In the standard Lambda cold dark matter paradigm, pure dark matter simulations predict dwarf galaxies should inhabit dark matter haloes with a centrally diverging density `cusp. This is in conflict with observations that typically favour a constant density `core. We investigate this `cusp-core problem in `ultra-faint dwarf galaxies simulated as part of the `Engineering Dwarfs at Galaxy formations Edge (EDGE) project. We find, similarly to previous work, that gravitational potential fluctuations within the central region of the simulated dwarfs kinematically heat the dark matter particles, lowering the dwarfs central dark matter density. However, these fluctuations are not exclusively caused by gas inflow/outflow, but also by impulsive heating from minor mergers. We use the genetic modification approach on one of our dwarfs initial conditions to show how a delayed assembly history leads to more late minor mergers and, correspondingly, more dark matter heating. This provides a mechanism by which even ultra-faint dwarfs ($M_* < 10^5,text{M}_{odot}$), in which star formation was fully quenched at high redshift, can have their central dark matter density lowered over time. In contrast, we find that late major mergers can regenerate a central dark matter cusp, if the merging galaxy had sufficiently little star formation. The combination of these effects leads us to predict significant stochasticity in the central dark matter density slopes of the smallest dwarfs, driven by their unique star formation and mass assembly histories.
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