### 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 Equations (97, 98). 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 Equation (97), 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 Equation (224); is made of the quadratic (multipolar) tail
integrals in Equation (98); is the square of the quadrupole tail in Equation (97); and
is the quadrupole tail of tail in Equation (97). 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 (97, 98), in which we make use of the explicit expressions for the source moments of compact
binaries as found in Section 10.1. Without going into accessory details (see Ref. [19]), let us
point out that following the general formalism of Part A, the total mass in front of the
tail integrals is the ADM mass of the binary which is given by the sum of the rest masses,
(which is the one appearing in the -parameter, Equation (188)), plus some
relativistic corrections. At the 2PN relative order needed here to compute the tail integrals we have

Let us give the two basic technical formulas needed when carrying out this reduction:
where and denotes the Euler constant [125]. The tail integrals are evaluated
thanks to these formulas for a fixed (non-decaying) circular orbit. Indeed it can be shown [50] 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
contributions
For the sum of squared tails and cubic tails of tails at 3PN, we get
By comparing Equations (224) and (229) we observe that the constants cleanly cancel out. Adding
together all these contributions we obtain
The gauge constant has not yet disappeared because the post-Newtonian expansion is still
parametrized by instead of the frequency-related parameter defined by Equation (192) - just as for
when it was given by Equation (191). After substituting the expression given by
Equation (193), 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:
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 [205]. In particular, the rational fraction
comes out exactly the same as in black-hole perturbations. On the other hand, the
ambiguity parameters and are part of the rational fraction , belonging to the coefficient
of the term at 3PN order proportional to (hence this coefficient cannot be computed by linear black hole
perturbations).