### 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 and
. We shall include in and 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 and
:

where and 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 oriented toward the ascending node . To the 2PN order
we have
The post-Newtonian terms are ordered by means of the frequency-related variable . They depend on the
binary’s 3.5PN-accurate phase through the auxiliary phase variable
where is a constant frequency that can conveniently be chosen to be the entry frequency
of a laser-interferometric detector (say Hz). We have, for the plus polarization,
and, for the cross polarization,
We use the shorthands and for the cosine and sine of the inclination angle
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
).
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 (171, 172), 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 (171, 172), notably those associated with the specific quadratic
and cubic non-linear tails exactly as they are predicted by Einstein’s theory [28, 29]. Indeed,
we don’t know of any other physical systems for which it would be possible to perform such
tests.