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

Luc Blanchet, Thibault Damour, Gilles Esposito-Farese, Bala R. Iyer

TL;DR

This work applies dimensional regularization to the gravitational-wave generation problem for two point masses at 3PN/3.5PN order, revealing a pole at $\\varepsilon=d-3$ and renormalizing it through shifts of the world-lines to yield finite, unambiguous results. By extending the multipolar post-Minkowskian formalism to $d$ dimensions and deriving the $d$-dimensional source terms, the authors determine the mass-type STF moments $I_L$ and fix the Hadamard ambiguities with unique values: $\xi=-9871/9240$, $\kappa=0$, and $\zeta=-7/33$, ensuring a consistent 3.5PN waveform. They validate DR through multiple checks, including the mass-dipole completeness with the center-of-mass, the boosted-point-particle analysis, and a diagrammatic renormalization picture that links poles to world-line shifts. The results support the effacement of internal structure at 3PN/3.5PN for non-spinning binaries and provide uniquely defined waveform templates essential for LIGO/VIRGO and LISA data analysis, while also clarifying debates about finite-size effects in the effective action.

Abstract

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), modelled by delta-function singularities. We find that the quadrupole moment of point-particle binaries in harmonic coordinates contains a pole when epsilon = d-3 -> 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 -> 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.

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

TL;DR

This work applies dimensional regularization to the gravitational-wave generation problem for two point masses at 3PN/3.5PN order, revealing a pole at and renormalizing it through shifts of the world-lines to yield finite, unambiguous results. By extending the multipolar post-Minkowskian formalism to dimensions and deriving the -dimensional source terms, the authors determine the mass-type STF moments and fix the Hadamard ambiguities with unique values: , , and , ensuring a consistent 3.5PN waveform. They validate DR through multiple checks, including the mass-dipole completeness with the center-of-mass, the boosted-point-particle analysis, and a diagrammatic renormalization picture that links poles to world-line shifts. The results support the effacement of internal structure at 3PN/3.5PN for non-spinning binaries and provide uniquely defined waveform templates essential for LIGO/VIRGO and LISA data analysis, while also clarifying debates about finite-size effects in the effective action.

Abstract

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), modelled by delta-function singularities. We find that the quadrupole moment of point-particle binaries in harmonic coordinates contains a pole when epsilon = d-3 -> 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 -> 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.

Paper Structure

This paper contains 18 sections, 183 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Multipole expansion of a scalar field in $d$ dimensions
  3. Multipole decomposition of the gravitational field
  4. $d$-dimensional generalization of the multipolar post-Minkowskian formalism
  5. Multipolar decomposition of the gravitational field in $d$ dimensions
  6. Symmetric-trace-free source multipole moments in $d$ dimensions
  7. Source terms for the 3PN moments in $d$ dimensions
  8. Difference between dimensional and pHS regularizations
  9. Difference for $d$-dimensional spatial integrals
  10. Proof that the outer-near-zone divergencies do not contribute to the difference
  11. Computation of the ambiguity parameters
  12. The 3PN mass-quadrupole moment
  13. The 3PN mass-dipole moment
  14. Renormalization and diagrammatic approach
  15. Diagrammatic interpretation of the poles in DR
  16. ...and 3 more sections

Figures (1)

  • 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 $\Box^{-1}$ 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 $\int d^d \mathbf{x} \cdots$.