When retaining the “even” relativistic corrections at the 1PN, 2PN and 3PN orders, and neglecting the “odd” radiation reaction term at the 2.5PN order, we find that the equations of motion admit a conserved energy (and a Lagrangian, as we shall see), and that energy can be straightforwardly obtained by guesswork starting from Eq. (
163 ), with the result
$$\begin{array}{ccc}E=...& & \end{array}$$ 
(165)

To the terms given above, we must add the terms corresponding to the relabelling
$1\leftrightarrow 2$
. Actually, this energy is not conserved because of the radiation reaction. Thus its time derivative, as computed by means of the 3PN equations of motion themselves (i.e., orderreducing all the accelerations), is purely equal to the 2.5PN effect,
$$\begin{array}{ccc}\frac{dE}{dt}& =& \frac{4}{5}\frac{{G}^{2}{m}_{1}^{2}{m}_{2}}{{c}^{5}{r}_{12}^{3}}[\left({v}_{1}{v}_{12}\right)\left({v}_{12}^{2}+2\frac{G{m}_{1}}{{r}_{12}}8\frac{G{m}_{2}}{{r}_{12}}\right)\end{array}$$  
$$\begin{array}{ccc}& & +\left({n}_{12}{v}_{1}\right)\left({n}_{12}{v}_{12}\right)\left(3{v}_{12}^{2}6\frac{G{m}_{1}}{{r}_{12}}+\frac{52}{3}\frac{G{m}_{2}}{{r}_{12}}\right)]\end{array}$$  
$$\begin{array}{ccc}& & +1\leftrightarrow 2+\mathcal{O}\left(\frac{1}{{c}^{7}}\right).\end{array}$$ 
(166)

The resulting “balance equation” can be better expressed by transfering to the lefthand side certain 2.5PN terms so that the righthand side takes the familiar form of a total energy flux. Posing
$$\begin{array}{c}\stackrel{~}{E}=E+\frac{4{G}^{2}{m}_{1}^{2}{m}_{2}}{5{c}^{5}{r}_{12}^{2}}\left({n}_{12}{v}_{1}\right)\left[{v}_{12}^{2}\frac{2G({m}_{1}{m}_{2})}{{r}_{12}}\right]+1\leftrightarrow 2,\end{array}$$ 
(167)

we find agreement with the standard Einstein quadrupole formula ( 4 , 5 ):
$$\begin{array}{c}\frac{d\stackrel{~}{E}}{dt}=\frac{G}{5{c}^{5}}\frac{{d}^{3}{Q}_{ij}}{d{t}^{3}}\frac{{d}^{3}{Q}_{ij}}{d{t}^{3}}+\mathcal{O}\left(\frac{1}{{c}^{7}}\right),\end{array}$$ 
(168)

where the Newtonian tracefree quadrupole moment is
${Q}_{ij}={m}_{1}({y}_{1}^{i}{y}_{1}^{j}\frac{1}{3}{\delta}^{ij}{\mathbf{y}}_{1}^{2})+1\leftrightarrow 2$
. As we can see, the 3PN equations of motion ( 163 ) are highly relativistic when describing the motion, but concerning the radiation they are in fact Newtonian, because they contain merely the “Newtonian” radiation reaction force at the 2.5PN order.
$$\frac{1}{\pi}\int \frac{{d}^{3}\mathbf{x}}{{r}_{1}^{2}{r}_{2}^{2}}=\frac{{\pi}^{2}}{{r}_{12}}.$$
9.2 Lagrangian and Hamiltonian formulations
The conservative part of the equations of motion in harmonic coordinates ( 163 ) is derivable from a generalized Lagrangian, depending not only on the positions and velocities of the bodies, but also on their accelerations:
${a}_{1}^{i}=d{v}_{1}^{i}/dt$
and
${a}_{2}^{i}=d{v}_{2}^{i}/dt$
. As shown by Damour and Deruelle [
55]
, the accelerations in the harmoniccoordinates Lagrangian occur already from the 2PN order. This fact is in accordance with a general result of Martin and Sanz [
100]
that
$N$
body equations of motion cannot be derived from an ordinary Lagrangian beyond the 1PN level, provided that the gauge conditions preserve the Lorentz invariance. Note that we can always arrange for the dependence of the Lagrangian upon the accelerations to be linear, at the price of adding some socalled “multizero” terms to the Lagrangian, which do not modify the equations of motion (see e.g. Ref. [
63]
). At the 3PN level, we find that the Lagrangian also depends on accelerations. It is notable that these accelerations are sufficient – there is no need to include derivatives of accelerations. Note also that the Lagrangian is not unique because we can always add to it a total time derivative
$dF/dt$
, where
$F$
depends on the positions and velocities, without changing the dynamics. We find [
66]
$$\begin{array}{ccc}{L}^{harm}=...& & \end{array}$$ 
(169)

