Leo C. Stein

Notes: Near-identity transformations to split fast and slow motion

2026-04-05T06:00:00+00:00

The name near-identity transformation (NIT) is just shorthand for a specific application of perturbation theory—one which is particularly useful in dynamical systems that have slow and fast timescales. This is not all they can be used for (see e.g. Fumagalli+¹ where a NIT is used to remove gauge dependence). What I write below is covered in some standard references but it’s easy enough to rederive it, so I’m writing it here so I can easily find my derivation.

$\newcommand{\pd}{\partial} \newcommand{\cd}{\nabla} \newcommand{\bs}{\boldsymbol} \newcommand{\nn}{\nonumber} \newcommand{\avg}[1]{\langle {#1} \rangle}$

The nicest possible setting where we can apply a NIT is close to an integrable Hamiltonian system. Say we start with a system we can put into action-angle variables,

\begin{align} \label{eq:AA-EOM} \dot{J}_a &= 0 , & \dot{\phi}^a &= \omega^a(\bs{J}) , \end{align}

where the $\phi$ ’s are all $2\pi$ -periodic, so the phase space is foliated by N-tori. This system already has a slow-fast split; in fact the action variables don’t evolve at all, which is the slowest possible! But, now we add a perturbation that breaks this structure and add forcing terms to the right hand sides,

\begin{align} \label{eq:pertEOMJ} \dot{J}_a &= \epsilon F_a(\bs{J}, \bs{\phi}) , \\ \label{eq:pertEOMphi} \dot{\phi}^a &= \omega^a(\bs{J}) + \epsilon f^a(\bs{J}, \bs{\phi}) . \end{align}

This doesn’t need to be a Hamiltonian system. (Here I will only do first order. This can be developed to arbitrary order; see e.g. Lynch+² for second order in the context of extreme mass-ratio inspirals, but beware of differences in notation).

The trouble with Eqs. \eqref{eq:pertEOMJ} and \eqref{eq:pertEOMphi} is that now both the $J$ ’s and the $\phi$ ’s can vary rapidly on the short timescale, while there is also (usually) a slow secular drift. A classic way to get a simpler dynamical system is to simply average the right hand sides. This is covered in most standard textbooks like Goldstein or Jose and Saletan. But, it turns out we can do much better. First, we give ourselves an infinite amount of additional freedom by doubling the number of functions we’re using,³

\begin{align} \label{eq:bar-tilde-split} \bs{J}(t) &= \bar{\bs{J}}(t) + \epsilon \tilde{\bs{J}}(t) , & \bs{\phi}(t) &= \bar{\bs{\phi}}(t) + \epsilon \tilde{\bs{\phi}}(t) . \end{align}

Well, obviously we’re going to need twice as many equations. But we basically get to pick whatever extra equations we want to close the system, which is why NITs are so powerful: you can use them to enforce a slow/fast split; or as in Fumagalli+,¹ pull out gauge dependence; or maybe something else!

Anyway, a convenient choice for the $\bar{\bs{J}}$ variables is that they just evolve on a slow timescale, which is $\mathcal{O}(\epsilon^{-1})$ longer than the fast time given by $|\bs{\omega}|^{-1}$ . Let’s decompose the $F$ ’s and $f$ ’s as Fourier series on their N-tori,

\begin{align} F_a(\bs{J}, \bs{\phi}) &= \sum_{\vec{n}\in \mathbb{Z}^N} F_{a,\vec{n}}(\bs{J}) \exp(i \vec{n}\cdot\vec{\phi} ) , \\ f^a(\bs{J}, \bs{\phi}) &= \sum_{\vec{n}\in \mathbb{Z}^N} f^a_{\vec{n}}(\bs{J}) \exp(i \vec{n}\cdot\vec{\phi} ) . \end{align}

The mode with $\vec{n}=\bs{0}$ is just the torus-average, which is $\bs{\phi}$ -independent, so we can write the split

\begin{align} f^a(\bs{J}, \bs{\phi}) &= \avg{f^a}(\bs{J}) + \sum_{\vec{n}\neq \bs{0}} f^a_{\vec{n}}(\bs{J}) e^{i \vec{n}\cdot\vec{\phi}} , \end{align}

where the average is simply

\begin{align} \avg{f^a}(\bs{J}) \equiv \int \frac{d^N\phi}{(2\pi)^N} f^a(\bs{J},\bs{\phi}) \end{align}

and similarly for the $F$ ’s.

