From Boundary Data to Bound States

Abstract: We introduce a -- somewhat holographic -- dictionary between gravitational observables for scattering processes (measured at the "boundary") and adiabatic invariants for bound orbits (in the "bulk"), to all orders in the Post-Minkowskian (PM) expansion. Our map relies on remarkable connections between the relative momentum of the two-body problem, the classical limit of the scattering amplitude and the deflection angle in hyperbolic motion. These relationships allow us to compute observables for generic orbits (such as the periastron advance $\Delta\Phi$) through analytic continuation, via a radial action depending only on boundary data. A simplified (more geometrical) map can be obtained for circular orbits, enabling us to extract the orbital frequency as a function of the (conserved) binding energy, $\Omega(E)$, directly from scattering information. As an example, using the results in Bern et al. [1901.04424, 1908.01493], we readily derive $\Omega(E)$ and $\Delta\Phi(J,E)$ to two-loop orders. We also provide closed-form expressions for the orbital frequency and periastron advance at tree-level and one-loop order, respectively, which capture a series of exact terms in the Post-Newtonian expansion. We then perform a partial PM resummation, using a "no-recoil" approximation for the amplitude. This limit is behind the map between the scattering angle for a test-particle and the two-body dynamics to 2PM. We show that it also captures a subset of higher order terms beyond the test-particle limit. While a (rather lengthy) Hamiltonian may be derived as an intermediate step, our map applies directly between gauge invariant quantities. Our findings provide a starting point for an alternative approach to the binary problem. We conclude with future directions and some speculations on the classical double copy.

• The paper introduces a novel analytical apparatus that directly maps gravitational scattering amplitudes to invariant bound state observables.
• It recalibrates Post-Minkowskian terms, accurately computing the scattering angle and periastron advance for binary systems.
• This approach bypasses traditional Hamiltonian techniques, offering enhanced precision in gravitational wave source modeling.

From Boundary Data to Bound States: An Overview of Gravitational Observables in Scattering and Bound Orbits

The paper under consideration presents a novel theoretical framework connecting scattering amplitudes in gravitational systems to observables for bound orbits, offering a bridge between the field of scattering processes and the requirements of accurate gravitational wave (GW) source parameter estimation. This is achieved through a comprehensive development of a dictionary linking the scattering data, particularly the amplitude and scattering angle, directly to adiabatic invariants for binary systems in bound orbits such as elliptical motion. The work introduces an analytical apparatus to facilitate this transformation, rather than relying on traditional Hamiltonian formulations, thereby enhancing computational efficiency and precision for taping into high-order Post-Minkowskian (PM) predictions.

Central to the discussion is the impetus formula, which relates the amplitude of classical scattering processes, accounting for IR divergences and matching conditions, to the relative momentum in the two-body problem,

$^2(r,E) = p_\infty^2(E) + \widetilde{\cal M}(r,E),$

where $\widetilde{\cal M}(r,E)$ is the Fourier transform of the scattering amplitude. This formula elegantly maps the gauge invariant scattering information directly to the gravitational observables, circumventing the need for intermediate Hamiltonian constructions that often carry cumbersome gauge dependencies.

The authors explore the implications of this revolutionary approach by recalibrating known PM terms within the scattering angle's expansion, thereby calculating the scattering angle to 3PM order. The innovative calculus extends to the computation of the periastron advance through analytic continuation, capturing contributions up to 2PN order, with forward path capabilities for higher PN orders. This procedure realistically accounts for all-order velocity effects, achieved through a direct analysis of the amplitude without recourse to effective Hamiltonian tactics, thus promising seamless integration with gravitational wave phenomenology.

A significant leap in understanding is facilitated by the mapping between scattering and the construction of invariant orbital elements for bound states, established using Firsov's inversion framework melded with an analytic continuation strategy, i.e., through transformations $\beta \to i\beta$ and $b \to ib$. This approach promises utility in efficiently computing scientific observables such as the orbital frequency, without recourse to iterative PN schemes, pivotal in analyzing circular orbits characteristic of inspiraling compact binaries detected by LIGO and other observatories.

In an attempt to harness the insight drawn from test-particle limits, the authors employ a no-recoil approximation to partial resummation of PM effects within the scattering process. Despite its approximation status, it intriguingly reproduces complete 2PM dynamical results while offering a versatile baseline for further higher-order corrections driven by intricate self-force effects.

Overall, this contribution provides a robust theoretical scaffold substantiating a direct gauge invariant approach, recasting binary inspiral dynamics via scattering amplitudes—a significant stride in gravitational physics theory poised to influence future precision computations pivotal for gravitational wave astronomy and the more comprehensive understanding of binary black hole mergers. The insights developed herein could further inspire novel exploitation of amplitude techniques, motivating multi-faceted applications in quantum gravity and classical general relativity. The classical double copy potential within these structures further ushers a rich interplay at the interface of gauge theory and gravity, foreseeing promising avenues for new theoretical developments.