Statistics of CDM substructure from strong gravitational lensing: quantifying the mass fraction and mass function

ABSTRACT A Bayesian statistical formalism is developed to quantify the level at which the mass-function parameters of (CDM) substructure in strong gravitational-lens galaxies can be recovered, as function of the lens-survey parameters and the detection threshold of the substructures mass. Results from multiple lenses are combined in order to constrain the substructure mass fraction ( f ) and their mass-function slope (α with dN/dm ∝ m−α). The method has been applied to different sets of mock data, under realistic conditions, in order to explore a range of different observational limits. In particular, we have tested the most important effects (i) the number of lens galaxies in the survey, (ii) the mass-detection threshold of the substructure Mlow and (iii) the error with which the substructure mass is measured. We have also explored two different priors on the mass function slope: a uniform prior and a Gaussian prior with α = 1.90 ± 0.1, as given by numerical simulations. With the nominal quality of the lens data (i.e. for a substructure detection threshold Mlow = 10M⊙), the number of lenses at the moment available (nl = 30) and a true dark-matter mass fraction in (CDM) substructure ≪ 1.0%, we find that the upper limit of f can be constrained down to a level 6 1.0% (99% CL), independently from the adopted prior. In the case of a uniform prior the complete substructure mass distribution (i.e. mass fraction and slope) can be characterised in a number of favorable cases, being either a larger number of lenses, a lower detection threshold, or a higher substructure mass fraction, which all lead to a larger number of detected substructures. We also find that lowering the detection threshold, by increasing the resolution and the signal-to-noise ratio of the data, has the largest impact on the ability to recover the mass-function, because of the (expected) steep mass-function slope. In the future, thanks to new survey and follow-up telescopes with high-fidelity data, such as e.g. SKA, LSST and JDEM, a significant increase in the number of known lenses (i.e. ≫104) will allows us to recover the dark satellite population in its completeness. For example, a sample of 200 lenses, equivalent in data-quality to SLACS and a detection threshold of 10M⊙, would allow one to determine f = 0.5 ± 0.1% (68% CL) and α = 1.90 ± 0.2 (68% CL).