  We investigate how strong lensing of dusty, star-forming galaxies by
foreground galaxies can be used as a probe of dark matter halo substructure. We
find that spatially resolved spectroscopy of lensed sources allows dramatic
improvements to measurements of lens parameters. In particular we find that
modeling of the full, three-dimensional (angular position and radial velocity)
data can significantly facilitate substructure detection, increasing the
sensitivity of observables to lower mass subhalos. We carry out simulations of
lensed dusty sources observed by early ALMA (Cycle 1) and use a Fisher matrix
analysis to study the parameter degeneracies and mass detection limits of this
method. We find that, even with conservative assumptions, it is possible to
detect galactic dark matter subhalos of ~ 10^8 M_{\odot} with high significance
in most lensed DSFGs. Specifically, we find that in typical DSFG lenses, there
is a ~ 55 % probability of detecting a substructure with M>10^8 M_{\odot} with
more than 5 sigma detection significance in each lens, if the abundance of
substructure is consistent with previous lensing results. The full ALMA array,
with its significantly enhanced sensitivity and resolution, should improve
these estimates considerably. Given the sample of ~100 lenses provided by
surveys like the South Pole Telescope, our understanding of dark matter
substructure in typical galaxy halos is poised to improve dramatically over the
next few years.
