Structure of the equivalent Newtonian systems in MOND N-body simulations. Density profiles and the core-cusp problem

We investigate the core-cusp problem of the $\mathrm{\Lambda }$ cold dark matter ($\mathrm{\Lambda }$CDM) scenario in the context of Modified Newtonian Dynamics (MOND) paradigm exploiting the concept of equivalent Newtonian system (ENS). By means of particle-mesh $N-$body simulations in MOND we explore processes of galaxy formation via cold dissipationless collapse or merging of smaller substructures. From the end states of our simulations we recover the associated ENS and study the properties of their dark matter halos. We compare the simulation results with simple analytical estimates with a family of $\gamma -$models. We find that the dark matter density of ENSs of most spherical cold collapses ha a markedly cored structure, in particular for the lowest values of the initial virial ratios. End states of some simulations with clumpy initial conditions have more complex profiles and some of their ENSs exhibit a moderate cusp, with logarithmic density slope always shallower than 1. These results seem to point towards the fact that the absence in most observed galaxies of a central DM cusp, at variance with what one would expect from theoretical and numerical arguments in $\mathrm{\Lambda }$CDM, would be totally consistent in a MONDian description.