Bayesian approach to gravitational lens model selection: constraining the Hubble constant with a selected sample of strong lenses

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.

Non-Gaussian Halo Mass Function and Non-Spherical Halo Collapse: Theory vs. Simulations

The statistical properties of the cosmic matter distribution carry unique information on the quantum mechanical processes that during the early inflationary era have generated the spectrum of primordial inhomogeneities. The standard model of inflation predicts a nearly Gaussian spectrum of primordial density fluctuations, while competing scenarios predict deviations from Gaussian statistics. Such deviations can be tested through a variety of observational probes among which the number counts of the most massive halos in the Universe is believed to be one of the most sensitive. Ixandra Achitouv and Pier Stefano Corasaniti have investigated how the imprint of the non-spherical gravitational collapse of dark matter, which leads to the formation of halos, alters the signature of primordial non-Gaussianity on the halo mass function. To this end the researchers have derived an analytical formulae using a path-integral approach to predict the halo counting statistics for a given type of primordial non-Gaussianity (PNG), while assuming simple model of the halo formation process which captures the main features of the non-spherical collapse of dark matter. The result of the computation shows that the PNG signal is entangled to that of the non-linear gravitational collapse, thus potentially diluted than previously thought. The authors found an unprecedented agreement comparing the analytical prediction against non-Gaussian N-body simulation results. More importantly, the comparative analysis indicated that deviations from the spherical collapse prediction increases for larger non-Gaussinity. In other words the larger is the deviation from non-Gaussian statistics and the greater is the departure from a simple spherical collapse model of halo formation. This explain why alternative approaches that have attempt to compute the non-Gaussian halo mass function require introducing an ad-hoc tuning parameter to recover non-Gaussian N-body simulation results.

The results have been published on the Journal of Cosmology and Astroparticle Physics.

Gaussian Halo Mass Function and Non-Spherical Collapse Model

Predicting the cosmic mass distribution of dark matter halos is a central problem in modern cosmology. The knowledge of this function is key to understanding the formation of the visible structures in the Universe. Numerical N-body simulations have been the primary tool for studying the properties of dark matter halos. Pier Stefano Corasaniti and Ixandra Achitouv from LUTH (CNRS, Observatoire de Paris & Université Paris Diderot) have tackled this challenge by developing a mathematical model in which the evolution of the matter density in halos resembles the time sequence of stock prices. Using a path-integral formulation of this stochastic model the researchers have derived a mathematical formula of the halo mass function which reproduces that inferred from N-body simulations with unprecedented accuracy. The results are published in a letter in the June edition of Physical Review Letter and an article in press on Physical Review D.

 

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