Now we plug in this Fourier decomposition and the bar/tilde split from Eq. \eqref{eq:bar-tilde-split} into the equations of motion,

\begin{align} \label{eq:J-bar-tilde-EOM} \dot{\bar{J}}_a + \epsilon \dot{\tilde{J}}_a &= \epsilon \avg{F_a}(\bs{J}) + \epsilon \sum_{\vec{n}\neq \bs{0}} F_{a,\vec{n}}(\bs{J}) e^{i \vec{n}\cdot\vec{\phi}} , \\ \label{eq:phi-bar-tilde-EOM} \dot{\bar{\phi}}^a + \epsilon \dot{\tilde{\phi}}^a &= \omega^a(\bar{\bs{J}}+\epsilon \tilde{\bs{J}}) + \epsilon \avg{f^a}(\bs{J}) + \epsilon \sum_{\vec{n}\neq \bs{0}} f^a_{\vec{n}}(\bs{J}) e^{i \vec{n}\cdot\vec{\phi}} . \end{align}

Note that I wrote $\omega(\bar{\bs{J}} + \epsilon\tilde{\bs{J}})$ . We can expand

\begin{align} \omega^a(\bar{\bs{J}} + \epsilon\tilde{\bs{J}}) = \omega^a(\bar{\bs{J}}) + \epsilon \tilde{J}_b \frac{\pd \omega^a}{\pd J_b}(\bar{\bs{J}}) . \end{align}

Everywhere else on the RHSs of Eqs. \eqref{eq:J-bar-tilde-EOM} and \eqref{eq:phi-bar-tilde-EOM}, the difference between $\bs{J}$ and $\bar{\bs{J}}$ is higher order than we need to track, so we can safely replace $\bs{J}$ with $\bar{\bs{J}}$ when convenient. Similarly, we can replace $\bs{\phi}$ with $\bar{\bs{\phi}}$ when convenient.

The first thing to notice is that if we choose the equation for $\dot{\bar{J}}$ to be the averaged one,

\begin{align} \label{eq:bar-J-dot} \dot{\bar{J}}_a = \epsilon \ \avg{F_a}(\bar{\bs{J}}) , \end{align}

then the $\bar{J}$ form an autonomous system that evolves only on the slow time—there is no short-time oscillatory $\phi$ dependence here. These are the “slow” variables with secular effects. Similarly, it would be convenient for the equation of motion for the $\bs{\bar{\phi}}$ variables to only depend on $\bar{\bs{J}}$ on the right hand side, similar to the action-angle equations of motion \eqref{eq:AA-EOM}. We are free to choose

\begin{align} \label{eq:bar-phi-dot} \dot{\bar{\phi}}^a = \omega^a(\bar{\bs{J}}) + \epsilon \ \avg{f^a}(\bar{\bs{J}}) . \end{align}

So the phases evolve with slowly-varying frequencies. The system of Eqs. \eqref{eq:bar-J-dot} and \eqref{eq:bar-phi-dot} can be easily evolved over long timescales, taking rather large timesteps.

Subtracting from the original equations of motion \eqref{eq:J-bar-tilde-EOM} and \eqref{eq:phi-bar-tilde-EOM}, the tilde variables must satisfy

\begin{align} \label{eq:tilde-J-dot} \dot{\tilde{J}}_a &= \sum_{\vec{n}\neq \bs{0}} F_{a,\vec{n}}(\bar{\bs{J}}) e^{i \vec{n}\cdot\bar{\phi}} , \\ \label{eq:tilde-phi-dot} \dot{\tilde{\phi}}^a &= \tilde{J}_b \frac{\pd \omega^a}{\pd \bar{J}_b}(\bs{\bar{J}}) + \sum_{\vec{n}\neq \bs{0}} f^{a}_{\vec{n}}(\bar{\bs{J}}) e^{i \vec{n}\cdot\bar{\phi}} , \end{align}

where we have replaced unbarred with barred variables on the right hand sides, because everything here is already $\mathcal{O}(\epsilon)$ . Observe that the RHS of \eqref{eq:tilde-J-dot} has zero mean, so except for a constant of integration—that we choose to vanish—we know that $\tilde{\bs{J}}$ are purely oscillatory, with no secular effects. Let’s perform a Fourier decomposition of $\tilde{\bs{J}}$ and compute its time derivative,