Witness the accelerations occuring at the 2PN and 3PN orders; see also the gaugedependent logarithms of
${r}_{12}/{r}_{1}^{\prime}$
and
${r}_{12}/{r}_{2}^{\prime}$
, and the single term containing the regularization ambiguity
$\lambda $
. We refer to [
66]
for the explicit expressions of the ten conserved quantities corresponding to the integrals of energy (also given in Eq. ( 165 )), linear and angular momenta, and centerofmass position. Notice that while it is strictly forbidden to replace the accelerations by the equations of motion in the Lagrangian, this can and should be done in the final expressions of the conserved integrals derived from that Lagrangian. Now we want to exhibit a transformation of the particles dynamical variables – or contact transformation, as it is called in the jargon – which transforms the 3PN harmoniccoordinates Lagrangian ( 169 ) into a new Lagrangian, valid in some ADM or ADMlike coordinate system, and such that the associated Hamiltonian coincides with the 3PN Hamiltonian that has been obtained by Damour, Jaranowski and Schäfer [
60]
. In ADM coordinates the Lagrangian will be “ordinary”, depending only on the positions and velocities of the bodies. Let this contact transformation be
${Y}_{1}^{i}\left(t\right)={y}_{1}^{i}\left(t\right)+\delta {y}_{1}^{i}\left(t\right)$
and
$1\leftrightarrow 2$
, where
${Y}_{1}^{i}$
and
${y}_{1}^{i}$
denote the trajectories in ADM and harmonic coordinates, respectively. For this transformation to be able to remove all the accelerations in the initial Lagrangian
${L}^{harm}$
up to the 3PN order, we determine [
66]
it to be necessarily of the form
$$\begin{array}{c}\delta {y}_{1}^{i}=\frac{1}{{m}_{1}}\left[\frac{\partial {L}^{harm}}{\partial {a}_{1}^{i}}+\frac{\partial F}{\partial {v}_{1}^{i}}+\frac{1}{{c}^{6}}{X}_{1}^{i}\right]+\mathcal{O}\left(\frac{1}{{c}^{8}}\right)\end{array}$$ 
(170)

(and idem
$1\leftrightarrow 2$
), where
$F$
is a freely adjustable function of the positions and velocities, made of 2PN and 3PN terms, and where
${X}_{1}^{i}$
represents a special correction term, that is purely of order 3PN. The point is that once the function
$F$
is specified there is a unique determination of the correction term
${X}_{1}^{i}$
for the contact transformation to work (see Ref. [
66]
for the details). Thus, the freedom we have is entirely coded into the function
$F$
, and the work then consists in showing that there exists a unique choice of
$F$
for which our Lagrangian
${L}^{harm}$
is physically equivalent, via the contact transformation ( 170 ), to the ADM Hamiltonian of Ref. [
60]
. An interesting point is that not only the transformation must remove all the accelerations in
${L}^{harm}$
, but it should also cancel out all the logarithms
$ln({r}_{12}/{r}_{1}^{\prime})$
and
$ln({r}_{12}/{r}_{2}^{\prime})$
, because there are no logarithms in ADM coordinates. The result we find, which can be checked to be in full agreement with the expression of the gauge vector in Eq. ( 164 ), is that
$F$
involves the logarithmic terms
$$\begin{array}{c}F=\frac{22}{3}\frac{{G}^{3}{m}_{1}{m}_{2}}{{c}^{6}{r}_{12}^{2}}\left[{m}_{1}^{2}\left({n}_{12}{v}_{1}\right)ln\left(\frac{{r}_{12}}{{r}_{1}^{\prime}}\right){m}_{2}^{2}\left({n}_{12}{v}_{2}\right)ln\left(\frac{{r}_{12}}{{r}_{2}^{\prime}}\right)\right]+...,\end{array}$$ 
(171)

together with many other nonlogarithmic terms (indicated by dots) that are entirely specified by the isometry of the harmonic and ADM descriptions of the motion. For this particular choice of
$F$
the ADM Lagrangian reads as
$$\begin{array}{c}{L}^{ADM}={L}^{harm}+\frac{\delta {L}^{harm}}{\delta {y}_{1}^{i}}\delta {y}_{1}^{i}+\frac{\delta {L}^{harm}}{\delta {y}_{2}^{i}}\delta {y}_{2}^{i}+\frac{dF}{dt}+\mathcal{O}\left(\frac{1}{{c}^{8}}\right).\end{array}$$ 
(172)

Inserting into this equation all our explicit expressions we find
$$\begin{array}{ccc}{L}^{ADM}=...& & \end{array}$$ 
(173)

The notation is the same as in Eq. ( 169 ), except that we use uppercase letters to denote the ADMcoordinates positions and velocities; thus, for instance
${\mathbf{N}}_{12}=({\mathbf{Y}}_{1}{\mathbf{Y}}_{2})/{R}_{12}$
and
$\left({N}_{12}{V}_{1}\right)={\mathbf{N}}_{12}.{\mathbf{V}}_{1}$
. The Hamiltonian is simply deduced from the latter Lagrangian by applying the usual Legendre transformation. Posing
${P}_{1}^{i}=\partial {L}^{ADM}/\partial {V}_{1}^{i}$
and
$1\leftrightarrow 2$
, we get [
87,
88,
89,
60,
66]
^{
$\text{21}$
}
$$\begin{array}{ccc}{H}^{ADM}=...& & \end{array}$$ 
(174)

