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Energy losses by gravitational radiation in inspiralling compact binaries to five halves post-Newtonian order

This paper derives the total power or energy loss rate generated in the form of gravitational waves by an inspiralling compact binary system to the five halves post-Newtonian (2.5PN) approximation of general relativity. Extending a recently developed gravitational-wave generation formalism valid for arbitrary (slowly-moving) systems, we compute the mass multipole moments of the system and the relevant tails present in the wave zone to 2.5PN order. In the case of two point-masses moving on a quasi-circular orbit, we find that the 2.5PN contribution in the energy loss rate is entirely due to tails. Relying on an energy balance argument we derive the laws of variation of the instantaneous frequency and phase of the binary. The 2.5PN order in the accumulated phase is significantly large, being grossly of the same order of magnitude as the previous 2PN order, but opposite in sign. However finite mass effects at 2.5PN order are small. The results of this paper should be useful when analyzing the data from inspiralling compact binaries in future gravitational-wave detectors like VIRGO and LIGO.


I. INTRODUCTION
Compact binaries in their late stage of evolution are very relativistic systems in which the two compact objects (neutron stars or black holes) orbit around eachother with velocities as large as 30 % of that of light. The gravitational radiation these systems emit during the inspiral phase preceding the coalescence of the two objects is expected to be routinely analyzed in future detectors like LIGO and VIRGO (see [1,2,3] for reviews). Hundreds to tens of thousands of gravitational-wave cycles should be monitored in the detectors' sensitive frequency bandwidth. The combination of high orbital velocities and a large number of observed rotations, together with the fact that the emitted waves are highly predictable, implies that high-order relativistic (or post-Newtonian) effects should show up in the gravitational signals observed by VIRGO and LIGO [4,5,6,7,8,9,10]. Alternatively, this means that high-order post-Newtonian effects should be known in advance so that they can be included in the construction of theoretical filters (templates) to be cross correlated with the outputs of the detectors.
The relevant model for describing most of the observed inspiral phase is a model of two point masses moving on a circular orbit. Radiation reaction forces tend to circularize very rapidly the orbit. On the other hand, point masses can be used in the case of non-rotating and (initially) spherically symmetric compact objects up to a very high precision [11]. This is due to a property owned by general relativity of "effacing" of the internal structure. Even in the case of stars with intrinsic rotations the dynamics of the binary is likely to be dominated by post-Newtonian gravitational effects [12].
High-order post-Newtonian effects that are measurable are mainly those affecting the orbital phase evolution of the binary, which in turn is determined using a standard energy balance argument by the total power emitted in the form of gravitational waves by the system at infinity, or total luminosity in the waves. To what level in a post-Newtonian expansion should we know the gravitational luminosity (or energy loss) in order to guarantee an optimal detection of the signal (given some power spectral density of the noise in a detector) is still unclear, but the theory of black-hole perturbations can be used to gain insights in this problem. Black-hole perturbations, which deal with the special case of a test mass orbiting a massive black hole, have been recently the focus of intense activity [13,14,15,16,17]. It emerges from this field that neglecting even a so high approximation as the third post-Newtonian (3PN) one, i.e. neglecting the relativistic corrections in the luminosity which are of relative order c −6 (or below), is likely to yield unacceptable systematic errors in the data analysis of binary signals [14,18,19]. This shows how relativistic are inspiralling compact binaries, as compared for instance to the binary pulsar for which the Newtonian approximation in the luminosity (Einstein quadrupole formula) is adequate. The post-Newtonian theory is presently completed through the second post-Newtonian (2PN) approximation, i.e., through relative order c −4 (both in the waveform and in the associated energy loss). Two computations were performed to this order, one by Blanchet, Damour and Iyer [20,21] based on a post-Minkowski matching formalism, and one by Will and Wiseman [22] using an approach initiated by Epstein and Wagoner [23] and generalized by Thorne [24]. The common result of these two computations for the energy loss was summarized in Ref. [12], and the waveform can be found in [25].
In the present paper we develop the post-Minkowski matching formalism one step beyond the work of Refs. [20,21] by computing the 2.5PN order in the energy loss of an inspiralling binary. This entails extending both Ref. [20] on the general formalism valid for an arbitrary (slowly-moving) source, and Ref. [21] dealing with the specific application to the binary.
The computation of the waveform of the binary to 2.5PN order (the square of which should give back the 2.5PN energy loss) will be left for future work.
The post-Minkowski matching formalism is a wave generation formalism which is especially suited for "semi relativistic" sources whose internal velocities can reach 0.3c at most (say), as in the case of inspiralling compact binaries (see [26] for a review). The formalism combines (i) an analytic post-Minkowskian approximation scheme for the computation of the gravitational field in the exterior of the source where multipole expansions can be used to simplify the problem, (ii) a direct post-Newtonian approximation scheme for the resolution of the field equations inside the near-zone of the source, and (iii) an asymptotic matching between both types of solutions which is performed in the exterior part of the near zone. The necessicity of using a post-Minkowskian approximation scheme first is because its validity extends up to the regions far away from the system where the observer is located, contrarily to the post-Newtonian approximation whose validity is limited to the near zone. The exterior field is computed using an algorithm developed in Ref. [27] which set on a general footing previous investigations by Thorne [24] and Bonnor [28]. The implementation of the wave generation formalism (i)-(iii) was done at first with 1PN accuracy in Ref. [29] which obtained the 1PN correction terms in the mass-type quadrupole moment of the source (and in fact in all the mass-type multipole moments). The dominant nonlinear contribution in the radiation field was added in Ref. [30] and shown to be due to the contribution in the wave zone of the well-known "tail" effect. The inclusion of this nonlinear contribution pushed the accuracy of the formalism to 1.5PN order in the energy loss. The 2PN precision both in the energy loss and waveform was reached in Ref. [20] where the second (2PN) correction terms in the mass-type multipole moments and the first (1PN) ones in the current-type multipole moments were obtained. An equivalent expression of the 1PN current-type moments had been derived earlier [31], however in a different form which is not amenable to generalization to higher order (but turned out to be useful in applications [21]).
