Pulsar timing arrays seek and study gravitational waves (GWs) through the angular two-point correlation function of timing residuals they induce in pulsars. The two-point correlation function induced by the standard transverse-traceless GWs is the famous Hellings-Downs curve, a function only of the angle between the two pulsars. Additional polarization modes (vector or scalar) that may arise in alternative-gravity theories have different angular correlation functions. Furthermore, anisotropy, linear, or circular polarization in the stochastic GW background gives rise to additional structure in the two-point correlation function that cannot be written simply in terms of the angular separation of the two pulsars. In this Letter, we provide a simple formula for the most general two-point correlation function-or overlap reduction function (ORF)-for a gravitational-wave background with an arbitrary polarization state, possibly containing anisotropies in its intensity and polarization (linear and/or circular). We provide specific expressions for the ORFs sourced by the general-relativistic transverse-traceless GW modes as well as vector (or spin-1) modes that may arise in alternative-gravity theories.
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