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The defining characteristic of the cold dark matter (CDM) hypothesis is thepresence of a very large number of low-mass haloes, too small to have made avisible galaxy. Other hypotheses for the nature of the dark matter, such aswarm dark matter (WDM), predict a much smaller number of such low-mass haloes.Strong lensing systems offer the possibility of detecting small-mass haloesthrough the distortions they induce in the lensed image. Here we show that themain contribution to the image distortions comes from haloes along the line ofsight rather than subhaloes in the lens as has normally been assumed so far.These interlopers enhance the differences between the predictions of CDM andWDM models. We derive the total perturber mass function, including bothsubhaloes and interlopers, and show that measurements of approximately 20strong lens systems with a detection limit of $M_{\rm low}=10^7 h^{-1}M_{\odot}$ would distinguish (at 3 sigma) between CDM and a WDM modelconsisting of 7 keV sterile neutrinos such as those required to explain therecently detected 3.5 keV X-ray emission line from the centres of galaxies andclusters.

Projection effects in the strong lensing study of subhaloes