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
(in a few hundred million years from now), the
eccentricity will be
- a value calculated from the Peters [177] 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,
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 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 [77
, 78
, 111
, 79
, 203
, 183
, 184
, 152
, 92
],
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 [47
, 48
].
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