A&A 414, 425-428 (2004)
DOI: 10.1051/0004-6361:20034184
P. Saha
Astronomy Unit,
Queen Mary and Westfield College,
University of London,
London E1 4NS, UK
Observatoire astronomique,
11 rue de l'Université,
67000 Strasbourg, France
Received 12 August 2003 / Accepted 18 September 2003
Abstract
We can think of a lensed quasar as taking the Hubble time,
shrinking it by 10^{-11}, and then presenting the result to us
as a time delay; the shrinking factor is of the order of fractional
sky-area that the lens occupies. This cute fact is a straightforward
consequence of lensing theory, and enables a simple rescaling of time
delays. Observed time delays have a 40-fold range, but after
rescaling the range reduces to 5-fold. The latter range depends on
details of the lens and lensing configuration - for example, quads
have systematically shorter rescaled time delays than doubles - and is
as expected from a simple model. The hypothesis that observed
time-delay lenses all come from a generalized-isothermal family can be
ruled out. But there is no indication of drastically different
populations either.
Key words: Gravitational lensing - galaxies: quasars: general
Most of the observables in gravitational lensing (image positions and magnifications) are intrinsically dimensionless. The exception is the time delay between images, which takes its dimensionality straight from the universe^{}: . This remarkable fact is the essential reason for much research effort going into measuring time delays. The observations have been increasingly successful - in 1995 there was but one controversial time delay, currently there are nine non-controversial ones. These are summarized in Table 1 below.
But curiously, even as the image and time delay data have improved, the error bars on the inferred H_{0} have not. As an example, consider 0957+561. Between Kundic et al. (1997) and Oscoz et al. (2001) the time-delay value changed by only 2%. But meanwhile, whereas Kundic et al. (1997) quote (95% confidence) in the usual units of , Bernstein & Fischer (1999) with more imaging and more modelling conclude that the data imply only 77^{+29}_{-24}, while Keeton et al. (2000) assert that further data on the lensed host galaxy invalidates all previously published models, and they decline to give an H_{0} estimate at all. Basically, the problem is that simple lens models are unable to fit the images to the mas-level demanded by current data, while more complicated models can fit the data but are non-unique and can produce identical observables from very different values of H_{0}.
Modellers have responded to this dilemma with two strategies. One is to try to identify simple models that both have enough parameters to fit or nearly fit the data and can be justified on galactic-structure grounds; Kochanek (2003) is typical of these. The other strategy is to try to explore the space of all plausible models allowed by the data; Raychaudhury et al. (2003) is a recent example. For a review by authors representing different points of view see Courbin et al. (2003).
In the current context of good data and active modelling but no consensus on models, it is interesting to step back and pose some questions that tend to get obscured in the details of modelling. First, we can think of the purpose of modelling time-delay lenses as being to discover one dimensionless number, the factor relating and H_{0}^{-1}. What contributions to this number are well-constrained and what are poorly constrained? What range of values do the data imply for the poorly-constrained part? Is that range systematically different for doubles and quads, and/or for isolated lensing galaxies versus interacting galaxies? And is that range consistent with what we expect from popular models? Nine systems is a small sample, but it is enough to provide preliminary answers to these questions, and to do so is the aim of this paper.
In lensing theory the arrival time can be written as
(1) |
(3) |
(6) |
For isothermal lenses,
ranges from 0 to 8, averaging
.
To see this, recall that for isothermals,
and note that
could be anywhere in
the Einstein ring. Hence
and using (7)
gives
Witt et al. (2000, hereafter WMK) show that is not restricted to isothermals but is valid for a large family of generalized-isothermal lenses, and argue that it will be generally applicable in nature. If so, could be eliminated altogether. We can readily test if this is the case.
We now present the obvious comparison of the scaled time delays with current data.
Table 1 lists the relevant quantities for the various time-delay systems. The time delays references are given in the table, and the other data are taken from the CASTLES survey and compilation by Kochanek et al. (1998). For quads, only the first and last images (that is, the longest time delay) are considered, to enable a simple comparison with doubles. There are some caveats to the values of and : for 1830 and 0218 the lens-centre is very uncertain and hence are especially uncertain, for 1608 the lens is apparently an interacting pair of galaxies, and 0957 and 0911 are in clusters and hence have large lensing contributions from other galaxies.
