How do scalaron modes of Schwarzschild black holes behave in Bondi-Sachs f(R) gravity?

The scalaron spectrum depends only on mu M

The paper studies linear perturbations of a Schwarzschild black hole in metric f(R) gravity, written in the Bondi-Sachs characteristic formulation. Expanding the action as f(R) = f1 R + (1/2) f2 R^2 + O(R^3) and linearising about the Ricci-flat Schwarzschild background, the trace equation reduces to a massive scalar equation for the Ricci scalar perturbation R, with an effective mass squared mu^2 = f1/(3 f2) = f'(0)/(3 f''(0)). The finite, non-tachyonic scalar sector adopted here requires f1 > 0 together with f2 > 0.

The key structural result is a degeneracy: after dividing the linearised vacuum system by the nonzero factor f1, the homogeneous equations depend on the theory only through the ratio f2/f1 = 1/(3 mu^2). Coefficients fk with k >= 3 first contribute beyond linear order, so a3, a4, ... do not modify the linear scalar quasinormal or quasi-bound frequencies on the Ricci-flat Schwarzschild background. At fixed geometric Schwarzschild mass M, two analytic models with the same value of f'(0)/f''(0) therefore share the same value of mu M and the same scalar spectrum, so that 2M omega_sc^(ln)[fA] = 2M omega_sc^(ln)[fB]. The authors describe this as a degeneracy of the homogeneous linear spectral problem, not an all-order equivalence of the underlying theories, and note it could be lifted on backgrounds with nonzero or non-constant Ricci curvature.

The tensor master equation keeps its GR form

The linearised Bondi-Sachs system yields two independent master equations: one for the scalar perturbations, encoded in the spin-0 variable R, and one for the gravitational perturbations, encoded in the spin-2 variable J. The trace equation is already a closed second-order ODE for R and directly supplies the scalar master equation. The tensor equation is less direct: the authors eliminate the auxiliary variables U, U,r and U,rr from three of the linearised equations, treating J, beta, R and their radial derivatives as known source functions, and show that the resulting tensor master equation takes the same form as in general relativity.

This matters for how the two sectors should be read. The scalar channel carries the extra degree of freedom of metric f(R) gravity and is where the mu M dependence lives; the tensor channel, at this linear order on a Ricci-flat Schwarzschild background, does not acquire a new f(R)-specific form. The paper uses the tensor recurrence only as a numerical consistency check of the continued-fraction solver and its Nollert termination, not as a physical analysis channel.

Quasinormal and quasi-bound branches are distinct conditions

The paper treats two spectral families that are easy to conflate. Quasinormal modes are the characteristic damped oscillations of a perturbed black hole, with complex frequencies fixed by the background geometry and field content. Quasi-bound states are distinguished from outgoing quasinormal modes by a spatially decaying condition at infinity. For the class of metric f(R) models considered here, the linear scalaron obeys a massive scalar equation, and the theory dependence enters through its effective mass mu.

Both branches are computed in the frequency domain using Leaver's continued-fraction method, with the scalar quasinormal and quasi-bound conditions expressed as branch-dependent Wronskians and implemented through continued-fraction equations. The authors report that the quasi-bound damping rates approach Detweiler's approximation at small coupling, recovering a smooth connection between the continued-fraction solutions and the known analytic limit. The computed spectra are compared against independent literature benchmarks.

Characteristic evolution independently recovers the frequencies

To complement the frequency-domain analysis, the authors evolve the scalaron perturbation equations with a fourth-order convergent characteristic scheme on a finite radial domain, using the Bondi-Sachs hierarchy and analysing the response at finite extraction radii. The characteristic approach foliates spacetime by outgoing null hypersurfaces and transports data to future null infinity, which the authors describe as a natural setting for radiation because null hypersurfaces align with the characteristics of the field equations.

Crucially, the initial-worldtube evolution does not impose an outgoing or decaying spectral branch at the outer edge, unlike the frequency-domain two-point boundary-value problems. Modal contributions are instead identified from their persistence across matrix-pencil stability scans, with the continued-fraction frequencies introduced only after the time-domain selection. The representative convergence tests recover the expected fourth-order behaviour, and the fundamental quasinormal frequencies extracted across the benchmark set agree with the continued-fraction values within the extraction-stability scales of the accepted clusters. For the representative finite-mass calculation, the extracted pole also remains robust under variations of the numerical setup and initial data.

