An improved, “phase-relaxed” $\mathcal{F}$-statistic for gravitational-wave data analysis

Rapidly rotating, slightly nonaxisymmetric neutron stars emit nearly periodic gravitational waves (GWs), quite possibly at levels detectable by ground-based GW interferometers. We refer to these sources as “GW pulsars.” For any given sky position and frequency evolution, the $\mathcal{F}$-statistic is the maximum likelihood statistic for the detection of GW pulsars. However, in “all-sky” searches for previously unknown GW pulsars, it would be computationally intractable to calculate the (fully coherent) $\mathcal{F}$-statistic at every point of (a suitably fine) grid covering the parameter space: the number of grid points is many orders of magnitude too large for that. Therefore, in practice some nonoptimal detection statistic is used for all-sky searches. Here we introduce a “phase-relaxed” $\mathcal{F}$-statistic, which we denote ${\mathcal{F}}_{\mathrm{pr}}$, for incoherently combining the results of fully coherent searches over short time intervals. We estimate (very roughly) that for realistic searches, our ${\mathcal{F}}_{\mathrm{pr}}$ is $\sim 10–15\%$ more sensitive than the “semicoherent” $\mathcal{F}$-statistic that is currently used. Moreover, as a by-product of computing ${\mathcal{F}}_{\mathrm{pr}}$, one obtains a rough determination of the time-evolving phase offset between one’s template and the true signal imbedded in the detector noise. Almost all the ingredients that go into calculating ${\mathcal{F}}_{\mathrm{pr}}$ are already implemented in the LIGO Algorithm Library, so we expect that relatively little additional effort would be required to develop a search code that uses ${\mathcal{F}}_{\mathrm{pr}}$.

Figure 1

Compares the FA probabilities for the two detection statistics, ${\mathcal{F}}_{4N}$ and ${\mathcal{F}}_{2N}$, as a function of FD probability, for $N=100$ and for several values of the total squared signal strength, ${\rho }^2\equiv \sum _{i=1}^N{\rho }_i^2$. The blue (dark) curves are for ${\mathcal{F}}_{4N}$ and the green (light) for ${\mathcal{F}}_{2N}$. From upper to lower, the squared signal strengths are ${\rho }^2=25$, 50, 75, 100.

Figure 2

Compares the FA probabilities for the two detection statistics, ${\mathcal{F}}_{\mathrm{sc}}$ and ${\mathcal{F}}_{\mathrm{pr}}$, as a function of FD probability, for several values of the (square of) the signal strength, ${\rho }^2$. The blue (dark) curves are for ${\mathcal{F}}_{\mathrm{sc}}$ and the green (light) for ${\mathcal{F}}_{\mathrm{pr}}$. From upper to lower, the signal strengths are ${\rho }^2=75$, 100, 125.

Figure 3

Same as in Fig. 2, except that, from the upper to lower curves, the signal strengths are ${\rho }^2=150$, 175, 200.

Figure 4

Similar to Figs. 2, 3 except that here the sensitivity of the ${\mathcal{F}}_{\mathrm{sc}}$ statistic is compared to that of ${\mathcal{F}}_{\mathrm{pr}}$ at lower ${\rho }^2$. From upper to lower, the signal strengths for ${\mathcal{F}}_{\mathrm{sc}}$ are for ${\rho }^2=140$, 160, 190.