Numerical investigation of lens models with substructures using the perturbative method

We present a statistical study of the effects induced by substructures on the deflection potential of dark matter haloes in the strong lensing regime. This investigation is based on the pertubative solution around the Einstein radius in which all the information on the deflection potential is specified by only a pair of 1D 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 per cent - 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 haloes 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.