Abstract
We study the propagation of scalar fields in the background of an asymptotically de Sitter black hole solution in f(R) gravity. The aim of this work is to analyze in modified theories of gravity the existence of an anomalous decay rate of the quasinormal modes (QNMs) of a massive scalar field which was recently reported in Schwarzschild black hole backgrounds, in which the longest-lived modes are the ones with higher angular number, for a scalar field mass smaller than a critical value, while that beyond this value the behavior is inverted. We study the QNMs for various overtone numbers and they depend on a parameter beta which appears in the metric and characterizes the f(R) gravity. For small beta , i.e. small deviations from the Schwarzschild–dS black hole the anomalous behavior in the QNMs is present for the photon sphere modes, and the critical value of the mass of the scalar field depends on the parameter beta while for large beta , i.e. large deviations, the anomalous behavior and the critical mass does not appear. Also, the critical mass of the scalar field increases when the overtone number increases until the f(R) gravity parameter beta approaches the near extremal limit at which the critical mass of the scalar field does not depend anymore on the overtone number. The imaginary part of the quasinormal frequencies is always negative leading to a stable propagation of the scalar fields in this background.
Highlights
One of the first modifications of the Einstein Lagrangian density was proposed in [19]
It was found that the same behavior is present in the Schwarzschild–de Sitter background, i.e., the absolute values of the imaginary part of the quasinormal frequencies (QNFs) decay when the angular harmonic numbers increase if the mass of the scalar field is smaller than the critical mass, and they grow when the angular harmonic numbers increase, if the mass of the scalar field is larger than the critical mass
For larger values of the parameter Mβ i.e large deviations from Schwarzschild–dS black hole, the quasinormal modes (QNMs) show a different behavior, where the anomalous behavior and the critical mass are not present, this cut off point could numerically occur for Mβ ≈ 0.10; see Fig. 2, where the QNFs present an inverted behavior with respect to Mβ for massless scalar field
Summary
In order to obtain the QNMs of scalar perturbations in the background of the metric (3) we consider the Klein–Gordon equation,. In order to solve numerically the differential equation (9) by using the pseudospectral Chebyshev method [82], it is convenient to perform the change of variable y = (r − rH )/(r − rH ). By inserting the above ansatz for R(y) in Eq (14), it is possible to obtain an equation for the function F(y). A system of algebraic equations is obtained, and it corresponds to a generalized eigenvalue problem, which is solved numerically to obtain the QNFs. It is worth mentioning that, for β = 0, the spacetime is described by the Schwarzschild–de Sitter black hole. It was shown that the frequencies all have a negative imaginary part, which means that the propagation of scalar field is stable in this background, and the presence of the cosmologicalconstant
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