On the angular localization of gravitational-wave signals by pulsar timing arrays

Abstract

We provide a complete study of the factors influencing gravitational-wave signal localization using pulsar timing arrays. We derive analytical expressions for the Cramér-Rao sky localization precision that delineate the impact of the angular proximity, $ξ$, between the pulsar and the gravitational wave source, and the precision, $σ_L$, with which pulsar distances are known. Interference between the Earth and pulsar terms creates rapid angular oscillations for sky-coordinate Fisher matrix elements that aids localization, which is complemented by more broadly varying antenna response gradient information. The relative importance of these factors depends on whether pulsar distances are known precisely [i.e., $σ_L\leqλ_\mathrm{GW}/(1-\cosξ)$] or imprecisely, respectively. If the former, tightening pulsar distance precisions improves signal localization according to $ΔΩ_\mathrm{sky}\proptoσ_L^2$ until the Earth-pulsar system reaches its diffraction limit. If the latter, localization precision is degraded, but more pulsars in close proximity to the source is the best means of improving. With $α$ indexing pulsars, this scales as $ΔΩ_\mathrm{sky}~\propto~(\sum_α\mathrm{SNR}_α^2/ξ_α^2)^{-1}$ in the small-angle limit of the unmarginalized Fisher matrix, and we derive the analytic generalization to any angle between a pulsar and the source. Finally, we study a scenario where pulsar-term phases are treated as nuisance variables that are unconnected to binary or PTA properties. This phase-decoupled scenario, which is how all PTA continuous wave searches are currently conducted, delivers localization performance similar to the antenna-response--driven case, and does not exhibit significant improvement as pulsar distance precisions are tightened.

Figures (9)

• Figure 1: As is convention, the direction of GW propagation is $\hat{\Omega}$ such that the unit vector pointing to the source at spherical polar coordinates $(\theta,\phi)$ is $-\hat{\Omega}$. We define a right-handed orthogonal basis triad such that $\hat{\Omega}=\hat{m}\times\hat{n}$, where $\hat{m}$ and $\hat{n}$ lie in the plane transverse to the direction of GW propagation.
• Figure 2: A polar-diagram visualization of the angular factors of $|\overline{\mathrm{SNR}}|\propto (1+\cos\xi)|\sin[\omega L (1-\cos\xi)]|$. The GW source position is kept fixed at $0^\circ$ while the pulsar is moved counter-clockwise to be at varying angles around the source. For visualization purposes, the pulsar distance has been reduced to $100$ pc. Inspired by a similar visualization in 2011MNRAS.414.3251L.
• Figure 3: We employ a coordinate system in which the unit vector pointing from the Earth (or Solar System Barycenter) to a pulsar, $\hat{p}$, is decomposed into components that are parallel or perpendicular to the direction of GW propagation, $\hat{\Omega}$. The angle between the GW source and the pulsar is $\xi$. The projection of $\hat{p}$ into the plane transverse to $\hat{\Omega}$ is $\hat{\rho}$, which also contains $\hat{m}$ and $\hat{n}$, such that $\hat{\Omega}=\hat{m}\times\hat{n}$. The angle between $\hat{\rho}$ and $\hat{m}$ is $\varphi$.
• Figure 4: A circular ring of $10\times1$-$\mathrm{kpc}$ pulsars (black stars) are arranged around a GW source (red cross) at $(\cos\theta=0,\phi=\pi/2)$. We visualize the marginalized GW localization precision in terms of ellipses around the source, doing so for two different precisions on pulsar distances: $\sigma_L=10^3\lambda_\mathrm{GW}\approx1\,\mathrm{kpc}$ (blue dashed), and $\sigma_L=10^2\lambda_\mathrm{GW}\approx0.1\,\mathrm{kpc}$ (blue solid).
• Figure 5: We systematically move our ring of $20\times 1$-$\mathrm{kpc}$ pulsars through varying angular radii to explore the influence on marginalized GW source localization precision. We show the impact of different precisions on pulsar distances as a fraction of the GW wavelength, and contrast with the unmarginalized localization precision (grey) and the form for the analytic antenna-response--driven localization (black dash-dot) given by \ref{['eq:analytic_response_omega']}.