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tokens: is much shorter than the dynamical time-scale of the system: To test this, we integrate equation ( A14 ) for Δ t = t 1 − t 0 = 1, and leave the rest of the parameters unchanged with respect to the fully adiabatic case shown in the left-hand panel of Fig. A3 . The right-hand panel of Fig. A3 shows the numerical solution in this case (solid black line). It is apparent that when the change in potential is impulsive, the solution to equation ( A14 ) obtained by numerical integration is no longer sensible, which confirms that we can no longer use the ansatz ( A9 ). Instead we need to resort to the arguments summarized in Appendix A1 and derived in Pontzen Governato ( 2012 ). A solution based on the latter is shown in the right-hand panel of Fig. A3 as a red dashed line. We caution the reader that a specific choice of phase underlies this solution, in particular one that maximizes the growth of the amplitude. APPENDIX B: STABILITY OF HALOES SIMULATED IN EQUILIBRIUM In order to assess the intrinsic differences between the adiabatic and impulsive core formation scenarios, we need to ensure that numerical artefacts are well under control in the central regions of the halo where core formation takes place. In particular, we want to ensure that two effects are negligible on the time-scales of our simulations within the region of interest for our work: (i) the growth of numerical errors coming from the initial set-up of a DM halo in equilibrium, and (ii) the development of collisionless relaxation due to particle discreteness. In order to assess the magnitude of these two effects, we look at the stability of the halo set-up in equilibrium in a self-consistent way in Section 3.1 (referred to as the equilibrium case throughout this work). Regarding (i) above, the method we have used (developed in Kazantzidis et al. 2004 ) has been used and tested extensively and we can be confident that numerical errors are under control as long as the amount of simulation particles sampled in the central region is large enough, which is connected to (ii), the issue of particle discreteness. Regarding collisionless relaxation, the relaxation time-scale is given by (Binney Tremaine 1987 ): where N is the number of collisionless particles and t cross is the crossing time. To ensure that an N -body representation of a DM halo does not suffer from gravitational two-body relaxation, the relaxation time needs to be longer than the simulation time. In order to achieve that, we need to avoid very large accelerations arising due to close encounters between simulation particles. Choosing an appropriate softening length ensures this (Power et al. 2003 ). In particular, by demanding that the maximal gravitational force due to two-body encounters (equal to the maximum stochastic softened force) is smaller than the minimum mean field force in the simulation (which occurs at the virial radius), we get From this, it follows that a lower limit to the softening length required to