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Axial quasi-normal modes of slowly rotating black holes in dynamical Chern-Simons gravity to second-order in spin and coupling

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Pith's one-line read Chern-Simons coupling shortens axial ringdown damping

desk verdict A useful next-step dCS ringdown calculation whose headline damping trend is probably right, but whose second-order coefficient omega22 rests on an unproven mode-truncation and is explicitly disputed. read the letter →

arxiv 2506.03600 v2 pith:ILPDYBOK submitted 2025-06-04 gr-qc hep-th

Tharaka Alapati , S. Shankaranarayanan (IIT Bombay) This is my paper
classification gr-qchep-th
keywords dynamicalChern-Simonsgravityquasinormalmodesslowlyrotatingblackholesaxialperturbationsringdowndampingtimemetricsecond-orderspincorrectionsparityviolation
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 computes the quasi-normal mode frequencies of the fundamental $n=0$, $l=m=2$ axial gravitational mode of a slowly rotating black hole in dynamical Chern-Simons (dCS) gravity, keeping corrections through second order in spin $\chi$ and second order in the dCS coupling $\alpha$. The central result is a fitted relation $M\omega = \omega_{00}+(\alpha/M^2)^2\omega_{02}+\chi(\omega_{10}+(\alpha/M^2)^2\omega_{12})+\chi^2(\omega_{20}+(\alpha/M^2)^2\omega_{22})$ whose dCS coefficients make the imaginary part more negative as $\alpha$ grows. In plain terms, increasing the dCS coupling lowers the oscillation frequency and shortens the damping time of the axial quadrupole mode, opposite to what earlier analytical work found for polar modes. A reader should care because this is a concrete, parity-dependent ringdown signature that distinguishes dCS gravity from general relativity and could be searched for in lower-mass merger remnants.

What carries the argument

The machinery is a metric-perturbation scheme in the Regge-Wheeler gauge applied to the slowly rotating dCS background, with a scalar-field perturbation included alongside the metric ones. Angular decomposition produces coupled radial equations for the primary axial $l=2$ mode and the polar $l\pm1$ modes; after using the ordering that back-reaction of $l\pm2$ axial modes enters only at $\mathcal{O}(\chi^4)$, the system truncates to eight first-order ODEs (six when the $l=1$ polar functions are dropped), solved by Runge-Kutta-Fehlberg integration with ingoing and outgoing wave boundary conditions and a determinant condition at a matching point. The load-bearing output is the fitted polynomial in $\chi$ and $(\alpha/M^2)^2$ for $M\omega$, which turns the numerical spectrum into a usable template.

What would settle it

Recompute the $n=0,l=m=2$ axial QNM keeping the full eight-equation system, with the $l=1$ polar functions $H_1^{1,m}$ and $K^{1,m}$ retained, at a representative point such as $\chi=0.1$, $\alpha/M^2=0.02$; if the resulting complex frequency differs from the six-equation value by more than the reported sub-percent accuracy, the fitted coefficients, especially $\mathrm{Im}(\omega_{22})$, would need to be revised.

Watch

Extended reading notes

Core claim

The paper claims that for the fundamental axial mode with $l=m=2$, dCS corrections at order $\alpha^2$ combine with spin corrections up to $\chi^2$ to shorten the ringdown: at fixed spin, $|\mathrm{Im}(\omega)|$ increases (damping time decreases) with $\alpha/M^2$, while rotation alone lengthens the damping time as in GR. This is expressed in the fitted formula of Eq. (46) with coefficients in Eq. (47), where the $\alpha^2$ terms $\omega_{02}=-0.769-0.389i$ and $\omega_{12}=-1.13-0.565i$ drive the faster damping, and the $\alpha^2\chi^2$ coefficient $\omega_{22}=-1.7+0.66i$ captures the coupled spin-coupling correction. The paper contrasts this with earlier analytical results on polar modes, which suggested dCS would increase the damping time, and interprets the difference as a parity-dependent effect of dCS gravity on black hole ringdowns.

Load-bearing premise

The load-bearing premise is that the $l=1$ polar perturbations can be set to zero by the gauge choice when solving for the $l=2$ axial mode, reducing the eight coupled equations to six; if those $l=1$ modes are physical in the rotating dCS background, the reported frequencies and fitting coefficients come from an incomplete system.

