Constraining the Geometry of Galactic Dark Matter with Gaia Data Release 3

We derive both the mid-plane and off-plane rotation curves, $v_c(R,z)$, and the vertical acceleration, $a_z(R,z)$, of the Milky Way (MW) using \textit{Gaia}~DR3 data over the ranges of vertical heights $z \in (-2,2)\,$ kpc and galactocentric distances $R \in (8.5,14)$ kpc where the velocity components are determined with high precision, i.e., with an error $< 5\%$. In contrast, the vertical acceleration $a_z(R,z)$ is dominated by model-dependent systematics, with uncertainties of up to $\sim 20\%$. This level of accuracy allows us to place stringent constraints on the geometry of the MW's dark matter (DM) distribution, as the vertical gradients of the gravitational potential attain their maximum within this range of radial and vertical distances corresponding to the characteristic scales of the disk. We find that models including the observed stellar components together with a spherical DM halo fail to reproduce both the pronounced variation of $v_c(R,z)$ with height and the observed behavior of $a_z(R,z)$. In particular, spherical halos with a scale radius of $r_s \sim 15$ kpc contribute negligibly to the off-plane rotation curve and vertical acceleration in the inner disk, leaving these features primarily determined by the stellar mass distribution. Conversely, models in which DM is confined to a flattened, disk-like configuration predict substantial contributions to both $v_c(R,z)$ and $a_z(R,z)$, resulting in a markedly better agreement with the data. We conclude that disk-like DM distributions are strongly favored over spherical halo models. Forthcoming Gaia data releases will enable even more stringent tests of the geometry and distribution of the MW's DM component.