Arguably, the results given by the ADMHamiltonian formalism (for the problem at hand) look simpler than their harmoniccoordinate counterparts. Indeed, the ADM Lagrangian is ordinary – no accelerations – and there are no logarithms nor associated gauge constants
${r}_{1}^{\prime}$
and
${r}_{2}^{\prime}$
. But of course, one is free to describe the binary motion in whatever coordinates one likes, and the two formalisms, harmonic ( 169 ) and ADM ( 173 , 174 ), describe rigorously the same physics. On the other hand, the higher complexity of the harmoniccoordinates Lagrangian ( 169 ) enables one to perform more tests of the computations, notably by inquiring about the future of the constants
${r}_{1}^{\prime}$
and
${r}_{2}^{\prime}$
, that we know must disappear from physical quantities such as the centerofmass energy and the total gravitationalwave flux.
$$\frac{{G}^{2}}{{c}^{6}{r}_{12}^{2}}\left(\frac{55}{12}{m}_{1}\frac{193}{48}{m}_{2}\right)\frac{({N}_{12}{P}_{2}{)}^{2}{P}_{1}^{2}}{{m}_{1}{m}_{2}}+1\leftrightarrow 2.$$
This misprint has been corrected in an Erratum [
60]
.
9.3 Equations of motion for circular orbits
Most inspiralling compact binaries will have been circularized by the time they become visible by the detectors LIGO and VIRGO. In the case of orbits that are circular – apart from the gradual 2.5PN radiationreaction inspiral – the quite complicated acceleration ( 163 ) simplifies drastically, since all the scalar products between
${\mathbf{n}}_{12}$
and the velocities are of small 2.5PN order: E.g.
$\left({n}_{12}{v}_{1}\right)=\mathcal{O}(1/{c}^{5})$
, and the remainder can always be neglected here. Let us translate the origin of coordinates to the binary's centerofmass by imposing that the binary's dipole
${I}_{i}=0$
(notation of Part aaa ). Up to the 2.5PN order, and in the case of circular orbits, this condition implies [
7]
^{
$\text{22}$
}
$$\begin{array}{ccc}m{y}_{1}^{i}& =& {y}_{12}^{i}\left[{m}_{2}+3{\gamma}^{2}\nu \delta m\right]\frac{4}{5}\frac{{G}^{2}{m}^{2}\nu \delta m}{{r}_{12}{c}^{5}}{v}_{12}^{i}+\mathcal{O}\left(\frac{1}{{c}^{6}}\right),\end{array}$$ 
(175)

$$\begin{array}{ccc}m{y}_{2}^{i}& =& {y}_{12}^{i}\left[{m}_{1}+3{\gamma}^{2}\nu \delta m\right]\frac{4}{5}\frac{{G}^{2}{m}^{2}\nu \delta m}{{r}_{12}{c}^{5}}{v}_{12}^{i}+\mathcal{O}\left(\frac{1}{{c}^{6}}\right).\end{array}$$ 
(176)

Mass parameters are the total mass
$m={m}_{1}+{m}_{2}$
(
$m\equiv M$
in the notation of Part aaa ), the mass difference
$\delta m={m}_{1}{m}_{2}$
, the reduced mass
$\mu ={m}_{1}{m}_{2}/m$
, and the very useful symmetric mass ratio
$$\begin{array}{c}\nu \equiv \frac{\mu}{m}\equiv \frac{{m}_{1}{m}_{2}}{({m}_{1}+{m}_{2}{)}^{2}}.\end{array}$$ 
(177)

The usefulness of this ratio lies in its interesting range of variation:
$0<\nu \le 1/4$
, with
$\nu =1/4$
in the case of equal masses, and
$\nu \to 0$
in the “testmass” limit for one of the bodies. To display conveniently the successive postNewtonian corrections, we employ the postNewtonian parameter
$$\begin{array}{c}\gamma \equiv \frac{Gm}{{r}_{12}{c}^{2}}=\mathcal{O}\left(\frac{1}{{c}^{2}}\right).\end{array}$$ 
(178)

Notice that there are no corrections of order 1PN in Eqs. ( 9.3 ) for circular orbits; the dominant term is of order 2PN, i.e. proportional to
${\gamma}^{2}=\mathcal{O}(1/{c}^{4})$
.
The relative acceleration
${a}_{12}^{i}\equiv {a}_{1}^{i}{a}_{2}^{i}$
of two bodies moving on a circular orbit at the 3PN order is then given by
$$\begin{array}{c}{a}_{12}^{i}={\omega}^{2}{y}_{12}^{i}\frac{32}{5}\frac{{G}^{3}{m}^{3}\nu}{{c}^{5}{r}_{12}^{4}}{v}_{12}^{i}+\mathcal{O}\left(\frac{1}{{c}^{7}}\right),\end{array}$$ 
(179)

where
${y}_{12}^{i}\equiv {y}_{1}^{i}{y}_{2}^{i}$
is the relative separation (in harmonic coordinates) and
$\omega $
denotes the angular frequency of the circular motion. The second term in Eq. ( 179 ), opposite to the velocity
${v}_{12}^{i}\equiv {v}_{1}^{i}{v}_{2}^{i}$
, is the 2.5PN radiation reaction force, which comes from the reduction of the coefficient of
$1/{c}^{5}$
in the expression ( 163 ). The main content of the 3PN equations ( 179 ) is the relation between the frequency
$\omega $
and the orbital separation
${r}_{12}$
, that we find to be given by the generalized version of Kepler's third law [
21,
22]
:
$$\begin{array}{ccc}{\omega}^{2}& =& \frac{Gm}{{r}_{12}^{3}}\{1+(3+\nu )\gamma +\left(6+\frac{41}{4}\nu +{\nu}^{2}\right){\gamma}^{2}\end{array}$$  
$$\begin{array}{ccc}& & +\left(10+\left[\frac{67759}{840}+\frac{41}{64}{\pi}^{2}+22ln\left(\frac{{r}_{12}}{{r}_{0}^{\prime}}\right)+\frac{44}{3}\lambda \right]\nu +\frac{19}{2}{\nu}^{2}+{\nu}^{3}\right){\gamma}^{3}\}\end{array}$$  
$$\begin{array}{ccc}& & +\mathcal{O}\left(\frac{1}{{c}^{8}}\right).\end{array}$$ 
(180)

