Go to previous page Go up Go to next page

10.2 Contribution of wave tails

To the “instantaneous” part of the flux, we must add the contribution of non-linear multipole interactions contained in the relationship between the source and radiative moments. The needed material has already been provided in Eqs. (91View Equation, 92View 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 Eq. (91View 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), (163)
where L inst has just been found in Eq. (162View Equation); L tail is made of the quadratic (multipolar) tail integrals in Eq. (92View Equation); L(tail)2 is the square of the quadrupole tail in Eq. (91View Equation); and Ltail(tail) is the quadrupole tail of tail in Eq. (91View 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 (91View Equation, 92View Equation). Without going into accessory details (see Ref. [10]), let us give the two basic technical formulas needed when carrying out this reduction:

integral + oo 1 0 dt ln t e-st = - --(C + lns), s integral [ 2 ] (164) + oo dt ln2t e-st = 1 p--+ (C + lns)2 , 0 s 6
where s (- C and C = 0.577 ... denotes the Euler constant [78]. The tail integrals are evaluated thanks to these formulas for a fixed (non-decaying) circular orbit. Indeed it can be shown [31] that the “remote-past” contribution to the tail integrals is negligible; the errors due to the fact that the orbit actually spirals in by gravitational radiation do not affect the signal before the 4PN order. We then find, for the quadratic tail term stricto sensu, the 1.5PN, 2.5PN and 3.5PN amounts UpdateJump To The Next Update Information
{ ( ) ( ) 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 . (165)
For the sum of squared tails and cubic tails of tails at 3PN, we get
{ ( ( ) ) 32c5 5 2 116761 16 2 1712 1712 r12 856 3 L(tail)2+tail(tail) = ----g n - -------+ ---p - -----C + -----ln --- - ----ln (16g) g 5G 36(75 )} 3 105 105 r0 105 1- +O c8 . (166)
By comparing Eqs. (162View Equation) and (166View Equation) we observe that the constants r0 cleanly cancel out. Adding together all these contributions we obtain UpdateJump To The Next Update Information
32c5 { ( 2927 5 ) (293383 380 ) ( 25663 125 ) L = ----g5n2 1 + - ------ -n g + 4pg3/2 + ------- + ----n g2 + - ------- ----n pg5/2 5G [ 336 4 9072 9 672 8 129386791 16p2 1712 856 + -----------+ ------ -----C - ----ln(16g) 776(1600 3 10(5 ) 105 ) ] 332051- 110- r12- 123p2- 88- 383- 2 3 + - 720 + 3 ln r'0 + 64 + 44c - 3 h n - 9 n g ( ) ( )} + 90205-+ 505747-n + 12809-n2 pg7/2 + O -1 . (167) 576 1512 756 c8
The gauge constant ' r0 has not yet disappeared because the post-Newtonian expansion is still parametrized by g instead of the frequency-related parameter x defined by Eq. (150View Equation) - just as for E when it was given by Eq. (149View Equation). After substituting the expression g(x) given by Eq. (151View 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: UpdateJump To The Next Update Information
5 { ( ) ( ) L = 32c-n2x5 1 + - 1247- - 35-n x + 4px3/2 + - 44711-+ 9271-n + 65-n2 x2 5G 336 12 9072 504 18 ( 8191 583 ) + - ----- - ----n px5/2 [ 672 24 6643739519-- 16-2 1712- 856- + 69854400 + 3 p - 105 C - 105 ln(16 x) ( ) ] + - 11497453- + 41-p2 + 176-c - 88-h n- 94403-n2- 775-n3 x3 272160 48 9 3 3024 324 ( ) ( )} + - 16285- + 214745-n + 193385-n2 px7/2 + O 1- . (168) 504 1728 3024 c8
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 [137]. In particular, the rational fraction 6643739519/69854400 comes out exactly the same as in black-hole perturbations.
  Go to previous page Go up Go to next page