tokens: by myriads of DM particles. This is the reason why two convergence criteria are applied to the CDM numerical simulations to declare their results reliable (e.g. Power et al. 2003 ). First, the finite mass of the DM particles used in the simulation artificially shortens the two-body relaxation time-scale; and this artificially-set time-scale is asked to be longer than the age of the Universe, t U . This requirement prevents the thermalization of the resulting DM haloes. Secondly, the artificial thermalization starts in the centre of the DM haloes so the simulated centres are discarded within the so-called convergence radius. However, when these two restrictions are relaxed then cores with polytropic shape appear in the DM haloes produced in the numerical simulation. In this paper, we analyse two types of numerical simulations showing such cores. If the DM is assumed to be self-interacting (SIDM) then the effective relaxation time-scale t SIDM is shortened by an amount that depends on the SIDM cross-section. As soon as , t SIDM ≲ t U and the simulations produce cores. We show in Section 4 how different SIDM from several authors produce cores which after normalization are very similar and equal to the core of a polytrope within error bars. The simulations comprise haloes that go all the way from dwarf galaxies to galaxy clusters. This agreement between simulated haloes and polytropes holds even when baryons are included in the simulation (Section 4.1 ). If standard CDM numerical simulations are analysed within the convergence radius, they also show cores (e.g. Wang et al. 2020 ). These cores are also indistinguishable from polytropes, a result which holds even with halo masses differing by as much as 10 orders of magnitude (Section 5 ). Outside the convergence radius, the profiles follow the canonical NFW or Einasto shape. Thus, because the profiles within the convergence radii are excluded from the standard analysis, it is easy to understand why the profiles resulting from these numerical simulations seem to be set by the initial conditions during the big bang (Cen 2014 ; Nipoti 2015 ; Brown et al. 2020 ). As a by-product of this work, we have shown how unresolved DM clump–clump collisions cannot thermalize the DM haloes within a time-scale comparable to the age of the Universe (Appendix A ). We conclude that the presence of cores in simulated or observed density profiles can used as evidence for systems in thermodynamic equilibrium. Cores are theoretically predicted and also generated in self-gravitating system known to be in thermodynamic equilibrium. APPENDIX A: TWO-BODY COLLISION CROSS-SECTION FOR A CONTINUOUS DISTRIBUTION OF UNRESOLVED DM CLUMPS Starting from the primordial density fluctuations, the gravitational collapse produces a full distribution of DM haloes with masses going from galaxy clusters to Earth-mass clumps and even smaller (e.g. Diemand, Moore Stadel 2005 ; Wang et al. 2020 ). However, numerical simulations cannot treat all
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