  The defining characteristic of the cold dark matter (CDM) hypothesis is the
presence of a very large number of low-mass haloes, too small to have made a
visible galaxy. Other hypotheses for the nature of the dark matter, such as
warm dark matter (WDM), predict a much smaller number of such low-mass haloes.
Strong lensing systems offer the possibility of detecting small-mass haloes
through the distortions they induce in the lensed image. Here we show that the
main contribution to the image distortions comes from haloes along the line of
sight rather than subhaloes in the lens as has normally been assumed so far.
These interlopers enhance the differences between the predictions of CDM and
WDM models. We derive the total perturber mass function, including both
subhaloes and interlopers, and show that measurements of approximately 20
strong 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 model
consisting of 7 keV sterile neutrinos such as those required to explain the
recently detected 3.5 keV X-ray emission line from the centres of galaxies and
clusters.
