Gravitational Wave Tests of General Relativity with Ground-Based Detectors and Pulsar Timing Arrays

TL;DR

This review surveys how gravitational waves from ground-based detectors and pulsar timing arrays enable tests of General Relativity in the non-linear, dynamical strong-field regime. It contrasts top-down direct tests of specific modified theories with bottom-up generic tests (e.g., ppE framework) to identify potential deviations from GR and quantify them with Bayesian and frequentist methods. It covers a broad landscape of alternative theories—scalar-tensor models, massive gravitons, modified quadratic gravity, variable G, non-commutative geometry, and parity-violating gravity—and discusses their predicted waveform modifications, polarization content, and propagation effects. The article also details detector responses, data-analysis techniques, and strategies for probing no-hair theorems through inspiral, merger, and ringdown phases, highlighting the complementary roles of ground-based interferometers and pulsar timing arrays. Overall, it emphasizes the need for robust theoretical modeling and cross-dataset analysis to leverage GW observations for rigorous tests of GR and potential new physics.

Abstract

This review is focused on tests of Einstein's theory of General Relativity with gravitational waves that are detectable by ground-based interferometers and pulsar timing experiments. Einstein's theory has been greatly constrained in the quasi-linear, quasi-stationary regime, where gravity is weak and velocities are small. Gravitational waves will allow us to probe a complimentary, yet previously unexplored regime: the non-linear and dynamical strong-field regime. Such a regime is, for example, applicable to compact binaries coalescing, where characteristic velocities can reach fifty percent the speed of light and compactnesses can reach a half. This review begins with the theoretical basis and the predicted gravitational wave observables of modified gravity theories. The review continues with a brief description of the detectors, including both gravitational wave interferometers and pulsar timing arrays, leading to a discussion of the data analysis formalism that is applicable for such tests. The review ends with a discussion of gravitational wave tests for compact binary systems.

Figures (5)

• Figure 1: Detector coordinate system and gravitational wave coordinate system.
• Figure 2: Antenna pattern response functions of an interferometer (see Eqs. \ref{['IFOresponse']}) for $\psi=0$. Panels (a) and (b) show the plus ($|F_+|$) and cross ($|F_{\times}|$) modes, panels (c) and (d) the vector x and vector y modes ($|F_{x}|$ and $|F_{y}|$), and panel (e) shows the scalar modes (up to a sign, it is the same for both breathing and longitudinal). Color indicates the strength of the response with red being the strongest and blue being the weakest. The black lines near the center give the orientation of the interferometer arms.
• Figure 3: Antenna patterns for the pulsar-Earth system. The plus mode is shown in (a), breathing modes in (b), the vector-x mode in (c), and longitudinal modes in (d), as computed from Eq. \ref{['PulsarResponse2']}. The cross mode and the vector-y mode are rotated versions of the plus mode and the vector-x mode, respectively, so we did not include them here. The GW propagates in the positive z-direction with the Earth at the origin, and the antenna pattern depends on the pulsar's location. The color indicates the strength of the response, red being the largest and blue the smallest.
• Figure 4: Schematic diagram of the projection of the data stream ${\boldsymbol{d}}$ orthogonal to the GR subspace spanned by $F^+$ and $F^\times$, along with a perpendicular subspace, for 3 detectors to build the GR null stream.
• Figure 5: (Top) Fitting curves (solid curve) and numerical results (points) of the universal I-Love (left) and Q-Love (right) relations for various equations of state, normalized as $\bar{I} = I/M_{\rm NS}^{3}$, $\bar{\lambda}^{(\rm tid)} = \lambda^{(\rm tid)}/M_{\rm NS}^{5}$ and $\bar{Q} = -Q^{(\rm rot)}/[M_{\rm NS}^{3} (S/M_{\rm NS}^{2})^{2}]$, $M_{\rm NS}$ is the neutron-star mass, $\lambda^\mathrm{(tid)}$ is the tidal Love number, $Q^\mathrm{(rot)}$ is the rotation-induced quadrupole moment, and $S$ is magnitude of the neutron star spin angular momentum. The neutron star central density is the parameter varied along each curve, or equivalently the NS compactness. The top axis shows the neutron star mass for the APR equation of state, with the vertical dashed line showing $M_{\rm NS} = 1 M_{\odot}$. (Bottom) Relative fractional errors between the fitting curve and the numerical results. Observe that these relations are essentially independent of the equation of state, with loss of universality at the 1% level.