Scalar perturbations on the background of Kerr black holes in the quadratic dynamical Chern-Simons gravity

We study the scalar perturbation on the background of a Kerr black hole in the dynamical Chern-Simons modified gravity with a quadratic coupling between the scalar field and Chern-Simons term. In particular, the late-time tails of scalar perturbations are investigated numerically in time domain by using the hyperboloidal foliation method. It is found that the Kerr black hole becomes unstable under linear perturbations in a certain region of the parameter space, which depends on the harmonic azimuthal index $m$ of the perturbation’s mode. This may indicate that some Kerr black holes in this theory will get spontaneously scalarized into a non-Kerr black hole.

Figure 1

The time-domain profiles of different modes of scalar perturbation on the Kerr BHs with spin parameter $a=0.9$ in the quadratic dCS gravity with coupling constant $\alpha =1$. In the left panel, the initial modes are chosen to be the modes with $m=0$ but $l=0$, 1, and 2, respectively, while the indices $m$ for the initial modes in the right panel are both $m=1$ but $l=1$ and 2, respectively. The observing location is at $r\to \infty$ ($R=1$), $\theta =\pi /2$.

Figure 2

Several modes with $m=0$ and different $l$ numbers are activated by the initial $l$ mode during the process of time evolution. From left to right, the initial mode is specified to be the one with $l=0$, 1, and 2, respectively. It is found that, at late enough time, the dominant mode is always the $l=m=0$ mode, whatever the initial mode is chosen.

Figure 3

The time-domain profiles of the dominant $m=0$ mode of scalar perturbation on the Kerr BHs with spin parameter $a=0.5$ (left panel) and 0.998 (right panel) in the quadratic dCS gravity with different values of coupling constant $\alpha$. Here, the initial mode is specified to be $l=2$. The observing location is at $r\to \infty$ ($R=1$), $\theta =\pi /2$.

Figure 4

The late-time evolution of the dominant $m=2$ mode on the background of the Kerr BH with spin parameter $a=0.9$ in theories with different values of coupling constant $\alpha$.

Figure 5

The real part (left panel) and imaginary part (right panel) of the frequency $\omega$ at the stage of the late-time tail of dominant $m=1$ ($l=1$) and $m=2$ ($l=2$) modes of the perturbation on the background of a Kerr BH with spin parameter $a=0.9$ in the quadratic dCS gravity with different values of $\alpha$.

Figure 6

The parameter space for the dominant $m=0$ mode ($l=0$). In the region above this line (shaded), there exist unstable perturbation modes, and then the BH becomes unstable, while in the region below the line (blank), there is no growing mode, and the BH is stable.

Figure 7

The boundary between stable and unstable regions in parameter space for different values of harmonic azimuthal index $m$.

Figure 8

The time-domain profiles of the dominant $m=2$ mode ($l=2$). Here, we choose three different radial grids ($N_R=200$, 400, and 800), and $a=0.9$, $\alpha =1$.

Figure 9

The time-domain profiles of the $m=0$ mode for initial wave packets with parameter $\sigma =0.01$, 0.1, 1, 10, and 100. Here, we set $a=0.9$, $\alpha =0.5$, and the initial mode $l=2$. Clearly, the rate of blowup does not change for a large range of $\sigma$.

Figure 10

The absolute value of relative difference between the auxiliary field $\mathrm{\Gamma }$ and ${\partial }_R\psi$. The parameters are selected to be the same as those in Fig. 8.