Pulsar Timing Response to Gravitational Waves with Spherical Wavefronts from a Massive Compact Source in the Quadrupole Approximation

Pulsar timing arrays (PTAs) are searching for nanohertz-frequency gravitational waves (GWs) through cross-correlation of pulse arrival times from a set of radio pulsars. PTAs have relied upon a frequency-shift formula of the pulse, where planar GWs are usually assumed. Phase corrections due to the wavefront curvature have been recently discussed. In this paper, frequency-shift and timing-residual formulae are derived for GWs with fully spherical wavefronts from a compact source such as a binary of supermassive black holes, where the differences in the GW amplitude and direction between the Earth and the pulsar are examined in the quadrupole approximation. By using the new formulae, effects beyond the plane-wave approximation are discussed, and a galactic-center PTA as well as nearby GW source candidates are also mentioned.