  Dark matter comprises the bulk of the matter in the universe but its particle
nature and cosmological origin remain mysterious. Knowledge of the dark matter
density distribution in the Milky Way Galaxy is crucial to both our
understanding of the standard cosmological model and for grounding direct and
indirect searches for the particles comprising dark matter. Current
measurements of Galactic dark matter content rely on model assumptions to infer
the forces acting upon stars from the distribution of observed velocities.
Here, we propose to apply the precision radial velocity method, optimized in
recent years for exoplanet astronomy, to measure the change in the velocity of
stars over time, thereby providing a direct probe of the local gravitational
potential in the Galaxy. Using numerical simulations, we develop a realistic
strategy to observe the differential accelerations of stars in our Galactic
neighborhood with next-generation telescopes, at the level of $10^{-8}$
cm/s$^{2}$. Our simulations show that detecting accelerations at this level
with an ensemble of $10^{3}$ stars requires the effect of stellar noise on
radial velocity measurements to be reduced to $<10$ cm/s. The measured stellar
accelerations may then be used to extract the local dark matter density and
morphological parameters of the density profile.
