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

The core-cusp problem is a discrepancy between simulations and observations of the slopes of inner dark matter density profiles. The two-population estimator, separating dwarf spheroidals into metallicity-split subcomponents, 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 Penarrubia (2011) two-population estimator, and a continuous $f(J,[\mathrm{F}\mathrm{e}\mathrm{/}\mathrm{H}])$ action-based distribution function. This work finds $\mathrm{\Gamma }=2.49_{-0.20}^{+0.23}$, with 99.6% of the posterior lying above $\mathrm{\Gamma }=2$ (84.4% after $\lambda$ calibration), robust to membership cuts and metal-poor contamination; GravSphere gives $\gamma =0.46_{-0.28}^{+0.40}$. Controlled mocks that hold tracer geometry fixed while breaking the metallicity-kinematics link attribute 77-83% of the recovered bias to geometry rather than decomposition itself; thus, mock-based claims about population splitting must control for geometry. 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 [Fe/H] declining by 0.3 dex across three decades in radial action. On the central-density diagnostic of Read et al. (2019), the same posteriors give ${\rho }_{DM}(150\mathrm{p}\mathrm{c})=1.77\times {10}^8{M}_{\odot }{\mathrm{k}\mathrm{p}\mathrm{c}}^{-3}$, 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, and the local slope at 150 pc is consistent with the Navarro-Frenk-White profile. Any tension with feedback-driven core formation therefore rests on the slope axis alone.