Energetics of two-body Hamiltonians in post-Minkowskian gravity

TL;DR

This paper investigates whether post-Minkowskian (PM) results, particularly at 3PM order, can enhance modeling of bound binary inspirals by embedding PM information into an effective-one-body (EOB) Hamiltonian and comparing the binding energy against numerical relativity (NR) data. It derives a 3PM EOB Hamiltonian from scattering-angle data and shows its agreement with the 3PM scattering results, connecting to Schwarzschild geodesics in the test-mass limit. Through NR comparisons for mass ratios $q=1$ and $q=10$, the study finds that 3PM improves over 2PM and PM-EOB generally aligns better with NR than plain PM, but PN-based EOB remains more accurate overall; an alternative 3PM EOB with tail terms can yield substantially better NR agreement. The results suggest that higher PM orders (4PM and beyond) and alternative EOB resummations are needed to surpass current LIGO/Virgo waveform models in the quasi-circular inspiral regime.

Abstract

Advanced methods for computing perturbative, quantum-gravitational scattering amplitudes show great promise for improving our knowledge of classical gravitational dynamics. This is especially true in the weak-field and arbitrary-speed (post-Minkowskian, PM) regime, where the conservative dynamics at 3PM order has been recently determined for the first time, via an amplitude calculation. Such PM results are most relevantly applicable to relativistic scattering (unbound orbits), while bound/inspiraling binary systems, the most frequent sources of gravitational waves for the LIGO and Virgo detectors, are most suitably modeled by the weak-field and slow-motion (post-Newtonian, PN) approximation. Nonetheless, it has been suggested that PM results can independently lead to improved modeling of bound binary dynamics, especially when taken as inputs for effective-one-body (EOB) models of inspiraling binaries. Here, we initiate a quantitative study of this possibility, by comparing PM, EOB and PN predictions for the binding energy of a two-body system on a quasi-circular inspiraling orbit against results of numerical relativity (NR) simulations. The binding energy is one of the two central ingredients (the other being the gravitational-wave energy flux) that enters the computation of gravitational waveforms employed by LIGO and Virgo detectors, and for (quasi-)circular orbits it provides an accurate diagnostic of the conservative sector of a model. We find that, whereas 3PM results do improve the agreement with NR with respect to 2PM (especially when used in the EOB framework), it is crucial to push PM calculations at higher orders if one wants to achieve better performances than current waveform models used for LIGO/Virgo data analysis.

Figures (7)

• Figure 1: NR simulations. In this paper the energetics of various approximants are compared against two NR simulations of non-spinning binary black holes produced by the Simulating eXtreme Spacetimes (SXS) collaboration Mroue:2013xnaChu:2015kft. The top (bottom) panel shows the waveform (more specifically the real part of the $l=m=2$ mode of the strain, $\mathfrak{R}(h_{22})$) of the simulation with mass-ratio $q=1$ ($q=10$), identified in the SXS catalog as SXS ID: 0180 (SXS ID: 0303). In both panels, the red shading shows the segment of the simulation used for the binding-energy's comparisons in all figures of this paper.
• Figure 2: Energetics of PM Hamiltonians. We compare to NR the binding energy as a function of orbital frequency $GM\Omega$ from both PM and PM-EOB Hamiltonians for a nonspinning binary black hole with mass ratio $q=1$ (left panel) and $q=10$ (right panel). The dots at the end of the curves mark the ISCOs, when present in the corresponding two-body dynamics. The NR binding energy and its error are in cyan. The top $x$-axis shows the number of orbits until merger. In the lower panel we show the fractional difference between the approximants and the NR result.
• Figure 3: Energetics of PM Hamiltonians. Same as in Fig. \ref{['fig:energywPM']} but versus the dimensionless angular momentum $l=L/(G \mu M)$. The cusps signal the presence of the ISCO, where the branches of stable and unstable circular-orbit solutions meet. Note that the orbital-frequency range in the plots ends about $1.4$ and $1.8$ GW cycles, for $q=1$ and $q=10$, respectively, before the two black holes merge.
• Figure 4: Energetics of PM Hamiltonians augmented by PN information. Same as in Fig. \ref{['fig:energywPM']} but now we compare to NR the binding energy of PM EOB Hamiltonians augmented by PN information. Notice that adding 3PM information at 3PN or above does not lead to a visible difference from plain PN EOB Hamiltonians (the 3PM-3PN and 3PN curves, as well as the 3PM-4PN and 4PN ones, are essentially on top of each other). Also included is a curve for an alternative 3PM EOB Hamiltonian, $H^{\rm EOB,\widetilde{PS}}_{3 \rm PM}$, derived in Appendix \ref{['appendixB']}.
• Figure 5: Energetics of PM EOB Hamiltonian and the EOB Hamiltonian used in LIGO/Virgo data-analysis. Same as in Fig. \ref{['fig:energywPM']} and Fig. \ref{['fig:energymixed']}, but now we show how the $H^{\rm EOB,PS}_{m \rm PM}$ Hamiltonian compares with the (original) $H^{\rm EOB}_{n \rm PN}$ Hamiltonian currently employed at 4PN order to build waveform models for LIGO/Virgo data-analysis. We observe that $H^{\rm EOB}_{n \rm PN}$ Hamiltonians still produce $e(\Omega)$-curves substantially closer to NR result than the 3PM approximant.