Orientation-dependent sensitivity of the all-sky all-frequency gravitational-wave radiometer search to continuous-wave sources

Abstract

The all-sky all-frequency (ASAF) gravitational-wave radiometer search cross-correlates the data of two detectors with equal weight on the $+$ and $\times$ tensor polarizations. This is optimal for an unpolarized stochastic background, but not for continuous-wave (CW) sources, whose polarization depends on the inclination angle $ι$ and the polarization angle $ψ$. Unless the source is seen face-on, the search measures less power than a face-on source with the same intrinsic amplitude, and underestimates its strength. The ratio of the two powers, the projection factor $μ^2(ι,ψ)$, is known from earlier radiometer studies. We show how $μ^2$ enters the ASAF statistic, derive its exact dependence on $ψ$, and prove that its average over orientations is exactly $2/5$, independent of sky position, baseline geometry, noise weighting, and frequency. We apply the formalism to four simulated pulsar signals at four sky positions, with polarization from circular to nearly linear, injected into the LIGO detectors during the second observing run (O2). A nearly edge-on injection is measured with a signal-to-noise ratio (SNR) of $5.20$, below the threshold of $6.19$, while the same signal seen face-on would have an SNR of $31.1 \pm 6.0$. Bayesian upper limits on the strain amplitude $h_0$, marginalized over orientation and calibration uncertainty, are $2.7$ to $2.9$ times the face-on values. For the $505$ sub-threshold candidates of the first part of the fourth LIGO-Virgo-KAGRA observing run (O4a), the upper limits increase by a factor of $2.5$ to $2.9$, and a source of the same strength would reach threshold only when inclined by less than $34.7^{\circ}$ to $50.3^{\circ}$ from face-on. The measured SNR and the significance of every candidate are unchanged.