3PN Dynamical Quantities in Gravitational Systems

• 3PN dynamical quantities are defined as (v/c)^6 corrections, providing refined models of energy, angular momentum, and orbital frequency in general relativistic systems.
• They encapsulate both instantaneous and hereditary effects including tail terms, quadratic-in-spin contributions, and eccentricity-dependent modulations in gravitational wave flux.
• Polynomial fittings and harmonic expansions at 3PN ensure high fidelity in gravitational wave templates, significantly improving parameter estimates in compact binary simulations.

3PN (third post-Newtonian) dynamical quantities refer to terms and observables computed at order $(v/c)^6$ in the post-Newtonian expansion of general relativistic two-body (and multi-body) dynamics. The 3PN approximation is central in modeling conservative and dissipative evolution in compact binary systems, including black holes and neutron stars, and is crucial for building gravitational wave templates, analyzing radiative outputs, and understanding the detailed structure of stellar and black hole interactions in both circular and eccentric binaries.

1. Formal Definition and Scope of 3PN Quantities

3PN dynamical quantities encompass orbital parameters (energy, angular momentum, orbital frequency), radiative outputs (energy flux, angular momentum loss), and post-Newtonian corrections to equations of motion, across various physical contexts:

• Binary and hierarchical triple systems (including cross terms (Will, 2014))
• Compact binaries (including spin and eccentricity effects (Bohé et al., 2015, Moore et al., 2016, Moore et al., 2019))
• Multi-body problems and restricted three-body systems (Dubeibe et al., 2016)
• Scalar–tensor theories (energy, angular momentum, Noether charges (Bernard, 2018))
• Binary black hole mergers (remnant quantities and correlations (Wang et al., 8 Jan 2025))

At 3PN order, corrections encapsulate both instantaneous and nonlocal (“tail”) effects, higher harmonics due to periastron precession, quadratic-in-spin contributions, and envelope behaviors for eccentric systems.

2. Mathematical Framework and Key Expressions

The post-Newtonian expansion for any dynamical quantity $Q$ is as follows:

$$Q = Q^\mathrm{N} + \frac{1}{c^2}Q^\mathrm{1PN} + \frac{1}{c^4}Q^\mathrm{2PN} + \frac{1}{c^6}Q^\mathrm{3PN}$$

Representative formulas include:

• Relative acceleration for scalar-tensor compact binaries (Bernard, 2018):

$$\mathbf{a} = -\frac{\tilde{G}\,\alpha\,m}{r^2}\left[(1+\mathcal{A})\mathbf{n} + \mathcal{B}\mathbf{v}\right]$$

Where $\mathcal{A}$, $\mathcal{B}$ admit expansions through 3PN order.

• Conserved energy for circular orbits:

$$E_{3\mathrm{PN}} = -\frac{1}{2} m\,\nu\,c^2\,x \left\{ 1 + E^{(1)}x + E^{(2)}x^2 + E^{(3)}x^3 + E^{\mathrm{tail}} \right\}$$

where $x = \left(\frac{G m \omega}{c^3}\right)^{2/3}$ and $E^{\mathrm{tail}}$ encodes nonlocal hereditary effects.

• GW energy flux (with spin-spin corrections) (Bohé et al., 2015):

$$\mathcal{F}_{SS} = \frac{G}{c^{5}}\left\{ \frac{1}{5}[I_{ij}^{(3)}]^2 + \frac{1}{c^{2}}\left[ \frac{1}{189}[I_{ijk}^{(4)}]^2 + \frac{16}{45}[J_{ij}^{(3)}]^2 \right] + \cdots \right\}_{SS}$$

For eccentric binaries, the phase evolution incorporates 3PN corrections and eccentricity-dependent terms:

$$\phi = \phi_c - \frac{1}{32\eta v^5} \left\{ 1 + \cdots - \frac{785}{272}e_0^2 \left(\frac{v_0}{v}\right)^{19/3} \left[ 1 + \left(\frac{6955261}{2215584} + \frac{436441}{79128}\eta\right)v^2 + \cdots \right] \right\}$$

3. 3PN Effects in Equations of Motion and Integrals of Motion

At this order, equations of motion include:

• Instantaneous contributions (higher powers and products of velocities, separation, spin variables)
• Nonlocal tail effects (arising from interaction of multipoles with past motion)
• Cross-terms in hierarchical systems: Explicit coupling between inner binary PN and Newtonian multi-body perturbations (Will, 2014)

The ten Noetherian integrals for compact binaries in scalar–tensor theories (energy, linear and angular momentum, center-of-mass vector) are all computed to 3PN including tail-type modifications (Bernard, 2018).