Editorial extensions

If this is right

  • At fixed spin, the axial $l=m=2$ ringdown damps more quickly when dCS coupling is nonzero, so the damping time itself carries a parity-dependent modified-gravity signature.
  • The fitted formula reproduces Kerr QNM frequencies in the $\alpha=0$ limit to within one percent for $\chi\le0.15$ and to sub-percent accuracy up to $\chi\le0.4$, extending earlier first-order-in-spin calculations.
  • The framework yields explicit $\mathcal{O}(\chi^2,\alpha^2)$ templates for the dominant quadrupole ringdown, making dCS tests with lower-mass mergers such as GW230529 more direct.
  • The reported axial damping trend is opposite to the polar-mode trend from earlier analytical work, so observations of both parities could distinguish the two sectors.

Reading between the lines

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

  • If the shorter-damping prediction survives a full eight-equation treatment, then measuring both real and imaginary parts of the $l=m=2$ mode as functions of effective spin could constrain $\alpha/M^2$ without needing higher overtones.
  • The noted disagreement in $\mathrm{Im}(\omega_{22})$ with the independent spectral computation suggests the $\alpha^2\chi^2$ coefficient is the least robust of the fitted numbers; a head-to-head comparison at identical parameter values and normalization conventions would settle which truncation is responsible.
  • The method's ordering argument implies the $l\pm2$ axial couplings matter only at $\mathcal{O}(\chi^4)$ for the primary mode, so extending the same calculation to higher spin would require resummation rather than simply keeping more terms in the hierarchy.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper computes the fundamental n=0, l=m=2 axial quasi-normal mode (QNM) frequency of slowly rotating black holes in dynamical Chern-Simons (dCS) gravity, including corrections to second order in the spin parameter chi and second order in the dCS coupling alpha. The authors use a metric perturbation approach in the Regge-Wheeler gauge, decompose into spherical harmonics, reduce the system to coupled radial ODEs for the primary axial l=2 mode and polar l=1 and l=3 modes, and solve the resulting eigenvalue problem numerically with ingoing/outgoing boundary conditions. They validate the pipeline against Schwarzschild QNMs and against Kerr QNMs from Leaver's method for chi <= 0.15, then present a fitting formula (Eq. 46) with coefficients in Eq. (47). The headline result is that the dCS correction makes the l=m=2 axial mode damp faster (shorter damping time) at fixed spin, in contrast to earlier polar-mode results, with the caveat that the fitted Im(omega22) disagrees with the METRICS result of Chung et al.

Significance. If correct, this is the first semi-analytic axial QNM computation for slowly rotating dCS black holes at O(chi^2, alpha^2), and it provides an explicit fitting formula and a falsifiable parity-dependent ringdown prediction: dCS shortens the axial l=m=2 damping time. The GR-limit validation against Leaver's continued-fraction Kerr results, with relative errors below 1% for chi <= 0.15 (Fig. 1), is a genuine strength and gives independent support to the numerical pipeline. It is also to the authors' credit that Eq. (46) is explicitly labeled as a fit rather than a derivation, and that the discrepancy in Im(omega22) relative to Ref. [50] is disclosed rather than hidden. However, the central coefficients omega02, omega12, and omega22 rest on a reduced ODE system that is not displayed, on a conditional gauge assumption that is not proven, and on a very small fit grid without reported uncertainties; these issues presently limit the reliability of the advertised quantitative predictions.

