An effective formalism for testing extensions to General Relativity with gravitational waves

TL;DR

This work constructs the most general effective field theory extension of General Relativity that can be tested with gravitational waves while avoiding new light degrees of freedom and preserving locality, causality, and unitarity. The leading corrections come from three curvature-based operators $\mathcal{C}^2$, $\tilde{\mathcal{C}}^2$, and $\mathcal{C}\tilde{\mathcal{C}}$, suppressing by scales $\Lambda$, $\tilde{\Lambda}$, and $\Lambda_-$, and they modify both the binary potential and the multipole radiation, thus altering the GW waveform in a controlled post-Newtonian expansion. The authors develop an NR-EFT framework to compute corrections to the potential and to radiative couplings, showing that the observable effects scale with $(\Lambda r)^{-6}$ and the orbital velocity $v$, enabling potential constraints from LIGO-Virgo events. They also discuss numerical strategies for simulating mergers within this EFT, and analyze constraints from weak-field tests, X-ray binaries, and cosmology under a UV-softness assumption, concluding that GW observations offer a promising probe of strong-field gravity within a broad and consistent theoretical setup.

Abstract

The recent direct observation of gravitational waves (GW) from merging black holes opens up the possibility of exploring the theory of gravity in the strong regime at an unprecedented level. It is therefore interesting to explore which extensions to General Relativity (GR) could be detected. We construct an Effective Field Theory (EFT) satisfying the following requirements. It is testable with GW observations; it is consistent with other experiments, including short distance tests of GR; it agrees with widely accepted principles of physics, such as locality, causality and unitarity; and it does not involve new light degrees of freedom. The most general theory satisfying these requirements corresponds to adding to the GR Lagrangian operators constructed out of powers of the Riemann tensor, suppressed by a scale comparable to the curvature of the observed merging binaries. The presence of these operators modifies the gravitational potential between the compact objects, as well as their effective mass and current quadrupoles, ultimately correcting the waveform of the emitted GW.

Figures (11)

• Figure 1: Radiative generation of $R_{\mu\nu\rho\sigma}^{2n}$ operator, in this case $R_{\mu\nu\rho\sigma}{}^{8}$, through ${\mathcal{C}}^2$ and $\tilde{\mathcal{C}}^2$.
• Figure 2: The two loop diagram with the "cross" topology. Arrows indicate the direction of the momentum flow. The $m$'s in the vertex specifies that we used the leading source-gravity vertex.
• Figure 3: The two loop diagram with the "peace/log" topology.
• Figure 4: One of the leading contributions to the effective potential from $\tilde{\mathcal{C }}^2$ with the "peace/log" topology. The dashed arrow indicates the contraction of tensor indices within one of the two $\tilde{\mathcal{C}}$, the other becoming at this point automatic.
• Figure 5: One of the other leading contributions to the effective potential from $\tilde{\mathcal{C }}^2$ with the "peace/log" topology. The dashed arrow indicates contraction of indices within $\tilde{\mathcal{C }}.$