Stationary phase approximation to binary inspiral chirps

Much of the literature on binary inspiral data analysis approximates chirps in the frequency domain by the method of stationary phase. The main reason for this approximation is the need to generate analytical expressions in the frequency domain, where almost all of the optimal filtering algorithm takes place. It is also in some sense more natural to generate waveforms in the frequency domain rather than the time domain because the post-Newtonian energy and flux functions used to construct even the time-domain waveforms are expanded in powers of the orbital frequency. A side benefit is that the post-Newtonian expansion seems better behaved in the frequency domain--that is, there is no nonmonotonic frequency evolution as depicted in figure .

Therefore, GRASP includes `sp_filters()`, a stationary phase
chirp generator similar to `make_filters()`. The advantage of
this function is a considerable savings in CPU time by avoiding FFTs
of time-domain chirps in the generation of matched filters. The
disadvantages are unknown--the question of which version of the
post-Newtonian expansion (time-domain or frequency-domain) is a better
approximation to the real thing is currently wide open.

The stationary phase approximation can be found in any textbook on mathematical methods in physics. An excellent discussion in the context of binary inspiral can be found in section II C of [21]. Another inspiral-related discussion can be found in [22], where it is shown that the errors induced by the stationary phase approximation itself [as opposed to differences between and ] are effectively fifth post-Newtonian order.

The stationary phase approximations to the Fourier transforms
of and
[Eqs. (,)]
are given in the restricted post-Newtonian approximation by

where is the gravitational wave frequency in Hz, is the total mass of the binary, and is the reduced mass. Note that have dimensions of 1/Hz. The instrument strain per Hz, , is obtained from a linear superposition of in exactly the same way as is obtained from . See the discussion following Eqs. (,).

The restricted post-Newtonian approximation assumes that the evolution
of the waveform amplitude is given by the 0'th-order post-Newtonian
expression, but that the phase evolution is accurate to higher order.
This phase is given by

where , the coalescence phase is determined by the binary ephemeris, and the coalescence time is the time at which the bodies collide. The chirps and are given and , respectively. All but the last term of () can be found in [23]; the last term was computed from [9] by Ben Owen and Alan Wiseman.

The chirps are set to zero for frequencies below the requested
starting frequency and above an upper cutoff (see below). This
square windowing in the frequency domain produces ringing at the
beginning and end of the waveform in the time domain (see
Fig. ). For data analysis purposes it appears
this ringing is not very important: it produces a mismatch (see
Sec. ) between waveforms generated by ` sp_filters()` and by `make_filters()` of a fraction of a
percent--*if* the stationary phase waveform is cut off at the
same frequency as the time-domain waveform.

The choice of the cutoff frequency is somewhat problematic.
Physically, should correspond to the epoch when orbital inspiral
turns to headlong plunge. The formula for currently is not
known for a pair of comparably massive objects, but in the limit of
extreme mass ratio (and no spins) it should be equivalent to the
well-known innermost stable circular orbit (ISCO) of Schwarzschild
geometry. The frequency of the Schwarzschild ISCO can be computed
exactly and is given in Hz by

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