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 PG 1115+80, SBS 1520+530, B1600+434, PKS 1830-211, and HE 2149-2745 have H-0 = A(1 - [kappa]) + B[kappa](eta - 1), where the two coefficients A similar or equal to 90 km s(-1) Mpc(-1) and B similar or equal to 10 km s(-1) Mpc(-1) depend on the measured delays and the observed image positions, [kappa] is the mean surface density in the annulus between the images, and there is a small correction from the logarithmic slope eta similar or equal to 2 of the density profile, kappa proportional to R1-eta, in the annulus. These five systems are very homogeneous since for fixed H-0 = 100 h km s(-1) Mpc(-1) they must have the same surface density, [kappa] = 1.11 - 1.22 h +/- 0.04, with an upper bound sigma(kappa) < 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 H0 ≃ 51 +/- 5 km s(-1) Mpc(-1), while if they have constant mass-to-light ratios, [κ] ≃ 0.1-0.2 and H-0 ≃ 73 +/- 8 km s(-1) Mpc(-1). More complicated lenses with multiple components or strong perturbations from nearby clusters, like RX J0911+0551 and Q0957+561, are easily recognized because they have significantly different coefficients.