The third post-Newtonian gravitational wave polarisations and associated spherical harmonic modes for inspiralling compact binaries in quasi-circular orbits (0802.1249v3)

Abstract: The gravitational waveform (GWF) generated by inspiralling compact binaries moving in quasi-circular orbits is computed at the third post-Newtonian (3PN) approximation to general relativity. Our motivation is two-fold: (i) To provide accurate templates for the data analysis of gravitational wave inspiral signals in laser interferometric detectors; (ii) To provide the associated spin-weighted spherical harmonic decomposition to facilitate comparison and match of the high post-Newtonian prediction for the inspiral waveform to the numerically-generated waveforms for the merger and ringdown. This extension of the GWF by half a PN order (with respect to previous work at 2.5PN order) is based on the algorithm of the multipolar post-Minkowskian formalism, and mandates the computation of the relations between the radiative, canonical and source multipole moments for general sources at 3PN order. We also obtain the 3PN extension of the source multipole moments in the case of compact binaries, and compute the contributions of hereditary terms (tails, tails-of-tails and memory integrals) up to 3PN order. The end results are given for both the complete plus and cross polarizations and the separate spin-weighted spherical harmonic modes.

Overview of 3PN Gravitational Wave Polarizations for Inspiralling Compact Binaries

The paper, authored by Blanchet et al., presents a thorough investigation into the gravitational waveforms emitted by inspiralling compact binaries in quasi-circular orbits at the third post-Newtonian (3PN) approximation within the framework of general relativity. This research has vital implications for enhancing gravitational wave signal analysis, particularly in the context of laser interferometric detectors like LIGO and Virgo, and in connecting analytical PN predictions with numerical relativity simulations focusing on binaries' merger and ringdown phases.

Theoretical Framework and Methodology

The paper employs the multipolar post-Minkowskian (MPM) formalism to achieve a 3PN level of precision. This involves extending the technique to compute the relations between radiative, canonical, and source multipole moments at 3PN order. In doing so, the researchers derive the full waveform — encompassing all higher-order amplitude corrections and harmonics related to the orbital frequency. The emphasis on spherical harmonic modes, including their spin-weighted version, offers a refined means for matching PN predictions to numerically simulated waveforms.

Core Numerical and Analytical Results

1. Radiative Moments: At 3PN order, the research details the radiative mass quadrupole moment $U_{ij}$, including all relevant subdominant contributions like tails and non-linear memory terms.
2. Canonical to Source Moments: The paper outlines corrections due to gauge terms affecting the canonical mass quadrupole, mass octupole, and current quadrupole, revealing interactions initiated at 2.5PN order.
3. Hereditary Contributions: The work extends upon prior understanding of hereditary effects such as tails and memory terms, analyzing their influence on the waveform and detailing the requisite integrations over their past evolution.
4. Polarization Waveforms: They provide explicit formulary expressions for the polarization states $h_+$ and $h_\times$, incorporating 3PN corrections in terms of the invariant PN parameter $x$. This precision enables a comparison with lower-order results and testing these findings against predictions from black hole perturbation theory.

Implications and Future Prospects

The formulations allow for optimizing gravitational wave templates used in data analysis, potentially improving detection accuracy and parameter estimation. The 3PN waveforms reduce uncertainties when comparing analytical templates to numerical relativity models, thus strengthening the fidelity of gravitational wave data interpretation.

Speculations on Future Developments

Given the theoretical depth achieved in this paper, future research could focus on extending these techniques to binary systems with spin or eccentric orbital configurations. Furthermore, adapting these forms for implementation in upcoming space-based detectors like LISA, which operate in varied frequency bands and are sensitive to different astrophysical sources, remains a pertinent objective. This could usher in advancements in the paper of supermassive black-hole coalescences, rendering them prime candidates for probing cosmological parameters and dark energy.

In conclusion, the research sets a benchmark for high-PN-order waveform generation and marks significant progress in corroborating analytical and numerical relativity frameworks. The implications of this work reach beyond binary black-hole systems, influencing broader applications in astrophysics and cosmology.