Fully coherent follow-up of continuous gravitational-wave candidates

The search for continuous gravitational waves from unknown isolated sources is computationally limited due to the enormous parameter space that needs to be covered and the weakness of the expected signals. Therefore, semicoherent search strategies have been developed and applied in distributed computing environments such as Einstein@Home, in order to narrow down the parameter space and identify interesting candidates. However, in order to optimally confirm or dismiss a candidate as a possible gravitational wave signal, a fully coherent follow-up using all the available data is required. We present a general method and implementation of a direct (two-stage) transition to a fully coherent follow-up on semicoherent candidates. This method is based on a gridless Mesh Adaptive Direct Search algorithm using the $\mathcal{F}$-statistic. We demonstrate the detection power and computing cost of this follow-up procedure using extensive Monte Carlo simulations on (simulated) semicoherent candidates from a directed, as well as from an all-sky search setup.

Figure 1

Two-dimensional search grid in $\{f,\dot{f}\}$ space. The black dots are the search templates, placed such that the loss of SNR on any putative signal ${\lambda }_s$ will be bounded by a maximal mismatch $m$, which defines the semicoherent isomismatch ellipses. The semicoherent Fisher ellipse centered on the MLE ${\widehat{\lambda }}_{\mathrm{MLE}}$ is used to constrain the zoom parameter space. The aim of the zoom stage is to find ${\tilde{\lambda }}_{\mathrm{MLE}}$.

Figure 2

Two-dimensional example: Fisher ellipse [Eq. (25)] defining the zoom search space, centered on the semicoherent MLE ${\widehat{\lambda }}_{\mathrm{MLE}}$. The extents $\{\Delta f,\Delta \dot{f}\}$ of the bounding box are given by Eq. (27).

Figure 3

MADS-based search algorithm with four passes, where ${\lambda }_0={\lambda }_c$ in the refinement stage and ${\lambda }_0={\widehat{\lambda }}_{\mathrm{MLE}}$ in the zoom stage.

Figure 4

Monte Carlo study of two-stage follow-up of candidates from a directed $\{f,\dot{f}\}$ semicoherent search pointed toward the Galactic center with $N=200$ segments of duration $\Delta T=1\mathrm{d}$. (a) SNR loss of the initial candidates $\check{\mu }$ versus semicoherent metric mismatch $\mu *$ to the closest template. (b) ${\widetilde{2\mathcal{F}}}_Z$ distribution after the fully coherent 2D $\{f,\dot{f}\}$ zoom stage of 5000 directed searches in pure Gaussian noise without injected signal. The maximal $2\mathcal{F}$ value found is ${\widetilde{2\mathcal{F}}}_Z^{\max }=51.61$. The mean value $⟨{\widetilde{2\mathcal{F}}}_Z⟩=29.00$ is plotted with a dotted line. (c) Percentage of the 5000 injected signals classified as recovered ($-S$) and of non-Gaussian origin ($\times \neg G$) as function of the noncentrality parameter ${\overline{{\rho }^2}}_s$, Eq. (7). The error bars are computed by using a Jackknife estimator. (d) Upper plot: computing cost of the semicoherent refinement stage. Middle plot: computing cost of the fully coherent zoom stage. Lower plot: total computing cost.

Figure 5

Monte Carlo study of two-stage follow-up of candidates from an all-sky $\{\alpha ,\delta ,f,\dot{f}\}$ semicoherent search with $N=200$ segments of duration $\Delta T=1\mathrm{d}$. (a) SNR loss of the initial candidates $\check{\mu }$ versus semicoherent metric mismatch $\mu *$. (b) ${\widetilde{2\mathcal{F}}}_Z$ distribution after the fully coherent 4D $\{\alpha ,\delta ,f,\dot{f}\}$ zoom stage of 7500 searches in pure Gaussian noise, without injected signal. The maximal $2\mathcal{F}$ value found is ${\widetilde{2\mathcal{F}}}_Z^{\max }=58.76$. The mean value $⟨{\widetilde{2\mathcal{F}}}_Z⟩=37.50$ indicated with dots. (c) Percentage of the 7500 injected signals classified as recovered ($-S$) and of non-Gaussian origin ($\times \neg G$) as function of the signal strength ${\overline{{\rho }^2}}_s$. The error bars are computed using a Jackknife estimator. (d) Upper plot: computing cost of the semicoherent refinement stage. Middle plot: computing cost of the fully coherent zoom stage. Lower plot: total computing cost.