Post-Newtonian Approximants

Post-Newtonian approximants reorganize the same finite set of weak-field coefficients into different evolution or frequency-domain formulas. They agree when re-expanded through their shared formal order, yet differ by uncontrolled higher-order terms. Near the end of inspiral those formally higher terms can accumulate substantial phase. An approximant label therefore identifies a construction, not merely a PN order. This page compares the constructions while accepting conservative energy and radiative flux from Relativistic Two-Body Dynamics; production simulation algorithms and detector-specific likelihood implementation remain outside its scope.

Helpful background. Einstein Equations supplies the weak-field field equations; Waves, Green Functions, and Geometric Optics supplies Fourier and stationary-phase methods; Relativistic Two-Body Dynamics supplies the binding energy and orbital invariants; Computational Relativity supplies late-inspiral numerical benchmarks; and Gravitational-Wave Astronomy treats detector-specific likelihood and inference verdicts.

One truncated dynamical input

For a nonprecessing, quasi-circular inspiral, let

$$x=(GM\Omega)^{2/3},$$

where $M$ is the physical total mass and $\Omega$ the orbital angular frequency. Write the conservative energy and positive outward luminosity as finite series,

$$E(x)=E_{\mathrm N}(x) \left[1+\sum_{k=1}^{N}e_kx^{k/2}\right],$$ $$\mathcal F(x)=\mathcal F_{\mathrm N}(x) \left[1+\sum_{k=2}^{N}f_kx^{k/2}\right].$$

Energy balance gives

$$\dot x=-\frac{\mathcal F(x)}{E'(x)}, \qquad \dot\Phi=\Omega=\frac{x^{3/2}}{GM}.$$

The finite coefficient lists and the assumptions behind balance are the common inputs. Blanchet organizes the high-order energy, flux, phase, and waveform expansions in Blanchet 2024, §§3.4.3–3.4.5, pp. 206–219, PDF. The choice of what to expand, integrate, or invert next defines the approximant.

At sufficiently high PN order the coefficients also contain powers of $\ln x$, inherited from hereditary propagation and conservative dynamics; the notation above permits $e_k=e_k(\ln x)$ and $f_k=f_k(\ln x)$. Any derivative such as $E'(x)$ acts on those logarithms too. A numerical approximant must also stop before a zero of the truncated $E'$ or another artifact makes the chosen evolution singular; algebraic re-expansion is not permission to evolve through a spurious pole.

Time-domain choices

TaylorT1 numerically integrates the ratio $-\mathcal F/E'$ while leaving the two truncated polynomials unexpanded. TaylorT4 first re-expands that ratio as a series in $x$ and truncates it at the declared order. TaylorT2 instead expands $dt/dx=-E'/\mathcal F$ and integrates $t(x)$ and $\Phi(x)$ parametrically. TaylorT3 further inverts the truncated $t(x)$ relation before constructing the phase.

The difference is visible in a one-line model. Suppose the common information is

$$-\frac{\mathcal F}{E'} =Kx^5\frac{1+f_1x}{1+e_1x}.$$

The unexpanded rational expression and its first-order re-expansion

$$Kx^5\left[1+(f_1-e_1)x\right]$$

agree through relative order $x$. Expanding the rational expression one order farther gives

$$Kx^5\left[ 1+(f_1-e_1)x+(e_1^2-e_1f_1)x^2+O(x^3) \right].$$

Thus their first difference is beyond the shared input order, exactly as required. It is not zero, and repeated integration can turn it into a significant accumulated phase.

Frequency-domain construction

A frequency-domain TaylorF2 waveform applies the stationary-phase approximation to a slowly chirping time-domain mode. With the volume convention $e^{-i\omega t}$ and $h=h_+-ih_\times$, a positive-frequency contribution has schematic form

$$\widetilde h(f) =\mathcal A(f)e^{i\Psi(f)}, \qquad f\gt 0,$$

where $\Psi$ contains the stationary-time phase and the Fourier stationary-point correction. Its coefficient series can be consistent with the same $E$ and $\mathcal F$ while differing from a time-domain approximant because stationary phase, re-expansion, truncation, and termination are additional operations.

