Gravitational waves from inspiralling compact binaries: Hexagonal template placement and its efficiency in detecting physical signals

Matched filtering is used to search for gravitational waves emitted by inspiralling compact binaries in data from the ground-based interferometers. One of the key aspects of the detection process is the design of a template bank that covers the astrophysically pertinent parameter space. In an earlier paper, we described a template bank that is based on a square lattice. Although robust, we showed that the square placement is overefficient, with the implication that it is computationally more demanding than required. In this paper, we present a template bank based on an hexagonal lattice, which size is reduced by 40% with respect to the proposed square placement. We describe the practical aspects of the hexagonal template bank implementation, its size, and computational cost. We have also performed exhaustive simulations to characterize its efficiency and safeness. We show that the bank is adequate to search for a wide variety of binary systems (primordial black holes, neutron stars, and stellar-mass black holes) and in data from both current detectors (initial LIGO, Virgo and GEO600) as well as future detectors (advanced LIGO and EGO). Remarkably, although our template bank placement uses a metric arising from a particular template family, namely, stationary phase approximation, we show that it can be used successfully with other template families (e.g., Padé resummation and effective one-body approximation). This quality of being effective for different template families makes the proposed bank suitable for a search that would use several of them in parallel (e.g., in a binary black hole search). The hexagonal template bank described in this paper is currently used to search for nonspinning inspiralling compact binaries in data from the Laser Interferometer Gravitational-Wave Observatory (LIGO).

Figure 1

Example of parameter space and hexagonal template bank placement. The parameter space is defined by the individual mass components (from $3M_{\odot }$ to $30M_{\odot }$) and the lower cutoff frequency ($f_L=40\mathrm{Hz}$). The bottom line corresponds to $m_1=m_2$ (i.e., $\eta =0.25$). The two other boundaries meet where $m_1=m_{\min }$ and $m_2=m_{\max }$. The bottom left point of the parameter space corresponds to $m_1=m_2=m_{\max }$ whereas the top right point corresponds to $m_1=m_2=m_{\min }$. The circles give the position of each template that is needed to cover the entire parameter space (black curves). Even though some templates lie outside the parameter space boundaries, these are required to fully cover the parameter space.

Figure 2

Two instances of template bank placements. In the two plots, we focus on a small area of the parameter space presented in Fig. 1. We used a square (top panel) and hexagonal (bottom panel) placement. For convenience, we rescale the metric components so that, $g_{00}\sim g_{11}$. Each template position is represented by a small circle. Around each template position, we plot an ellipse that represents an isomatch contour of $MM=0.95\%$. Each ellipse contains an inscribed square or hexagon which emphasizes how ellipses overlap each other. We can see that squares (top) slightly overlap each other. This is because templates are laid along the ${\tau }_3$ equal constant line and not along the eigenvector directions, which change over the parameter space. In the hexagonal placement, we take care of this problem shortcoming, and therefore hexagons are perfectly adjacent to each other: the placement is optimal.

Figure 3

Efficiencies of the square template bank. For convenience we remind the reader of some results of the square template bank provided in 12. In the simulations, we used stationary phase approximant models for both injections and templates. Injections consist of binary neutron stars. We used 4 design sensitivity curves (LIGO-I, advanced LIGO, VIRGO and GEO), and for each of them we performed 10 000 injections. In the top panel, we show all the results together: all injections are recovered with a match higher than 95%, as requested. In the bottom panel, we decomposed the 4 simulations and show that all of them behave similarly. Most of the injections are recovered with even higher matches (above 97%) showing the overefficiency of the placement.

Figure 4

Hexagonal template bank placement. Using the terminology that is introduced in the text, we can describe the template bank placement algorithm as follows. First, in subfigure (a), a cell/template is arbitrary placed in the parameter space. Its coordinates correspond to $m_1=m_{\min }$ and $m_2=m_{\max }$, and its ID is 1. In subfigure (b), the cell with $\mathrm{ID}=1$, which is also a mother cell, reproduces 6 new cells according to an optimal hexagonal lattice that takes into account its metric components. In subfigure (c) the connection between the offsprings and the mother cell are created (solid arrows). Cells that belong to the same generation (white ellipses) are also connected if they are adjacent to each other (dashed arrows). In subfigure (d), the new cells can start to reproduce. However, the reproduction is exclusive: reproduction takes place cell after cell, and the cell with lowest ID is chosen to reproduce first. In subfigure (e), therefore, the cell with ID equals 2 starts to reproduce. Because connections already exist with other cells, this cell will reproduce only 3 directions (i.e., 8, 9 and 10). In subfigure (f), the cell with ID equals 3 starts to populate. The 3 new cells (IDs 11,12 and 13) created by the cell with $\mathrm{ID}=2$ have to wait for the current generation (IDs 3 to 7) to fully reproduce. The cells spread until the boundary of the parameter space is reached.

Figure 5

Cumulative efficiencies of the hexagonal template bank. Both template and signal are based on TaylorF2 model. From top left to bottom right (clockwise), injection and template bank cover the PBH binary, BNS, BHNS, and BBH inspiralling compact binaries.

Figure 6

Efficiencies of the hexagonal template bank. Both template and signal are based on TaylorF2 model. From top left to bottom right (clockwise), injection and template bank cover the PBH binary, BNS, BHNS, and BBH inspiralling compact binaries.

Figure 7

Hexagonal template bank efficiencies using TaylorT1 model. From left to right, results of the BNS, BHNS, and BBH injections.

Figure 8

Hexagonal template bank efficiencies using TaylorT3 model. From left to right, results of the BNS, BHNS, and BBH injections.

Figure 9

Hexagonal template bank efficiencies using PadeT1 model. From left to right, results of the BNS, BHNS, and BBH injections.

Figure 10

Hexagonal template bank efficiencies using EOB model. From left to right, results of the BNS, BHNS, and BBH injections.

Figure 11

Flow chart of the hexagonal placement algorithm. See the text for detailed description of the initialization, reproduction, and connection process.