Constraining Alternative Theories of Gravity Using Pulsar Timing Arrays

Efficient Computation of Overlap Reduction Functions for Pulsar Timing Arrays

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.