A novel Lagrange-multiplier approach to the effective-one-body dynamics of binary systems in post-Minkowskian gravity

TL;DR

This work introduces a Lagrange-EOB (LEOB) formalism that uses a Lagrange multiplier to impose the EOB mass-shell constraint, enabling a PM-infused yet nonrecursive description of conservative binary dynamics. By working in a geodesic-like, energy-dependent metric within a Lagrange framework, LEOB blends PM and PN information while avoiding the problematic recursive Hamiltonian construction of previous HEOB approaches. The authors develop local and nonlocal PM contributions up to 4PM/4PN, incorporate spinning terms via a spin-orbit sector, and build two NR-informed spin-aligned waveforms, LEOB-PM$_{a_0}$ and LEOB-PM$_{ t SS}$. NR-calibrated performance shows sub-per-mille unfaithfulness in many cases and favorable comparisons with SEOBNR-PM and TEOBResumS-Dalí, highlighting LEOB as a robust, flexible pathway for PM-informed waveform modeling.

Abstract

We present a new approach to the conservative dynamics of binary systems, within the effective one-body (EOB) framework, based on the use of a Lagrange multiplier to impose the mass-shell constraint. When applied to the post-Minkowskian (PM) description of the two-body problem in Einsteinian gravity, this Lagrange-EOB (LEOB) approach allows for a new formulation of the conservative dynamics that avoids the drawbacks of the recursive definition of EOB-PM Hamiltonians. Using state-of-the-art expressions of the resummed waveform and radiation reaction, we apply our new formalism to the construction of an aligned-spin, quasi-circular, inspiraling EOB waveform model, called {\tt LEOB-PM}, that incorporates analytical information up to the 4PM level, completed by 4PN contributions up to the sixth order in eccentricity, in the orbital sector, and by 4.5PN contributions, in the spin-orbit sector. In the nonspinning case, we find that an uncalibrated LEOB-PM model delivers maximum EOB/NR unfaithfulness ${\bar{F}}_{\rm EOBNR}$ (with the Advanced LIGO noise in the total mass range $10-200M_\odot$) varying between $0.2\%$ and $1\%$ over all the nonspinning dataset of the Simulating eXtreme Spacetime (SXS) Numerical Relativity (NR) catalog up to mass ratio $q=15$. It also delivers excellent phasing agreement with the $q=32$ configuration of the RIT catalog. We also found consistency between binding energies within a few percent at the NR merger location. Then, when NR-informing the dynamics of the model (both orbital and spinning sectors) by using 17 SXS dataset, we find that the EOB/NR unfaithfulness (compared to 530 spin-aligned SXS waveforms) has a median value of $5.39\times 10^{-4}$, or $6.13\times 10^{-4}$ (depending on the spin-spin interactions), reaching at most $\sim 1\%$ in some of the high-spin corners.

Figures (14)

• Figure 1: Radial potentials for inspiraling orbits against the associated radial coordinate for the PM-informed orbital dynamics determined by our new LEOB approach (upper panel) and the SEOB-PM model of Ref. Buonanno:2024byg (bottom panel). For each curve, obtained at different fixed values of the angular momentum $j$, the colored dots indicate the location of the stable circular orbits. We also show, as black dots, the actual radial position of the system as it moves along the full radiation-reacted dynamics. See text for more details.
• Figure 2: Uncalibrated nonspinning model. Performance of a 4PM-4PN, purely analytic, dynamics on a sample of SXS nonspinning waveform with $1\leq q \leq 15$. The LEOB waveform is completed only by NR-informed NQC corrections and ringdown, without any NR-calibrated effective parameter entering the dynamics. Left panel: illustrative time-domain phasing for $q=4.5$. where the vertical, dot-dashed, lines indicate the alignment interval during the inspiral. The vertical dashed line marks the NR merger time, corresponding to the waveform amplitude peak. Middle panel: time-evolution of the waveform amplitude and frequency to illustrate the contributions of the NQC corrections and ringdown (see the text for precise description). Twice the orbital frequency $\Omega$ is represented with a gray line. Right panel EOB/NR unfaithfulness with the advanced LIGO power spectral density.
• Figure 3: Uncalibrated, nonspinning model. Time domain phasing comparison for $q=32$ with the NR waveform of Ref. Lousto:2020tnb. The EOB and NR waveforms are aligned around merger time.
• Figure 4: NR-informed, nonspinning case: performance with the 5PM-5PN parameter $a_{52}^{\rm NR}$ informed by NR simulations. Left and middle panels: time-domain phasings for two configurations: compare the $q=4.5$ one with Fig. \ref{['fig:s0_leob_nocalibration']}. Rightmost panel: $\bar{\cal F}_{\rm EOBNR}(M)$ for the same nonspinning NR datasets considered in Fig. \ref{['fig:s0_leob_nocalibration']}. The values of $\bar{\cal F}_{\rm EOBNR}^{\rm max}$ are listed in Table \ref{['tab:nonspinning_datasets']}. For nearly equal-mass configurations the NR-informed, effective, $a_{52}^{\rm NR}(\nu)$ function yields an improvement of almost two orders of magnitude.
• Figure 5: Illustrative EOB/NR time-domain phasing comparison for a selected sample of spin-aligned configurations using the LEOB-PM$_{a_0}$ model. The vertical lines in the left subpanels mark the alignment frequency interval, while the one in the right bottom panels identifies the NR merger time. The rightmost panel refers to the SXS:BBH:1445 NR simulation, that is also explicitly used in Ref. Buonanno:2024byg to show the performance of the SEOBNR-PM model developed there. The EOB/NR phasing agreement also in this case is rather good even if the spin-sector of the model was NR-informed, through the function $\hat{g}_{32}^{\rm NR}$, by using only the equal-mass, equal-spin SXS dataset listed in Table \ref{['tab:g42_eqmass']} in Appendix \ref{['app:data']}.