\begin{align} \tilde{J}_a &= \sum_{\vec{n}\neq \bs{0}} \tilde{J}_{a,\vec{n}}(\bar{\bs{J}}) e^{i \vec{n}\cdot\bar{\phi}} , \\ \label{eq:tilde-J-Fourier-dot} \frac{d}{dt}\tilde{J}_a &= \sum_{\vec{n}\neq \bs{0}} \left[ \frac{\pd \tilde{J}_{a,\vec{n}}}{\pd \bar{J}_b} \dot{\bar{J}}_b + \tilde{J}_{a,\vec{n}}(\bar{\bs{J}}) (i \vec{n}\cdot \dot{\bar{\bs{\phi}}}) \right]e^{i \vec{n}\cdot\bar{\phi}} , \\ \label{eq:tilde-J-Fourier-dot-plug-in} \frac{d}{dt}\tilde{J}_a &= \sum_{\vec{n}\neq \bs{0}} \tilde{J}_{a,\vec{n}}(\bar{\bs{J}}) (i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}})) e^{i \vec{n}\cdot\bar{\phi}} + \mathcal{O}(\epsilon) . \end{align}

In going from Eq. \eqref{eq:tilde-J-Fourier-dot} to \eqref{eq:tilde-J-Fourier-dot-plug-in}, we plugged in the equations of motion for $\dot{\bar{\bs{J}}}$ and $\dot{\bar{\bs{\phi}}}$ , keeping only the $\epsilon^0$ piece. Now we simply match coefficients in this Fourier expansion and the RHS of Eq. \eqref{eq:tilde-J-dot} to find

\begin{align} \label{eq:tilde-J-sol} \tilde{J}_{a,\vec{n}}(\bar{\bs{J}}) = \frac{F_{a,\vec{n}}(\bar{\bs{J}})}{i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}})} . \end{align}

Notice that this will fail near resonances, where $\tilde{\bs{J}}$ might no longer be small relative to $\bar{\bs{J}}$ ! But anyway, far from (important) resonances, if you know the Fourier decomposition of the forcing functions $F_a$ on your tori, then you know the solution for $\tilde{\bs{J}}$ (once you plug in a (possibly numerical) solution for $\bar{\bs{J}}$ ).

The same approach works for $\tilde{\bs{\phi}}$ , except there is one more term in Eq. \eqref{eq:tilde-phi-dot}. In the term $\tilde{J}_b \partial\omega^a/\partial\bar{J}_b$ , it is important that $\partial\omega^a/\partial\bar{J}_b$ depends only on $\bar{\bs{J}}$ , so that its Fourier expansion is purely “DC”. This means we don’t have to re-expand a product of Fourier expansions. Fourier-expanding $\tilde{\bs{\phi}}$ , taking a time derivative, again using the time derivatives $\dot{\bar{\bs{J}}}$ and $\dot{\bar{\bs{\phi}}}$ , and equating with the RHS of Eq. \eqref{eq:tilde-phi-dot}, we eventually find

\begin{align} \tilde{\phi}^a_{\vec{n}}(\bar{\bs{J}}) (i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}})) &= \tilde{J}_{b,\vec{n}}(\bar{\bs{J}}) \frac{\pd \omega^a}{\pd \bar{J}_b}(\bar{\bs{J}}) + f^a_\vec{n}(\bar{\bs{J}}) , \\ \tilde{\phi}^a_{\vec{n}}(\bar{\bs{J}}) &= \frac{F_{b,\vec{n}}(\bar{\bs{J}})}{(i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}}))^2} \frac{\pd \omega^a}{\pd \bar{J}_b}(\bar{\bs{J}}) + \frac{f^a_\vec{n}(\bar{\bs{J}})}{i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}})} , \end{align}

where in going to the second line we plugged in the solution for $\tilde{\bs{J}}$ from Eq. \eqref{eq:tilde-J-sol}. Here of course we still have the problem of small denominators near resonances; but other than that, the solution is just given in terms of Fourier coefficients, frequencies, and (background) gradients of the frequencies, all evaluated upon the slow solution for $\bar{\bs{J}}$ .

