  Using a simple analytic formalism, we demonstrate that significant dark
matter self-interactions produce halo cores that obey scaling relations nearly
independent of the underlying particle physics parameters such as the
annihilation cross section and the mass of the dark matter particle. For dwarf
galaxies, we predict that the core density $\rho_c$ and the core radius $r_c$
should obey $\rho_c r_c \approx 41 \,\text{M}_\odot \text{pc}^{-2}$ with a weak
mass dependence $\sim M^{0.2}$. Remarkably, such a scaling relation has
recently been empirically inferred. Scaling relations involving core mass, core
radius, and core velocity dispersion are predicted and agree well with
observational data. By calibrating against numerical simulations, we predict
the scatter in these relations and find them to be in excellent agreement with
existing data. Future observations can test our predictions for different halo
masses and redshifts.
