Quantifying the Line-of-Sight Halo Contribution to the Dark Matter
  Convergence Power Spectrum from Strong Gravitational Lenses
  Galaxy-galaxy strong gravitational lenses have become a popular probe of dark
matter (DM) by providing a window into structure formation on the smallest
scales. In particular, the convergence power spectrum of subhalos within
lensing galaxies has been suggested as a promising observable to study DM.
However, the distances involved in strong-lensing systems are vast, and we
expect the relevant volume to contain line-of-sight (LOS) halos that are not
associated with the main lens. We develop a formalism to calculate the effect
of LOS halos as an effective convergence power spectrum. The multi-lens plane
equation couples the angular deflections of consecutive lens planes, but by
assuming that the perturbations due to the LOS halos are small, we show that
they can be projected onto the main-lens plane as effective subhalos. We test
our formalism by simulating lensing systems using the full multi-plane lens
equation and find excellent agreement. We show how the relative contribution of
LOS halos and subhalos depends on the source and lens redshift, as well as the
assumed halo and subhalo mass functions. For a fiducial system with fraction of
DM halo mass in substructure $f_{\rm sub}=0.4\%$ for subhalo masses
$[10^5-10^8]\rm{M}_{\odot}$, the interloper contribution to the power spectrum
is at least several times greater than that of subhalos for source redshifts
$z_s\gtrsim0.5$. Furthermore, it is likely that for the SLACS and BELLS lenses
the interloper contribution dominates: $f_{\rm sub}\gtrsim2\%$ ($4\%$) is
needed for subhalos to dominate in SLACS (BELLS), which is higher than current
upper bounds on $f_{\rm sub}$ for our mass range. Since the halo mass function
is better understood from first principles, the dominance of interlopers in
galaxy-galaxy lenses with high-quality imaging can be seen as a significant
advantage when translating this observable into a constraint on DM.
