Fast Self-forced Inspirals

by Niels Warburton and Maarten van de Meent

LISA will fly. Since being given the green light by the European Space Agency a year ago, the scientific consortium around the Laser Interferometer Space Antenna (LISA) has been reorganising as it gears up to meet the challenge of building and operating a gravitational wave detector in space. This process has led to a renewed focus on the waveform templates that will be needed to extract the signals and estimate source parameters.

One of the key sources for LISA are extreme mass-ratio inspirals (EMRIs). In these binaries a stellar mass compact object (such as a black hole or neutron star) spirals into a massive black hole. Emitting hundreds of thousands of gravitational wave cycles in the millihertz band, LISA will detect individual EMRIs for months or even years. The low instantaneous signal-to-noise-ratio of the gravitational waves necessitates accurate waveform templates that can be used with matched filtering techniques to extract the signal from the detectors data stream. Coherently matching a signal over months or even years requires going beyond leading-order, flux-based black hole perturbation models and calculating the so-called ‘self-force’ that drives the inspiral [1]. Roughly, one can think of this self-force as arising from the smaller orbiting body interacting with its own perturbation to the metric of the massive black hole. To this end the recent “LISA Data Analysis Work Packages” document defined a number of source-modelling challenges that must be overcome before LISA flies [2]. One of these requires the community to:

Design and implement a framework for incorporating self-force-based numerical calculations, as they become available, into a flexible semi-analytical Kludge model that enables fast production of waveform templates

Our work [3], “Fast Self-forced Inspirals”, is a response to this challenge.

Accurate, astrophysical, and fast

These are the three things required of a future EMRI waveform template generator. Let’s briefly consider the requirements this places on our models.

Accurate: The waveform templates should track the phase of the physical waveform to better than one radian. Reaching this goal necessitates knowledge of the self-force [4]. Understanding the self-force has required a great deal of theoretical development in order to model the relativistic small mass-ratio two-body a problem. Actually computing the self-force is no less demanding. The “Capra” community of researchers devotes itself to these tasks.

Astrophysical: Unlike compact binary sources for ground-based detectors, EMRIs are also expected to be highly eccentric with eccentricities as large as e~0.8. Both the primary and secondary are expected to be spinning and there is no reason to believe their spins will be aligned. This means the model parameter space is very large.

Fast: The full EMRI waveform parameter space is 17 dimensional. Covering this large parameter space will require the development of new data analysis techniques. It will also require the waveform templates to be evaluated on sub-second timescales. Seven of EMRI waveform parameters, such as sky location and binary orientation, are extrinsic meaning they relate to the projection of the incoming waveform onto the LISA detector. This leaves 10 intrinsic parameters, such as mass ratio, spins, eccentricity, etc, that the self-force models must incorporate. Until our work, self-force methods for computing the waveforms were slow, taking minutes to hours for a single waveform. Our work speeds this up by many orders of magnitude.

Two timescales

In the physics of EMRIs there are at least two [5] widely disparate timescales. These are the (short) orbital timescale, the time between successive periastron passages, and the (long) radiation-reaction timescale that characterises the changes in, e.g., the secondary’s energy and angular momentum due to the emission of gravitational waves — see animation. As with any physical system with multiple timescales, capturing the effects from physics on each scale is a challenging task for modellers. For modelling EMRIs the main goal is to ensure our waveform templates faithfully capture the long-term phase evolution of the binary (and the associated waveform). Our current numerical models have to take very short time-steps in order to capture the effects the physics on the orbital timescale. In an EMRI the compact object will orbit the massive black hole hundreds of thousand of times whilst emitting gravitational waves in the LISA band. Resolving all these short timescale oscillations greatly slows down the rate at which our current models can be evaluated.

Fast Self-forced Inspirals_pic_2018

The two timescales are clearly visible in this animation of an EMRI by Steve Drasco. The motion of the secondary around is shown by the black line that traces out the recent motion of the secondary. The short orbital timescale is apparent as the rapid motion of the secondary leads its trace to look like a ‘ball of string’. Over the course of the animation the (long) radiation reaction timescale manifests itself as the ball of string becomes less eccentric and decreases in size. The video pauses as the binary goes through orbital resonances [5]. Image credited to Steve Drasco.

To overcome this issue our goal was to average over the short-timescale physics to produce an equation of motion which captures the long-term secular evolution of the system without the need to resolve the shorter timescale. There are a number of ways this can be tackled, and we turned to an approach called near-identity transformations. This is not a new technique, these transformations have a rich history being applied to dynamical systems, and in particular planetary dynamics, stretching back more than a century. Our goal was to apply this successful old idea to modelling EMRIs.

