Inspiral tests of general relativity and waveform geometry

TL;DR

This work reframes tests of general relativity with gravitational-wave signals using a geometric view of the waveform manifold, focusing on how beyond-GR deviations bias parameter estimation via the Cutler-Vallisneri formalism and how the metric structure governs detectability and Bayes factors.It demonstrates that parameterized post-Einsteinian templates capture generic phase deviations because the perpendicular component to the GR manifold, $\Delta h_{\perp}$, drives evidence for deviations, and the overlap between different deviations explains observed degeneracies.The authors show that the leading ppE terms often suffice to capture a wide class of deviations, but that multiparameter tests suffer from strong degeneracies that inflate uncertainties; to address this, they introduce an SVD-based method to extract orthogonal, information-rich directions in the perpendicular ppE space and construct robust, data-driven tests using a packed frequency-domain representation.Overall, the paper provides a principled framework linking waveform geometry, bias, Bayes factors, and dimensionality reduction to improve GR tests in current and future GW detector networks, with practical implications for disentangling spin, eccentricity, systematics, glitches, and genuine new physics.

Abstract

The phase evolution of gravitational waves encodes critical information about the orbital dynamics of binary systems. In this work, we test the robustness of parameterized tests against unmodeled deviations from general relativity. We demonstrate that these parameterized tests are flexible and sensitive in detecting generic deviations in the waveform using the Cutler-Vallisneri bias formalism. This universality arises from examining the inherent geometry of the waveform signal and understanding how biases manifest. We show how Bayes factors are governed by the intrinsic geometry of the waveform signal manifold when parameterized tests are used to approximate generic violations of GR. We use the singular value decomposition to propose templates that are orthogonal to parameterized tests, identifying degeneracies and enhancing the detection of potential deviations. More broadly, the geometric framework developed here clarifies -- at a fundamental level -- how subtle waveform effects (including orbital eccentricity, spin precession, waveform systematics, and instrumental glitches) can mimic one another in data, and when they are intrinsically distinguishable.

Figures (11)

• Figure 1: Illustration of degeneracy when testing GR. We show the injected signal (blue) which depends on the true GR parameters $\bf\theta_\mathrm{t}$ and the beyond GR parameters $\bf\lambda_\mathrm{t}$. The model signal at the true GR parameters $\bf\theta_\mathrm{t}$ (red) is shown and the best fit signal is at the maximum likelihood point $\bf\theta_\mathrm{ML}$ (black). The GR waveform is modified by $\Delta h$ which causes biases to the GR waveform, thus residual signal to measure beyond GR deviations is given by the perpendicular signal $\Delta h^\perp$. Note that this is a high dimensional manifold where $(d_1, d_2, d_3)$ are the values of the signal at particular frequency bins.
• Figure 2: In this plot, we visually show how the ppE tests of GR can capture generic bGR deviations. The GR manifold (black) is a line that is at the intersection between the true bGR manifold (red) and the ppE manifold (blue). One can see that the perpendicular part of the signal from GR is $\rho_\perp = \Vert \Delta h_\perp\Vert$ which is the residual SNR after allowing the GR parameters to be biased $h_\mathrm{GR}^\mathrm{ML} = h_\mathrm{GR}(\theta_\mathrm{t} + \Delta\theta_\mathrm{bias})$. One can see that the best fit ppE parameter is located at the blue mark and has residual SNR $\rho_\perp^\mathrm{ppE} = \mathcal{O} \rho_\perp$ as given in Eq. \ref{['eq:captured-residual']}. Finally, the brown line is the missed signal from our ppE model. One can see that the best fit for the ppE model corresponds to a value such that the residual (brown line) is perpendicular to the ppE manifold. With this picture in mind, we can explore how well tests of GR capture generic deviations.
• Figure 3: Residual amplitude for ppE injected deviations from GR for a GW150914-like detection. On the left and right, we include the GR waveform (black) and O3 Livingston ASD (grey). On the left, we show the waveform residuals that would be caused by injection what we show in Fig. \ref{['fig:dephasing-PN-150914']} from parameterized tests. On the right, we show what the residual deviation is after the stealth biases in the GR parameters are accounted for. This plot is made by tuning the ppE coefficients so that their residuals $\Vert\Delta h^{\perp \mathrm{GR}}\Vert=1$ are of unit size. This figure illustrates that the primary observable in these parameterized tests is the magnitude of the residual deviation (right), which is markedly reduced compared to the nominal deviations (left). https://github.com/BrianCSeymour/waveform-geometry-testing-gr/blob/main/ppe-perpendicular-plots.ipynb
• Figure 4: This figure shows how the ppE behave after GR parameter projection. On the left, we show the ppE dephasing for a GW150914-like detection and with $\delta\varphi_k$ normalized equivalently to Fig. \ref{['fig:amplitude-PN-150914']}. On the right, we show the residual (perpendicular) phase deviation in the injection after the GR biases are taken into account. The total dephasing of the perpendicular waveforms is noticeably smaller with the residual phase deviations $\Delta \Psi^{\perp \mathrm{GR}}$ displaying multiple zero crossings, akin to the behavior of an oscillatory polynomial. The near-identical appearance of the ppE dephasing residuals on the right visually demonstrates why leading-order deviations from GR appear similar to one another. https://github.com/BrianCSeymour/waveform-geometry-testing-gr/blob/main/ppe-perpendicular-plots.ipynb
• Figure 5: The overlap between deviations injected (y-axis) and the recovery model (x-axis). We use GW150914 event parameters with three detectors at O3 Livingston sensitivity, all extrinsic parameters measured and intrinsic parameters $(\mathcal{M}_c,q,\chi_\mathrm{eff},\chi_\mathrm{p})$. One can see that injecting a deviation at $k/2$-th PN order fractionally from GR is perfectly captured on the diagonal but has less overlap as the injection and recovery order grow. This means that if there is a true deviation, the significance drops slowly as you search for the wrong PN order $k_\mathrm{rec}\ne k_\mathrm{inj}$ because the intrinsic parameters will capture the difference. Note that we do not include $k=5$ in either of these plots because it is nearly completely degenerate with phase of coalescence. https://github.com/BrianCSeymour/waveform-geometry-testing-gr/blob/main/overlap-plots.ipynb