Effect of small interpulsar distances in stochastic gravitational wave background searches with pulsar timing arrays

One of the primary objectives for pulsar timing arrays (PTAs) is to detect a stochastic background generated by the incoherent superposition of gravitational waves (GWs), in particular from the cosmic population of supermassive black hole binaries. Current stochastic background searches assume that pulsars in a PTA are separated from each other and the Earth by many GW wavelengths. As more millisecond pulsars are discovered and added to PTAs, some may be separated by only a few radiation wavelengths or less, resulting in correlated GW phase changes between close pulsars in the array. Here we investigate how PTA overlap reduction functions (ORFs), up to quadrupole order, are affected by these additional correlated phase changes, and how they are in turn affected by relaxing the assumption that all pulsars are equidistant from the solar system barycenter. We find that in the low-frequency GW background limit of f10^-9 Hz, and for pulsars at varying distances from the Earth, these additional correlations only affect the ORFs by a few percent for pulsar pairs at large angular separations, as expected. However, when nearby (order 100 pc) pulsars are separated by less than a few degrees, the correlated phase changes can introduce variations of a few tens of percent in the magnitude of the isotropic ORF, and much larger fractional differences in the anisotropic ORFs---up to 188 in the m=0, l=2 ORF for equidistant pulsars separated by 3^. In fact, the magnitude of most of the anisotropic ORFs is largest at small, but nonzero, pulsar separations. Finally, we write down a small angle approximation for the correlated phase changes which can easily be implemented in search pipelines, and for completeness, examine the behavior of the ORFs for pulsars which lie at a radiation wavelength from the Earth.