Ratio Waveforms in Gravitational and ISAC Applications

• Ratio waveforms are defined by a constitutive ratio (e.g. mass ratio or PAPR) that governs waveform scaling, generation, and feasibility across different applications.
• In gravitational-wave modeling, conditional autoencoders use mass ratio as a key coordinate, achieving over 97% fitting-factor accuracy and enabling efficient, near-extrapolative waveform generation.
• In RIS-aided ISAC, ratio constraints via peak-to-average power ratio (PAPR) are used to optimize waveforms for multi-user interference reduction and radar beampattern fidelity under hardware limits.

“Ratio waveforms” is a useful organizing label for waveform families in which a ratio variable plays a constitutive role in generation, scaling, matching, or hardware feasibility. In the literature considered here, the dominant instances are gravitational waveforms indexed by binary mass ratio—either $q \equiv \frac{m_2}{m_1}$ in binary black hole coalescence or $\nu=\frac{m}{M}\ll 1$ in extreme-mass-ratio systems—and transmit waveforms constrained by peak-to-average power ratio (PAPR) in RIS-aided integrated sensing and communication. Across these settings, the ratio is not a passive descriptor: it determines conditioning inputs for generative models, asymptotic rescalings, perturbative phase structure, or admissible waveform amplitudes under implementation constraints (Liao et al., 2021, Strusberg et al., 12 May 2025, Wu et al., 10 Apr 2025).

1. Mass ratio as a waveform coordinate

In gravitational-wave modeling, the mass ratio appears as a primary coordinate on waveform space. In the conditional-autoencoder framework of Han and collaborators, the model is conditioned on the binary masses $(m_1,m_2)$, from which the mass ratio is derived as

$q \equiv \frac{m_2}{m_1}, \qquad 1 \le q \le 10.$

The training architecture uses one encoder for waveform input, one encoder for the labels $(m_1,m_2)$, and a decoder that reconstructs the waveform; after training, the waveform encoder is removed and the remaining network acts as a generator from labels to strain. The latent representation of the waveform is tied to the label representation through a latent loss, so the model is trained to learn the source-to-waveform relation rather than merely compressing an unconditioned signal. For the AE, the latent loss is based on the mismatch between latent vectors, typically MSE; for the VAE, latent distributions are matched via KL divergence. The VAE objective is written as

$-\log p_{\theta}(x)\le \mathbf{E}_{z\sim q_{\phi}(z|x)}[-\log p_{\theta}(\tilde{x}|z)] + D_{\mathrm{KL}\!\left[q_{\phi}(z|x)\,\|\,\mathcal{N}(0,1)\right],$

with the first term the reconstruction loss and the second the KL regularization; in the conditional setting, the same idea is applied with labels as additional inputs (Liao et al., 2021).

In perturbative and asymptotic descriptions of extreme or intermediate mass-ratio inspirals, the relevant parameter is instead

$\nu=\frac{m}{M}\ll 1$

or, in related notation,

$\varepsilon:=\mu/M\ll 1.$

Here the ratio governs not only parameter labeling but also the ordering scheme of the dynamics and phase. For quasi-circular self-force inspirals in Schwarzschild spacetime, the phase expansion is

$$\Phi=\varepsilon^{-1}\left[\Phi^{(0)}+\varepsilon\,\Phi^{(1)}+O(\varepsilon^2)\right],$$

so that terms entering at different orders in the force may contribute at the same observable order $O(\varepsilon^0)$ in the waveform phase (Burko et al., 2015, Burko et al., 2013).

A distinct waveform-design use of a ratio variable appears in RIS-aided ISAC, where the relevant quantity is the peak-to-average power ratio. There, the waveform is not indexed by astrophysical parameters but is optimized under the hardware-facing constraint

$\nu=\frac{m}{M}\ll 1$0

with $\nu=\frac{m}{M}\ll 1$1 a tunable threshold. This use of a ratio variable is structurally analogous in that it shapes the admissible waveform family, but it acts through transmitter feasibility rather than source dynamics (Wu et al., 10 Apr 2025).

