PN Convergence, Resummation, and Error Estimates

A post-Newtonian formula is useful only together with a statement of where it is accurate and how its omitted terms are controlled. The formal small parameter is often clear, but a finite list of coefficients does not by itself establish convergence, and a resummation can improve one observable while degrading another. This page develops a conservative procedure: estimate uncertainty for a specified invariant observable and parameter domain, compare genuinely different approximants, and test the estimate against independent strong-field information whenever such information exists.

Evidence cutoff: 2026-08-12. The assessments below include literature available through this date. They are conclusions about finite-order performance, not a proof that the PN expansion converges.

Required background. Weak Metric Perturbations and Gauge Freedom supplies perturbative gauge transformations; Presymplectic Potentials and Currents supplies the constrained and canonical mechanics behind invariant comparisons. Helpful background. The Einstein Equation supplies the parent dynamics; Gravitational Radiation and Waveforms supplies the fluxes and phases used in strong-field comparisons.

What convergence can mean

For a circular, nonspinning binary, let

$$x=\left(\frac{GM\Omega}{c^3}\right)^{2/3}, \qquad \nu=\frac{m_1m_2}{(m_1+m_2)^2}.$$

A dimensionless invariant $Q$ may have the schematic truncation

$$Q_N(x,\nu) =\sum_{n=0}^{N} x^{n/2} \sum_{k=0}^{k_{\max}(n)} q_{n,k}(\nu)(\ln x)^k.$$

Half-integer powers can arise from hereditary effects, while logarithms record scale mixing rather than a failure of power counting. Three distinct questions must be kept separate.

Mathematical convergence asks whether $Q_N$ approaches a unique limit as $N\to\infty$ at fixed $x$ and $\nu$. Establishing it would require information about arbitrarily high orders or about the analytic structure of the exact function. Present comparable-mass results provide neither.

Asymptotic usefulness asks whether the first neglected term becomes smaller over the orders actually known. An asymptotic series can give excellent predictions before its terms eventually grow; the best truncation then depends on $x$.

Phenomenological accuracy asks whether the finite approximant meets a specified tolerance for a specified observable—for example, a binding energy, periastron advance, accumulated phase, or waveform mismatch. It is this last question that comparison with self-force theory or numerical relativity can answer directly.

The test-mass circular energy gives a clean warning against conflating these meanings. In Schwarzschild spacetime,

$$\frac{E_{\mathrm{Schw}}}{\mu c^2} =\frac{1-2x}{\sqrt{1-3x}}-1.$$

Its Taylor series about $x=0$ encounters a branch point at the light ring, $x=1/3$, even though stable circular motion ends earlier at the innermost stable circular orbit, $x=1/6$. Thus a series may have a nonzero radius of convergence yet become practically inefficient before reaching its nearest complex-plane singularity. The exact result and its PN expansion are displayed in Blanchet 2006, § 9.3, eqs. (208)–(209). For finite $\nu$, it is reasonable to use the test-mass singularity as a structural clue, but it is not legitimate to assume without further evidence that the comparable-mass analytic structure is identical.

Estimating a finite-order remainder

The exact test-mass result also supplies a small calculation that can be repeated without waveform software. Expanding the energy gives

$$e_{\rm 3PN}(x)\equiv\frac{E_{\rm 3PN}}{\mu c^2} =-\frac x2+\frac38x^2+\frac{27}{16}x^3+\frac{675}{128}x^4.$$

Here 1PN retains the $x^2$ term and 2PN retains the $x^3$ term. Direct evaluation of the exact expression and this polynomial gives:

$x$	Exact $E/(\mu c^2)$	3PN $E/(\mu c^2)$	Relative 3PN energy error
$0.01$	$-0.00496075817$	$-0.00496075977$	$3.22\times10^{-7}$
$0.10$	$-0.04381711253$	$-0.04403515625$	$4.98\times10^{-3}$
$1/6$	$-0.05719095842$	$-0.06103515625$	$6.72\times10^{-2}$

The last row is the marginally stable circular orbit, approached from the stable branch. The omitted remainder there is about $0.945$ times the last retained correction, even though the Taylor series converges. At $x=0.01$ that ratio is only $0.0303$. Thus the last-term estimate can be conservative in one domain and barely cover the error in another. These numbers concern only the test-mass energy, not the flux, phase, or comparable-mass dynamics.

Write the newest correction to one invariant as

$$\Delta_N Q=Q_N-Q_{N-1}.$$

The elementary estimate $\delta_{\mathrm{next}}=\lvert\Delta_N Q\rvert$ is dimensionally and parametrically sensible when consecutive terms decrease regularly. It is not a confidence interval: an accidentally small coefficient, a logarithmic cancellation, or a change of sign can make it severely optimistic. A ratio estimate,

$$\delta_{\mathrm{ratio}} =\lvert\Delta_N Q\rvert \frac{\lvert\Delta_N Q\rvert} {\lvert\Delta_{N-1}Q\rvert},$$

is useful only when the denominator is not anomalously small and the last several ratios are stable. Even then, it is an extrapolation of observed coefficient behavior.

