Preparation of calibration curves

While the perturbative (self-force) approach has proven highly effective to date, there remain several important and challenging areas for further development. Here, we list a number of the most important future challenges and prospects for the GSF programme.

2.2.1. Efficient incorporation of self-force information into waveform models There are at least two key aspects to producing an EMRI model: (i) computing the GSF; and (ii) using the GSF to actually drive an inspiral. Unfortunately, despite the substantial

advances in calculational approaches, GSF calculations are still much too slow to be useful on their own as a means for producing LISA gravitational waveforms. Existing work has been able to produce first-order GSF driven inspirals for a small number of

cases [764, 784, 792, 879] but when one takes into account the large parameter space

of EMRI systems it is clear that these existing methods are inadequate. It is therefore important to develop efficient methods for incorporating GSF information into EMRI models and waveforms. There has been some promising recent progress in this direction, with fast “kludge” codes producing approximate (but not sufficiently accurate) inspirals

[711, 713, 716, 717, 880–882], and with the emergence of mathematical frameworks

based on near-identity transformations [883], renormalization group methods [884] and

two-timescale expansions [885].

To complicate matters, inspirals generically go through a number of transient resonances, when the momentary radial and polar frequencies of the orbit occur in a small rational ratio. During such resonances, approximations based on adiabaticity

break down [886–890]. Works so far have mapped the locations of resonances in the

inspiral parameter space, studied how the orbital parameters (including energy and angular momentum) experience a “jump” upon resonant crossing, and illustrated how the magnitude of the jump depends sensitively on the precise resonant phase (the relative phase between the radial and polar motions at resonance). The impact of

resonances on the detectability of EMRIs with LISA was studied in Refs. [891, 892]

using an approximate model of the resonant crossing. But so far there has been no actual calculation of the orbital evolution through a resonance. Now that GSF codes for generic orbits are finally at hand, such calculations become possible, in principle. There is a vital need to perform such calculations, in order to allow orbital evolution methods to safely pass through resonances without a significant loss in the accuracy of the inspiral model.

2.2.2. Producing accurate waveform models: self-consistent evolution and second-order

gravitational self-force Possibly the most challenging outstanding obstacle to reaching

the sub-radian phase accuracy required for LISA data analysis is the fact that the first-order GSF on its own is insufficient and one must also incorporate information

at second perturbative order [702]. There are ongoing efforts to develop tools for

computing the second order GSF [727, 734, 735, 769, 806, 885, 893–896], but, despite

significant progress, a full calculation of the second-order metric perturbation has yet to be completed.

One of the challenges of the GSF problem when considered through second order is that it is naturally formulated as a self-consistent problem, whereby the coupled equations for the metric perturbation and for the particle worldline are evolved simultaneously. Indeed, this self-consistent evolution has yet to be completed even for the first-order GSF (it has, however, been done for the toy-model scalar charge case

[764]). Even when methods are developed for computing the second order GSF, it will

scheme.

2.2.3. Gravitational Green Function One of the first proposals for a practical method

for computing the GSF was based on writing the regularized metric perturbation in terms of a convolution, integrating the Green function for the wave equation along the

worldline of the particle [740, 743]. It took several years for this idea to be turned

into a complete calculation of the GSF, and even then the results were restricted to

the toy-model problem of a scalar charge moving in Schwarzschild spacetime [768,771].

Despite this deficiency, the Green function approach has produced a novel perspective on the GSF problem.

In principle, the methods used for a scalar field in Schwarzschild spacetime should be applicable to the GSF problem in Kerr. The challenge is two-fold: (i) to actually adapt the methods to the Kerr problem and to the relevant wave equation, which is the linearized Einstein equation in the Lorenz gauge; and (ii) to explore whether and how one can instead work with the much simpler Teukolsky wave equation.

2.2.4. Internal-structure effects The vast majority of GSF calculations to date have

been based on the assumption that the smaller object is spherically symmetric and non-spinning. This idealization ignores the possibility that the smaller BH may be spinning, or more generally that other internal-structure effects may be relevant. Unfortunately, this picture is inadequate; certain internal-structure effects can make important contributions to the equations of motion. For example, the coupling of the small body’s spin to the larger BH’s spacetime curvature (commonly referred to as the Mathisson-Papapetrou force) is expected to contribute to the phase evolution of a

typical EMRI at the same order as the conservative piece of the first-order GSF [897].

Furthermore, finite internal-structure is likely to be even more important in the case that the smaller body is a NS.

While there has been some progress in assessing the contribution from the smaller

body’s spin to the motion [879, 897–903], existing work has focused on flux-based

calculations or on the PN regime. It remains an outstanding challenge to determine the influence of internal-structure effects on the GSF, especially at second order.