The Fourier sign can be checked without memorizing a template. For $h(t)=A(t)e^{-i\varphi(t)}$, the forward transform uses $e^{+2\pi ift}$. Its stationary point $t_f$ satisfies $\dot\varphi(t_f)=2\pi f$; if $\ddot\varphi(t_f)\gt0$, then

$$\widetilde h(f)\simeq A(t_f)\sqrt{\frac{2\pi}{\ddot\varphi(t_f)}} \exp\!\left\{i\left[2\pi ft_f-\varphi(t_f)-\frac\pi4\right]\right\}.$$

The $-\pi/4$ comes from a negative second derivative of the Fourier exponent. This approximation requires slowly varying amplitude and chirp over the stationary region and a stationary point sufficiently far from the waveform endpoints; abrupt termination introduces additional spectral structure.

Resummed frequency-domain approximants may replace a Taylor polynomial by a Padé or other rational form. Damour, Iyer, and Sathyaprakash compare such resummations and make clear that effectualness and faithfulness depend on the historical approximant order and the stated noise model Damour, Iyer, and Sathyaprakash 1998, §§II–IV, PDF. No resummation creates unknown PN coefficients; it encodes an assumption about their continuation.

Comparison axes and limits

Approximants should be compared along explicit axes:

• the PN orders retained separately in energy, flux, phase, and amplitude;
• whether ratios are kept rational or re-expanded;
• time-domain integration, parametric inversion, or stationary phase;
• spin, eccentricity, precession, and mode content;
• starting and termination conditions;
• source, inertial, and Bondi frames and the $s=-2$ harmonic phases.

Every construction must recover the weak-field limit $x\to0$. When its ingredients admit a test-mass limit, it should also reproduce the corresponding perturbative coefficients. Agreement at those limits checks the shared physics but does not select a unique continuation into strong field.

If a match is quoted, this chapter uses the one-sided-PSD inner product

$$(a|b)=4\operatorname{Re} \int_{f_{\min}}^{f_{\max}} \frac{\widetilde a(f)\widetilde b(f)^*}{S_n(f)}df,$$

for real projected strain signals $a,b$, with PSD, band, modes, frame, and maximized time/phase shifts declared. A complex mode-space norm needs its own polarization or frequency prescription. An approximant spread under one such norm is a sensitivity diagnostic, not a confidence interval and not an inference verdict.

Common pitfalls

Treating formal agreement as numerical identity. Equal coefficients through order $x^{N/2}$ permit different terms at all higher orders. Late-inspiral phase agreement must be checked, not inferred from the label.

Mixing amplitude and phase orders. A restricted-amplitude waveform can use a high-order phase. Its label must not imply that all modes and amplitudes are known to the same order.

Calling spread an uncertainty distribution. Several approximants can share the same missing physics and bias. Their envelope has no probabilistic coverage without an additional error model.

Exercise

Take $e_1=-1/2$ and $f_1=1/3$ in the model ratio above. Find the relative $x$ and $x^2$ coefficients of the unexpanded ratio, and compare them with a TaylorT4-style truncation after the $x$ term.

Solution

The shared first-order coefficient is

$$f_1-e_1=\frac13+\frac12=\frac56.$$

The next coefficient generated by the rational form is

$$e_1^2-e_1f_1 =\frac14+\frac16=\frac{5}{12}.$$

The unexpanded ratio therefore has $1+(5/6)x+(5/12)x^2+O(x^3)$, whereas the re-expanded truncation retains only $1+(5/6)x$. They agree through the declared relative order $x$ and first differ at $x^2$; that difference is not an independently known PN coefficient.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024): 4. DOI. Open PDF.
• Damour, Thibault, Bala R. Iyer, and B. S. Sathyaprakash. “Improved Filters for Gravitational Waves from Inspiralling Compact Binaries.” Physical Review D 57 (1998): 885–907. DOI. Open PDF.