2.6.2. Gravitational Lensing Time Delay

When quasar image is split into two or more by gravitational lensing, we expect the time delay among images, arising from different path lengths and gravitational potentials among image positions. The time delay between images A and B takes the form

(9)

where is the angular difference between the source and image, is the difference in the potential and D_IJ is the angular diameter distances. The time delay is observable if the source is variable, and can be used to infer H₀ (Refsdal 1964). Crucial in this argument is a proper modelling of the mass distribution of the deflector. The DD/D factor depends on only weakly; its dependence is even weaker.

The first case where H₀ is derived is with the 0957+561 lens system. The deflector is complicated by the fact that a giant elliptical galaxy is embedded into a cluster. Falco, Gorenstein & Shapiro (1991) noted an ambiguity associated with a galaxy mass - cluster mass separation, which does not change any observed lens properties but affects the derived Hubble constant. One way to resolve this degeneracy is to use the velocity dispersion of the central galaxy (Falco et al. 1991; Grogin & Narayan 1996). Kundic et al. (1997b), having resolved a long-standing uncertainty about the time delay, obtained H₀ = 64 ± 13 employing the Grogin-Narayan model. Tonry & Franx (1999) revised it to 71 ± 7 with their new velocity dispersion measurement near the central galaxy. More recently, Bernstein & Fischer (1999) searched a wider variety of models, also using weak lensing information to constrain the mass surface density of the cluster component, and concluded H₀ = 77⁺²⁹_-24, the large error representing uncertainties associated with the choice of models.

The second example, PG1115+080, is again an unfortunate case. The deflector is elliptical galaxy embedded in a Hickson-type compact group of galaxies (Kundic et al. 1997a). Keeton & Kochanek (1997) and Courbin et al. (1997) derived (51-53) ± 15 from the time delay measured by Schechter et al. (1997). Impey et al. (1998) examined the dependence of the derived H₀ on the assumption for the dark matter distribution, and found it to vary from 44 ± 4 (corresponding to M/L linearly increasing with the distance) to 65 ± 5 (when M/L is constant over a large scale). The latter situation may sound strange, but it seems not too unusual for elliptical galaxies, a typical example being seen in NGC5128 (Peng et al. 1998).

Recently, time delays have been measured for three more lenses, B0218+357, B1608+656 and PKS1830-211. B0218+357 is a rather clean, isolated spiral galaxy lens, and Biggs et al. (1999) derived H₀ = 69⁺¹³_-19 (the central value will be 74 if = 0.3) with a simple galaxy model of a singular isothermal ellipsoid. For B1608+656, Koopmans & Fassnacht (1999) obtained 64 ± for = 0.3 (59 ± 7 for EdS). For PKS1830-211, they gave 75⁺¹⁸_-10 for EdS and 85⁺²⁰_-11 for = 0.3 from the time delay measured by Lovell et al. (1998). More work is clearly needed to exhaust the class of models, but these three lens systems seem considerably simpler than the first two examples. Koopmans & Fassnacht concluded 74 ± 8 for low density cosmologies (69 ± 7 for EdS) from four (excluding the second) lensing systems using the simplest model of deflectors. It is encouraging to find a good agreement with the values from the ladder argument, though the current results from lenses are still less accurate than the ladder value. It would be important to ask whether H₀ < 60 or > 80 is possible within a reasonable class of deflector models.