The main result of the present paper is the expression of the 2.5PN-accurate energy loss by gravitational radiation from a general (semi relativistic) source, and from an inspiralling compact binary. In the latter case of application, the 2.5PN contribution in the energy loss is found to be entirely due to tails in the wave zone (this is like the 1.5PN contribution), and to reduce in the test-body limit to the known result of perturbation theory [15,16,17].
With the energy loss one can derive the laws of variation of the inspiralling binary's orbital frequency and phase using an energy balance equation. However note that this is a weak point of the analysis because the latter energy balance equation has been proved to hold only at the Newtonian order (see [32,33,34,35,36] for general systems, and [37] for binary systems), and more recently at the 1PN order [38,39]. It is also known to hold for the specific effects of tails at 1.5PN order [30,40]. To prove this equation at the 2.5PN order as we would need below, one should in principle obtain the equations of motion of the binary up to the very high 5PN order (or order c −10 ) beyond the Newtonian acceleration. Indeed the radiation reaction forces which are responsible for the decrease of the binding energy of the binary are themselves dominantly of order 2.5PN beyond the Newtonian term. The only method which is available presently in order to deal with this problem is to assume that the 2.5PN-accurate radiation reaction forces are such that there is exact agreement between the loss of 2.5PN-accurate binding energy of the binary (as computed from the Damour-Deruelle equations of motion [37]) and the 2.5PN-accurate energy flux we shall compute below. This assumption is verified at the 1.5PN order and sounds reasonable, but will have to be justified in future work.
As an indication of the quantitative importance of the 2.5PN approximation in the or-bital phase of the binary, we compute the contribution of the 2.5PN term to the number of gravitational-wave cycles between the entry and exit frequencies of some detector. Essentially we find that the 2.5PN approximation is of the same order of magnitude as the 2PN term (computed in [12]), but opposite in sign. In the case of two neutron stars of mass 1.4M ⊙ and of the frequency bandwidth [10 Hz, 1000 Hz], the 2PN term contributed to +9 units to the total number of cycles [12]. We find here that the 2.5PN term contributes to −11 cycles in the same conditions. This shows the importance of the 2.5PN approximation for constructing accurate theoretical templates. [However, we find that the contribution of the finite mass effects in the 2.5PN term (which cannot be obtained in perturbation theory) is numerically small.] In the present paper we shall make a thorough investigation (see Sections II, III and IV below) of all the relativistic corrections in the multipole moments of the system which contribute to the 2.5PN-accurate energy loss. However when we are interested only in the application to inspiralling compact binaries, this investigation can be seen a posteriori to be unnecessary. Indeed the orbit of an inspiralling binary is circular, and we shall prove in this case that the 2.5PN relativistic corrections in the multipole moments give in fact no contribution in the energy loss. As we said above, the only contribution is that of the tails present at this order (see Section VI). A simple argument (concerning the result at 2.5PN order of a contracted product of tensors made of the relative separation and velocity of the bodies) could be used beforehand to see that this is true. But because inspiralling compact binaries may not constitute the only sources for which the 2.5PN approximation in the energy loss is needed, or simply because one may need in the future to consider the case of a binary moving on an eccentric orbit, we have chosen in this paper to compute systematically all the terms which enter the 2.5PN-accurate energy loss for general systems.
This permits us to show explicitly that all the terms but the tail terms give zero in the energy loss for (circular) inspiralling binaries. The simple argument mentioned above may be used in future work to simplify the investigation of higher post-Newtonian orders.
The plan of this paper is as follows. In the following section II and in the next one III we follow step by step the derivation done in Ref. [20] of the near-zone gravitational field and the corresponding matching equation, and show how this derivation can be extended to the 2.5PN order. In section IV we obtain the explicit 2.5PN corrections arising in the mass-type multipole moments. Section V deals with the derivation of the energy loss formula valid for general systems (however some coefficients are left un-specified in the formula). Finally these results are applied in Section VI to inspiralling compact binaries. Appendix A derives a useful integration formula and Appendix B presents a relevant summary of the 2.5PN equations of motion.
Throughout this paper we refer to Ref. [20] as paper I and to Ref. [21] as paper II.

II. THE GRAVITATIONAL FIELD IN THE NEAR ZONE
Following the plan of paper I we first investigate the gravitational field generated by a slowly moving isolated source in its near zone, which is defined in the usual way as being a zone whose size is of small extent with respect to a typical wavelength of the emitted radiation. Two distinctive methods are used. The first one is a direct post-Newtonian iteration (speed of light c → +∞) of the field equations inside the source, and is valid allover the near zone. The second method consists of re-expanding when c → +∞ a solution of the vacuum field equations obtained by means of the multipolar and post-Minkowskian iteration scheme of Ref. [27], and is valid only in the exterior part of the near zone.
