Limits of dynamical stability in the (M_core, M) diagram for Pop III hylotropic SMSs (μ = 0.6, T_c = 1.8 × 10⁸ K) in the presence of a homogeneous dark matter distribution with various densities ρ_χ. The red curves correspond to ρ_χ = 0 and to log ρ_χ [cgs] = −10, −9, …, −3 (from left to right); the orange curves to log ρ_χ/ρ_c = −10, −9, …, −3 (from left to right). The black, blue, and cyan lines are GENEC tracks for the indicated accretion rates (Haemmerlé et al. 2018a, 2019). The grey lines correspond to constant mass fractions of the convective core.