[Paper Review] Energetics and scattering of gravitational two-body systems at fourth post-Minkowskian order

Mohammed Khalil, Alessandra Buonanno+2 more|arXiv (Cornell University)|Apr 11, 2022

Pulsars and Gravitational Waves ResearchPhysics and Astronomy144 references71 citations

TL;DR

This paper advances post-Minkowskian (PM) gravity by analyzing fourth post-Minkowskian (4PM) conservative dynamics for compact binary systems, comparing its predictions for binding energy and scattering angle against numerical relativity (NR) simulations. It demonstrates that 4PM dynamics outperforms 3PM and matches or exceeds third post-Newtonian (3PN) accuracy for scattering, and when embedded in effective-one-body (EOB) models, reduces disagreement with NR from ~40% to ~10–3%, significantly improving waveform modeling for gravitational wave astronomy.

ABSTRACT

Upcoming observational runs of the LIGO-Virgo-KAGRA collaboration, and future gravitational-wave (GW) detectors on the ground and in space, require waveform models more accurate than currently available. High-precision waveform models can be developed by improving the analytical description of compact binary dynamics and completing it with numerical-relativity (NR) information. Here, we assess the accuracy of the recent results for the fourth post-Minkowskian (PM) conservative dynamics through comparisons with NR simulations for the circular-orbit binding energy and for the scattering angle. We obtain that the 4PM dynamics gives better agreement with NR than the 3PM dynamics, and that while the 4PM approximation gives comparable results to the third post-Newtonian (PN) approximation for bound orbits, it performs better for scattering encounters. Furthermore, we incorporate the 4PM results in effective-one-body (EOB) Hamiltonians, which improves the disagreement with NR over the 4PM-expanded Hamiltonian from $\sim 40\%$ to $\sim 10\%$, or $\sim 3\%$ depending on the EOB gauge, for an equal-mass binary, two GW cycles before merger. Finally, we derive a 4PN-EOB Hamiltonian for hyperbolic orbits, and compare its predictions for the scattering angle to NR, and to the scattering angle of a 4PN-EOB Hamiltonian computed for elliptic orbits.

Motivation & Objective

• To assess the accuracy of fourth post-Minkowskian (4PM) conservative dynamics for compact binary systems using numerical relativity (NR) as a benchmark.
• To compare 4PM results with third post-Newtonian (3PN) and 3PM approximations for both bound (circular orbits) and unbound (scattering) configurations.
• To incorporate 4PM results into effective-one-body (EOB) Hamiltonians to enhance their agreement with NR simulations.
• To derive and test a 4PN-EOB Hamiltonian for hyperbolic orbits and compare its scattering angle predictions to NR and 4PN-EOB models for elliptical orbits.

Proposed method

• Derive and apply 4PM Hamiltonians for conservative two-body dynamics using recent results from amplitude-based and effective field theory methods.
• Construct two variants of 4PM-EOB Hamiltonians by embedding 4PM conservative dynamics into the EOB formalism with different gauge choices.
• Compute the binding energy for circular orbits and the scattering angle for hyperbolic trajectories using the 4PM and 4PM-EOB Hamiltonians.
• Compare these analytical predictions directly with high-precision numerical relativity (NR) simulations from the SXS catalog.
• Use the EOB energy map to relate the EOB effective energy to physical observables like binding energy and scattering angle.
• Evaluate the accuracy of the models by computing relative deviations from NR results at two GW cycles before merger for equal-mass binaries.

Experimental results

Research questions

• RQ1 How does the 4PM approximation for conservative dynamics compare to 3PM and 3PN in predicting the binding energy of circular orbits?
• RQ2 Does the 4PM approximation improve the prediction of the scattering angle compared to 3PM and 3PN for hyperbolic encounters?
• RQ3 To what extent does embedding 4PM dynamics into EOB Hamiltonians reduce the discrepancy with NR simulations compared to 4PM-expanded Hamiltonians?
• RQ4 How do the predictions of a 4PN-EOB Hamiltonian for hyperbolic orbits compare to NR and to the same model applied to elliptical orbits?

Key findings

• The 4PM conservative dynamics yields better agreement with numerical relativity (NR) for both binding energy in circular orbits and scattering angle in hyperbolic encounters than the 3PM approximation.
• For bound orbits, the 4PM approximation performs comparably to the 3PN approximation, but for scattering encounters, it significantly outperforms 3PN and 3PM.
• When incorporated into EOB Hamiltonians, the 4PM results reduce the disagreement with NR from approximately 40% to about 10% or 3%, depending on the EOB gauge choice, for equal-mass binaries two GW cycles before merger.
• The 4PN-EOB Hamiltonian for hyperbolic orbits produces scattering angle predictions that are in good agreement with NR, validating its use for unbound configurations.
• The radiative contribution to the scattering angle is found to be much smaller than the conservative contribution, indicating that the dominant physics is captured by the conservative 4PM dynamics.
• The results demonstrate that the 4PM approximation is a viable and more accurate alternative to PN methods for high-velocity scattering events and can be systematically embedded into EOB models to improve waveform accuracy.