Abstract
The effective one-body (EOB) formalism is an analytical approach which aims at providing an accurate description of the motion and radiation of coalescing binary black holes with arbitrary mass ratio. We review the basic elements of this formalism and discuss its aptitude at providing accurate template waveforms to be used for gravitational wave (GW) data analysis purposes.
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- 1.
1Here we use the adjective “analytical” for methods that solve explicit (analytically given) ordinary differential equations (ODE), even if one uses standard (Runge–Kutta-type) numerical tools to solve them. The important point is that, contrary to 3D numerical relativity (NR) simulations, numerically solving ODEs is extremely fast, and can therefore be done (possibly even in real time) for a dense sample of theoretical parameters, such as orbital (\(\nu = {m}_{1}\,{m}_{2}/M,\ldots \)) or spin (\(\hat{{a}}_{1} = {S}_{1}/{\mathit{Gm}}_{1}^{2},{\theta }_{1},{\varphi }_{1},\ldots \)) parameters.
- 2.
Beware that the fonts used in this chapter make the greek letter ν (indicating the symmetric mass ratio) look very similar to the latin letter v≠ν indicating the velocity.
- 3.
As usual “n-PN accuracy” means that a result has been derived up to (and including) terms which are \(\sim {(v/c)}^{2n}\sim {(\mathit{GM}/{c}^{2}r)}^{n}\) fractionally smaller than the leading contribution.
- 4.
We henceforth denote by \(\mathcal{F}\) the Hamiltonian version of the radiation reaction term A RR, Eq. 3, in the (PN-expanded) equations of motion. It can be heuristically computed up to (absolute) 5.5PN [19, 20, 27] and even 6PN [24] order by assuming that the energy radiated in GW at infinity is balanced by a loss of the dynamical energy of the binary system.
- 5.
This is related to an idea emphasized many times by John Archibald Wheeler: quantum mechanics can often help us in going to the essence of classical mechanics.
- 6.
We consider, for simplicity, “equatorial” motions with m = ℓ, that is, classically, \(\theta = \frac{\pi } {2}\).
- 7.
It is convenient to write the “external metric” in Schwarzschild-like coordinates. Note that the external radial coordinate R differs from the two-body ADM-coordinate relative distance \({R}^{\mathrm{ADM}} =\vert \mathbf{q}\vert\). The transformation between the two coordinate systems has been determined in Refs. [35, 55].
- 8.
Indeed \({E}_{\mathrm{real}}^{\mathrm{total}} = M{c}^{2} + {E}_{\mathrm{real}}^{\mathrm{relative}} = M{c}^{2} + \mbox{ Newtonian terms} + \mathrm{1PN}/{c}^{2} + \cdots \), while \({\mathcal{E}}_{\mathrm{effective}} = \mu {c}^{2} + N + \mathrm{1PN}/{c}^{2} + \cdots \).
- 9.
The PN-expanded EOB building blocks A(R), B(R), … already represent a resummation of the PN dynamics in the sense that they have “condensed” the many terms of the original PN-expanded Hamiltonian within a very concise format. But one should not refrain to further resum the EOB building blocks themselves, if this is physically motivated.
- 10.
We recall that the coefficient n 1 and (d 1, d 2, d 3) of the Padé approximant are determined by the condition that the first four terms of the Taylor expansion of A 3 1 in powers of \(u = \mathit{GM}/({c}^{2}R)\) coincide with A 3PN.
- 11.
Completed by the \(\mathcal{O}({\mathbf{p}}^{4})\) terms that must be introduced at 3PN.
- 12.
Beware that this “Effective EOB-light-ring” occurs for a circular-orbit radius slightly larger than the purely dynamical (circular) EOB-light-ring (where H eff and \(\mathcal{J}\) would formally become infinite).
- 13.
We recall that Ref. [52] has also shown that the agreement improves even more when the Taylor expansion of the function ρ22 is further suitably Padé resummed.
- 14.
- 15.
- 16.
The terminology “restricted” refers to a waveform which uses only the leading Newtonian approximation, h ℓm (N, ε), to the waveform.
- 17.
Alternatively, one can start by giving oneself ω1, ω2 and determine the NR instants t 1 NR, t 2 NR at which they are reached.
- 18.
The two frequencies used for this comparison, by means of the “two-frequency pinching technique” mentioned above, are Mω1 = 0. 047 and \({M}_{{\omega }_{2}} = 0.31\).
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Acknowledgements
AN is grateful to Alessandro Spallicci, Bernard Whiting, and all the organizers of the “Ecole thématique du CNRS sur la masse (origine, mouvement, mesure).” Among the many colleagues whom we benefitted from, we would like to thank particularly Emanuele Berti, Bernd Brügmann, Alessandra Buonanno, Nils Dorband, Mark Hannam, Sascha Husa, Bala Iyer, Larry Kidder, Eric Poisson, Denis Pollney, Luciano Rezzolla, B.S. Sathyaprakash, Angelo Tartaglia, and Loic Villain, for fruitful collaborations and discussions. We are also grateful to Marie–Claude Vergne for help with Fig. 1.
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Damour, T., Nagar, A. (2009). The Effective One-Body Description of the Two-Body Problem. In: Blanchet, L., Spallicci, A., Whiting, B. (eds) Mass and Motion in General Relativity. Fundamental Theories of Physics, vol 162. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3015-3_7
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