Subject headings: cosmology:dark matter — galaxies:formation — galaxies:halos — methods:N-body simulations accepted by the Astrophysical Journal A series of high-resolution numerical simulations were performed to study the structure and substructure of Milky Way (MW)- and cluster-sized halos in a Λ−Cold Dark Matter (CDM) cosmology with self-interacting (SI) dark particles. The cross section per unit of particle mass has the form σDM = σ0(1/v100) α, where σ0 is a constant in units of cm 2 gr −1 and v100 is the relative velocity in units of 100 kms −1. Different values for σ0 with α = 0 or 1 were used. For small values of σDM= const. ( ∼ < 0.5, α = 0), the core density of the halos at z = 0 is typically higher at a given mass for lower values of σ0 or, at a given σ0, for lower masses. For values of σ0 as high as 3.0, both cluster- and MW-sized halos may undergo the gravothermal catastrophe before z = 0. The core expansion occurs in a stable regime because the heat capacity, C, is positive in the center. After the maximum expansion, the isothermal core is hotter than the periphery and C < 0. Then, the gravothermal catastrophe triggers. The instability onset can be delayed by both the dynamical heating of the halo by major mergers an
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