Measuring angular diameter distances of strong gravitational lenses

TL;DR

This paper presents a method to determine the angular diameter distance of strong gravitational lenses by combining measurements of image positions, time delays, and velocity dispersions, improving cosmological distance estimates.

Contribution

It extends previous models by allowing arbitrary mass profile slopes and accounts for external convergence and anisotropic velocity dispersion, providing a more general framework for lens distance measurement.

Findings

Current data yields about 16% uncertainty in $D_A$ per lens.

Measuring velocity dispersion at the sweet-spot reduces uncertainty to 13%.

Achieving 7% uncertainty requires 5% precision in velocity dispersion measurements.

Abstract

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.