The mass quadrupole moment of compact binary systems at the fourth post-Newtonian order

Abstract

The mass-type quadrupole moment of inspiralling compact binaries (without spins) is computed at the fourth post-Newtonian (4PN) approximation of general relativity. The multipole moments are defined by matching between the field in the exterior zone of the matter system and the PN field in the near zone, following the multipolar-post-Minkowskian (MPM)-PN formalism. The matching implies a specific regularization for handling infra-red (IR) divergences of the multipole moments at infinity, based on the Hadamard finite part procedure. On the other hand, the calculation entails ultra-violet (UV) divergences due to the modelling of compact objects by delta-functions, that are treated with dimensional regularization (DR). In future work we intend to systematically study the IR divergences by means of dimensional regularization as well. Our result constitutes an important step in the goal of obtaining the gravitational wave templates of inspiralling compact binary systems with 4PN/4.5PN accuracy.

The fifth-order post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach: Potential contributions

“…To 5PN their structure was derived by classical methods in [34,45], see also [15], and using EFT methods in [16,28,46]. Both methods lead to the same results for the electric quadrupole moment, EQ ij Q ji , [37], the octupole moment, O ijk O ijk , and the magnetic quadrupole moment, EJ ij J ji , which develop logarithmic and pole contributions in the EFT-approach. Their normalization coefficients are the same as for their imaginary part, contributing to dE/dt, [34], Eq.…”

“…[16]. 3 There we also described how the singularities in the potential and tail terms [7,15,19,28,[34][35][36][37] are canceling, together with an additional canonical transformation. The whole 5PN calculation is performed starting in the Lagrange formalism and finally deriving the Hamiltonian.…”

“…terms contribute at O(ν) and O(ν 2 ). They are due to the 1PN correction to the electric quadrupole moment, written symbolically, EQ ij Q ji , with E the energy, [37], the octupole moment, EO ijk O ijk , the magnetic quadrupole moment, EJ ij J ji , the angular momentum failed tail, L k ε ijk Q il Q jl , with L the angular momentum, and the memory terms, Q ij Q jk Q ki , which have also been considered in Ref. [28].…”

Gravitational radiation from inspiralling compact objects: Spin-spin effects completed at the next-to-leading post-Newtonian order

“…In particular, the effective field theory (EFT) approach introduced in [56], and later extended in [57] to incorporate rotational degrees of freedom, has been instrumental to reach the present state of the art. 1 However, in the radiation sector, while the source multipoles needed to obtain the GW fluxes in an adiabatic expansion have been computed in some cases to fourth order in the PN expansion for nonspinning bodies [89,90], the spin-dependent counterparts are known to next-to-next-to-leading order (N 2 LO) at linear order in spin [91][92][93], and only to NLO for bilinear in spins contributions [93][94][95]. 2 This begs for more accurate computations of radiative observables in the case of spinning bodies.…”

Fourth post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach

“…[9,11] before. In the case of the tail terms one first applies the multi-pole expansion valid for the far zone [3,13,15,25,27,[31][32][33][34][35][36][37][38][39] to the respective post-Newtonian order and then applies EFT methods to calculate their contribution, cf. [40].…”