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

A Bayesian statistical formalism is developed to quantify the level at which the mass function (dN/dm proportional to m-alpha) and the projected cumulative mass fraction (f) of [cold dark matter (CDM)] substructure in strong gravitational lens galaxies, with arcs or Einstein rings, can be recovered as function of the lens survey parameters and the detection threshold of the substructure mass. The method is applied to different sets of mock data to explore a range of observational limits: (i) the number of lens galaxies in the survey; (ii) the mass threshold, M(low), for the detection of substructures and (iii) the uncertainty of the measured substructure masses. We explore two different priors on the mass function slope: a uniform prior and a Gaussian prior with alpha = 1.90 +/- 0.1. With a substructure detection threshold M(low) = 3 x 108 M(circle dot), the number of lenses available now (n(l) = 30), a true dark matter mass fraction in (CDM) substructure <1.0 per cent and a prior of alpha = 1.90 +/- 0.1, we find that the upper limit of f can be constrained down to a level <1.0 per cent [95 per cent confidence level (CL)]. In the case of a uniform prior, the complete substructure mass distribution (i.e. mass fraction and slope) can only be characterized in a number of favourable cases with a large number of detected substructures. This can be achieved by an increase of the resolution and the signal-to-noise ratio of the lensed images. In the case of a Gaussian prior on alpha, instead, it is always possible to set stringent constraints on both parameters. We also find that lowering the detection threshold has the largest impact on the ability to recover alpha, because of the (expected) steep mass function slope. In the future, thanks to new surveys with telescopes, such as Square Kilometre Array (SKA), Large Synoptic Survey Telescope (LSST) and Joint Dark Energy Mission (JDEM) and follow-up telescopes with high-fidelity data, a significant increase in the number of known lenses (i.e. 104) will allow us to recover the satellite population in its completeness. For example, a sample of 200 lenses, equivalent in data quality to the Sloan Lens ACS Survey and a detection threshold of 108 M(circle dot), allows one to determine f = 0.5 +/- 0.1 per cent (68 per cent CL) and alpha = 1.90 +/- 0.2 (68 per cent CL).