The length scale
${r}_{0}^{\prime}$
is given in terms of the two gaugeconstants
${r}_{1}^{\prime}$
and
${r}_{2}^{\prime}$
by
$$\begin{array}{c}ln{r}_{0}^{\prime}=\frac{{m}_{1}}{m}ln{r}_{1}^{\prime}+\frac{{m}_{2}}{m}ln{r}_{2}^{\prime}.\end{array}$$ 
(181)

As for the energy, it is immediately obtained from the circularorbit reduction of the general result ( 165 ).
We have
$$\begin{array}{ccc}E& =& \frac{\mu {c}^{2}\gamma}{2}\{1+\left(\frac{7}{4}+\frac{1}{4}\nu \right)\gamma +\left(\frac{7}{8}+\frac{49}{8}\nu +\frac{1}{8}{\nu}^{2}\right){\gamma}^{2}\end{array}$$  
$$\begin{array}{ccc}& & +(\frac{235}{64}+\left[\frac{106301}{6720}\frac{123}{64}{\pi}^{2}+\frac{22}{3}ln\left(\frac{{r}_{12}}{{r}_{0}^{\prime}}\right)\frac{22}{3}\lambda \right]\nu \end{array}$$  
$$\begin{array}{ccc}& & +\frac{27}{32}{\nu}^{2}+\frac{5}{64}{\nu}^{3}){\gamma}^{3}\}+\mathcal{O}\left(\frac{1}{{c}^{8}}\right).\end{array}$$ 
(182)

This expression is that of a physical observable
$E$
; however, it depends on the choice of a coordinate system, as it involves the postNewtonian parameter
$\gamma $
defined from the harmoniccoordinate separation
${r}_{12}$
. But the numerical value of
$E$
should not depend on the choice of a coordinate system, so
$E$
must admit a frameinvariant expression, the same in all coordinate systems. To find it we reexpress
$E$
with the help of a frequencyrelated parameter
$x$
instead of the postNewtonian parameter
$\gamma $
. Posing
$$\begin{array}{c}x\equiv {\left(\frac{Gm\omega}{{c}^{3}}\right)}^{2/3}=\mathcal{O}\left(\frac{1}{{c}^{2}}\right),\end{array}$$ 
(183)

we readily obtain from Eq. ( 180 ) the expression of
$\gamma $
in terms of
$x$
at 3PN order,
$$\begin{array}{ccc}\gamma & =& x\{1+\left(1\frac{\nu}{3}\right)x+\left(1\frac{65}{12}\nu \right){x}^{2}\end{array}$$  
$$\begin{array}{ccc}& & +\left(1+\left[\frac{10151}{2520}\frac{41}{192}{\pi}^{2}\frac{22}{3}ln\left(\frac{{r}_{12}}{{r}_{0}^{\prime}}\right)\frac{44}{9}\lambda \right]\nu +\frac{229}{36}{\nu}^{2}+\frac{1}{81}{\nu}^{3}\right){x}^{3}\end{array}$$  
$$\begin{array}{ccc}& & +\mathcal{O}\left(\frac{1}{{c}^{8}}\right)\},\end{array}$$ 
(184)

that we substitute back into Eq. ( 182 ), making all appropriate postNewtonian reexpansions. As a result, we gladly discover that the logarithms together with their associated gauge constant
${r}_{0}^{\prime}$
have cancelled out.
Therefore, our result is
$$\begin{array}{ccc}E& =& \frac{\mu {c}^{2}x}{2}\{1+\left(\frac{3}{4}\frac{1}{12}\nu \right)x+\left(\frac{27}{8}+\frac{19}{8}\nu \frac{1}{24}{\nu}^{2}\right){x}^{2}\end{array}$$  
$$\begin{array}{ccc}& & +\left(\frac{675}{64}+\left[\frac{209323}{4032}\frac{205}{96}{\pi}^{2}\frac{110}{9}\lambda \right]\nu \frac{155}{96}{\nu}^{2}\frac{35}{5184}{\nu}^{3}\right){x}^{3}\}\end{array}$$  
$$\begin{array}{ccc}& & +\mathcal{O}\left(\frac{1}{{c}^{8}}\right).\end{array}$$ 
(185)

