Next to Leading Order Spin(1)Spin(1) Effects in the Motion of Inspiralling Compact Binaries

TL;DR

This work completes the next-to-leading order ($S^2$) contributions to the 3PN spin potential for inspiralling compact binaries using NRGR and a covariant spin supplementary condition (SSC). The calculation separates non-linear gravitational effects, finite-size corrections, and Routhian contributions, assembling the full $V_{3PN}^{s^2}$ in the covariant SSC and verifying consistency with the equations of motion, including a 3PN precession equation as a concrete example. Divergences are handled via standard EFT regularization, with pure counterterms renormalizing masses and quadrupole couplings, while tidal logs appear at higher PN order. The results, cross-checked against the extreme mass ratio Kerr limit, complete the spin sector of the 3PN dynamics and pave the way for NNLO spin terms and spin-driven radiation calculations within NRGR.

Abstract

Using effective field theory techniques we compute the next to leading order Spin(1)Spin(1) terms in the potential of spinning compact objects at third Post-Newtonian order, including sub-leading self-induced finite size effects. This result represents the last ingredient to complete the relevant spin potentials to 3PN order from which the equations of motion follow via a canonical formalism. As an example we include the precession equation.

Figures (5)

• Figure 1: Non--linear $S^2$ terms. The blob represents a spin insertion.
• Figure 2: Leading order finite size spin$^2$--spin contributions. The black square represents an insertion of the finite size operator
• Figure 3: Diagrams contributing to the 3PN which do not involve non-linearities. The cross corresponds to a propagator correction, i.e. a retardation effect.
• Figure 4: Non--linear finite size contributions to the 3PN spin--orbit potential.
• Figure 5: Divergent diagrams which renormalize the mass and quadrupole couplings.