Constraining the Inner Dark-Matter Slope of Sculptor: A Comparative Analysis of Dynamical Methods

Abstract

The core-cusp problem is a discrepancy between simulations and observations of the slopes of inner dark matter density profiles. Splitting dwarf spheroidals by metallicity is a standard method to break the mass-anisotropy degeneracy. Four dynamical methods are applied to the 1339-star Tolstoy et al. (2023) VLT/FLAMES Sculptor catalog: spherical Jeans, GravSphere, the Walker and Pe\~narrubia (2011) two-population estimator, and a continuous $f(\boldsymbol{J},[\text{Fe}/\text{H}])$ action-based distribution function. This work finds $\Gamma=2.49^{+0.23}_{-0.20}$, with 99.6\% of the posterior lying above $\Gamma=2$ (84.4\% after $\lambda$ calibration), robust to membership cuts and metal-poor contamination; GravSphere gives $\gamma=0.46^{+0.40}_{-0.28}$. Controlled mocks that hold tracer geometry fixed while breaking the metallicity-kinematics link attribute $\sim77$--$83\%$ of the recovered bias to geometry rather than decomposition itself. The continuous distribution function encodes a metallicity--action coupling $k_J$ inaccessible to any split-based method; a direct action-space computation on 575 stars with Gaia proper motions finds $[\text{Fe}/\text{H}]$ declining by $\sim0.3\,\text{dex}$ across three decades in radial action. On the central-density diagnostic of Read et al. (2019), the same posteriors give $\rho_\text{DM}(150\,\mathrm{pc})=1.77\times10^8\,M_\odot\,\mathrm{kpc^{-3}}$, agreeing with their independent Sculptor measurement at $0.74\sigma$ and placing Sculptor in their cusp-like class. The shallow $\gamma$ of this work is an asymptotic quantity; the local slope at $150\,\mathrm{pc}$ is consistent with the Navarro--Frenk--White profile under the spherical models used here, though a recent axisymmetric analysis using resolved 3D velocities finds a shallow slope over that range and identifies an inclination degeneracy to which spherical models are blind.