Numerical challenges for energy conservation in N-body simulations of collapsing self-interacting dark matter halos

Dark matter (DM) halos can be subject to gravothermal collapse if the DM is not collisionless, but engaged in strong self-interactions. When the scattering can efficiently transfer heat from the centre to the outskirts, the central region of the halo collapses and reaches densities much higher than those for collisionless DM. This phenomenon is potentially observable in studies of strong lensing. Current theoretical efforts are motivated by observations of surprisingly dense substructures. A comparison with observations requires accurate predictions. One method to obtain such predictions is to use N-body simulations. Collapsed halos are extreme systems that pose challenges when applying state-of-the-art codes to model self-interacting dark matter (SIDM). We investigate the root of such problems, with a focus on energy non-conservation and discuss possible strategies to avoid them. We ran N-body simulations, with and without SIDM, of an isolated DM-only halo and we adjusted the numerical parameters to check the accuracy of the simulation. We find that not only the numerical scheme for SIDM can lead to energy non-conservation, but also the modelling of gravitational interaction and the time integration are problematic. The issues we find are: (a) particles changing their time step in a non-time-reversible manner; (b) the asymmetry in the tree-based gravitational force evaluation; and (c) SIDM velocity kicks breaking the time symmetry. Tuning the parameters of the simulation allows us to conserve energy not only at early stages of the evolution, but also later on. However, the cost of the simulations becomes prohibitively large. Some of the problems that make the simulations of the gravothermal collapse phase inaccurate can be overcome by choosing appropriate numerical schemes. However, other issues still pose a challenge. Our findings motivate further works on addressing these challenges.