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.

Post-Newtonian templates for binary black-hole inspirals: the effect of the horizon fluxes and the secular change in the black-hole masses and spins

“…imply that the BH masses m i and spins S i in E and F ∞ are no longer secularly constants during the inspiral phase, but they rather slowly evolve as a function of t at 3.5PN order for m i and 2PN order for S i : we recall that the spin effects to the orbital phase first appear at 1.5PN order [20]. Such a 3.5PN order contribution therefore alters the expressions for E and F ∞ †.…”

“…Given that the gravitational radiation causes the orbits of isolated binary systems to circularize [52,53], we will consider only the PN-inspirals in quasicircular orbits with masses m i (i = 1, 2) and (the magnitude of) spins S i that are (anti-)aligned and normal to the orbital plane, but they have an arbitrary mass ratio. (All throughout, we use geometric units, where G = c = 1, with the useful conversion factor 1M ⊙ = 1.477 km = 4.926 × 10 −6 s.) In this adiabatic setup, the GW phase of the dominant harmonic is twice the orbital phase [20]. The orbital phase φ(t) in terms of the PN barycentric time t can be computed by the center-of-mass binding energy E(t ; m i , S i ) and the energy flux of the gravitational radiation carried out to infinity F ∞ (t ; m i , S i ); the state-of-art of their PN approximations including spin effects are reviewed in [20,54] (see also section 2).…”

“…(All throughout, we use geometric units, where G = c = 1, with the useful conversion factor 1M ⊙ = 1.477 km = 4.926 × 10 −6 s.) In this adiabatic setup, the GW phase of the dominant harmonic is twice the orbital phase [20]. The orbital phase φ(t) in terms of the PN barycentric time t can be computed by the center-of-mass binding energy E(t ; m i , S i ) and the energy flux of the gravitational radiation carried out to infinity F ∞ (t ; m i , S i ); the state-of-art of their PN approximations including spin effects are reviewed in [20,54] (see also section 2). Motivated by the Bondi-Sachs massloss formula [55,56] in full GR, the (orbital-averaged) change rate of E is assumed to be related with F ∞ through the balance equation,…”

“…We shall import many relevant results from the review [20] and references [62,67,109] as well as literatures cited in these references. We then compute the LO PN expressions for the secular change in the BH mass and spin during the inspiral phase, which are at 3.5PN and 2PN order, respectively.…”

“…Following the notation used in [20], we define the projected value of the spin vectors S i along the unit normal ℓ to the orbital plane by S i ≡ S i · ℓ, and introduce two combinations of them:…”