Self-Force-Based Beyond-GR Waveform Model

• Self-force-based beyond-GR waveform models are constructed using small-mass-ratio expansions to introduce modified gravity effects into binary merger dynamics.
• They build on GR reference models by incorporating extra fields and effective-field-theory corrections to compute precise merger and ringdown phase details.
• The modular phase-space dynamics framework facilitates systematic waveform assembly, enabling quantification of observable shifts such as dephasing and amplitude variations.

Searching arXiv for papers on self-force-based beyond-GR waveform modeling and closely related self-force waveform frameworks. A self-force-based beyond-GR waveform model is a waveform construction in which binary dynamics and radiation are organized in a small-mass-ratio expansion, while departures from general relativity are introduced through modified backgrounds, extra fields, altered regular fields and fluxes, and theory-dependent merger-ringdown structure. Within the literature represented here, the first explicit realization of this program for merger and ringdown is the small-mass-ratio beyond-GR framework of “Black hole mergers beyond general relativity: a self-force approach” (Roy et al., 13 Oct 2025). At the same time, much of the practical architecture derives from earlier GR self-force models that define the reusable pipeline: specify a background spacetime, compute or fit a regularized self-force or equivalent invariant inputs, evolve the worldline or phase-space variables, and generate radiation from the resulting trajectory (Burko et al., 2012).

1. Definition and boundaries

The topic sits at the intersection of three distinct literatures: GR self-force waveform modeling, beyond-GR perturbation theory, and phenomenological merger deformations. These should not be conflated.

A useful classification is the following.

Work	Status relative to the topic	Main role
(Roy et al., 13 Oct 2025)	Actual self-force-based beyond-GR waveform model	First-principles merger-ringdown beyond GR in the small-mass-ratio regime
(Burko et al., 2012)	Not beyond GR	GR reference model for orbit integration, conservative effects, and waveform generation
(Wardell et al., 2021)	Not beyond GR	Second-order self-force inspiral waveform model through 1PA
(Küchler et al., 2 Jun 2025)	Not beyond GR	Offline/online phase-space IMR architecture through plunge and ringdown
(Upton et al., 31 Jul 2025)	Not beyond GR	Second-order effective-source infrastructure
(Oltean et al., 2017)	Not a waveform model	Frequency-domain PwP source-handling method for scalar self-force
(Watarai et al., 2023)	Beyond-GR, but not self-force-based	Phenomenological merger deformation with flux-consistency closure

A recurrent misconception is to label any self-force waveform as “beyond-GR.” The Schwarzschild self-force model “Self Force Orbit-Integrated gravitational waveforms for E(I)MRIs” is explicitly a GR construction and is best understood as a GR baseline rather than a beyond-GR model (Burko et al., 2012). Conversely, the merger-ringdown framework of (Roy et al., 13 Oct 2025) is genuinely beyond GR, but only in the small-mass-ratio regime and only for a restricted class of effective-field-theory extensions. A separate misconception is to treat phenomenological merger deformations as self-force models; the physically consistent merger parameterization of (Watarai et al., 2023) is beyond GR in intent but not derived from self-force theory.

2. General-relativistic reference architecture

The baseline GR scaffold is particularly clear in the Schwarzschild quasi-circular EMRI/IMRI model of (Burko et al., 2012). The system is a point particle of mass $\mu$ inspiraling into a Schwarzschild black hole of mass $M$, with $\mu \ll M$, and in the explicit numerical example $\mu=10^{-2}M$. The inspiral runs from $r_0=10M$ to near the ISCO at $r=6M$. The worldline is evolved with the local first-order self-force according to

$$u^{\beta}\nabla_{\beta}u^{\alpha}=\mu^{-1}f^{\alpha}_{\rm SF},$$

with the orthogonality condition

$f^{\rm SF}_{\mu}u^{\mu}=0.$

The working force model uses fitted Lorenz-gauge circular-orbit data from Barack and Sago, with $f^t$ encoding the dissipative sector and $f^r$ encoding the conservative correction. Turning off the conservative part is implemented by setting $M$0, and in that limit the local self-force inspiral reproduces the energy-balance inspiral (Burko et al., 2012).

