Topological black holes with curvature induced scalarization in the extended scalar-tensor theories

We study the stability of topological black holes in the presence of a cosmological constant and a scalar field coupled to the Gauss-Bonnet (GB) term in the extended scalar-tensor theories. We find two competing effects. As the strength of the coupling $\lambda$ of the scalar field to the GB term is increasing, the matter is interacting more strongly with gravity while as the hyperbolicity $\xi$ of spacetime is getting larger the kinetic effects of matter tend to dominate. Calculating both analytically and numerically the quasinormal modes (QNMs) we found for each $\lambda$ a critical value of $\xi$, below which there is an instability. When the coupling constant $\lambda$ is getting very large, all of the QNMs develop a positive imaginary part indicating an instability. This behavior indicates a phase transition to a scalarized topological black hole induced by curvature effects.

Figure 1

${\omega }_R$ (left) and ${\omega }_I$ (right) versus $\lambda$ for $\xi =0$ and $n=0$, 1 and 2. The value $n=0$ corresponds to the uppermost curve.

Figure 2

$X^{*}(u)X(u)$ for the scalar field versus $u$ for $\lambda =0.5$ and $\xi =0$.

Figure 3

$X^{*}(u)X(u)$ for the scalar field versus $u$ for $\lambda =2.0$ and either $\xi =0$ (left) or $\xi =10$ (right).

Figure 4

$\lambda =0.5$, $\xi =0$, $r_{+}=1.10$, $r_{+}=1.60$.

Figure 5

$\lambda =1.5$, $\xi =0.0$, and $r_{+}=1.10$, 2.00.

Figure 6

$\xi =0.0$, $r_{+}=1.10$, and $\lambda =1.5$ (left), $\lambda =3.0$ (right).

Figure 7

$\lambda =0.50$, $r_{+}=1.10$, and $\xi =0.0$ (left), $\xi =5.0$ (right).

Figure 8

$\lambda =1.50$, $r_{+}=1.10$, and $\xi =5.0$ (left), $\xi =30.0$ (right).