Search for continuous gravitational waves: Optimal StackSlide method at fixed computing cost

Coherent wide parameter-space searches for continuous gravitational waves are typically limited in sensitivity by their prohibitive computing cost. Therefore, semicoherent methods (such as StackSlide) can often achieve a better sensitivity. We develop an analytical method for finding optimal StackSlide parameters at fixed computing cost under ideal conditions of gapless data with Gaussian stationary noise. This solution separates two regimes: an unbounded regime, where it is always optimal to use all the data, and a bounded regime with a finite optimal observation time. Our analysis of the sensitivity scaling reveals that both the fine- and coarse-grid mismatches contribute equally to the average StackSlide mismatch, an effect that had been overlooked in previous studies. We discuss various practical examples for the application of this optimization framework, illustrating the potential gains in sensitivity compared to previous searches.

Figure 1

Mismatch triangle formed by the signal point ${\lambda }_{\mathrm{s}}$, closest fine-grid template $\widehat{\lambda }$, and the coarse-grid template ${\tilde{\lambda }}_k(\widehat{\lambda })$ closest to $\widehat{\lambda }$ in segment $k$.

Figure 2

$N$-scaling coefficient $w$ defined in Eq. (40) as a function of $N$, for false-dismissal probability $p_{\mathrm{fD}}^{*}=0.1$, and different false-alarm probabilities $p_{\mathrm{fA}}^{*}\in [{10}^{-10},{10}^{-5},{10}^{-2}]$. Solid lines show the scaling obtained from the exact solution Eq. (27), while dashed lines refer to the Gauss approximation of Eq. (36). The WSG approximation corresponds to $w=1$.

Figure 3

Numerical optimal fixed-$T$ solution for a directed Cas-A search as a function of $T$. The dashed vertical line indicates the analytical WSG-optimal solution of Eq. (118), while the dotted vertical line corresponds to the exact optimal solution. Panel (a) shows the weakest detectable signal strength compared to the reference value $h_{\mathrm{th},\mathrm{ref}}$ of Eq. (114), for the exact $h_{\mathrm{th}}$ and for the WSG-approximated $h_{\mathrm{th}}^{\mathrm{WSG}}$ (using $w=1$). (b) shows the optimal mismatch parameters ${\tilde{m}}_{\mathrm{opt}}(T)$ and ${\widehat{m}}_{\mathrm{opt}}(T)$, (c) shows the optimal computing-cost ratio ${\varkappa }_{\mathrm{opt}}(T)$, and (d) the optimal number of segments $N_{\mathrm{opt}}(T)$ (treated as continuous).