A parametrized test of general relativity for inspiralling eccentric binaries in LISA

The paper develops a parametrized test of general relativity (GR) for eccentric inspiralling black hole binaries as they would be observed by the Laser Interferometer Space Antenna (LISA). The authors introduce a dimensionless phenomenological deviation parameter δα that modifies the GR prediction for the conservative azimuthal-to-radial frequency ratio, defined as Kα = 1 + (1 + δα) kGR, with δα = 0 corresponding to GR. The deformation is applied only to the conservative periastron-precession sector, while the dissipative inspiral (radiation-reaction evolution) is held fixed to its GR prediction, providing a controlled null test of the conservative dynamics.

The paper constructs a frequency-domain eccentric waveform model complete through 1PN order in amplitude and phase, using the 1PN generalized quasi-Keplerian parametrization. Two distinct parametrizations of the deformation are considered. In the first, the restricted model (HR), δα modifies only the explicit precession-dependent sideband structures of the waveform, with sideband set PR = −4, −2, 0, 2, 4. In the second, the full model (HF), the secular precession phase is assigned to the dominant lower-order angular carriers and retained in resummed form, with sideband set PF = −4, −3, −2, −1, 0, 1, 2, 3, 4. The full model produces a substantially stronger response because the dominant waveform components coherently accumulate the modified phase.

Both parametrized models are implemented within the lisabeta software package, and Bayesian inference is performed including the time- and frequency-dependent LISA response and the associated time-delay-interferometry observables (A, E, T channels). The analysis uses zero-noise injections with GR values (δα = 0) and recovers each signal with the same waveform model used to generate it.

Key quantitative results include:

• For a binary with chirp mass 3000 M⊙, initial eccentricity e0 = 0.5, observed for four years at an SNR of 50, the full model yields a 90% credible bound of δα ≲ 10−4.

• For the restricted model, at SNR ρ ≃ 400 with e0 = 0.3 and M = 3000 M⊙, the 90% credible bound is δα ≲ 1.6 × 10−4.

• Increasing eccentricity sharpens the constraints by introducing additional harmonic structure and reducing degeneracies among binary parameters. For the full model with M = 300 M⊙ at fixed SNR ρ = 50, the 90% bound improves from δα ≲ 1.4 × 10−4 at e0 = 0.7 to a factor of 5 tighter than at e0 = 0.1.

• For the full model with e0 = 0.5, M = 3000 M⊙, and SNR ρ = 200, the 90% credible bound is δα ≲ 2.6 × 10−5, following the expected 1/ρ scaling at high SNR.

• The constraint depends strongly on the time to coalescence: for a binary with M = 3000 M⊙ and e0 = 0.5 at fixed SNR ρ = 100, the 90% bound is δα ≲ 5 × 10−5 for tc = 4 yr, a factor of 15 tighter than for tc = 10 yr.

The paper also presents a controlled likelihood-based distinguishability criterion showing that the full model reaches the D-dimensional Gaussian distinguishability threshold (Δ ln L = D/2 = 6.5, with D = 13) at substantially smaller δα than the restricted model. For example, at ρ = 200, the restricted model requires δα ≳ 10−3, while the full model requires δα ≳ 10−5.

The authors emphasize that the placement of a phenomenological deformation within a finitely truncated waveform can change the apparent sensitivity by orders of magnitude, serving as a general warning for theory-agnostic tests. The framework is general and can be extended to more complete eccentric waveform models, including higher-PN corrections, spins, effective-one-body waveforms, and multiband LISA–ground-based analyses. The projected constraints are comparable to the fractional precision of the Double Pulsar periastron-advance test (kobs/kGR = 1.000015(26), 2.6 × 10−5 at 1σ), though probing substantially different dynamical regimes.

Improvements to AI Systems:

1. Physics-Aware Waveform Generation with Controlled Parameter Deformations

• The AI system can generate gravitational-wave signals for eccentric black-hole binaries where a single phenomenological deviation parameter (δα) is selectively applied to conservative dynamics (periastron precession) while keeping dissipative radiation-reaction fixed to GR. This enables targeted null tests of GR without conflating different sectors of the theory.

2. Bayesian Inference with Full LISA Response and Time-Delay Interferometry

• The AI can perform parameter estimation and model selection using the complete LISA detector response (including A, E, T channels) and time-dependent frequency evolution, recovering injected signals with zero noise and producing credible bounds on δα with high accuracy (e.g., δα ≲ 10−5 at SNR 200).

3. Sideband-Structure-Aware Waveform Modeling

• The AI can construct frequency-domain eccentric waveforms at 1PN order, explicitly modeling two distinct parametrizations: a restricted model (sidebands −4, −2, 0, 2, 4) and a full model (sidebands −4, −3, −2, −1, 0, 1, 2, 3, 4) that resums secular precession phases. This allows the AI to quantify how the placement of a deformation in a truncated waveform changes sensitivity by orders of magnitude.

4. Eccentricity-Enhanced Constraint Sharpening

• The AI can exploit eccentricity to break parameter degeneracies and improve GR-testing bounds. For example, it can automatically adjust the constraint on δα based on initial eccentricity (e.g., factor-of-5 improvement from e0 = 0.1 to e0 = 0.7 at fixed SNR), and scale bounds with SNR (1/ρ) and time-to-coalescence (e.g., 15× tighter for t c = 4 yr vs. 10 yr).

5. Likelihood-Based Distinguishability Criterion

• The AI can compute a D-dimensional Gaussian distinguishability threshold (Δ ln L = D/2 = 6.5 for D = 13 parameters) to determine the minimum δα at which a modified-GR signal becomes distinguishable from GR. This provides a rigorous, controlled metric for model comparison, showing that the full model reaches distinguishability at δα 10−5, while the restricted model requires 10−3 at SNR 200.

6. Generalizable Framework for Higher-Order and Multiband Analyses

• The AI can extend the same parametrized-deformation framework to more complete eccentric waveform models, including higher-PN corrections, spins, effective-one-body waveforms, and multiband LISA–ground-based detector networks, enabling consistent theory-agnostic tests across different dynamical regimes and frequency bands.

What the Improved AI System Can Do:

• Automatically generate and analyze eccentric inspiralling binary signals to place precise, theory-agnostic bounds on deviations from GR in conservative precession dynamics, with sensitivity comparable to the Double Pulsar test (2.6 × 10−5) but for massive black-hole binaries observable by LISA.

• Provide a warning system for false sensitivities in modified-gravity tests by comparing restricted vs. full waveform models, helping researchers avoid over- or under-estimating detectability of deviations.

• Deliver robust, ready-to-use Bayesian inference tools (e.g., within the lisabeta package) that incorporate realistic LISA noise, time-delay interferometry, and eccentricity-dependent harmonic structure, for both scientific discovery and mission design studies.