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Structure: struct Template

0 The structure used to describe the ``chirp" signals from coalescing binary systems is: struct Template {
int num: In order to deal with templates ``wholesale" it is useful to number them. The numbering system is up to you; we typically give each template a number, starting from 0 and going up to the number of templates minus one!
float f_lo: This is the starting (low) frequency $f_0$ of template, in units of ${\rm sec}^{-1}$.
float f_hi: This is the ending (high) frequency of the template, in units of ${\rm sec}^{-1}$
float tau0: The Newtonian time $\tau_0$ to coalescence, in seconds, starting from the moment when the frequency of the waveform is f_lo.
float tau1: First post-Newtonian correction $\tau_1$ to $\tau_0$.
float tau15: 3/2 PN correction
float tau20: second order PN correction
float pha0: Newtonian phase to coalescence, radians
float pha1: First post-Newtonian correction to pha0
float pha15: 3/2 PN correction
float pha20: second order PN correction
float mtotal: total mass $m_1+m_2$, in solar masses
float mchirp: chirp mass $\mu \eta^{-2/5}$, in solar masses
float mred: the reduced mass $\mu=m_1 m_2/(m_1+m_2)$, in solar masses
float eta: reduced mass/total mass $\eta=m_1 m_2/(m_1+m_2)^2$ , dimensionless
float m1: the smaller of the two masses, in solar masses.
float m2: the larger of the two masses, in solar masses.
};

One may use the technique of matched filtering to search for chirps. The (noisy) signal is compared with templates, each formed from a chirp with a particular values of $m_1$, $m_2$, and a ``start frequency" $f_0$ of the waveform at the time that it enters the bandpass of the gravitational wave detector. Several theoretical studies [4,5] have shown how the template filtering technique performs when the detector is not ideal, but is contaminated by instrument noise.

In the presence of detector noise, one can never be entirely certain that a given chirp (determined by $m_1,m_2$) will be detected by a particular template, even one with the exact same mass parameters. However one can make statistical statements about a template, such as ``if the masses $m_1$ and $m_2$ of the chirp lie in region $R$ of parameter space, then with 97% probability, they will be detected if their amplitude exceeds value $h$". Thus, associated with each chirp, and a specified level of uncertainty, is a region of parameter space.

It turns out that if we use the correct choice of coordinates on the parameter space $(m_1,m_2)$ then these regions $R$ are quite simple. If we demand that the uncertainty associated with each template be fairly small, then these regions are ellipses. Moreover, to a good approximation, the shape of the ellipses is determined only by the noise power spectrum of the detector, and does not change significantly as we move about in the parameter space. These ``nice" coordinates $(\tau_0,\tau_1)$ have units of time, and are defined by

$\displaystyle \tau_0$ $\textstyle =$ $\displaystyle {5 \over 256} \left({ G M \over c^3}\right)^{-5/3} \eta^{-1}
(\pi f_0)^{-8/3}$ (9.1.190)
  $\textstyle =$ $\displaystyle {5 \over 256} \left({ M \over M_\odot}\right)^{-5/3}\eta^{-1} (\pi
f_0)^{-8/3} T_\odot^{-5/3}$  

and
$\displaystyle \tau_1$ $\textstyle =$ $\displaystyle {5 \over 192} \left({c^3 \over G \eta M }\right) \left({743
\over 336} + {11 \over 4} \eta \right) (\pi f_0)^{-2}$ (9.1.191)
  $\textstyle =$ $\displaystyle {5 \over 192} \left( {M_\odot \over M }\right) \left({743 \over
336} \eta^{-1} + {11 \over 4} \right) (\pi f_0)^{-2} T_\odot^{-1}.$  

The symbol
\begin{displaymath}
M \equiv m_1+m_2
\end{displaymath} (9.1.192)

denotes the total mass of the binary system, and
\begin{displaymath}
\eta \equiv {m_1 m_2 \over (m_1 + m_2)^2}
\end{displaymath} (9.1.193)

is the ratio of the reduced mass to $M$. Notice that $\eta$ is always (by definition) less than or equal to $1/4$.

We are generally interested in a region of parameter space corresponding to binary systems, each of whose masses lie in some given range, say from $1/2$ to $3$ solar masses. The region of parameter space is determined by a minimum and maximum mass; we show an example of this in Figure [*].

Figure: The set of binary stars with masses lying between set minimum and maximum values defines the interior of a triangle in parameter space
\begin{figure}\begin{center}
\epsfig{file=Figures/figure1.ps,height=6cm,bbllx=72pt,bblly=160pt,
bburx=540pt,bbury=650pt}\end{center}\end{figure}

Since we may take $m_2 \le m_1$ without loss of generality, the region of interest is triangular rather than rectangular. The three lines on this diagram are:
(1) The equal mass line. Along this line $\eta=1/4$.
(2) The minimum mass line. Along this line, one of the masses has its smallest value.
(3) The maximum mass line. Along this line, one of the masses has its largest value.
This triangular region is mapped into the $(\tau_0,\tau_1)$ plane as shown in Figure [*] In this diagram, the lower curve $\tau_1 \propto \tau_0^{3/5}$ is the equal mass line (1). The upper curve, to the right of the ``kink" is the minimum mass line (2). The upper curve, to the left of the ``kink" is the maximum mass line (3).

Figure: The triangular region of the previous figure is mapped into a distorted triangle in the $(\tau_0,\tau_1)$ plane. Here $f_0$ is 120 Hz.
\begin{figure}\begin{center}
\epsfig{file=Figures/figure2.ps,height=5cm,bbllx=72pt,bblly=250pt,
bburx=540pt,bbury=540pt}\end{center}\end{figure}


next up previous contents
Next: Structure: struct Scope Up: GRASP Routines: Template Bank Previous: GRASP Routines: Template Bank   Contents
Bruce Allen 2000-11-19