ClearPotential: 3D Gravitational Mapping

• ClearPotential is a data-driven framework that estimates the Milky Way’s 3D gravitational potential and local dark matter density using neural networks and Gaia DR3 data.
• It leverages normalizing flows to solve the collisionless Boltzmann equation with minimal assumptions, accurately correcting for observational biases.
• The framework produces high-resolution maps revealing non-axisymmetric features and constrains dark matter disk properties through precise gravitational field measurements.

ClearPotential is a data-driven framework for measuring the three-dimensional gravitational potential of the local Milky Way using unsupervised machine learning, sidestepping the symmetry assumptions, parametric forms, and binning required by previous work. The method, as introduced in "ClearPotential: Revealing Local Dark Matter in Three Dimensions" (Putney et al., 10 Dec 2025), employs neural networks and normalizing flows to infer the local potential, acceleration, and dark matter distribution in the vicinity of the Sun from Gaia DR3 astrometric data. By solving the collisionless Boltzmann equation at high spatial resolution and correcting directly for observation selection effects, ClearPotential provides unprecedented local maps of both the gravitational field and the dark matter density, revealing structural features such as a tilted oblate halo and constraining the properties of putative dark matter disks.

1. Scientific Objectives

ClearPotential targets the reconstruction of the smooth 3D gravitational potential $\Phi(\vec{x})$, the acceleration field $\vec{a}(\vec{x}) = -\nabla \Phi(\vec{x})$, and the total mass density $\rho(\vec{x}) = \nabla^2 \Phi(\vec{x})/(4\pi G)$ within a $\sim 4$ kpc sphere around the Sun. The central aim is to provide a point-wise, unbiased measurement of the local dark matter density by separating baryonic and non-baryonic components without enforcing symmetry (e.g., axisymmetry), parametric halo models, or spatial binning, and to robustly correct for selection biases induced by dust extinction and stellar crowding in Gaia observations (Putney et al., 10 Dec 2025).

2. Data Sources and Preprocessing

The principal data set comprises approximately 5.8 million Gaia DR3 stars, with full six-dimensional phase-space information—positions and velocities—after performing quality cuts on parallax and magnitude (e.g., $\Delta\pi/\pi<1/3,\;G_\text{RVS}<14$ at 4 kpc). The analysis focuses primarily on the Red Clump (RC) and Red Giant Branch (RGB) stellar populations, which are kinematically older and closer to dynamical equilibrium. The spatial coordinates are transformed into a right-handed Galactocentric Cartesian system with the Sun at $(x,y,z) = (8.122, 0, 0.0208)\;\text{kpc}$. To correct for position-dependent incompleteness due to dust extinction or stellar crowding, the method trains a dedicated "efficiency" network $\varepsilon(\vec{x})$ alongside the potential network (Putney et al., 10 Dec 2025).

3. Mathematical Formulation

ClearPotential enforces the steady-state collisionless Boltzmann equation (CBE) as the core physical constraint: $$\vec{\nabla}_x \cdot (f \vec{v}) + \vec{\nabla}_v \cdot (f \vec{a}) = 0,$$ which can be recast as

$$\vec{v}\cdot\nabla_x \ln f - \vec{a} \cdot \nabla_v \ln f = 0,$$

where $f(\vec{x}, \vec{v})$ is the phase-space density. The observed density $f_\text{obs}$ is incomplete due to selection effects and is thus modeled as $$f_\text{obs}(\vec{x},\vec{v}) = \varepsilon(\vec{x}) f_\text{corr}(\vec{x},\vec{v})$$. Substituting, the constraint becomes

$$\vec{v}\cdot\nabla \ln f_\text{obs} - \vec{v}\cdot\nabla \ln \varepsilon(\vec{x}) - \nabla \Phi(\vec{x})\cdot\partial_v \ln f_\text{obs} = 0.$$

ClearPotential models both $f_\text{obs}$ and $p_\text{obs}(v|x)$ using Masked Autoregressive Flows (MAFs), while $\Phi(\vec{x})$ and $\varepsilon(\vec{x})$ are each parameterized as fully-connected neural networks. The loss function minimized during training is

$$L_\text{CBE} = \big\langle [\vec{v}\cdot\nabla \ln f_\text{obs} - \vec{v}\cdot\nabla \ln \varepsilon(\vec{x}) - \nabla \Phi_\theta(\vec{x})\cdot\partial_v \ln f_\text{obs}]^2 \big\rangle,$$

with regularization terms ensuring physical plausibility (e.g., penalizing negative mass densities and constraining $\varepsilon$) (Putney et al., 10 Dec 2025).