For spin–orbit and spin–spin effects, 3PN corrections incorporate precession equations for the conserved spin vector:

$$\frac{d\mathbf{S}_i}{dt} = \epsilon_{ijk}\Omega_j S_k$$

with $\Omega_j$ containing explicit next-to-leading order spin-spin contributions (Bohé et al., 2015).

4. Impact of Eccentricity and Periastron Precession

Eccentricity complicates GW signals, producing envelopes in radiative outputs (energy, angular momentum, recoil), and continuous domains in remnant quantities (Wang et al., 8 Jan 2025, Moore et al., 2019). The phase function in eccentric binaries receives major corrections:

• At each initial mean anomaly $l_0$, radiative quantities oscillate, but the envelope over $l_0$ defines maximal/minimal boundaries for observables.
• Modeling these envelope domains provides robust constraints for remnant parameters over mass ratio $q$.

GW templates for eccentric binaries exploit (at 3PN):

• Quasi-Keplerian formalism for multiple eccentricities (time, radial, phi)
• Harmonic expansion for signal: $$\tilde{h}_{+,\times} \sim \sum_j A^{(j)}_{+,\times}(f)\,e^{i\psi_j(f)}$$
• Stationary phase approximation and truncated harmonic sums for Fourier-domain models.

5. Polynomial Modeling and Correlations in BBH Mergers

For circular binary black hole mergers, remnant mass, peak luminosity, final spin, and recoil velocity are accurately fit as polynomials in mass ratio:

$$A(q) \approx a_0 + a_1 q + a_2 q^2 + a_3 q^3 + a_4 q^4$$

Residuals $(res = (A_{poly} - A)/A \times 100\%)$ confirm model accuracy ($\lesssim 1\%$ for remnant mass and spin, $\lesssim 5\%$ for peak luminosity and recoil velocity) (Wang et al., 8 Jan 2025).

For eccentric mergers, domains constructed from minimum and maximum envelopes over $l_0$ and $e_0$ define admissible parameter regions and tightly constrain correlations across dynamical quantities (spiral/complex correlations in parameter space for non-circular mergers).

6. Spin Alignment, Precession, and Universal Envelope Features

Spin effects (alignment or precession) universally manifest oscillatory amplitude modulations in the waveform. While precessing binaries introduce further periodicities, the eccentricity–induced envelope persists in both radiative and dynamical outcomes (Wang et al., 8 Jan 2025). These envelope domains remain foundational in constraining GW observables, regardless of spin configuration.

7. Practical and Computational Considerations

Implementing 3PN dynamical quantities requires:

• Careful scheme selection for harmonic coordinates vs ADM/EFT formalisms (Bohé et al., 2015)
• Inclusion of nonlocal tail terms via Fourier expansion with appropriate regularization (Bernard, 2018)
• Interpolation and polynomial fitting for remnant property domains (Wang et al., 8 Jan 2025)
• Truncation and convergence checks for harmonic expansions in waveform models (Moore et al., 2019)

Numerical simulations verify analytic 3PN models across parameter spaces (matches $97\%–99\%$ (Moore et al., 2019)). For data analysis, inclusion of 3PN corrections and eccentric envelope effects is mandatory to reduce systematic error in parameter estimation.

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In summary, 3PN dynamical quantities comprise a critical layer in the analytic and numerical modeling of gravitational systems, yielding precise predictions for energy, angular momentum, waveform phase evolution, and remnant properties, both in circular and eccentric binaries. Envelope constructions for eccentric systems and polynomial modeling for circular orbits robustly articulate the complex dependencies on initial conditions, mass ratios, and spin configurations. These tools and results are foundational for current and future gravitational wave astronomy, waveform modeling, and tests of strong-field gravity.