2. Conditional generation across finite mass-ratio waveform space

The conditional autoencoder and conditional variational autoencoder constructions of Han and collaborators treat inspiral-merger waveforms of binary black hole coalescence as a conditional generative problem. The central claim is that, once trained, the decoder can generate the waveform associated with a specified label $\nu=\frac{m}{M}\ll 1$2 without running a costly physical solver. The full dataset spans

$\nu=\frac{m}{M}\ll 1$3

and is explicitly partitioned into low-mass-ratio and high-mass-ratio regions:

• LMR: $\nu=\frac{m}{M}\ll 1$4
• HMR: $\nu=\frac{m}{M}\ll 1$5

The data split is intentionally asymmetric. For LMR, the training/validation/test proportions are $\nu=\frac{m}{M}\ll 1$6; for HMR, they are $\nu=\frac{m}{M}\ll 1$7. The HMR fraction in the total training+validation set is only about $\nu=\frac{m}{M}\ll 1$8. This construction is used to probe whether a model trained mainly on low mass ratios can nevertheless generate accurate high-mass-ratio waveforms (Liao et al., 2021).

The best conditional autoencoder models achieve average fitting-factor accuracies above $\nu=\frac{m}{M}\ll 1$9 for both low- and high-mass-ratio systems. Specifically, the best reported cAE results are:

• 1c2E1D: $(m_1,m_2)$0 for LMR, $(m_1,m_2)$1 for HMR
• 2c2E1D: $(m_1,m_2)$2 for LMR, $(m_1,m_2)$3 for HMR

For the best 2c2E1D model, the reported fitting-factor distribution is also explicit:

• minimum FF: $(m_1,m_2)$4 (LMR), $(m_1,m_2)$5 (HMR)
• median FF: $(m_1,m_2)$6 (LMR), $(m_1,m_2)$7 (HMR)
• maximum FF: $(m_1,m_2)$8 (LMR), $(m_1,m_2)$9 (HMR)

The paper evaluates waveform quality with the standard matched-filter overlap

$q \equiv \frac{m_2}{m_1}, \qquad 1 \le q \le 10.$0

and the fitting factor

$q \equiv \frac{m_2}{m_1}, \qquad 1 \le q \le 10.$1

using flat noise, $q \equiv \frac{m_2}{m_1}, \qquad 1 \le q \le 10.$2. The average FF over the test set is reported as the “accuracy” (Liao et al., 2021).

A central implication of these results is that mass ratio can be learned as a conditioning axis of a waveform manifold. The paper explicitly states that “with the mainly low-mass-ratio training set, the resultant trained model is capable of generating large amount of accurate high-mass-ratio waveforms.” Because the HMR templates are only about $q \equiv \frac{m_2}{m_1}, \qquad 1 \le q \le 10.$3 of the total training set yet the HMR average overlap remains $q \equiv \frac{m_2}{m_1}, \qquad 1 \le q \le 10.$4, the result suggests a near-extrapolative generative capability within the tested range. The same work reports a generation time of about one millisecond per waveform, approximately $q \equiv \frac{m_2}{m_1}, \qquad 1 \le q \le 10.$5 to $q \equiv \frac{m_2}{m_1}, \qquad 1 \le q \le 10.$6 times faster than the EOBNR algorithm on the same computing facility, and proposes progressive self-training as a possible route toward mass ratios greater than $q \equiv \frac{m_2}{m_1}, \qquad 1 \le q \le 10.$7 in future work (Liao et al., 2021).

3. Universal rescaling in the extreme-mass-ratio limit

In the extreme-mass-ratio literature, the waveform can be organized around self-similar scaling in $q \equiv \frac{m_2}{m_1}, \qquad 1 \le q \le 10.$8. The “universal waveform” construction for spinless binary black hole merger in the extreme mass-ratio limit treats the smaller black hole as a test particle orbiting a Schwarzschild black hole. The orbital dynamics are split into two stages: adiabatic inspiral outside the ISCO and geodesic plunge inside the ISCO. The effective potential is

$q \equiv \frac{m_2}{m_1}, \qquad 1 \le q \le 10.$9

with radial equation

$(m_1,m_2)$0

For Schwarzschild,

$(m_1,m_2)$1

Outside the ISCO, the inspiral is approximated as a sequence of nearly circular geodesics satisfying

$(m_1,m_2)$2

and evolved in terms of rescaled variables

$(m_1,m_2)$3

The adiabatic equations are

$(m_1,m_2)$4

with

$(m_1,m_2)$5

The associated waveform is

$(m_1,m_2)$6

This expression makes the universality precise: time appears as $(m_1,m_2)$7, phase as $(m_1,m_2)$8, and amplitude with an overall factor of $(m_1,m_2)$9 (Strusberg et al., 12 May 2025).

