Close Hyperbolic Encounters In f(R) Gravity

Alvaro de la Cruz-Dombriz; Jenna Bruton; Peter Dunsby

REVIEW 4 major objections 4 minor 119 references

Close Hyperbolic Encounters In f(R) Gravity

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper argues that close hyperbolic black-hole encounters in f(R) gravity emit scalar gravitational waves with amplitudes comparable to the tensor modes, producing time delays of up to roughly a hundred seconds, making such bursts a…

desk verdict A recognizable extension of existing f(R) scalar-quadrupole work to hyperbolic encounters, with clean waveform algebra but a load-bearing mistake: the scalar 'radiation' is computed with a static Yukawa Green's function, so the advertised order-unity amplitude ratios do not follow. read the letter →

arxiv 2506.20787 v1 pith:4CS3RQ67 submitted 2025-06-25 gr-qc astro-ph.CO

Jenna Bruton , Peter Dunsby , Alvaro de la Cruz-Dombriz This is my paper

classification gr-qcastro-ph.CO MSC 83C3583D05 PACS 04.30.-w04.50.Kd

keywords closehyperbolicencountersf(R)gravityscalargravitationalwavesscalar-to-tensoramplituderatiowavetimedelayquadrupoleradiationorbitalprecessionmodified

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper derives the gravitational-wave signal from black holes on unbound hyperbolic orbits—close hyperbolic encounters—in $f(R)$ gravity, where the theory's extra scalar polarization mode radiates alongside the two tensor modes. The central output is a set of quadrupole waveforms for both non-precessing and precessing orbits, from which the authors compute two observables: the arrival-time delay between scalar and tensor bursts, and the scalar-to-tensor amplitude ratio. Across scalar masses typical of viable $f(R)$ models and for source distances representing galactic centers, globular clusters, and primordial black-hole populations, the paper finds time delays from about $10^{-26}$ s to about $10^2$ s and scalar-to-tensor ratios of order unity (roughly 0.5 to 0.7). It concludes that close hyperbolic encounters are likely to generate detectable scalar-mode radiation for all parameter combinations considered, and that orbital eccentricities near $e \approx 1.4$ with low precession give the best prospects. If this holds, unbound black-hole scattering events would supply a new, strong-field test of modified gravity that does not rely on binary coalescences.

What carries the argument

The load-bearing object is the scalar quadrupole radiation formula, equation (2.48): $\Phi = \frac{4G\,e^{-m_S R}}{R c^2} M + \frac{2G\,e^{-m_S R}}{R c^4}\ddot{Q}$, which comes from the retarded Green's function of a massive scalar field in equation (2.36). Here $m_S$ is the mass of the scalar mode, $M$ is the total mass, $Q$ is the trace of the quadrupole moment, and the exponential is a Yukawa-like cutoff. Feeding this with Newtonian hyperbolic orbits, and with orbits modified by a precession parameter $\alpha$, yields the waveform expressions (3.10) and (3.17). The $\alpha$-dependent factor $(\alpha-1)^2$ in the scalar formula is what makes the scalar mode decrease as precession increases, and the $r^{-1}e^{-m_S R}$ structure determines both the amplitude ratio and the time-delay estimates. The paper notes in a footnote that different gravitational potentials give different precession relations, and it restricts itself to precession values $\alpha < 3/4$, where the periastron stays outside the Schwarzschild radius.

What would settle it

Re-evaluate the scalar amplitude using the exact retarded Green's function of equation (2.29) (or equation 2.36) at the specific $(m_S, R)$ pairs in Tables I and II, without the constant-propagation-speed approximation. At the galactic-center pair ($R = 10\,\mathrm{kpc}$, $m_S = 10^{-22}\,\mathrm{eV}/c^2$) the factor $e^{-m_S R}$ is astronomically small, so the scalar-to-tensor ratio would drop far below the tabulated 0.5–0.7; reproducing the tabulated values would require a derivation of why the exponential drops out for radiative waves, or a corrected radiative formula.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the scalar gravitational-wave mode of $f(R)$ gravity is not negligible in close hyperbolic encounters: the scalar strain peaks at periastron like the tensor modes, and the averaged scalar-to-tensor amplitude ratio $R_{ST}$ ranges between about 0.51 and 0.70 for eccentricities $1.1 \le e \le 1.7$ and precession parameters $0 \le \alpha \le 0.1$, values that are large compared with earlier $f(R)$ expectations of $10^{-3}$ to $10^{-1}$. The authors also derive time delays between the tensor and scalar arrivals that can be as large as $1.2 \times 10^2$ s for the largest scalar mass scale they consider at cosmological distances, and they map how these observables depend on eccentricity and precession: higher eccentricity suppresses all modes, while higher precession suppresses the scalar mode specifically, through the $(\alpha-1)^2$ prefactor in the scalar quadrupole formula. The paper states that these results suggest close hyperbolic encounters are likely to generate detectable $f(R)$ scalar-mode radiation for all combinations of parameter values considered.

