Optimal strategies for gravitational wave stochastic background searches in pulsar timing data

A low frequency stochastic background of gravitational waves may be detected by pulsar timing experiments in the next 5 to 10 yr. Using methods developed to analyze interferometric gravitational wave data, in this paper we lay out the optimal techniques to detect a background of gravitational waves using a pulsar timing array. We show that for pulsar distances and gravitational wave frequencies typical of pulsar timing experiments, neglecting the effect of the metric perturbation at the pulsar does not result in a significant deviation from optimality. We discuss methods for setting upper limits using the optimal statistic, show how to construct skymaps using the pulsar timing array, and consider several issues associated with realistic analysis of pulsar timing data.

Figure 1

Plot of the full overlap reduction, Eq. (38), along with the approximation Eq. (39) for two pulsars a distance $L$ from the Solar System barycenter. The overlap reduction function is a function of $fL$. The top two (blue) curves show Eq. (38) with $\beta =3/4\pi$ (solid line) and Eq. (39) (dashed line) for two pulsars at an angle $\xi =\pi /8$ as a function of $fL$. The middle (red) and bottom (green) curves show the same quantities for two pulsars at $\xi \approx 0.86$ and $\xi =\pi /2$ respectively. The smallest value of the frequency $f_{\min }\sim 0.1{\mathrm{yr}}^{-1}$ and the closest pulsars used in timing experiments are at a distance of $L_{\min }\sim 100\mathrm{ly}$ so that $fL\gtrsim 10$. This range of $fL$ puts pulsar timing experiments in the regime where Eq. (39) is an excellent approximation to Eq. (38) and we are justified in throwing out the pulsar term while remaining close to optimal.

Figure 2

Plots of $|{}^{(ij)}\Gamma _{\widehat{\Omega }}|$ from Eq. (92) for two pulsars with $\xi =\pi /2$ (top panel) and $\xi =\pi$ degrees (bottom panel).

Figure 3

Skymap of the sensitivity, Eq. (94), for the Parkes pulsar timing array.

Figure 4

Plot of the point spread function Eq. (95) for the Parkes pulsar timing array for a source at 6 h RA 45° Dec (top panel) and 18 h RA $-45°$ Dec (bottom panel).