Quasistationary solutions of self-gravitating scalar fields around black holes

Recent perturbative studies have shown the existence of long-lived, quasistationary configurations of scalar fields around black holes. In particular, such configurations have been found to survive for cosmological time scales, which is a requirement for viable dark matter halo models in galaxies based on such types of structures. In this paper we perform a series of numerical relativity simulations of dynamical nonrotating black holes surrounded by self-gravitating scalar fields. We solve numerically the coupled system of equations formed by the Einstein and the Klein-Gordon equations under the assumption of spherical symmetry using spherical coordinates. Our results confirm the existence of oscillating, long-lived, self-gravitating scalar field configurations around nonrotating black holes in highly dynamical spacetimes with a rich scalar field environment. Our numerical simulations are long-term stable and allow for the extraction of the resonant frequencies to make a direct comparison with results obtained in the linearized regime. A by-product of our simulations is the existence of a degeneracy in plausible long-lived solutions of Einstein equations that would induce the same motion of test particles, either with or without the existence of quasibound states.

Figure 1

Time evolution of the L2 norm of the Hamiltonian constraint for an initial scalar field pulse with $A_0=0.1$ and mass $M\mu =0.1$. The plot shows results for three different resolutions, rescaled by the factors corresponding to second-order convergence. The inset shows a magnified view of the initial $100M$ in the evolution. At $t=0$ the convergence is to fourth order.

Figure 2

Upper panel: Time evolution of the scalar field with mass $M\mu =0.1$ and $A_0=0.0005$ corresponding to model 3 for an observer at $r_{\text{ext}}=100M$. The inset shows a magnified view of the initial $5000M$ in the evolution. Lower panel: Fourier transform of the evolution of the scalar field. $M\omega$ refers to the real part of the oscillation frequencies. The units in the vertical axis are arbitrary; we have normalized them by the amplitude of the fundamental mode.

Figure 3

Same as Fig. 2 but for model 4.

Figure 4

Same as Fig. 2 but for model 5.

Figure 5

Same as Fig. 2 but for model 6.

Figure 6

Frequency $\omega$ as a function of the scalar field mass $\mu$ for the models shown in Table 1. The semianalytic relation (41) for several values of $M=\{1.1,1.5,1.6,2.0,2.3,2.7,2.9\}$ is also plotted. For models with small initial $A_0$ the numerical frequencies match the values of the semianalytical values for $M=1$ showing consistency with the test field approximation and the limit $M\mu \ll 1$.

Figure 7

Radial profiles of the Weyl invariants $I$ (left) and $J$ (right) for model 6 in Table 1 at $t=9000M$. The red curve corresponds to our numerically evolved spacetime while the green and blue curves correspond to analytic (static) Schwarzschild spacetimes of mass $M_{\mathrm{AH}}=2.18$ and 4.25, respectively. The inset shows a region close to the black hole horizon where the numerical solution differs considerably from a stationary Schwarzschild black hole.