Generalized F-statistic : Multiple detectors and multiple gravitational wave pulsars

The $\mathcal{F}$-statistic, derived by Jaranowski, Krolak and Schutz (1998), is the optimal (frequentist) statistic for the detection of nearly periodic gravitational waves from known neutron stars, in the presence of stationary, Gaussian detector noise. The $\mathcal{F}$-statistic was originally derived for the case of a single detector, whose noise spectral density was assumed constant in time, and for a single known neutron star. Here we show how the $\mathcal{F}$-statistic can be straightforwardly generalized to the cases of (1) a network of detectors with time-varying noise curves, and (2) a collection of known sources (e.g., all known millisecond pulsars within some fixed distance). Fortunately, all the important ingredients that go into our generalized $\mathcal{F}$-statistics are already calculated in the single-source/single-detector searches that are currently implemented, e.g., in the Laser Interferometer Gravitational-Wave Observatory software library, so implementation of optimal multidetector, multisource searches should require negligible additional cost in computational power or software development. This paper also includes an analysis of the likely efficacy of a collection-type search, and derives criteria for deciding which candidate sources should be included in a collection, if one is trying to maximize the detectability of the whole. In particular we show that for sources distributed uniformly in a thin disk, the strongest source in the collection should have signal-to-noise-squared $\ensuremath{\sim}5$ times larger than weakest source, for an optimized collection. We show that gravitational waves from collection of the few brightest (in gravitational waves) neutron stars could perhaps be detected before the single brightest source, but that this is far from guaranteed. Once gravitational waves from the few brightest neutron stars have been discovered, grouping more distant (individually undetectable) pulsars into collections, and then searching for those collections, should be an effective way of measuring the average gravitational-wave strengths of those more distant pulsars.