Summary

Summarizing, the full solution—with both secular drifts and oscillations on short timescales—is reconstructed from the sums

\begin{align} \bs{J}(t) &= \bar{\bs{J}}(t) + \epsilon \tilde{\bs{J}}(t) , & \bs{\phi}(t) &= \bar{\bs{\phi}}(t) + \epsilon \tilde{\bs{\phi}}(t) , \end{align}

where the barred (slow) variables solve the system

\begin{align} \dot{\bar{J}}_a &= \epsilon \ \avg{F_a}(\bar{\bs{J}}) , \\ \dot{\bar{\phi}}^a &= \omega^a(\bar{\bs{J}}) + \epsilon \ \avg{f^a}(\bar{\bs{J}}) , \end{align}

while the tilded (fast) variables have the Fourier expansions

\begin{align} \tilde{J}_a &= \sum_{\vec{n}\neq \bs{0}} \tilde{J}_{a,\vec{n}}(\bar{\bs{J}}) e^{i \vec{n}\cdot\bar{\phi}} , \\ \tilde{\phi}^a &= \sum_{\vec{n}\neq \bs{0}} \tilde{\phi}^{a}_{\vec{n}}(\bar{\bs{J}}) e^{i \vec{n}\cdot\bar{\phi}} , \end{align}

where their Fourier coefficients are found from

\begin{align} \tilde{J}_{a,\vec{n}}(\bar{\bs{J}}) &= \frac{F_{a,\vec{n}}(\bar{\bs{J}})}{i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}})} , \\ \tilde{\phi}^a_{\vec{n}}(\bar{\bs{J}}) &= \frac{F_{b,\vec{n}}(\bar{\bs{J}})}{(i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}}))^2} \frac{\pd \omega^a}{\pd \bar{J}_b}(\bar{\bs{J}}) + \frac{f^a_\vec{n}(\bar{\bs{J}})}{i \vec{n}\cdot \vec{\omega}(\bar{\bs{J}})} . \end{align}

References

Fumagalli et al., Nonadiabatic dynamics of eccentric black-hole binaries in post-Newtonian theory, Phys. Rev. D 112, 024012 (2025) ↩ ↩²
Lynch et al., Eccentric self-forced inspirals into a rotating black hole, Class. Quantum Grav. 39 145004 (2022) ↩
Beware: much of the literature uses different conventions for what tilde denotes. I like a bar to denote something average, and a tilde to denote something oscillatory. ↩

Fixing the center-of-mass frame of numerical relativity waveforms using the post-Newtonian center-of-mass charge

2026-03-26T00:00:00+00:00

The Bondi–van der Burg–Metzner–Sachs (BMS) frame of gravitational waves produced by numerical relativity (NR) simulations is crucial for building accurate waveform models. A proper comparison of NR waveforms with other models requires fixing the arbitrary BMS frame. In this work we improve the center-of-mass (CoM) frame fixing for quasicircular, nonprecessing binary systems. Past work approximated the CoM motion with just a linear fit. We compute a post-Newtonian result of the boosted CoM charge to also capture its physical out-spiraling oscillations. We show that using the analytical results improves the robustness of the fit parameters – translation and boost vectors – to the choice of duration and time of the fitting window. Our analysis demonstrates a maximum improvement in robustness when the window is placed at the center of the inspiral. We quantified this improvement by computing the ratio of variances of fit parameters when the fit window size is varied. The largest improvement in robustness of parameters is by a factor of ∼25 for the boost vector and ∼20 for the translation vector. Finally, we incorporate this method into the BMS frame-fixing routine of the python package 𝚜𝚌𝚛𝚒 for waveforms produced with Cauchy-characteristic evolution.

Chaos and fractals of the black hole photon ring

2026-03-10T00:00:00+00:00

The photon ring of a Kerr black hole decomposes into a self-similar hierarchy of subrings. Here, we show that this self-similar structure persists in phase space. Moreover, near the photon shell of bound photon orbits, dynamics are controlled by a Lyapunov exponent γ, whose role we highlight by computing the first-return map for light rays close to an unstably bound orbit. Despite an exponential $e^\gamma$ sensitivity to initial conditions, nearly bound rays do not exhibit chaotic behavior. However, as the background spacetime is deformed away from the Kerr geometry, chaos sets in, with its first onset most visible near strongly resonant bound orbits in the photon shell. We display two animations: one illustrating the emergence of chaos near the photon shell, which results in a fractal phase-space structure, and another exhibiting how this chaotic, fractal, self-similar structure is encoded in the first-return map.

