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 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. Based on these motivations, in this manuscript we present a systematic numerical analysis of the reaction of equilibrium configurations to various axisymmetric perturbations with modes $l=0$, 1, 2, 3, 4. We then calculate the first few oscillation frequencies of equilibrium configurations for each mode.

Figure 1

Central value of the density in a subset of the time domain and its FT. In the top panels we show the results for the case with perturbation parameters $k_r=0$, 1, 2, ${\sigma }_p=1$, ${\varrho }_{\max }=20$ and $\delta M=0.1\%$. As a consistency check, in the bottom panels we show the time series of the central value of the density calculated using a spherically symmetric code with three very high resolutions $\mathrm{\Delta }{r}_3=\mathrm{\Delta }{r}_2/2$, $\mathrm{\Delta }{r}_1/4=2.5\times {10}^{-3}{r}_{95}$, where the density oscillations are triggered by the numerical truncation errors only.

Figure 2

Spectrum of the perturbations with the $l=1$ mode in the frequency domain. Shown are the results measured by $D_1$ located on the $\varrho$ axis (left) and $D_2$ located on the $z$ axis (right).

Figure 3

Time series of the density at detector $D_1$ (left) and $D_2$ (right), for the case $l=1$, ${\sigma }_p=1$, ${\varrho }_{\max }=20$ and $\delta M=0.1\%$. There is a clear low-frequency mode measured by $D_2$.

Figure 4

Spectrum of the perturbations for the $l=2$ mode with ${\sigma }_p=1$, ${\rho }_{\max }=20$, the three values of the wave number $k_r$ and two out of the four mass contributions of the perturbation $\delta M=0.1\%$, 0.5%. Shown are the results measured by detector $D_1$ (left) and detector $D_2$ (right).

Figure 5

Spectrum of the perturbations for the $l=3$ mode for ${\sigma }_p=1$, ${\rho }_{\max }=20$, the three values of the wave number $k_r$ and two out of the four mass contributions of the perturbation $\delta M=0.1\%$, 0.5%. Shown are the results measured by detector $D_1$ (left) and detector $D_2$ (right).

Figure 6

Spectrum of the perturbations for the $l=4$ mode for ${\sigma }_p=1$, ${\rho }_{\max }=20$, the three values of the wave number $k_r$ and two out of the four mass contributions of the perturbation $\delta M=0.1\%$, 0.5%. Shown are the results measured by detector $D_1$ (left) and detector $D_2$ (right).