Detecting a stochastic gravitational-wave background: The overlap reduction function

Detection of a gravitational-wave stochastic background via ground or space-based gravitational-wave detectors requires the cross correlation of the response of two or more independent detectors. The cross correlation involves a frequency-dependent factor---the so-called overlap reduction function or Hellings-Downs curve---that depends on the relative geometry of each detector pair, i.e., the detector separations and the relative orientation of their antenna patterns (beams). An incorrect formulation of this geometrical factor has appeared in the literature, leading to incorrect conclusions regarding the sensitivity of proposed detectors to a stochastic gravitational-wave background. To rectify these errors and as a reference for future work we provide here a complete, first-principles derivation of the overlap reduction function and assess the nature of the errors associated with the use of the incorrect expression that has appeared in the literature. We describe the behavior of the overlap reduction function in different limiting regimes, and show how the difference between the correct and incorrect expressions can be understood physically.