Load-bearing premise

The whole detectability prediction rests on one approximation: in the scalar quadrupole formula, the retarded massive-scalar response is replaced by a static Yukawa factor $e^{-m_S R}$, with a constant propagation speed and a source that does not depend on the scalar speed. If that factor really applies across the distances used in the tables, the galactic-center and globular-cluster scalar amplitudes would be exponentially tiny, not comparable to tensor waves; if it does not apply to radiating waves, then the formula used to compute those amplitudes is not the correct radiative formula.

Editorial extensions

If this is right

If the scalar quadrupole formula is correct, close hyperbolic encounters add a new class of burst sources whose scalar-mode amplitude rivals the tensor strain, so detectors that see these bursts could measure $f(R)$ deviations from general relativity without waiting for a merger.
The predicted time delays, up to about $10^2$ s at cosmological distances for the heaviest scalar masses considered, give an observable signature that could separate a massive scalar mode from a luminal tensor mode in the same event.
The finding that the scalar-to-tensor ratio peaks near $e \approx 1.4$ and drops with increasing precession gives an orbital-shape preference: searches should target intermediate-eccentricity, low-precession encounters.
Higher eccentricity suppresses all three polarization amplitudes, making detection most likely for encounters with $e$ close to $1.1$\textendash$1.3$, where the interaction time is longer.
The three source environments—galactic center, globular clusters, and primordial black-hole populations—yield a wide spread of expected time delays, so a non-detection in one environment but detection in another could already constrain the scalar mass scale.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the paper leaves implicit is to convert the time-domain waveforms into a frequency-domain power spectrum; if the radiation is dominated by low frequencies, space-based observatories rather than ground-based ones would be the natural follow-up.
Because the same quadrupole machinery works for any theory with one massive scalar polarization, the waveform templates could be adapted to scalar-tensor and other modified-gravity models with minimal changes, giving a common framework for burst searches.
The paper's $e \approx 1.4$ optimum is a population-level prediction: if $f(R)$ scalar bursts exist, the detected close-encounter population should show a detection-efficiency peak at intermediate eccentricities and low precession, a statistical signature that general-relativity-only templates would not produce.
The scalar-mode memory effect and total radiated energy, both flagged as future work, would provide independent observables; a memory measurement could distinguish the massive scalar's long-range behavior from a massless one.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a legitimate new combination—f(R) scalar-mode waveforms for close hyperbolic encounters, both non-precessing and precessing—and the algebraic machinery in Section III is mostly coherent. The authors are honest about where they import the scalar quadrupole formula [60] and the precessed orbit parameterization [22]. The precessing waveforms (3.15)–(3.17) and the parameter scans are new, and the time-delay table follows from the dispersion relation. The citation pattern is fine; the main formula is attributed to the right source, and self-citations are only background. That is real credit and should be said first.

The soft spot is not in the algebra but in the physics of Eq. (2.47). The scalar amplitude is obtained by assuming the source is independent of cosh ξ, which reduces the retarded massive Green's function to a static Yukawa e^(−m_S R). A close hyperbolic encounter is not static: the source varies on the orbital timescale, and for the frequencies and masses considered here the radiative part of the massive Green's function propagates as 1/R with no exponential suppression. The e^(−m_S R) factor in (3.17) is therefore not the radiative amplitude. This is not a minor detail: the paper's central observable, the scalar-to-tensor ratio, is order unity in Table II precisely because the exponential factor is not applied. Indeed, the RST values are nearly identical across the three mass columns, even though plugging the stated (m_S, R) into (3.17) for the galactic-centre row gives e^(−1.5×10^5) suppression. Either the exponential was silently dropped or Eq. (2.47) is being used outside its validity; either way, the central detectability claim is unsupported.

