Post-Newtonian Inspiral Trajectory

• Post-Newtonian inspiral motion trajectory is a detailed analytic description of compact binary evolution, achieving 5PN accuracy through the effective one-body formalism.
• It employs five-loop effective field theory and high-order multipolar analyses to incorporate strong-field effects and precise gravitational-wave flux corrections.
• The formalism improves waveform phasing by refining the innermost stable circular orbit and enabling precise integration of orbital motion using Hamilton’s equations.

A post-Newtonian (PN) inspiral motion trajectory describes the time-dependent orbital evolution of compact-object binaries governed by the post-Newtonian expansion of general relativity. Such expansions provide an asymptotic, velocity- and field-strength-ordered framework accurately modeling binary motion during the inspiral phase, where velocities $v \ll c$ and $GM/rc^2 \ll 1$ but compactness and strong-field effects can be captured at systematically increasing orders. The state of the art, as advanced by five-loop effective field theory techniques and high-order multipolar analyses, is the construction of fully analytic trajectories accurate through 5PN (and partially 6PN) order, efficiently encoded in the Effective One-Body (EOB) formalism and coupled to 5PN gravitational-wave flux for both phase and amplitude evolution (Blümlein et al., 2022).

1. Effective-One-Body Hamiltonian and 5PN Dynamics

The post-Newtonian dynamics to 5PN are most systematically implemented in the EOB framework, allowing a resummed, canonical Hamiltonian treatment of binary motion. The EOB Hamiltonian for two nonspinning compact bodies of masses $m_1$, $m_2$ is

$$H_{\rm EOB}^{\rm loc} = M\sqrt{1+2\nu(\hat{H}_{\rm eff}-1)}, \qquad \hat{H}_{\rm eff} = \sqrt{A(u)\left(1+p_\varphi^2 u^2\right) + A(u)D(u)p_r^2 + Q(r,p_r)},$$

where $u = 1/r$, $M = m_1 + m_2$, $\mu = m_1 m_2/M$, $\nu = \mu/M$, and $(r, p_r, \varphi, p_\varphi)$ are canonical phase-space variables. The EOB metric potentials are Taylor-expanded in $u$ as: $$\begin{aligned} A(u) &= 1 + a_1 u + \cdots + a_6(\nu) u^6, \ D(u) &= 1 + d_2 u^2 + \cdots + d_5(\nu) u^5, \ Q(r,p_r) &= p_r^4(q_{42}(\nu) u^2 + q_{43}(\nu) u^3 + q_{44}(\nu) u^4) + p_r^6(q_{62}(\nu) u^2 + q_{63}(\nu) u^3) + p_r^8 q_{82}(\nu) u^2. \end{aligned}$$ The new 5PN terms computed in (Blümlein et al., 2022) are: $$\begin{aligned} q_{82} &= \tfrac{6}{7}\nu + \tfrac{18}{7}\nu^2 + \tfrac{24}{7}\nu^3-6\nu^4, \ q_{63} &= \tfrac{123}{10}\nu-\tfrac{69}{5}\nu^2+116\nu^3-14\nu^4, \ q_{44} &= (\tfrac{1580641}{3150}-\tfrac{93031\pi^2}{1536})\nu + (-\tfrac{3670222}{4725}+\tfrac{31633\pi^2}{512})\nu^2 + (640-\tfrac{615\pi^2}{32})\nu^3, \ d_{5} &= (\tfrac{331054}{175}-\tfrac{63707\pi^2}{512})\nu + (-\tfrac{31295104}{4725}+\tfrac{306545\pi^2}{512})\nu^2 + (\tfrac{1069}{3}-\tfrac{205\pi^2}{16})\nu^3, \ a_{6} &= (-\tfrac{1026301}{1575}+\tfrac{246367\pi^2}{3072})\nu + (-\tfrac{1749043}{1575}+\tfrac{25911\pi^2}{256})\nu^2 + 4\nu^3. \end{aligned}$$

All coefficients at lower PN orders are imported from the 4PN literature, permitting a recursive extension in accuracy.

2. Orbital Evolution via Hamilton’s Equations

Conservative orbital motion is propagated by Hamilton’s equations: $$\dot r = \frac{\partial H_{\rm EOB}^{\rm loc}}{\partial p_r}, \quad \dot p_r = -\frac{\partial H_{\rm EOB}^{\rm loc}}{\partial r}, \quad \dot\varphi = \frac{\partial H_{\rm EOB}^{\rm loc}}{\partial p_\varphi}, \quad \dot p_\varphi = 0,$$ with $p_\varphi = j = J/(M\mu)$ constant. For circular orbits ($p_r=0$), the orbital frequency is given by

$$\Omega = \dot\varphi = \frac{\partial H_{\rm EOB}^{\rm loc}}{\partial j}.$$

The circular condition $\partial H_{\rm EOB}^{\rm loc}/\partial r=0$ at fixed $j$ yields $r(j)$ and thus $\Omega(j)$.