major comments (3)
  1. [Sec. V.B, Eq. (47)] The reduction from the eight-field system (43)-(44) to the six-field system is justified only by the conditional statement in Sec. IV.B that 'if the l-1=1 polar modes are absent due to the gauge choice, the system can simplify.' No gauge transformation is exhibited that sets H_1^{1,m}=0 and K^{1,m}=0 on the rotating dCS background. The polar l=1 sector is sourced at O(chi) by rotation and at O(alpha^2 chi) by the background scalar hair (Eq. (12)); if it is not pure gauge, its feedback through the chi-coupling reaches the axial eigenvalue at O(alpha^2 chi^2), which is exactly the order of the omega22 coefficient in Eq. (47). The GR-limit validation in Fig. 1 only tests the truncation at alpha=0, so it cannot certify the omitted alpha^2 chi^2 terms. This is also the sector where Sec. V.B concedes that Im(omega22)=+0.66i disagrees with Ref. [50] beyond a simple normalization. The authors should either prove the gauge elimination on the rotating dCS background or integrate the full eight-field system and demonstrate that the l=1 polar modes decouple.
  2. [Sec. V.B, Eq. (47)] The central numerical result depends on the 8x8 matrix \hat A_l in Eq. (43), but this matrix and the explicit radial coefficients of the reduced ODE system are not displayed; Appendix B lists angular integrals but not the radial operators. No code, data tables, or boundary-convergence tests are provided, and the fitted coefficients in Eq. (47) are quoted without error bars or fit residuals. The fit uses only nine grid points, namely chi and alpha/M^2 in {0, 0.01, 0.02}, and the sub-percent accuracy statement for chi <= 0.4 refers to the Kerr (alpha=0) coefficients only, not to the dCS coefficients. To make omega02, omega12, and omega22 usable and checkable, the authors should provide the full ODE system and numerical implementation, and report uncertainties and cross-validation for the fitted coefficients.
  3. [Sec. VI] The paper's GW230529 relevance discussion lies outside the stated domain of validity of the calculation. The abstract and Sec. I restrict the computation to alpha/M^2 <= 0.05 and chi <= 0.15, while the NICER bound quoted in Sec. II.B gives alpha/M^2 ~ 33 (M_sun/M)^2; for a remnant of a few solar masses this is O(1)-O(10), far beyond the fitted range. It may be qualitatively plausible that lower-mass remnants have larger dCS corrections, but Eq. (46)-(47) cannot be used to quantify that statement at these masses. The authors should explicitly state this limitation when discussing GW230529 and next-generation detectors.
minor comments (5)
  1. [Sec. III.B] The text contains a typo: 'gauge dedundancy' should read 'gauge redundancy'.
  2. [Eqs. (32) and (40)] The general structure in Eq. (32) contains a term chi^2 m^2 \bar{\bar A}_{lm}, while the truncated system in Eq. (40) writes the corresponding term as chi^2 \bar{\bar A}_{lm} without the m^2 factor; please clarify whether this is a typo or a different definition.
  3. [Sec. I and Sec. V.B] The introduction states that the results 'show good agreement with those presented in Ref. [50]', but Sec. V.B concedes a substantial discrepancy in Im(omega22); the introduction should be qualified to reflect the partial agreement.
  4. [Sec. I] The statement that 'all calculations leading up to this numerical integration step are analytical and exact' is unverifiable as written because the reduced ODE system is not displayed; please specify which steps are exact, which are numerical, and which are fitting.
  5. [Fig. 2] The right panel of Fig. 2 appears to have garbled tick labels on the vertical axis; please check that the plotted values are correctly rendered.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QNM frequencies are obtained by solving the perturbed field equations, the dCS damping-time trend is read from those eigenvalues, and the analytic formula is explicitly a fit rather than a derived prediction.

full rationale

The derivation chain is self-contained. The central quantities, Mω(χ, α), are eigenvalues of the coupled system of linearized dCS perturbation equations (Eqs. (40), (43)-(44)) obtained by numerical integration with ingoing/outgoing boundary conditions (Eq. (38)); the dCS-induced decrease in damping time is a property of those numerically computed eigenvalues, not an input to the calculation. The analytic formula (46) is introduced as 'a numerical fit' to the computed frequencies over χ∈{0,0.01,0.02} and α/M2∈{0,0.01,0.02}, so the fit coefficients being derived from the same dataset they represent is disclosed and does not convert the fit into a prediction. The GR-sector coefficients are checked against independent Leaver continued-fraction results (Fig. 1) and Schwarzschild QNMs, providing external validation of the numerical framework in the α→0 limit. Citations of the authors' earlier work (Refs. [18,24,47]) are contextual or comparative (e.g., the polar-mode damping-time contrast in [47]) and do not supply the load-bearing equations or the uniqueness of the dCS metric perturbation system. The assumption in Sec. IV.B that l−1=1 polar modes vanish 'due to the gauge choice' is an unproven truncation and is a correctness/robustness risk, especially for the disputed Im(ω22) coefficient, but it is not circular: the eigenvalues are not defined in terms of that assumption, and the paper does not use the assumption to enforce the damping-time sign. No step in the paper reduces, by construction or by self-citation, to its own output.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The computation inherits the dCS scalar field from the existing theory; no new particles, forces, or dimensions are introduced. The main free inputs are the six fit coefficients and the numerical boundaries; six perturbative and background assumptions frame the calculation.

