Quasinormal modes of black holes in scalar-tensor theories with nonminimal derivative couplings

We study the quasinormal modes of asymptotically anti–de Sitter black holes in a class of shift-symmetric Horndeski theories where a gravitational scalar is derivatively coupled to the Einstein tensor. The spacetime differs from exact Schwarzschild–anti–de Sitter, resulting in a different effective potential for the quasinormal modes and a different spectrum. We numerically compute this spectrum for a massless test scalar coupled both minimally to the metric, and nonminimally to the gravitational scalar. We find interesting differences from the Schwarzschild–anti–de Sitter black hole found in general relativity.

Figure 1

Effective potential for a scalar perturbation given in Eq. (12). The tortoise coordinate $r^{*}$ takes values from $-\infty$ to $\pi /2$. $z=1/3$ and $j=0$.

Figure 2

Real (a) and imaginary (b) parts of the QNMs (${\omega }_{ST}$), as a function of the BH horizon radius $r_h$. Here $z=1/3$. The data points are the principal QNM (black circles), and the first (red squares) and second (blue diamonds) overtones. The solid continuous curves are $(7.75-11.16i)T$ (principal), $(13.24-20.59i)T$ (first overtone) and $(18.70-30.02i)T$ (second overtone), where $T$ is the Hawking temperature of the BH horizon. The dashed continuous curves in (b) are $2.66r_h$ (principal), $4.98r_h$ (first overtone) and $7.18r_h$ (second overtone). The units are chosen by setting $m_p=1$ and $l=1$.

Figure 3

Contour plot of the principal QNMs for different values of $j$ and $r_h$. Different $j$’s are denoted by different marker shapes. Different $r_h$’s are represented by different marker colors. Here $z=1/3$.

Figure 4

The real (a) and imaginary (b) parts of the QNMs as a function of the constant $z$. The dashed lines are given by Eq. (17) with the corresponding BH horizon radius and QNM order. Different $r_h$’s are represented by different colors. The integers between data points are the orders of QNMs: “0” for principal mode, “1” for the first overtone and “2” for the second overtone.

Figure 5

Imaginary part of QNMs as a function of the BH horizon radius, for $z=1/3$ (blue) and $z=2$ (red). The filled circles are data from our calculation, and the solid/dashed curves are the analytic functions shown in the legend.

Figure 6

Contour plot of the principal QNMs for different $j$’s and $z$’s for $r_h=10$. Different $j$’s are denoted by different marker shapes shown in the figure. For each $j$, $z$ takes values uniformly from $1/3$ to $20/9$, from top to bottom.

Figure 7

The effective potential for a nonminimally coupled scalar model [Eq. (21)] with different values of the coupling constant $\xi$, for $r_h=10$, $j=0$ and $z=2/3$. The critical value of $\xi$ is ${\xi }_c=-1/2$.

Figure 8

Real part (upper half-plane) and imaginary part (lower half-plane) of the principal QNMs as a function of $z$ for different values of the coupling constant $\xi$ [defined in Eq. (19)] given in the figure. Here $r_h=10$ and $j=0$. Neighboring data points are joined by line segments for better illustration.