Oscillation modes of ultralight BEC dark matter cores

Structure formation simulations of ultralight bosonic dark matter, ruled by the Gross-Pitaevskii-Poisson (GPP) system of equations, show that dark matter clumps to form structures with density profiles consisting of a core surrounded by a power-law distribution. The core has a density profile similar to that of spherically symmetric equilibrium configurations of the GPP system. These configurations have been shown to be stable under a variety of perturbations and to have attractor properties. It is interesting to know the dominant frequencies of oscillation of these configurations. One reason is that in formed galaxies the effects of perturbations can trigger observable effects for specific frequency modes. Another reason is that during the process of structure formation, the oscillation modes can help to characterize a core. In this manuscript we present a systematic numerical analysis of the reaction of equilibrium configurations to various axi-symmetric perturbations with modes $l=0,1,2$, $m=0$. We then calculate the first few oscillation frequencies of equilibrium configurations for each mode.