free parameters (7)
  • Fitting coefficient omega00 = 0.37367 - 0.088962i
    Fit coefficient from Eq. (47); reproduces the Schwarzschild l=2 axial QNM frequency.
  • Fitting coefficient omega02 = -0.769 - 0.389i
    Fit coefficient for the alpha^4 term from Eq. (47); the negative imaginary part drives the dCS damping decrease.
  • Fitting coefficient omega10 = 0.126 + 0.00192i
    Fit coefficient for the linear spin term; matches the Kerr first-order rotational splitting.
  • Fitting coefficient omega12 = -1.13 - 0.565i
    Fit coefficient for the chi alpha^2 cross term from Eq. (47).
  • Fitting coefficient omega20 = 0.078 + 0.016i
    Fit coefficient for the chi^2 term; matches the Kerr second-order correction.
  • Fitting coefficient omega22 = -1.7 + 0.66i
    Fit coefficient for the chi^2 alpha^2 term; the imaginary part disagrees with Ref. [50].
  • Numerical boundary locations delta and r_infinity = delta=0.002M, r_infinity=40M
    Chosen integration boundaries in Section IV.B.1; no convergence study is reported, so these choices could affect the computed eigenvalues.
assumptions (6)
  • domain assumption The O(chi^2, alpha^2) slowly rotating dCS background metric of Yagi, Yunes and Tanaka (Ref. [31]) is correct and complete.
    Section II.B and Eqs. (10)-(13); the perturbation equations and horizon radius use this metric. If this background is incomplete, all QNM results shift.
  • domain assumption The truncation to O(chi^2, alpha^2) in the perturbed equations is valid; terms of O(chi^3), O(alpha^3) and higher do not affect the QNM frequency at the stated order.
    Section III.A step 3; standard perturbative counting but not proven for the coupled dCS system.
  • ad hoc to paper Axial l to l+/-2 modes backreact on the primary axial l-mode only at O(chi^4), so they can be omitted for an O(chi^2) frequency.
    Section IV.B item 4; this ordering justifies solving only l and l+/-1 modes. If the backreaction is O(chi^2), the truncation is invalid.
  • ad hoc to paper For l=2, the polar l=1 modes H_1^{1,m} and K^{1,m} vanish due to the Regge-Wheeler gauge choice.
    Section IV.B; the paper states this conditionally ('if') and uses it to reduce the 8-equation system to 6 equations. No proof is given for the rotating dCS background.
  • domain assumption The tortoise coordinate F(r) defined in Eq. (39) correctly decouples the asymptotic wave equations.
    Section IV.A; used to impose ingoing and outgoing boundary conditions. F(r) is a constructed function rather than derived from the background metric.
  • domain assumption The ADM mass M and ADM angular momentum J define chi=J/M^2 in dCS, matching Kerr parameters in the alpha to 0 limit.
    Section II.B; needed so comparisons with Kerr QNMs are meaningful.

