Effective Field Theories of Post-Newtonian Gravity: A comprehensive review

TL;DR

This review articulates how effective field theories provide a universal, multi-scale framework for post-Newtonian gravity and gravitational-wave modeling. It details a three-stage tower (removing $r_s$, then $r$, then $\lambda$) that recasts the two-body problem into a sequence of EFTs—one-particle, composite, and radiative multipoles—utilizing worldline DOFs, KK metric reduction, and a closed-time-path formalism to handle dissipation. Key results include high-precision PN results for spin and finite-size effects (up to 4PN in the conservative sector and beyond for dissipative effects), as well as the public EFTofPNG code that automates these calculations. The framework connects GR with quantum-field-theory techniques, exposes classical renormalization-group flows in gravitational radiation, and offers a path toward integrating advanced field-theory tools into GW data analysis and potentially into modified theories of gravity.

Abstract

[Abridged] This review article presents the progress made over the last decade, since the introduction of effective field theories (EFTs) into post-Newtonian (PN) gravity. These have been put forward in the context of gravitational waves (GWs) from the compact binary inspiral. The mature development of this interdisciplinary field has resulted in significant advances of wide interest to physics at several levels serving various purposes. The field has firmly demonstrated, that seemingly disparate physical domains, such as quantum field theory and classical gravity, are related, and that the EFT framework is a universal one, where it has been proven to supply a robust methodology to boost progress in the development of PN theory. The review is aimed at a broad audience, from general readers new to the field, to specialists and experts in related subjects. The review begins with an overview of the introduction of EFTs into classical gravity and their development. Then, the basic ideas, which form the conceptual foundation of EFTs, are provided, and the strategy of a multi-stage EFT framework, which is utilized for the PN binary inspiral problem, is outlined. The main body of the review is then dedicated to presenting in detail the study of each of the effective theories at each of the intermediate scales in the problem, up to the actual GW observables. The review is concluded with the multiple prospects of building on the progress in the field, and using further modern field theory insights and tools, to specifically address the study of GWs, as well as to broadly expand our fundamental understanding of gauge and gravity theories across the classical and quantum regimes.

Figures (21)

• Figure 1: The various elementary analytical and numerical methodologies used to study CBCs, depending on the mass ratio, $0<m_1/m_2\le1$, of the components of the binary, and the compactness of the system, evaluated by the parameter $M/r$, where $M=m_1+m_2$ is the total mass, and $r$ the typical size of the system. However, in order to model the complete CBC signal, one needs to resort to the comprehensive EOB framework, which enables their integration into the full theoretical GW templates, currently used in the GW detectors. Note that BH perturbation theory also accounts for the case of EMRIs, which will be targeted by the future space-based GW detector LISA. Reproduced with permission from Le Tiec Tiec:2014lba.
• Figure 2: The hierarchy of scales in the binary inspiral problem: $r_s$, the scale of the single compact object; $r$, the scale of the orbital separation between the components of the binary; $\lambda$, the wavelength of radiation, emitted from the inspiraling binary; it holds that $r_s\lll r\ll \lambda$. To eliminate the smallest scale of the single object, we construct the one-particle EFT. Next, we explicitly integrate out the field modes at the orbital separation scale, to be subsequently matched to the EFT of the composite particle, that is, the binary system. Finally, we eliminate the scale of radiation by explicitly integrating out the radiation field modes.
• Figure 3: A $\phi^4$ EFT through Wilson's approach: ${\hat{\phi}}$, the high momentum field component is represented by a double line, while $\phi$, the low momentum component, is represented by a single line. Note that the former will always appear as an internal line, whereas the latter will always appear as external. Diagram (a) is evaluated as $\frac{1}{2}\rho\phi^2$, where the new coefficient, $\rho$, gives a correction to the mass parameter, $m^2$. Diagram (b) is evaluated as $\frac{1}{4!}\zeta\phi^4$ at LO, where $\zeta$ is a correction to the coupling $\lambda$. Diagram (c) generates a new $\phi^6$ interaction with a new coupling constant, $\kappa$; and so on.
• Figure 4: Graph topologies that are excluded from the diagrammatic expansion of the two-particle interaction for the EFT of the composite particle Goldberger:2004jtGoldberger:2007hy: (a) Graph with more than a single connectivity component (where worldlines are stripped off); (b) Graph containing a graviton loop; (c) Graph renormalizing the UV divergence related with the Wilson coefficients of the one-particle EFT, e.g. the mass. We note that the bold vertical lines represent the worldlines, where time flows up, rather than from left to right as is customary in particle physics.
• Figure 5: The single topology at ${\cal{O}}(G^1)$: One-graviton exchange with no gravitational self-interaction.