ClearPotential: Revealing Local Dark Matter in Three Dimensions

TL;DR

ClearPotential delivers the first fully data-driven, three-dimensional map of the local Galactic potential by solving the equilibrium collisionless Boltzmann equation with neural-network potentials and dust-correction learned directly from Gaia DR3 RC/RGB stars within $4\,\mathrm{kpc}$. The method employs Masked Autoregressive Flows to model the observed phase-space density and neural networks for the potential and extinction, producing continuous 3D maps of $\Phi$, $\vec{a}$, and $\rho$ while accounting for selection effects. The results reveal a predominantly axisymmetric potential with a tilted oblate dark matter halo, hints of a cored inner profile, and the strongest constraints to date on a dark matter disk, alongside mild disequilibrium evidenced by pulsar timing tests. This work demonstrates a powerful, data-driven framework for mapping the Milky Way’s mass distribution and DM structure, underscoring the need for improved baryonic mass models and setting the stage for future Gaia data releases to sharpen these inferences.

Abstract

We present ClearPotential, a data-driven, three-dimensional measurement of the gravitational potential of the local Milky Way using unsupervised machine learning, without the symmetry assumptions, specific functional forms, and binning required in previous work. The potential is modeled as a neural network, optimized to solve the equilibrium collisionless Boltzmann equation for the observed phase space density of Gaia DR3 Red Clump stars within 4 kpc of the Sun. This density is obtained from data using normalizing flows, and our unsupervised solution to the Boltzmann equation automatically corrects for selection effects from crowding and the dust-driven extinction of starlight. Our fully-differentiable model of the gravitational potential allows us to map the acceleration and mass density of the Galaxy in the volume around the Sun, including in the dust-obscured disk towards the Galactic Center. We determine the dark matter density at the Solar radius to be $(0.84 \pm 0.08)\times 10^{-2}\,{M}_\odot/{\rm pc}^3$, and analyze the structure of the dark matter halo. We find strong evidence for a tilted oblate halo, weak preference for a cored inner profile, and the strongest constraints to date on a possible dark matter disk. We place a bound on the timescale of disequilibrium in the local Milky Way, and find mild evidence for disequilibrium using independent acceleration measurements from timings of binary pulsar systems. This work provides the clearest map of the local Galactic potential to date and marks an important step in the era of data-driven astrometry.

Figures (12)

• Figure 1: Left: Contours of the total gravitational potential $\Phi$ estimated in this work (solid) compared to MWP14 (dashed) in the midplane ($z=0$). Middle: Contours of the azimuthally averaged $\Phi$ in the $R-z$ plane. Right: 3-dimensional isocontours of $\Phi$ within $3.8$ kpc of the Solar location ($\star$).
• Figure 2: Azimuthally averaged radial $R$ (left), aziumuthal $\phi$ (center), and vertical $z$ (right) accelerations in the $R-z$ plane, compared to MWP14 (dashed). MWP14 predicts $a_\phi=0$ due to the assumption of azimuthal symmetry.
• Figure 3: Left column: Measured relative line-of-sight (LOS) accelerations of nearby binary pulsar systems as measured by this work (black), Donlon et al. (blue, Ref. donlon), Moran et al. (red, Ref. moran), and as predicted by MWP14 (grey). Center column: $\sigma$-tension between the LOS pulsar acceleration measurements and this work. Ordering of pulsars is sorted from left to right by lowest to highest level of mutual agreement with this work. Right column: Non-stationarity metric (inverse dynamic timescale of disequilibrium) given the LOS accelerations estimated in this work, Donlon, or Moran. Transparent contours denote the uncertainty in non-stationarity given the LOS acceleration uncertainties.
• Figure 4: Left column: Non-stationarity (NS) map in the $x-y$ (top) and $x-z$ (bottom) plane, using our flow-based gradients of $f$ and assuming the neural-network-based acceleration $\vec{a}$ satisfies the CBE. Center column: NS map assuming the accelerations in MWP14 satisfy the CBE. Right column: Locations of the pulsars in Galactocentric coordinates, with overlapping blue (Donlon) and red (Moran) circles of area proportional to the increase in NS assuming $\vec{a}_{\rm LOS}=\vec{a}_{\rm Pulsar,\:LOS}$. A black dot denotes the location of the proposed subhalo (labeled C25) in Ref. 2025arXiv250716932C.
• Figure 5: The inferred azimuthally-averaged total Galactic mass density field $\rho(\vec{x})$ (black) in the $R$–$z$ plane, compared to MWP14 (white dashed). The inset shows the average pull relative to MWP14 (light/dark red for $2\sigma$–$5\sigma$/$>5\sigma$ higher, light/dark blue for lower; grey indicates $\sigma<2$ agreement).