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Cite this review

Pith. "Pith review of Axial quasi-normal modes of slowly rotating black holes in dynamical Chern-Simons gravity to second-order in spin and coupling." pith.science (2026). https://pith.science/paper/ILPDYBOK

@misc{pith2026250603600,
  author       = {Pith},
  title        = {Pith review of: Axial quasi-normal modes of slowly rotating black holes in dynamical Chern-Simons gravity to second-order in spin and coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ILPDYBOK}},
  note         = {Machine review of arXiv:2506.03600}
}
abstract

We compute the quasi-normal mode (QNM) frequencies of slowly rotating black holes in dynamical Chern-Simons (dCS) gravity, including corrections up to second order in the black hole's dimensionless spin parameter $\chi = J/M^2$ and second order in the dCS coupling parameter ($\alpha$). Due to the complexities of constructing a Newman-Penrose tetrad at this order, we employ a metric perturbation approach. We derive a system of coupled ordinary differential equations for the primary axial $l$-mode and the polar $l\pm 1$ modes, which is then solved numerically with appropriate ingoing and outgoing wave boundary conditions. Our numerical framework is validated in the General Relativistic limit against known Schwarzschild QNMs and highly accurate Kerr QNM results for $\chi \leq 0.15$. For the fundamental $n=0, l=m=2$ axial mode, we present detailed numerical results illustrating the dependence of QNM frequencies on both $\chi$ and $\alpha$. We observe that while rotation generally increases the damping time, increasing the dCS coupling parameter significantly reduces the damping time of the axial mode. This finding contrasts with previous analytical work on polar modes, which suggested an increase in damping time due to dCS effects, highlighting a crucial parity-dependent difference in how dCS gravity impacts black hole ringdowns. Furthermore, we provide an analytical fitting formula for this mode. These results, incorporating coupled spin and dCS effects at second order, provide more accurate theoretical predictions for testing dCS gravity with gravitational wave observations of black hole ringdowns. The refined QNM calculations are particularly relevant for lower-mass black hole merger events, such as GW230529, where dCS corrections may be more prominent and their distinct damping signatures could be observable. [Abridged Version]

Figures

Figures reproduced from arXiv: 2506.03600 by the authors.

Figure 1
Figure 1. FIG. 1: The relative error between QNM frequencies [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Real and Imaginary parts of the [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Real and Imaginary parts of the [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Parity violating spectral dynamics of black holes in dynamical Chern-Simons gravity

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    In dynamical Chern-Simons gravity, an environmental potential bump reshapes black hole quasinormal-mode spectra, producing branch reconnections, a delayed overtaking instability, and scalar-mode-dominated ringdown tha...

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Works this paper leans on

93 extracted references · 32 canonical work pages · cited by 1 Pith paper

  1. [50]

    Alexander, G

    S. Alexander, G. Gabadadze, L. Jenks, and N. Yunes, Chern-Simons caps for rotating black holes, Phys. Rev. D 104, 064033 (2021), arXiv:2104.00019 [hep-th]

    arXiv 2021

  2. [1]

    The Regge-Wheeler gauge [38, 42] is a convenient choice for handling perturbations of spheri- cally symmetric backgrounds, and its generalization is adapted here (19)

    Metric perturbations in the Regge-Wheeler gauge and obtain the perturbation equations: We start by perturbing the dCS corrected slowly rotating BH metric (¯gµν) as gµν = ¯gµν + ϵ hµν and ϑ = ¯ϑ + ϵ δϑ (14) where ϵ is a book keeping parameter, hµν and δϑ are perturbations. The Regge-Wheeler gauge [38, 42] is a convenient choice for handling perturbations o...

  3. [2]

    Angular decomposition of the perturbation equations: To separate the angular dependence, we ex- pand the metric perturbations hµν and the pseudoscalar perturbation δϑ in terms of spherical harmonics. For a slowly rotating background, the equations are typically decomposed using scalar, vector, and tensor spherical harmonics appropriate for the spin-weight...

  4. [3]

    Terms of O(χ3), O(α3), O(χ3α2), O(χ2α3) can be dropped

    Simplify the equations by eliminating terms that do not contribute to the QNM frequency at O(χ2, α2): As we are interested in QNM frequency corrections δω = δωχ + δωχ2 + δωα + δωα2 + δωχα + ..., we consistently keep all terms in the perturbation equations that can contribute to the desired order. Terms of O(χ3), O(α3), O(χ3α2), O(χ2α3) can be dropped

  5. [4]

    As we show, many of these functions will be related through algebraic constraint equations arising from the gauge choice or certain combinations of the field equations

    Eliminate the dependent functions by express- ing them in terms of independent functions: After angular decomposition, we will have a system of coupled ordinary differential equations (ODEs) for various radial functions representing the components of hµν (e.g., H0(r), H1(r), H2(r), K(r), h0(r), h1(r)) [41, 63] and the pseudoscalar perturbation R(r). As we...

