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

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.

💡 Research Summary

The paper introduces a new formulation of the conservative dynamics of binary black‑hole systems within the Effective‑One‑Body (EOB) framework, called Lagrange‑EOB (LEOB). The key innovation is the insertion of a Lagrange multiplier into the EOB action to enforce the mass‑shell constraint directly, thereby avoiding the need to solve the constraint recursively for an explicit Hamiltonian. This approach allows the post‑Minkowskian (PM) information—up to the 4‑PM level (with partial 5‑PM contributions) and the corresponding post‑Newtonian (PN) corrections (4PN for the orbital sector, 4.5PN for spin‑orbit)—to be incorporated as deformations of an energy‑dependent effective metric (functions A, B, C, and the spin‑orbit coupling G).

The authors first review the standard EOB construction, emphasizing the difficulties of the Hamiltonian‑based PM (HEOB) method: the recursive solution of the mass‑shell condition leads to highly involved expressions, and the resulting effective radial potentials develop singular behavior at the Schwarzschild horizon (r = 2 GM). By contrast, the LEOB action
S = ∫dτ