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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 (97View Equation, 98View Equation). 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 (97View Equation), 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
L = Linst + Ltail + L(tail)2 + Ltail(tail), (225)
where Linst has just been found in Equation (224View Equation); Ltail is made of the quadratic (multipolar) tail integrals in Equation (98View Equation); L(tail)2 is the square of the quadrupole tail in Equation (97View Equation); and Ltail(tail) is the quadrupole tail of tail in Equation (97View Equation). We find that Ltail contributes at the half-integer 1.5PN, 2.5PN, and 3.5PN orders, while both L(tail)2 and Ltail(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 (97View Equation, 98View Equation), 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. [19Jump To The Next Citation Point]), let us point out that following the general formalism of Part A, the total mass M in front of the tail integrals is the ADM mass of the binary which is given by the sum of the rest masses, m = m1 + m2 (which is the one appearing in the g-parameter, Equation (188View Equation)), plus some relativistic corrections. At the 2PN relative order needed here to compute the tail integrals we have

[ ( )] M = m 1 - n-g + n-(7- n)g2 + O -1 . (226) 2 8 c6
Let us give the two basic technical formulas needed when carrying out this reduction:
integral + oo -st 1- dt lnt e = - s (C + ln s), 0 (227) integral + oo 1 [p2 ] dt ln2t e-st = -- ---+ (C + ln s)2 , 0 s 6
where s (- C and C = 0.577 ... 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 contributions39
{ ( ) ( ) 32c5 5 2 3/2 25663 125 5/2 90205 505747 12809 2 7/2 Ltail = ----g n 4pg + - ------- ----n pg + ------+ -------n + ------n pg 5G ( )} 672 8 576 1512 756 1- + O c8 . (228)
For the sum of squared tails and cubic tails of tails at 3PN, we get
5 { ( ( ) ) L 2 = 32c-g5n2 - 116761-+ 16-p2 - 1712-C + 1712-ln r12 - 856-ln (16g) g3 (tail)+tail(tail) 5G 3675 3 105 105 r0 105 ( )} +O 1- . (229) c8
By comparing Equations (224View Equation) and (229View Equation) we observe that the constants r 0 cleanly cancel out. Adding together all these contributions we obtain
{ ( ) ( ) ( ) 32c5 5 2 2927 5 3/2 293383 380 2 25663 125 5/2 L = ----g n 1 + - ------ -n g + 4pg + ------- + ----n g + - ------- ----n pg 5G [ 336 4 9072 9 672 8 129386791-- 16p2- 1712- 856- + 7761600 + 3 - 105 C - 105 ln(16g) ( ( ) 2) ] + - 50625- + 110-ln r12 + 123p-- n - 383-n2 g3 112 3 r'0 64 9 (90205 505747 12809 ) ( 1)} + ------+ -------n + ------n2 pg7/2 + O -8 . (230) 576 1512 756 c
The gauge constant r' 0 has not yet disappeared because the post-Newtonian expansion is still parametrized by g instead of the frequency-related parameter x defined by Equation (192View Equation) - just as for E when it was given by Equation (191View Equation). After substituting the expression g(x) given by Equation (193View Equation), we find that r'0 does cancel as well. Because the relation g(x) 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:
32c5 { ( 1247 35 ) ( 44711 9271 65 ) L = ----n2x5 1 + - ------ ---n x + 4px3/2 + - ------+ ----n + --n2 x2 5G ( 336 12 ) 9072 504 18 8191 583 5/2 + - ----- - ----n px [ 672 24 6643739519-- 16-2 1712- 856- + 69854400 + 3 p - 105 C - 105 ln(16 x) ( ) ] + - 134543- + 41-p2 n - 94403-n2 - 775-n3 x3 7776 48 3024 324 ( 16285 214745 193385 ) ( 1 )} + - ------ + -------n + -------n2 px7/2 + O -8 . (231) 504 1728 3024 c
In the test-mass limit n --> 0 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 6643739519/69854400 comes out exactly the same as in black-hole perturbations. On the other hand, the ambiguity parameters c and h are part of the rational fraction - 134543/7776, belonging to the coefficient of the term at 3PN order proportional to n (hence this coefficient cannot be computed by linear black hole perturbations)40.
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