Method to detect gravitational waves from an ensemble of known pulsars

Combining information from weak sources, such as known pulsars, for gravitational wave detection, is an attractive approach to improve detection efficiency. We propose an optimal statistic for a general ensemble of signals and apply it to an ensemble of known pulsars. Our method combines $\mathcal{F}$-statistic values from individual pulsars using weights proportional to each pulsar’s expected optimal signal-to-noise ratio to improve the detection efficiency. We also point out that to detect at least one pulsar within an ensemble, different thresholds should be designed for each source based on the expected signal strength. The performance of our proposed detection statistic is demonstrated using simulated sources, with the assumption that all pulsar ellipticities belong to a common (yet unknown) distribution. Comparing with an equal-weight strategy and with individual source approaches, we show that the weighted combination of all known pulsars, where weights are assigned based on the pulsars’ known information, such as sky location, frequency and distance, as well as the detector sensitivity, always provides a more sensitive detection statistic.

Figure 1

Detection probability for the model given by Eq. (39), using the optimal combined statistic (com-opt) and individual thresholds (ind-subopt for equal thresholds, see Sec. 3e1, and ind-opt for optimal thresholding, see Sec. 3e2). We have fixed $P_{\mathrm{FA}}=0.01$.

Figure 2

Detection probability of linear combination statistics [Eq. (38)] compared to the optimal detection probability. Different traces correspond to ${\lambda }_0$ ranging from 3 to 10, and we have fixed $P_{\mathrm{FA}}=0.01$.

Figure 3

Detection probability for models with ${\lambda }_j$ following exponential distributions with mean value given by Eq. (39). We have fixed $P_{\mathrm{FA}}=0.01$. Shown here are from the optimal statistic (com-opt), optimal statistic scaling prior distributions by 10 (com-opt-10p) and by $1/10$ (com-opt-01p), linear-combination statistic with $\beta =1/2$ (com-lin), individual pulsar detection with optimal thresholding on each $X_j$ (ind-opt) and individual pulsar detection with common threshold (ind-subopt).

Figure 4

Values of signal strength (${\lambda }_0$) at which each detection strategy can yield detection probability of 50% (red symbols) and 90% (blue symbols), respectively, as a function of the number of sources, $N$. We have used the optimal statistic (stars), linear-combination statistic with $\beta =0.5$ (square), uniform threshold for all sources (hollow circles) and optimal thresholding (solid circles). Increase of $\lambda$ with $N$ in the uniform threshold case indicates that including more sources introduces contamination from weaker sources. We assumed exponential distributions for ${\lambda }_j$ in this plot and have fixed $P_{\mathrm{FA}}=0.01$.

Figure 5

ROC curves for different detection methods. WA (EA), W3 (E3), W6 (E6), W50 (E50), N1 and M1 correspond to weighted (equal) combinations of all, the expected brightest three, the expected brightest six, the expected brightest fifty, the expected brightest and the measured brightest pulsar(s) respectively. The ${\epsilon }^2$ in injections were drawn from an exponential distribution with rate parameter $2\times {10}^{-16}$.

Figure 6

Same as Fig. 5 but with $\epsilon$ following the Gaussian distribution with mean value of $1.5\times {10}^{-8}$, and half of mean values as its variance value.

Figure 7

ROC curves of the weighted-combination (WA), equal-combination (EA), the expected brightest (N1) and the measured brightest (M1) detection methods for $10^5$ signal injection simulations and $10^5$ noise only simulations. WA and EA correspond to weighted and equal combinations of all pulsars, respectively. The intrinsic parameter in injections was drawn from two distributions: (i) ${\epsilon }^2$ follows an exponential distribution with rate parameter $2\times {10}^{-16}$ (top-left panel) and $4\times {10}^{-16}$ (bottom-left panel), and (ii) $\epsilon$ follows a Gaussian distribution with mean value of $1.5\times {10}^{-8}$ (top right) and $2\times {10}^{-8}$ (bottom right) with standard deviations equal to half of the mean value. Detection probability ratios (dashed lines) are shown in each panel.

Figure 8

The same as for the top panels of Fig. 7 but taking into account the measured distance error. The measured distance error effect is included by drawing “true” pulsar distances from a normal distribution with mean value equal to their current best estimates and a standard deviation of 20% of those mean values.