Search for continuous gravitational waves: Metric of the multidetector<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">F</mml:mi></mml:math>-statistic

We develop a general formalism for the parameter-space metric of the multidetector $\mathcal{F}$-statistic, which is a matched-filtering detection statistic for continuous gravitational waves. We find that there exists a whole family of $\mathcal{F}$-statistic metrics, parametrized by the (unknown) amplitude parameters of the gravitational wave. The multidetector metric is shown to be expressible in terms of noise-weighted averages of single-detector contributions, which implies that the number of templates required to cover the parameter space does not scale with the number of detectors. Contrary to using a longer observation time, combining detectors of similar sensitivity is therefore the computationally cheapest way to improve the sensitivity of coherent wide-parameter searches for continuous gravitational waves. We explicitly compute the $\mathcal{F}$-statistic metric family for signals from isolated spinning neutron stars, and we numerically evaluate the quality of different metric approximations in a Monte Carlo study. The metric predictions are tested against the measured mismatches and we identify regimes in which the local metric is no longer a good description of the parameter-space structure.