The orbit of most inspiralling compact binaries can be considered to be circular, apart from the gradual inspiral, because the gravitational radiation reaction forces tend to circularize the motion rapidly. For instance, the eccentricity of the orbit of the Hulse-Taylor binary pulsar is presently . At the time when the gravitational waves emitted by the binary system will become visible by the detectors, i.e. when the signal frequency reaches about 10 Hz (in a few hundred million years from now), the eccentricity will be - a value calculated from the Peters [116] law, which is itself based on the quadrupole formula (2).

The main point about modelling the inspiralling compact binary is that a model made of two structureless point particles, characterized solely by two mass parameters and (and possibly two spins), is sufficient. Indeed, most of the non-gravitational effects usually plaguing the dynamics of binary star systems, such as the effects of a magnetic field, of an interstellar medium, and so on, are dominated by gravitational effects. However, the real justification for a model of point particles is that the effects due to the finite size of the compact bodies are small. Consider for instance the influence of the Newtonian quadrupole moments and induced by tidal interaction between two neutron stars. Let and be the radius of the stars, and the distance between the two centers of mass. We have, for tidal moments,

where and are the star’s dimensionless (second) Love numbers [103], which depend on their internal structure, and are, typically, of the order unity. On the other hand, for compact objects, we can introduce their “compactness”, defined by the dimensionless ratios which equal for neutron stars (depending on their equation of state). The quadrupoles and will affect both sides of Eq. (5), i.e. the Newtonian binding energy of the two bodies, and the emitted total gravitational flux as computed using the Newtonian quadrupole formula (4). It is known that for inspiralling compact binaries the neutron stars are not co-rotating because the tidal synchronization time is much larger than the time left till the coalescence. As shown by Kochanek [92] the best models for the fluid motion inside the two neutron stars are the so-called Roche-Riemann ellipsoids, which have tidally locked figures (the quadrupole moments face each other at any instant during the inspiral), but for which the fluid motion has zero circulation in the inertial frame. In the Newtonian approximation we find that within such a model (in the case of two identical neutron stars) the orbital phase, deduced from Eq. (5), reads where is a standard dimensionless post-Newtonian parameter ( is the orbital frequency), and where is the Love number and is the compactness of the neutron star. The first term in the right-hand side of (8) corresponds to the gravitational-wave damping of two point masses; the second term is the finite-size effect, which appears as a relative correction, proportional to , to the latter radiation damping effect. Because the finite-size effect is purely Newtonian, its relative correction should not depend on ; and indeed the factors cancel out in the ratio . However, the compactness of compact objects is by Eq. (7) of the order unity (or, say, one half), therefore the it contains should not be taken into account numerically in this case, and so the real order of magnitude of the relative contribution of the finite-size effect in Eq. (8) is given by alone. This means that for compact objects the finite-size effect should be comparable, numerically, to a post-Newtonian correction of order 5PN or (see Ref. [52] for the proof in the context of relativistic equations of motion). This is a much higher post-Newtonian order than the one at which we shall investigate the gravitational effects on the phasing formula. Using and for neutron stars (and the bandwidth of a VIRGO detector between 10 Hz and 1000 Hz), we find that the cumulative phase error due to the finite-size effect amounts to less that one orbital rotation over a total of produced by the gravitational-wave damping of point masses. The conclusion is that the finite-size effect can in general be neglected in comparison with purely gravitational-wave damping effects. But note that for non-compact or moderately compact objects (such as white dwarfs for instance) the Newtonian tidal interaction dominates over the radiation damping.The inspiralling compact binaries are ideally suited for application of a high-order post-Newtonian wave generation formalism. The main reason is that these systems are very relativistic, with orbital velocities as high as 0.3 in the last rotations (as compared to for the binary pulsar), and it is not surprising that the quadrupole-moment formalism (2, 3, 4, 5) constitutes a poor description of the emitted gravitational waves, since many post-Newtonian corrections play a substantial role. This expectation has been confirmed in recent years by several measurement-analyses [48, 49, 72, 50, 135, 121, 122, 96, 59], which have demonstrated that the post-Newtonian precision needed to implement successively the optimal filtering technique in the LIGO/VIRGO detectors corresponds grossly, in the case of neutron-star binaries, to the 3PN approximation, or beyond the quadrupole moment approximation. Such a high precision is necessary because of the large number of orbital rotations that will be monitored in the detector’s frequency bandwidth ( in the case of neutron stars), giving the possibility of measuring very accurately the orbital phase of the binary. Thus, the 3PN order is required mostly to compute the time evolution of the orbital phase, which depends, via the energy equation (5), on the center-of-mass binding energy and the total gravitational-wave energy flux .

In summary, the theoretical problem posed by inspiralling compact binaries is two-fold: On the one hand , and on the other hand , are to be deduced from general relativity with the 3PN precision or better. To obtain we must control the 3PN equations of motion of the binary in the case of general, not necessarily circular, orbits. As for it necessitates the application of a 3PN wave generation formalism (actually, things are more complicated because the equations of motion are also needed during the computation of the flux). It is quite interesting that such a high order approximation as the 3PN one should be needed in preparation for LIGO and VIRGO data analysis. As we shall see, the signal from compact binaries contains at the 3PN order the signature of several non-linear effects which are specific to general relativity. Therefore, we have here the possibility of probing, experimentally, some aspects of the non-linear structure of Einstein’s theory [28, 29].

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