The constant
$\lambda $
is the one introduced in Eq. ( 160 ). For circular orbits one can check that there are no terms of order
${x}^{7/2}$
in Eq. ( 185 ), so our result for
$E$
is actually valid up to the 3.5PN order. In the testmass limit
$\nu \to 0$
, we recover the energy of a particle with mass
$\mu $
in a Schwarzschild background of mass
$m$
, i.e.
${E}_{test}=\mu {c}^{2}\left[(12x)(13x{)}^{1/2}1\right]$
, when developed to 3.5PN order.
10 Gravitational Waves from Compact Binaries
We pointed out that the 3PN equations of motion, Eqs. ( 179 , 180 ), are merely Newtonian as regards the radiative aspects of the problem, because with that precision the radiation reaction force is at the lowest 2.5PN order. A solution would be to extend the precision of the equations of motion so as to include the full relative 3PN or 3.5PN precision into the radiation reaction force, but, needless to say, the equations of motion up to the 5.5PN or 6PN order are quite impossible to derive with the present technology. The much better alternative solution is to apply the wavegeneration formalism described in Part aaa , and to determine by its means the work done by the radiation reaction force directly as a total energy flux at future null infinity. In this approach, we replace the knowledge of the higherorder radiation reaction force by the computation of the total flux
$\mathcal{\mathcal{L}}$
, and we apply the energy balance equation as in the test of the
$\dot{P}$
of the binary pulsar (see Eqs. ( 4 , 5 )):
$$\begin{array}{c}\frac{dE}{dt}=\mathcal{\mathcal{L}}.\end{array}$$ 
(186)

Therefore, the result ( 185 ) that we found for the 3.5PN binary's centerofmass energy
$E$
constitutes only “half ” of the solution of the problem. The second “half ” consists of finding the rate of decrease
$dE/dt$
, which by the balance equation is nothing but finding the total gravitationalwave flux
$\mathcal{\mathcal{L}}$
at the 3.5PN order.
Because the orbit of inspiralling binaries is circular, the balance equation for the energy is sufficient (no need of a balance equation for the angular momentum). This all sounds perfect, but it is important to realize that we shall use the equation (
186 ) at the very high 3.5PN order, at which order there are no proofs (following from first principles in general relativity) that the equation is correct, despite its physically obvious character.
Nevertheless, Eq. (
186 ) has been checked to be valid, both in the cases of pointparticle binaries [
85,
86]
and extended weakly selfgravitating fluids [
5,
9]
, at the 1PN order and even at 1.5PN (the 1.5PN approximation is especially important for this check because it contains the first wave tails).
Obtaining
$\mathcal{\mathcal{L}}$
can be divided into two equally important steps: (1) the computation of the source multipole moments
${I}_{L}$
and
${J}_{L}$
of the compact binary and (2) the control and determination of the tails and related nonlinear effects occuring in the relation between the binary's source moments and the radiative ones
${U}_{L}$
and
${V}_{L}$
(cf. the general formalism of Part aaa ).
10.1 The binary's multipole moments
The general expressions of the source multipole moments given by Theorem 6 (Eqs. ( 88 )) are first to be worked out explicitly for general fluid systems at the 3PN order. For this computation one uses the formula ( 98 ), and we insert the 3PN metric coefficients (in harmonic coordinates) expressed in Eq. ( 133 , 134 , 135 ) by means of the retardedtype elementary potentials ( 140 , 143 , 146 ). Then we specialize each of the (quite numerous) terms to the case of pointparticle binaries by inserting, for the matter stressenergy tensor
${T}^{\alpha \beta}$
, the standard expression made out of Dirac deltafunctions. The infinite selffield of pointparticles is removed by means of the Hadamard regularization (see Section 8 ). This computation has been performed by Blanchet and Schäfer [
30]
at the 1PN order, and by Blanchet, Damour and Iyer [
18]
at the 2PN order; we report below the most accurate 3PN results obtained by Blanchet, Iyer and Joguet [
26]
.
The difficult part of the analysis is to find the closedform expressions, fully explicit in terms of the particle's positions and velocities, of many nonlinear integrals. We refer to [
26]
for full details; nevertheless, let us give a few examples of the type of technical formulas that are employed in this calculation. Typically we have to compute some integrals like
$$\begin{array}{c}\stackrel{(n,p)}{{Y}_{L}}({\mathbf{y}}_{1},{\mathbf{y}}_{2})=\frac{1}{2\pi}\mathcal{\mathcal{F}}\mathcal{P}\int {d}^{3}\mathbf{x}{\hat{x}}_{L}{r}_{1}^{n}{r}_{2}^{p},\end{array}$$ 
(187)

where
${r}_{1}=\mathbf{x}{\mathbf{y}}_{1}$
and
${r}_{2}=\mathbf{x}{\mathbf{y}}_{2}$
. When
$n>3$
and
$p>3$
, this integral is perfectly welldefined (recall that the finite part
$\mathcal{\mathcal{F}}\mathcal{P}$
deals with the bound at infinity). When
$n\le 3$
or
$p\le 3$
, our basic ansatz is that we apply the definition of the Hadamard partie finie provided by Eq. ( 151 ). Two examples of closedform formulas that we get, which do not necessitate the Hadamard partie finie, are (quadrupole case
$l=2$
)
$$\begin{array}{ccc}\stackrel{(1,1)}{{Y}_{ij}}& =& \frac{{r}_{12}}{3}\left[{y}_{1}^{\langle ij\rangle}+{y}_{1}^{\langle i}{y}_{2}^{j\rangle}+{y}_{2}^{\langle ij\rangle}\right],\end{array}$$ 
(188)

$$\begin{array}{ccc}\stackrel{(2,1)}{{Y}_{ij}}& =& {y}_{1}^{\langle ij\rangle}\left[\frac{16}{15}ln\left(\frac{{r}_{12}}{{r}_{0}}\right)\frac{188}{225}\right]+{y}_{1}^{\langle i}{y}_{2}^{j\rangle}\left[\frac{8}{15}ln\left(\frac{{r}_{12}}{{r}_{0}}\right)\frac{4}{225}\right]\end{array}$$  
$$\begin{array}{ccc}& & +{y}_{2}^{\langle ij\rangle}\left[\frac{2}{5}ln\left(\frac{{r}_{12}}{{r}_{0}}\right)\frac{2}{25}\right].\end{array}$$ 
(189)

