Over the last five years or so, significant progress on PN modelling of compact binary systems has been achieved on multiple fronts, including (i) the extension of the EOM to 4PN order for non-spinning bodies (with partial results also obtained for aspects of the two-body dynamics at the 5PN order [818, 827, 959]), (ii) the inclusion of spin effects in the binary dynamics and waveform, (iii) the comparison of several PN predictions to those from GSF theory, and (iv) the derivation of general laws controlling the mechanics of compact binaries.
3.2.1. 4PN equations of motion for non-spinning compact-object binaries Recently,
the computation of the two-body EOM has been extended to 4PN order, by using both the canonical Hamiltonian framework in ADM-TT coordinates [960–965] and a Fokker Lagrangian approach in harmonic coordinates [966–970]. Partial results at 4PN order have also been obtained using EFT techniques [971–973]. All of those high-order PN calculations resort to a point-particle model for the (non-spinning) compact objects, and rely on dimensional regularization to treat the local ultraviolet (UV) divergences
∗ By convention, “nPN” refers to EOM terms that are O(1/c2n) smaller than the Newtonian acceleration,
that are associated with the use of point particles. The new 4PN results have been used to inform the EOB framework [974], a semi-analytic model of the binary dynamics and wave emission (see Sec. 6below).
The occurrence at the 4PN order of infrared (IR) divergences of spatial integrals led to the introduction of several ambiguity parameters; one in the ADM Hamiltonian approach and two in the Fokker Lagrangian approach. One of those IR ambiguity parameters was initially fixed by requiring agreement with an analytical GSF calculation [975] of the so-called Detweiler redshift along circular orbits [816, 826]. Recently, however, Marchand et al. [969] gave the first complete (i.e., ambiguity-free) derivation of the 4PN EOM. The last remaining ambiguity parameter was determined from first principles, by resorting to a matching between the near-zone and far-zone fields, together with a computation of the conservative 4PN tail effect in d dimensions, allowing to treat both UV and IR divergences using dimensional regularization.
Another interesting (and related) feature of the binary dynamics at the 4PN order is that it becomes non-local in time [962, 966], because of the occurence of a GW tail effect at that order: gravitational radiation that gets scattered off the background spacetime curvature backreacts on the orbital motion at later times, such that the binary’s dynamics at a given moment in time depends on its entire past history [976– 978].
3.2.2. Spin effects in the binary dynamics and gravitational waveform Since stellar-
mass and/or supermassive BHs may carry significant spins [979, 980], much effort has recently been devoted to include spin effects in PN template waveforms. In particular, spin-orbit coupling terms linear in either of the two spins have been computed up to the next-to-next-to leading order, corresponding to 3.5PN order in the EOM, using the ADM Hamiltonian framework [981, 982], the PN iteration of the Einstein field equations in harmonic coordinates [983, 984], and EFT techniques [985]. Spin-spin coupling terms proportional to the product of the two spins have also been computed to the next-to-next-to leading order, corresponding to 4PN order in the EOM, using the ADM Hamiltonian and EFT formalisms [986–990]. The leading order 3.5PN cubic-in- spin and 4PN quartic-in-spin contributions to the binary dynamics are also known for generic compact bodies [991–995], as well as all higher-order-in-spin contributions for BBHs (to leading PN order) [996]. All these results are summarized in Fig. 9. 2PN BH binary spin precession was recently revisited using multi-timescale methods [997, 998], uncovering new phenomenology such as precessional instabilities [999] and nutational resonances [1000].
Spin-related effects on the far-zone field have also been computed to high orders, for compact binaries on quasi-circular orbits. To linear order in the spins, those effects are known up to the relative 4PN order in the GW energy flux and phasing [1001,1002], and to 2PN in the wave polarizations [1003, 1004]. At quadratic order in the spins, the contributions to the GW energy flux and phasing have been computed to 3PN order [1005, 1006], and partial results were derived for amplitude corrections to 2.5PN order
[1007]. The leading 3.5PN cubic-in-spin effects in the GW energy flux and phasing are known as well [993]. ������� �������� ��� ������ ������� ��� � ��� ��� ��� ��� ����� ������ ������� �������� ������ ������� �������� ��������� ������ ������� �������� ������ ������� �������� ������ ������� ������ ������� ������ ��� ������� ������ ���������
Figure 9. Contributions to the two-body Hamiltonian in the PN spin expansion, for
arbitrary-mass-ratio binaries with spin induced multipole moments. Contributions in red are yet to be calculated. LO stands for “leading order”, NLO for “next-to-leading order”, and so on. SO stands for “spin-orbit”. Figure from Ref. [996].
