Scalaron Quasinormal and Quasi-Bound Modes of Schwarzschild Black Holes in the Bondi-Sachs Formulation of Metric $f(R)$ Gravity
We study scalaron quasinormal modes and quasi-bound states of Schwarzschild black holes directly within the Bondi-Sachs formulation of metric $f(R)$ gravity, combining frequency-domain analyses with characteristic time evolution. Starting from the linearised characteristic equations, we obtain the massive scalar master equation and show that the tensor master equation takes the same form as in general relativity. For the analytic models considered here, the linear scalar spectra depend on the $f(R)$ theory only through the dimensionless combination $μM$. We compute both quasinormal and quasi-bound spectral branches using Leaver's continued-fraction method and compare them with independent literature benchmarks. To complement the frequency domain analysis, we evolve the scalaron perturbation equations using a fourth order convergent characteristic scheme. Applying matrix-pencil stability scans to the resulting waveforms, we independently recover fundamental quasinormal frequencies consistent with the continued-fraction results within the measured extraction-stability scales. Finally, for a selected finite-mass case, we demonstrate the extraction of both oscillation frequencies and damping rates of the fundamental quasinormal mode and quasi-bound state from separate time windows of the same waveform at one extraction radius, without imposing either spectral branch at the finite outer boundary.