A universal scaling relation incorporating the cusp-to-core transition of dark matter haloes

The dark matter haloes associated with galaxies have hitherto established strong correlations within a range of observed parameters, known as scaling relations of dark matter haloes. The origin of these scaling relations still contains significant ambiguities and requires comprehensive exploration for complete understanding. Utilising the correlation between the concentration and mass of dark matter haloes inferred from cosmological $N$-body simulations based on the cold dark matter paradigm ($c$-$M$ relation), we derive theoretical scaling relations among other physical quantities such as the surface mass density, the maximum circular velocity, and the scale radius of the dark matter halo. By comparing theoretical and observed scaling relations at various mass scales, it is found that the scaling relations observed in dwarf galaxies and galaxies originate in the $c$-$M$ relation of the dark matter halo. We predict that this theoretical scaling relation is also established in galaxy clusters. Moreover, we propose a novel theoretical scaling relation that incorporates the effects of the cusp-to-core transition, which is supposed to occur in cold dark matter haloes. Our discussion concludes with the exploration of potential observational verification of the cusp-to-core transition process in dark matter haloes.