Efficient gravitational wave template bank generation with differentiable waveforms

The most sensitive search pipelines for gravitational waves from compact binary mergers use matched filters to extract signals from the noisy data stream coming from gravitational wave detectors. Matched-filter searches require banks of template waveforms covering the physical parameter space of the binary system. Unfortunately, template bank construction can be a time-consuming task. Here we present a new method for efficiently generating template banks that utilizes automatic differentiation to calculate the parameter space metric. Principally, we demonstrate that automatic differentiation enables accurate computation of the metric for waveforms currently used in search pipelines, whilst being computationally cheap. Additionally, by combining random template placement and a Monte Carlo method for evaluating the fraction of the parameter space that is currently covered, we show that search-ready template banks for frequency-domain waveforms can be rapidly generated. Finally, we argue that differentiable waveforms offer a pathway to accelerating stochastic placement algorithms. We implement all our methods into an easy-to-use python package based on the jax framework, diffbank, to allow the community to easily take advantage of differentiable waveforms for future searches.

Figure 1

Comparison of the metric computed using automatic differentiation with an analytic reference metric [50], parametrized in terms of dimensionless chirp times [45]. Both were computed using 64-bit precision floating point numbers, the precision of which is about 15 significant digits. The noise model is described in the text, and the frequency grid spans 10–512 Hz with a spacing of 0.1 Hz. The colored regions are a Voronoi tessellation.

Figure 2

The scalar curvature for the waveform from Ref. [50], parametrized in terms of dimensionless chirp times [45]. The configuration for the metric calculation is described in the caption for Fig. 1.

Figure 3

A schematic illustration of our template bank generation procedure. The bank generation starts with $n_{\mathrm{eff}}$ live effectualness points which are sampled according to the metric density $p(\mathit{\theta })\propto \sqrt{g(\mathit{\theta })}$ (illustrated by the filled red stars in the left panel). The solid (green) curves in the shape of a distorted triangle illustrate the boundary of the parameter space. Templates (illustrated by the blue points with accompanying purple ellipses with radial scale $\sqrt{{\mathrm{m}}_{\mathrm{mis},*}}$) are then added sequentially where the template’s position is also sampled from the metric density. Each time a template is added, the match is calculated for all live effectualness points. If the match is greater than $1-{\mathrm{m}}_{\mathrm{mis},*}$ for some effectualness point, then that point is indeed successfully described by the new template; that effectualness point “dies” (illustrated by the empty stars in the central panel which are covered by at least one template). This sampling continues until $\lceil \eta n_{\mathrm{eff}}\rceil$ effectualness points are covered (illustrated in the right panel for $\eta =0.9$ and $n_{\mathrm{eff}}=10$). Hence the choice of $\eta$ determines our stopping criterion. Note here we have taken the orientation and size of the template ellipses to be constant. In general both will vary over the parameter space, further complicating bank generation.

Figure 4

Left panel: covering fraction $\eta$ of random banks as a function of the number of effectualness points $n_{\mathrm{eff}}$ using the model configuration explained in Sec. 3b. The blue points indicate estimates of $\widehat{\eta }$ calculated using an additional set of randomly injected points (i.e., not the effectualness points used to generate the bank). The gray bands show the one and two ${\sigma }_{\widehat{\eta }}$ error bands on $\widehat{\eta }$. Note that for $n_{\mathrm{eff}}\sim \mathcal{O}(10)$, the error extends to the unphysical region $\eta \ge 1$. This could be corrected by using an asymmetric estimate of ${\sigma }_{\widehat{\eta }}$ such as the Jeffreys interval [56]). Right panel: the number of templates in the bank required to meet the stopping criterion as a function of $n_{\mathrm{eff}}$. For comparison, we also plot the number of templates in a stochastic bank (orange line) which was generated with $\eta =0.9$. Note that in this example we find that $N_T^{\mathcal{R}}>n_{\mathrm{eff}}$ for $n_{\mathrm{eff}}\sim \mathcal{O}({10}^4)$. In this case, the scaling estimates discussed in Sec. 3b will be inaccurate since a single template is likely to cover multiple effectualness points.

Figure 5

Scaling of bank size and cost of generation as a function of minimum match and target covering fraction. The analytic estimates (orange points) come from Eq. (15) (top row) and Eqs. (16) and (17) (bottom row). Note that our analytic estimates start to differ from our real banks for $(1-{\mathrm{m}}_{\mathrm{mis},*})\to 1$ and $\eta \to 1$. This is due to the growing importance of a spatially dependent $q$ (see text for more details). In the bottom row, one can see that the cost of generating a stochastic bank is significantly greater than a random bank. This difference continues to grow with larger template banks. All plots are based on the two-dimensional 3.5PN model and analytic model for the noise power spectral density described in the text.

Figure 6

Cumulative distribution function for effectualnesses of each of the three different template banks as labeled in the legend. In each case the effectualnesses were computed at 1000 points sampled from the metric density. The vertical lines show the target values of $1-{\mathrm{m}}_{\mathrm{mis},*}$ for each bank; note that the blue and orange lines overlap. The bands show the CDF corresponding to the target value of $\eta$ plus and minus $2{\sigma }_{\widehat{\eta }}$, Eq. (13); the blue and green bands overlap.