An Effective Field Theory of Gravity for Extended Objects

TL;DR

This work formulates an effective field theory (NRGR) for non-relativistic, extended objects interacting with gravity to compute gravitational radiation spectra from binaries. By separating scales $r_s$, $r$, and $r/v$ and employing a worldline EFT with non-minimal curvature couplings, the authors achieve manifest velocity power counting and a clean decoupling between near-zone dynamics and far-zone radiation. They demonstrate that ${ m O}(v^6)$ finite-size effects can be removed by field redefinitions, while genuine tidal effects appear at ${ m O}(v^{10})$ and require matching to short-distance physics; logarithmic enhancements are handled via renormalization group flows of worldline couplings. The NRGR framework yields familiar PN results (e.g., EIH) and provides a systematic path to higher-order predictions, with clear extensions to include spins, absorptive effects, and potential deviations from general relativity.

Abstract

We present an Effective Field Theory (EFT) formalism which describes the dynamics of non-relativistic extended objects coupled to gravity. The formalism is relevant to understanding the gravitational radiation power spectra emitted by binary star systems, an important class of candidate signals for gravitational wave observatories such as LIGO or VIRGO. The EFT allows for a clean separation of the three relevant scales: r_s, the size of the compact objects, r the orbital radius and r/v, the wavelength of the physical radiation (where the velocity v is the expansion parameter). In the EFT radiation is systematically included in the v expansion without need to separate integrals into near zones and radiation zones. We show that the renormalization of ultraviolet divergences which arise at v^6 in post-Newtonian (PN) calculations requires the presence of two non-minimal worldline gravitational couplings linear in the Ricci curvature. However, these operators can be removed by a redefinition of the metric tensor, so that the divergences at arising at v^6 have no physically observable effect. Because in the EFT finite size features are encoded in the coefficients of non-minimal couplings, this implies a simple proof of the decoupling of internal structure for spinless objects to at least order v^6. Neglecting absorptive effects, we find that the power counting rules of the EFT indicate that the next set of short distance operators, which are quadratic in the curvature and are associated with tidal deformations, do not play a role until order v^10. These operators, which encapsulate finite size properties of the sources, have coefficients that can be fixed by a matching calculation. By including the most general set of such operators, the EFT allows one to work within a point particle theory to arbitrary orders in v.

Figures (11)

• Figure 1: The incoming potential graviton with momentum ${\bf p}$ absorbs a radiation graviton with momentum $k$.
• Figure 2: Diagrams contributing to $Lv^0$ potentials. The self-energy graphs in (b), (c) are pure counterterm and have no physical effect.
• Figure 3: NRGR diagram whose imaginary part gives rise to the quadrupolar gravitational radiation power spectrum.
• Figure 4: Diagrams contributing to $Lv^2$ terms in the two-body potential. The $\otimes$ in (a) denotes an insertion of the potential graviton kinetic term. A similar diagram to (c) with $1\leftrightarrow 2$ is not shown.
• Figure 5: Diagrams with two and three potential graviton lines that also contribute to the $Lv^2$potentials. Similar diagrams with $1\leftrightarrow 2$ not shown.