An Enhanced Isothermal Jeans Approach to Constraining Dark Matter Self-Interactions from Galactic Kinematics

We present an improved semi-analytical model to predict density profiles of self-interacting dark matter (SIDM) halos and apply it to constrain the self-scattering cross section using SPARC galaxy rotation curves. Building on the isothermal Jeans approach, our model incorporates (i) velocity-dependent cross sections, (ii) an empirical treatment of core collapse, and (iii) enhanced robustness for identifying solutions. These advances allow us to fit a large sample of galaxies, including systems with baryon-dominated centers often excluded in earlier studies. We find that roughly 1/6 of galaxies admit both a core-growth and a core-collapse solution, while the rest favor a unique evolutionary state. Joint constraints across the sample reveal clear velocity dependence: the allowed parameter space forms an L-shaped degeneracy, where both nearly constant, low cross sections ($\sigma_0\sim2\,{\rm cm}^2$/g, $\omega\gtrsim500\,$km/s) and strongly velocity-dependent models ($\sigma_0\sim100\,{\rm cm}^2$/g, $\omega\sim60\,$km/s) are viable. Adopting the core-growth interpretation yields best-fit values $\sigma_0\simeq5\,{\rm cm}^2$/g and $\omega\simeq250\,$km/s. Our constraints are remarkably consistent with previous results derived from a variety of independent probes. Compared to cold dark matter (CDM) models, SIDM outperforms simple adiabatic-contraction profiles and rivals empirical feedback-based CDM profiles, yet shows no correlation with stellar-to-halo mass ratio, a proxy for feedback strength, offering a distinct explanation for dwarf galaxy diversity. Moreover, SIDM does not affect galaxy-halo scaling relations significantly and makes concentration systematically lower. Our results highlight SIDM as a compelling framework for small-scale structure, while future low-mass kinematic data will be crucial for breaking degeneracies in velocity-dependent cross-section models.