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10.4 The two polarization wave-forms

The theoretical templates of the compact binary inspiral follow from insertion of the previous solutions for the 3.5PN-accurate orbital frequency and phase into the binary’s two polarization wave-forms h+ and h ×. We shall include in h+ and h× all the harmonics, besides the dominant one at twice the orbital frequency, up to the 2PN order, as they have been calculated by Blanchet, Iyer, Will and Wiseman [27].

The polarization wave-forms are defined with respect to two polarization vectors p = (pi) and q = (qi):

h = 1-(p p - q q)hTT , + 2 ij i j ij (173) 1- TT h × = 2 (piqj + pjqi)h ij ,
where p and q are chosen to lie along the major and minor axis, respectively, of the projection onto the plane of the sky of the circular orbit, with p oriented toward the ascending node N. To the 2PN order we have
{ ( )} 2Gmx-- (0) 1/2 (1/2) (1) 3/2 (3/2) 2 (2) 1- h+,× = c2R H +,× + x H +,× + xH +,× + x H +,× + x H +,×+ O c5 . (174)
The post-Newtonian terms are ordered by means of the frequency-related variable x. They depend on the binary’s 3.5PN-accurate phase f through the auxiliary phase variable
( ) 2Gmw--- -w- y = f - c3 ln w0 , (175)
where w0 is a constant frequency that can conveniently be chosen to be the entry frequency of a laser-interferometric detector (say w0/p = 10Hz). We have, for the plus polarization,
H(0) = - (1 + c2)cos2y, + i [ ] (1/2) sidm-- 2 2 H + = - 8 m (5 + ci) cosy - 9(1 + ci)cos 3y , (1) 1-[ 2 4 2 4] 4-2 2 H + = 6 19 + 9ci- 2ci- n(19 - 11ci- 6c i) cos2y - 3si(1 + ci)(1 - 3n) cos4y, (3/2) si dm { [ ] H + = -------- 57 + 60c2i - c4i- 2n(49 - 12c2i - c4i) cosy 192 m 27[ ] - ---73 + 40c2i- 9c4i - 2n(25 - 8c2i - 9c4i) cos 3y 2 } + 625(1 - 2n)s2i(1 + c2i)cos 5y - 2p(1 + c2i)cos2y, 2 [ (176) H(2) = --1- 22 + 396c2 + 145c4 - 5c6 + 5n(706 - 216c2- 251c4+ 15c6) + 120 i i i 3 i i i ] - 5n2(98 - 108c2 + 7c4+ 5c6) cos2y i i i [ ] -2- 2 2 4 5- 2 4 2 2 4 + 15 si 59 + 35ci - 8ci - 3n(131 + 59ci- 24ci) + 5n (21 - 3c i- 8ci) cos4y 81 2 4 2 - ---(1- 5n + 5n )si(1 + ci) cos6y 40 s dm { [ ] + -i----- 11 + 7c2i + 10(5 + c2i) ln 2 siny - 5p(5 + c2i)cos y 40 m [ ] } - 27 7 - 10ln(3/2) (1 + c2i)sin 3y + 135p(1 + c2i) cos3y ,
and, for the cross polarization,
H(0)= - 2c sin 2y, × i (1/2) 3- dm-- H × = - 4sicim [sin y - 3 sin 3y] , (1) ci[ 2 2 ] 8- 2 H× = 3 17 - 4ci- n(13 - 12ci) sin2y - 3(1 - 3n)cisi sin 4y, (3/2) sicidm { [ ] 27 [ ] H × = -------- 63 - 5c2i- 2n(23 - 5c2i) sin y - --- 67 - 15c2i - 2n(19 - 15c2i) sin 3y 96 m 2 625 } + ---(1 - 2n)s2i sin 5y - 4pcisin2y, (177) 2 [ ] H(2)= ci- 68 + 226c2 - 15c4 + 5n(572 - 490c2+ 45c4)- 5n2(56 - 70c2+ 15c4) sin2y × 60 i i 3 i i i i [ ] 4-- 2 2 5- 2 2 2 + 15cisi 55 - 12c i- 3 n(119 - 36ci) + 5n (17 - 12ci) sin 4y 81 2 4 - 20(1 - 5n + 5n )cisi sin6y 3 dm - --sici---{[3 + 10 ln 2]cosy + 5p sin y - 9 [7 - 10 ln(3/2)] cos3y - 45p sin 3y}. 20 m
We use the shorthands ci = cosi and si = sini for the cosine and sine of the inclination angle i between the direction of the detector as seen from the binary’s center-of-mass, and the normal to the orbital plane (we always suppose that the normal is right-handed with respect to the sense of motion, so that 0 < i < p).

To conclude, the use of theoretical templates based on the preceding 2PN wave forms, and having their frequency evolution built in via the 3.5PN phase evolution (171View Equation, 172View Equation), should yield some accurate detection and measurement of the binary signals. Interestingly, it should also permit some new tests of general relativity, because we have the possibility of checking that the observed signals do obey each of the terms of the phasing formulas (171View Equation, 172View Equation), notably those associated with the specific quadratic and cubic non-linear tails exactly as they are predicted by Einstein’s theory [2829]. Indeed, we don’t know of any other physical systems for which it would be possible to perform such tests.


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