Inside the ISCO, the plunge is approximated by a geodesic universal infall trajectory and solved with the Regge–Wheeler–Zerilli equation. The plunge waveform is written in retarded time

$-\log p_{\theta}(x)\le \mathbf{E}_{z\sim q_{\phi}(z|x)}[-\log p_{\theta}(\tilde{x}|z)] + D_{\mathrm{KL}\!\left[q_{\phi}(z|x)\,\|\,\mathcal{N}(0,1)\right],$0

and shifted so that its peak occurs at $-\log p_{\theta}(x)\le \mathbf{E}_{z\sim q_{\phi}(z|x)}[-\log p_{\theta}(\tilde{x}|z)] + D_{\mathrm{KL}\!\left[q_{\phi}(z|x)\,\|\,\mathcal{N}(0,1)\right],$1. The complete waveform is then assembled by sharply connecting the adiabatic inspiral for $-\log p_{\theta}(x)\le \mathbf{E}_{z\sim q_{\phi}(z|x)}[-\log p_{\theta}(\tilde{x}|z)] + D_{\mathrm{KL}\!\left[q_{\phi}(z|x)\,\|\,\mathcal{N}(0,1)\right],$2 to the plunge waveform for $-\log p_{\theta}(x)\le \mathbf{E}_{z\sim q_{\phi}(z|x)}[-\log p_{\theta}(\tilde{x}|z)] + D_{\mathrm{KL}\!\left[q_{\phi}(z|x)\,\|\,\mathcal{N}(0,1)\right],$3 (Strusberg et al., 12 May 2025).

The near-ISCO transition is controlled by the Ori–Thorne formalism. With

$-\log p_{\theta}(x)\le \mathbf{E}_{z\sim q_{\phi}(z|x)}[-\log p_{\theta}(\tilde{x}|z)] + D_{\mathrm{KL}\!\left[q_{\phi}(z|x)\,\|\,\mathcal{N}(0,1)\right],$4

$-\log p_{\theta}(x)\le \mathbf{E}_{z\sim q_{\phi}(z|x)}[-\log p_{\theta}(\tilde{x}|z)] + D_{\mathrm{KL}\!\left[q_{\phi}(z|x)\,\|\,\mathcal{N}(0,1)\right],$5

and $-\log p_{\theta}(x)\le \mathbf{E}_{z\sim q_{\phi}(z|x)}[-\log p_{\theta}(\tilde{x}|z)] + D_{\mathrm{KL}\!\left[q_{\phi}(z|x)\,\|\,\mathcal{N}(0,1)\right],$6, the expanded radial equation implies a transition region scaling

$-\log p_{\theta}(x)\le \mathbf{E}_{z\sim q_{\phi}(z|x)}[-\log p_{\theta}(\tilde{x}|z)] + D_{\mathrm{KL}\!\left[q_{\phi}(z|x)\,\|\,\mathcal{N}(0,1)\right],$7

hence a characteristic timescale

$-\log p_{\theta}(x)\le \mathbf{E}_{z\sim q_{\phi}(z|x)}[-\log p_{\theta}(\tilde{x}|z)] + D_{\mathrm{KL}\!\left[q_{\phi}(z|x)\,\|\,\mathcal{N}(0,1)\right],$8

From this, the near-ISCO phase error obeys

$-\log p_{\theta}(x)\le \mathbf{E}_{z\sim q_{\phi}(z|x)}[-\log p_{\theta}(\tilde{x}|z)] + D_{\mathrm{KL}\!\left[q_{\phi}(z|x)\,\|\,\mathcal{N}(0,1)\right],$9

The paper refines this to

$\nu=\frac{m}{M}\ll 1$0

and concludes that the waveform accumulates more than one radian of phase error near the ISCO only for approximately

$\nu=\frac{m}{M}\ll 1$1

The circular-orbit flux $\nu=\frac{m}{M}\ll 1$2 is computed to relative accuracy $\nu=\frac{m}{M}\ll 1$3, sufficient for waveforms down to about $\nu=\frac{m}{M}\ll 1$4. Comparison with FEW gives phase error below $\nu=\frac{m}{M}\ll 1$5 radian up to twice the ISCO, while comparison with TEOBResumS shows a phase mismatch reaching about one radian around $\nu=\frac{m}{M}\ll 1$6, about one gravitational radius outside the ISCO (Strusberg et al., 12 May 2025).