One waveform can carry both mode families

For a selected finite-mass case with l = 2 and mu M = 0.575, the paper demonstrates extraction of both the fundamental quasinormal mode and the quasi-bound state from separate time windows of the same waveform at one extraction radius, using a single initial-worldtube evolution. Early-time fitting windows capture the quasinormal response, while later, longer windows capture the quasi-bound contribution. Neither spectral branch is imposed at the finite outer boundary.

The authors present this as evidence that the characteristic evolution and extraction procedure can resolve both the relatively rapid quasinormal damping and the much slower quasi-bound decay within a single waveform. This is a demonstration of feasibility for the chosen case, not a claim that the same separation works uniformly across the parameter space.

Quasi-bound damping rates are harder to extract than frequencies

The time-domain quasi-bound results illustrate the limits of the current extraction procedure. At mu M = 0.25, both the oscillation frequencies and damping rates are recovered for l = 0 and l = 1, whereas for l = 2 only the oscillation frequency is constrained and the damping rate remains unresolved. The authors draw the explicit lesson that a recoverable quasi-bound oscillation does not necessarily imply a reliable measurement of its decay rate.

Two further qualifications sit alongside these numbers. The reported cluster MADs measure sensitivity to the extraction choices, not total numerical uncertainty, and should be interpreted accordingly. The robustness tests concern numerical pole recovery for a fixed evolution equation; they do not constitute a pseudospectral analysis of perturbations to the underlying operator. The tensor continued-fraction check is likewise a numerical validation: for l = 2, the n = 10 result differs from a recent high-precision value by 3.44874 x 10^-15 with a conservative truncation-stability measure of 1.37346 x 10^-12, while the n = 29 and n = 59 roots are treated as deep-overtone checkpoints rather than precision benchmarks.

The degeneracy is linear-order and Ricci-flat specific

The mu M degeneracy holds for the analytic models considered here, on a Ricci-flat Schwarzschild background, at linear order in perturbation theory. The authors state that the specific degeneracy based on the values of the theory at R = 0 can be lifted on backgrounds with nonzero or non-constant Ricci curvature, and that a change to another Ricci-flat background does not by itself reintroduce dependence on fk with k >= 3 at linear order. Higher-order perturbations and backgrounds with non-constant Ricci curvature could probe derivatives of f(R) that are absent from the current eigenvalue problem.

The mode decoupling used throughout is also tied to spherical symmetry: the linearised equations decouple for each (l, m) mode on Schwarzschild, whereas for rotating Kerr black holes, which are only axisymmetric, a decomposition in this form generically mixes different l. The authors identify rotating Kerr black holes as a worthwhile extension, along with waveform extraction adapted to massive fields and the question of whether long-lived scalaron configurations can source a delayed tensor response through scalaron-tensor coupling beyond linear order.

Read the benchmark set as a consistency check

The paper's most decision-relevant structure is the pairing of two independent routes to the same scalar spectrum. The frequency-domain route imposes explicit spectral conditions through branch-dependent Wronskians and continued fractions; the time-domain route imposes no such branch condition at the finite outer boundary and instead selects modes by persistence across matrix-pencil scans. Agreement between them, within the extraction-stability scales of the accepted clusters, is the paper's central quantitative check on the fundamental quasinormal frequencies.

The practical priority that follows is to treat the two branches differently when reading the results. Fundamental quasinormal frequencies are recovered consistently across the benchmark set and are the better-supported output. Quasi-bound oscillation frequencies are recovered at mu M = 0.25 for l = 0, 1 and 2, but the corresponding damping rate is only resolved for l = 0 and l = 1, so a quasi-bound frequency alone should not be read as a measured decay rate. And because the linear scalar spectrum depends on the theory only through mu M, any two analytic f(R) models sharing f'(0)/f''(0) are indistinguishable in this linear, Ricci-flat setting, which is exactly the degeneracy that higher-order perturbations or non-constant-Ricci-curvature backgrounds would need to break.

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