Dimensional regularization of the third post-Newtonian gravitational wave generation from two point masses

Dimensional regularization is applied to the computation of the gravitational wave field generated by compact binaries at the third post-Newtonian (3PN) approximation. We generalize the wave generation formalism from isolated post-Newtonian matter systems to $d$ spatial dimensions, and apply it to point masses (without spins), modeled by delta-function singularities. We find that the quadrupole moment of point-particle binaries in harmonic coordinates contains a pole when $\epsilon \equiv d-3\to 0$ at the 3PN order. It is proved that the pole can be renormalized away by means of the same shifts of the particle world lines as in our recent derivation of the 3PN equations of motion. The resulting renormalized (finite when $\epsilon \to 0$) quadrupole moment leads to unique values for the ambiguity parameters $\xi$, $\kappa$, and $\zeta$, which were introduced in previous computations using Hadamard’s regularization. Several checks of these values are presented. These results complete the derivation of the gravitational waves emitted by inspiralling compact binaries up to the 3.5PN level of accuracy which is needed for detection and analysis of the signals in the gravitational wave antennas LIGO/VIRGO and LISA.

Figure 1

Dangerously divergent diagrams contributing to the 3PN multipole moments. The world lines of particles 1 and 2 are represented by vertical solid lines, the propagator ${\square }^{-1}$ by dotted lines, the source points by bullets, and the $\otimes$ symbol means a multiplication by a multipolar factor, such as ${\widehat{x}}_L$, together with a spatial integration $\int d^d\mathbf{x}\cdots$.