Parameter matching between horizon quasi-local and point-particle definitions at 1PN for quasi-circular and non spinning BBH systems in harmonic gauge

2025-10-29T00:00:00+00:00

We investigate how commonly used parameter definitions in Post-Newtonian (PN) theory compare with those from Numerical Relativity (NR) for binary black hole (BBH) systems. In NR, masses and spins of each companion are measured quasi-locally from apparent horizon geometry, whereas in PN they are attributes of point particles defined via asymptotic matching in body zones. Although these definitions coincide in the infinite-separation limit, they could differ by finite-separation corrections that matter for precision modeling. Working entirely in harmonic gauge, we perform asymptotic matching between each companion’s inner zone metric – obtained from black hole perturbation theory – and the PN two-body metric, and construct the coordinate transformation that preserves the gauge in the strong field region. We solve perturbatively for the apparent horizon (AH) on a group of harmonic inertial time slice and compute its quasi-local areal mass from the horizon geometry. Then we establish the leading order matching between quasi-local (AH based) and PN (point-particle) parameter definitions in harmonic gauge. We find that on a horizon penetrating harmonic slicing, the AH quasi-local mass agrees with the PN point-particle mass at 1PN order. For generic harmonic slicings that deviate from the horizon penetrating condition by a 1PN order perturbation, the AH mass differs from the PN mass also by a 1PN correction. This parameter matching is crucial for hybridizing PN and NR waveforms and for providing better initial conditions in NR and Cauchy-Characteristic Evolution (CCE) simulations. The framework provides a bridge between different descriptions of BBH systems, and it can be extended to spinning and eccentric cases and more general NR gauges.

Black hole spectroscopy: from theory to experiment

2025-06-02T00:00:00+00:00

The “ringdown” radiation emitted by oscillating black holes has great scientific potential. By carefully predicting the frequencies and amplitudes of black hole quasinormal modes and comparing them with gravitational-wave data from compact binary mergers we can advance our understanding of the two-body problem in general relativity, verify the predictions of the theory in the regime of strong and dynamical gravitational fields, and search for physics beyond the Standard Model or new gravitational degrees of freedom. We summarize the state of the art in our understanding of black hole quasinormal modes in general relativity and modified gravity, their excitation, and the modeling of ringdown waveforms. We also review the status of LIGO-Virgo-KAGRA ringdown observations, data analysis techniques, and the bright prospects of the field in the era of LISA and next-generation ground-based gravitational-wave detectors.

The SXS Collaboration’s third catalog of binary black hole simulations

2025-05-19T00:00:00+00:00

We present a major update to the Simulating eXtreme Spacetimes (SXS) Collaboration’s catalog of binary black hole simulations. Using highly efficient spectral methods implemented in the Spectral Einstein Code SpEC, we have nearly doubled the total number of binary configurations from 2,018 to 3,756. The catalog now densely covers the parameter space with precessing simulations up to mass ratio $q=8$ and dimensionless spins up to $|\vec{\chi}|\le 0.8$ with near-zero eccentricity. The catalog also includes some simulations at higher mass ratios with moderate spin and more than 250 eccentric simulations. We have also deprecated and rerun some simulations from our previous catalog (e.g., simulations run with a much older version of SpEC or that had anomalously high errors in the waveform). The median waveform difference (which is similar to the mismatch) between resolutions over the simulations in the catalog is $4\times10^{-4}$ . The simulations have a median of 22 orbits, while the longest simulation has 148 orbits. We have corrected each waveform in the catalog to be in the binary’s center-of-mass frame and exhibit gravitational-wave memory. We estimate the total CPU cost of all simulations in the catalog to be 480,000,000 core-hours. We find that using spectral methods for binary black hole simulations is over 1,000 times more efficient than much shorter finite-difference simulations of comparable accuracy. The full catalog is publicly available through the sxs Python package and at https://data.black-holes.org.

