ClearPotential: Revealing Local Dark Matter in Three Dimensions

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.

💡 Research Summary

The authors present ClearPotential, a novel, fully data‑driven framework that reconstructs the three‑dimensional gravitational potential of the Milky Way within a 4 kpc sphere around the Sun, without imposing axisymmetry, analytic functional forms, or spatial binning. Using Gaia DR3, they select ~5.8 million well‑measured Red Clump and Red Giant Branch stars, applying strict parallax‑error and completeness cuts to ensure a high‑quality sample that includes dust‑obscured regions.

First, they model the observed phase‑space density f_obs(x,v) with Masked Autoregressive Flows (MAFs), a type of normalizing flow that provides a smooth, differentiable probability density and can be trained on the full six‑dimensional data. To correct for dust extinction and other selection effects, they introduce a position‑dependent efficiency function ε(x) such that f_obs = ε · f_corr, where f_corr satisfies the collisionless Boltzmann equation (CBE) in equilibrium.

Both the gravitational potential Φ(x) and the efficiency ε(x) are represented by fully‑connected neural networks with parameters θ and ϑ. The equilibrium CBE, written as
v·∇_x ln f_obs − v·∇_x ln ε − ∇_x Φ·∂_v ln f_obs = 0,
is turned into a mean‑squared‑error loss that is evaluated over a large set of (x,v) pairs sampled from the learned number‑density flow n_obs(x) and the conditional velocity flow p_obs(v|x). For each sampled position they draw 16 velocities, ensuring that the six components of ∇_x ln ε and ∇_x Φ are well constrained.

Two regularization terms are added: (i) λ_ε |ln ε|² to fix the absolute scale of the dust correction, and (ii) λ_Φ max(0, −∇²Φ) to penalize negative mass densities, thereby enforcing ρ ≥ 0. The authors adopt λ_ε = 10⁻¹ and λ_Φ = 10, a compromise that stabilizes training while keeping the physical constraint strong.

Uncertainty quantification is performed in three layers. Instrumental uncertainties are propagated by perturbing the Gaia astrometric observables with their reported Gaussian errors and retraining on ten such realizations. Statistical uncertainties are estimated via ten bootstrap resamplings of the stellar catalogue. Finally, model variance is captured by training 100 independent neural‑network initializations on the original data. The spread among these ensembles provides 1σ error bands for Φ, the acceleration a = −∇Φ, and the derived mass density ρ.

The resulting potential matches the widely used MilkyWayPotential2014 model in the mid‑plane and extends smoothly into the dust‑blocked central disk. The acceleration field exhibits the expected radial and vertical symmetries, with a measured vertical acceleration at the Sun of a_z ≈ −(2.3 ± 0.1) km s⁻¹ kpc⁻¹, consistent with traditional Jeans analyses but with markedly smaller uncertainties. By applying the Poisson equation, the authors obtain a three‑dimensional mass‑density map; after subtracting an analytic baryonic model (disk, bulge, gas), they isolate the dark‑matter density distribution. Assuming spherical symmetry for the purpose of a global value, they find a local dark‑matter density ρ_⊙ = (0.84 ± 0.08) × 10⁻² M_⊙ pc⁻³ (≈ 0.32 ± 0.03 GeV cm⁻³).

Fitting the inferred dark‑matter profile to NFW, generalized NFW, and triaxial halo models yields a preference for short scale radii (r_s ≈ 10 kpc), a markedly oblate shape (axis ratio ≈ 0.7), and a tilt of roughly 30° relative to the Galactic plane. There is weak evidence for a central core, and the analysis places the strongest to‑date upper limit on a co‑rotating dark‑matter disk, constraining its surface density to less than ~5 % of the baryonic disk.

To probe departures from equilibrium, the authors compare their acceleration field with line‑of‑sight accelerations inferred from timing measurements of nearby binary pulsars. A modest (~2σ) discrepancy suggests a non‑stationarity timescale of order 0.5 Gyr, compatible with known vertical oscillations and spiral‑arm perturbations.

In summary, ClearPotential demonstrates that modern deep‑learning tools—normalizing flows for density estimation and neural‑network potentials for solving the CBE—can be combined to produce a high‑resolution, physically consistent map of the Milky Way’s gravitational field. The method automatically corrects for dust extinction, enforces positivity of the mass density, and provides robust uncertainty estimates. This work sets a new benchmark for local dark‑matter measurements, opens the door to precise studies of halo shape, possible dark disks, and dynamical disequilibrium, and offers a scalable framework that can be applied to future Gaia data releases or to other galaxies with rich kinematic surveys.