The previous definition of the ICO is motivated by our comparison with the results of numerical relativity. Indeed we shall confront the prediction of the standard (Taylor-based) post-Newtonian approach with a recent result of numerical relativity by Gourgoulhon, Grandclément, and Bonazzola [123, 126]. These authors computed numerically the energy of binary black holes under the assumptions of conformal flatness for the spatial metric and of exactly circular orbits. The latter restriction is implemented by requiring the existence of an “helical” Killing vector, which is time-like inside the light cylinder associated with the circular motion, and space-like outside. In the numerical approach [123, 126] there are no gravitational waves, the field is periodic in time, and the gravitational potentials tend to zero at spatial infinity within a restricted model equivalent to solving five out of the ten Einstein field equations (the so-called Isenberg-Wilson-Mathews approximation; see Ref.  for a discussion). Considering an evolutionary sequence of equilibrium configurations Refs. [123, 126] obtained numerically the circular-orbit energy and looked for the ICO of binary black holes (see also Refs. [52, 124, 154] for related calculations of binary neutron and strange quark stars).
Since the numerical calculation [123, 126] has been performed in the case of corotating black holes, which are spinning with the orbital angular velocity , we must for the comparison include within our post-Newtonian formalism the effects of spins appropriate to two Kerr black holes rotating at the orbital rate. The total relativistic mass of the Kerr black hole is given by34[123, 126]. In the limit of slow rotation we get
To take into account the spin effects our first task is to replace all the masses entering the energy function (194) by their equivalent expressions in terms of and the two irreducible masses. It is clear that the leading contribution is that of the spin kinetic energy given by Equation (199), and it comes from the replacement of the rest mass-energy (where ). From Equation (199) this effect is of order in the case of corotating binaries, which means by comparison with Equation (194) that it is equivalent to an “orbital” effect at the 2PN order (i.e. ). Higher-order corrections in Equation (199), which behave at least like , will correspond to the orbital 5PN order at least and are negligible for the present purpose. In addition there will be a subdominant contribution, of the order of equivalent to 3PN order, which comes from the replacement of the masses into the “Newtonian” part, proportional to , of the energy (see Equation (194)). With the 3PN accuracy we do not need to replace the masses that enter into the post-Newtonian corrections in , so in these terms the masses can be considered to be the irreducible ones.
Our second task is to include the specific relativistic effects due to the spins, namely the spin-orbit (SO) interaction and the spin-spin (SS) one. In the case of spins and aligned parallel to the orbital angular momentum (and right-handed with respect to the sense of motion) the SO energy reads[146, 144] (based on seminal works of Barker and O’Connell [7, 8]) who have computed the SO contribution and expressed it by means of the orbital frequency . The derivation of Equation (200) in Ref. [146, 144] takes into account the fact that the relation between the orbital separation (in the harmonic coordinate system) and the frequency depends on the spins. We immediately infer from Equation (200) that in the case of corotating black holes the SO effect is equivalent to a 3PN orbital effect and thus must be retained with the present accuracy (with this approximation, the masses in Equation (200) are the irreducible ones). As for the SS interaction (still in the case of spins aligned with the orbital angular momentum) it is given by 
In conclusion, we find that the location of the ICO as computed by numerical relativity, under the helical-symmetry and conformal-flatness approximations, is in good agreement with the post-Newtonian prediction. See also Ref.  for the results calculated within the effective-one-body approach method [60, 61] at the 3PN order, which are close to the ones reported in Figure 1. This agreement constitutes an appreciable improvement of the previous situation, because the earlier estimates of the ICO in post-Newtonian theory  and numerical relativity [180, 9] strongly disagreed with each other, and do not match with the present 3PN results. The numerical calculation of quasi-equilibrium configurations has been since then redone and refined by a number of groups, for both corotational and irrotational binaries (see in particular Ref. ). These works confirm the previous findings.
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