We denote the small post-Newtonian parameter by ε ∼ v/c, where v is a typical velocity in the source (e.g., the relative orbital velocity of the two bodies in the case of a binary system). A remainder term of order O(ε n ) is abbreviated by O(n). In a vector A µ or a tensor B µν , the remainder term is denoted by O(n, p) or O(n, p, q), by which we mean a term of order O(n) in A 0 or B 00 , of order O(p) in A i or B 0i or B i0 , and of order O(q) in B ij . [Greek indices range from 0 to 3, and Latin indices range from 1 to 3.] Most of the notations used here are like in paper I.
A. The inner gravitational field T αβ are the contravariant components of the stress-energy tensor (with dimension of an energy density) of the material source in some inner coordinate system (x, t). The densities of mass σ, of current σ i and of stress σ ij in the source are defined by where T ii denotes the spatial trace Σδ ij T ij . These definitions are such that σ, σ i , σ ij have a finite non-zero limit as c → +∞. From these matter densities one defines the retarded where G is Newton's constant, and where −1 R denotes the retarded integral operator Contrarily to the sources of V and V i which are of compact support: V = −4πGσ and Indeed we have included in W ij the stress density of the (Newtonian) gravitational field itself since it is of the same order as σ ij when c → +∞. [Note that paper II used the notation W ij for a closely related but different potential; here we do not follow paper II but stick to the notation of paper I.] To Newtonian order the densities and potentials so defined satisfy the equations of continuity and of motion From these dynamical equations one deduces the differential identities With the introduction in paper I of the retarded potentials V , V i , W ij , a simple expression of the gravitational field h αβ inside the source which is valid to some intermediate accuracy O (6,5,6) was written, namely The field variable is h αβ ≡ √ −gg αβ − η αβ , where g and g αβ are the determinant and inverse of the usual covariant metric g αβ , and where η αβ is the Minkowski metric (signature −+++).
where λ and Λ αβ denote some explicit combinations of the inner gravitational potentials V , V i , W ij and their derivatives. First we have which represents in fact the post-Newtonian expansion of (minus) the determinant of the metric, with terms O(6) suppressed. Second the components of Λ αβ read As we shall see the post-Newtonian accuracy of the inner field (2.7)-(2.10) is sufficient for our purpose.

B. The external gravitational field
In the exterior we use a solution of the vacuum field equations which has in principle sufficient generality for dealing with an arbitrary source of gravitational radiation. This solution is given as a (nonlinear) functional of two infinite sets of time-varying multipole moments, M L (t) and S L (t). The index L carried by these moments represents a multi-index formed with ℓ spatial indices: L = i 1 i 2 . . . i ℓ . The "order of multipolarity" ℓ goes from zero to infinity for the "mass-type" moments M L (t) and from one to infinity for the "current- (2.11c) The notation ∂ L is shorthand for a product of partial derivatives, Having defined these potentials one introduces a more complicated potential W ext ij by the formula The second term appearing here involves the retarded integral (2.3) but regularized by means of the analytic continuation process defined in Ref. [27]. One multiplies the source by r B where r = |x| and B is a complex number. Then applying the operator −1 R defines a function of B which can be analytically continued all-over the complex plane deprived of the integers, but admitting there a Laurent expansion with only some simple poles [27]. With the external potentials V ext , V ext i and W ext ij one has a result similar to Eq. (2.6), namely the expression of the external field h µν can (where the notation "can" means that we are considering specifically the "canonical" construction of the external field as defined in Sect. 4.3 of [27]) up to the intermediate accuracy O (6,5,6) : Because this expression has the same form as Eq. (2.6), the nonlinearities in the exterior field will have in turn the same form as Eq. (2.9). Up to order O(8, 7, 8) we obtain the "canonical" field This term is the solution of the linearized (vacuum) equations on which is based the post-Minkowskian algorithm for the construction of the canonical metric in Ref. [27]. The powers of c −1 in Eqs. (2.17) are such that h µν can(1) = O(2, 3, 4). Finally the term G 2 q µν can(2) in (2.15) is a particular solution of the source-free wave equation which has to be added in order that the harmonic gauge condition (2.16) be satisfied. It was proved in the Appendix A of paper I that this term is of order  (2) ). Relying on our previous papers we know that this sum is made of some retarded waves of the type ∂ L (r −1 X(t − r/c)) with scalar or dipolar multipolarity only (ℓ = 0 or 1). Furthermore we know that the dependence on c −1 of such a wave is O(5 + ℓ 1 + ℓ 2 − ℓ) where ℓ 1 and ℓ 2 are the multipolarities of the two interacting moments composing the wave (we use the same notation as e.g. the Appendix A of paper I; see also after Eq. (5.5) below). With ℓ = 0 or 1 a term of order O(7) necessarily has ℓ 1 + ℓ 2 = 2 or 3. The term in question is made of the product of the mass M with the quadrupoles M ij and S ij , or of the product of the mass dipole M i with the quadrupole M ij . The first possibility is excluded because M ij and S ij are STF (thus no scalar or dipole wave with no free index can be formed), and the second possibility does not exist in a mass centred-frame where M i = 0. So we have proved that in a mass-centred frame we have which is all what will be needed in the following.

A. Relations between inner and outer potentials
Our matching requirement is that there exists a change of coordinates valid in the external near zone and transforming the inner gravitational field h αβ (x) given by Eq. (2.7) into the outer field h αβ can (x can ) given by Eq. (2.15). Let this change of coordinates be where x µ are the inner coordinates used in section II.A and x µ can the outer coordinates used in section II.B. The vector ϕ µ has been shown in paper I to be of order The matching equations consistent with the order O(8, 7, 8) read from paper I as where both the inner and outer fields are expressed in the same (inner) coordinate system x µ . We denote by ∂ϕ µν the linear part of the coordinate transformation, The nonlinear part of the transformation enters to this order only the 00 component (3.4a).