This universal construction gives a distinct sense in which a ratio waveform becomes reusable: the expensive numerical ingredients are computed once in normalized form, and application to any specific EMRI proceeds by rescaling time, phase, and amplitude rather than recomputing the full problem (Strusberg et al., 12 May 2025).

4. Perturbative and hybrid waveform banks at intermediate and large mass ratio

A separate strand of mass-ratio waveform research constructs waveform banks by combining sparse high-fidelity information with perturbative evolution. For initially nonspinning black-hole binaries in the range

$\nu=\frac{m}{M}\ll 1$7

Nakano, Lousto, and Zlochower model the small body’s trajectory in two pieces: a modified PN-inspired quasicircular inspiral and a Schwarzschild plunging geodesic attached at a matching radius near the light ring,

$\nu=\frac{m}{M}\ll 1$8

The perturbative waveform is then obtained from a spin-regge-wheeler-zzerilli framework driven by the particle trajectory $\nu=\frac{m}{M}\ll 1$9. At large extraction radius, the strain is reconstructed from the perturbative master functions via

$\varepsilon:=\mu/M\ll 1.$0

Using the Advanced LIGO noise curve, the reported $\varepsilon:=\mu/M\ll 1.$1-mode matches are

• $\varepsilon:=\mu/M\ll 1.$2: $\varepsilon:=\mu/M\ll 1.$3
• $\varepsilon:=\mu/M\ll 1.$4: $\varepsilon:=\mu/M\ll 1.$5
• $\varepsilon:=\mu/M\ll 1.$6: $\varepsilon:=\mu/M\ll 1.$7

Nonleading modes are also reported to have high matches, including $\varepsilon:=\mu/M\ll 1.$8 and $\varepsilon:=\mu/M\ll 1.$9, though the $$\Phi=\varepsilon^{-1}\left[\Phi^{(0)}+\varepsilon\,\Phi^{(1)}+O(\varepsilon^2)\right],$$0 mode is somewhat worse. The central conclusion is that a sparse set of full numerical-relativity simulations, together with fitted trajectories and perturbation theory, can generate a waveform bank with overlaps around $$\Phi=\varepsilon^{-1}\left[\Phi^{(0)}+\varepsilon\,\Phi^{(1)}+O(\varepsilon^2)\right],$$1 in the intermediate-mass-ratio regime (Nakano et al., 2011).

In the larger-mass-ratio limit treated in black-hole perturbation theory, Bernuzzi and collaborators compute inspiral-plunge-merger-ringdown waveforms at future null infinity for

$$\Phi=\varepsilon^{-1}\left[\Phi^{(0)}+\varepsilon\,\Phi^{(1)}+O(\varepsilon^2)\right],$$2

The dynamics are driven by a leading-order-in-$$\Phi=\varepsilon^{-1}\left[\Phi^{(0)}+\varepsilon\,\Phi^{(1)}+O(\varepsilon^2)\right],$$3, 5PN-resummed EOB radiation reaction, while the waveform is computed by solving the RWZ equations on a spacelike foliation with a hyperboloidal layer so that $$\Phi=\varepsilon^{-1}\left[\Phi^{(0)}+\varepsilon\,\Phi^{(1)}+O(\varepsilon^2)\right],$$4 is included in the computational domain. A major result is that finite-radius waveforms differ measurably from null-infinity waveforms. For $$\Phi=\varepsilon^{-1}\left[\Phi^{(0)}+\varepsilon\,\Phi^{(1)}+O(\varepsilon^2)\right],$$5, extractions at $$\Phi=\varepsilon^{-1}\left[\Phi^{(0)}+\varepsilon\,\Phi^{(1)}+O(\varepsilon^2)\right],$$6 differ from $$\Phi=\varepsilon^{-1}\left[\Phi^{(0)}+\varepsilon\,\Phi^{(1)}+O(\varepsilon^2)\right],$$7 by phase shifts of order $$\Phi=\varepsilon^{-1}\left[\Phi^{(0)}+\varepsilon\,\Phi^{(1)}+O(\varepsilon^2)\right],$$8 rad at $$\Phi=\varepsilon^{-1}\left[\Phi^{(0)}+\varepsilon\,\Phi^{(1)}+O(\varepsilon^2)\right],$$9, decreasing to about $O(\varepsilon^0)$0 rad at $O(\varepsilon^0)$1, while amplitude differences are about $O(\varepsilon^0)$2 at $O(\varepsilon^0)$3 and about an order of magnitude smaller at $O(\varepsilon^0)$4. These are larger than the numerical error bars $O(\varepsilon^0)$5 and are therefore identified as physical finite-radius effects rather than numerical noise (Bernuzzi et al., 2011).

