Full-Stokes response to a stochastic gravitational-wave background in pulsar timing arrays: Fully analytic overlap reduction functions at finite distance and coherence of close pulsar pairs

Abstract

We derive fully analytic overlap reduction functions for the full-Stokes response of a pulsar timing array to an anisotropic tensor gravitational-wave background in general relativity. The response retains both pulsar distances and their complete propagation phases. At each fixed finite sky multipole, the intensity and circular polarization responses reduce to finite combinations involving exponential integrals and logarithms, while the electric and magnetic linear polarization responses require only elementary functions. We evaluate all components through $\ell=8$. Symmetry separates exact nulls from additional Earth-term nulls that propagation can lift. Even an unpolarized background can produce an imaginary cross-spectrum. For a pair with three-dimensional separation $d$, the pulsar terms can remain coherent when $2πfd/c\lesssim1$, even if both pulsars are many wavelengths from Earth. A uniform approximation retaining all four Earth and pulsar terms connects this regime to the short-wavelength limit, and a rotationally invariant relation fixes the leading transverse magnetic response. Common-distance errors cancel from the aligned pair phase, while relative-depth errors directly affect coherence. Under Gaussian differential averaging, fixed low-multipole norms and the squared norm summed over all multipoles decay at different rates as the response shifts to higher multipoles. Published pulsar configurations illustrate the roles of distance accuracy, bandwidth, and timing fits. At fixed pointwise intensity, a polarization covariance bound and a relative-depth noise budget quantify when differencing suppresses the background. Independent sky integration and matched-accuracy benchmarks test the analytic expressions over the parameter ranges studied.