3. Gauge-Invariant Observables and 5PN Series

One introduces the PN frequency parameter $x = (M\Omega)^{2/3}$ and expands the gauge-invariant binding energy per unit reduced mass as

$$\frac{E(x)}{\mu} = -\frac{1}{2} x \left[ 1 + \left(-\frac{3}{4} - \frac{\nu}{12}\right)x + \left(-\frac{27}{8} + \frac{19}{8}\nu - \frac{\nu^2}{24}\right)x^2 + \cdots + e_{5\mathrm{PN}}(\nu)x^5 \right],$$

where $e_{5\mathrm{PN}}(\nu)$ is an explicit polynomial in $\nu$ from the EOB expansion. Similarly, the dimensionless angular momentum is

$$\frac{J(x)}{M\mu} = x^{-1/2} \left[ 1 + j_{1\mathrm{PN}}(\nu)x + \cdots + j_{5\mathrm{PN}}(\nu)x^5 \right].$$

Explicit closed-form $e_{5\mathrm{PN}}(\nu)$ and $j_{5\mathrm{PN}}(\nu)$ polynomials are assembled by inserting the EOB potentials into the orbit condition and expanding in $x$.

4. Time-Domain Phasing and Energy Balance

To obtain $r(t)$ and $\varphi(t)$, the conservative EOB evolution is coupled to the 5PN-accurate gravitational-wave flux $\mathcal{F}(x)$ via the balance law: $$\frac{dx}{dt} = -\frac{\mathcal{F}(x)}{dE(x)/dx}, \qquad \frac{d\varphi}{dt} = \frac{x^{3/2}}{M}.$$ Analytic integration yields

$$t(x) = t_0 + \int^{x}\frac{dE/dx'}{\mathcal{F}(x')}\, dx', \qquad \varphi(x) = \varphi_0 + \int^x \frac{x'^{3/2}}{M}\frac{dE/dx'}{\mathcal{F}(x')}\, dx'.$$

Both integrands are expanded as series up to $x^5$, allowing fully analytic 5PN $t(x)$ and $\varphi(x)$ expressions. Inversion of $t(x)$ provides $x(t)$, whence $r(t) = Mx(t)^{-1}$.

5. Structure and Impact of the 5PN EOB Metric Potentials

For rapid implementation, the relevant EOB metric functions are: $$\begin{aligned} A(u) &= 1 - 2u + 2\nu u^3 + \cdots + a_6(\nu)u^6, \ D(u) &= 1 + 6\nu u^2 + \cdots + d_5(\nu)u^5, \ Q(r,p_r) &= p_r^4(q_{42}u^2 + q_{43}u^3 + q_{44}u^4) + p_r^6(q_{62}u^2 + q_{63}u^3) + p_r^8 q_{82}u^2, \end{aligned}$$ with $\{a_6, d_5, q_{44}, q_{63}, q_{82}\}$ as above and all other coefficients available from the preceding 4PN computations.

6. Quantitative Improvements from 5PN Terms

The passage from 4PN to 5PN order achieves the following quantitative advancements in inspiral trajectory modeling:

• The truncation error in $E(x)$ is reduced from $\mathcal{O}(x^5)$ to $\mathcal{O}(x^6)$, improving the accuracy throughout the inspiral.
• The ISCO (innermost stable circular orbit) frequency shifts by $\sim 10^{-3}$ upon inclusion of 5PN terms, refining the mapping of inspiral endpoint in the conservative sector.
• For waveform phasing $\varphi(f)$, the 5PN corrections account for $\sim 1$–$2$ radians of accumulated phase at $f\sim 100$ Hz for a $1.4 + 1.4 M_\odot$ binary. This is crucial for achieving sub-rad accuracy demanded by 3rd-generation detectors.
• Nonlocal (tail) 5PN effects enter as logarithmic terms $\propto x^5\ln x$ in both $E(x)$ and $\varphi(f)$, incorporating leading-order tail–of–tail hereditary interactions.

7. Domain of Applicability and Summary Prescription

The 5PN-accurate inspiral trajectory constructed in this framework provides:

• Trajectories $r(t), \varphi(t)$ consistent to $\mathcal{O}(v^{10}/c^{10})$ in both conservative and dissipative sectors for nonspinning, nonprecessing, and quasi-circular binaries in the regime $v/c \lesssim 0.4$.
• All input quantities for constructing 5PN-accurate time-domain waveform models (or frequency-domain Taylor expansion models via SPA phasing).
• Ingredients for extending the Hamiltonian or EOB-PN representation to include late-inspiral and (with additional, calibrated EOB potentials or merger-ringdown attachments) merger dynamics.

Implementation simply requires:

1. Inserting the explicit 5PN $A(u)$, $D(u)$, $Q(r,p_r)$ as above into the EOB Hamiltonian.
2. Evaluating the gauge-invariant $E(x), J(x)$ series through $x^5$ using the PN expansions.
3. Integrating the coupled ODE system using the analytic 5PN energy and GW flux, optionally expanding and inverting $t(x)$ and $\varphi(x)$.
4. Applying the additional 5PN hereditary (logarithmic) flux contributions to ensure consistency with full 5PN waveform phasing.

This formalism currently yields the most accurate analytic inspiral description compatible with first principles up to the 5PN order and provides all necessary structure for future integration with numerical relativity or higher-order resummation (Blümlein et al., 2022).