  The vast majority of the mass in the Milky Way (MW) is in dark matter (DM);
we therefore cannot directly observe the MW mass distribution, and have to use
tracer populations in order to infer properties of the MW DM halo. However, MW
halo tracers do not only feel the gravitational influence of the MW itself.
Tracers can also be affected by MW satellites; Garavito-Camargo et al. (2019)
(hereafter GC19) demonstrate that the Large Magellanic Cloud (LMC) induces a
density wake in the MW DM, resulting in large scale kinematic patterns in the
MW stellar halo. In this work, we use spherical harmonic expansion (SHE) of the
velocity fields of simulated stellar halos in an effort to disentangle
perturbations on large scales (e.g., due to the LMC itself as well as the
LMC-induced DM wake) and small scales (due to substructure). Using the GC19
simulations, we demonstrate how the different terms in the SHE of the stellar
velocity field reflect the different wake components, and show that these
signatures are a strong function of the LMC mass. An exploration of model halos
built from accreted dwarfs Bullock & Johnston (2005) suggests that stellar
debris from massive, recent accretion events can produce much more power in the
velocity angular power spectra than the perturbation from the LMC-induced wake.
We therefore consider two models for the Sagittarius (Sgr) stream -- the most
recent, massive accretion event in the MW apart from the LMC -- and find that
the angular power on large scales is generally dominated by the LMC-induced
wake, even when Sgr is included. We conclude that SHE of the MW stellar halo
velocity field may therefore be a useful tool in quantifying the response of
the MW DM halo to the LMC's infall.