The matching equations were used in paper I first to obtain the relations between the external potentials and the multipole expansions of the corresponding inner potentials. The insertion of the intermediate expressions of h µν and h µν can valid up to O(6, 5, 6) and given by (2.6) and (2.14) yields The notationx L is for the tracefree projection of the product of ℓ spatial vectors x L ≡ Some explanations on the expressions (3.7)-(3.9) are in order. First notice that the expressions of the multipole expansions of the potentials V and V i whose sources have a compact support are quite standard. They can be found in this form in the Appendix B of Ref. [29] (but were derived earlier in an alternative form [41]). Notably the presence of the function δ ℓ (z) is due to the time delays of the propagation of the waves with finite velocity c inside the source. By contrast the expression of the multipole expansion of the potential W ij whose source extends up everywhere in space is more complicated. The second term in Eq. (3.7c) This second term involves the multipole expansion M(V ) of the inner potential and not V itself. This is in conformity with the use of the regularized operator FP −1 R which is defined only when acting on multipole expansions (like in Eq. (2.13)). On the other hand the integrand of (3.8c) involves the non-compact supported source of W ij where appears the potential V itself, i.e. not in multipole expanded form. Thus the integrand of (3.8c) is valid everywhere inside and outside the source. Very far from the source it diverges because of the presence of the productx L of ℓ spatial vectors, behaving like |x| ℓ at spatial infinity. The well-definiteness of the integral results from the presence of the regularization factor |x| B and the finite part symbol. We notice that no ad hoc prescription is necessary in order to obtain the multipole moments in a well-defined form even in the case of a non-compact supported source. This is proved in paper I.

B. The matching equation
The relations (3.6) linking V ext , V ext i , W ext ij to the multipole expansions M(V ), M(V i ), M(W ij ) serve us with re-expressing the non-linearities in the external metric in terms of the potentials belonging to the inner metric. The result of paper I is where the components of the tensor Ω µν are given by The only non-zero component of this tensor is the 00 component which is of order . Both sides of Eq. (3.10) are multiplied by r B , and one applies afterwards the retarded integral −1 R and take the finite part at B = 0. We find where one must be careful that an extra term X µν with respect to paper I arises whose components are given by The only non-zero component of this term is the 00 component which satisfies The fact that X 00 is a (retarded) solution of the source-free d'Alembert equation is clear from its definition (3.14a) and the main property of the operator FP −1 R which is the inverse of (when acting on multipolar sources). The fact that X 00 is of order O (7), that is exactly one order in c −1 smaller than the order of the corresponding Ω 00 , is not immediately obvious but will be proved below. The quantity X 00 was rightly neglected in paper I but has to be considered here since it will contribute to the mass moments at the 2.5PN order.
We are now in a position to write down a matching equation valid up to the neglect of O (8,7,8)

terms. The equation is obtained by insertion into Eqs. (3.4) of both the inner field
h µν given by (2.7) and the outer field h µν can given by (2.15). Use is made of the link derived in (3.13) between the external and internal non-linearities. We find that Ω µν cancels to the required order the nonlinear part of the coordinate transformation so that only remains the linear part ∂ϕ µν given by Eq. (3.5). The resulting matching equation (extending the less accurate equation (3.35) of paper I) reads as There are two new terms with respect to (3.35) in paper I : G 2 q µν can (2) in the left side and −X µν in the right side which both are terms of 2.5PN order. Now, by exactly the same reasoning as in paper I one can transform the difference between the two retarded integrals in the right side into an explicit multipole expansion parametrized by some moments T µν L (t), namely These moments are given by where τ µν denotes the total stress-energy tensor of the material and gravitational fields (valid up to the considered precision) which is conserved in the sense It is important to note that the effective stress-energy tensor τ µν is a functional of the potentials V , V i and W ij valid everywhere inside and outside the source. There is no contribution of the multipole expansions of the potentials in the final result (see paper I).
The left side of the matching equation (3.17) is a functional of the original multipole moments M L , S L parametrizing the exterior metric. On the other hand the right side is a functional of the actual densities of mass, current and stress of the material fields in the source. To find the explicit expressions of M L and S L in terms of these source densities we decompose the right side into irreducible multipole moments. Inspection of the reasoning done in section IV.A of paper I shows that this reasoning is still valid in the present more accurate case. As a result we find where the last term is a linear coordinate transformation associated with the vector ξ µ = ϕ µ + ω µ where ω µ is the same as in Eq. (4.4) of paper I. The linearized metric Gh µν can(1) in the right side takes the same expression as in the left side but is parametrized instead of M L and S L by the (STF) source multipole moments I L and J L given by We have posed The equation (3.21) can be solved uniquely for the multipole moments M L and S L . To do so it suffices to notice that for any gauge term ∂ξ µν ≡ ∂ µ ξ ν + ∂ ν ξ µ − η µν ∂ λ ξ λ the identity holds. This is nothing but the vanishing of the 0i0j component of the linearized Riemann tensor when computed with g gauge µν = ∂ µ ξ ν + ∂ ν ξ µ . Applying (3.24) to the gauge term of (3.21) and using the form of the multipole moment decomposition of h µν can(1) one finds where δI L in the mass moments is of 2.5PN order and comes from the decomposition of X 00 (which we recall is a retarded solution of the wave equation) into multipole moments according to There is no contribution coming from G 2 q µν can (2) in the left side of (3.21) thanks to the result proved in Eq. (2.19) in the case where we are using a mass-centred frame. The result (3.25a) generalizes to 2.5PN order the result (4.7a) of paper I. In a future work we shall investigate more systematically the relations linking the exterior moments M L and S L to the source moments I L and J L as they have been defined here.

