Bayesian model compar ison of dark matter density profiles in the Milky Way: SIDM versus CDM

We perform a Bayesian model comparison of dark matter (DM) density profiles in the Milky Way using five rotation curve compilations based on Gaia DR3 (D1--D5, covering $5.24$--$27.5$ kpc, including the new classical Cepheid curve) and three baryonic models. Marginal likelihoods are computed with dynamic nested sampling (nₗᵢᵥₑ=200) under a unified, physically motivated log-uniform prior, for eight DM profiles: NFW, Einasto, Burkert, coreNFW, gNFW, a simplified isothermal SIDM, a self-consistent SIDM isothermal solver based on the Jeans--Poisson equations with baryon coupling (SIDM_Jeans), and the two-component SIDM parametrisation of Yang et al. (SIDM2c). The Einasto profile is decisively preferred on four of the five datasets (ln B = 3.3--$30.0$ over the competing profiles), and NFW is never preferred. The sole exception is D2, whose inner point at $R = 5.24$ kpc decisively favours SIDM_Jeans over every analytic profile (ln B > 27), with a very tight cross-section constraint σ/m = 0.097 ± 0.007\ cm²\,g⁻¹. Across datasets the physical Jeans solver yields σ/m = 0.1--0.9\ cm²\,g⁻¹, consistently below ~ 1\ cm²\,g⁻¹. The gNFW inner slope is α 0.4--$0.6$ (84 per cent upper limits $0.68$--$1.06$), disfavouring the NFW cusp; SIDM2c, whose mass-segregation effect is calibrated on dwarf-galaxy scales, is indistinguishable from NFW at the Milky Way scale. Our results support a cored (rather than cuspy) central DM profile and demonstrate that rotation curve data can constrain σ/m at the sub-cm²\,g⁻¹ level, consistent with dwarf-galaxy constraints for velocity-dependent cross-sections.