There are two smaller issues. The paper asserts detectability without SNR estimates or sensitivity curves, which is stronger than the calculations justify. And the fiducial parameters used for Fig. 2 do not satisfy the paper's own eccentricity relation: with M = 60 M☉, v0 = 0.007c and b = 3.39×10^(−14) Mpc, Eq. (3.2) gives e ≈ 1.15, not e = 1.3.

Who should read this? Someone working on modified-gravity burst signals or on the distinction between static Yukawa fields and massive radiative Green's functions. The paper is worth a serious referee: the formalism is recognizable, the authors know the existing literature, and the flaw is central but potentially fixable. I would send it to review with instructions to redo the scalar radiation amplitude using the full retarded integral, and to re-examine the detectability claims afterwards. As submitted, the conclusion that CHEs produce detectable f(R) scalar modes does not follow.

Referee Report

4 major / 4 minor

Summary. The paper studies gravitational-wave emission from close hyperbolic encounters (CHEs) of black holes in f(R) gravity. It derives quadrupole-order waveforms for the tensor polarizations and the scalar longitudinal polarization, first for Newtonian hyperbolic orbits and then for orbits modified by a precession parameter. Using masses associated with Starobinsky, Hu-Sawicki, and Exponential f(R) models, it estimates scalar-to-tensor time delays and amplitude ratios for galactic-centre, globular-cluster, and primordial-black-hole distances, and concludes that CHEs are a promising probe of f(R) gravity.

Significance. The topic is timely: CHEs are plausible burst sources, and the f(R) scalar polarization is a concrete beyond-GR prediction. The paper's main conceptual contribution would be a first systematic treatment of scalar-mode bremsstrahlung from unbound encounters, with explicit waveform formulas and tabulated observables. However, the central quantitative claims rest on an incorrect treatment of the massive-scalar radiative Green's function, and the tables are internally inconsistent with the stated formulas. As it stands, the detectability conclusions are not supported; the useful parts are the tensor waveform algebra and the outline of a framework that could be repaired with a proper radiative derivation.

major comments (4)

[II.B.2, Eq. (2.47)] Equation (2.47) is obtained from the retarded solution (2.36) by assuming that T is independent of cosh ξ and that the scalar propagation speed cS is constant. This is the static/Yukawa limit, not the radiative limit. For a CHE source varying on the orbital timescale, the relevant Fourier components satisfy ω ≫ mS c^2/ħ for all masses in Tables I and II, so the massive Green's function (2.25) gives a propagating wave with amplitude proportional to 1/R and phase kR with k = sqrt(ω^2/c^2 - mS^2 c^2/ħ^2), with no exponential factor e^{-mS R}. Consequently the scalar waveforms (3.10) and (3.17), and every observable derived from them, do not follow from the equations presented.
[Table II, Eqs. (3.17) and (5.1)] Table II is internally inconsistent with the stated scalar amplitude. For mS = 10^-22 eV/c^2 and R = 10 kpc, the factor e^{-mS R} equals e^{-1.5×10^5}; for mS = 10^-24 eV/c^2 and R = 100 kpc, it equals e^{-1.5×10^4}. The table reports RST ≈ 0.69-0.70 in both columns. If the exponential factor is retained, these entries should be astronomically small; if it is omitted, then Eq. (2.47) is not being used and the tabulated values are not derived from the paper's formulas. Either way, the order-unity RST values that support the detectability claim are unsupported.
[§5, Eq. (5.2)] The definition of RST is incomplete. It is written as <|Q|>/<|h|>, but the relation of these averages to the scalar and tensor strains in Eqs. (3.15)-(3.17) is not shown, and it is unclear how the constant M/R term in Eq. (2.48), which is not radiative, is treated. Without this derivation the reader cannot verify the tabulated ratios, especially their near-independence of mS.
[§IV, Eqs. (4.7)-(4.8)] The time-delay estimates assume a monochromatic wave at f = 1 Hz and a single group velocity. A CHE burst has a broadband spectrum, and for a massive scalar the group delay is frequency dependent; for frequencies characteristic of the encounters considered here, the delay can be orders of magnitude larger than the f = 1 Hz estimate. The detectability statement should be based on a frequency-dependent arrival-time analysis over the relevant signal band.

minor comments (4)