IV. EXPLICIT EXPRESSIONS OF THE MULTIPOLE MOMENTS
Two things must be done in order to obtain the expressions of the moments M L and S L of Eqs. (3.25). Firstly one must expand when c → +∞ the source moments I L and J L of Eqs. (3.22) up to consistent order. Secondly one must evaluate the 2.5PN modification δI L entering the mass moments (3.25a).

A. The source multipole moments
The post-Newtonian expansion of the source moments (3.22) is straighforwardly performed using a formula which was given in Eq. (B.14) of Ref. [29], namely  (4.13) of paper I for the source moments in raw form : where the Newtonian-like potentials U, X, U i and P ij are defined as in paper I by and where the new terms involve the tracefree quadrupole moment Q ij and moment of inertia K associated with the mass distribution σ, namely where I L and J L are the 2PN-accurate moments which were obtained in paper I, and where δ I L denotes a 2.5PN correction term (which is distinct from δI L found in Eq. (3.25a)). After the transformation of I L as in section IV.B of paper I one can write I L and J L in the form The moments I L and J L constituted the central result of paper I and they were at the basis of the application to inspiralling compact binaries in paper II. On the other hand the 2.5PN correction term δ I L is obtained by simple inspection of Eq. (4.3a). A simplifying fact in obtaining δ I L is that the moments Q ij and K are only functions of time so that their spatial gradients vanish. We obtain The second term is an integral having a priori a non-compact support. However it can be transformed into a manifestly compact-support form by means of the formula This formula is proved by noticing that the integral defined by I B = d 3 x|x| Bx L |x − y| −1 is proportional to a prefactor |y| B and satisfies ∆ y I B = −4π|y| Bŷ L . These two facts imply that I B = ∆ −1 y (−4π|y| Bŷ L ) where ∆ −1 y is defined as in Eq. (3.9) of Ref. [27], and thus that I B = −4π|y| B+2ŷ L /(B + 2)(B + 2ℓ + 3) which yields (4.10) after taking the finite part. The more complicated formula (4.23) in paper I can interestingly be compared with (4.10).
Thanks to (4.10) we can write Expanding the spatial derivatives in the second term yields finally where we have posed The powers of 1/c in front of these terms are such that W L , · · · , Z L have a non-zero limit when c → +∞. We compute these tensors to the lowest order in 1/c. To do this let us recall the relations between V ext i and W ext ij and the multipole expansions M(V i ) and M(W ij ) as obtained in Eqs. (3.6b) and (3.6c). We have where we have transformed the relation for W ext ij into a simpler relation for V ext ij , and where the moments take to lowest order the form [see Eqs. (3.8b) and (3.8c)] : The (non-compact-supported) stress density Σ ij is defined by Eq. pieces with respect to their ℓ + 1 and ℓ + 2 indices. We do not detail this computation but simply give the result, which is The tensor W L is manifestly of compact-supported form. There is agreement for this tensor with the previous result obtained in Eqs. (2.22a) and (2.19c) of Ref. [29]. As they are written the other tensors X L , Y L , Z L do not have a compact-supported form. However Y L can be re-written equivalently in such a form: The transformation of Y L into the form (4.17e) is done using the results (4.2) and (4.18) of paper I.
The quantities Ω 00 and X 00 can now be evaluated. As is clear from its structure and the form of ϕ µ and M(V ), M(V i ), the quantity Ω 00 is made up of a sum of quadratic products of retarded waves. We shall write where F and G are some functions of the retarded time symbolizing some derivatives of the functions V L , V L i and W L , X L , Y L , Z L (all indices suppressed). We assume (as can always be done) that the derivative operators are trace-free:∂ P ≡ ∂ <i 1 ∂ i 2 · · · ∂ ip> and∂ Q ≡ ∂ <j 1 ∂ j 2 · · · ∂ jq> . The power of 1/c in front indicates the true order of magnitude of Ω 00 when c → +∞ [see (3.12)], and the remainder O(10) comes from the un-controlled remainder terms in (4.14). To evaluate X 00 we need to know the action of the operator FP B=0 −1 R r B − 1 1, where 1 1 denotes the unit operator, on the generic term composing Ω 00 . Actually we shall be interested only in that part of X 00 which is strictly larger than the remainder O(8) we neglect in (3.16). In that case, a useful formula shows that all the terms in Ω 00 which are composed of the product of two waves with multipolarities p ≥ 1 and q ≥ 1 yield negligible terms in X 00 . This formula, which is proved in Appendix A, reads (δ p,q is the Kronecker symbol). When both p ≥ 1 and q ≥ 1 the right side of (4.19) is of order O(3) relatively to the left side and the corresponding term in X 00 is of negligible order O(9). We can thus limit our consideration to the terms in Ω 00 involving at least one monopolar wave p = 0 or q = 0. We insert into Eq. (3.11a) the multipole expansions (3.7a) and (3.7b) together with those of ϕ 0 , ϕ i given by (4.14). By straightforward application of (4.19) to each of the resulting terms one finds where Q L (t) is the moment defined in Eq. (4.13a). On the other hand W (t) satisfies where K(t) is the moment of inertia (4.6b). Similarly one finds where the vector G i (t) reads With these notations we end up with the 2.5PN correction term δI L [compare (3.26) and (4.20)], . where I L is given by Eq. (4.8a) (this was the result of paper I), and where ∆I L = δ I L + δI L is given by .
