Universal calculation approach of overlap reduction function for pulsar timing array and laser interferometer detector

Stochastic gravitational-wave background (SGWB) is one of the major targets for contemporary gravitational wave (GW) observations, including pulsar timing array (PTA), ground-based laser interferometer (GbLI), and space-based laser interferometer (SbLI). The data analysis of SGWB involves cross-correlating the outputs of a pair of detectors to find the functional form of the correlations, which is referred to as the overlap reduction function (ORF). While conventional approaches employ detector-specific approximations for ORF calculations, in this paper, we present a universal formula to calculate the precise ORF for isotropic SGWB detection with PTA and laser interferometer detectors. Since no approximations are employed, the exact value of the ORF can be obtained for all three types of detectors.