10.2 Contribution of wave tails
To the “instantaneous” part of the flux, we must add the contribution of non-linear multipole
interactions contained in the relationship between the source and radiative moments. The needed material
has already been provided in Eqs. (91, 92). Up to the 3.5PN level we have the dominant quadratic-order
tails, the cubic-order tails or tails of tails, and the non-linear memory integral. We shall see that the tails
play a crucial role in the predicted signal of compact binaries. By contrast, the non-linear memory effect,
given by the integral inside the 2.5PN term in Eq. (91), does not contribute to the gravitational-wave
energy flux before the 4PN order in the case of circular-orbit binaries (essentially because the memory
integral is actually “instantaneous” in the flux), and therefore has rather poor observational consequences
for future detections of inspiralling compact binaries. We split the energy flux into the different terms
where
has just been found in Eq. (162);
is made of the quadratic (multipolar) tail integrals in
Eq. (92);
is the square of the quadrupole tail in Eq. (91); and
is the quadrupole tail of
tail in Eq. (91). We find that
contributes at the half-integer 1.5PN, 2.5PN and 3.5PN orders, while
both
and
appear only at the 3PN order. It is quite remarkable that so small an effect as
a “tail of tail” should be relevant to the present computation, which is aimed at preparing the ground for
forthcoming experiments.
The results follow from the reduction to the case of circular compact binaries of the general
formulas (91, 92). Without going into accessory details (see Ref. [10]), let us give the two basic technical
formulas needed when carrying out this reduction:
where
and
denotes the Euler constant [78]. The tail integrals are evaluated thanks
to these formulas for a fixed (non-decaying) circular orbit. Indeed it can be shown [31] that the
“remote-past” contribution to the tail integrals is negligible; the errors due to the fact that the orbit
actually spirals in by gravitational radiation do not affect the signal before the 4PN order.
We then find, for the quadratic tail term stricto sensu, the 1.5PN, 2.5PN and 3.5PN amounts
Update
For the sum of squared tails and cubic tails of tails at 3PN, we get
By comparing Eqs. (162) and (166) we observe that the constants
cleanly cancel out.
Adding together all these contributions we obtain Update
The gauge constant
has not yet disappeared because the post-Newtonian expansion is still
parametrized by
instead of the frequency-related parameter
defined by Eq. (150) - just as for
when it was given by Eq. (149). After substituting the expression
given by Eq. (151), we
find that
does cancel as well. Because the relation
is issued from the equations
of motion, the latter cancellation represents an interesting test of the consistency of the two
computations, in harmonic coordinates, of the 3PN multipole moments and the 3PN equations
of motion. At long last we obtain our end result: Update
In the test-mass limit
for one of the bodies, we recover exactly the result following from linear
black-hole perturbations obtained by Tagoshi and Sasaki [137]. In particular, the rational fraction
comes out exactly the same as in black-hole perturbations.