Testing general relativity with amplitudes of subdominant gravitational-wave modes

TL;DR

We present an extended subdominant-mode amplitude (SMA) test of general relativity that probes amplitude deviations in higher-order GW modes while fixing the dominant (2,2) mode, now including the (3,2) and (4,4) modes with the IMRPhenomXPHM model. The framework is rigorously benchmarked against Gaussian noise, numerical-relativity simulations, precession, and eccentricity, and shown to recover injected deviations while remaining null for GR-consistent signals; however, strong precession, high mass, and eccentricity can induce apparent deviations due to waveform systematics and unmodeled physics. The SMA test also responds coherently to phase perturbations, demonstrating sensitivity to a broader class of beyond-GR effects via parameter correlations. Applied to LVK O4 detections, the method yields the strongest current constraints on $δA_{33}$ and $δA_{44}$ for the corresponding HOMs, while highlighting cases where waveform modeling biases or low HOM content limit interpretability. Overall, SMA provides a robust, broadly sensitive, and discriminating framework for testing GR in the strong-field regime and a principled blueprint for assessing the robustness of GW-based tests.

Abstract

We present an improved subdominant-mode amplitude (SMA) test of general relativity (GR), which probes amplitude-level deviations in the higher-order modes of gravitational-wave (GW) signals from binary black hole mergers while keeping the dominant quadrupole mode fixed. Using a comprehensive parameter-estimation campaign, we benchmark the test against Gaussian noise fluctuations, waveform modeling systematics, and physical effects such as spin precession and orbital eccentricity. When applied to numerical-relativity simulations, the SMA test performs reliably for aligned-spin and mildly precessing systems but exhibits measurable biases for strongly precessing or eccentric binaries. Although designed to detect amplitude deviations, the test also responds coherently to phase perturbations, yielding apparent GR violations when applied to phase-modified waveforms. Applied to recent GW detections, we report the strongest constraint on the hexadecapolar (4,4) mode amplitude deviation, $δA_{44} = -0.30^{+1.16}_{-3.45}$, consistent with GR. With these results, this work establishes the SMA test as a robust and broadly sensitive framework for probing strong-field gravity and demonstrates a systematic approach for assessing the robustness of GW tests of GR.

Figures (16)

• Figure 1: Absolute value of $Y^{-2}_{\ell m}$ as a function of $\theta_{JN}$ for different $(\ell,m)$ modes. Among the listed modes, only $(2,2)$ and $(3,2)$ contribute for face-on binaries, and other HOMs become relatively more important as inclination increases.
• Figure 2: The ratio of the orthogonal , $\rho_{\ell m}^{\perp}$, in the $(\ell,m)$ modes compared to the $(2,2)$ mode, as a function of the total mass $M_{\rm tot}$ and the mass ratio $q$ of the binary.
• Figure 3: P-P plot for the deviation parameters in a $(32\mathrm{\, M_\odot},8\mathrm{\, M_\odot})$ BBH across 20 Gaussian noise realizations. The shaded regions indicate the $1\sigma$, $2\sigma$, and $3\sigma$ confidence intervals (CI).
• Figure 4: Probability distributions for $\delta A_{33}$ across 20 parameter estimation runs in different Gaussian noise realizations. Several runs show significant bimodality in the $\delta A_{33}$ posteriors, which is attributed to the feect of Gaussian noise and parameter degeneracies (see Appendix \ref{['app:degeneracy']} for more details).
• Figure 5: Posterior distributions for $\delta A_{\ell m}$ from the SMA test on SXS simulations listed in Tab. \ref{['tab:nr_runs']}. The black dashed line marks $\delta A_{\ell m} = 0$, while the dotted lines denote the $90\%$ credible intervals. $\log \mathcal{B}$ represents the logarithm of the Bayes factor in favor of the non-GR hypothesis.