[Section IV heading] The heading 'CONSTRAINS ON SCALAR MASSIVE MODE TIME DELAY' should read 'CONSTRAINTS ON SCALAR MASSIVE MODE TIME DELAY'.
[Eq. (2.21)] The Fourier representation of the tensor Green's function appears to contain misplaced 1/√π^3 factors; the equation should be cleaned up.
[Fig. 5 caption] The caption states that the cross-polarization strain is scaled by Gµv0^2/R, but Eq. (3.9) has prefactor Gµv0^2/(R c^4); the c^4 is missing.
[Eq. (3.12) and surrounding text] The statement that valid precession values satisfy α < 3/4 is loose: the condition Rs < rmin gives α < 3/[2(e+1)], which depends on eccentricity and is tighter for large e. The text should state this dependence explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: scalar-mode inputs (mS) come from external f(R) models and observables are computed from stated formulae.

full rationale

The paper's derivation chain uses as inputs the f(R) action (2.1), the scalar-mass definition (2.16), external mS values from Starobinsky, Hu-Sawicki and Exponential models, and orbital parameters. The tensor and scalar waveforms are obtained by substituting a Newtonian or precessing hyperbolic trajectory into the quadrupole formulae (2.41)-(2.43) and (2.48). No parameter is fitted to the quantity it is later claimed to predict: time delays in Table I follow directly from the dispersion relation (4.3) and the distance R, while the RST values in Table II follow from the amplitude ratio (5.1) applied to the derived strains. The scalar formula (2.47) is an explicitly stated approximation drawn from Ref. [60] (Inagaki and Taniguchi), an external work rather than a self-citation; the assumptions of constant scalar propagation speed and a source independent of cosh xi are stated openly in the text. Self-citations [38,41,61,62] appear only for background f(R) perturbation theory and are not load-bearing for the observational conclusions. The possible physical objection that Eq. (2.47) replaces a retarded massive Green's function with a static Yukawa factor, or that Table II appears not to include e^{-mS R}, is a correctness and validity concern rather than circularity: it does not make the output equivalent to the input by construction. No uniqueness theorem is imported from the authors' prior work, and no known result is merely renamed. Accordingly, the circularity burden is not met, and the paper is assigned score 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The ledger is small: no new particles or forces are introduced; the scalar mode is the standard f(R) longitudinal polarization. The key burden falls on the choice of mS (a free parameter of each f(R) model) and on the static-Yukawa approximation used for the scalar Green's function. The RST table effectively drops the Yukawa factor, which is the main inconsistency.

free parameters (2)

scalar mode mass mS = case values 10^-22, 10^-24, 10^-33 eV/c^2
Chosen by hand as representative values from Starobinsky, Hu-Sawicki and Exponential f(R) models; every observable in Tables I-III depends on it.
effective gravitational constant G/f'(0) = G with f'(0) implicitly set to 1
The amplitude formulas use G/f'(0), but f'(0) is never specified; numerical amplitudes implicitly set it to 1, which is a choice that would be constrained by solar-system tests.

assumptions (5)

domain assumption Linearization around Minkowski with R=0, f(R)=f(0)+f'(0)δR
Section II, Eqs. (2.4)-(2.6). Assumes background curvature is negligible; cosmologically motivated f(R) models have nonzero background R, and environmental/chameleon effects are absent.
domain assumption Non-relativistic quadrupole approximation and distant-observer limit |Δx|~R
Section II.B, Eq. (2.37). Defensible for v0=0.007c, but periastron values near the Schwarzschild radius, corresponding to α close to 3/4, would require post-Newtonian or numerical-relativity corrections.
domain assumption Scalar Green's function approximated by a static Yukawa e^{-mS R}/R with constant cS and T independent of cosh ξ
Eqs. (2.36)-(2.47). This is the key fragility: for ω>mS the retarded propagator oscillates as e^{ikR}/R rather than decaying exponentially, and the paper does not justify the static limit for gravitational-wave frequencies.
domain assumption Precession parameter α = 3G^2M^2/(c^2 L^2) with orbit r_pr(φ), Eqs. (3.11)-(3.12), taken from [22]
Imports a GR precession relation into f(R); the paper itself footnotes that different gravitational potentials yield different precession relations.
standard math Trace source decomposition T = -T^0_0 + T^i_i and ∫ T^0_0 d^3x = M c^2
Used to derive Eq. (2.48). Standard, but the monopole term is non-radiative for constant M and is retained in the scalar waveform.

