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Stationary phase approximation to binary inspiral chirps

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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 $t(f)$ and $f(t)$] are effectively fifth post-Newtonian order.

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

$\displaystyle \tilde{h}_c(f)$ $\textstyle =$ $\displaystyle \left(\frac{5\mu}{96M_\odot}\right)^{1/2}
\left(\frac{M}{\pi^2M_\odot}\right)^{1/3}f^{-7/6}T_\odot^{-1/6}
\exp\,[i\Psi(f)],$ (6.11.55)
$\displaystyle \tilde{h}_s(f)$ $\textstyle =$ $\displaystyle i\tilde{h}_c(f),$ (6.11.56)

where $f$ is the gravitational wave frequency in Hz, $M$ is the total mass of the binary, and $\mu$ is the reduced mass. Note that $\tilde{h}_{c,s}(f)$ have dimensions of 1/Hz. The instrument strain per Hz, $\tilde{h}(f)$, is obtained from a linear superposition of $\tilde{h}_{c,s}(f)$ in exactly the same way as $h(t)$ is obtained from $h_{c,s}(t)$. 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

$\displaystyle \Psi(f)$ $\textstyle =$ $\displaystyle 2\pi ft_c-2\phi_c-\pi/4$  
    $\displaystyle +\frac{3}{128\eta}\biggl[x^{-5}+
\left(\frac{3715}{756}+\frac{55}{9}\eta\right)x^{-3}
-16\pi x^{-2}$  
    $\displaystyle +\left(\frac{15\,293\,365}{508\,032}+\frac{27\,145}{504}\eta
+\frac{3085}{72}\eta^2\right)x^{-1}$  
    $\displaystyle +\left(\frac{38\,645}{252}+5\eta\right)\pi\ln{x}\biggr],$ (6.11.57)

where $x=(\pi MfT_\odot/M_\odot)^{1/3}$, the coalescence phase $\phi_c$ is determined by the binary ephemeris, and the coalescence time $t_c$ is the time at which the bodies collide. The chirps $\tilde{h}_c$ and $\tilde{h}_s$ are given $\phi_c=0$ and $\phi_c=-\pi/4$, 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 $f_c$ (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 $f_c$ as the time-domain waveform.

The choice of the cutoff frequency $f_c$ is somewhat problematic. Physically, $f_c$ should correspond to the epoch when orbital inspiral turns to headlong plunge. The formula for $f_c$ 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

\begin{displaymath}
f_c=\frac{M_\odot}{6^{3/2}\pi MT_\odot}.
\end{displaymath} (6.11.58)

Use of the Schwarzschild $f_c$ for all binaries is a kludge which seems to work surprisingly well in the sense that it yields an $f_c$ close to that at which the time-domain waveforms cut off due to $df/dt$ going negative (see the compare_chirps program).


next up previous contents
Next: Function: sp_filters() Up: GRASP Routines: Gravitational Radiation Previous: Function: make_filters()   Contents
Bruce Allen 2000-11-19