GWSurrogate: A Python package for gravitational wave surrogate models

2025-03-29T00:00:00+00:00

Fast and accurate waveform models are fundamentally important to modern gravitational wave astrophysics, enabling the study of merging compact objects like black holes and neutron stars. However, generating high-fidelity gravitational waveforms through numerical relativity simulations is computationally intensive, often requiring days to months of computation time on supercomputers. Surrogate models provide a practical solution to dramatically accelerate waveform evaluations (typically tens of milliseconds per evaluation) while retaining the accuracy of computationally expensive simulations. The GWSurrogate Python package provides easy access to these gravitational wave surrogate models through a user-friendly interface. Currently, the package supports 16 surrogate models, each varying in duration, included physical effects (e.g., nonlinear memory, tidal forces, harmonic modes, eccentricity, mass ratio range, precession effects), and underlying solution methods (e.g., Effective One Body, numerical relativity, black hole perturbation theory). GWSurrogate models follow the waveform model conventions used by the LIGO-Virgo-Kagra collaboration, making the package immediately suitable for both theoretical studies and practical gravitational wave data analysis. By enabling rapid and precise waveform generation, GWSurrogate serves as a production-level tool for diverse applications, including parameter estimation, template bank generation, and tests of general relativity.

Modeling the BMS transformation induced by a binary black hole merger

2025-03-13T00:00:00+00:00

Understanding the characteristics of the remnant black hole formed in a binary black hole merger is crucial for conducting gravitational wave astronomy. Typically, models of remnant black holes provide information about their mass, spin, and kick velocity. However, other information related to the supertranslation symmetries of the BMS group, such as the memory effect, is also important for characterizing the final state of the system. In this work, we build a model of the BMS transformation that maps a binary black hole’s inspiral frame to the remnant black hole’s canonical rest frame. Training data for this model are created using high-precision numerical relativity simulations of quasi-circular systems with mass ratios $q \le 8$ and spins parallel to the orbital angular momentum with magnitudes $\chi_{1}, \chi_{2} \le 0.8$ . We use Gaussian Process Regression to model the BMS transformations over the three-dimensional parameter space $\left(q, \chi_{1}^{z}, \chi_{2}^{z}\right)$ . The physics captured by this model is strictly non-perturbative and cannot be obtained from post-Newtonian approximations alone, as it requires knowledge of the strong nonlinear effects that are sourced during the merger. Apart from providing the first model of the supertranslation induced by a binary black hole merger, we also find that the kick velocities predicted using Cauchy-characteristic evolution waveforms are, on average, $\sim5\%$ larger than the ones obtained from extrapolated waveforms. Our work has broad implications for improving gravitational wave models and studying the large-scale impact of memory, such as on the cosmological background. The fits produced in this work are available through the Python package surfinBH under the name NRSur3dq8BMSRemnant.

Named a Kavli Fellow

2025-03-08T00:00:00+00:00

I am honored to have been selected as one of this year’s Kavli Fellows! The National Academy of Sceinces names these fellows annually (since 1989). This year, eighty-eight early-career scientists were named Kavli Fellows, from across industry, academia, and government. We were invited to attend the 3-day symposium, the 2025 U.S. Kavli Frontiers of Science. This is the broadest-scope conference I’ve ever attended, and it was refreshing to hear so much science from outside my tiny niche. Thanks to the NAS and the Kavli Foundation for including me!

Length dependence of waveform mismatch: a caveat on waveform accuracy

2025-02-21T00:00:00+00:00

The Simulating eXtreme Spacetimes Collaboration’s code SpEC can now routinely simulate binary black hole mergers undergoing $\sim25$ orbits, with the longest simulations undergoing nearly $\sim180$ orbits. While this sounds impressive, the mismatch between the highest resolutions for this long simulation is $\mathcal{O}(10^{-1})$ . Meanwhile, the mismatch between resolutions for the more typical simulations tends to be $\mathcal{O}(10^{-4})$ , despite the resolutions being similar to the long simulations’. In this note, we explain why mismatch alone gives an incomplete picture of code—and waveform—quality, especially in the context of providing waveform templates for LISA and 3G detectors, which require templates with $\mathcal{O}(10^{3}) - \mathcal{O}(10^{5})$ orbits. We argue that to ready the GW community for the sensitivity of future detectors, numerical relativity groups must be aware of this caveat, and also run future simulations with at least three resolutions to properly assess waveform accuracy.