# Towards 3.5PN accurate polarizations for compact binaries

Guillaume Faye is Chargé de Recherche (researcher)

The gravitational-wave observatory Advanced LIGO is now in its commissioning stage and preparing for its first scientific runs in early 2015. It will be soon followed by the Advanced Virgo detector. Being reasonably optimistic, one can expect the first detections to occur by 2018. The most likely sources to be observed are coalescences of two compact objects. As both detection and
parameter estimation rely on matched filtering techniques, several programs to compute accurate waveform models using various approximation techniques are being pursued. Notably, the post-Newtonian perturbative approach, where all quantities of interest are expanded in powers of $(v/c)$, with $v$ being the relative velocity and $c$ the speed of light, is best suited to describe the inspiralling orbits of the binary companions before the merger.

In this context, we have been working recently in improving the knowledge of the gravitational-wave polarization amplitudes. This should enhance the angular resolution and the distance measurement for massive enough systems. With our recent paper, we took a significant step towards the computation of the waveform at the third-and-a-half post-Newtonian (PN) order, which means including corrections of order $(v/c)^7$ beyond the Einstein quadrupole formula.

Bala Iyer is currently Visiting Professor at ICTS-TIFR, Bangalore

The observable signal generated by an arbitrary source can be parameterized by
two infinite sets of so-called “radiative” multipole moments, either of mass or of current type. On the other hand, the gravitational field outside the matter source may be parameterized by six sets of “source” moments. As a first task, we obtained the explicit expression of the radiative moments in
terms of the source moments at order 3.5PN. Notably, we computed the lengthy contribution of the instantaneous quadratic interactions, which are purely local in time. They add to the hereditary contributions, which, by contrast,
depend on the past history of the system through a backward time integration.
The hereditary contributions comprise: tail effects, produced by the interaction of one monopole with one non-static source moment; quadratic memory effects, arising from the interaction between two non-static moment; tail-of-tail effects, which come from the cubic interaction of two monopoles and one non-static moment. The latter piece is new with our work at this level of accuracy. It depends on two cutoff constants, one of which is related to the choice of date origin in the coordinate grid we use to describe the waveform near future radiative infinity, whereas the other is associated to the appearance of ultraviolet logarithms.

Luc Blanchet is a Directeur de Recherche (senior researcher) at the Institut d’Astrophysique de Paris (IAP)

As a second task, we calculated the source mass octupole for a binary system of compact objects at order 3PN, starting from its known expression in the form of some integral over an unspecified effective source. In our approach,
motivated by the effacement principle in general relativity, the bodies are represented as point particles and the self-field divergencies are taken care
of by dimensional regularization. The third time derivative of the octupole in the center-of-mass frame entering the radiative octupole depends on the same ultraviolet cutoff, as do the tail-of-tail terms that arise there, but those occurrences compensate, so that the physical observable does not contain any nonphysical parameter.

With the radiative mass octupole in hands, we were in position to derive the gravitational-wave modes (3,3) and (3,1) for binaries of non-spinning objects in a spherical-harmonic-type decomposition of the polarizations at order 3.5PN. The gravitational-wave modes are of direct use both by the numerical relativity and data analysis communities. In addition we have recast our result into a form that is appropriate for effective-one-body approaches to model semi-analytically the gravitational-wave signal. This will improve by half a post-Newtonian order the accuracy that is currently reached for the mass octupolar contribution to the signal amplitude.

Non-linear multipole interactions and gravitational-wave octupole modes for inspiralling compact binaries to third-and-a-half post-Newtonian order
G Faye, Luc Blanchet and Bala Iyer
Guillaume Faye et al 2015 Class. Quantum Grav. 32 045016