Dark Matter in Draco and Boötes I: Hints of a Core in an Ultra-Faint Dwarf from Simulation-Based Inference

Technical Summary: Dark Matter in Draco and Boötes I: Hints of a Core in an Ultra-Faint Dwarf from Simulation-Based Inference

Problem Statement
The density profiles of dwarf spheroidal galaxies (dSphs) serve as critical probes for distinguishing between collisionless Cold Dark Matter (CDM) models, which predict "cuspy" inner density profiles ($\rho \propto r^{-1}$), and alternative models (e.g., self-interacting or dissipative DM) that predict "cored" profiles ($\rho \propto r^0$). However, extracting these profiles from noisy stellar kinematics is hindered by systematic uncertainties, most notably the mass-anisotropy degeneracy. Traditional methods relying solely on the line-of-sight velocity dispersion (LOSV), $\sigma_{\text{los}}$, often fail to break this degeneracy and are biased toward cuspy solutions, even in the absence of the degeneracy, due to the intrinsic insensitivity of $\sigma_{\text{los}}$ to inner slopes in the cored regime and the influence of sampling noise. Furthermore, existing inference pipelines often neglect measurement uncertainties and spectroscopic selection functions (e.g., magnitude limits, crowding), which can introduce significant biases.

Methodology
The authors present GraphNPE (Graph Neural Posterior Estimation), an extension of a simulation-based inference (SBI) framework designed to infer dynamical mass models directly from kinematic samples.

• Forward Model: The model generates synthetic dwarf galaxy kinematics using the Agama library. It assumes a Zhao profile for the dark matter (DM) density, a Plummer profile for the stellar tracer density, and a Cuddeford-Osipkov-Merritt (COM) profile for velocity anisotropy.
• Incorporation of Observational Effects: Unlike previous iterations, this framework explicitly incorporates:
  • Measurement Uncertainties: Per-star LOSV uncertainties are modeled via convolution with Gaussian kernels. The uncertainties are drawn from a Jeffreys prior to cover the range of values found in spectroscopic catalogs.
  • Selection Functions: Radial dropout schemes (simulating magnitude limits in outer regions and crowding in inner regions) are applied to mock data to mimic the spatial incompleteness of real surveys.
  • Graph Construction: The data is represented as a graph where nodes are stars and edges connect nearest neighbors. Node features include projected radius, LOSV, and LOSV uncertainty. The number of neighbors is set adaptively (20% of the total tracer count) to maintain connectivity across varying sample sizes.
• Inference: GraphNPE learns the posterior distribution $p(\theta|d)$ by minimizing the Kullback-Leibler divergence between the true posterior and a neural density estimator (normalizing flow) conditioned on the graph representation of the data. This allows for amortized inference, providing rapid posterior evaluation without retraining.
• Comparison Baseline: The authors compare GraphNPE against second-order Jeans modeling, which fits the LOSV dispersion profile $\sigma_{\text{los}}(R)$ using a Gaussian likelihood. This baseline serves as a control to isolate the impact of higher-order velocity moments.

Key Contributions

1. Framework Extension: The integration of heterogeneous measurement uncertainties and spectroscopic selection functions into the GraphNPE forward model, bridging the gap between idealized simulations and real observational data.
2. Demonstration of Higher-Order Moment Utility: The paper demonstrates that relying solely on $\sigma_{\text{los}}$ leads to biased inferences. By operating on the full phase-space distribution function (DF), GraphNPE implicitly accesses higher-order moments, specifically the LOSV kurtosis ($\kappa_{\text{los}}$). The authors show that $\kappa_{\text{los}}$ carries complementary information to $\sigma_{\text{los}}$, particularly in distinguishing cored profiles, thereby breaking the mass-anisotropy degeneracy and mitigating biases caused by sampling noise.
3. Diagnostic Tool: The authors propose LOSV kurtosis as a standard posterior diagnostic. They show that a model can fit $\sigma_{\text{los}}$ well while failing to reproduce $\kappa_{\text{los}}$, indicating a biased recovery of the underlying density profile.

Results
The framework was applied to two Milky Way satellites: Draco (a classical dSph) and Boötes I (an ultra-faint dwarf), using multiple spectroscopic datasets (MMT/Hectochelle, DESI, S5, and archival data).

• Mock Validation: On mock datasets matching the observational properties of the targets, GraphNPE recovered true density and anisotropy profiles with less bias than second-order Jeans modeling. Crucially, while Jeans modeling often inferred biased cuspy profiles for cored mocks (due to noise in $\sigma_{\text{los}}$), GraphNPE remained robust.
• Draco:
  • GraphNPE yielded consistent density profiles across the MMT and DESI datasets, marginally preferring a cuspy inner profile ($\rho_{150} \sim 1.6\text{--}1.9 \times 10^8 M_\odot \text{kpc}^{-3}$), consistent with literature.
  • Second-order Jeans modeling on the DESI dataset failed to reproduce the observed kurtosis, inferring a significantly lower density normalization and a cored profile. This discrepancy highlights that the DESI DESI-only Jeans inference was biased by undersampling in the inner regions, a bias corrected by GraphNPE's use of higher-order moments.
• Boötes I:
  • Statistical power was limited by small sample sizes and large measurement uncertainties relative to the intrinsic dispersion.
  • While the asymptotic inner slope $\gamma$ could not be definitively determined, GraphNPE recovered a significantly lower inner DM density than previous non-spherical Jeans analyses: $\rho_{150} = 0.36^{+0.15}_{-0.11} \times 10^8 M_\odot \text{kpc}^{-3}$ (for the combined sample).
  • This value places Boötes I among the lowest density dwarfs of comparable stellar mass and is consistent with a cored inner profile.
  • The authors note that while the inner slope remains ambiguous, the low $\rho_{150}$ is robust across different modeling assumptions.

Significance and Claims
The paper claims that the combination of low inner density and shallow inner slope in Boötes I provides the first hint of a core in an ultra-faint dwarf. The authors discuss four potential physical origins for this signature:

1. Feedback-driven core formation: Disfavored due to Boötes I's low stellar-to-halo mass ratio and its formation in a single short burst of star formation, which is insufficient for sustained feedback.
2. Tidal stripping: Plausible given the satellite's small pericenter, but simulations suggest tides alone cannot transform a primordial cusp into a core without destroying the stellar component.
3. Low-concentration halo: Statistically unlikely but possible.
4. Alternative DM models: Self-interacting DM (SIDM) is a viable candidate, as it can generate cores via gravothermal heating. Fuzzy DM is disfavored by existing constraints.

The authors conclude that distinguishing between these scenarios requires a larger statistical sample of ultra-faint dwarfs. They emphasize that GraphNPE, by leveraging the full kinematic distribution function, offers a robust framework for extracting maximum information from sparse kinematics, a capability essential for future studies of ultra-faint dwarfs where traditional dispersion-based methods are prone to bias. The work underscores the necessity of incorporating higher-order moments (like kurtosis) or distribution-function-based methods to avoid the systematic biases inherent in dispersion-only analyses.