A more defensible error budget uses several diagnostics:

1. Order variation. Compare $Q_N$, $Q_{N-1}$, and, when available, $Q_{N-2}$. A nonmonotone sequence is information, not a reason to discard an inconvenient order.
2. Representation variation. Compare Taylor, inverse-Taylor, Padé, factorized, or effective-one-body representations built from the same known coefficients. Their spread probes uncomputed terms only imperfectly because the approximants are correlated.
3. Scale and logarithm variation. If the separation of local and hereditary pieces introduces an intermediate scale, vary it before cancellation at finite order. The variation must be performed consistently across all terms that share that scale.
4. Independent benchmarks. Compare invariant quantities with black-hole perturbation theory, gravitational self-force results, or numerical relativity in their overlap domain. Coordinate accelerations or gauge-dependent orbital radii are not suitable comparison targets.

For practical use, one may define an envelope

$$\delta Q =\max\!\left( \lvert\Delta_N Q\rvert,\, \frac{1}{2}\max_{a,b} \lvert Q_N^{(a)}-Q_N^{(b)}\rvert,\, \delta_{\mathrm{bench}} \right),$$

where $a,b$ label defensible representations and $\delta_{\mathrm{bench}}$ includes both reference uncertainty and observed discrepancy. This prescription is deliberately cautious. It should be recomputed as a function of $x$, $\nu$, eccentricity, and spins; quoting one number for an entire inspiral conceals the growth of strong-field error.

The observable also matters. An energy can agree well while the flux does not, and a small instantaneous flux error can integrate into a consequential phase error. If $dE/dt=-\mathcal F$, then

$$\frac{d\phi}{dx} =-\Omega(x)\, \frac{dE/dx}{\mathcal F(x)}.$$

Errors in $E'$ and $\mathcal F$ are therefore combined through a ratio and can cancel or reinforce. A waveform-accuracy statement should propagate them through the phasing construction rather than compare their coefficient sizes in isolation.

What resummation changes

A resummation uses the known PN coefficients to define a new function whose re-expansion reproduces those coefficients but whose higher-order terms are chosen implicitly. For a Taylor polynomial $T_N(x)$, common choices include a Padé rational function, a factorization into physically motivated pieces, or a mapping to effective-one-body potentials. None adds new perturbative information at orders already fixed; each adds a hypothesis about the uncomputed completion.

Padé approximants can represent a pole or branch-like trend more economically than a polynomial. They can also produce a pole in the physical domain that is absent from the true observable. Such a pole is a failure of that approximant, not evidence for new binary dynamics. Factorized waveforms separate a Newtonian multipole, an effective source, tail factors, phases, and residual amplitudes. In the extreme-mass-ratio comparisons of Damour, Iyer, and Nagar 2009, § II, eqs. (1)–(7), and § V, this structure substantially improved many multipolar amplitudes relative to direct Taylor expansion. The result is evidence for that factorization in the tested regime, not a theorem that every factorized comparable-mass model is superior.

The effective-one-body construction similarly repackages the conservative and dissipative PN information into deformed test-body dynamics. Exact test-mass limits, known logarithms, and independently calculated self-force coefficients can constrain the repackaging. Calibration to numerical relativity can improve a model further, but then the error statement must distinguish analytically predicted information from fitted information and must test configurations outside the calibration set.

Evidence and credible competing positions

The available comparisons support several positions that are credible because they answer different questions.

Use the strict Taylor sequence as the baseline. This position emphasizes transparency: the last known term and order-to-order change are visible, and no artificial rational pole is introduced. It is especially compelling at small $x$, where several decreasing terms are available. A classic equal-mass, nonspinning comparison found that generic Taylor approximants accumulated several radians late in a 30-cycle simulation, while the exceptional agreement of 3.5PN TaylorT4 was judged likely coincidental Boyle et al. 2007, §§ V–VI, especially figs. 22–24. The lesson is not that TaylorT4 always fails; it is that one unusually successful approximant cannot by itself establish a reliable remainder model.

Use structure-informed resummation for late inspiral. This position emphasizes exact limits and known nonperturbative organization. Factorized multipoles and effective-one-body dynamics often extend useful agreement toward merger. A 350-cycle mass-ratio-seven numerical simulation found close agreement with an effective-one-body model across a long overlap, while comparisons among uncalibrated PN approximants depended on the chosen phasing prescription Szilágyi et al. 2015, pp. 2–5, figs. 2–3. This supports resummation as a powerful modeling method, while leaving its uncertainty configuration-dependent.

Decide observable by observable and regime by regime. High-order perturbation theory makes this pluralist view unavoidable. For eccentric Schwarzschild extreme-mass-ratio fluxes, a 19PN calculation found generally strong agreement with numerical data, but performance degraded for small semilatus rectum and high eccentricity, and different modewise or full-flux resummations behaved differently Munna 2020, § VI, especially figs. 2–8. In quasi-circular Kerr extreme-mass-ratio tests, the region satisfying a chosen error criterion generally expanded with PN order but had local nonmonotonicities at lower orders Sago, Fujita, and Nakano 2016, §§ III–IV, figs. 1–6. Neither result transfers automatically to comparable masses, generic spins, or waveform modes that were not tested.

