Gravitational lenses with anomalous flux ratios are often cited as possible evidence for dark matter satellites predicted by simulations of hierarchical merging in cold dark matter cosmogonies .
We show that the fraction of quads with anomalous flux ratios depends primarily on the total mass and spatial extent of the satellites , and the characteristic lengthscale d _ { 1 / 2 } of their distribution .
If d _ { 1 / 2 } \sim 100 \mathrm { kpc } , then for a moderately elliptical galaxy with a line-of-sight velocity dispersion of \sim 250 \mathrm { km } \mathrm { s } ^ { -1 } , a mass of \sim 3 \times 10 ^ { 9 } M _ { \odot } in highly-concentrated ( Plummer model ) satellites is needed for 20 % of quadruplets to show anomalous flux ratios , rising to \sim 1.25 \times 10 ^ { 10 } M _ { \odot } for 50 % .
Several times these masses are required if the satellites have more extended Hernquist profiles .
Compared to a typical elliptical , the flux ratios of quads formed by typical edge-on disc galaxies with maximum discs are significantly less susceptible to changes through substructure – three times the mass in satellite galaxies is needed to affect 50 % of the systems .

In many of the lens systems with anomalous flux ratios , there is evidence for visible satellites ( e.g. , B2045+265 or MG0414+0534 ) .
We show that if the anomaly is produced by substructure with properties similar to the simulations , then optically identified substructure should not be preponderant among lens systems with anomalies .
There seem to be two possible resolutions of this difficulty .
First , in some cases , visible substructure may be projected within or close to the Einstein radius and wrongly ascribed as the culprit , whereas dark matter substructure is causing the flux anomaly .
Second , bright satellites , in which baryon cooling and condensation has taken place , may have higher central densities than dark satellites , rendering them more efficient at causing flux anomalies .