  The density of dark matter near the Sun is important for experiments hunting
for dark matter particles in the laboratory, and for constraining the local
shape of the Milky Way's dark matter halo. Estimates to date have typically
assumed that the Milky Way's stellar disc is axisymmetric and in a
steady-state. Yet the Milky Way disc is neither, exhibiting prominent spiral
arms and a bar, and vertical and radial oscillations. We assess the impact of
these assumptions on determinations of the local dark matter density by
applying a free-form, steady-state, Jeans method to two different N-body
simulations of Milky Way-like galaxies. In one, the galaxy has experienced an
ancient major merger, similar to the hypothesized Gaia-Sausage-Enceladus; in
the other, the galaxy is perturbed more recently by the repeated passage and
slow merger of a Sagittarius-like dwarf galaxy. We assess the impact of each of
the terms in the Jeans-Poisson equations on our ability to correctly extract
the local dark matter density from the simulated data. We find that common
approximations employed in the literature - axisymmetry and a locally flat
rotation curve - can lead to significant systematic errors of up to a factor
~1.5 in the recovered surface mass density ~2kpc above the disc plane, implying
a fractional error on the local dark matter density of order unity. However,
once we add in the tilt term and the rotation curve term in our models, we
obtain an unbiased estimate, consistent with the true value within our 95%
confidence intervals for realistic 20% uncertainties on the baryonic surface
density of the disc. Other terms - the axial tilt, 2:nd Poisson and time
dependent terms - contribute less than 10% to the local dark matter density
(given current data) and can be safely neglected for now. In the future, as
more data become available, these terms will need to be included in the
analysis.
