Template Placement for Binary Inspiral Searches

Template placement is a problem of populating the binary parameter space (masses, spins, etc.) with as small a number of templates as possible, subject to the constraint that every signal that lies within the space has an overlap greater than or equal to a certain minimal match with at least one template in the grid. Past studies [1,2,3] have shown that this is most easily achieved by using the space of chirp-times to lay templates rather than the space of component masses.

Figure: The parameter space binary masses and corresponding chirp-times. Chirp times are computed using an $f_0$ of 40 Hz. Note that our parameter is specified by and $M=m_1+m_2 < M_{\rm max}$ rather than by $m_{\rm min} < m_1,m_2 < m_{\rm max}.$ (We use capital $M$ to denote the total mass and lower-case $m$ to denote the component masses.) In the above example $m_{\rm min} = 0.2 M_\odot$ and $M_{\rm max} = 100 M_\odot.$

The number of chirp times that one can define is determined by the post-Newtonian order one is working with. At the second post-Newtonian order (i.e. an approximation accurate to order^38.1 $v^4$) and for a binary consisting of two non-spinning compact objects in a quasi-circular orbit, there are four chirp-times $\tau_k,$ $k=0, 2, 3, 4,$ of which we can choose any two to characterize the binary:

$$\tau_{0} = \frac{5M}{256 \eta v_{0}^{8}}$$ (38.2)

$$\tau_{2} = \frac{5M}{192 \eta v_{0}^{6}} \left( \frac{743}{336} + \frac{11}{4} \eta \right)$$ (38.3)

$$\tau_{3} = \frac{\pi M}{8 \eta v_{0}^{5}}$$ (38.4)

$$\tau_{4} = \frac{5M}{128 \eta v_{0}^4} \left( \frac{3 058 ... ...4} + \frac{5429}{1008} \eta + \frac{617}{144} \eta^{2} \right)$$ (38.5)

where $m$ is the total mass of the binary, $\eta=m_1 m_2/m^2$ is the symmetric mass ratio and $v_0 = (\pi m f_0)^{1/3}$ is a fiducial velocity parameter corresponding to a fiducial frequency $f_0,$ usually chosen to be the lower frequency cutoff of the detector sensitivity.

This algorithm allows one to choose a coarse grid of templates either in the $\tau_0$-$\tau_2$ or $\tau_0$-$\tau_3$ space depending on the value of the enum CoordinateSpace, which can take one of two values: Tau0Tau2 or Tau0Tau3. The shape of the coordinate spaces for some interesting range of masses is shown in Fig. . The important point to note in these figures is that the $\eta=1/4$ curve spans from the minimum to the maximum value of the Newtonian chirp-time $\tau_0.$ This feature will be used in the construction of the grid. Note that the minimum (maximum) value of the Newtonian chirp-time occurs when the two masses are equal and the total mass is a maximum (minimum).