We denote for example
${y}_{1}^{\langle ij\rangle}={y}_{1}^{\langle i}{y}_{1}^{j\rangle}$
; the constant
${r}_{0}$
is the one pertaining to the finitepart process (see Eq. ( 39 )). One example where the integral diverges at the location of the particle 1 is
$$\begin{array}{c}\stackrel{(3,0)}{{Y}_{ij}}=\left[2ln\left(\frac{{s}_{1}}{{r}_{0}}\right)+\frac{16}{15}\right]{y}_{1}^{\langle ij\rangle},\end{array}$$ 
(190)

where
${s}_{1}$
is the Hadamardregularization constant introduced in Eq. ( 151 )^{
$\text{23}$
}
.
The crucial input of the computation of the flux at the 3PN order is the mass quadrupole moment
${I}_{ij}$
, since this moment necessitates the full 3PN precision. The result of Ref. [
26]
for this moment (in the case of circular orbits) is
$$\begin{array}{c}{I}_{ij}=\mu \left(A{x}_{\langle ij\rangle}+B\frac{{r}_{12}^{3}}{Gm}{v}_{\langle ij\rangle}+\frac{48}{7}\frac{{G}^{2}{m}^{2}\nu}{{c}^{5}{r}_{12}}{x}_{\langle i}{v}_{j\rangle}\right)+\mathcal{O}\left(\frac{1}{{c}^{7}}\right),\end{array}$$ 
(191)

where we pose
${x}_{i}={x}^{i}\equiv {y}_{12}^{i}$
and
${v}_{i}={v}^{i}\equiv {v}_{12}^{i}$
. The third term is the 2.5PN radiationreaction term, which does not contribute to the energy flux for circular orbits. The two important coefficients are
$A$
and
$B$
, whose expressions through 3PN order are
$$\begin{array}{ccc}A& =& 1+\gamma \left(\frac{1}{42}\frac{13}{14}\nu \right)+{\gamma}^{2}\left(\frac{461}{1512}\frac{18395}{1512}\nu \frac{241}{1512}{\nu}^{2}\right)\end{array}$$  
$$\begin{array}{ccc}& & +{\gamma}^{3}(\frac{395899}{13200}\frac{428}{105}ln\left(\frac{{r}_{12}}{{r}_{0}}\right)+\left[\frac{139675}{33264}\frac{44}{3}\xi \frac{88}{3}\kappa \frac{44}{3}ln\left(\frac{{r}_{12}}{{r}_{0}^{\prime}}\right)\right]\nu \end{array}$$  
$$\begin{array}{ccc}& & +\frac{162539}{16632}{\nu}^{2}+\frac{2351}{33264}{\nu}^{3}),\end{array}$$ 
(192)

$$\begin{array}{ccc}B& =& \gamma \left(\frac{11}{21}\frac{11}{7}\nu \right)+{\gamma}^{2}\left(\frac{1607}{378}\frac{1681}{378}\nu +\frac{229}{378}{\nu}^{2}\right)\end{array}$$  
$$\begin{array}{ccc}& & +{\gamma}^{3}\left(\frac{357761}{19800}+\frac{428}{105}ln\left(\frac{{r}_{12}}{{r}_{0}}\right)+\left[\frac{75091}{5544}+\frac{44}{3}\zeta \right]\nu +\frac{35759}{924}{\nu}^{2}+\frac{457}{5544}{\nu}^{3}\right).\end{array}$$  
$$\begin{array}{ccc}& & \end{array}$$ 
(193)

