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Detection of a gravitational-wave stochastic background via ground orspace-based gravitational-wave detectors requires the cross-correlation of theresponse of two or more independent detectors. The cross-correlation involves afrequency-dependent factor -- the so-called overlap reduction function orHellings-Downs curve -- that depends on the relative geometry of each detectorpair: i.e., the detector separations and the relative orientation of theirantenna patterns (beams). An incorrect formulation of this geometrical factorhas appeared in the literature, leading to incorrect conclusions regarding thesensitivity of proposed detectors to a stochastic gravitational-wavebackground. To rectify these errors and as a reference for future work weprovide here a complete, first-principles derivation of the overlap reductionfunction and assess the nature of the errors associated with the use of theincorrect expression that has appeared in the literature. We describe thebehavior of the overlap reduction function in different limiting regimes, andshow how the difference between the correct and incorrect expressions can beunderstood physically.

Detecting a Stochastic Gravitational-Wave Background: The Overlap Reduction Function