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Cite this review

Pith. "Pith review of Close Hyperbolic Encounters In f(R) Gravity." pith.science (2026). https://pith.science/paper/4CS3RQ67

@misc{pith2026250620787,
  author       = {Pith},
  title        = {Pith review of: Close Hyperbolic Encounters In f(R) Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4CS3RQ67}},
  note         = {Machine review of arXiv:2506.20787}
}

read the original abstract

We explore the dynamics and gravitational-wave emission from black hole pairs on unbound orbits undergoing close hyperbolic encounters (CHEs) in dense astrophysical environments. While General Relativity predicts gravitational Bremsstrahlung radiation occurring at periastron, the contribution of the scalar gravitational-wave mode in fully general f(R) gravity remains largely unexplored. We characterize the f(R) scalar mode gravitational radiation for both non-precessing and precessing hyperbolic orbits, identifying potential detection signatures for advanced gravitational wave observatories. By systematically varying orbital precession and eccentricity, we examine their influence on the emitted gravitational waves. We derive potentially detectable time delay and scalar-to-tensor amplitude ratio estimates for representative astrophysical environments and determine optimal orbital configurations for the detection of f(R) scalar gravitational waves from hyperbolic encounters. Our results provide a theoretical framework for scalar-mode signals and observables, establishing CHEs as a promising probe of f(R) gravity with future detectors.

Figures

Figures reproduced from arXiv: 2506.20787 by the authors.

**Figure 3.** FIG. 3. Gravitational-wave strain of the scalar radiation [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 2.** FIG. 2. Gravitational-wave strain of the tensor mode po [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 9.** Figure 9: FIG. 9. Scalar strain Φ [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

**Figure 10.** Figure 10: FIG. 10. Taylor Expansion of Φ [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

**Figure 11.** Figure 11: FIG. 11. Variation of [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

**Figure 12.** Figure 12: FIG. 12. Percentage change of [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]

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(2.18) The Fourier representation of ˜hµν is given by ˜hµν(x, t) = 1√ 2π Z dk0 ˜hµν(x, k0) e−ik0t, (2.19) where k0 ≡ ω c

Solution of the tensor mode wave equation We begin by introducing the energy-momentum tensor as a source to (2.13) [64], □˜hµν = 8πG c4 Tµν , (2.17) where G is the effective f (R) gravitational constant defined as G/f ′(0), and introduce a Green’s function as the solution to this sourced wave equation, □GT(x) = δ(x) . (2.18) The Fourier representation of ...

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(2.23) To solve this wave equation we use the retarded Green’s function, GS, satisfying (□ − m2 S) GS(x) = δ(x)

Solution of the scalar mode wave equation Similarly, (2.15) is sourced by the trace of the energy-momentum tensor, (□ − m2 S)Φ = 8πG c4 T . (2.23) To solve this wave equation we use the retarded Green’s function, GS, satisfying (□ − m2 S) GS(x) = δ(x) . (2.24) Note that the inclusion of the mass mS in (2.24) differentiates this Green’s function from that ...

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This assumption leads to the simplification |∆x| ∼ |x| ≡ R, (2.37) Thus, to leading order, (2.22) can be expressed as ˜hij(x, t) ∼ 4G Rc4 Z T ij(x′, t− R) d3x′

Tensor mode quadrupole radiation We begin by integrating (2.22), assuming that the source and observer are significantly distant and that the source is non-relativistic. This assumption leads to the simplification |∆x| ∼ |x| ≡ R, (2.37) Thus, to leading order, (2.22) can be expressed as ˜hij(x, t) ∼ 4G Rc4 Z T ij(x′, t− R) d3x′ . (2.38) Using conservation...

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Scalar mode quadrupole radiation For the scalar mode formula, we begin with a variable change that simplifies the integration of the Bessel function in (2.36) [60], w ≡ mSR sinh ξ . (2.44) 5 It follows then that the second integration term be- comes Z ∞ 0 dξ mSJ1(mSR sinh ξ) = Z ∞ 0 dw mSJ1(w)p w2 + (mSR)2 , (2.45) with Z ∞ 0 dw mSJ1(w)p w2 + (mSR)2 = mS ...

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