Statistics of magnification perturbations by substructure in the cold dark matter cosmological model

We study the statistical properties of magnification perturbations by substructures in strong lensed systems using linear perturbation theory and an analytical substructure model including tidal truncation and a continuous substructure mass spectrum. We demonstrate that magnification perturbations are dominated by perturbers found within roughly a tidal radius of an image and that sizable magnification perturbations may arise from small, coherent contributions from several substructures within the lens halo. The rms fluctuation of the magnification perturbation is similar to 10% - 20%, and both the average and rms perturbations are sensitive to the mass spectrum and density profile of the perturbers. Interestingly, we find that relative to a smooth model of the same mass, the average magnification in clumpy models is lower ( higher) than that in smooth models for positive- parity ( negative- parity) images. This is opposite from what is observed if one assumes that the image magnification predicted by the best- fit smooth model of a lens is a good proxy for what the observed magnification would have been if substructures were absent. While it is possible for this discrepancy to be resolved via nonlinear perturbers, we argue that a more likely explanation is that the assumption that the best- fit lens model is a good proxy for the magnification in the absence of substructure is not correct. We conclude that a better theoretical understanding of the predicted statistical properties of magnification perturbations by CDM substructure is needed in order to affirm that CDM substructures have been unambiguously detected.