Finally, some recent works have uncovered remarkable relationships between the PN [996] and PM [1008] dynamics of a binary system of spinning BHs with an arbritrary mass ratio on the one hand, and that of a test BH in a Kerr background spacetime on the other hand. Those results are especially relevant for the ongoing development of EOB models for spinning BH binaries (see Sec.6), and in fact give new insight into the energy map at the core of such models.
3.2.3. Comparisons to perturbative gravitational self-force calculations The GWs
generated by a coalescing compact binary system are not the only observable of interest. As we have described in Sec. 2, over recent years, several conservative effects on the orbital dynamics of compact-object binaries moving along quasi-circular orbits have been used to compare the predictions of the PN approximation to those of the GSF framework, by making use of gauge-invariant quantities such as (i) the Detweiler redshift [816,827,828,831,1009,1010], (ii) the relativistic periastron advance [818, 819, 842, 844], (iii) the geodetic spin precession frequency [820], and (iv) various tidal invariants [824, 825], all computed as functions of the circular-orbit frequency of
the binary. Some of these comparisons were extended to generic bound (eccentric) orbits [822, 836, 847]. All of those comparisons showed perfect agreement in the common domain of validity of the two approximation schemes, thus providing crucial tests for both methods. Building on recent progress on the second-order GSF problem [729, 734, 735, 796, 895, 896], we expect such comparisons to be extended to second order in the mass ratio, e.g. by using the redshift variable [893].
Independently, the BH perturbative techniques of Mano, Suzuki and Takasugi [857, 1011] have been applied to compute analytically, up to very high orders, the PN expansions of the GSF contributions to the redshift for circular [838, 862, 1012, 1013] and eccentric [832, 833, 837, 839, 840] orbits, the geodetic spin precession frequency [841, 863,1014], and various tidal invariants [821,830, 834]. Additionally, using similar techniques, some of those quantities have been computed numerically, with very high accuracy, allowing the extraction of the exact, analytical values of many PN coefficients [854, 859, 861].
3.2.4. First law of compact binary mechanics The conservative dynamics of a binary
system of compact objects has a fundamental property now known as the first law of
binary mechanics [1015]. Remarkably, this variational formula can be used to relate local
physical quantities that characterize each body (e.g. the redshift) to global quantites that characterize the binary system (e.g. the binding energy). For point-particle binaries moving along circular orbits, this law is a particular case of a more general result, valid for systems of BHs and extended matter sources [1016].
Using the ADM Hamiltonian formalism, the first law of [1015] was generalized to spinning point particles, for spins (anti-)aligned with the orbital angular momentum [1017], and to non-spinning binaries moving along generic bound (eccentric) orbits [1018]. The derivation of the first law for eccentric motion was then extended to account for the non-locality in time of the orbital dynamics due to the occurence at the 4PN order of a GW tail effect [1019]. These various laws were derived on general grounds, assuming only that the conservative dynamics of the binary derives from an autonomous canonical Hamiltonian. (First-law-type relationships have also been derived in the context of linear BH perturbation theory and the GSF framework [1020–1022].) Moreover, they have been checked to hold true up to 3PN order, and even up to 5PN order for some logarithmic terms.
So far the first laws have been applied to (i) determine the numerical value of the aforementioned ambiguity parameter appearing in derivations of the 4PN two-body EOM [975], (ii) calculate the exact linear-in-the-mass-ratio contributions to the binary’s binding energy and angular momentum for circular motion [843], (iii) compute the shift in the frequencies of the Schwarzschild and Kerr innermost stable circular orbits induced by the (conservative) GSF [782, 790, 817, 819, 829, 843, 1009], (iv) test the weak cosmic censorship conjecture in a scenario where a massive particle subject to the GSF falls into a nonrotating BH along unbound orbits [876, 877], (v) calibrate the effective potentials that enter the EOB model for circular [829, 1023] and mildly
eccentric orbits [833,835, 839], and spin-orbit couplings for spinning binaries [832], and (vi) define the analogue of the redshift of a particle for BHs in NR simulations, thus allowing further comparisons to PN and GSF calculations [845, 846].