4.3. Modeling and Influence of the Astrometry
Using a simple isothermal model plus external shear (induced by the cluster) Schechter et al. (1997) infered H_{0} = 42 km s^{-1} Mpc^{-1}, but no detailed observations of the lensing galaxy was available at that time. In fact, Schechter et al. (1997) found that a value as high as 84 was possible as well. Keeton & Kochanek (1997) investigated analytical models for the lensing galaxy, including isothermal profiles and softened power laws with core radius. They found that the potential well in PG 1115+080 could not be modeled using a single lensing galaxy, whatever its mass profile. External shear by the nearby group had to be invoked. A combination of models were tested for the main lens and for the cluster, and analyzed in a statistical way in order to derive a probability density for H_{0}. The experiment is done in two ways: by taking into account only one time-delay in PG 1115+080, and by using the two measured time-delays.
The models by Keeton & Kochanek are based on astrometry obtained with HST data, before the optics was refurbished (Kristian et al. 1993), and with rather large astrometric errors. As systems with symmetric image configurations about the lens are more sensitive to errors on the astrometry than assymetric ones, the result by Keeton & Kochanek could be improved further, simply by improving the HST astrometry. This was done by applying deconvolution techniques to ground based images of PG 1115+080. Fig. 8 illustrates the gain in the accuracy. With the new astrometry, not only the width of the probability distributions is decreased, but there is also better agreement between the curves obtained for one and for two time-delays. The combined probability for the pre-refurbishment HST astrometry is H_{0} of 51^{+14}_{-10} km s^{-1} Mpc^{-1}while H_{0} = 53^{+10}_{-7} km s^{-1} Mpc^{-1}with the improved astrometry.
Figure 8. Top: probability density for H_{0} using the models of Keeton & kochanek (1997), for various combinations of models for the main lens and intervening group and using one time-delay (top panel) or two time-delays (bottom panel). Bottom: Same as above but using improved astrometry (Courbin et al. 1997). |
In parallel with the efforts to improve the observational constraints for PG 1115+080, new models were developed, with the aim of exploring more of the parameter space defined by the slope of the mass profile of the lens, its ellipticity and its position angle. One approach to the problem is to consider a family of non-parametric models, where the lensing galaxy is decomposed on a grid of mass "pixels" or tiles. The astrometry, flux ratios, and time-delays are given as observational constraints, and the output of the procedure is a pixelized mass map of the lens, as well as its slope. As is done for parametric models, one can have a statistical approach and run many models in order to infer a probability density for H_{0} (Williams & Saha 2000). As the range of possible models is much broader with the non-parametric approach than with the parametric one, it was found that H_{0} could be anything between 42 and 84, given the observations available for PG 1115+080 at that time (Saha & Williams 1997) !