[Paper Review] Quasinormal modes of a black hole with a cloud of strings in Einstein-Gauss-Bonnet gravity

J. P. Morais Graça, Ines G. Salako+1 more|arXiv (Cornell University)|Apr 16, 2016

Black Holes and Theoretical Physics23 references3 citations

TL;DR

This paper calculates quasinormal modes (QNMs) of a black hole with a cloud of strings in Einstein-Gauss-Bonnet gravity using the WKB approximation. It reveals that increasing the string cloud parameter $η^2$ reduces both the real and imaginary parts of QNMs, slowing wave decay and lowering emission frequency, while also inducing near-isospectrality between scalar and tensor modes at high $η^2$, and shows that the critical $α_0$ value for QNM stability depends on the string cloud energy.

ABSTRACT

The quasinormal modes for a scalar field in the background spacetime corresponding to a black hole, with a cloud of strings, in Einstein-Gauss-Bonnet gravity, and the tensor quasinormal modes corresponding to perturbations in such spacetime, were both calculated using the WKB approximation. In the obtained results we emphasize the role played by the parameter associated with the string cloud, comparing them with the results already obtained for the Boulware-Deser metric. We also study how the Gauss-Bonnet correction to general relativity affects the results for the quasinormal modes, comparing them with the same background in general relativity.

Motivation & Objective

• To investigate the impact of a cloud of cosmic strings on quasinormal modes (QNMs) in higher-dimensional Einstein-Gauss-Bonnet gravity.
• To analyze how the Gauss-Bonnet parameter $\alpha$ modifies QNM spectra compared to general relativity.
• To compare scalar and tensor QNM behavior, particularly assessing whether isospectrality emerges under specific conditions.
• To determine the dependence of the critical $\alpha_0$ value (where QNM damping transitions) on the string cloud energy parameter $\eta^2$.
• To explore the interplay between modified gravity and extended matter configurations in black hole perturbation theory.

Proposed method

• Employed the 3rd-order WKB approximation to compute quasinormal modes for scalar and tensor perturbations in a black hole spacetime with a cloud of strings.
• Used the Einstein-Gauss-Bonnet action with a dimensionally extended Lovelock framework, focusing on $d=7$ and $d=9$ spacetimes.
• Constructed the effective potential for scalar and tensor perturbations using the metric derived from the cloud of strings and Gauss-Bonnet corrections.
• Varied the string cloud parameter $\eta^2$ (related to string energy density) and the Gauss-Bonnet coupling $\alpha$ to analyze their effects on QNM frequencies and damping rates.
• Compared results with the Boulware-Deser metric ($\alpha \neq 0, \eta^2 = 0$) and Schwarzschild limit ($\alpha = 0, \eta^2 = 0$) for reference.
• Analyzed the behavior of QNM spectra as $\eta^2$ increased, particularly focusing on convergence between scalar and tensor modes.

Experimental results

Research questions

• RQ1 How does the presence of a cloud of strings affect the quasinormal mode frequencies and damping rates in Einstein-Gauss-Bonnet gravity?
• RQ2 How does the Gauss-Bonnet coupling parameter $\alpha$ influence the stability and spectral characteristics of QNMs compared to general relativity?
• RQ3 Does isospectrality emerge between scalar and tensor quasinormal modes in the presence of a string cloud, and if so, under what conditions?
• RQ4 How does the critical $\alpha_0$ value—where QNM damping rate transitions from decreasing to increasing—depend on the string cloud energy parameter $\eta^2$?
• RQ5 What is the interplay between the string cloud energy and the higher-curvature corrections in shaping the gravitational wave emission profile of black holes?

Key findings

• Increasing the string cloud parameter $\eta^2$ reduces both the real and imaginary parts of quasinormal modes, indicating lower emission frequency and slower decay rates.
• For $d=7$, $l=2$, $n=0$, the real part of the QNM frequency decreases from 2.088 to 0.982 and the imaginary part from 0.515 to 0.215 as $\eta^2$ increases from 0.1 to 30 at $\alpha = 0.1$.
• The tensor QNM frequency for $d=7$, $l=2$, $n=0$ decreases from 2.223 to 1.023 and the damping rate from 0.435 to 0.181 as $\eta^2$ increases from 0.1 to 30 at $\alpha = 5.0$.
• As $\eta^2$ increases, scalar and tensor quasinormal modes converge, approaching isospectrality, especially when $\eta^2 \gg \alpha$.
• The critical $\alpha_0$ value, where the imaginary part of QNMs stops decreasing and begins increasing, increases with $\eta^2$, indicating that higher string cloud energy delays the onset of instability in the damping profile.
• The results differ from prior work [11] due to a distinct parametrization of $\alpha$, with the authors justifying their choice based on consistency with the tensor potential in [27].