That paper also fixes the operational comparison standard that later beyond-GR work can inherit. It contrasts three evolutions: an energy-balance inspiral, a self-force inspiral with the conservative part turned off, and a full first-order self-force inspiral including the conservative piece. Waveform generation is not a kludge prescription; after orbit construction, the source trajectory is used to evolve the linearized Einstein equations via a sourced Teukolsky equation with hyperboloidal slicing, and the displayed waveform is the real part of $M$1. The main quantitative result is a cumulative dephasing

$M$2

roughly $M$3 cycles out of $M$4 cycles, between the full-self-force and energy-balance waveforms over the inspiral from $M$5 to near $M$6. For the overlap study, the threshold window length is

$M$7

at which the overlap drops to $M$8, interpreted as a $M$9 event-rate loss. The same model shows that the gauge-invariant relation $\mu \ll M$0 is unchanged within numerical accuracy by the conservative force, while the time taken to traverse that curve is changed (Burko et al., 2012).

The later second-order inspiral model of (Wardell et al., 2021) extends this architecture through first post-adiabatic order. There the inspiral is built from a two-timescale expansion of the Einstein equations through second order in the mass ratio, yielding waveform production in tens of milliseconds. The mode structure is written as

$\mu \ll M$1

with the orbital evolution controlled by

$\mu \ll M$2

This provides the GR inspiral-side template for any beyond-GR generalization that can supply analogous invariant functions (Wardell et al., 2021).

3. First-principles beyond-GR construction

The explicitly beyond-GR construction in (Roy et al., 13 Oct 2025) is formulated in an EFT language,

$\mu \ll M$3

with the common case of an extra scalar $\mu \ll M$4. The regime is a small or intermediate mass ratio,

$\mu \ll M$5

with formal validity at $\mu \ll M$6. The primary background is Schwarzschild, the motion is quasicircular leading to plunge, and the implementation is restricted to a nonspinning primary. The secondary is skeletonized as a point particle carrying a scalar monopole charge $\mu \ll M$7, with charge-to-mass ratio

$\mu \ll M$8

The perturbative content is organized as

$\mu \ll M$9

$\mu=10^{-2}M$0

The leading beyond-GR merger corrections therefore appear at order $\mu=10^{-2}M$1. Varying the action yields a coupled metric-scalar system, and the small body obeys

$\mu=10^{-2}M$2

$\mu=10^{-2}M$3

with

$\mu=10^{-2}M$4

At the order actually computed, the force decomposes as

$\mu=10^{-2}M$5

with

$\mu=10^{-2}M$6

and the mass correction is

$\mu=10^{-2}M$7

The beyond-GR content is modular in a precise sense. In the current implementation, the leading correction arises from the scalar monopole charge $\mu=10^{-2}M$8; theory-dependent curvature couplings such as $\mu=10^{-2}M$9 enter only at higher order in the small-$r_0=10M$0 regime. This is why the paper can treat a broad class of EFT extensions while computing a specific leading correction. It also explains the caveat that in dynamical Chern-Simons gravity, where the leading scalar is dipolar rather than monopolar, the theory is indistinguishable from GR at this order (Roy et al., 13 Oct 2025).

4. Phase-space dynamics and waveform assembly

The beyond-GR merger-ringdown model of (Roy et al., 13 Oct 2025) is built by extending the GR post-geodesic plunge formalism into beyond GR. The worldline is parameterized as

$r_0=10M$1

and the reduced plunge dynamics is written as

$r_0=10M$2

$r_0=10M$3

The geodesic-order terms are algebraic in $r_0=10M$4, while the first post-geodesic corrections satisfy linear ODEs driven by $r_0=10M$5. The plunge solution is not stitched arbitrarily; it is fixed by asymptotic matching to a transition-to-plunge regime using

$r_0=10M$6

This supplies the boundary data that determine $r_0=10M$7 and $r_0=10M$8 (Roy et al., 13 Oct 2025).

