4.1 Gravitational Lenses
Refsdael (1964, 1966) noted that the arrival times for the light from two gravitationally lensed images of a background point source are dependent on the path lengths and the gravitational potential traversed in each case. Hence, a measurement of the time delay and the angular separation for different images of a variable quasar can be used to provide a measurement of H0. This method offers tremendous potential because it can be applied at great distances and it is based on very solid physical principles (Blandford & Kundic 1997).
There are of course difficulties with this method as there are with any other. Astronomical lenses are galaxies whose underlying (luminous or dark) mass distributions are not independently known, and furthermore they may be sitting in more complicated group or cluster potentials. A degeneracy exists between the mass distribution of the lens and the value of H0 (e.g., Keeton and Kochanek 1997; Schechter et al. 1997). Ideally velocity dispersion measurements as a function of position are needed (to constrain the mass distribution of the lens). Such measurements are very difficult (and generally have not been available). Perhaps worse yet, the distribution of the dark matter in these systems is unknown.
Unfortunately, to date, there are very few systems known which have both a favorable geometry (for providing constraints on the lens mass distribution) and a variable background source (so that a time delay can be measured). The two systems to date that have been well-studied yield values of H0 in the approximate range of 40-70 km/sec/Mpc (Schechter et al. 1997; Impey et al. 1998) with an uncertainty of ~ 20-30%. These values assume a value of = 1, and rise by 10% for low . Tonry & Franx (1998) have recently reported an accurate new velocity dispersion of = 288 ± 9 km/sec for the lensing galaxy in 0957+561, based on data obtained at the Keck 10m telescope. Adopting m = 0.25 and the model of Grogin & Narayan (1996) for the mass distribution of the lens yields a value of H0 = 72 ± 7 (1- statistical) ± 15% (systematic).
As the number of favorable lens systems increases (as further lenses are discovered that have measurable time delays), the prospects for measuring H0 and its uncertainty using this technique are excellent. Schechter (private communication) reports that there are now 6 lenses with measured time delays, but perhaps only half of these will be useful for H0 determinations due to the difficulty of modelling the galaxies.