4. Optimization and Inference Workflow

The method proceeds through three stages:

1. Normalizing flow fitting: Independent MAFs are trained for the 3D spatial distribution $n_\text{obs}(\vec{x})$ and the conditional velocity distribution $p_\text{obs}(\vec{v}|\vec{x})$, maximizing the corresponding likelihoods.
2. Neural network initialization: The potential network $\Phi_\theta(\vec{x})$ and the efficiency network $\varepsilon_\vartheta(\vec{x})$ are initialized.
3. Joint CBE-constrained training: At each step, anchor points $\vec{x}$ are sampled from $n_\text{obs}$, with associated velocity draws. Gradients are propagated through both flow and network modules to minimize the combined loss $L=L_\text{CBE}+L_\text{reg}$, using the Adam optimizer. Regularization hyperparameters (e.g., $\lambda_\varepsilon\simeq 0.1$, $\lambda_\Phi\simeq 10$) are empirically tuned. Training halts when validation CBE loss plateaus (Putney et al., 10 Dec 2025).

Because the model is fully differentiable, all physical constraints and selection effects propagate end-to-end through the optimizer, enabling direct estimation of point-wise potential, acceleration, and density fields.

5. Principal Results

Potential and Acceleration:

The reconstructed gravitational potential $\Phi(\vec{x})$ closely matches the analytic MilkyWayPotential2014 standard model to 1–2%. Solar-neighborhood acceleration is measured as $$\vec{a}_\odot\simeq(-6.56\pm0.04, 0.31\pm0.03, -0.12\pm0.03)$$ mm/s/yr. Small non-axisymmetric features (azimuthal accelerations $\lesssim10\%$ of $|a_R|$) are detected.

Density Estimation:

The inferred total density $\rho(\vec{x})$ exhibits correlation lengths $\sim50$ pc in the disk and $\sim300$ pc in the halo. Overall agreement with analytic models is strong except for deviations $>2\sigma$ near the Galactic Center and $R\sim9$–$11.5$ kpc.

Local Dark Matter Determination:

Subtracting a baryonic model from the total density yields a local dark matter density at the Solar radius of $$(0.84 \pm 0.08)\times10^{-2}\,M_\odot\text{ pc}^{-3}\ (\approx 0.32\,\text{GeV cm}^{-3})$$. The uncertainty budget includes Gaia measurement errors (via Monte Carlo simulation), statistical resampling, and model initialization variance. This result is consistent with and more precise than nearly 100 prior literature estimates (Putney et al., 10 Dec 2025).

Halo and Disk Structure:

Fitting spherical NFW and generalized NFW profiles yields $r_s\sim 2-3$ kpc, with hints of a cored inner profile ($\beta\sim 0$), and strong evidence for a tilted, oblate halo (triaxial fit axis ratios $\xi_1\approx0.89$, $\xi_2\approx0.33$; yaw $\psi\approx -89^\circ$, pitch $\theta\approx52^\circ$). Constraints on a putative dark matter disk improve previous limits by an order of magnitude for scale heights $h_\text{DD}\lesssim 50$ pc (e.g., $\Sigma_\text{DD}<0.38\,M_\odot\mathrm{pc}^{-2}$ for $h_\text{DD}=20$ pc at 95% CL).

6. Disequilibrium Diagnostics and Model Validation

Agreement between modeled line-of-sight accelerations and independent acceleration measurements from 24 binary pulsar systems is achieved within $2\sigma$ for most pulsars; mild $2$–$3\sigma$ discrepancies arise for a subset. The framework defines a non-stationarity metric,

$$\text{NS}(\vec{x};\vec{a})^2 = \text{Var}_{\vec{v}\sim f_\text{obs}} [\vec{v}\cdot\nabla \ln f_\text{obs} - \vec{a}\cdot\nabla_v \ln f_\text{obs}],$$

whose inverse yields the local disequilibrium timescale; mean timescales near the Sun are $\sim$29 Myr, with indications that some disk regions (notably toward the Galactic Center) deviate more strongly from equilibrium, corroborating vertical oscillatory phenomena described in the literature. Systematic uncertainties are dominated by the baryonic model.

7. Systematic Uncertainties and Future Directions

The principal systematic arises from uncertainties in the baryonic mass, especially disk scale height and gas distribution (10% propagated uncertainty). In regions of high extinction ($|l| \lesssim 10^\circ$), dust correction becomes less reliable and may bias results. While the dynamical equilibrium assumption is valid at the $\lesssim10\%$ level over Myr timescales, further work is needed on transient phenomena (e.g., bar buckling or satellite impacts). Foreseeable improvements include: exploitation of Gaia DR4/DR5 for extended phase-space coverage; adoption of more expressive probabilistic flows (e.g., continuous normalizing flows or diffusion models); and fully joint training of baryonic and dark matter components within a single differentiable architecture (Putney et al., 10 Dec 2025).

ClearPotential establishes a new paradigm for data-driven Galactic dynamics, providing high-resolution, selection-corrected maps of the local gravitational field and dark matter, free from strong prior assumptions. This approach opens the path for rapid, precision mapping of Galactic structure as astrometric data sets expand.