  6. [5]

    Obtain the final set of linear coupled differen- tial equations: By linearly combining the simplified and reduced equations from the previous steps, we obtain a final system of coupled linear homogeneous ordinary differential equations for the chosen master functions

  7. [6]

    Numerically integrate the final set of linear cou- pled differential equations to obtain QNM fre- quencies: With the system of ODEs at hand, QNM frequencies ω are found by imposing appropriate bound- ary conditions: purely outgoing waves at spatial infinity (r∗ → ∞) and purely ingoing waves at the event horizon (r∗ → −∞). B. Regge-Wheeler Gauge As mention...

  8. [7]

    Slow rotation introduces terms proportional to χ (and an- gular derivatives like χ sin θ or χ cos θ), which couple an l-mode primarily to its neighbors, l ± 1 [59, 65, 66]

    Essential l ↔ l ± 1 Mode Couplings: In a spher- ically symmetric background, perturbations with dif- ferent angular momentum numbers l decouple. Slow rotation introduces terms proportional to χ (and an- gular derivatives like χ sin θ or χ cos θ), which couple an l-mode primarily to its neighbors, l ± 1 [59, 65, 66]. While these couplings are formally O(χ)...

Show all 93 references
  1. [8]

    These l ↔ l ± 2 couplings are typically absent at O(χ) [47] but are intrinsic to the second-order rotational effects

    Emergence of l ↔ l ± 2 Couplings at O(χ2): At second order in spin, new coupling structures arise in the perturbation equations, including direct cou- plings between l-modes and l±2-modes [65, 66]. These l ↔ l ± 2 couplings are typically absent at O(χ) [47] but are intrinsic t...

  2. [9]

    However, for dCS perturbations at O(χ2, α2), we find that no such transformation can simultane- ously eliminate all couplings between l, l ± 1, and l ± 2 modes

    Obstacles to decoupling via basis transforma- tions: A common strategy for simplifying coupled systems is to seek a basis transformation that di- agonalizes the equations, effectively decoupling the modes. However, for dCS perturbations at O(χ2, α2), we find that no such trans...

  3. [10]

    In the non-rotating limit ( χ = 0), all polar par- ity perturbations ( Z l′m pol ) are zero, and axial parity perturbations (Z l′m ax ) are zero for l′ ̸= l

    Assume the system is initially excited by a purely axial perturbation with a specific harmonic index l. In the non-rotating limit ( χ = 0), all polar par- ity perturbations ( Z l′m pol ) are zero, and axial parity perturbations (Z l′m ax ) are zero for l′ ̸= l. 11

  4. [11]

    Thus, the amplitudes of these modes are Z l±1m pol = O(χ)

    At O(χ), rotation induces couplings that excite po- lar parity functions with harmonic indices l ± 1. Thus, the amplitudes of these modes are Z l±1m pol = O(χ)

  5. [12]

    Consequently, the amplitudes of these modes are Z l±2m ax = O(χ2)

    At O(χ2), rotation can also induce couplings be- tween the primary axial l-mode and axial l ± 2 modes. Consequently, the amplitudes of these modes are Z l±2m ax = O(χ2)

  6. [13]

    Therefore, for determining the QNM fre- quency of Z lm ax up to O(χ2), the explicit inclusion of Z l±2m ax modes and the axial l ↔ l ± 2 coupling terms is not necessary

    The crucial point for simplification is that the back- reaction of these O(χ2)-amplitude Z l±2m ax modes onto the QNM frequency of the primary Z lm ax mode will be of O(χ4) (since the coupling term itself is O(χ2)). Therefore, for determining the QNM fre- quency of Z lm ax up ...

  7. [14]

    We numerically integrate the sys- tem of equations using Runge-Kutta-Fehlberg method

    Numerical method and boundary conditions As mentioned earlier, obtaining QNMs is an eigen- value problem for the complex frequency ω, subject to the boundary conditions of purely ingoing waves at the horizon ( r∗ → −∞) and purely outgoing waves at in- finity ( r∗ → +∞). We num...

