Antenna patterns of interferometric detectors of gravitational waves - I. Linearly polarized waves

We consider the response of a free-mass interferometric gravitational wave detector to plane gravitational waves arriving from an arbitrary direction with an arbitrary linear polarization in the long-wavelength approximation. After deriving the well-known single-detector antenna pattern η, we address the problem of a detector fixed on the Earth observing gravitational wave bursts that arrive from the direction of the Virgo cluster with random polarization and random arrival time. First we calculate the rms sensitivity of a single detector $$\langle{\eta }^{2}\rangle^{1/2}$$ as a function of its latitude and orientation on Earth. Then we consider coincidences between two fixed detectors. Let each detector have a threshold X, being the minimum detectable value of η². The coincidence probability C clearly depends upon the thresholds X₁ and X₂ of the two detectors. However, we are able to prove a remarkable result for random burst of gravitational waves, that the mean squared product of the antenna patterns $$\langle \eta _{1}^{2}\eta _{2}^{2}\rangle$$ equals the average of the coincidence probability of the two detectors over all thresholds, $$\int_{0}^{1}\int_{0}^{1}C({X}_{1*,}{X}_{2*}){dX}_{1*,}{dX}_{2*})$$. It is therefore possible to extract meaningful information about coincidences from the purely geometrical function $$\langle \eta _{1}^{2}\eta _{2}^{2}\rangle$$. We argue that this function probably underestimates realistic coincidence probabilities, but does so uniformly, so it allows comparisons to be made between different sites and different orientations at the same site. By plotting this function for several pairs of likely detector locations in the USA and Europe for random waves from Virgo, we find a number of interesting results, among which are: (i) coincidences between detectors in the USA are very sensitive to small changes in their relative orientations, and (ii) the coincidence probability between a detector in the USA and one in Europe is generally a factor of about 2 smaller than probabilities within America or Europe. We also perform similar calculations for sources randomly distributed on the sky. Finally, we discuss the implications of these results for the choice of orientation of the planned detectors and for the numbers of detectors world-wide.