 3.1. The range of andThe mass clustered with galaxies on smaller scales, , is 0.1 within a factor of two (e.g. Peebles 1993). This lower limit is small compared to our full range so we do not assume any prior lower limit on except, of course, 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, is a 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 (Czoske 1995). Since then, these methods have started to indicate smaller values of , (e.g. da Costa et al. 1998) more in line with both a long tradition of low 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 per se. This has two reasons: -
`Direct' measurements of (as opposed to measurements of a combination of parameters involving ) are virtually nonexistent. -
We obtain a small enough range in from the values obtained from joint constraints on the range of and .
Historically, positive 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 and are correlated, so that increasing can be compensated for in some sense by increasing as well. Thus, effects of negative values of for a given value of are hard to differentiate from the effects of larger values of for larger (less negative) values of or even . Here, we consider negative values of
as well. There is no a priori reason
why they cannot exist. In spatially closed () models, the
antipode is required to be at , the
redshift of the most redshifted multiply imaged object currently known
(Gott et al. 1989; Park & Gott
1997).
The age of the universe in units of the Hubble time, , is where is given by Eq. (2) and thus depends on and . (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 as discussed above and are thus not relevant for this work.) Clearly, in any physically realistic world model, exceeds the age of the oldest galactic globular clusters: Following Carroll et al. (1992), we take a robust lower limit on
from conservative lower limits on the
Hubble constant and age of the universe. This gives a lower limit on
from the value at
; at larger values of
the constraint on
is not as strict-by assuming the
lower limit of independent of
we are being conservative. We choose
instead of
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
## 3.2. Prior probability for andWe have assumed no prior knowledge of per se, apart from the upper and lower limits discussed above. This has three reasons: -
`Direct' measurements of (as opposed to measurements of a combination of parameters involving ) are virtually nonexistent. -
Based on general knowledge from the literature and our own low-resolution calculations, we expect lens statistics itself to constrain quite well. -
Although recent measurements are encouraging (see Sect. 5), the value of is observationally not as well established as that of .
Regarding and as independent random quantities with known prior probability density functions and , the probability that Eq. (11) is satisfied is 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 The best estimate of the absolute age of the oldest galactic globular clusters currently is (Chaboyer et al. 1998). We choose to formulate this prior information in the form of a lognormal distribution that meets these statistics Similarly, we roughly estimate and choose to formulate this prior information in form of a normal distribution where the notation for This estimate is compatible with `small' values of the Hubble constant, which is conservative in the sense that it restricts our region of the - 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 und . Although observational evidence has always indicated a low value of
(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
, has created a prejudice in favour of
, A prior constraint on is useful since lensing statistics alone, as expected and as our results show, cannot usefully constrain . In addition, we also consider the product of with the age constraint , ## 3.3. General discussion of prior informationUsing 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, © European Southern Observatory (ESO) 1999 Online publication: March 29, 1999 |