  8. [15]

    Schwarzschild Limit: Setting α = 0 and χ = 0, our code reproduced the well-established QNM frequencies for a Schwarzschild BH with excellent agreement with previous results, such as those in [43, 44]

  9. [16]

    These were then compared against highly accurate “exact” QNM frequencies obtained using Leaver’s continued fraction method [68]

    Slow rotation (Kerr) Limit: To assess the ac- curacy of our second-order-in-spin computation, we set α = 0 and calculated QNM frequencies for a Kerr BH. These were then compared against highly accurate “exact” QNM frequencies obtained using Leaver’s continued fraction method [...

  10. [17]

    Analytic Fitting Formula To provide a convenient analytical representation of our results for small χ and α/M2, we performed a nu- merical fit to the QNM frequencies obtained for χ ∈ {0, 0.01, 0.02} and α/M2 ∈ {0, 0.01, 0.02}. For the n = 0, l= m = 2 axial QNM frequency, we ob...

  11. [18]

    Our numerical code was validated against known results for Schwarzschild BHs and, for the Kerr limit (α = 0) in GR

    Numerical computation up to second order in χ: We have successfully computed the QNM frequen- cies for the fundamental n = 0, l= m = 2 axial mode of slowly rotating dCS BHs. Our numerical code was validated against known results for Schwarzschild BHs and, for the Kerr limit (α...

  12. [19]

    We observed that both the real and imaginary parts of the QNM frequency increase with spin, similar to GR

    dCS effects on QNM frequencies: We presented numerical results illustrating the dependence of QNM frequencies on both the spin χ and the dCS coupling α/M2. We observed that both the real and imaginary parts of the QNM frequency increase with spin, similar to GR. Our results im...

  13. [20]

    These coefficients quantify the intricate interplay be- tween BH spin and dCS modifications

    Analytical Fitting F ormula: We derived an ana- lytical fitting formula (46) for the n = 0, l = m = 2 axial QNM frequency, explicitly providing the coef- ficients ω00, ω02, ω10, ω12, ω20, and ω22 that capture the zeroth, first, and second-order spin effects, each including zer...

  14. [21]

    Group 1 C1fl = Qlfl−1 + Ql+1fl+1 C2fl = QlQl−1fl−2 + (Q2 l+1 + Q2 l )fl + Ql+1Ql+2fl+2 S1fl = (l − 1)Qlfl−1 − (l + 2)Ql+1fl+1 S2fl = (l − 2)Ql−1Qlfl−2 + (lQ2 l+1 − (l + 1)Q2 l )fl − (l + 3)Ql+2Ql+1fl+2 ¯S1fl = lQl+1fl+1 − (l + 1)Qlfl−1 ¯S2fl = −(l + 1)QlQl−1fl−2 + (lQ2 l+1 − (...

  15. [22]

    Group 2 A0fl = l(l + 1)fl 17 A1fl = (l2 − 1)Qlfl−1 + l(l + 2)Ql+1fl+1 A2fl = (l + 1)(l − 2)QlQl−1fl−2 + l(l + 3)Ql+1Ql+2fl+2 + (Q2 l+1l(l + 3) + Q2 l (l + 1)(l − 2))fl B0fl = 0 B1fl = imfl B2fl = 2im Qlfl−1 + Ql+1fl+1 X0fl = im(l − 1)(l + 2)fl X1fl = im(l − 2)(l + 3)(Qlfl−1 + ...

  16. [23]

    Group 3 F0fl = −im(l2 + l − 2)fl F1fl = −im (l − 1)(l + 4)Ql+1fl+1 + (l − 3)(l + 2)Qlfl−1 G0fl = l(l − 1)(l + 2)Ql+1fl+1 − (l − 1)(l + 1)(l + 2)Qlfl−1 G1fl = Ql+1Ql+2fl+2(l − 1)l(l + 3) − Ql−1Qlfl−2(l + 1)(l − 2)(l + 2) + fl Q2 l (l + 1)(l2 − l + 4) − Q2 l+1l(l2 + 3l + 6) + 2m...

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