What Do Gravitational Lens Time Delays Measure?

Gravitational lens time delays depend on the Hubble constant, the observed image positions, and the surface mass density of the lens in the annulus between the images. Simple time delay lenses like PG1115+080, SBS1520+530, B1600+434, PKS1830–211 and HE2149–2745 have H 0 = A(1 − κ) + Bκ(η − 1) where the two coefficients A ≃ 90 km/s Mpc and B ≃ 10 km/s Mpc depend on the measured delays and the observed image positions, κ is the mean surface density in the annulus between the images, and there is a small correction from the logarithmic slope η ≃ 2 of the surface density profile, κ ∝ R 1−η , in the annulus. These 5 systems are very homogeneous, since for fixed H 0 = 100h km/s Mpc they must have the same surface density, κ = 1.11 − 1.22h ± 0.04, with an upper bound of σ κ < 0.07 on any dispersion in κ beyond those due to the measurement errors. If the lenses have their expected dark halos, κ ≃ 0.5 and H 0 ≃ 51 ± 5 km/s Mpc, while if they have constant mass-to-light ratios, κ ≃ 0.1–0.2 and H 0 ≃ 73 ± 8 km/s Mpc. More complicated lenses with multiple components or strong perturbations from nearby clusters, like RXJ0911+0551 and Q0957+561, are easily recognized because they have significantly different coefficients.