i + O (7) , where M is the total mass such that Q = M + O(2), and where we have used a frame such that Q i = O(2). The quadrupolar correction term (4.27c) will contribute to the asymptotic waveform at the 2.5PN order. The dipolar correction term (4.27b) will be used to determine the center of mass of the system at this order. precision one should consider also the next-order post-Newtonian term in Eq. (5.1) ; however this term will not be needed in the following.] It is sufficient to control the transverse and tracefree projection of the leading-order term ∼ R −1 in the spatial metric. A multipole decomposition yields a parametrization into two and only two sets of STF moments U L and V L which depend on T R and can be referred to as the "radiative" or "observable" mass and current moments. These are chosen so that they reduce in the limit c → +∞ to the ℓth Because the powers of 1/c go by steps of two in L, this expression is in fact the same as already used in paper I.
All the problem is to find the relations between the radiative moments U L , V L and the moments M L , S L we have previously determined. We rely on previous papers ( [29,30] and paper I) having written the general form of these relations as where the functions X nL and Y nL represent some nonlinear (and in general non-local) func- With the latter equation it is simple to control the type of nonlinearities which are present in the radiative moments to 2.5PN order. Since the reasoning has already been done in paper I to 2PN order we consider only the case which is further needed, that of a term of pure 2.5PN order in the mass-type quadrupole moment U ij (having ℓ = 2). By Eqs. (5.4a) and (5.6) this case corresponds to 3(n − 1) + 2k = 5. The only solution is n = 2 (quadratic nonlinearity) and k = 1 (one contraction of indices between the moments). With two moments, one contraction and ℓ = 2 one has ℓ 1 + ℓ 2 = 4. Furthermore one of the two moments is nonstatic, hence ℓ 2 ≥ 2 say, so we obtain only two possibilities (ℓ 1 , ℓ 2 ) = (1, 3) or (2,2). The first possibility is excluded because the moment having ℓ 1 = 1 is necessarily the constant mass dipole M i which has been set to zero. Only remains the second possibility ℓ 1 = ℓ 2 = 2 which corresponds either to the interaction between two mass-type quadrupole moments M ij or to the interaction of the (constant) current-type dipole S i with M ij .
We combine these facts with the study done in Ref. [30] of the occurence of "hereditary" terms in the asymptotic metric at the quadratic approximation n = 2. Two and only two types of hereditary terms were found: the "tail" terms coming from the interaction between the monopole M and non-static multipoles, and the nonlinear "memory" term which is made of the interaction between two non-static multipoles. The tail terms are of order c −3 and have been included in paper I, but the memory term arises at the order c −5 (in the radiative quadrupole U ij ). The latter term can be straightforwardly computed from Eqs. (2.42a), (2.21) and (2.11a) in Ref. [30]. An equivalent result can be found in Ref. [43]. For discussions on the memory term see Refs. [44,45]. It is clear by the previous reasoning that the memory term represents the hereditary part of the 2.5PN contribution in the radiative moment U ij , corresponding to the interaction of two moments M ij . Associated with this term there are also some instantaneous terms having the same structure and exhausting (a priori) the possibilities of sharing time-derivatives between the two moments.
Gathering these results with the results of Ref. [30] and paper I we obtain the expression of the radiative quadrupole U ij to 2.5PN order as follows : The constant b in the term of order c −3 (tail integral) is the same as in Eq. (5.1). The memory term is the integral in the brackets of order c −5 . The qualitatively different nature of these two integrals can be clearly understood when taking the limit T R → +∞ corresponding to very late times after the system has ceased to emit radiation. In this limit the third and higher time derivatives of M ij are expected to tend to zero, so the tail term tends to zero while by contrast the memory term tends to the finite limit −(2G/7c 5 ) (see Refs. [44,45]). The coefficients α, β, γ and λ are some purely numerical coefficients in front of instantaneous terms (which depend on T R only). These coefficients can be obtained by a long computation using the algorithm of Ref. [27], however we shall not need them in the application below (they will be computed in a future work). The higher-order radiative moments take similar expressions but are needed only to a lower precision. The relevant expressions for U ijk and V ij have been written in paper I, and read while the other needed moments are given by The expressions (5.7)-(5.9) of the radiative moments are to be inserted into the gravitational luminosity (5.3). This leads to a natural (though not unique) decomposition of L into instantaneous and tail contributions, The contribution L inst depends only on the instant T R and is given by mi ε jkm + O(6) . (5.11) Note that this involves the term coming from the nonlinear memory which is instantaneous in the energy loss (the memory effect exists only in the waveform). The tail contribution depends on all instants T R − τ anterior to T R and reads where we have posed for simplicity The luminosity (5.10)-(5.13) in which the moments have been determined in Sect. IV is our final result for the general case of a (semi relatistic) isolated system.

VI. APPLICATION TO INSPIRALLING COMPACT BINARIES
The authors of Ref. [21] (paper II) applied the results of paper I to an inspiralling compact binary system modelled by two point-masses moving on a circular orbit. Here we do the same for the results derived previously and obtain the energy loss rate and associated laws of variation of the frequency and phase of the binary to 2.5PN order.

