Numerical investigation of lenses with substructures using the perturbative method

15 pages, 9 figures. Major changes. Accepted for publication in MNRAS.We present a statistical study of the effects induced by substructures on the deflection potential of dark matter halos in the strong lensing regime. This investigation is based on the pertubative solution around the Einstein radius (Alard 2007) in which all the information on the deflection potential is specified by only a pair of one-dimensional functions on this ring. Using direct comparison with ray-tracing solutions, we found that the iso-contours of lensed images predicted by the pertubative solution is reproduced with a mean error on their radial extension of less than 1% - in units of the Einstein radius, for reasonable substructure masses. It demonstrates the efficiency of the approximation to track possible signatures of substructures. We have evaluated these two fields and studied their properties for different lens configurations modelled either through massive dark matter halos from a cosmological N-body simulation, or via toy models of Monte Carlo distribution of substructures embedded in a triaxial Hernquist potential. As expected, the angular power spectra of these two fields tend to have larger values for larger harmonic numbers when substructures are accounted for and they can be approximated by power-laws, whose values are fitted as a function of the profile and the distribution of the substructures