Performance of random template banks

When searching for new gravitational-wave or electromagnetic sources, the $n$ signal parameters (masses, sky location, frequencies, etc.) are unknown. In practice, one hunts for signals at a discrete set of points in parameter space, called a template bank. These may be constructed systematically as a lattice or, alternatively, by placing templates at randomly selected points in parameter space. Here, we calculate the fraction of signals lost by an $n$-dimensional random template bank (compared to a very finely spaced bank). This fraction is compared to the corresponding loss fraction for the best possible lattice-based template banks containing the same number of grid points. For dimensions $n<4$, the lattice-based template banks significantly outperform the random ones. However, remarkably, for dimensions $n>8$, the difference is negligible. In high dimensions, random template banks outperform the best known lattices.

Figure 1

The current record-holding (smallest $G$) lattice template banks (Ref. [23], Table 1) (blue points) lie above the conjectured Conway and Sloane [28] lower bound (cyan curve). The random template bank $G$ (orange) has its performance given by the Zador upper bound (3.8). The remaining colored curves show the well-known classical lattices $A_n$ and $D_n$ and their duals. For a fixed number of grid points, in dimensions $n>8$, a random template bank has a performance (detection loss) which is within 10% of the theoretically best possible template bank (see Table 1). In many higher dimensions (for example, 15 or 19), the random template bank outperforms any known lattice.

Figure 2

A random template bank for a two-dimensional parameter space, with an (area) template point density $\rho =100$. The upper bar has a length ${⟨r^2⟩}^{1/2}=(\pi \rho {)}^{-1/2}$, which is the root-mean-square (rms) distance to the closest template point. The middle of the parameter space has a region which is “thin” compared to this distance, with effective dimension 1. To obtain the same average mismatch $⟨r^2⟩$ as the two-dimensional part of parameter space, templates must be placed along this one-dimensional “line” with characteristic separation $1/{\rho }^{\prime }=(\pi \rho /2{)}^{-1/2}$ corresponding to the lower bar.

Figure 3

The fraction of signals which are lost by a random template bank as a function of the grid spacing $\mathrm{\Delta }$. These are computed using the spherical ansatz for the mismatch, for a $D=3-\mathrm{dimensional}$ source distribution; the curves show parameter-space dimensions $n=2,4,\dots ,10$. The dashed lines show the quadratic approximation for the mismatch Eq. (6.2), which is accurate at small grid spacings.