  Measurements of the accelerations of stars enabled by time-series
extreme-precision spectroscopic observations, from pulsar timing, and from
eclipsing binary stars in the Solar Neighborhood offer insights into the mass
distribution of the Milky Way that do not rely on traditional equilibrium
modeling. Given the measured accelerations, we can determine a total mass
density, and from this, by accounting for the mass in stars, gas, and dust, we
can infer the amount of dark matter. Leveraging the FIRE-2 simulations of Milky
Way-mass galaxies, we compare vertical acceleration profiles between cold dark
matter (CDM) and self-interacting dark matter (SIDM) with constant
cross-section of 1 cm$^2$ g$^{-1}$ across three halos with diverse assembly
histories. Notably, significant asymmetries in vertical acceleration profiles
near the midplane at fixed radii are observed in both CDM and SIDM,
particularly in halos recently affected by mergers with satellites of
Sagittarius/SMC-like masses or greater. These asymmetries offer a unique window
into exploring the merger history of a galaxy. We show that SIDM halos
consistently exhibit higher local stellar and dark matter densities and steeper
vertical acceleration gradients, up to 30% steeper near the Solar Neighborhood.
SIDM halos also manifest a more oblate halo shape in the Solar Neighborhood.
Furthermore, enhanced precision in acceleration measurements and larger
datasets promise to provide better constraints on the local dark matter
density, complementing our understanding from kinematic analysis of their
distribution within galaxies.