Waveform assembly is likewise phase-space based. The asymptotic strain is extracted via

$r_0=10M$9

and the mode amplitudes are written as

$r=6M$0

Operationally, waveform generation proceeds by solving for $r=6M$1 and $r=6M$2, evaluating precomputed $r=6M$3, and multiplying by the phase factor $r=6M$4 (Roy et al., 13 Oct 2025).

A closely related GR development makes the organizational principle explicit: the same offline/online phase-space paradigm used for inspiral can be continued through transition, plunge, merger, and ringdown (Küchler et al., 2 Jun 2025). In that framework, expensive field solutions are precomputed as functions on orbital phase space, while online waveform generation reduces to integrating a few ODEs and evaluating interpolants. The paper shows that this structure survives even when there is no separation of timescales during plunge. This suggests that the most portable component of self-force waveform modeling is not a particular GR equation, but the phase-space organization of the waveform problem itself (Küchler et al., 2 Jun 2025).

5. Invariant structure and computational infrastructure

Invariant diagnostics play a dual role: they calibrate conservative dynamics and they separate physical content from gauge artifacts. In the Schwarzschild first-order model, the redshift invariant $r=6M$5 is unchanged in shape, within numerical accuracy, by inclusion of the conservative force; the change lies in the rate at which the inspiral moves along the curve (Burko et al., 2012). This suggests a two-layer validation strategy: first match invariant relations, then match timing and phase accumulation along them.

The invariant pseudo-Hamiltonian framework of (Lewis et al., 10 Jul 2025) systematizes that idea at 1PA order. It reformulates the multiscale self-force problem on a six-dimensional phase space and writes each waveform mode as

$r=6M$6

In its final invariant form, the dynamics becomes

$r=6M$7

$r=6M$8

A central byproduct is the identification

$r=6M$9

which leads to the statement that the on-shell conservative Hamiltonian equals the mechanical energy predicted by the first law of binary black-hole mechanics (Lewis et al., 10 Jul 2025).

Practical second-order waveform construction requires effective-source technology. The technical infrastructure paper (Upton et al., 31 Jul 2025) constructs the multiscale second-order effective source on Schwarzschild and shows that it contains four essential pieces: quadratic coupling of first-order field modes, slow evolution of first-order fields, quadratic products of a first-order puncture field, and the second-order puncture field. Mode by mode, the second-order source is written as

$$u^{\beta}\nabla_{\beta}u^{\alpha}=\mu^{-1}f^{\alpha}_{\rm SF},$$0

providing the regularized second-order input that underlies post-adiabatic self-force waveforms (Upton et al., 31 Jul 2025).

At the source-handling level, the frequency-domain particle-without-particle method of (Oltean et al., 2017) is not a waveform model and not beyond GR, but it is structurally relevant. Its master scalar equation,

$$u^{\beta}\nabla_{\beta}u^{\alpha}=\mu^{-1}f^{\alpha}_{\rm SF},$$1

is turned into homogeneous left/right subdomain problems plus jump conditions by writing

$$u^{\beta}\nabla_{\beta}u^{\alpha}=\mu^{-1}f^{\alpha}_{\rm SF},$$2

This is a computational template for localized worldline sources in any theory that reduces to wave-type equations with point-particle support (Oltean et al., 2017).

