Strong gravitational lenses are spectacular manifestations of the influence of gravity on the light emitted by far distant galaxies and quasars. When photons cross the gravitational potential generated by a mass distribution such as that associated with galaxy clusters or single galaxies, then their trajectory is deviated over different paths. This results in multiple images that can take the forms of arcs as in the case of extended sources (e.g. spiral galaxies) or point like images as in the case of quasars. If the source luminosity varies over time then as photons travel over different paths the luminosity variation will occur in the images at different times. This is the gravitational lens time-delay effect.
The measurements of time-delays in strong lens systems can provide information on the cosmic expansion. In particular the time-delay is proportional to the inverse of the Hubble constant, hence its measure provides an independent estimation of H0 that does not rely on the distance ladder calibration methods. However, such measurement is degenerate with the lens mass distribution, since the same time-delay corresponding to a different value of H0 can be produced by a different projected lens mass. The standard approach is therefore to assume a model of the lens and constraining it using as many lens observable as possible. The selection of the lens model is then purely driven by the level of goodness-of-fit. However, determining the most likely model allowed by the data is a problem of model selection and not one of parameter fitting. As such, it should be approached as a Bayesian model selection problem. In fact it is only in this framework that is possible to fairly compare competing models by confronting their probabilities given the available data. This is done by computing the Bayesian evidence for each model under consideration to construct the so called Bayes ratios. The method is particularly suited to select on the base of the Bayesian evidence homogeneous lens samples from future large lens catalog for which the data favorite the same lens model.
Irene Balmes and Pier Stefano Corasaniti have tested the application of this approach to strong lenses by simulating a synthetic catalog of double image lens systems and shown that on the Bayesian analysis can lead to the selection of homogeneous lens sample that then can be used to infer an unbiased value of the Hubble constant. The level of bias depends on the purity of the sample and it can be controlled by using additional lens data. Due to the current paucity of time-delay lens measurements, the application of this method to available lens data does not provide competitive constraints on H0. However, this should improve in the future with the arrival of larger dataset for which Bayesian model selection analysis will be especially useful.
The results are published on the May 2013 edition of Monthly Notice of Royal Astronomical Society.
