Sensitivity of the Laser Interferometer Gravitational Wave Observatory to a stochastic background, and its dependence on the detector orientations

We analyze the sensitivity of a network of interferometer gravitational-wave detectors to the gravitational-wave stochastic background, and derive the dependence of this sensitivity on the orientations of the detector arms. We build on and extend the recent work of Christensen, but our conclusion for the optimal choice of orientations of a pair of detectors differs from his. For a pair of detectors (such as LIGO) that subtends an angle at the center of the Earth of ≲70°, we find that the optimal configuration is for each detector to have its arms make an angle of 45° (modulo 90°) with the arc of the great circle that joins them. For detectors that are farther separated, each detector should instead have one arm aligned with this arc. We show that the broadband sensitivity to the stochastic background of a detector pair which are ≲3000 km apart is essentially determined by their relative rotation. Their average rotation with respect to the arc joining them is unimportant. We also describe in detail the optimal data-analysis algorithm for searching for the stochastic background with a detector network, which is implicit in earlier work of Michelson. The LIGO pair of detectors will be separated by ∼3000 km. The minimum detectable stochastic energy density for these detectors with their currently planned orientations is ∼3% greater than what it would be if the orientations were optimal, and ∼4 times what it would be if their separation were ≲ a few kilometers. (The detectors are chosen to be far apart so that their sources of noise will be uncorrelated, and in order to improve the angular resolution of the determinations of positions of burst sources.)