Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries

Abstract

To be observed and analyzed by the network of gravitational wave detectors on ground (LIGO, VIRGO, etc.) and by the future detectors in space (eLISA, etc.), inspiralling compact binaries - binary star systems composed of neutron stars and/or black holes in their late stage of evolution - require high-accuracy templates predicted by general relativity theory. The gravitational waves emitted by these very relativistic systems can be accurately modelled using a high-order post-Newtonian gravitational wave generation formalism. In this article, we present the current state of the art on post-Newtonian methods as applied to the dynamics and gravitational radiation of general matter sources (including the radiation reaction back onto the source) and inspiralling compact binaries. We describe the post-Newtonian equations of motion of compact binaries and the associated Lagrangian and Hamiltonian formalisms, paying attention to the self-field regularizations at work in the calculations. Several notions of innermost circular orbits are discussed. We estimate the accuracy of the post-Newtonian approximation and make a comparison with numerical computations of the gravitational self-force for compact binaries in the small mass ratio limit. The gravitational waveform and energy flux are obtained to high post-Newtonian order and the binary's orbital phase evolution is deduced from an energy balance argument. Some landmark results are given in the case of eccentric compact binaries - moving on quasi-elliptical orbits with non-negligible eccentricity. The spins of the two black holes play an important role in the definition of the gravitational wave templates. We investigate their imprint on the equations of motion and gravitational wave phasing up to high post-Newtonian order (restricting to spin-orbit effects which are linear in spins), and analyze the post-Newtonian spin precession equations as well as the induced precession of the orbital plane.

Keywords: Gravitational radiation; Inspiralling compact binary; Multipolar expansion; Post-Newtonian approximations.

Figure 1

The binding energy E_ICO versus Ω_ICO in the equal-mass case (ν = 1/4). Left panel: Comparison with the numerical relativity result of Gourgoulhon, Grandclément et al. [228, 232] valid in the corotating case (marked by a star). Points indicated by nPN are computed from the minimum of Eq. (232), and correspond to irrotational binaries. Points denoted by nPN^corot come from the minimum of the sum of Eqs. (232) and (250), and describe corotational binaries. Note the very good convergence of the standard (Taylor-expanded) PN series. Right panel: Numerical relativity results of Cook, Pfeiffer et al. [133, 121] for quasi-equilibrium (QE) configurations and various boundary conditions for the lapse function, in the non-spinning (NS), leading-order non spinning (LN) and corotating (CO) cases. The point from [228, 232] (HKV-GGB) is also reported as in the left panel, together with IVP, the initial value approach with effective potential [132, 342], as well as standard PN predictions from the left panel and non-standard (EOB) ones. The agreement between the QE computation and the standard non-resummed 3PN point is excellent especially in the irrotational NS case.

Figure 2

Different analytical approximation schemes and numerical techniques to study black hole binaries, depending on the mass ratio q = m₁/m₂ and the post-Newtonian parameter ε² ∼ v²/c² ∼ Gm/(c²r₁₂). Post-Newtonian theory and perturbative self-force analysis can be compared in the post-Newtonian regime (ε ≪ 1 thus r₁₂ ≫ Gm/c²) of an extreme mass ratio (m₁ ≪ m₂) binary.

Figure 3

Variation of the enhancement factor φ(e) with the eccentricity e. This function agrees with the numerical calculation of Ref. [87] modulo a trivial rescaling with the Peters-Mathews function (356a). The inset graph is a zoom of the function at a smaller scale. The dots represent the numerical computation and the solid line is a fit to the numerical points. In the circular orbit limit we have φ(0) = 1.

Figure 4

Geometric definitions for the precessional motion of spinning compact binaries [54, 306]. We show (i) the source frame defined by the fixed orthonormal basis {x, y, z}; (ii) the instantaneous orbital plane which is described by the orthonormal basis {x_ℓ, y_ℓ, ℓ}; (iii) the moving triad {n, λ, ℓ} and the associated three Euler angles α, ι and Φ; (v) the direction of the total angular momentum J which coincides with the z-direction. Dashed lines show projections into the x-y plane.