A. The equations of motion
Obviously the equations of motion of two point-masses at the 2.5PN approximation are needed for this application. These equations have been obtained in the same coordinates as used here by Damour and Deruelle [37] studying the dynamics of the binary pulsar. For inspiralling compact binaries one needs only to specialize these equations to the case of an orbit which is circular (apart from the gradual inspiral). A relevant summary of the Damour-Deruelle equations of motion is presented for the reader's convenience in Appendix B. Here we quote the results valid for circular orbits, following mostly the notation of paper II. Mass parameters are denoted by (the total mass is henceforth denoted by m to conform with paper II). The individual positions of the two bodies in harmonic coordinates are y i 1 and y i 2 . Their relative separation and relative velocity are A small ordering post-Newtonian parameter is defined to be γ = Gm rc 2 (6.3) with r = |x|. Next we use the fact that the origin of the coordinate system is located at the center of mass of the binary. This means that M i = 0 where M i is the dipole mass moment of the external field. By Eqs. (4.25) and (4.27b) this means I i +∆I i = O (6) where I i is the dipole moment which was computed in paper II (before expressing it in the relative frame) and where ∆I i = (4G/5c 5 )mG . For circular orbits one finds G i = mν(X 2 −X 1 )r 2 v i +O(2) and thus ∆I i = (4G/5c 5 )Gm 3 ν(X 1 −X 2 )(v i /r)+O (7).
This readily shows how the relations (3.7) of paper II are to be extended to 2.5PN order.
We find This result is in agreement with the 2.5PN-accurate center of mass theorem of Refs. [37].
The assumption that the orbit is circular apart from the adiabatic inspiral due to reaction where we have introduced the angular frequency ω 2PN defined by There are no terms of order γ 5/2 = O(5) for circular orbits because the term O(5) in Eq. (B11) is proportional to (nv) and thus vanishes in this case.

B. The energy loss rate
The gravitational luminosity L of a general source was split into two contributions, an instantaneous one L inst given by Eq. (5.11) and a tail one L tail given by Eq. (5.12). We shall basically show that only L tail contributes to the 2.5PN order in the case (but only in this case) of a binary system moving on a circular orbit.
Let us consider first the contribution L inst . The only moment it contains which is required with full 2.5PN accuracy is the mass quadrupole moment where I ij results from Eq. (4.8a) and ∆I ij is given by Eq. (4.27c). For a circular orbit, the moment of inertia K is constant [indeed K = mνr 2 + O(2)], hence ∆I ij reduces to two terms only: k<i Q j>k + O(7) . (6.10) We prove that the contribution in L inst which is due to ∆I ij is in fact zero. Indeed this contribution is made of the contracted product between Q ij and ∆I (3) ij (recall that M ij = Q ij + O(2)), hence of the contracted products Q jk . But in the circular case an odd number of time-derivatives of Q ij yields a term proportional to x <i v j> while an even number yields either x <i x j> or v <i v j> . Thus a contracted product of the type ij where n + p is odd necessarily involves one scalar product (xv) and is thus zero; similarly a product of the type Q jk where n + p + q is odd is also zero. The same is true of a product like ε ijk Q However we need the third time derivative of (6.12) which does involve, because of the more precise equations of motion, some new terms with respect to paper II. We find The quadrupole moment includes 1PN terms while the other moments are Newtonian. The computation of L tail involves many scalar products between x i or v i given at the current , with the same vectors but evaluated at an arbitrary . Relevant formulas for these scalar products are (xx ′ ) = r 2 cos(ωτ ) , (6.17a) (vv ′ ) = r 2 ω 2 cos(ωτ ) , (6.17b) where ω is the current value of the frequency. The reduction of L tail is quite straightforward.
We need to use ω 2 = (Gm/r 3 )[1 − (3 − ν)γ + O(γ 2 )] (the 1PN correction in ω 2 is sufficient) and the formula (6.15) where Ω = ω, 2ω or 3ω. One uses also (X 2 − X 1 ) 2 = 1 − 4ν. The result reads The complete 2.5PN-accurate gravitational luminosity generated by a inspiralling compact binary moving on a quasi-circular orbit is therefore obtained as the sum of Eqs. (6.14) and ( where we recall that the post-Newtonian ordering parameter is γ = Gm/rc 2 with r being the harmonic-coordinate orbital separation. This expression was already obtained to 1PN order in Refs. [23,47,48], to 1.5PN order in Refs. [13,30,46,49], and to 2PN order in Refs. [12,20,21,22]. The 2.5PN order added in this paper is like the 1.5PN order due to the presence of the radiation tails in the wave zone, as indicated by the irrational number π in factor coming from the formula (6.15). Hence there is up to 2.5PN order a clean separation between the integer post-Newtonian approximations which come from instantaneous relativistic effects in the source multipole moments and the half-integer approximations which are due to hereditary effects in the wave zone. However the next post-Newtonian approximation (3PN) is expected to involve both types of effects.

