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 ($n_{\mathrm{live}}=200$) 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 $\sigma/m = 0.097 \pm 0.007\ \mathrm{cm^2\,g^{-1}}$. Across datasets the physical Jeans solver yields $\sigma/m = 0.1$--$0.9\ \mathrm{cm^2\,g^{-1}}$, consistently below $\sim 1\ \mathrm{cm^2\,g^{-1}}$. The gNFW inner slope is $\alpha \simeq 0.4$--$0.6$ (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 $\sigma/m$ at the $\mathrm{sub\text{-}cm^2\,g^{-1}}$ level, consistent with dwarf-galaxy constraints for velocity-dependent cross-sections.