The angular diameter distance to the lens galaxy measured in this way can be expressed in terms of several observables such as the time delay, velocity dispersion, lens redshift, and image positions. It also depends on the steepness of the mass profile of the lens, which can be determined by analysing the shape and brightness distributions of the lens and images.
One big advantage of this new method, which was found by scientists at MPA, is that the effect of the external mass that lies along the line-of-sight between the observer and the source, which is called the 'external convergence', cancels out. In other words, the angular diameter distance does not change under the presence of external mass that causes extra bending of light rays. In principle, the external convergence acts like another lens added to the lens system, and adds uncertainty to the traditional cosmological measurements using strong lenses. A good analogy exists in the classical optics: It is impossible to tell the difference between a single lens system with a given focal length, and a multiple lens system consisting of lenses with different focal lengths which collectively form a system with the same effective focal length as that of the single lens. Likewise, gravitational lens systems with a single lens and multiple lenses cannot be distinguished by observables.
However, in our case the resulting angular diameter distance does not depend on the external convergence, since the time delay and the velocity dispersion are both determined solely by the properties of the lens. Applying this method to one of the existing lens systems, B1608+656 (Figure 1), the researchers find that the distance can be measured with 15% accuracy (Figure 3).
This new method requires precise estimation of the mass and the potential of the system from data. Challenges lie both in the observation and the modelling of the lens: Assuming that the lens galaxy (which is a massive elliptical galaxy in most cases) has reached dynamical equilibrium, the random motion of matter particles inside the galaxy counteracts the gravitational force so that the galaxy neither collapses nor expands. This is measured by the velocity dispersion of stars with respect to the centre of the galaxy. The Jeans equation relates the radial component of the velocity dispersion to the gravitational potential; however, it is impossible to observe just the radial component of the velocity dispersion. The measurement is done using the Doppler shift of stellar light, which means that only the line-of-sight component of the velocity dispersion is measurable. The fact that only the luminous tracers (such as stars) can be measured results in the observation of projected, luminosity weighted velocity dispersion. When the velocity dispersion is anisotropic, the situation becomes even more complicated. Moreover, the measured velocity dispersion is usually a quantity that is averaged over an aperture of size few tenth of arc seconds. In the end, it is not easy to relate the value we observe to the potential itself.
To overcome this complication, we use spatially resolved spectroscopy of the lens galaxy to obtain the radial profile of the velocity dispersion. Then, we use a radius at which the scatter between different anisotropic profiles is minimised, the "sweet spot", which was derived previously by other researchers at MPA (Churazov et al.). With this method, the uncertainty from the anisotropic velocity dispersion is minimised and the accuracy in the determination of the angular diameter distance improves to 12%.
This study demonstrates that, using a strong lens galaxy with measured time delays, the angular diameter distance can be measured accurately, providing a powerful, new way to chart our universe.
Inh Jee, Eiichiro Komatsu and Sherry H. Suyu(ASIAA)