Ringing of a black hole in a dark matter halo

Recently, we obtained the simple metrics of a spherically symmetric black hole in a dark matter halo, and extended to the case of rotation. As the characteristic sound of black holes, quasinormal modes (QNMs) are one of the important means to understand black holes currently. Based on these two metrics of a spherically symmetric black hole, we study the QNMs of cold dark matter (CDM) and scalar field dark matter (SFDM) models using the methods of the material field perturbations and the gravitational perturbation, and make comparisons with the Schwarzschild black hole. Our results show that black hole QNMs of CDM and SFDM in a dark matter halo are different from the Schwarzschild black hole, unlike a Schwarzschild black hole with a prominent power-law tail. The different kinds of models of dark matter can be distinguished by their QNMs. The time of QNMs ringing and frequencies increase with increasing parameter $l$. The overall QNMs of CDM are stronger than that of SFDM in the same condition, which is easier to be detected. In addition, QNM frequencies using the sixth-order WKB method and the Prony method are in good agreement.

Figure 1

The effective potentials of the scalar field with the different $l$. The three panels, from left to right, are CDM, SFDM, SCHW. The parameters we used are $M=0.5,R_{\mathrm{c}}=6,{\rho }_{\mathrm{c}}=0.001,R_{\mathrm{s}}=3,{\rho }_{\mathrm{s}}=0.01$.

Figure 2

The effective potentials of the electromagnetic field with the different $l$. The three panels, from left to right, are CDM, SFDM, SCHW. The parameters we used are $M=0.5,R_{\mathrm{c}}=6,{\rho }_{\mathrm{c}}=0.001,R_{\mathrm{s}}=3,{\rho }_{\mathrm{s}}=0.01$.

Figure 3

The effective potentials of the gravitational field with the different $l$. The three panels, from left to right, are CDM, SFDM, SCHW. The parameter we used is $M=0.5$.

Figure 4

The effective potentials of scalar field with the different space-times. The three panels, from left to right, are $l=0$, 1, 2.

Figure 5

The effective potentials of electromagnetic field with the different space-times. The two panels, from left to right, are $l=1$, 2.

Figure 6

The effective potentials of gravitational field with the different space-times. The two panels, from left to right, are $l=2$, 3.

Figure 7

The dynamical evolutions in the scalar perturbation with the different space-time ($M=0.5$, $l=1$, $s=0$).

Figure 8

The dynamical evolutions in the electromagnetic field with the different space-time ($M=0.5$, $l=1$, $s=1$).

Figure 9

The dynamical evolutions in the gravitational perturbation with the different space-time ($M=0.5$, $l=2$).

Figure 10

Quasinormal modes in the scalar field with the different $l$. The ringing time of the QNMs increases with increasing parameter $l$ in each panel. The parameters we used: $M=0.5$, ${\nu }_0=10$, $\sigma =3$.

Figure 11

Quasinormal modes in the electromagnetic field with the different $l$. The time of QNMs ringing increases with increasing parameter $l$ in each panel.

Figure 12

Quasinormal modes in the gravitational perturbation with the different $l$.

Figure 13

Comparisons of quasinormal modes in the scalar fields with the different space-times.

Figure 14

Comparisons of quasinormal modes in the electromagnetic fields with the different space-times.

Figure 15

Comparisons of quasinormal modes in the gravitational perturbation with the different space-times.

Figure 16

Comparisons of quasinormal modes with the different backgrounds ($l=2$).