Table 1: Summary of time-delay data.
Figure 1 shows against for the currently known time-delay systems. Since error bars on time delays are typically a few percent they are not shown here. We notice three things:
Figure 1: Plot of the scaled time delay defined in Eq. (5) against the observed time delay. The various lenses are labelled by their short names: quads are labelled below, doubles above. | |
Open with DEXTER |
Figure 2: As in Fig. 1, but omitting the D factor in the scaled time delay. | |
Open with DEXTER |
We can also compare against the data to test whether the lenses belong to the generalized isothermal family studied by WMK. Figure 3 shows against for the same systems. We notice the following
Whereas is rejected, are other scalings possible that improve upon ? L.L.R. Williams (personal communication) points out that the definition (Eq. 5) of considers the size of the lens but not its asymmetry, and that if we multiply in the definition by a further factor of as a measure of asymmetry, then the scaled time delays would range over a factor of only 2.5, with no significant trend. But the meaning of such an asymmetry correction in terms of lensing theory is not known.
Figure 3: as defined in Eq. (9) against the observed time delay. The non-physical trend is significant (see text), and hence the generalized isothermal models are rejected. | |
Open with DEXTER |
From the above, it appears that the scatter in reflects a range of mass profiles and source positions, and that its value must be inferred for each lens by detailed modelling. But without going into detailed models for nine lenses, we can at least check whether the observed range of is plausible.
Figure 4 shows such a check. The main plot is of against the area for an example model (an elliptical isothermal potential plus external shear.) The value of is shown for different source positions, the two loops corresponding to source positions along the two caustics (actually just inside the caustics, to avoid computational problems). Quads are below the lower loop, with . Doubles are between the two loops, with ^{}. The values are model-dependent - for example, a steeper model will have both loops somewhat higher. Also, the value of depends on the source position: smaller for sources along the long axis of the potential, larger for sources perpendicular to that axis. But with these qualifications, Figure 4 shows that the general ranges of , including the separation of quads and doubles, is just as it is in the data, and there is no evidence that the observed systems come from drastically different populations of lenses.
Figure 4: Computation of values from a simple model of 1115+080, taken from Saha & Williams (2003). The top two panels show an image morphology similar to 1115+080, and the corresponding source position. The lower panel shows against for source positions along the two caustics. (The horizontal axis is not labelled because has arbitrary units: arcsec^{2}, steradians, etc.) The lower loop corresponds to the diamond caustic and the upper loop corresponds to the outer caustic. Hence quads are below the lower loop and doubles are between the two loops. | |
Open with DEXTER |
We see in this paper a new interpretation of lensing time delays: is H_{0}^{-1} shrunk by the lens's covering factor on the sky, times a number of the order of unity. On separating off a redshift dependent-term (also of order unity) we are left with a number (say) that summarizes the dependence on details of the lens and lens configuration.
Using these ideas, we can rescale the observed time delays for the nine currently-measured systems. The observed time delays range over a factor of 40, but the rescaled delays range over a factor of 5. The latter is the inferred range of , and moreover it appears that for quads and . Reassuringly, the same spread in is reproduced by a simple model.
Using rescaled time-delays we can also test the hypothesis that the observed lenses all belong to a generalized-isothermal family. This hypothesis is ruled out: it over-predicts time delaysfor large lenses. On the other hand, there is no indication that the known time-delay systems come from drastically different types of lenses.
In Figs. 1 to 3 we have some points (x_{i},y_{i}) and we want to know whether there is any trend in the scatter. There are many statistical tests relating to the significance of trends in data, but none of the standard ones address quite this question. However, it is not difficult to design a suitable statistical test. Let us pose the question: what is the probability of improving the fit to by shuffling the y_{i}? If nearly all shufflings reduce the we would conclude that the data have a trend.
In the familiar straight-line fit, the slope is monotonic in . Hence as a statistic, is equivalent to the slope.
In the main text, I use the phrase "significant at the 95% level'' to mean that 5% of shufflings increase the . Statisticians might use a phrase like "p-value of 95%''.
Acknowledgements
I am grateful to Rodrigo Ibata and Liliya Williams, who contributed some fruitful suggestions.