Direction-of-arrival estimation of a gravitational wave by correlations between quadrupole moments of pulsar timings

Abstract

Can we estimate the direction of arrival (DOA) of a gravitational wave (GW) signal from pulsar timing array observations? The present paper addresses the inverse problem, for which we consider quadrupole moments of pulsar timings due to GWs from a dominant isolated source such as a binary of supermassive black holes over an isotropic stochastic background. Correlations between the quadrupole moments are discussed, where the correlations between pulsar pairs over the full sky are taken into account. The correlations turn out to be in the form of a three-dimensional traceless matrix with rank 2 that can be closely related with a projection tensor for the GW. Thereby, we demonstrate that the rank-2 matrix allows to estimate the DOA of the GW. In expectation of the forthcoming Square Kilometer Array, angular resolutions as well as DOA estimation errors are also examined.

Figures (3)

• Figure 1: DOA estimation errors limited by the numbers of observed pulsars and observation epochs. The vertical axis denotes the DOA estimation error for the GW, while the horizontal axis means the number of observation epochs $N_{obs}$. The solid (red in color) and dashed (blue in color) curves are theoretical curves for $N_p = 64$ and $N_p = 256$, respectively, where $h^+ = h^{\times} = 1$ and $\sigma = 10$ are chosen, and the DOA estimation error is estimated as a sum of $\delta_p \hat{\Omega}_{GW}$ and $\delta_n \hat{\Omega}_{GW}$ in Eqs. (\ref{['deltap2']}) and (\ref{['deltan2']}), respectively. The filled (red in color) and empty (blue in color) squares denote the mean of the error for for $N_p = 64$ and $N_p = 256$, respectively, and the error bar corresponds to one sigma error, where 100 runs are numerically performed.
• Figure 2: Scatter plots of DOA estimations for a single GW. $h^+ = h^{\times} = 1$, and $\sigma = 10$ are chosen for $N_{obs} = 256$, and 100 runs are numerically performed. Here, $\hat{\Omega}_{GW} = (x, y, z)$ is chosen as $\theta_{GW} = \phi_{GW} = 45$ deg. Two panels (left: $N_p = 64$, right: $N_p = 256$) correspond to two squares in Figure \ref{['fig-curves']}, where one filled square in Figure \ref{['fig-curves']} is for $N_p =64$ and $N_{obs} = 256$, and the other empty square in the same figure is for $N_p =256$ and $N_{obs} = 256$.
• Figure 3: The polar coordinates associated with Eq. (\ref{['e']}) and $\hat{\Omega}_{GW} = (0, 0, 1)$.