Spatial correlations between pulsars for interfering gravitational-wave sources in pulsar timing array

Pulsar timing arrays (PTAs) have recently made significant strides in detecting a gravitational-wave background (GWB). The Hellings--Downs (HD) correlation, traditionally considered the hallmark of a GWB, has been observed with compelling evidence in a common-spectrum signal. However, the HD curve is derived under the assumption of discrete and noninterfering sources---an assumption unlikely to hold in realistic scenarios involving interference among numerous GW sources, such as supermassive black hole binaries. In this work, we demonstrate how such interference generates measurable, frequency-dependent spatial correlations that deviate from the HD curve. The spatial correlations for interfering sources (referred to as ISC) still exhibit contributions in the quadrupole and higher multipoles, resembling the HD correlation and encoding the characteristic nature of gravitational-wave radiation. We apply the ISC framework to search for a GWB in the NANOGrav 15-yr dataset. Although current sensitivity limits the performance of the multiparameter ISC model in direct model comparison, a marked improvement in statistical significance is achieved when the parameters are fixed to their best-fit values. While this improvement may be overly optimistic at present, it highlights the potential for significantly enhancing GWB detection by accounting for interference effects, particularly as PTA sensitivity improves in future observations.