The same study provides an internal self-consistency test by comparing angular-momentum loss from the EOB radiation reaction with the gravitational-wave angular-momentum flux computed at $O(\varepsilon^0)$6. For $O(\varepsilon^0)$7, the relative difference is about $O(\varepsilon^0)$8 initially, rises only to about $O(\varepsilon^0)$9 at LSO crossing, and stays small deep into plunge. This supports the use of the 5PN-resummed EOB radiation reaction as a driver for high-mass-ratio waveform generation in the perturbative regime (Bernuzzi et al., 2011).

5. Mass-ratio-independent dephasing and the structure of phase corrections

Mass ratio governs not only the location of a system in parameter space but also the perturbative hierarchy of phase corrections. In the self-force framework for extreme and intermediate mass-ratio inspirals, the force expansion is

$\nu=\frac{m}{M}\ll 1$00

while the waveform phase expansion is

$\nu=\frac{m}{M}\ll 1$01

This means that the first-order conservative self-force and the second-order dissipative self-force, although different in force order, both contribute at $\nu=\frac{m}{M}\ll 1$02 to the observable dephasing (Burko et al., 2013).

Warburton, Akcay, Barack, Gair, and Sago analyze quasi-circular Schwarzschild inspirals evolved from $\nu=\frac{m}{M}\ll 1$03 down to the ISCO at $\nu=\frac{m}{M}\ll 1$04, including first- and second-order dissipative self-force, first-order conservative self-force, and spin-orbit coupling. The spin pseudovector is taken as

$\nu=\frac{m}{M}\ll 1$05

and the equations of motion include explicit $\nu=\frac{m}{M}\ll 1$06-dependent spin-orbit terms. Their dephasing analysis shows that for $\nu=\frac{m}{M}\ll 1$07, the total dephasing due to spin-orbit coupling ranges over

$\nu=\frac{m}{M}\ll 1$08

while the self-force-only dephasing at $\nu=\frac{m}{M}\ll 1$09 is

$\nu=\frac{m}{M}\ll 1$10

dominated by the first-order conservative self-force. The paper further notes, using results from the preceding study, that the second-order dissipative self-force contributes about $\nu=\frac{m}{M}\ll 1$11 of the first-order conservative dephasing (Burko et al., 2015).

A particularly sharp result is the near-cancellation between self-force and spin-orbit dephasing. The dephasing as a function of spin is fitted by

$\nu=\frac{m}{M}\ll 1$12

Equating this to the self-force-only dephasing yields a cancellation point at

$\nu=\frac{m}{M}\ll 1$13

At this spin value, the waveform including spin-orbit coupling and self-force is almost indistinguishable from the waveform with

$\nu=\frac{m}{M}\ll 1$14

the overlap differs from unity by less than $\nu=\frac{m}{M}\ll 1$15, and the phase difference never exceeds about

$\nu=\frac{m}{M}\ll 1$16

The stated implication is that omitting mass-ratio-independent dephasing terms can produce a non-perturbative error in parameter estimation even when overlaps remain extremely high (Burko et al., 2015).

The preceding paper in the same series isolates the second-order dissipative contribution by comparing three waveform models: WF-I with only first-order dissipative terms, WF-II adding the first-order conservative self-force, and WF-III adding the second-order dissipative self-force. For inspirals from $\nu=\frac{m}{M}\ll 1$17 down to near the ISCO, the dephasing between WF-III and WF-II is about $\nu=\frac{m}{M}\ll 1$18 of the dephasing between WF-II and WF-I, with negative sign, and the overlap changes only at roughly the $\nu=\frac{m}{M}\ll 1$19 level when the second-order dissipative term is added. This establishes the second-order dissipative effect as systematic but substantially smaller than the first-order conservative contribution in the regime studied (Burko et al., 2013).

6. Ratio-constrained waveform design beyond gravitational astronomy

A different but structurally related use of a ratio parameter appears in RIS-aided integrated sensing and communication waveform design. The problem is to jointly optimize the transmit waveform $\nu=\frac{m}{M}\ll 1$20 and RIS passive beamforming phase shifts $\nu=\frac{m}{M}\ll 1$21 so that the system reduces multi-user interference, approximates a desired radar beampattern, and satisfies a tunable PAPR constraint. The communication model is

$\nu=\frac{m}{M}\ll 1$22

where $\nu=\frac{m}{M}\ll 1$23 is the MUI term. The beampattern is

$\nu=\frac{m}{M}\ll 1$24

and the joint design problem is formulated as

$\nu=\frac{m}{M}\ll 1$25

Here $\nu=\frac{m}{M}\ll 1$26 controls the sensing/communication trade-off, while $\nu=\frac{m}{M}\ll 1$27 controls the allowable ratio of peak to average power (Wu et al., 10 Apr 2025).

