Quasinormal modes and their anomalous behavior for black holes in $f(R)$ gravity

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 holes 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 behaviour 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 behaviour 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.