### 10.4 The two polarization waveforms

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 waveforms and
. We shall include in and all the harmonics, besides the dominant one at twice the
orbital frequency, up to the 2.5PN order, as they have been calculated in Refs. [46, 4]. The
polarization waveforms 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 the ADM mass (cf. Equation (226)), and where
is a constant frequency that can conveniently be chosen to be the entry frequency
of a laser-interferometric detector (say ). For the plus polarization we
have
For the cross polarization, we have
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
). Finally, the more recent calculation of the 2.5PN order in Ref. [4] is reported here:
The practical implementation of the theoretical templates in the data analysis of detectors follows the
standard matched filtering technique. The raw output of the detector consists of the superposition of
the real gravitational wave signal and of noise . The noise is assumed to be a stationary
Gaussian random variable, with zero expectation value, and with (supposedly known) frequency-dependent
power spectral density . The experimenters construct the correlation between and a filter
, i.e.
and divide by the square root of its variance, or correlation noise. The expectation value of this ratio
defines the filtered signal-to-noise ratio (SNR). Looking for the useful signal in the detector’s
output , the experimenters adopt for the filter
where and are the Fourier transforms of and of the theoretically computed template
. By the matched filtering theorem, the filter (243) maximizes the SNR if . The
maximum SNR is then the best achievable with a linear filter. In practice, because of systematic
errors in the theoretical modelling, the template will not exactly match the real signal
, but if the template is to constitute a realistic representation of nature the errors will be
small. This is of course the motivation for computing high order post-Newtonian templates, in
order to reduce as much as possible the systematic errors due to the unknown post-Newtonian
remainder.
To conclude, the use of theoretical templates based on the preceding 2.5PN wave forms, and having their
frequency evolution built in via the 3.5PN phase evolution (234, 235), 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 (234, 235), e.g., those associated with the
specific non-linear tails, exactly as they are predicted by Einstein’s theory [47, 48, 5]. Indeed,
we don’t know of any other physical systems for which it would be possible to perform such
tests.