Near-identity transformations

The crucial feature of the EMRI equations of motion that makes numerically finding their solution slow is that they depend explicitly upon the orbital phase. We circumvent this problem by transforming the equations of motion to a new set of variables via a near identity transformation (NIT). This transformation has two important properties: i) the resulting equations of motion no longer depend explicitly on the orbital phase and ii) the transformation is small (hence ‘near identity’) such that the solution to the transformed equations of motion remains always close to the solution to the original equations of motion. The first of these properties allows the transformed equations of motion to be numerically solved in milliseconds, rather than minutes or hours as for the original equations. The second property ensures that the resulting solution encapsulates all the self-force physics that the original, slow to compute, solution did.

In our work we wrote down the NITs to the EMRI equations of motion in a general form before specialising to the case where the massive black hole is not rotating. We then implemented these equations into a numerical scheme and showed that our new scheme recovered all the previously known physics but hundreds to hundreds of thousands of times faster — see Table.


Table showing the speed up of the calculation of EMRI phase space trajectories [7]. Here eta is the mass ratio = (secondary mass)/(primary mass). Old self-force codes took seconds to hours to compute what our new approach can compute in milliseconds. Depending on the mass ratio, this represents a speed up of ~1000’s to hundreds of 1000’s.

Future Work

Our work provides one possible answer [8] to the challenge to “Design and implement a framework for incorporating self-force-based numerical calculations … that enables fast production of waveform templates“. Of the three requirements of such a framework that it be accurate, astrophysical and fast our work deals with the latter: it is fast to evaluate. Crucially though, it is also a framework which can easily incorporate more physics in a systematic way to produce a model that meets all three requirements.

Our work provides a proof of principle code that covers the small eccentricity (e <= 0.2) part of the parameter space for a non-rotating black hole. The code can be made more astrophysically relevant by incorporating the results of, e.g., [12] to extend the eccentricity to cover the expected range. The extension to a rotating primary involves writing down near identity transformed equations of motion (something which we are actively working on) and computing the self-force for generic orbits about a Kerr black hole [13].

Finally, to reach the desired accuracy goal it is necessary for the self-force community to go beyond linear (in the mass) ratio perturbation theory and consider second-order corrections. A great deal of theoretical work, beginning with [14,15], has gone into this goal over the last 5 years with the first concrete results beginning to emerge [16].

Niels Warburton is a Royal Society – Science Foundation Ireland University Research Fellow at University College Dublin in Ireland. His website is

Maarten van de Meent is a Marie Skłodowska-Curie Fellow at the Albert Einstein institute in Potsdam, Germany.

Read the full paper in CQG here.

The CQG paper is part of the Focus Issue: Approaches to the two-body problem. Read more papers from this Focus Issue here.

The code developed as part of this work is publicly available as part of the Black Hole Perturbation Toolkit at

References and footnotes:

[1] A recent review of the self-force, aimed at non-specialists with some knowledge of General Relativity, can be found at A much more technical review can be found here:
[2] LISA Data Analysis Work Packages: LISA-LCST-SGS-WPD-001, Section 1.2
[3] M. van de Meent, N. Warburton, arXiv:1802.05281
[4] T. Hinderer, E. E. Flanagan, Phys. Rev. D78:064028, (2008), arXiv:0805.3337
[5] When the secondary is spinning there is also a precession timescale. Another timescale also enters when the EMRI configuration evolves through a radial-polar orbital resonance [5].
[6] T. Hinderer, E. E. Flanagan, Phys. Rev. Lett. 109.071102 (2010), arXiv:1009.4923
[7] There are a number of steps in producing an EMRI waveform. Generally, these are i) compute the phase space trajectory, ii) compute the physical trajectory, and iii) compute the waveform. In the past is was step i) that was very slow in self-force calculations. The time spent on the other steps depend on how often you wish to sample the EMRI waveform, which in turn depends upon your data analyse approach.
[8] Another well developed approach are so-called “Kludge” models [9,10,11]. These were designed to scope out the data analysis task and so are quick to evaluate. Over the years they had added additional physics though to date have not included self-force corrections. One advantage to our approach is that we can systematically add self-force corrections as they become available.
[9] L. Barack, C. Cutler, Phys.Rev. D69 (2004) 082005, arXiv:gr-qc/0310125
[10] S. Babak, H. Fang, J. R. Gair, K. Glampedakis, S. A. HughesPhys.Rev.D75:024005 (2007), arXiv:gr-qc/0607007
[11] A. J. K. Chua, C. J. Moore, J. R. Gair, Phys. Rev. D 96, 044005 (2017), arXiv:1705.04259
[12] T. Osburn, N. Warburton, C. R. Evans, Phys. Rev. D 93, 064024 (2016), arXiv:1511.01498
[13] M. van de Meent, Phys. Rev. D 97, 104033 (2018), arXiv:1711.09607
[14] A. Pound, Phys. Rev. Lett. 109, 051101 (2012), arXiv:1201.5089
[15] S. E. Gralla, Phys. Rev. D 85, 124011 (2012), arXiv:1203.3189
[16] A. Pound, B. Wardell, N. Warburton, J. Miller, “Second-order self-force calculation of gravitational binding energy”, In preparation

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