These expressions are valid in harmonic coordinates via the postNewtonian parameter
$\gamma $
given by Eq. ( 178 ).
As we see, there are two types of logarithms in the moment: One type involves the length scale
${r}_{0}^{\prime}$
related by Eq. ( 181 ) to the two gauge constants
${r}_{1}^{\prime}$
and
${r}_{2}^{\prime}$
present in the 3PN equations of motion; the other type contains the different length scale
${r}_{0}$
coming from the general formalism of Part aaa – indeed, recall that there is a
$\mathcal{\mathcal{F}}\mathcal{P}$
operator in front of the source multipole moments in Theorem 6 . As we know, that
${r}_{0}^{\prime}$
is pure gauge; it will disappear from our physical results at the end. On the other hand, we have remarked that the multipole expansion outside a general postNewtonian source is actually free of
${r}_{0}$
, since the
${r}_{0}$
's present in the two terms of Eq. ( 74 ) cancel out. We shall indeed find that the constants
${r}_{0}$
present in Eqs. ( 193 ) are compensated by similar constants coming from the nonlinear wave “tails of tails”. More seriously, in addition to the harmless constants
${r}_{0}$
and
${r}_{0}^{\prime}$
, there are three unknown dimensionless parameters in Eqs. ( 193 ), called
$\xi $
,
$\kappa $
and
$\zeta $
. These parameters reflect some incompleteness of the Hadamard selffield regularization (see the discussion in Section 8.2 ).
Besides the 3PN mass quadrupole (
191 , 193 ), we need also the mass octupole moment
${I}_{ijk}$
and current quadrupole moment
${J}_{ij}$
, both of them at the 2PN order; these are given by [
26]
$$\begin{array}{ccc}{I}_{ijk}& =& \mu \frac{\delta m}{m}{\hat{x}}_{ijk}\left[1+\gamma \nu +{\gamma}^{2}\left(\frac{139}{330}+\frac{11923}{660}\nu +\frac{29}{110}{\nu}^{2}\right)\right]\end{array}$$  
$$\begin{array}{ccc}& & +\mu \frac{\delta m}{m}{x}_{\langle i}{v}_{jk\rangle}\frac{{r}_{12}^{2}}{{c}^{2}}\left[1+2\nu +\gamma \left(\frac{1066}{165}+\frac{1433}{330}\nu \frac{21}{55}{\nu}^{2}\right)\right]\end{array}$$  
$$\begin{array}{ccc}& & +\mathcal{O}\left(\frac{1}{{c}^{5}}\right),\end{array}$$ 
(194)

$$\begin{array}{ccc}{J}_{ij}& =& \mu \frac{\delta m}{m}{\varepsilon}_{ab\langle i}{x}_{j\rangle a}{v}_{b}\left[1+\gamma \left(\frac{67}{28}+\frac{2}{7}\nu \right)+{\gamma}^{2}\left(\frac{13}{9}+\frac{4651}{252}\nu +\frac{1}{168}{\nu}^{2}\right)\right]\end{array}$$  
$$\begin{array}{ccc}& & +\mathcal{O}\left(\frac{1}{{c}^{5}}\right).\end{array}$$ 
(195)

Also needed are the 1PN mass
${2}^{4}$
pole, 1PN current
${2}^{3}$
pole (octupole), Newtonian mass
${2}^{5}$
pole and Newtonian current
${2}^{4}$
pole:
$$\begin{array}{ccc}{I}_{ijkl}& =& \mu {\hat{x}}_{ijkl}\left[13\nu +\gamma \left(\frac{3}{110}\frac{25}{22}\nu +\frac{69}{22}{\nu}^{2}\right)\right]\end{array}$$  
$$\begin{array}{ccc}& & +\frac{78}{55}\mu {x}_{\langle ij}{v}_{kl\rangle}\frac{{r}_{12}^{2}}{{c}^{2}}(15\nu +5{\nu}^{2})+\mathcal{O}\left(\frac{1}{{c}^{3}}\right),\end{array}$$ 
(196)

$$\begin{array}{ccc}{J}_{ijk}& =& \mu {\varepsilon}_{ab\langle i}{x}_{jk\rangle a}{v}_{b}\left[13\nu +\gamma \left(\frac{181}{90}\frac{109}{18}\nu +\frac{13}{18}{\nu}^{2}\right)\right]\end{array}$$  
$$\begin{array}{ccc}& & +\frac{7}{45}\mu (15\nu +5{\nu}^{2}){\varepsilon}_{ab\langle i}{v}_{jk\rangle b}{x}_{a}\frac{{r}_{12}^{2}}{{c}^{2}}+\mathcal{O}\left(\frac{1}{{c}^{3}}\right),\end{array}$$ 
(197)

$$\begin{array}{ccc}{I}_{ijklm}& =& \mu \frac{\delta m}{m}(1+2\nu ){\hat{x}}_{ijklm}+\mathcal{O}\left(\frac{1}{c}\right),\end{array}$$ 
(198)

$$\begin{array}{ccc}{J}_{ijkl}& =& \mu \frac{\delta m}{m}(1+2\nu ){\varepsilon}_{ab\langle i}{x}_{jkl\rangle a}{v}_{b}+\mathcal{O}\left(\frac{1}{c}\right).\end{array}$$ 
(199)