These positions are not mutually exclusive. A sound workflow keeps the Taylor result as a reproducible reference, uses resummed models when benchmark evidence supports them, and assigns uncertainty from independent comparisons rather than from preference for a particular representation.

Uncertainties and open questions

Several uncertainties remain material. Only finitely many comparable-mass coefficients are known. Logarithms and half-integer terms complicate simple ratio tests. Coordinate transformations can make gauge-dependent coefficients look better or worse without changing the physics. Numerical-relativity comparisons carry truncation, extraction, eccentricity, and alignment errors. Self-force data are exceptionally precise in small mass ratio but require a controlled extrapolation in $\nu$. Calibrated waveform models can interpolate accurately within their training domain yet have larger errors for high eccentricity, large precession, or extreme mass ratio.

This assessment should be revisited when any of the following occurs:

• a new PN, post-Minkowskian, or self-force coefficient changes the observed large-order pattern;
• a long, higher-accuracy numerical-relativity simulation probes mass ratios, spin precession, or eccentricities outside existing calibration sets;
• analytic work identifies the relevant complex-plane singularities or Borel structure for a gauge-invariant comparable-mass observable;
• a proposed Padé or other rational approximant develops a spurious pole in the physical inspiral domain;
• independent calculations disagree on a regularization constant, tail term, or logarithmic coefficient;
• detector accuracy requirements make a previously negligible phasing or amplitude discrepancy consequential.

Until such evidence is incorporated, the honest output is not “the PN series converges” or “resummation solves convergence,” but a dated, observable-specific error envelope and a clearly stated domain of validity.

Exercise

At one frequency, three successive approximants to an invariant are $Q_1=1.100$, $Q_2=1.145$, and $Q_3=1.151$. Two resummations that reproduce the same known PN coefficients give $Q_3^{(A)}=1.149$ and $Q_3^{(B)}=1.162$. An independent calculation gives $Q_{\mathrm{ref}}=1.150\pm0.002$. Compute the last-term estimate, half the representation spread, and the discrepancy from the reference central value. Apply the envelope above, taking $\delta_{\mathrm{bench}}$ to be the larger discrepancy plus the reference uncertainty. Explain why the result is local rather than a guarantee for the rest of the inspiral.

Solution

The last correction is

$$\delta_{\mathrm{next}} =\lvert Q_3-Q_2\rvert =0.006.$$

Across the Taylor value and the two resummations, the largest separation is $1.162-1.149=0.013$, so half the representation spread is $0.0065$. The largest central-value discrepancy is $\lvert1.162-1.150\rvert=0.012$; adding the reference uncertainty gives $\delta_{\mathrm{bench}}=0.014$. The envelope is therefore

$$\delta Q=\max(0.006,0.0065,0.014)=0.014.$$

The benchmark, not the small final Taylor correction, controls the estimate. This conclusion applies only at the stated frequency and parameters. As $x$ grows, coefficient ratios, cancellations, and the relative performance of the representations can change, so the calculation must be repeated along the inspiral.

References

• Blanchet, Luc. “Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries.” Living Reviews in Relativity 9, 4 (2006). doi:10.12942/lrr-2006-4.
• Boyle, Michael, Duncan A. Brown, Lawrence E. Kidder, Abdul H. Mroué, Harald P. Pfeiffer, Mark A. Scheel, Gregory B. Cook, and Saul A. Teukolsky. “High-Accuracy Comparison of Numerical Relativity Simulations with Post-Newtonian Expansions.” Physical Review D 76, 124038 (2007). doi:10.1103/PhysRevD.76.124038.
• Damour, Thibault, Bala R. Iyer, and Alessandro Nagar. “Improved Resummation of Post-Newtonian Multipolar Waveforms from Circularized Compact Binaries.” Physical Review D 79, 064004 (2009). doi:10.1103/PhysRevD.79.064004.
• Munna, Christopher. “Analytic Post-Newtonian Expansion of the Energy and Angular Momentum Radiated to Infinity by Eccentric-Orbit Nonspinning Extreme-Mass-Ratio Inspirals to the 19th Order.” Physical Review D 102, 124001 (2020). doi:10.1103/PhysRevD.102.124001.
• Sago, Norichika, Ryuichi Fujita, and Hiroyuki Nakano. “Accuracy of the Post-Newtonian Approximation for Extreme Mass Ratio Inspirals from a Black-Hole Perturbation Approach.” Physical Review D 93, 104023 (2016). doi:10.1103/PhysRevD.93.104023.
• Szilágyi, Béla, Jonathan Blackman, Alessandra Buonanno, Andrea Taracchini, Harald P. Pfeiffer, Mark A. Scheel, Tony Chu, Lawrence E. Kidder, and Yi Pan. “Approaching the Post-Newtonian Regime with Numerical Relativity: A Compact-Object Binary Simulation Spanning 350 Gravitational-Wave Cycles.” Physical Review Letters 115, 031102 (2015). doi:10.1103/PhysRevLett.115.031102.