The spatial correlations between pulsars for interfering sources in Pulsar Timing Array and evidence for gravitational-wave background in NANOGrav 15-year data set

Pulsar timing arrays (PTAs), aimed at detecting gravitational waves (GWs) in the $1\sim 100$ nHz range, have recently made significant strides. Compelling evidence has emerged for a common spectrum signal spatially correlated among pulsars, following a Hellings-Downs (HD) pattern, which is crucial for detecting a gravitational-wave background (GWB). However, the HD curve is expected for discrete and non-interfering sources, which is unlikely to hold in realistic scenarios with potential interference among numerous GW sources, such as the supermassive black-hole binaries. Incorporating interference was previously expected to introduce an irreducible uncertainty (known as "cosmic variance") in discerning the HD correlation; however, our work reveals how this interference generates measurable frequency-dependent spatial correlations distinct from the HD curve. The spatial correlations for interfering sources (referred to as "ISC") still exhibit contributions in the quadrupole and higher orders, resembling the HD correlation and encoding the nature of GW radiations. We apply these novel correlations to search for a GWB in the NANOGrav 15-year data set. In an optimistic estimation, our findings show a Bayes factor of $33.7\pm 3.2$ comparing ISC to the HD correlation, and an improvement in optimal statistic signal-to-noise ratio from $4.9\pm 1.1$ for the HD correlation to $6.6\pm 1.7$ for the ISC, highlighting the significant enhancement in evidence for detecting a GWB through incorporating interference.