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

Understand

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.