C. The orbital phase
Once the energy loss has been derived in a particular coordinate system it can be reexpressed in a coordinate-independent way by using the directly observable frequency ω ≡ ω 2PN instead of the parameter γ. Defining x = (Gmω/c 3 ) 2/3 , we obtain the inverse of Eq. (6.6) as The laws of variation of the frequency ω and phase φ of the binary during the orbital decay will be computed using the energy balance equation where T R is the retarded time in the wave zone. Note that we meet here the remarks made in the introduction. Up to now the analysis has been rigorous within the framework of post-Newtonian theory. Namely, L has been computed from a well-defined formalism based on convergent integrals for the multipole moments (see Section IV). The reduction of L to binary systems using delta functions to describe the compact objects can probably be justified at the 2.5PN order by using the results of Ref. [11] (see paper II). Furthermore E is computed directly from the Damour-Deruelle equations of motion [37]. We now postulate the validity of the energy balance equation (6.23) where both E and L take their full 2.5PN accuracy. This equation has been proved here (in Eq. (B9) of Appendix B) only when L takes its Newtonian value L N . It was proved before to Newtonian order in Refs. [32,33,34,35,36,37]. On the other hand the balance equation is also known to hold at 1PN order [38,39], and even for tails at 1.5PN order [30,40]. To prove it at 2.5PN order like in (6.23) would mean knowing all the radiation reaction effects in the binary's equations of motion up to the 5PN order.
To palliate this it has been customary in this field (see Refs. [7,8,9,13,14,18,19,21]) to compute the binary's orbital decay assuming that the energy balance equation is valid to high order (the angular momentum balance equation is unnecessary for circular orbits).
Adopting the same approach to the problem, we introduce like in paper II the adimensional time where Θ 0 denotes some constant.
The presence of the logarithm of the frequency or coalescing time in the phase is a novel feature of the 2.5PN approximation. As a result the phase φ tends formally to infinity at the time of coalescence. In other words, the number of cycles left till the time of coalescence starting from an initial frequency is always infinite. This is in contrast with the previous 2PN approximation where φ was tending to a finite value φ c . Of course this peculiarity is not relevant to the actual physics of the binary because the post-Newtonian approximation breaks down before the coalescence, at some finite value of the frequency. The fact that φ tends to infinity is merely a consequence of our idealized model assuming that the two bodies have no internal structure or horizons, which is valid only during the inspiral phase preceding the coalescence. However, during this phase the model is expected to be very precise and in particular the logarithmic 2.5PN term in the phase has to be taken numerically into account. This logarithm makes in fact the contribution of the 2.5PN term to the total number of cycles in the detector's bandwidth somewhat more important than what could be deduced from a simple order of magnitude estimate.
Let us be more precise and evaluate the contribution of each post-Newtonian terms in the phase (6.28) to the accumulated number N of gravitational-wave cycles between some initial and final frequencies ω i and ω f . Note that such a computation can only be indicative of the relative orders of magnitude of the different terms in the phase. A full analysis would require the knowledge of the power spectral density of the noise in a detector, and a complete simulation of the parameter estimation using matched filtering [7,8,9,50,51].
The contribution to N due to the 2.5PN term in (6.28) is In the case of the inspiral of two neutron stars of mass 1.4M ⊙ , and with the values ω i /π = 10Hz (set by seismic noise) and ω f /π = 1000Hz (set by photon shot noise), we find from Ref. [12] and Eq. (6.30) which permits the transformation of the left side of (4.19), namely into Because of the presence of the explicit factor B in the integrand, the equation (A3) appears to be the residue of the Laurent expansion of the retarded integral near B = 0 (we have discarded a term proportional to B 2 which is known to give zero contribution). Residues of retarded integrals have been investigated in Eq.
Here, [(p + q)/2] is the integer part of (p + q)/2,∂ L denotes a tracefree derivative operator composed of ℓ spatial derivatives where ℓ = p + q − 2k, and the number of time derivatives on F and G is a + b = 2k + 1. Omitted in (A4) are pure dimensionless coefficients and numerous Kronecker symbols. The explicit dependence on 1/c present in Eq. (A4) shows that if we are interested only in the leading-order term when c → +∞ we can limit us to the computation of the terms having the maximum number of space derivatives, i.e. ℓ max = p+q (corresponding to k = 0). The errors made in keeping only such terms are of relative order O(3). We use to compute (A3) the formulâ where the numerical coefficient is (see Eq. (A35a) in [27]). This yields a double sum of terms having in front the product of the tracefree tensorsn P andn Q . This product can be decomposed into tracefree tensors of multipolarities ℓ = p + q − 2k with k = 0, ..., [(p + q)/2], however if we are looking only for the term having ℓ max = p + q one can simply replace the productn PnQ by the tracefree tensorn P Q (with one as a coefficient in front). Hence where the remainder O(3) comes from the errors made in the replacementn PnQ →n P Q . The integrand in (A7) is easily transformed into a sum of terms of the form r B−Kn P Q H(t − r/c) times the factor B, where K is an integer. Using the fact that the residues of such terms exist only when K ≥ p + q + 3 (see Eq. (4.26) in [40]), one finds that the summation indices i and j in (A7) must satisfy i + j = p + q or p + q − 1. This simplifies much the computation of (A7) which is done using the expression (A6) of the coefficients a p i and the equation (4.26) in [40]. The result is P X Q =∂ P Q 1 rc δ p,0 F (1) G + δ 0,q F G (1) where δ p,0 denotes the usual Kronecker symbol.
where m 1 and m 2 are the two masses, v i and in the right side the expression It is simple to re-write the latter expression in the well-known form ij + O(7) .
As this is the standard (Newtonian) quadrupole formula we conclude that Eq. (B10) proves the energy balance between the loss of orbital energy E 2.5PN of the binary and the Newtonian energy flux carried out by the gravitational waves. Note that it is important that the energy which enters the left side of the balance equation is given as an instantaneous functional of the two world-lines of the binary (see Damour [11] for a discussion).