Numerical Kludge Waveforms

• Numerical kludge waveforms are approximate EMRI models that combine exact curved-spacetime geodesics with weak-field radiation formulas for computational efficiency.
• They capture key strong-field orbital features such as zoom–whirl behavior, secular phase accumulation, and precession effects in waveform morphology.
• Their modular design allows adaptation to non-Kerr and modified spacetimes, facilitating exploratory gravitational-wave studies without full relativistic perturbation theory.

Searching arXiv for the core papers on numerical kludge waveforms and closely related kludge frameworks. Numerical kludge waveforms are approximate extreme-mass-ratio inspiral waveform models that combine relativistic orbital dynamics in the background spacetime with flat-space multipolar radiation formulas. In the literature represented here, the defining construction is: solve the motion of the small body as a genuine geodesic, or as an adiabatically evolving sequence of geodesics, in the curved spacetime of interest; then interpret the resulting trajectory as motion in flat-space spherical coordinates and compute the emitted radiation with weak-field multipole expressions. This makes numerical kludge waveforms substantially more faithful to strong-field orbital structure than analytic kludges, while remaining far cheaper than fully relativistic Teukolsky- or self-force-based waveform models (Deng et al., 28 Oct 2025, Yang et al., 2024, Chua, 2016).

1. Taxonomic position within EMRI waveform modeling

In the EMRI literature summarized here, “kludge” denotes semi-relativistic waveform constructions that trade formal consistency for speed and modularity. The principal distinction is between analytic kludges, which evolve post-Newtonian Keplerian ellipses with phenomenological precession, and numerical kludges, which keep relativistic geodesic motion but still use flat-space radiation formulas. Augmented analytic kludges occupy an intermediate position by importing key Kerr frequency information into an AK-like structure, while the “new kludge” scheme of the MPM-based approach combines Kerr geodesics with a local radiation-reaction prescription and higher multipoles (Chua, 2016, Chua et al., 2017, Sopuerta et al., 2011).

Family	Orbital dynamics	Radiation construction
Analytic kludge (AK)	PN-evolving Keplerian ellipses in flat space	Peters–Mathews-type harmonic sums
Numerical kludge (NK)	Exact curved-spacetime geodesics, or adiabatically evolving geodesics	Flat-space quadrupole or quadrupole–octupole formulas
Augmented analytic kludge (AAK)	AK dynamics corrected by frequency matching to Kerr geodesic combinations	AK/Peters–Mathews structure with improved phasing
“New kludge”	Sequence of self-adjusting Kerr geodesics with local radiation reaction	MPM waveform generation up to mass hexadecapole and current octopole

A persistent point of confusion is the identification of all “kludge” models with NK. This is incorrect. Recent computational infrastructures may advertise a “kludge scheme” while actually implementing only the Barack–Cutler analytic kludge. A clear example is EMRI_MC, whose authors state that the only EMRI waveform model actually implemented is the Analytic Kludge; NK is listed only as a possible future replacement (Saltas et al., 2023).

2. Core construction of numerical kludge waveforms

The papers considered here describe the numerical kludge recipe in a common sequence. First, one computes the worldline of the small body in the target spacetime. In the periodic-orbit studies this is a bound geodesic with constant energy $E$ and angular momentum $L$ during waveform generation; in inspiral-oriented constructions the trajectory is an adiabatically evolving sequence of geodesics (Deng et al., 28 Oct 2025, Yang et al., 2024).

Second, one embeds the curved-space trajectory into flat space by interpreting the spherical coordinates as ordinary Minkowski spherical coordinates,

$$x = r \sin\theta\cos\phi,\qquad y = r \sin\theta\sin\phi,\qquad z = r \cos\theta.$$

The small body is then treated as a point mass in flat spacetime, with stress-energy

$$T^{tt}(t,\mathbf{x}) = m\, \delta^3(\mathbf{x} - \mathbf{Z}(t)),$$

and STF mass quadrupole

$I^{ij}(t)=m\left[x^i x^j\right]_{\mathrm{STF}}.$

Third, one evaluates the weak-field radiation formula. In the leading mass-quadrupole implementations,

$$h_{ij}(t)=\frac{2}{D_L}\frac{d^2 I_{ij}}{dt^2} =\frac{2m}{D_L}\,\big(a_i x_j + a_j x_i + 2 v_i v_j\big),$$

where $v_i=\dot x_i$ and $a_i=\dot v_i$. The plus and cross polarizations are then obtained by projection onto a detector-adapted basis. Several of the recent non-Kerr studies retain only this leading mass quadrupole, explicitly excluding current multipoles and higher mass multipoles; by contrast, the MPM-based “new kludge” extends waveform generation to mass hexadecapole and current octopole order (Deng et al., 28 Oct 2025, Yang et al., 2024, Sopuerta et al., 2011).

This construction should not be conflated with a fully relativistic perturbative treatment. Numerical kludge waveforms do not solve a curved-space perturbation equation. Their defining approximation is precisely that the radiation is computed as if the motion occurred in flat spacetime, even when the trajectory itself is a strong-field relativistic geodesic (Deng et al., 28 Oct 2025).

3. Orbital structure, periodicity, and zoom–whirl morphology

A major virtue of numerical kludge waveforms is that they inherit the strong-field orbital structure of the underlying spacetime. In the spherically symmetric examples discussed here, bound periodic orbits are classified by the rational frequency parameter

$$q \equiv \frac{\omega_\varphi}{\omega_r} - 1 = w + \frac{v}{z},$$

with $(z,w,v)$ the Levin–Perez-Giz integers. The zoom number $L$0 counts the distinct leaves, the whirl number $L$1 counts the near-circular revolutions near periapsis, and the vertex index $L$2 specifies how the leaves are connected (Deng et al., 28 Oct 2025, Yang et al., 2024, Gong et al., 27 Sep 2025).

These orbital labels map directly into waveform morphology. Large $L$3 produces multiple tight loops near periapsis and therefore concentrated, high-frequency bursts in $L$4 and $L$5. Large $L$6 produces multi-leaf orbital patterns and repeated sub-bursts within one full waveform cycle. The recent periodic-orbit studies emphasize that each zoom phase yields relatively lower amplitude and lower instantaneous frequency, whereas each whirl phase generates the most prominent high-frequency, high-amplitude structure (Yang et al., 2024, Gong et al., 27 Sep 2025).

This feature is one of the principal reasons NK models remain attractive even when their radiation sector is approximate. A purely Newtonian quadrupole model would miss the relativistic orbital content that generates periapsis precession and zoom–whirl behavior. By contrast, NK captures these effects because they are already present in the geodesic worldline; only the radiation formula is weak-field (Deng et al., 28 Oct 2025).

4. Accuracy, limitations, and relation to Teukolsky and self-force models

The central limitation of numerical kludge waveforms is conceptually simple: exact or near-exact orbital motion does not imply exact radiation. Teukolsky-based EMRI waveforms solve perturbations of the curved spacetime and include the correct curved-space Green’s functions and frequency-domain structure, whereas NK waveforms do not solve any curved-space perturbation equation (Deng et al., 28 Oct 2025).

At the same time, the literature summarized here shows that the amplitude-level approximation can be remarkably effective when the worldline is accurate. In the Schwarzschild, highly eccentric, non-spinning setting, kludge waveforms built on identical self-forced inspiral worldlines compare favorably to Teukolsky waveforms. One representative case yields a fractional overlap $L$7, and a systematic study reports overlaps generally $L$8 for non-spinning primaries over a range of eccentricities; the largest discrepancies arise near plunge (McCart et al., 2021). This supports a useful separation of errors: when the inspiral model is accurate enough, the dominant residual in NK can be an amplitude-model error rather than catastrophic phase failure.

The comparison with analytic kludges is different. AK is substantially cheaper, but its mixed PN and flat-space frequency prescription causes rapid dephasing against more accurate models. Augmented analytic kludges were introduced precisely to mitigate this by forcing the AK frequencies to match Kerr geodesic combinations. In the examples quoted in the literature, original AK and NK can be a full cycle out of phase within $L$9 hours, whereas the augmented model remains phase-coherent with NK over much longer durations; for one-year inspirals, phase coherence extends from less than $$x = r \sin\theta\cos\phi,\qquad y = r \sin\theta\sin\phi,\qquad z = r \cos\theta.$$0 hour in AK to more than 2 months in augmented AK (Chua, 2016).

The limitations of NK remain those repeatedly acknowledged across the cited work: no self-force in the short periodic-orbit implementations, no radiation reaction over the modeled orbital cycle, weak-field radiation formulas in the strong field, and omission of higher multipoles in the simplest constructions. The MPM-based “new kludge” may be viewed as a response to some of these limitations, since it uses local-in-time radiation reaction and a more elaborate far-zone multipolar expansion, but it remains a kludge in the same broad sense (Sopuerta et al., 2011).

5. Numerical kludge waveforms in non-Kerr and modified spacetimes

One of the most important developments in recent work is the export of the NK recipe to non-Kerr backgrounds. Because the waveform generator requires only a worldline and a flat-space multipole prescription, it is straightforward to replace Kerr geodesics by geodesics of a modified spacetime while leaving the radiation machinery essentially unchanged. This metric-agnostic character is explicit in studies of a charged black hole with scalar hair, a loop-quantum-gravity-inspired quantum-corrected black hole, and a regular black hole with a Minkowski core (Deng et al., 28 Oct 2025, Yang et al., 2024, Gong et al., 27 Sep 2025).

In the scalar-hairy charged black-hole analysis, the scalar-hair parameter $$x = r \sin\theta\cos\phi,\qquad y = r \sin\theta\sin\phi,\qquad z = r \cos\theta.$$1 shifts the effective potential and the locations of the MBO and ISCO. The resulting NK waveforms show visibly modified zoom–whirl structure and a cumulative phase shift; as $$x = r \sin\theta\cos\phi,\qquad y = r \sin\theta\sin\phi,\qquad z = r \cos\theta.$$2 increases, the waveforms advance in phase relative to the $$x = r \sin\theta\cos\phi,\qquad y = r \sin\theta\sin\phi,\qquad z = r \cos\theta.$$3 case (Deng et al., 28 Oct 2025). In the quantum-corrected black-hole analysis, the dimensionless parameter $$x = r \sin\theta\cos\phi,\qquad y = r \sin\theta\sin\phi,\qquad z = r \cos\theta.$$4 changes the effective potential, the periodic-orbit families, and the associated waveform morphology; the most robust imprint is again a phase advance, while the spectra remain in the mHz band and the characteristic strain crosses the LISA sensitivity curve for the chosen masses and distances (Yang et al., 2024). In the regular-black-hole study, the deviation parameter $$x = r \sin\theta\cos\phi,\qquad y = r \sin\theta\sin\phi,\qquad z = r \cos\theta.$$5 produces phase shifts and amplitude modulations, and the LISA-noise-weighted faithfulness decreases both with increasing $$x = r \sin\theta\cos\phi,\qquad y = r \sin\theta\sin\phi,\qquad z = r \cos\theta.$$6 and with increasing rational parameter $$x = r \sin\theta\cos\phi,\qquad y = r \sin\theta\sin\phi,\qquad z = r \cos\theta.$$7; large-$$x = r \sin\theta\cos\phi,\qquad y = r \sin\theta\sin\phi,\qquad z = r \cos\theta.$$8 periodic orbits therefore act as especially sensitive probes of departures from Schwarzschild (Gong et al., 27 Sep 2025).

The broader modified-gravity literature represented here points in the same direction even when the implemented waveform family is not yet NK. Parameterized modified-gravity AK models already provide deformed geodesics, modified orbital frequencies, and flux corrections in bumpy spacetimes, and these ingredients are described as exactly the inputs a numerical kludge would require in the same background (Gair et al., 2011). This suggests that the recent non-Kerr NK studies are not isolated applications but part of a general methodological pattern.

6. Data-analysis role, computational infrastructures, and future extensions

Numerical kludge waveforms occupy a strategic position in EMRI data analysis because they are substantially more accurate than AK while remaining far more tractable than self-force or Teukolsky pipelines. Augmented kludge and Gaussian-process approaches make this role explicit: a fast approximate model can be statistically corrected using sparse training from a more accurate model, and the quality of that strategy improves as the approximate model becomes closer to NK (Chua, 2016). This is one route by which NK-like structure continues to influence current EMRI inference even when the final online template is not itself a conventional NK waveform.

Contemporary computational infrastructures are increasingly modular in a way that favors NK substitution. EMRI_MC is a clear example: it currently implements only the Analytic Kludge of Barack & Cutler, but the code and paper explicitly identify the Numerical Kludge as a more accurate waveform generator that could replace AK. In the implementation discussion, the corresponding replacement is described schematically as: replace the AK ODEs with a geodesic- or self-force-driven evolution in Kerr, while keeping the same FFT and likelihood machinery; the source model is therefore treated as swappable within a GPU-accelerated Bayesian pipeline (Saltas et al., 2023).

Recent AK studies of additional physics reinforce the same modular lesson. In the tidal-Love-number model, the conservative dynamics are unchanged at leading order in the mass ratio and the tidal effect enters through the induced quadrupole moment and flux corrections; the authors explicitly note that these ingredients can be ported to NK by replacing PN orbits and fluxes with geodesic-based ones (Zi et al., 2023). In the charged Kerr–Newman analysis, the analytic kludge framework isolates three physically distinct charge effects—Coulomb interaction, dipole electromagnetic radiation, and metric deformation—and the paper closes by identifying a charged NK model as the natural next step, with the same charge-dependent orbital frequencies, flux corrections, and ISCO shift as transferable inputs (Zi et al., 2022).

The resulting picture is stable across the cited literature. Numerical kludge waveforms are best understood as a reusable architecture rather than a single fixed implementation: exact or nearly exact orbital dynamics in the target spacetime, approximate radiation generation in flat space, and optional adiabatic evolution of the constants of motion. Their principal scientific value lies in capturing the geodesic imprint of strong-field gravity—especially zoom–whirl structure, secular phase accumulation, and the deformation of characteristic orbits—at a computational cost compatible with large parameter surveys, forecasting, and exploratory inference. Their principal limitation is equally clear: they are not a substitute for curved-spacetime perturbation theory when final-precision EMRI parameter estimation is required.