6. Physical content, limitations, and adjacent directions

The beyond-GR merger-ringdown model of (Roy et al., 13 Oct 2025) already produces concrete waveform observables. For the $$u^{\beta}\nabla_{\beta}u^{\alpha}=\mu^{-1}f^{\alpha}_{\rm SF},$$3 mode and the example considered there, the scalar self-force shifts the invariant peak orbital frequency to

$$u^{\beta}\nabla_{\beta}u^{\alpha}=\mu^{-1}f^{\alpha}_{\rm SF},$$4

changes the peak amplitude to

$$u^{\beta}\nabla_{\beta}u^{\alpha}=\mu^{-1}f^{\alpha}_{\rm SF},$$5

and modifies the fundamental ringdown excitation amplitude according to

$$u^{\beta}\nabla_{\beta}u^{\alpha}=\mu^{-1}f^{\alpha}_{\rm SF},$$6

Ringdown is beyond GR only through excitation in the current implementation; QNM frequency corrections are not yet computed because the quadratic scalar stress-energy term in the metric perturbation equation is omitted (Roy et al., 13 Oct 2025).

The principal limitations are explicit. The present beyond-GR realization is formally valid at small mass ratio, assumes a Schwarzschild primary, treats quasicircular plunge only, computes only a partial leading $$u^{\beta}\nabla_{\beta}u^{\alpha}=\mu^{-1}f^{\alpha}_{\rm SF},$$7 correction, and retains an effectively GR-like remnant/QNM spectrum. It does not cover Kerr spin, eccentricity, inclination, strong-coupling departures from the EFT hierarchy, or the dCS case at leading order because there the secondary has no scalar monopole (Roy et al., 13 Oct 2025).

A nearby but distinct direction is finite-size response of non-black-hole central objects. The compact-star self-force study (Feng et al., 2024) shows that, relative to a Schwarzschild black hole at the same orbital frequency, the additional self-force can be represented approximately by a universal frequency-dependent function multiplied by the dynamical tidal deformability. For the conservative radial sector,

$$u^{\beta}\nabla_{\beta}u^{\alpha}=\mu^{-1}f^{\alpha}_{\rm SF},$$8

and for the dissipative sector,

$$u^{\beta}\nabla_{\beta}u^{\alpha}=\mu^{-1}f^{\alpha}_{\rm SF},$$9

In the illustrative EMRI estimate with $f^{\rm SF}_{\mu}u^{\mu}=0.$0, a four-year inspiral, and a supermassive star-like central object, the tide-induced orbital phase shift reaches $f^{\rm SF}_{\mu}u^{\mu}=0.$1 rad, while the abstract summarizes the generic scale as $f^{\rm SF}_{\mu}u^{\mu}=0.$2 rad (Feng et al., 2024). This is not a beyond-GR field-theory model, but it provides a response-theory template for exotic compact objects.

Phenomenological merger deformations remain an adjacent literature rather than part of the self-force line. The model of (Watarai et al., 2023) introduces a compact beyond-GR merger parameterization on top of IMRPhenomD and enforces physical consistency by feeding the extra radiated energy and angular momentum back into the remnant spin and ringdown frequencies. Its relevance here is methodological: it supplies a flux-consistency closure principle for merger deformations, but it does not derive those deformations from self-force theory (Watarai et al., 2023).

Taken together, these works define the present meaning of a self-force-based beyond-GR waveform model. In the strict sense, it is presently exemplified by a first-principles small-mass-ratio merger-ringdown framework with scalar-field corrections, asymptotic matching through plunge, and modular phase-space waveform assembly (Roy et al., 13 Oct 2025). In the broader methodological sense, it is supported by a GR ecosystem that already supplies orbit-integrated first-order baselines, second-order post-adiabatic inspirals, effective-source regularization, invariant action-angle dynamics, and offline/online IMR organization (Burko et al., 2012, Wardell et al., 2021, Upton et al., 31 Jul 2025, Lewis et al., 10 Jul 2025, Küchler et al., 2 Jun 2025). The plausible implication is that future beyond-GR extensions will be won less by replacing the overall scaffold than by replacing its physics modules: background spacetime, regular fields, fluxes, waveform amplitudes, and remnant spectra.