The waveform-design method is MO-ADMM, combining ADMM for waveform optimization under PAPR and power constraints, a closed-form update for the radar template $\nu=\frac{m}{M}\ll 1$28, and manifold optimization for the RIS phase shifts under unit-modulus constraints. The full procedure alternates: update $\nu=\frac{m}{M}\ll 1$29 by ADMM, update $\nu=\frac{m}{M}\ll 1$30 in closed form, and update $\nu=\frac{m}{M}\ll 1$31 by manifold optimization until convergence (Wu et al., 10 Apr 2025).

The reported simulations use $\nu=\frac{m}{M}\ll 1$32 BS antennas, $\nu=\frac{m}{M}\ll 1$33 users, $\nu=\frac{m}{M}\ll 1$34 RIS elements, frame length $\nu=\frac{m}{M}\ll 1$35, transmit power $\nu=\frac{m}{M}\ll 1$36 dBm, and targets at $\nu=\frac{m}{M}\ll 1$37. The paper states that the algorithm drives the waveform PAPR to the prescribed threshold $\nu=\frac{m}{M}\ll 1$38, with smaller $\nu=\frac{m}{M}\ll 1$39 requiring more iterations. Under high SNR conditions, compared to systems without RIS, the RIS-aided ISAC system achieves approximately $\nu=\frac{m}{M}\ll 1$40 communication-rate improvement and at least $\nu=\frac{m}{M}\ll 1$41 dB improvement in beampattern error (Wu et al., 10 Apr 2025).

Within a broader taxonomy of ratio waveforms, this case shows that ratio variables need not parameterize source physics. They may instead parameterize hardware admissibility, with the waveform family shaped by power-amplifier compatibility rather than dynamical scaling. That distinction is not explicit in the cited paper, but it is a plausible implication of placing PAPR-constrained waveform design alongside mass-ratio-dependent gravitational-wave generation.

7. Synthesis and scope

Across the cited literature, ratio-dependent waveform construction falls into three technically distinct categories. First, there are label-conditioned waveform manifolds, where the ratio enters as a conditioning parameter and the generator learns a source-to-waveform map, as in conditional autoencoders over $\nu=\frac{m}{M}\ll 1$42 (Liao et al., 2021). Second, there are asymptotically universal waveforms in which the ratio determines the rescaling of time, phase, amplitude, and phase-error structure, as in the EMRI construction with $\nu=\frac{m}{M}\ll 1$43 (Strusberg et al., 12 May 2025). Third, there are perturbative and hybrid waveform banks in which the ratio fixes the validity regime of a trajectory model, self-force expansion, or null-infinity extraction strategy, as seen in intermediate-mass-ratio perturbative templates and large-mass-ratio RWZ–EOB calculations (Nakano et al., 2011, Bernuzzi et al., 2011).

The self-force literature adds an important caution: a waveform can have excellent overlap while still omitting physically necessary mass-ratio-independent dephasing terms. The cancellation at $\nu=\frac{m}{M}\ll 1$44 illustrates that high overlap alone does not guarantee faithful parameter inference when different $\nu=\frac{m}{M}\ll 1$45 phase contributions compensate each other (Burko et al., 2015). Likewise, the finite-radius versus null-infinity comparison shows that waveform differences of $\nu=\frac{m}{M}\ll 1$46 to $\nu=\frac{m}{M}\ll 1$47 rad can persist even when extraction is performed at large radii, so “where” the waveform is defined is itself part of ratio-sensitive modeling at high precision (Bernuzzi et al., 2011).

Taken together, these works depict waveform space as strongly structured by ratio variables. In gravitational-wave science, mass ratio controls conditioning, interpolation, asymptotic universality, perturbative ordering, and dephasing systematics. In ISAC waveform design, PAPR controls feasibility under nonlinear hardware. The common theme is that a ratio parameter does not merely annotate a waveform family; it organizes how that family is generated, constrained, compared, and ultimately used in inference or system design.