These results permit one to control what can be called the “instantaneous” part, say
${\mathcal{\mathcal{L}}}_{inst}$
, of the total energy flux, by which we mean that part of the flux that is generated solely by the source multipole moments, i.e. not counting the “noninstantaneous” tail integrals. The instantaneous flux is defined by the replacement into the general expression of
$\mathcal{\mathcal{L}}$
given by Eq. ( 67 ) of all the radiative moments
${U}_{L}$
and
${V}_{L}$
by the corresponding (
$l$
th time derivatives of the) source moments
${I}_{L}$
and
${J}_{L}$
. Actually, we prefer to define
${\mathcal{\mathcal{L}}}_{inst}$
by means of the intermediate moments
${M}_{L}$
and
${S}_{L}$
. Up to the 3.5PN order we have
$$\begin{array}{ccc}{\mathcal{\mathcal{L}}}_{inst}& =& \frac{G}{{c}^{5}}\{\frac{1}{5}{M}_{ij}^{\left(3\right)}{M}_{ij}^{\left(3\right)}+\frac{1}{{c}^{2}}\left[\frac{1}{189}{M}_{ijk}^{\left(4\right)}{M}_{ijk}^{\left(4\right)}+\frac{16}{45}{S}_{ij}^{\left(3\right)}{S}_{ij}^{\left(3\right)}\right]\end{array}$$  
$$\begin{array}{ccc}& & +\frac{1}{{c}^{4}}\left[\frac{1}{9072}{M}_{ijkm}^{\left(5\right)}{M}_{ijkm}^{\left(5\right)}+\frac{1}{84}{S}_{ijk}^{\left(4\right)}{S}_{ijk}^{\left(4\right)}\right]\end{array}$$  
$$\begin{array}{ccc}& & +\frac{1}{{c}^{6}}\left[\frac{1}{594000}{M}_{ijkmn}^{\left(6\right)}{M}_{ijkmn}^{\left(6\right)}+\frac{4}{14175}{S}_{ijkm}^{\left(5\right)}{S}_{ijkm}^{\left(5\right)}\right]\end{array}$$  
$$\begin{array}{ccc}& & +\mathcal{O}\left(\frac{1}{{c}^{8}}\right)\}.\end{array}$$ 
(200)

The time derivatives of the source moments ( 191 , 193 , 195 , 199 ) are computed by means of the circularorbit equations of motion given by Eq. ( 179 , 180 ); then we substitute them into Eq. ( 200 ) (for circular orbits there is no difference at this order between
${I}_{L}$
,
${J}_{L}$
and
${M}_{L}$
,
${S}_{L}$
). The net result is
$$\begin{array}{ccc}{\mathcal{\mathcal{L}}}_{inst}& =& \frac{32{c}^{5}}{5G}{\nu}^{2}{\gamma}^{5}\{1+\left(\frac{2927}{336}\frac{5}{4}\nu \right)\gamma +\left(\frac{293383}{9072}+\frac{380}{9}\nu \right){\gamma}^{2}\end{array}$$  
$$\begin{array}{ccc}& & +(\frac{53712289}{1108800}\frac{1712}{105}ln\left(\frac{{r}_{12}}{{r}_{0}}\right)\end{array}$$  
$$\begin{array}{ccc}& & +\left[\frac{332051}{720}+\frac{123}{64}{\pi}^{2}+\frac{110}{3}ln\left(\frac{{r}_{12}}{{r}_{0}^{\prime}}\right)+44\lambda \frac{88}{3}\theta \right]\nu \end{array}$$  
$$\begin{array}{ccc}& & \frac{383}{9}{\nu}^{2}){\gamma}^{3}\end{array}$$  
$$\begin{array}{ccc}& & +\mathcal{O}\left(\frac{1}{{c}^{8}}\right)\}.\end{array}$$ 
(201)

The Newtonian approximation,
${\mathcal{\mathcal{L}}}_{N}=\left(32{c}^{5}\right)/\left(5G\right)\cdot {\nu}^{2}{\gamma}^{5}$
, is the prediction of the Einstein quadrupole formula ( 4 ), as computed by Landau and Lifchitz [
97]
. The selffield regularization ambiguities arising at the 3PN order are the equationofmotionrelated constant
$\lambda $
and the multipolemomentrelated constant
$\theta =\xi +2\kappa +\zeta $
(see Section 8.2 ).
10.2 Contribution of wave tails
To the “instantaneous” part of the flux, we must add the contribution of nonlinear multipole interactions contained in the relationship between the source and radiative moments. The needed material has already been provided in Eqs. ( 101 , 6 ). Up to the 3.5PN level we have the dominant quadraticorder tails, the cubicorder tails or tails of tails, and the nonlinear memory integral. We shall see that the tails play a crucial role in the predicted signal of compact binaries. By contrast, the nonlinear memory effect, given by the integral inside the 2.5PN term in Eq. ( 101 ), does not contribute to the gravitationalwave energy flux before the 4PN order in the case of circularorbit 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
$$\begin{array}{c}\mathcal{\mathcal{L}}={\mathcal{\mathcal{L}}}_{inst}+{\mathcal{\mathcal{L}}}_{tail}+{\mathcal{\mathcal{L}}}_{(tail{)}^{2}}+{\mathcal{\mathcal{L}}}_{tail(tail)},\end{array}$$ 
(202)

where
${\mathcal{\mathcal{L}}}_{inst}$
has just been found in Eq. ( 201 );
${\mathcal{\mathcal{L}}}_{tail}$
is made of the quadratic (multipolar) tail integrals in Eq. ( 6 );
${\mathcal{\mathcal{L}}}_{(tail{)}^{2}}$
is the square of the quadrupole tail in Eq. ( 101 ); and
${\mathcal{\mathcal{L}}}_{tail(tail)}$
is the quadrupole tail of tail in Eq. ( 101 ). We find that
${\mathcal{\mathcal{L}}}_{tail}$
contributes at the halfinteger 1.5PN, 2.5PN and 3.5PN orders, while both
${\mathcal{\mathcal{L}}}_{(tail{)}^{2}}$
and
${\mathcal{\mathcal{L}}}_{tail(tail)}$
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 (
101 , 6 ). Without going into accessory details (see Ref. [
10]
), let us give the two basic technical formulas needed when carrying out this reduction: