Scalar field evolution in Gauss-Bonnet black holes

It is presented a thorough analysis of scalar perturbations in the background of Gauss-Bonnet, Gauss-Bonnet-de Sitter and Gauss-Bonnet-anti-de Sitter black hole spacetimes. The perturbations are considered both in frequency and time domain. The dependence of the scalar field evolution on the values of the cosmological constant $\Lambda$ and the Gauss-Bonnet coupling $\alpha$ is investigated. For Gauss-Bonnet and Gauss-Bonnet-de Sitter black holes, at asymptotically late times either power-law or exponential tails dominate, while for Gauss-Bonnet-anti-de Sitter black hole, the quasinormal modes govern the scalar field decay at all times. The power-law tails at asymptotically late times for odd-dimensional Gauss-Bonnet black holes does not depend on $\alpha$, even though the black hole metric contains $\alpha$ as a new parameter. The corrections to quasinormal spectrum due to Gauss-Bonnet coupling is not small and should not be neglected. For the limit of near extremal value of the (positive) cosmological constant and pure de Sitter and anti-de Sitter modes in Gauss-Bonnet gravity we have found analytical expressions.

Figure 1

Field decay in the Gauss-Bonnet black holes, for $d=5$ and $d=7$. It is observed a quasinormal mode dominated region. Asymptotically, the field decays as a power-law tail (dashed lines). The parameters in this graph are $\alpha =1$, $\mu =1.0$ and $\ell =0$.

Figure 2

Power-law tails in the Gauss-Bonnet black holes. The estimated power-law are $R_{\ell }\propto t^{-5.076}$, $R_{\ell }\propto t^{-5.082}$ and $R_{\ell }\propto t^{-5.068}$ for $\alpha =0$, $\alpha =5$ and $\alpha =10$ respectively. The predicted power for $\alpha =0$ is $-5$. The parameters in this graph are $d=7$, $\mu =1.0$ and $\ell =0$.

Figure 3

Graph of $\mathrm{Re}(\omega {)}^2$ as a function of $\ell (\ell +d-3)$, in the near-extreme limit positive $\Lambda$ limit. The bullets are the values calculated from the time-evolution profiles, and the dashed lines are values obtained from the expression (4b). The parameters in this graph are ${\kappa }_{+}=0.01$, $\alpha =1$, $\mu =1.0$, and the differences between the analytical and numerical results are under $2\%$.

Figure 4

Field decay in the Gauss-Bonnet-de Sitter black holes, with $\ell =0$. A quasinormal mode dominated region is observed (above), and asymptotically the field decays to a constant (below). The fundamental mode, calculated with the WKB method and directly from the characteristic data are $0.8356-0.2935i$ and $0.8011-0.2608i$. The parameters in this graph are $d=6$, $\alpha =3.0$, $\Lambda =0.1$ and $\mu =1.0$.

Figure 5

Field decay in the Gauss-Bonnet-de Sitter black holes, for $d=5,6,7,8$. It is observed a quasinormal mode dominated region. Asymptotically, the field decays as an exponential tail. The parameters in this graph are $\alpha =0.1$, $\mu =1.0$ and $\ell =1$.

Figure 6

Field decay in the Gauss-Bonnet anti de Sitter black holes, for several values of the Gauss-Bonnet coupling. It is observed a quasinormal mode dominated region. Asymptotically, the field decays in quasinormal modes. The parameters in this graph are $d=5$, $\mu =1.0$ $\Lambda =-0.1$ and $\ell =0$.