On the Power Spectrum of Dark Matter Substructure in Strong
  Gravitational Lenses
  Studying the smallest self-bound dark matter structure in our Universe can
yield important clues about the fundamental particle nature of dark matter.
Galaxy-scale strong gravitational lensing provides a unique way to detect and
characterize dark matter substructures at cosmological distances from the Milky
Way. Within the cold dark matter (CDM) paradigm, the number of low-mass
subhalos within lens galaxies is expected to be large, implying that their
contribution to the lensing convergence field is approximately Gaussian and
could thus be described by their power spectrum. We develop here a general
formalism to compute from first principles the substructure convergence power
spectrum for different populations of dark matter subhalos. As an example, we
apply our framework to two distinct subhalo populations: a truncated
Navarro-Frenk-White subhalo population motivated by standard CDM, and a
truncated cored subhalo population motivated by self-interacting dark matter
(SIDM). We study in detail how the subhalo abundance, mass function, internal
density profile, and concentration affect the amplitude and shape of the
substructure power spectrum. We determine that the power spectrum is mostly
sensitive to a specific combination of the subhalo abundance and moments of the
mass function, as well as to the average tidal truncation scale of the largest
subhalos included in the analysis. Interestingly, we show that the asymptotic
slope of the substructure power spectrum at large wave number reflects the
internal density profile of the subhalos. In particular, the SIDM power
spectrum exhibits a characteristic steepening at large wave number absent in
the CDM power spectrum, opening the possibility of using this observable, if at
all measurable, to discern between these two scenarios.
