### 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.