Identifying the Quadrupolar Nature of Gravitational Wave Background through Space-based Missions

The stochastic gravitational wave background (SGWB) consists of an incoherent collection of waves from both astrophysical and cosmological sources. To distinguish the SGWB from noise, it is essential to verify its quadrupolar nature, exemplified by the cross-correlations among pairs of pulsars within a pulsar timing array, commonly referred to as the Hellings-Downs curve. We study how this quadrupolar signature manifests in correlations between general GW detector pairs, characterized by their antenna responses. Focusing on space-based missions including laser interferometers (LISA, Taiji, TianQin) and atom interferometers (AEDGE/MAGIS), we demonstrate how orbital motion dynamically modulates detector correlations through time-dependent separations and relative orientation shifts. These modulations encode unique statistical features that serve as definitive markers of the SGWB's quadrupolar nature. Our findings identify optimal configurations for these missions, offer forecasts for the time needed to identify the quadrupolar nature of the SGWB, and are applicable to both space-space and space-terrestrial correlations.