A team has derived fully analytic overlap reduction functions describing the response of a pulsar timing array to an anisotropic tensor stochastic gravitational-wave background within general relativity. The calculation retains both pulsar distances and their complete propagation phases, something previously handled in approximate or numerical form. At each fixed sky multipole, the intensity and circular polarization responses reduce to finite combinations of exponential integrals and logarithms, while the electric and magnetic linear polarization responses require only elementary functions. The authors evaluate all components through ℓ = 8.

Symmetry separates exact nulls from additional Earth-term nulls that propagation can lift. A striking result is that even an unpolarized background can produce an imaginary cross-spectrum. For a pulsar pair with three-dimensional separation d, the pulsar terms can remain coherent when 2πfd/c ≲ 1, 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.

The work also quantifies the role of distance errors: 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 effects 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 validate the analytic expressions over the parameter ranges studied, with implications for the search for gravitational-wave backgrounds with pulsar timing arrays such as NANOGrav, EPTA, or IPTA.