Mapping Dark Matter in the Milky Way using Normalizing Flows and Gaia DR3

TL;DR

We address the Milky Way's local gravity and dark matter content by learning the six-dimensional phase-space density $f(\vec{x},\vec{v})$ from Gaia DR3 with normalizing flows, solving the equilibrium collisionless Boltzmann equation to obtain unbinned accelerations $-\nabla\Phi$ and total mass density $\rho$ within $\sim$3 kpc of the Sun without imposing symmetries. The method models $f$ as a product $f(\vec{x},\vec{v})=n(\vec{x})\,p(\vec{v}|\vec{x})$ using two Masked Autoregressive Flows, and derives $\vec{a}=-\nabla\Phi$ by minimizing a Boltzmann-residual loss, with mass density from Poisson's equation via Gaussian smoothing. After subtracting a baryonic mass model (McKee et al.), we infer a local dark matter density $\rho_{\rm DM,\odot}=0.47\pm 0.05$ GeV cm$^{-3}$ under the assumption of spherical symmetry, and find a generalized NFW profile consistent with recent analyses. This data-driven framework provides a model-free map of Galactic gravity, enabling tests of equilibrium and symmetry and offering a path to significant improvements with future Gaia releases and refined error modeling.

Abstract

We present a novel, data-driven analysis of Galactic dynamics, using unsupervised machine learning -- in the form of density estimation with normalizing flows -- to learn the underlying phase space distribution of 6 million nearby stars from the Gaia DR3 catalog. Solving the equilibrium collisionless Boltzmann equation, we calculate -- for the first time ever -- a model-free, unbinned estimate of the local acceleration and mass density fields within a 3 kpc sphere around the Sun. As our approach makes no assumptions about symmetries, we can test for signs of disequilibrium in our results. We find our results are consistent with equilibrium at the 10% level, limited by the current precision of the normalizing flows. After subtracting the known contribution of stars and gas from the calculated mass density, we find clear evidence for dark matter throughout the analyzed volume. Assuming spherical symmetry and averaging mass density measurements, we find a local dark matter density of $0.47\pm 0.05$ GeV/cm$^3$. We compute the dark matter density at four radii in the stellar halo and fit to a generalized NFW profile. Although the uncertainties are large, we find a profile broadly consistent with recent analyses.

Figures (17)

• Figure 1: Schematic representation of the Solar location (red dot) relative to the Galactic Center (black dot). The 4 kpc observation volume is shown as a transparent grey sphere. The three coordinate systems used in this work are shown: the Galactocentric Cartesian coordinates $(x,y,z)$, the spherical coordinates $(r,\theta,\phi)$, and the cylindrical coordinates $(R,\phi,z)$. The lines through the observational volume with low dust extinction along which we measure accelerations and mass densities are shown in color. In orange, we show two lines at $z=+1.5$ kpc, one varying $r$ and another $\phi$. The two corresponding lines at $z=-1.5$ kpc are shown in purple. The line parameterized by polar arclength $s = r_\odot \times (\pi/2-\theta)$ passing through the Solar location is shown in green.
• Figure 2: Density plots of the stars with full 6-dimensional kinematic information available from Gaia within 4 kpc of the Solar location in the $x-y$ (top row) and $x-z$ (bottom row) planes. The left column shows all 24,789,061 fully-characterized stars. The middle column shows the 5,811,956 remaining stars after applying the selection criteria described in the text. The right column applies the additional requirement of $|z|>1$ kpc, resulting in 470,702 stars.
• Figure 3: Absolute magnitude $M_G$ and color $BP-RP$ of all 24,789,061 stars within 4 kpc of the Sun with 6-dimensional kinematic information measured by Gaia. The horizontal dashed white line denotes the magnitude completeness criteria Eq. \ref{['eq:magcut']}. All 5,811,956 stars above the dashed white line are bright enough to be observable for Gaia regardless of position in the 4 kpc sphere. The visible peak in the white box is the red clump. All features in the space of uncorrected $M_G$ and $BP-RP$ will appear to be smeared towards the bottom-right of this figure, as dust extinction both dims and reddens stars. While a fraction of red clump stars are smeared beyond the magnitude cut threshold, all of these stars reside in the disk and will not significantly influence solutions to the CBE in the stellar halo.
• Figure 4: The cumulative histogram of distance to a star, $d$. Black points are the cumulative counts of selected stars given $d$, with error bars indicating $1\sigma$ statistical uncertainty. The red line represents the expected cumulative histogram for a uniform distribution, normalized to match the observed cumulative count at $d = 200$ pc. The gray dashed line shows the expected cumulative histogram of the flow-generated dataset, normalized to match the observed cumulative count at $d = 200$ pc. The lower panel shows the pull distributions, i.e., the difference between the uniform distribution histogram and the corresponding histogram divided by the $1\sigma$ statistical uncertainty of the dataset.
• Figure 5: Stellar number density from Gaia data (left) and MAF-learned number density (right) $n(\vec{r})$ on Galactic longitude ($\ell$) and latitude ($b$) planes at a distance (top) 1 kpc, (center) 2 kpc, and (bottom) 3 kpc. The stellar number densities are obtained by directly counting the number of stars within 0.1 kpc from the center of the pixel. The MAF-learned number densities are described in Section \ref{['sec:methods']}. The $K_s$-band extinction maps at a given distance (obtained from the dustmaps package 2018JOSS....3..695M) are shown as contours. The maps are Gaussian-kernel smoothed with a bandwidth of $8^\circ$. We show four extinction value contours from white to blue: 0.15, 0.3, 0.45, and 0.60.