2.2.5. EMRIs in alternative theories of gravity In all of the discussion so far, we have

made one overarching assumption: that BHs behave as described by General Relativity (GR). However, with EMRIs we have the exciting prospect of not simply assuming this fact, but of testing its validity with exquisite precision. There is initial work on the

self-force in the context of scalar-tensor gravity [904], but much more remains to be

done to establish exactly what EMRIs can do to test the validity of GR (and how) when pushed to its most extreme limits. Much more on this in Chapter III (see, in particular,

2.2.6. Open tools and datasets While there has been significant progress in developing tools for computing the GSF, much of it has been ad-hoc, with individual groups developing their own private tools and codes. Now that a clear picture has emerged of exactly which are the most useful methods and tools, the community has begun to combine their efforts. This has lead to the development of a number of initiatives, including (i) tabulated results for Kerr quasinormal modes and their excitation

factors [905, 906]; (ii) open source “kludge” codes for generating an approximate

waveform for EMRIS [881, 907]; and (ii) online repositories of self-force results [908].

It is important for such efforts to continue, so that the results of the many years of development of GSF tools and methods are available to the widest possible user base. One promising initiative in this direction is the ongoing development of the Black Hole

Perturbation Toolkit [909], a free and open source set of codes and results produced by

the GSF community.

3. Post-Newtonian and Post-Minkowskian Methods

Contributor: A. Le Tiec

3.1. Background

The PN formalism is an approximation method in GR that is well suited to describe the orbital motion and the GW emission from binary systems of compact objects, in a regime where the orbital velocity is small compared to the speed of light and the gravitational fields are weak. This approximation method has played a key role in the recent detections, by the LIGO and Virgo observatories, of GWs generated by

inspiralling and merging BH and NS binaries [1, 2, 11, 18, 21], by providing accurate

template waveforms to search for those signals and to interpret them. Here we give a brief overview of the application of the PN approximation to binary systems of compact objects, focusing on recent developments and future prospects. See the review articles

[910–917] and the textbooks [631,918] for more information.

In PN theory, relativistic corrections to the Newtonian solution are incorporated in a systematic manner into the equations of motion (EOM) and the radiation field,

order by order in the small parameter v2_/c2 _{∼ Gm/(c}2_{r), where v and r are the typical}

relative orbital velocity and binary separation, m is the sum of the component masses,

and we used the fact that v2 _{∼ Gm/r for bound motion. (The most promising sources}

for current and future GW detectors are bound systems of compact objects.)

Another important approximation method is the post-Minkowskian (PM) approximation, or non-linearity expansion in Newton’s gravitational constant G, which

assumes weak fields (Gm/c2_{r  1) but unrestricted speeds (v}2_/c2 . 1), and perturbs

about the limit of special relativity. In fact, the construction of accurate gravitational waveforms for inspiralling compact binaries requires a combination of PN and PM techniques in order to solve two coupled problems, namely the problem of motion and

that of wave generation.

The two-body EOM have been derived in a PN framework using three well- developed sets of techniques in classical GR: (i) the PN iteration of the Einstein

field equations in harmonic coordinates [919–923], (ii) the Arnowitt-Deser-Misner

(ADM) canonical Hamiltonian formalism [924–926], and (iii) a surface integral approach

pioneered by Einstein, Infeld and Hoffmann [927–929]. By the early 2000s, each of these

approaches has independently produced a computation of the EOM for binary systems

of non-spinning compact objects through the 3rd PN order (3PN).∗ More recently, the

application of effective field theory (EFT) methods [930], inspired from quantum field

theory, has provided an additional independent derivation of the 3PN EOM [931]. All

of those results were shown to be in perfect agreement. Moreover, the 3.5PN terms— which constitute a 1PN relative correction to the leading radiation-reaction force—are

also known [932–937].

At the same time, the problem of radiation (i.e., computing the field in the far/wave zone) has been extensively investigated within the multipolar PM wave generation

formalism of Blanchet and Damour [938–940], using the “direct integration of the relaxed

Einstein equation” approach of Will, Wiseman and Pati [941, 942], and more recently

with EFT techniques [943, 944]. The application of these formalisms to non-spinning

compact binaries has, so far, resulted in the computation of the GW phase up to the

relative 3.5PN (resp. 3PN) order for quasi-circular (resp. quasi-eccentric) orbits [945–

952], while amplitude corrections in the GW polarizations are known to 3PN order, and

even to 3.5PN order for the quadrupolar mode [953–958].