Measuring angular diameter distances of strong gravitational lenses
  The distance-redshift relation plays a fundamental role in constraining
cosmological models. In this paper, we show that measurements of positions and
time delays of strongly lensed images of a background galaxy, as well as those
of the velocity dispersion and mass profile of a lens galaxy, can be combined
to extract the angular diameter distance of the lens galaxy. Physically, as the
velocity dispersion and the time delay give a gravitational potential ($GM/r$)
and a mass ($GM$) of the lens, respectively, dividing them gives a physical
size ($r$) of the lens. Comparing the physical size with the image positions of
a lensed galaxy gives the angular diameter distance to the lens. A mismatch
between the exact locations at which these measurements are made can be
corrected by measuring a local slope of the mass profile. We expand on the
original idea put forward by Paraficz and Hjorth, who analyzed singular
isothermal lenses, by allowing for an arbitrary slope of a power-law spherical
mass density profile, an external convergence, and an anisotropic velocity
dispersion. We find that the effect of external convergence cancels out when
dividing the time delays and velocity dispersion measurements. We derive a
formula for the uncertainty in the angular diameter distance in terms of the
uncertainties in the observables. As an application, we use two existing strong
lens systems, B1608+656 ($z_{\rm L}=0.6304$) and RXJ1131$-$1231 ($z_{\rm
L}=0.295$), to show that the uncertainty in the inferred angular diameter
distances is dominated by that in the velocity dispersion, $\sigma^2$, and its
anisotropy. We find that the current data on these systems should yield about
16% uncertainty in $D_A$ per object. This improves to 13% when we measure
$\sigma^2$ at the so-called sweet-spot radius. Achieving 7% is possible if we
can determine $\sigma^2$ with 5% precision.
