  We study the abundance of substructure in the matter density near galaxies
using ALMA Science Verification observations of the strong lensing system
SDP.81. We present a method to measure the abundance of subhalos around
galaxies using interferometric observations of gravitational lenses. Using
simulated ALMA observations, we explore the effects of various systematics,
including antenna phase errors and source priors, and show how such errors may
be measured or marginalized. We apply our formalism to ALMA observations of
SDP.81. We find evidence for the presence of a $M=10^{8.96\pm 0.12} M_{\odot}$
subhalo near one of the images, with a significance of $6.9\sigma$ in a joint
fit to data from bands 6 and 7; the effect of the subhalo is also detected in
both bands individually. We also derive constraints on the abundance of dark
matter subhalos down to $M\sim 2\times 10^7 M_{\odot}$, pushing down to the
mass regime of the smallest detected satellites in the Local Group, where there
are significant discrepancies between the observed population of luminous
galaxies and predicted dark matter subhalos. We find hints of additional
substructure, warranting further study using the full SDP.81 dataset
(including, for example, the spectroscopic imaging of the lensed carbon
monoxide emission). We compare the results of this search to the predictions of
$\Lambda$CDM halos, and find that given current uncertainties in the host halo
properties of SDP.81, our measurements of substructure are consistent with
theoretical expectations. Observations of larger samples of gravitational
lenses with ALMA should be able to improve the constraints on the abundance of
galactic substructure.
