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. 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 Peñarrubia (2011) two-population estimator, and a continuous f(,[Fe/H]) action-based distribution function. This work finds Γ=2.49+0.23-0.20, with 99.6% of the posterior lying above Γ=2 (84.4% after λ calibration), robust to membership cuts and metal-poor contamination; GravSphere gives γ=0.46+0.40-0.28. Controlled mocks that hold tracer geometry fixed while breaking the metallicity-kinematics link attribute about77 -- 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 [Fe/H] declining by about0.3, dex across three decades in radial action. On the central-density diagnostic of Read et al. (2019), the same posteriors give ρ DM(150, pc)=1.77 times10 8,M, kpc-3, agreeing with their independent Sculptor measurement at 0.74σ and placing Sculptor in their cusp-like class. The shallow γ of this work is an asymptotic quantity; the local slope at 150, 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.